Subgroups — C₅₃×#61234 #27268746

Order 13568.

1874 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3475	C₂	1	2	yes	maximal, complemented
2	C₅₃×#3342	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅₃×#3341	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#3340	C₂	1	2	yes	maximal, complemented
5	C₅₃×#3339	C₂	1	2	yes	maximal, complemented
6	C₅₃×#3338	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#3337	C₂	1	4	yes	maximal, complemented
8	C₅₃×#3336	C₂	1	2	yes	maximal, complemented
9	C₅₃×#3335	C₂	1	2	yes	maximal, complemented
10	C₅₃×#3334	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#3091	C₂	1	4	yes	maximal, complemented
13	C₅₃×#3076	C₂	1	2	yes	maximal, complemented
14	C₅₃×#3050	C₂	1	2	yes	maximal, complemented
15	C₄×#135×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₈×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
17	C₂×C₅₃×#427	C₂	1	2	yes	maximal, complemented
18	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
20	C₅₃×#556	C₂²	1	2	yes	—
21	C₅₃×#555	C₂²	1	2	yes	—
22	C₅₃×#554	C₂²	1	2	yes	—
23	C₅₃×#553	C₂²	1	4	yes	—
24	C₅₃×#552	—	2	2	no	—
25	C₅₃×#552	C₂²	1	4	yes	—
26	C₅₃×#551	—	2	4	no	—
27	C₅₃×#547	C₂²	1	2	yes	—
28	C₅₃×#546	—	2	2	no	—
29	C₅₃×#533	C₂²	1	2	yes	—
30	C₅₃×#529	—	2	2	no	—
31	C₅₃×#529	—	2	4	no	—
32	C₅₃×#525	C₂²	1	4	yes	—
33	C₄×#41×C₅₃	—	2	2	no	—
34	C₄×#41×C₅₃	C₂²	1	2	yes	—
35	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
36	C₂×#117×C₅₃	C₂²	1	4	yes	—
37	C₅₃×#500	C₂²	1	1	yes	characteristic, complemented
38	C₅₃×#500	C₂²	1	1	yes	characteristic, complemented
39	C₅₃×#499	C₂²	1	2	yes	complemented
40	C₅₃×#499	C₂²	1	2	yes	complemented
41	C₅₃×#498	C₂²	1	2	yes	complemented
42	C₅₃×#485	C₂²	1	4	yes	complemented
43	C₅₃×#485	C₂²	1	4	yes	complemented
44	C₅₃×#484	C₂²	1	4	yes	complemented
45	C₅₃×#483	C₂²	1	4	yes	complemented
46	C₅₃×#482	C₂²	1	4	yes	complemented
47	C₅₃×#482	C₂²	1	4	yes	complemented
48	C₅₃×#481	C₂²	1	4	yes	complemented
49	C₅₃×#480	C₂²	1	4	yes	complemented
50	C₅₃×#479	C₂²	1	4	yes	complemented
51	C₅₃×#478	C₂²	1	4	yes	complemented
52	C₅₃×#477	C₂²	1	2	yes	complemented
53	C₅₃×#476	C₂²	1	2	yes	complemented
54	C₅₃×#475	C₂²	1	2	yes	complemented
55	C₅₃×#474	C₂²	1	2	yes	complemented
56	C₅₃×#467	C₂²	1	2	yes	complemented
57	C₅₃×#467	C₂²	1	2	yes	complemented
58	C₅₃×#466	C₂²	1	2	yes	complemented
59	C₅₃×#466	C₂²	1	2	yes	complemented
60	C₅₃×#465	C₂²	1	4	yes	complemented
61	C₅₃×#465	C₂²	1	4	yes	complemented
62	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
63	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic, complemented
64	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	complemented
65	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	complemented
66	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
67	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
68	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
69	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
70	C₈×D₄×C₅₃	C₂²	1	2	yes	—
71	C₈×D₄×C₅₃	C₂²	1	2	yes	—
72	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
73	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
74	C₅₃×#427	C₂²	1	1	yes	characteristic
75	C₅₃×#427	C₂²	1	2	yes	—
76	C₅₃×#427	C₂²	1	4	yes	—
77	C₅₃×#427	C₂²	1	2	yes	—
78	C₅₃×#427	C₂²	1	2	yes	complemented
79	C₅₃×#427	C₂²	1	4	yes	complemented
80	C₅₃×#427	C₂²	1	2	yes	complemented
81	C₂×#107×C₅₃	C₂²	1	2	yes	complemented
82	C₂×#107×C₅₃	C₂²	1	2	yes	complemented
83	C₂×#106×C₅₃	C₂²	1	4	yes	complemented
84	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
85	C₅₃×#416	C₂²	1	2	yes	—
86	C₅₃×#416	C₂²	1	4	yes	—
87	C₂×C₂²⋊C₈×C₅₃	C₂²	1	2	yes	—
88	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
89	C₂×#41×C₅₃	—	2	2	no	—
90	C₂×#41×C₅₃	C₂³	1	2	yes	—
91	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
92	C₂³×C₄×C₅₃	C₂³	1	2	yes	—
