Subgroups — C₅₃×#60682 #27268194

Order 13568.

1878 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₃×#3455	C₂	1	2	yes	maximal, complemented
2	C₅₃×#3177	C₂	1	2	yes	maximal, complemented
3	C₅₃×#3176	C₂	1	2	yes	maximal, complemented
4	C₅₃×#3175	C₂	1	4	yes	maximal, complemented
5	C₅₃×#3174	C₂	1	4	yes	maximal, complemented
6	C₅₃×#3173	C₂	1	4	yes	maximal, complemented
7	C₅₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₃×#3147	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#3135	C₂	1	2	yes	maximal, complemented
10	C₅₃×#3054	C₂	1	2	yes	maximal, complemented
11	C₅₃×#3050	C₂	1	2	yes	maximal, complemented
12	C₅₃×#2943	C₂	1	2	yes	maximal, complemented
13	C₄×#135×C₅₃	C₂	1	2	yes	maximal, complemented
14	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#135×C₅₃	C₂²	1	2	yes	—
16	C₂×#135×C₅₃	C₂²	1	2	yes	—
17	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#542	C₂²	1	8	yes	—
20	C₅₃×#541	C₂²	1	4	yes	—
21	C₅₃×#540	C₂²	1	4	yes	—
22	C₅₃×#539	—	2	4	no	—
23	C₅₃×#538	—	2	4	no	—
24	C₅₃×#533	C₂²	1	2	yes	—
25	C₅₃×#532	C₂²	1	2	yes	—
26	C₅₃×#529	—	2	8	no	—
27	C₅₃×#525	C₂²	1	4	yes	—
28	C₄×#41×C₅₃	C₂²	1	4	yes	—
29	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
30	C₅₃×#500	C₂²	1	2	yes	complemented
31	C₅₃×#499	C₂²	1	2	yes	complemented
32	C₅₃×#499	C₂²	1	2	yes	complemented
33	C₅₃×#498	C₂²	1	2	yes	complemented
34	C₅₃×#495	C₂²	1	2	yes	complemented
35	C₅₃×#495	C₂²	1	2	yes	complemented
36	C₅₃×#494	C₂²	1	1	yes	characteristic, complemented
37	C₅₃×#493	C₂²	1	1	yes	characteristic, complemented
38	C₅₃×#492	C₂²	1	2	yes	complemented
39	C₅₃×#491	C₂²	1	4	yes	complemented
40	C₅₃×#490	C₂²	1	4	yes	complemented
41	C₅₃×#489	C₂²	1	8	yes	complemented
42	C₅₃×#470	C₂²	1	4	yes	—
43	C₅₃×#469	C₂²	1	4	yes	—
44	C₅₃×#468	C₂²	1	8	yes	—
45	C₅₃×#467	C₂²	1	4	yes	—
46	C₅₃×#466	C₂²	1	4	yes	—
47	C₅₃×#465	C₂²	1	8	yes	—
48	C₅₃×#442	C₂²	1	4	yes	complemented
49	C₅₃×#441	C₂²	1	4	yes	complemented
50	C₅₃×#440	C₂²	1	8	yes	complemented
51	C₄×Q₁₆×C₅₃	C₂²	1	4	yes	complemented
52	C₄×SD₁₆×C₅₃	C₂²	1	8	yes	complemented
53	C₄×D₈×C₅₃	C₂²	1	4	yes	complemented
54	C₅₃×#427	C₂²	1	2	yes	—
55	C₅₃×#427	C₂²	1	2	yes	—
56	C₅₃×#416	C₂²	1	4	yes	—
57	C₅₃×#416	C₂²	1	4	yes	—
58	C₅₃×#405	C₂²	1	2	yes	—
59	C₅₃×#405	C₂²	1	2	yes	—
60	C₅₃×#405	C₂²	1	2	yes	—
61	C₅₃×#405	C₂²	1	2	yes	—
62	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
63	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
64	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
65	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
66	C₂×#41×C₅₃	C₂³	1	4	yes	—
67	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
68	#135×C₅₃	—	2	8	no	—
69	#135×C₅₃	—	2	8	no	—
70	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
