Subgroups — C₅₃×#61260 #27268772

Order 13568.

1782 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3493	C₂	1	2	yes	maximal, complemented
2	C₅₃×#3349	C₂	1	4	yes	maximal, complemented
3	C₅₃×#3347	C₂	1	4	yes	maximal, complemented
4	C₅₃×#3340	C₂	1	4	yes	maximal, complemented
5	C₅₃×#3337	C₂	1	4	yes	maximal, complemented
6	C₅₃×#3163	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#3098	C₂	1	2	yes	maximal, complemented
8	C₅₃×#3093	C₂	1	2	yes	maximal, complemented
9	C₅₃×#3083	C₂	1	2	yes	maximal, complemented
10	C₅₃×#3060	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#3050	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅₃×#2952	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅₃×#2929	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅₃×#2912	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₅₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#135×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#556	—	2	2	no	—
20	C₅₃×#556	C₂²	1	4	yes	—
21	C₅₃×#555	—	2	2	no	—
22	C₅₃×#555	C₂²	1	4	yes	—
23	C₅₃×#553	—	2	2	no	—
24	C₅₃×#553	C₂²	1	4	yes	—
25	C₅₃×#552	—	2	2	no	—
26	C₅₃×#552	C₂²	1	4	yes	—
27	C₅₃×#544	C₂²	1	2	yes	—
28	C₅₃×#538	—	2	4	no	—
29	C₅₃×#537	C₂²	1	2	yes	—
30	C₅₃×#529	—	2	4	no	—
31	C₅₃×#525	C₂²	1	2	yes	—
32	C₅₃×#525	C₂²	1	1	yes	characteristic
33	C₅₃×#520	C₂²	1	2	yes	—
34	C₅₃×#520	C₂²	1	2	yes	—
35	C₂×#117×C₅₃	C₂²	1	2	yes	—
36	C₅₃×#501	C₂²	1	4	yes	complemented
37	C₅₃×#499	C₂²	1	2	yes	complemented
38	C₅₃×#498	C₂²	1	2	yes	complemented
39	C₅₃×#485	C₂²	1	4	yes	—
40	C₅₃×#485	C₂²	1	4	yes	—
41	C₅₃×#485	C₂²	1	4	yes	complemented
42	C₅₃×#484	C₂²	1	4	yes	complemented
43	C₅₃×#482	C₂²	1	4	yes	—
44	C₅₃×#482	C₂²	1	4	yes	—
45	C₅₃×#482	C₂²	1	4	yes	complemented
46	C₅₃×#480	C₂²	1	4	yes	complemented
47	C₅₃×#479	C₂²	1	4	yes	complemented
48	C₅₃×#478	C₂²	1	4	yes	complemented
49	C₅₃×#477	C₂²	1	4	yes	complemented
50	C₅₃×#474	C₂²	1	4	yes	complemented
51	C₅₃×#470	C₂²	1	2	yes	complemented
52	C₅₃×#469	C₂²	1	2	yes	complemented
53	C₅₃×#467	C₂²	1	2	yes	—
54	C₅₃×#466	C₂²	1	2	yes	—
55	C₅₃×#465	C₂²	1	2	yes	—
56	C₅₃×#465	C₂²	1	2	yes	—
57	C₅₃×#465	C₂²	1	4	yes	complemented
58	C₅₃×#442	C₂²	1	2	yes	complemented
59	C₅₃×#441	C₂²	1	2	yes	complemented
60	C₄×SD₁₆×C₅₃	C₂²	1	4	yes	complemented
61	C₅₃×#435	C₂²	1	4	yes	—
62	C₈×D₄×C₅₃	C₂²	1	4	yes	—
63	C₅₃×#428	C₂²	1	2	yes	—
64	C₅₃×#428	C₂²	1	2	yes	—
65	C₅₃×#428	C₂²	1	4	yes	complemented
66	C₅₃×#427	C₂²	1	1	yes	characteristic
67	C₅₃×#427	C₂²	1	4	yes	—
68	C₅₃×#427	C₂²	1	4	yes	—
69	C₅₃×#427	C₂²	1	2	yes	complemented
70	C₂×#107×C₅₃	C₂²	1	2	yes	—
71	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
72	C₂×#106×C₅₃	C₂²	1	1	yes	characteristic
73	C₂×#106×C₅₃	C₂²	1	2	yes	complemented
74	C₅₃×#424	C₂²	1	1	yes	characteristic
75	C₅₃×#417	C₂²	1	2	yes	—
76	C₅₃×#417	C₂²	1	2	yes	—
77	C₅₃×#416	C₂²	1	2	yes	—
78	C₅₃×#408	C₂²	1	1	yes	characteristic
79	C₅₃×#407	C₂²	1	1	yes	characteristic
80	C₅₃×#405	C₂²	1	1	yes	characteristic
81	C₂×#41×C₅₃	C₂³	1	2	yes	—
82	C₂×#41×C₅₃	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