93	#135×C₅₃	—	2	4	no	—
94	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
95	C₂×SD₁₆×C₅₃	C₂³	1	2	yes	—
96	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
97	C₂²×C₈×C₅₃	—	2	8	no	—
98	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
99	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
100	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
101	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
102	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
103	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
104	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
105	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
106	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
107	#126×C₅₃	—	2	4	no	—
108	#126×C₅₃	—	2	4	no	—
109	#126×C₅₃	—	2	4	no	—
110	#125×C₅₃	—	2	4	no	—
111	#125×C₅₃	—	2	2	no	—
112	#125×C₅₃	C₂³	1	4	yes	—
113	#124×C₅₃	—	2	4	no	—
114	#124×C₅₃	—	2	2	no	—
115	#123×C₅₃	—	2	4	no	—
116	#123×C₅₃	—	2	4	no	—
117	#123×C₅₃	—	2	4	no	—
118	#123×C₅₃	—	2	4	no	—
119	C₂²⋊Q₈×C₅₃	—	2	2	no	—
120	C₂²⋊Q₈×C₅₃	—	2	4	no	—
121	C₂²⋊Q₈×C₅₃	—	2	2	no	—
122	C₂²⋊Q₈×C₅₃	—	2	2	no	—
123	C₂²⋊Q₈×C₅₃	C₂³	1	4	yes	—
124	C₂²⋊Q₈×C₅₃	C₂³	1	4	yes	—
125	#121×C₅₃	—	2	4	no	—
126	#121×C₅₃	—	2	2	no	—
127	#121×C₅₃	C₂³	1	4	yes	—
128	#121×C₅₃	C₂³	1	4	yes	—
129	#120×C₅₃	—	2	4	no	—
130	C₄×Q₈×C₅₃	—	2	2	no	—
131	C₄×Q₈×C₅₃	—	2	2	no	—
132	C₄×Q₈×C₅₃	—	2	2	no	—
133	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
134	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
135	C₄×D₄×C₅₃	—	2	4	no	—
136	C₄×D₄×C₅₃	—	2	2	no	—
137	C₄×D₄×C₅₃	—	2	4	no	—
138	C₄×D₄×C₅₃	—	2	2	no	—
139	C₄×D₄×C₅₃	—	2	2	no	—
140	C₄×D₄×C₅₃	—	2	4	no	—
141	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
142	C₄×D₄×C₅₃	D₄	1	2	yes	—
143	C₄×D₄×C₅₃	D₄	1	1	yes	characteristic
144	C₄×D₄×C₅₃	C₂³	1	2	yes	—
145	C₄×D₄×C₅₃	C₂³	1	2	yes	—
146	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
147	C₄×D₄×C₅₃	C₂³	1	2	yes	—
148	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
149	#117×C₅₃	—	2	2	no	—
150	#117×C₅₃	—	2	4	no	—
151	#117×C₅₃	—	2	2	no	—
152	#117×C₅₃	—	2	4	no	—
153	#117×C₅₃	—	2	4	no	—
154	#117×C₅₃	C₂³	1	4	yes	—
155	#117×C₅₃	C₂³	1	4	yes	—
156	#117×C₅₃	C₂³	1	2	yes	—
157	#117×C₅₃	C₂³	1	4	yes	—
158	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
159	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
160	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
161	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
162	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
163	C₂×C₄²×C₅₃	—	2	2	no	—
164	C₂×C₄²×C₅₃	—	2	4	no	—
165	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
166	#107×C₅₃	C₂³	1	2	yes	—
167	#107×C₅₃	C₂³	1	4	yes	—
168	#107×C₅₃	C₂³	1	2	yes	—
169	#107×C₅₃	C₂³	1	2	yes	—
170	#107×C₅₃	C₂³	1	2	yes	—
171	#107×C₅₃	C₂³	1	1	yes	characteristic
172	#107×C₅₃	C₂³	1	1	yes	characteristic
173	#107×C₅₃	C₂³	1	4	yes	—
174	#106×C₅₃	C₂³	1	4	yes	—
175	#106×C₅₃	C₂³	1	4	yes	—
176	#106×C₅₃	C₂³	1	4	yes	—
177	#106×C₅₃	C₂³	1	2	yes	—
178	#106×C₅₃	C₂³	1	4	yes	—
179	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
180	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
181	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
182	#103×C₅₃	C₂³	1	2	yes	—
183	#103×C₅₃	C₂³	1	2	yes	—
184	#103×C₅₃	C₂³	1	4	yes	—
185	#103×C₅₃	C₂³	1	4	yes	—
186	#102×C₅₃	C₂³	1	2	yes	—
187	#102×C₅₃	C₂³	1	2	yes	—
188	#102×C₅₃	C₂³	1	4	yes	—
189	#102×C₅₃	C₂³	1	4	yes	—
190	C₂²⋊C₈×C₅₃	C₂³	1	2	yes	—
191	C₂²⋊C₈×C₅₃	C₂³	1	2	yes	—
192	C₂²⋊C₈×C₅₃	C₂³	1	2	yes	—
193	C₂²⋊C₈×C₅₃	C₂³	1	2	yes	—
194	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
195	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
196	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
197	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
198	C₂⁴×C₅₃	C₂×D₄	1	2	yes	—
199	#41×C₅₃	—	2	4	no	—