71	C₂×Q₁₆×C₅₃	C₂³	1	2	yes	—
72	C₂×SD₁₆×C₅₃	C₂³	1	4	yes	—
73	C₂×SD₁₆×C₅₃	C₂³	1	4	yes	—
74	C₂×D₈×C₅₃	C₂³	1	2	yes	—
75	C₂×D₈×C₅₃	C₂³	1	2	yes	—
76	C₂×M₁₆×C₅₃	—	2	4	no	—
77	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
78	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
79	C₂²×C₈×C₅₃	—	2	4	no	—
80	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
82	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
83	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
84	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
85	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
86	C₄⋊Q₈×C₅₃	C₂³	1	4	yes	—
87	C₄⋊Q₈×C₅₃	C₂³	1	2	yes	—
88	#127×C₅₃	C₂³	1	2	yes	—
89	#126×C₅₃	—	2	8	no	—
90	#126×C₅₃	—	2	8	no	—
91	#125×C₅₃	—	2	8	no	—
92	#125×C₅₃	C₂³	1	4	yes	—
93	#124×C₅₃	—	2	8	no	—
94	#124×C₅₃	C₂³	1	4	yes	—
95	#123×C₅₃	—	2	8	no	—
96	#123×C₅₃	—	2	8	no	—
97	#123×C₅₃	—	2	4	no	—
98	C₂²⋊Q₈×C₅₃	—	2	8	no	—
99	C₂²⋊Q₈×C₅₃	—	2	4	no	—
100	C₂²⋊Q₈×C₅₃	C₂³	1	8	yes	—
101	#121×C₅₃	—	2	8	no	—
102	#121×C₅₃	—	2	4	no	—
103	#121×C₅₃	C₂³	1	8	yes	—
104	#120×C₅₃	—	2	4	no	—
105	C₄×Q₈×C₅₃	—	2	4	no	—
106	C₄×Q₈×C₅₃	C₂³	1	8	yes	—
107	C₄×D₄×C₅₃	—	2	4	no	—
108	C₄×D₄×C₅₃	—	2	8	no	—
109	C₄×D₄×C₅₃	—	2	8	no	—
110	C₄×D₄×C₅₃	C₂³	1	8	yes	—
111	#117×C₅₃	—	2	4	no	—
112	#117×C₅₃	D₄	1	4	yes	—
113	#117×C₅₃	C₂³	1	4	yes	—
114	#117×C₅₃	C₂³	1	2	yes	—
115	#117×C₅₃	C₂³	1	2	yes	—
116	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
117	C₂×C₂²⋊C₄×C₅₃	D₄	1	2	yes	—
118	C₂×C₄²×C₅₃	—	2	4	no	—
119	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
121	#107×C₅₃	C₂³	1	4	yes	—
122	#107×C₅₃	C₂³	1	4	yes	—
123	#106×C₅₃	C₂³	1	4	yes	—
124	#106×C₅₃	C₂³	1	4	yes	—
125	#103×C₅₃	C₂³	1	8	yes	—
126	#103×C₅₃	C₂³	1	8	yes	—
127	#102×C₅₃	C₂³	1	8	yes	—
128	#102×C₅₃	C₂³	1	8	yes	—
129	#97×C₅₃	C₂³	1	2	yes	—
130	#97×C₅₃	C₂³	1	2	yes	—
131	#97×C₅₃	C₂³	1	2	yes	—
132	#97×C₅₃	C₂³	1	2	yes	—
133	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
134	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
135	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
136	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
137	C₄×C₈×C₅₃	C₂³	1	2	yes	—
138	C₄×C₈×C₅₃	C₂³	1	2	yes	—
139	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
140	#41×C₅₃	—	2	8	no	—
141	#41×C₅₃	—	2	8	no	—
142	C₂×Q₈×C₅₃	—	2	4	no	—
143	C₂×Q₈×C₅₃	C₂⁴	1	4	yes	—
144	C₂×D₄×C₅₃	—	2	4	no	—
145	C₂×D₄×C₅₃	—	2	8	no	—
146	C₂×D₄×C₅₃	—	2	4	no	—
147	C₂×D₄×C₅₃	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₅₃	—	2	2	no	—
149	C₂²×C₄×C₅₃	—	2	2	no	—
150	C₂²×C₄×C₅₃	—	2	4	no	—
151	C₂²×C₄×C₅₃	—	2	8	no	—
152	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
154	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
155	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