84	#135×C₅₃	—	2	4	no	—
85	C₂×Q₁₆×C₅₃	C₂³	1	1	yes	characteristic
86	C₂×SD₁₆×C₅₃	C₂³	1	2	yes	—
87	C₂×D₈×C₅₃	C₂³	1	1	yes	characteristic
88	#131×C₅₃	—	2	4	no	—
89	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
90	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
91	C₂×M₁₆×C₅₃	C₂³	1	1	yes	characteristic
92	C₂²×C₈×C₅₃	—	2	4	no	—
93	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
94	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
95	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
96	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
97	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
98	C₄⋊Q₈×C₅₃	—	2	2	no	—
99	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
100	#126×C₅₃	—	2	4	no	—
101	#126×C₅₃	—	2	4	no	—
102	#126×C₅₃	—	2	4	no	—
103	#126×C₅₃	—	2	4	no	—
104	#125×C₅₃	—	2	4	no	—
105	#125×C₅₃	—	2	2	no	—
106	#125×C₅₃	C₂³	1	4	yes	—
107	#124×C₅₃	—	2	4	no	—
108	#124×C₅₃	—	2	4	no	—
109	#124×C₅₃	—	2	4	no	—
110	#123×C₅₃	—	2	4	no	—
111	#123×C₅₃	—	2	4	no	—
112	#123×C₅₃	—	2	2	no	—
113	#123×C₅₃	D₄	1	2	yes	—
114	C₂²⋊Q₈×C₅₃	—	2	4	no	—
115	C₂²⋊Q₈×C₅₃	—	2	2	no	—
116	C₂²⋊Q₈×C₅₃	D₄	1	2	yes	—
117	C₂²⋊Q₈×C₅₃	C₂³	1	2	yes	—
118	C₂²⋊Q₈×C₅₃	C₂³	1	2	yes	—
119	C₂²⋊Q₈×C₅₃	C₂³	1	4	yes	—
120	#121×C₅₃	—	2	4	no	—
121	#121×C₅₃	—	2	2	no	—
122	#121×C₅₃	—	2	4	no	—
123	#121×C₅₃	—	2	4	no	—
124	#121×C₅₃	D₄	1	2	yes	—
125	#121×C₅₃	C₂³	1	2	yes	—
126	#121×C₅₃	C₂³	1	2	yes	—
127	#121×C₅₃	C₂³	1	4	yes	—
128	#120×C₅₃	—	2	2	no	—
129	#120×C₅₃	D₄	1	2	yes	—
130	C₄×Q₈×C₅₃	—	2	4	no	—
131	C₄×Q₈×C₅₃	—	2	2	no	—
132	C₄×Q₈×C₅₃	C₂³	1	4	yes	—
133	C₄×D₄×C₅₃	—	2	4	no	—
134	C₄×D₄×C₅₃	—	2	4	no	—
135	C₄×D₄×C₅₃	—	2	4	no	—
136	C₄×D₄×C₅₃	—	2	2	no	—
137	C₄×D₄×C₅₃	—	2	2	no	—
138	C₄×D₄×C₅₃	—	2	2	no	—
139	C₄×D₄×C₅₃	C₂³	1	4	yes	—
140	C₄×D₄×C₅₃	C₂³	1	4	yes	—
141	#117×C₅₃	—	2	2	no	—
142	#117×C₅₃	—	2	4	no	—
143	#117×C₅₃	—	2	4	no	—
144	#117×C₅₃	—	2	2	no	—
145	#117×C₅₃	—	2	2	no	—
146	#117×C₅₃	C₂³	1	4	yes	—
147	#117×C₅₃	C₂³	1	4	yes	—
148	#117×C₅₃	C₂³	1	4	yes	—
149	#117×C₅₃	C₂³	1	2	yes	—
150	#117×C₅₃	C₂³	1	1	yes	characteristic
151	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
152	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
153	C₂×C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
154	C₂×C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
155	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
156	C₂×C₄²×C₅₃	—	2	2	no	—
157	#107×C₅₃	C₂³	1	4	yes	—
158	#107×C₅₃	C₂³	1	4	yes	—
159	#107×C₅₃	C₂³	1	4	yes	—
160	#107×C₅₃	C₂³	1	2	yes	—
161	#107×C₅₃	C₂³	1	4	yes	—
162	#106×C₅₃	C₂³	1	4	yes	—
163	#106×C₅₃	C₂³	1	4	yes	—
164	#106×C₅₃	C₂³	1	1	yes	characteristic
165	#106×C₅₃	C₂³	1	2	yes	—
166	#106×C₅₃	C₂³	1	1	yes	characteristic
167	#106×C₅₃	C₂³	1	2	yes	—
168	#106×C₅₃	C₂³	1	2	yes	—
169	#106×C₅₃	C₂³	1	2	yes	—
170	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
171	#103×C₅₃	C₂³	1	4	yes	—
172	#103×C₅₃	C₂³	1	4	yes	—
173	#103×C₅₃	C₂³	1	2	yes	—
174	#103×C₅₃	C₂³	1	2	yes	—
175	#102×C₅₃	C₂³	1	2	yes	—
176	#102×C₅₃	C₂³	1	2	yes	—