200	#41×C₅₃	—	2	4	no	—
201	#41×C₅₃	—	4	8	no	—
202	C₂×Q₈×C₅₃	—	2	2	no	—
203	C₂×Q₈×C₅₃	—	2	2	no	—
204	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
205	C₂×D₄×C₅₃	—	2	4	no	—
206	C₂×D₄×C₅₃	—	2	2	no	—
207	C₂×D₄×C₅₃	—	2	4	no	—
208	C₂×D₄×C₅₃	—	2	4	no	—
209	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
210	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
211	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
212	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
213	C₂²×C₄×C₅₃	—	2	4	no	—
214	C₂²×C₄×C₅₃	—	2	4	no	—
215	C₂²×C₄×C₅₃	—	2	4	no	—
216	C₂²×C₄×C₅₃	—	2	4	no	—
217	C₂²×C₄×C₅₃	—	2	2	no	—
218	C₂²×C₄×C₅₃	—	2	2	no	—
219	C₂²×C₄×C₅₃	—	2	2	no	—
220	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
221	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
222	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
223	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
224	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
225	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
226	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
227	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
228	Q₁₆×C₅₃	—	2	2	no	—
229	SD₁₆×C₅₃	—	2	4	no	—
230	D₈×C₅₃	—	2	2	no	—
231	C₂×C₈×C₅₃	—	2	4	no	—
232	C₂×C₈×C₅₃	—	2	2	no	—
233	C₂×C₈×C₅₃	—	2	4	no	—
234	C₂×C₈×C₅₃	—	2	4	no	—
235	C₂×C₈×C₅₃	—	2	4	no	—
236	C₂×C₈×C₅₃	—	2	4	no	—
237	C₂×C₈×C₅₃	#41	1	4	yes	complemented
238	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
239	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
240	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
241	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
242	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
243	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
244	C₄⋊C₄×C₅₃	—	2	4	no	—
245	C₄⋊C₄×C₅₃	—	2	4	no	—
246	C₄⋊C₄×C₅₃	—	2	2	no	—
247	C₄⋊C₄×C₅₃	—	2	4	no	—
248	C₄⋊C₄×C₅₃	—	2	4	no	—
249	C₄⋊C₄×C₅₃	—	2	2	no	—
250	C₄⋊C₄×C₅₃	—	2	2	no	—
251	C₄⋊C₄×C₅₃	—	2	4	no	—
252	C₄⋊C₄×C₅₃	—	2	4	no	—
253	C₄⋊C₄×C₅₃	—	2	2	no	—
254	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
255	C₄⋊C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
256	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
257	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
258	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
259	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
260	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
261	C₂²⋊C₄×C₅₃	—	2	2	no	—
262	C₂²⋊C₄×C₅₃	—	2	4	no	—
263	C₂²⋊C₄×C₅₃	—	2	4	no	—
264	C₂²⋊C₄×C₅₃	—	2	4	no	—
265	C₂²⋊C₄×C₅₃	—	2	4	no	—
266	C₂²⋊C₄×C₅₃	—	2	4	no	—
267	C₂²⋊C₄×C₅₃	—	2	4	no	—
268	C₂²⋊C₄×C₅₃	—	2	2	no	—
269	C₂²⋊C₄×C₅₃	—	2	2	no	—
270	C₂²⋊C₄×C₅₃	—	2	4	no	—
271	C₂²⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
272	C₂²⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
273	C₂²⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
274	C₂²⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
275	C₄²×C₅₃	—	2	4	no	—
276	C₄²×C₅₃	—	2	2	no	—
277	C₄²×C₅₃	—	2	4	no	—
278	C₄²×C₅₃	—	2	2	no	—
279	C₄²×C₅₃	—	2	2	no	—
280	C₄²×C₅₃	—	2	4	no	—
281	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
282	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
283	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
284	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
285	C₂³×C₅₃	—	2	4	no	—
286	C₂³×C₅₃	—	2	4	no	—
287	C₂³×C₅₃	—	2	2	no	—
288	C₂³×C₅₃	—	4	2	no	—
289	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
290	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
291	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
292	Q₈×C₅₃	—	2	2	no	—
293	Q₈×C₅₃	—	2	2	no	—
294	Q₈×C₅₃	—	4	4	no	—
295	Q₈×C₅₃	—	8	2	no	—
296	D₄×C₅₃	—	2	4	no	—
297	D₄×C₅₃	—	2	2	no	—
298	D₄×C₅₃	—	2	2	no	—
299	D₄×C₅₃	—	2	2	no	—
300	D₄×C₅₃	—	4	4	no	—