156	C₂²×C₄×C₅₃	C₂⁴	1	2	yes	—
157	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
158	Q₁₆×C₅₃	—	2	4	no	—
159	Q₁₆×C₅₃	—	2	4	no	—
160	SD₁₆×C₅₃	—	2	8	no	—
161	SD₁₆×C₅₃	—	2	8	no	—
162	D₈×C₅₃	—	2	4	no	—
163	D₈×C₅₃	—	2	4	no	—
164	M₁₆×C₅₃	—	2	4	no	—
165	M₁₆×C₅₃	—	2	4	no	—
166	C₂×C₈×C₅₃	—	2	2	no	—
167	C₂×C₈×C₅₃	—	2	2	no	—
168	C₂×C₈×C₅₃	—	2	4	no	—
169	C₂×C₈×C₅₃	—	2	4	no	—
170	C₂×C₈×C₅₃	#41	1	4	yes	—
171	C₂×C₈×C₅₃	#41	1	4	yes	—
172	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
173	C₂×C₈×C₅₃	C₂⁴	1	2	yes	—
174	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
175	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
176	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
177	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
178	C₄⋊C₄×C₅₃	—	2	8	no	—
179	C₄⋊C₄×C₅₃	—	2	4	no	—
180	C₄⋊C₄×C₅₃	—	2	8	no	—
181	C₄⋊C₄×C₅₃	—	2	8	no	—
182	C₄⋊C₄×C₅₃	—	2	4	no	—
183	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
184	C₄⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
185	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
186	C₂²⋊C₄×C₅₃	—	2	4	no	—
187	C₂²⋊C₄×C₅₃	—	2	8	no	—
188	C₂²⋊C₄×C₅₃	—	2	8	no	—
189	C₂²⋊C₄×C₅₃	—	2	8	no	—
190	C₂²⋊C₄×C₅₃	—	2	8	no	—
191	C₂²⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
192	C₂²⋊C₄×C₅₃	C₂×D₄	1	4	yes	—
193	C₄²×C₅₃	—	2	8	no	—
194	C₄²×C₅₃	—	2	4	no	—
195	C₄²×C₅₃	C₂×D₄	1	2	yes	—
196	C₄²×C₅₃	C₂×D₄	1	2	yes	—
197	C₄²×C₅₃	C₂⁴	1	2	yes	—
198	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
199	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
200	C₂³×C₅₃	—	2	2	no	—
201	C₂³×C₅₃	—	2	4	no	—
202	C₂³×C₅₃	—	4	2	no	—
203	C₂³×C₅₃	C₂²×D₄	1	2	yes	—
204	C₂³×C₅₃	C₂²×D₄	1	1	yes	characteristic
205	Q₈×C₅₃	—	2	4	no	—
206	Q₈×C₅₃	—	2	4	no	—
207	Q₈×C₅₃	—	8	4	no	—
208	D₄×C₅₃	—	2	4	no	—
209	D₄×C₅₃	—	2	4	no	—
210	D₄×C₅₃	—	4	8	no	—
211	D₄×C₅₃	—	8	4	no	—
212	D₄×C₅₃	—	8	8	no	—
213	C₂×C₄×C₅₃	—	2	2	no	—
214	C₂×C₄×C₅₃	—	2	4	no	—
215	C₂×C₄×C₅₃	—	2	2	no	—
216	C₂×C₄×C₅₃	—	2	2	no	—
217	C₂×C₄×C₅₃	—	2	8	no	—
218	C₂×C₄×C₅₃	—	2	4	no	—
219	C₂×C₄×C₅₃	—	4	2	no	—
220	C₂×C₄×C₅₃	—	4	2	no	—
221	C₂×C₄×C₅₃	—	4	8	no	—
222	C₂×C₄×C₅₃	—	8	8	no	—
223	C₂×C₄×C₅₃	C₂²×D₄	1	4	yes	—
224	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
225	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
226	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
227	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
229	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
230	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
231	C₈×C₅₃	—	2	2	no	—
232	C₈×C₅₃	—	2	2	no	—
233	C₈×C₅₃	C₂×#41	1	2	yes	—
234	C₈×C₅₃	C₂×#41	1	2	yes	—
235	C₈×C₅₃	C₂×#41	1	2	yes	—
236	C₈×C₅₃	C₂×#41	1	2	yes	—
237	#60682	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