177	#102×C₅₃	C₂³	1	4	yes	—
178	#102×C₅₃	C₂³	1	4	yes	—
179	C₂²⋊C₈×C₅₃	C₂³	1	4	yes	—
180	C₂²⋊C₈×C₅₃	C₂³	1	4	yes	—
181	#97×C₅₃	C₂³	1	2	yes	—
182	C₄×C₈×C₅₃	C₂³	1	2	yes	—
183	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
184	#41×C₅₃	—	2	2	no	—
185	#41×C₅₃	—	2	2	no	—
186	#41×C₅₃	—	2	4	no	—
187	#41×C₅₃	—	2	4	no	—
188	C₂×Q₈×C₅₃	—	2	4	no	—
189	C₂×Q₈×C₅₃	C₂×D₄	1	1	yes	characteristic
190	C₂×Q₈×C₅₃	C₂⁴	1	2	yes	—
191	C₂×D₄×C₅₃	—	2	2	no	—
192	C₂×D₄×C₅₃	—	2	2	no	—
193	C₂×D₄×C₅₃	—	2	2	no	—
194	C₂×D₄×C₅₃	—	2	4	no	—
195	C₂×D₄×C₅₃	C₂×D₄	1	2	yes	—
196	C₂×D₄×C₅₃	C₂×D₄	1	1	yes	characteristic
197	C₂×D₄×C₅₃	C₂×D₄	1	4	yes	—
198	C₂×D₄×C₅₃	C₂⁴	1	2	yes	—
199	C₂²×C₄×C₅₃	—	2	2	no	—
200	C₂²×C₄×C₅₃	—	2	2	no	—
201	C₂²×C₄×C₅₃	—	2	2	no	—
202	C₂²×C₄×C₅₃	—	2	2	no	—
203	C₂²×C₄×C₅₃	—	2	4	no	—
204	C₂²×C₄×C₅₃	—	2	2	no	—
205	C₂²×C₄×C₅₃	—	2	4	no	—
206	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
207	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
208	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
209	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
210	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
211	Q₁₆×C₅₃	—	2	2	no	—
212	SD₁₆×C₅₃	—	2	4	no	—
213	D₈×C₅₃	—	2	2	no	—
214	M₁₆×C₅₃	—	2	2	no	—
215	M₁₆×C₅₃	—	2	2	no	—
216	M₁₆×C₅₃	—	2	4	no	—
217	C₂×C₈×C₅₃	—	2	2	no	—
218	C₂×C₈×C₅₃	—	2	4	no	—
219	C₂×C₈×C₅₃	—	2	4	no	—
220	C₂×C₈×C₅₃	—	2	4	no	—
221	C₂×C₈×C₅₃	#41	1	4	yes	—
222	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
223	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
224	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
225	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
226	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
227	C₄⋊C₄×C₅₃	—	2	2	no	—
228	C₄⋊C₄×C₅₃	—	2	4	no	—
229	C₄⋊C₄×C₅₃	—	2	2	no	—
230	C₄⋊C₄×C₅₃	—	2	4	no	—
231	C₄⋊C₄×C₅₃	—	2	2	no	—
232	C₄⋊C₄×C₅₃	—	2	4	no	—
233	C₄⋊C₄×C₅₃	—	2	4	no	—
234	C₄⋊C₄×C₅₃	—	2	2	no	—
235	C₄⋊C₄×C₅₃	—	2	4	no	—
236	C₄⋊C₄×C₅₃	—	2	4	no	—
237	C₄⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
238	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
239	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
240	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
241	C₄⋊C₄×C₅₃	C₂⁴	1	4	yes	—
242	C₄⋊C₄×C₅₃	C₂⁴	1	2	yes	—
243	C₂²⋊C₄×C₅₃	—	2	4	no	—
244	C₂²⋊C₄×C₅₃	—	2	4	no	—
245	C₂²⋊C₄×C₅₃	—	2	4	no	—
246	C₂²⋊C₄×C₅₃	—	2	2	no	—
247	C₂²⋊C₄×C₅₃	—	2	4	no	—
248	C₂²⋊C₄×C₅₃	—	2	2	no	—
249	C₂²⋊C₄×C₅₃	—	2	2	no	—
250	C₂²⋊C₄×C₅₃	—	2	2	no	—
251	C₂²⋊C₄×C₅₃	—	2	4	no	—
252	C₂²⋊C₄×C₅₃	—	2	4	no	—
253	C₂²⋊C₄×C₅₃	—	2	2	no	—
254	C₂²⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
255	C₂²⋊C₄×C₅₃	C₂×D₄	1	2	yes	—
256	C₂²⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
257	C₄²×C₅₃	—	2	4	no	—
258	C₄²×C₅₃	—	2	4	no	—
259	C₄²×C₅₃	—	2	4	no	—
260	C₄²×C₅₃	—	2	4	no	—
261	C₄²×C₅₃	—	2	2	no	—
262	C₄²×C₅₃	C₂⁴	1	2	yes	—
263	C₂³×C₅₃	—	2	2	no	—
264	C₂³×C₅₃	—	2	2	no	—
265	C₂³×C₅₃	—	4	2	no	—
266	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
267	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