301	D₄×C₅₃	—	4	8	no	—
302	D₄×C₅₃	—	8	4	no	—
303	D₄×C₅₃	#135	1	4	yes	complemented
304	C₂×C₄×C₅₃	—	2	4	no	—
305	C₂×C₄×C₅₃	—	2	4	no	—
306	C₂×C₄×C₅₃	—	2	4	no	—
307	C₂×C₄×C₅₃	—	2	2	no	—
308	C₂×C₄×C₅₃	—	2	2	no	—
309	C₂×C₄×C₅₃	—	2	2	no	—
310	C₂×C₄×C₅₃	—	2	2	no	—
311	C₂×C₄×C₅₃	—	2	2	no	—
312	C₂×C₄×C₅₃	—	2	2	no	—
313	C₂×C₄×C₅₃	—	2	2	no	—
314	C₂×C₄×C₅₃	—	2	4	no	—
315	C₂×C₄×C₅₃	—	2	4	no	—
316	C₂×C₄×C₅₃	—	2	2	no	—
317	C₂×C₄×C₅₃	—	4	1	no	—
318	C₂×C₄×C₅₃	—	4	8	no	—
319	C₂×C₄×C₅₃	—	4	4	no	—
320	C₂×C₄×C₅₃	—	4	4	no	—
321	C₂×C₄×C₅₃	—	8	2	no	—
322	C₂×C₄×C₅₃	—	8	4	no	—
323	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
324	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
325	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
326	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
327	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
328	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
329	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
330	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
331	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
332	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
333	C₈×C₅₃	—	2	2	no	—
334	C₈×C₅₃	—	2	2	no	—
335	C₈×C₅₃	—	2	1	no	—
336	C₈×C₅₃	—	2	1	no	—
337	C₈×C₅₃	C₂×#41	1	2	yes	—
338	C₈×C₅₃	C₂×#41	1	2	yes	—
339	#61234	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
340	C₂²×C₅₃	—	2	1	no	—
341	C₂²×C₅₃	—	2	2	no	—
342	C₂²×C₅₃	—	2	2	no	—
343	C₂²×C₅₃	—	2	1	no	—
344	C₂²×C₅₃	—	2	2	no	—
345	C₂²×C₅₃	—	2	2	no	—
346	C₂²×C₅₃	—	2	2	no	—
347	C₂²×C₅₃	—	4	8	no	—
348	C₂²×C₅₃	—	4	2	no	—
349	C₂²×C₅₃	—	4	2	no	—
350	C₂²×C₅₃	—	4	2	no	—
351	C₂²×C₅₃	#547	1	2	yes	—
352	C₂²×C₅₃	C₂×#135	1	4	yes	—
353	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
354	C₄×C₅₃	—	2	1	no	—
355	C₄×C₅₃	—	2	1	no	—
356	C₄×C₅₃	—	2	2	no	—
357	C₄×C₅₃	—	2	2	no	—
358	C₄×C₅₃	—	2	1	no	—
359	C₄×C₅₃	—	4	2	no	—
360	C₄×C₅₃	—	4	4	no	—
361	C₄×C₅₃	—	4	2	no	—
362	C₄×C₅₃	—	4	2	no	—
363	C₄×C₅₃	—	4	4	no	—
364	C₄×C₅₃	#547	1	2	yes	—
365	C₄×C₅₃	C₂×#135	1	2	yes	—
366	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
367	C₄×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
368	#3475	C₂×C₅₃	1	2	yes	complemented
369	#3342	C₂×C₅₃	1	1	yes	characteristic, complemented
370	#3341	C₂×C₅₃	1	1	yes	characteristic, complemented
371	#3340	C₂×C₅₃	1	2	yes	complemented
372	#3339	C₂×C₅₃	1	2	yes	complemented
373	#3338	C₂×C₅₃	1	1	yes	characteristic, complemented
374	#3337	C₂×C₅₃	1	4	yes	complemented
375	#3336	C₂×C₅₃	1	2	yes	complemented
376	#3335	C₂×C₅₃	1	2	yes	complemented
377	#3334	C₂×C₅₃	1	1	yes	characteristic, complemented
378	#3160	C₂×C₅₃	1	1	yes	characteristic, complemented
379	#3091	C₂×C₅₃	1	4	yes	complemented
380	#3076	C₂×C₅₃	1	2	yes	complemented
381	#3050	C₂×C₅₃	1	2	yes	complemented
382	C₄×#135	C₂×C₅₃	1	1	yes	characteristic, complemented
383	C₂×C₈×D₄	C₂×C₅₃	1	1	yes	characteristic, complemented
384	C₂×#427	C₂×C₅₃	1	2	yes	complemented
385	C₂×C₅₃	—	2	2	no	—
386	C₂×C₅₃	—	2	4	no	—
387	C₂×C₅₃	—	4	2	no	—
388	C₂×C₅₃	—	8	2	no	—
389	C₂×C₅₃	C₂×#547	1	1	yes	characteristic
390	C₂×C₅₃	C₂²×#135	1	1	yes	characteristic
391	C₂×C₅₃	#3584	1	1	yes	characteristic
392	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
393	C₂³×C₈	C₂²×C₅₃	1	2	yes	—
394	#556	C₂²×C₅₃	1	2	yes	—
395	#555	C₂²×C₅₃	1	2	yes	—
396	#554	C₂²×C₅₃	1	2	yes	—
397	#553	C₂²×C₅₃	1	4	yes	—
398	#552	—	2	2	no	—
399	#552	C₂²×C₅₃	1	4	yes	—
400	#551	—	2	4	no	—
401	#547	C₂²×C₅₃	1	2	yes	—
402	#546	—	2	2	no	—
403	#533	C₂²×C₅₃	1	2	yes	—
404	#529	—	2	4	no	—
405	#529	—	2	2	no	—
406	#525	C₂²×C₅₃	1	4	yes	—
407	C₄×#41	—	2	2	no	—
408	C₄×#41	C₂²×C₅₃	1	2	yes	—
409	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
410	C₂×#117	C₂²×C₅₃	1	4	yes	—
411	#500	C₂²×C₅₃	1	1	yes	characteristic, complemented
412	#500	C₂²×C₅₃	1	1	yes	characteristic, complemented