268	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
269	Q₈×C₅₃	—	2	2	no	—
270	Q₈×C₅₃	—	2	2	no	—
271	Q₈×C₅₃	—	2	2	no	—
272	Q₈×C₅₃	—	8	4	no	—
273	D₄×C₅₃	—	2	4	no	—
274	D₄×C₅₃	—	2	2	no	—
275	D₄×C₅₃	—	2	2	no	—
276	D₄×C₅₃	—	2	2	no	—
277	D₄×C₅₃	—	4	4	no	—
278	D₄×C₅₃	—	4	4	no	—
279	D₄×C₅₃	—	8	2	no	—
280	D₄×C₅₃	—	8	4	no	—
281	D₄×C₅₃	—	8	2	no	—
282	C₂×C₄×C₅₃	—	2	2	no	—
283	C₂×C₄×C₅₃	—	2	4	no	—
284	C₂×C₄×C₅₃	—	2	2	no	—
285	C₂×C₄×C₅₃	—	2	2	no	—
286	C₂×C₄×C₅₃	—	2	2	no	—
287	C₂×C₄×C₅₃	—	2	2	no	—
288	C₂×C₄×C₅₃	—	2	4	no	—
289	C₂×C₄×C₅₃	—	2	2	no	—
290	C₂×C₄×C₅₃	—	2	4	no	—
291	C₂×C₄×C₅₃	—	2	2	no	—
292	C₂×C₄×C₅₃	—	2	2	no	—
293	C₂×C₄×C₅₃	—	4	4	no	—
294	C₂×C₄×C₅₃	—	8	4	no	—
295	C₂×C₄×C₅₃	—	8	2	no	—
296	C₂×C₄×C₅₃	—	8	4	no	—
297	C₂×C₄×C₅₃	—	8	2	no	—
298	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
299	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
300	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
301	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
302	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
303	C₂×C₄×C₅₃	2₋¹⁺⁴	1	2	yes	—
304	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
305	C₈×C₅₃	—	2	2	no	—
306	C₈×C₅₃	—	2	2	no	—
307	C₈×C₅₃	—	2	2	no	—
308	C₈×C₅₃	C₂×#41	1	2	yes	—
309	C₈×C₅₃	C₂×#41	1	2	yes	—
310	#61260	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
311	C₂²×C₅₃	—	2	2	no	—
312	C₂²×C₅₃	—	2	4	no	—
313	C₂²×C₅₃	—	2	2	no	—
314	C₂²×C₅₃	—	2	2	no	—
315	C₂²×C₅₃	—	2	2	no	—
316	C₂²×C₅₃	—	2	2	no	—
317	C₂²×C₅₃	—	4	4	no	—
318	C₂²×C₅₃	—	4	2	no	—
319	C₂²×C₅₃	—	4	2	no	—
320	C₂²×C₅₃	—	4	2	no	—
321	C₂²×C₅₃	#547	1	2	yes	—
322	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
323	C₄×C₅₃	—	2	4	no	—
324	C₄×C₅₃	—	2	2	no	—
325	C₄×C₅₃	—	2	2	no	—
326	C₄×C₅₃	—	4	4	no	—
327	C₄×C₅₃	—	4	2	no	—
328	C₄×C₅₃	—	4	4	no	—
329	C₄×C₅₃	—	4	2	no	—
330	C₄×C₅₃	—	4	2	no	—
331	C₄×C₅₃	#547	1	2	yes	—
332	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
333	C₄×C₅₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
334	#3493	C₂×C₅₃	1	2	yes	complemented
335	#3349	C₂×C₅₃	1	4	yes	complemented
336	#3347	C₂×C₅₃	1	4	yes	complemented
337	#3340	C₂×C₅₃	1	4	yes	complemented
338	#3337	C₂×C₅₃	1	4	yes	complemented
339	#3163	C₂×C₅₃	1	1	yes	characteristic, complemented
340	#3098	C₂×C₅₃	1	2	yes	complemented
341	#3093	C₂×C₅₃	1	2	yes	complemented
342	#3083	C₂×C₅₃	1	2	yes	complemented
343	#3060	C₂×C₅₃	1	1	yes	characteristic, complemented
344	#3050	C₂×C₅₃	1	1	yes	characteristic, complemented
345	#2952	C₂×C₅₃	1	1	yes	characteristic, complemented
346	#2929	C₂×C₅₃	1	1	yes	characteristic, complemented
347	#2912	C₂×C₅₃	1	1	yes	characteristic, complemented
348	C₂×#427	C₂×C₅₃	1	1	yes	characteristic, complemented
349	C₂×C₅₃	—	2	2	no	—
350	C₂×C₅₃	—	4	2	no	—
351	C₂×C₅₃	—	4	2	no	—
352	C₂×C₅₃	—	8	2	no	—
353	C₂×C₅₃	C₂×#547	1	1	yes	characteristic
354	C₂×C₅₃	#3584	1	2	yes	—
355	C₂×#135	C₂²×C₅₃	1	1	yes	characteristic
356	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
357	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
358	#556	—	2	2	no	—
359	#556	C₂²×C₅₃	1	4	yes	—
360	#555	—	2	2	no	—
361	#555	C₂²×C₅₃	1	4	yes	—
362	#553	—	2	2	no	—
363	#553	C₂²×C₅₃	1	4	yes	—
364	#552	—	2	2	no	—
365	#552	C₂²×C₅₃	1	4	yes	—
366	#544	C₂²×C₅₃	1	2	yes	—
367	#538	—	2	4	no	—
368	#537	C₂²×C₅₃	1	2	yes	—
369	#529	—	2	4	no	—
370	#525	C₂²×C₅₃	1	2	yes	—
371	#525	C₂²×C₅₃	1	1	yes	characteristic
372	#520	C₂²×C₅₃	1	2	yes	—
373	#520	C₂²×C₅₃	1	2	yes	—
374	C₂×#117	C₂²×C₅₃	1	2	yes	—
375	#501	C₂²×C₅₃	1	4	yes	complemented
376	#499	C₂²×C₅₃	1	2	yes	complemented
377	#498	C₂²×C₅₃	1	2	yes	complemented
378	#485	C₂²×C₅₃	1	4	yes	—
379	#485	C₂²×C₅₃	1	4	yes	—
380	#485	C₂²×C₅₃	1	4	yes	complemented
381	#484	C₂²×C₅₃	1	4	yes	complemented
382	#482	C₂²×C₅₃	1	4	yes	—
383	#482	C₂²×C₅₃	1	4	yes	—
384	#482	C₂²×C₅₃	1	4	yes	complemented
385	#480	C₂²×C₅₃	1	4	yes	complemented
386	#479	C₂²×C₅₃	1	4	yes	complemented
387	#478	C₂²×C₅₃	1	4	yes	complemented
388	#477	C₂²×C₅₃	1	4	yes	complemented
389	#474	C₂²×C₅₃	1	4	yes	complemented
390	#470	C₂²×C₅₃	1	2	yes	complemented
391	#469	C₂²×C₅₃	1	2	yes	complemented
392	#467	C₂²×C₅₃	1	2	yes	—
393	#466	C₂²×C₅₃	1	2	yes	—
394	#465	C₂²×C₅₃	1	2	yes	—
395	#465	C₂²×C₅₃	1	2	yes	—
396	#465	C₂²×C₅₃	1	4	yes	complemented
397	#442	C₂²×C₅₃	1	2	yes	complemented
398	#441	C₂²×C₅₃	1	2	yes	complemented
399	C₄×SD₁₆	C₂²×C₅₃	1	4	yes	complemented
400	#435	C₂²×C₅₃	1	4	yes	—
401	C₈×D₄	C₂²×C₅₃	1	4	yes	—
402	#428	C₂²×C₅₃	1	2	yes	—
403	#428	C₂²×C₅₃	1	2	yes	—
404	#428	C₂²×C₅₃	1	4	yes	complemented
405	#427	C₂²×C₅₃	1	4	yes	—
406	#427	C₂²×C₅₃	1	4	yes	—
407	#427	C₂²×C₅₃	1	1	yes	characteristic
408	#427	C₂²×C₅₃	1	2	yes	complemented
409	C₂×#107	C₂²×C₅₃	1	2	yes	—
410	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
411	C₂×#106	C₂²×C₅₃	1	1	yes	characteristic
412	C₂×#106	C₂²×C₅₃	1	2	yes	complemented
413	#424	C₂²×C₅₃	1	1	yes	characteristic
414	#417	C₂²×C₅₃	1	2	yes	—
415	#417	C₂²×C₅₃	1	2	yes	—
416	#416	C₂²×C₅₃	1	2	yes	—
417	#408	C₂²×C₅₃	1	1	yes	characteristic
418	#407	C₂²×C₅₃	1	1	yes	characteristic
419	#405	C₂²×C₅₃	1	1	yes	characteristic
420	C₅₃	#61260	1	1	yes	characteristic, direct product factor, Sylow
421	C₂×#41	C₂³×C₅₃	1	2	yes	—
422	C₂×#41	C₂³×C₅₃	1	1	yes	characteristic
423	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
424	#135	—	2	4	no	—
425	C₂×Q₁₆	C₂³×C₅₃	1	1	yes	characteristic
426	C₂×SD₁₆	C₂³×C₅₃	1	2	yes	—
427	C₂×D₈	C₂³×C₅₃	1	1	yes	characteristic
428	#131	—	2	4	no	—
429	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
430	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
431	C₂×M₁₆	C₂³×C₅₃	1	1	yes	characteristic
432	C₂²×C₈	—	2	4	no	—
433	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
434	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
435	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
436	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
437	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
438	C₄⋊Q₈	—	2	2	no	—
439	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
440	#126	—	2	4	no	—
441	#126	—	2	4	no	—
442	#126	—	2	4	no	—
443	#126	—	2	4	no	—