413	#499	C₂²×C₅₃	1	2	yes	complemented
414	#499	C₂²×C₅₃	1	2	yes	complemented
415	#498	C₂²×C₅₃	1	2	yes	complemented
416	#485	C₂²×C₅₃	1	4	yes	complemented
417	#485	C₂²×C₅₃	1	4	yes	complemented
418	#484	C₂²×C₅₃	1	4	yes	complemented
419	#483	C₂²×C₅₃	1	4	yes	complemented
420	#482	C₂²×C₅₃	1	4	yes	complemented
421	#482	C₂²×C₅₃	1	4	yes	complemented
422	#481	C₂²×C₅₃	1	4	yes	complemented
423	#480	C₂²×C₅₃	1	4	yes	complemented
424	#479	C₂²×C₅₃	1	4	yes	complemented
425	#478	C₂²×C₅₃	1	4	yes	complemented
426	#477	C₂²×C₅₃	1	2	yes	complemented
427	#476	C₂²×C₅₃	1	2	yes	complemented
428	#475	C₂²×C₅₃	1	2	yes	complemented
429	#474	C₂²×C₅₃	1	2	yes	complemented
430	#467	C₂²×C₅₃	1	2	yes	complemented
431	#467	C₂²×C₅₃	1	2	yes	complemented
432	#466	C₂²×C₅₃	1	2	yes	complemented
433	#466	C₂²×C₅₃	1	2	yes	complemented
434	#465	C₂²×C₅₃	1	4	yes	complemented
435	#465	C₂²×C₅₃	1	4	yes	complemented
436	C₄×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic, complemented
437	C₄×Q₁₆	C₂²×C₅₃	1	1	yes	characteristic, complemented
438	C₄×SD₁₆	C₂²×C₅₃	1	2	yes	complemented
439	C₄×SD₁₆	C₂²×C₅₃	1	2	yes	complemented
440	C₄×D₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
441	C₄×D₈	C₂²×C₅₃	1	1	yes	characteristic, complemented
442	C₈×D₄	C₂²×C₅₃	1	1	yes	characteristic
443	C₈×D₄	C₂²×C₅₃	1	1	yes	characteristic
444	C₈×D₄	C₂²×C₅₃	1	2	yes	—
445	C₈×D₄	C₂²×C₅₃	1	2	yes	—
446	C₈×D₄	C₂²×C₅₃	1	1	yes	characteristic
447	C₈×D₄	C₂²×C₅₃	1	1	yes	characteristic
448	#427	C₂²×C₅₃	1	4	yes	—
449	#427	C₂²×C₅₃	1	2	yes	—
450	#427	C₂²×C₅₃	1	2	yes	—
451	#427	C₂²×C₅₃	1	1	yes	characteristic
452	#427	C₂²×C₅₃	1	2	yes	complemented
453	#427	C₂²×C₅₃	1	4	yes	complemented
454	#427	C₂²×C₅₃	1	2	yes	complemented
455	C₂×#107	C₂²×C₅₃	1	2	yes	complemented
456	C₂×#107	C₂²×C₅₃	1	2	yes	complemented
457	C₂×#106	C₂²×C₅₃	1	4	yes	complemented
458	C₂×C₄⋊C₈	C₂²×C₅₃	1	1	yes	characteristic
459	#416	C₂²×C₅₃	1	4	yes	—
460	#416	C₂²×C₅₃	1	2	yes	—
461	C₂×C₂²⋊C₈	C₂²×C₅₃	1	2	yes	—
462	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
463	C₅₃	#61234	1	1	yes	characteristic, direct product factor, Sylow
464	C₂×#41	—	2	2	no	—
465	C₂×#41	C₂³×C₅₃	1	2	yes	—
466	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
467	C₂³×C₄	C₂³×C₅₃	1	2	yes	—
468	#135	—	2	4	no	—
469	C₂×Q₁₆	C₂³×C₅₃	1	1	yes	characteristic
470	C₂×SD₁₆	C₂³×C₅₃	1	2	yes	—
471	C₂×D₈	C₂³×C₅₃	1	1	yes	characteristic
472	C₂²×C₈	—	2	8	no	—
473	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
474	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
475	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
476	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
477	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
478	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
479	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
480	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
481	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
482	#126	—	2	4	no	—
483	#126	—	2	4	no	—
484	#126	—	2	4	no	—
485	#125	—	2	4	no	—
486	#125	—	2	2	no	—
487	#125	C₂³×C₅₃	1	4	yes	—
488	#124	—	2	2	no	—
489	#124	—	2	4	no	—
490	#123	—	2	4	no	—
491	#123	—	2	4	no	—
492	#123	—	2	4	no	—
493	#123	—	2	4	no	—
494	C₂²⋊Q₈	—	2	2	no	—
495	C₂²⋊Q₈	—	2	2	no	—
496	C₂²⋊Q₈	—	2	2	no	—
497	C₂²⋊Q₈	—	2	4	no	—
498	C₂²⋊Q₈	C₂³×C₅₃	1	4	yes	—
499	C₂²⋊Q₈	C₂³×C₅₃	1	4	yes	—
500	#121	—	2	4	no	—
501	#121	—	2	2	no	—
502	#121	C₂³×C₅₃	1	4	yes	—
503	#121	C₂³×C₅₃	1	4	yes	—
504	#120	—	2	4	no	—
505	C₄×Q₈	—	2	2	no	—
506	C₄×Q₈	—	2	2	no	—
507	C₄×Q₈	—	2	2	no	—
508	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
509	C₄×Q₈	C₂³×C₅₃	1	2	yes	—
510	C₄×D₄	—	2	2	no	—
511	C₄×D₄	—	2	4	no	—
512	C₄×D₄	—	2	2	no	—
513	C₄×D₄	—	2	2	no	—
514	C₄×D₄	—	2	4	no	—
515	C₄×D₄	—	2	4	no	—
516	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
517	C₄×D₄	D₄×C₅₃	1	1	yes	characteristic
518	C₄×D₄	D₄×C₅₃	1	2	yes	—
519	C₄×D₄	C₂³×C₅₃	1	2	yes	—
520	C₄×D₄	C₂³×C₅₃	1	2	yes	—
521	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
522	C₄×D₄	C₂³×C₅₃	1	2	yes	—
523	C₄×D₄	C₂³×C₅₃	1	1	yes	characteristic
524	#117	—	2	4	no	—
525	#117	—	2	4	no	—
526	#117	—	2	2	no	—
527	#117	—	2	4	no	—
528	#117	—	2	2	no	—
529	#117	C₂³×C₅₃	1	2	yes	—
530	#117	C₂³×C₅₃	1	4	yes	—
531	#117	C₂³×C₅₃	1	4	yes	—
532	#117	C₂³×C₅₃	1	4	yes	—
533	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
534	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
535	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
536	C₂×C₂²⋊C₄	—	2	4	no	—
537	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
538	C₂×C₄²	—	2	4	no	—
539	C₂×C₄²	—	2	2	no	—
540	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
541	#107	C₂³×C₅₃	1	1	yes	characteristic
542	#107	C₂³×C₅₃	1	1	yes	characteristic
543	#107	C₂³×C₅₃	1	2	yes	—
544	#107	C₂³×C₅₃	1	2	yes	—
545	#107	C₂³×C₅₃	1	4	yes	—
546	#107	C₂³×C₅₃	1	4	yes	—
547	#107	C₂³×C₅₃	1	2	yes	—
548	#107	C₂³×C₅₃	1	2	yes	—
549	#106	C₂³×C₅₃	1	2	yes	—
550	#106	C₂³×C₅₃	1	4	yes	—
551	#106	C₂³×C₅₃	1	4	yes	—
552	#106	C₂³×C₅₃	1	4	yes	—
553	#106	C₂³×C₅₃	1	4	yes	—
554	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
555	C₄⋊C₈	C₂³×C₅₃	1	1	yes	characteristic
556	C₄⋊C₈	C₂³×C₅₃	1	2	yes	—
557	#103	C₂³×C₅₃	1	2	yes	—
558	#103	C₂³×C₅₃	1	2	yes	—
559	#103	C₂³×C₅₃	1	4	yes	—
560	#103	C₂³×C₅₃	1	4	yes	—
561	#102	C₂³×C₅₃	1	2	yes	—
562	#102	C₂³×C₅₃	1	2	yes	—
563	#102	C₂³×C₅₃	1	4	yes	—
564	#102	C₂³×C₅₃	1	4	yes	—
565	C₂²⋊C₈	C₂³×C₅₃	1	2	yes	—
566	C₂²⋊C₈	C₂³×C₅₃	1	2	yes	—
567	C₂²⋊C₈	C₂³×C₅₃	1	2	yes	—
568	C₂²⋊C₈	C₂³×C₅₃	1	2	yes	—
569	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
570	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
571	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
572	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
573	C₂⁴	C₂×D₄×C₅₃	1	2	yes	—
574	#41	—	2	4	no	—
575	#41	—	2	4	no	—
576	#41	—	4	8	no	—
577	C₂×Q₈	—	2	2	no	—
578	C₂×Q₈	—	2	2	no	—
579	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
580	C₂×D₄	—	2	4	no	—
581	C₂×D₄	—	2	2	no	—
582	C₂×D₄	—	2	4	no	—
583	C₂×D₄	—	2	4	no	—
584	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
585	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
586	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
587	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
588	C₂²×C₄	—	2	4	no	—
589	C₂²×C₄	—	2	4	no	—
590	C₂²×C₄	—	2	4	no	—
591	C₂²×C₄	—	2	4	no	—
592	C₂²×C₄	—	2	2	no	—
593	C₂²×C₄	—	2	2	no	—
594	C₂²×C₄	—	2	2	no	—
595	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
596	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
597	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
598	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
599	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
600	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
601	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
602	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
603	Q₁₆	—	2	2	no	—
604	SD₁₆	—	2	4	no	—
605	D₈	—	2	2	no	—
606	C₂×C₈	—	2	4	no	—
607	C₂×C₈	—	2	2	no	—
608	C₂×C₈	—	2	4	no	—
609	C₂×C₈	—	2	4	no	—
610	C₂×C₈	—	2	4	no	—
611	C₂×C₈	—	2	4	no	—
612	C₂×C₈	#41×C₅₃	1	4	yes	complemented
613	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
614	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
615	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
616	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
617	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
618	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
619	C₄⋊C₄	—	2	4	no	—
620	C₄⋊C₄	—	2	4	no	—
621	C₄⋊C₄	—	2	2	no	—
622	C₄⋊C₄	—	2	4	no	—
623	C₄⋊C₄	—	2	4	no	—
624	C₄⋊C₄	—	2	2	no	—
625	C₄⋊C₄	—	2	2	no	—
626	C₄⋊C₄	—	2	4	no	—
627	C₄⋊C₄	—	2	4	no	—
628	C₄⋊C₄	—	2	2	no	—