444	#125	—	2	4	no	—
445	#125	—	2	2	no	—
446	#125	C₂³×C₅₃	1	4	yes	—
447	#124	—	2	4	no	—
448	#124	—	2	4	no	—
449	#124	—	2	4	no	—
450	#123	—	2	4	no	—
451	#123	—	2	2	no	—
452	#123	—	2	4	no	—
453	#123	D₄×C₅₃	1	2	yes	—
454	C₂²⋊Q₈	—	2	2	no	—
455	C₂²⋊Q₈	—	2	4	no	—
456	C₂²⋊Q₈	D₄×C₅₃	1	2	yes	—
457	C₂²⋊Q₈	C₂³×C₅₃	1	2	yes	—
458	C₂²⋊Q₈	C₂³×C₅₃	1	2	yes	—
459	C₂²⋊Q₈	C₂³×C₅₃	1	4	yes	—
460	#121	—	2	4	no	—
461	#121	—	2	2	no	—
462	#121	—	2	4	no	—
463	#121	—	2	4	no	—
464	#121	D₄×C₅₃	1	2	yes	—
465	#121	C₂³×C₅₃	1	2	yes	—
466	#121	C₂³×C₅₃	1	2	yes	—
467	#121	C₂³×C₅₃	1	4	yes	—
468	#120	—	2	2	no	—
469	#120	D₄×C₅₃	1	2	yes	—
470	C₄×Q₈	—	2	2	no	—
471	C₄×Q₈	—	2	4	no	—
472	C₄×Q₈	C₂³×C₅₃	1	4	yes	—
473	C₄×D₄	—	2	4	no	—
474	C₄×D₄	—	2	2	no	—
475	C₄×D₄	—	2	2	no	—
476	C₄×D₄	—	2	2	no	—
477	C₄×D₄	—	2	4	no	—
478	C₄×D₄	—	2	4	no	—
479	C₄×D₄	C₂³×C₅₃	1	4	yes	—
480	C₄×D₄	C₂³×C₅₃	1	4	yes	—
481	#117	—	2	2	no	—
482	#117	—	2	2	no	—
483	#117	—	2	2	no	—
484	#117	—	2	4	no	—
485	#117	—	2	4	no	—
486	#117	C₂³×C₅₃	1	4	yes	—
487	#117	C₂³×C₅₃	1	4	yes	—
488	#117	C₂³×C₅₃	1	2	yes	—
489	#117	C₂³×C₅₃	1	4	yes	—
490	#117	C₂³×C₅₃	1	1	yes	characteristic
491	C₂×C₄⋊C₄	—	2	2	no	—
492	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
493	C₂×C₄⋊C₄	C₂³×C₅₃	1	2	yes	—
494	C₂×C₄⋊C₄	C₂³×C₅₃	1	4	yes	—
495	C₂×C₂²⋊C₄	—	2	2	no	—
496	C₂×C₄²	—	2	2	no	—
497	#107	C₂³×C₅₃	1	4	yes	—
498	#107	C₂³×C₅₃	1	4	yes	—
499	#107	C₂³×C₅₃	1	2	yes	—
500	#107	C₂³×C₅₃	1	4	yes	—
501	#107	C₂³×C₅₃	1	4	yes	—
502	#106	C₂³×C₅₃	1	2	yes	—
503	#106	C₂³×C₅₃	1	2	yes	—
504	#106	C₂³×C₅₃	1	2	yes	—
505	#106	C₂³×C₅₃	1	2	yes	—
506	#106	C₂³×C₅₃	1	1	yes	characteristic
507	#106	C₂³×C₅₃	1	4	yes	—
508	#106	C₂³×C₅₃	1	1	yes	characteristic
509	#106	C₂³×C₅₃	1	4	yes	—
510	C₄⋊C₈	C₂³×C₅₃	1	4	yes	—
511	#103	C₂³×C₅₃	1	2	yes	—
512	#103	C₂³×C₅₃	1	2	yes	—
513	#103	C₂³×C₅₃	1	4	yes	—
514	#103	C₂³×C₅₃	1	4	yes	—
515	#102	C₂³×C₅₃	1	2	yes	—
516	#102	C₂³×C₅₃	1	2	yes	—
517	#102	C₂³×C₅₃	1	4	yes	—
518	#102	C₂³×C₅₃	1	4	yes	—
519	C₂²⋊C₈	C₂³×C₅₃	1	4	yes	—
520	C₂²⋊C₈	C₂³×C₅₃	1	4	yes	—
521	#97	C₂³×C₅₃	1	2	yes	—
522	C₄×C₈	C₂³×C₅₃	1	2	yes	—
523	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
524	#41	—	2	2	no	—
525	#41	—	2	2	no	—
526	#41	—	2	4	no	—
527	#41	—	2	4	no	—
528	C₂×Q₈	—	2	4	no	—
529	C₂×Q₈	C₂×D₄×C₅₃	1	1	yes	characteristic
530	C₂×Q₈	C₂⁴×C₅₃	1	2	yes	—
531	C₂×D₄	—	2	2	no	—
532	C₂×D₄	—	2	2	no	—
533	C₂×D₄	—	2	2	no	—
534	C₂×D₄	—	2	4	no	—
535	C₂×D₄	C₂×D₄×C₅₃	1	2	yes	—
536	C₂×D₄	C₂×D₄×C₅₃	1	1	yes	characteristic
537	C₂×D₄	C₂×D₄×C₅₃	1	4	yes	—
538	C₂×D₄	C₂⁴×C₅₃	1	2	yes	—
539	C₂²×C₄	—	2	2	no	—
540	C₂²×C₄	—	2	2	no	—
541	C₂²×C₄	—	2	2	no	—
542	C₂²×C₄	—	2	2	no	—
543	C₂²×C₄	—	2	4	no	—
544	C₂²×C₄	—	2	2	no	—
545	C₂²×C₄	—	2	4	no	—
546	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
547	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
548	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
549	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
550	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
551	Q₁₆	—	2	2	no	—
552	SD₁₆	—	2	4	no	—
553	D₈	—	2	2	no	—
554	M₁₆	—	2	2	no	—
555	M₁₆	—	2	2	no	—
556	M₁₆	—	2	4	no	—
557	C₂×C₈	—	2	2	no	—
558	C₂×C₈	—	2	4	no	—
559	C₂×C₈	—	2	4	no	—
560	C₂×C₈	—	2	4	no	—
561	C₂×C₈	#41×C₅₃	1	4	yes	—
562	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
563	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
564	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
565	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
566	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
567	C₄⋊C₄	—	2	2	no	—
568	C₄⋊C₄	—	2	4	no	—
569	C₄⋊C₄	—	2	2	no	—
570	C₄⋊C₄	—	2	4	no	—
571	C₄⋊C₄	—	2	2	no	—
572	C₄⋊C₄	—	2	4	no	—
573	C₄⋊C₄	—	2	4	no	—
574	C₄⋊C₄	—	2	2	no	—
575	C₄⋊C₄	—	2	4	no	—
576	C₄⋊C₄	—	2	4	no	—
577	C₄⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
578	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
579	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
580	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
581	C₄⋊C₄	C₂⁴×C₅₃	1	4	yes	—
582	C₄⋊C₄	C₂⁴×C₅₃	1	2	yes	—
583	C₂²⋊C₄	—	2	4	no	—
584	C₂²⋊C₄	—	2	4	no	—
585	C₂²⋊C₄	—	2	4	no	—
586	C₂²⋊C₄	—	2	2	no	—
587	C₂²⋊C₄	—	2	4	no	—
588	C₂²⋊C₄	—	2	2	no	—
589	C₂²⋊C₄	—	2	2	no	—
590	C₂²⋊C₄	—	2	2	no	—
591	C₂²⋊C₄	—	2	4	no	—
592	C₂²⋊C₄	—	2	4	no	—
593	C₂²⋊C₄	—	2	2	no	—
594	C₂²⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
595	C₂²⋊C₄	C₂×D₄×C₅₃	1	2	yes	—
596	C₂²⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
597	C₄²	—	2	4	no	—
598	C₄²	—	2	4	no	—
599	C₄²	—	2	4	no	—
600	C₄²	—	2	4	no	—
601	C₄²	—	2	2	no	—
602	C₄²	C₂⁴×C₅₃	1	2	yes	—
603	C₂³	—	2	2	no	—
604	C₂³	—	2	2	no	—
605	C₂³	—	4	2	no	—
606	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
607	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
608	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
609	Q₈	—	2	2	no	—
610	Q₈	—	2	2	no	—
611	Q₈	—	2	2	no	—
612	Q₈	—	8	4	no	—
613	D₄	—	2	4	no	—
614	D₄	—	2	2	no	—
615	D₄	—	2	2	no	—
616	D₄	—	2	2	no	—
617	D₄	—	4	4	no	—
618	D₄	—	4	4	no	—
619	D₄	—	8	2	no	—
620	D₄	—	8	4	no	—
621	D₄	—	8	2	no	—
622	C₂×C₄	—	2	2	no	—
623	C₂×C₄	—	2	4	no	—
624	C₂×C₄	—	2	2	no	—
625	C₂×C₄	—	2	2	no	—
626	C₂×C₄	—	2	2	no	—
627	C₂×C₄	—	2	2	no	—
628	C₂×C₄	—	2	4	no	—
629	C₂×C₄	—	2	2	no	—
630	C₂×C₄	—	2	2	no	—
631	C₂×C₄	—	2	4	no	—
632	C₂×C₄	—	2	2	no	—
633	C₂×C₄	—	4	4	no	—
634	C₂×C₄	—	8	4	no	—
635	C₂×C₄	—	8	2	no	—
636	C₂×C₄	—	8	4	no	—
637	C₂×C₄	—	8	2	no	—
638	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
639	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
640	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
641	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
642	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
643	C₂×C₄	2₋¹⁺⁴×C₅₃	1	2	yes	—
644	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
645	C₈	—	2	2	no	—
646	C₈	—	2	2	no	—
647	C₈	—	2	2	no	—
648	C₈	C₂×#41×C₅₃	1	2	yes	—
649	C₈	C₂×#41×C₅₃	1	2	yes	—
650	C₂²	—	2	2	no	—
651	C₂²	—	2	2	no	—
652	C₂²	—	2	2	no	—
653	C₂²	—	2	4	no	—
654	C₂²	—	2	2	no	—
655	C₂²	—	2	2	no	—
656	C₂²	—	4	4	no	—
657	C₂²	—	4	2	no	—
658	C₂²	—	4	2	no	—
659	C₂²	—	4	2	no	—