629	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
630	C₄⋊C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
631	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
632	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
633	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
634	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
635	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
636	C₂²⋊C₄	—	2	2	no	—
637	C₂²⋊C₄	—	2	4	no	—
638	C₂²⋊C₄	—	2	4	no	—
639	C₂²⋊C₄	—	2	4	no	—
640	C₂²⋊C₄	—	2	4	no	—
641	C₂²⋊C₄	—	2	4	no	—
642	C₂²⋊C₄	—	2	4	no	—
643	C₂²⋊C₄	—	2	2	no	—
644	C₂²⋊C₄	—	2	2	no	—
645	C₂²⋊C₄	—	2	4	no	—
646	C₂²⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
647	C₂²⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
648	C₂²⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
649	C₂²⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
650	C₄²	—	2	2	no	—
651	C₄²	—	2	4	no	—
652	C₄²	—	2	2	no	—
653	C₄²	—	2	4	no	—
654	C₄²	—	2	4	no	—
655	C₄²	—	2	2	no	—
656	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
657	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
658	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
659	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
660	C₂³	—	2	4	no	—
661	C₂³	—	2	4	no	—
662	C₂³	—	2	2	no	—
663	C₂³	—	4	2	no	—
664	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
665	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
666	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
667	Q₈	—	2	2	no	—
668	Q₈	—	2	2	no	—
669	Q₈	—	4	4	no	—
670	Q₈	—	8	2	no	—
671	D₄	—	2	4	no	—
672	D₄	—	2	2	no	—
673	D₄	—	2	2	no	—
674	D₄	—	2	2	no	—
675	D₄	—	4	4	no	—
676	D₄	—	4	8	no	—
677	D₄	—	8	4	no	—
678	D₄	#135×C₅₃	1	4	yes	complemented
679	C₂×C₄	—	2	4	no	—
680	C₂×C₄	—	2	4	no	—
681	C₂×C₄	—	2	4	no	—
682	C₂×C₄	—	2	2	no	—
683	C₂×C₄	—	2	2	no	—
684	C₂×C₄	—	2	2	no	—
685	C₂×C₄	—	2	2	no	—
686	C₂×C₄	—	2	2	no	—
687	C₂×C₄	—	2	2	no	—
688	C₂×C₄	—	2	2	no	—
689	C₂×C₄	—	2	2	no	—
690	C₂×C₄	—	2	4	no	—
691	C₂×C₄	—	2	4	no	—
692	C₂×C₄	—	4	1	no	—
693	C₂×C₄	—	4	8	no	—
694	C₂×C₄	—	4	4	no	—
695	C₂×C₄	—	4	4	no	—
696	C₂×C₄	—	8	2	no	—
697	C₂×C₄	—	8	4	no	—
698	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
699	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
700	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
701	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
702	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
703	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
704	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
705	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
706	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
707	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
708	C₈	—	2	2	no	—
709	C₈	—	2	2	no	—
710	C₈	—	2	1	no	—
711	C₈	—	2	1	no	—
712	C₈	C₂×#41×C₅₃	1	2	yes	—
713	C₈	C₂×#41×C₅₃	1	2	yes	—
714	C₂²	—	2	2	no	—
715	C₂²	—	2	1	no	—
716	C₂²	—	2	1	no	—
717	C₂²	—	2	2	no	—
718	C₂²	—	2	2	no	—
719	C₂²	—	2	2	no	—
720	C₂²	—	2	2	no	—
721	C₂²	—	4	8	no	—
722	C₂²	—	4	2	no	—
723	C₂²	—	4	2	no	—
724	C₂²	—	4	2	no	—
725	C₂²	C₅₃×#547	1	2	yes	—
726	C₂²	C₂×#135×C₅₃	1	4	yes	—
727	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
728	C₄	—	2	2	no	—
729	C₄	—	2	2	no	—
730	C₄	—	2	1	no	—
731	C₄	—	2	1	no	—
732	C₄	—	2	1	no	—
733	C₄	—	4	2	no	—
734	C₄	—	4	2	no	—
735	C₄	—	4	4	no	—
736	C₄	—	4	2	no	—
737	C₄	—	4	4	no	—
738	C₄	C₅₃×#547	1	2	yes	—
739	C₄	C₂×#135×C₅₃	1	2	yes	—
740	C₄	C₂²×#41×C₅₃	1	1	yes	characteristic
741	C₄	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
742	C₂	—	2	2	no	—
743	C₂	—	2	4	no	—
744	C₂	—	4	2	no	—
745	C₂	—	8	2	no	—
746	C₂	C₂×C₅₃×#547	1	1	yes	characteristic
747	C₂	C₂²×#135×C₅₃	1	1	yes	characteristic
748	C₂	C₅₃×#3584	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
socle	C₂²×C₅₃	C₂³×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×C₄×D₄×C₅₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₅₃×#3475	C₂	semidirect
2. C₅₃×#3342	C₂	semidirect
3. C₅₃×#3341	C₂	semidirect
4. C₅₃×#3340	C₂	semidirect
5. C₅₃×#3339	C₂	semidirect
6. C₅₃×#3338	C₂	semidirect
7. C₅₃×#3337	C₂	semidirect
8. C₅₃×#3336	C₂	semidirect
9. C₅₃×#3335	C₂	semidirect
10. C₅₃×#3334	C₂	semidirect
11. C₅₃×#3160	C₂	semidirect
12. C₅₃×#3091	C₂	semidirect
13. C₅₃×#3076	C₂	semidirect
14. C₅₃×#3050	C₂	semidirect
15. C₄×#135×C₅₃	C₂	semidirect
16. C₂×C₈×D₄×C₅₃	C₂	semidirect
17. C₂×C₅₃×#427	C₂	semidirect
37. C₅₃×#500	C₂²	semidirect
38. C₅₃×#500	C₂²	semidirect
39. C₅₃×#499	C₂²	semidirect
40. C₅₃×#499	C₂²	semidirect
41. C₅₃×#498	C₂²	semidirect
42. C₅₃×#485	C₂²	semidirect
43. C₅₃×#485	C₂²	semidirect
44. C₅₃×#484	C₂²	semidirect
45. C₅₃×#483	C₂²	semidirect
46. C₅₃×#482	C₂²	semidirect
47. C₅₃×#482	C₂²	semidirect
48. C₅₃×#481	C₂²	semidirect
49. C₅₃×#480	C₂²	semidirect
50. C₅₃×#479	C₂²	semidirect
51. C₅₃×#478	C₂²	semidirect
52. C₅₃×#477	C₂²	semidirect
53. C₅₃×#476	C₂²	semidirect
54. C₅₃×#475	C₂²	semidirect
55. C₅₃×#474	C₂²	semidirect
56. C₅₃×#467	C₂²	semidirect
57. C₅₃×#467	C₂²	semidirect
58. C₅₃×#466	C₂²	semidirect
59. C₅₃×#466	C₂²	semidirect
60. C₅₃×#465	C₂²	semidirect
61. C₅₃×#465	C₂²	semidirect
62. C₄×Q₁₆×C₅₃	C₂²	semidirect
63. C₄×Q₁₆×C₅₃	C₂²	semidirect
64. C₄×SD₁₆×C₅₃	C₂²	semidirect
65. C₄×SD₁₆×C₅₃	C₂²	semidirect
66. C₄×D₈×C₅₃	C₂²	semidirect
67. C₄×D₈×C₅₃	C₂²	semidirect
78. C₅₃×#427	C₂²	semidirect
79. C₅₃×#427	C₂²	semidirect
80. C₅₃×#427	C₂²	semidirect
81. C₂×#107×C₅₃	C₂²	semidirect
82. C₂×#107×C₅₃	C₂²	semidirect
83. C₂×#106×C₅₃	C₂²	semidirect
237. C₂×C₈×C₅₃	#41	semidirect
303. D₄×C₅₃	#135	semidirect
339. #61234	C₅₃	direct (N × H)
368. #3475	C₂×C₅₃	semidirect
369. #3342	C₂×C₅₃	semidirect
370. #3341	C₂×C₅₃	semidirect
371. #3340	C₂×C₅₃	semidirect
372. #3339	C₂×C₅₃	semidirect
373. #3338	C₂×C₅₃	semidirect
374. #3337	C₂×C₅₃	semidirect
375. #3336	C₂×C₅₃	semidirect
376. #3335	C₂×C₅₃	semidirect
377. #3334	C₂×C₅₃	semidirect
378. #3160	C₂×C₅₃	semidirect
379. #3091	C₂×C₅₃	semidirect
380. #3076	C₂×C₅₃	semidirect
381. #3050	C₂×C₅₃	semidirect
382. C₄×#135	C₂×C₅₃	semidirect
383. C₂×C₈×D₄	C₂×C₅₃	semidirect
384. C₂×#427	C₂×C₅₃	semidirect
411. #500	C₂²×C₅₃	semidirect
412. #500	C₂²×C₅₃	semidirect
413. #499	C₂²×C₅₃	semidirect
414. #499	C₂²×C₅₃	semidirect
415. #498	C₂²×C₅₃	semidirect
416. #485	C₂²×C₅₃	semidirect
417. #485	C₂²×C₅₃	semidirect
418. #484	C₂²×C₅₃	semidirect
419. #483	C₂²×C₅₃	semidirect
420. #482	C₂²×C₅₃	semidirect
421. #482	C₂²×C₅₃	semidirect
422. #481	C₂²×C₅₃	semidirect
423. #480	C₂²×C₅₃	semidirect
424. #479	C₂²×C₅₃	semidirect
425. #478	C₂²×C₅₃	semidirect
426. #477	C₂²×C₅₃	semidirect
427. #476	C₂²×C₅₃	semidirect
428. #475	C₂²×C₅₃	semidirect
429. #474	C₂²×C₅₃	semidirect
430. #467	C₂²×C₅₃	semidirect
431. #467	C₂²×C₅₃	semidirect
432. #466	C₂²×C₅₃	semidirect
433. #466	C₂²×C₅₃	semidirect
434. #465	C₂²×C₅₃	semidirect
435. #465	C₂²×C₅₃	semidirect
436. C₄×Q₁₆	C₂²×C₅₃	semidirect
437. C₄×Q₁₆	C₂²×C₅₃	semidirect
438. C₄×SD₁₆	C₂²×C₅₃	semidirect
439. C₄×SD₁₆	C₂²×C₅₃	semidirect
440. C₄×D₈	C₂²×C₅₃	semidirect
441. C₄×D₈	C₂²×C₅₃	semidirect
452. #427	C₂²×C₅₃	semidirect
453. #427	C₂²×C₅₃	semidirect
454. #427	C₂²×C₅₃	semidirect
455. C₂×#107	C₂²×C₅₃	semidirect
456. C₂×#107	C₂²×C₅₃	semidirect
457. C₂×#106	C₂²×C₅₃	semidirect
463. C₅₃	#61234	direct (N × H)
612. C₂×C₈	#41×C₅₃	semidirect
678. D₄	#135×C₅₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅₃×#3475	2
C₅₃×#3342	2
C₅₃×#3341	2
C₅₃×#3340	2
C₅₃×#3339	2
C₅₃×#3338	2
C₅₃×#3337	2
C₅₃×#3336	2
C₅₃×#3335	2
C₅₃×#3334	2
C₅₃×#3160	2
C₅₃×#3091	2
C₅₃×#3076	2
C₅₃×#3050	2
C₄×#135×C₅₃	2
C₂×C₈×D₄×C₅₃	2
C₂×C₅₃×#427	2
#61234	53

Subgroups — C53×#61234 #27268746

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#61234 #27268746