660	C₂²	C₅₃×#547	1	2	yes	—
661	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
662	C₄	—	2	2	no	—
663	C₄	—	2	2	no	—
664	C₄	—	2	4	no	—
665	C₄	—	4	2	no	—
666	C₄	—	4	4	no	—
667	C₄	—	4	2	no	—
668	C₄	—	4	4	no	—
669	C₄	—	4	2	no	—
670	C₄	C₅₃×#547	1	2	yes	—
671	C₄	C₂²×#41×C₅₃	1	1	yes	characteristic
672	C₄	C₂×2₋¹⁺⁴×C₅₃	1	1	yes	characteristic
673	C₂	—	2	2	no	—
674	C₂	—	4	2	no	—
675	C₂	—	4	2	no	—
676	C₂	—	8	2	no	—
677	C₂	C₂×C₅₃×#547	1	1	yes	characteristic
678	C₂	C₅₃×#3584	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₅₃×#525 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3493	C₂	semidirect
2. C₅₃×#3349	C₂	semidirect
3. C₅₃×#3347	C₂	semidirect
4. C₅₃×#3340	C₂	semidirect
5. C₅₃×#3337	C₂	semidirect
6. C₅₃×#3163	C₂	semidirect
7. C₅₃×#3098	C₂	semidirect
8. C₅₃×#3093	C₂	semidirect
9. C₅₃×#3083	C₂	semidirect
10. C₅₃×#3060	C₂	semidirect
11. C₅₃×#3050	C₂	semidirect
12. C₅₃×#2952	C₂	semidirect
13. C₅₃×#2929	C₂	semidirect
14. C₅₃×#2912	C₂	semidirect
15. C₂×C₅₃×#427	C₂	semidirect
36. C₅₃×#501	C₂²	semidirect
37. C₅₃×#499	C₂²	semidirect
38. C₅₃×#498	C₂²	semidirect
41. C₅₃×#485	C₂²	semidirect
42. C₅₃×#484	C₂²	semidirect
45. C₅₃×#482	C₂²	semidirect
46. C₅₃×#480	C₂²	semidirect
47. C₅₃×#479	C₂²	semidirect
48. C₅₃×#478	C₂²	semidirect
49. C₅₃×#477	C₂²	semidirect
50. C₅₃×#474	C₂²	semidirect
51. C₅₃×#470	C₂²	semidirect
52. C₅₃×#469	C₂²	semidirect
57. C₅₃×#465	C₂²	semidirect
58. C₅₃×#442	C₂²	semidirect
59. C₅₃×#441	C₂²	semidirect
60. C₄×SD₁₆×C₅₃	C₂²	semidirect
65. C₅₃×#428	C₂²	semidirect
69. C₅₃×#427	C₂²	semidirect
73. C₂×#106×C₅₃	C₂²	semidirect
310. #61260	C₅₃	direct (N × H)
334. #3493	C₂×C₅₃	semidirect
335. #3349	C₂×C₅₃	semidirect
336. #3347	C₂×C₅₃	semidirect
337. #3340	C₂×C₅₃	semidirect
338. #3337	C₂×C₅₃	semidirect
339. #3163	C₂×C₅₃	semidirect
340. #3098	C₂×C₅₃	semidirect
341. #3093	C₂×C₅₃	semidirect
342. #3083	C₂×C₅₃	semidirect
343. #3060	C₂×C₅₃	semidirect
344. #3050	C₂×C₅₃	semidirect
345. #2952	C₂×C₅₃	semidirect
346. #2929	C₂×C₅₃	semidirect
347. #2912	C₂×C₅₃	semidirect
348. C₂×#427	C₂×C₅₃	semidirect
375. #501	C₂²×C₅₃	semidirect
376. #499	C₂²×C₅₃	semidirect
377. #498	C₂²×C₅₃	semidirect
380. #485	C₂²×C₅₃	semidirect
381. #484	C₂²×C₅₃	semidirect
384. #482	C₂²×C₅₃	semidirect
385. #480	C₂²×C₅₃	semidirect
386. #479	C₂²×C₅₃	semidirect
387. #478	C₂²×C₅₃	semidirect
388. #477	C₂²×C₅₃	semidirect
389. #474	C₂²×C₅₃	semidirect
390. #470	C₂²×C₅₃	semidirect
391. #469	C₂²×C₅₃	semidirect
396. #465	C₂²×C₅₃	semidirect
397. #442	C₂²×C₅₃	semidirect
398. #441	C₂²×C₅₃	semidirect
399. C₄×SD₁₆	C₂²×C₅₃	semidirect
404. #428	C₂²×C₅₃	semidirect
408. #427	C₂²×C₅₃	semidirect
412. C₂×#106	C₂²×C₅₃	semidirect
420. C₅₃	#61260	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅₃×#3493	2
C₅₃×#3349	2
C₅₃×#3347	2
C₅₃×#3340	2
C₅₃×#3337	2
C₅₃×#3163	2
C₅₃×#3098	2
C₅₃×#3093	2
C₅₃×#3083	2
C₅₃×#3060	2
C₅₃×#3050	2
C₅₃×#2952	2
C₅₃×#2929	2
C₅₃×#2912	2
C₂×C₅₃×#427	2
#61260	53

Subgroups — C53×#61260 #27268772

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#61260 #27268772