238	C₂²×C₅₃	—	2	2	no	—
239	C₂²×C₅₃	—	2	4	no	—
240	C₂²×C₅₃	—	2	2	no	—
241	C₂²×C₅₃	—	4	4	no	—
242	C₂²×C₅₃	—	4	8	no	—
243	C₂²×C₅₃	—	4	4	no	—
244	C₂²×C₅₃	#532	1	2	yes	—
245	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
246	C₄×C₅₃	—	2	4	no	—
247	C₄×C₅₃	—	2	2	no	—
248	C₄×C₅₃	—	2	2	no	—
249	C₄×C₅₃	—	2	2	no	—
250	C₄×C₅₃	—	4	8	no	—
251	C₄×C₅₃	—	4	4	no	—
252	C₄×C₅₃	#532	1	2	yes	—
253	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
254	C₄×C₅₃	C₂²×#41	1	1	yes	characteristic
255	#3455	C₂×C₅₃	1	2	yes	complemented
256	#3177	C₂×C₅₃	1	2	yes	complemented
257	#3176	C₂×C₅₃	1	2	yes	complemented
258	#3175	C₂×C₅₃	1	4	yes	complemented
259	#3174	C₂×C₅₃	1	4	yes	complemented
260	#3173	C₂×C₅₃	1	4	yes	complemented
261	#3160	C₂×C₅₃	1	1	yes	characteristic, complemented
262	#3147	C₂×C₅₃	1	1	yes	characteristic, complemented
263	#3135	C₂×C₅₃	1	2	yes	complemented
264	#3054	C₂×C₅₃	1	2	yes	complemented
265	#3050	C₂×C₅₃	1	2	yes	complemented
266	#2943	C₂×C₅₃	1	2	yes	complemented
267	C₄×#135	C₂×C₅₃	1	2	yes	complemented
268	C₂×#405	C₂×C₅₃	1	1	yes	characteristic, complemented
269	C₂×C₅₃	—	2	2	no	—
270	C₂×C₅₃	—	4	2	no	—
271	C₂×C₅₃	—	8	4	no	—
272	C₂×C₅₃	C₂×#532	1	1	yes	characteristic
273	C₂×C₅₃	#3584	1	2	yes	—
274	C₂×#135	C₂²×C₅₃	1	2	yes	—
275	C₂×#135	C₂²×C₅₃	1	2	yes	—
276	C₂²×M₁₆	C₂²×C₅₃	1	1	yes	characteristic
277	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
278	#542	C₂²×C₅₃	1	8	yes	—
279	#541	C₂²×C₅₃	1	4	yes	—
280	#540	C₂²×C₅₃	1	4	yes	—
281	#539	—	2	4	no	—
282	#538	—	2	4	no	—
283	#533	C₂²×C₅₃	1	2	yes	—
284	#532	C₂²×C₅₃	1	2	yes	—
285	#529	—	2	8	no	—
286	#525	C₂²×C₅₃	1	4	yes	—
287	C₄×#41	C₂²×C₅₃	1	4	yes	—
288	C₂×#117	C₂²×C₅₃	1	1	yes	characteristic
289	#500	C₂²×C₅₃	1	2	yes	complemented
290	#499	C₂²×C₅₃	1	2	yes	complemented
291	#499	C₂²×C₅₃	1	2	yes	complemented
292	#498	C₂²×C₅₃	1	2	yes	complemented
293	#495	C₂²×C₅₃	1	2	yes	complemented
294	#495	C₂²×C₅₃	1	2	yes	complemented
295	#494	C₂²×C₅₃	1	1	yes	characteristic, complemented
296	#493	C₂²×C₅₃	1	1	yes	characteristic, complemented
297	#492	C₂²×C₅₃	1	2	yes	complemented
298	#491	C₂²×C₅₃	1	4	yes	complemented
299	#490	C₂²×C₅₃	1	4	yes	complemented
300	#489	C₂²×C₅₃	1	8	yes	complemented
301	#470	C₂²×C₅₃	1	4	yes	—
302	#469	C₂²×C₅₃	1	4	yes	—
303	#468	C₂²×C₅₃	1	8	yes	—
304	#467	C₂²×C₅₃	1	4	yes	—
305	#466	C₂²×C₅₃	1	4	yes	—
306	#465	C₂²×C₅₃	1	8	yes	—
307	#442	C₂²×C₅₃	1	4	yes	complemented
308	#441	C₂²×C₅₃	1	4	yes	complemented
309	#440	C₂²×C₅₃	1	8	yes	complemented
310	C₄×Q₁₆	C₂²×C₅₃	1	4	yes	complemented
311	C₄×SD₁₆	C₂²×C₅₃	1	8	yes	complemented
312	C₄×D₈	C₂²×C₅₃	1	4	yes	complemented
313	#427	C₂²×C₅₃	1	2	yes	—
314	#427	C₂²×C₅₃	1	2	yes	—
315	#416	C₂²×C₅₃	1	4	yes	—
316	#416	C₂²×C₅₃	1	4	yes	—
317	#405	C₂²×C₅₃	1	2	yes	—
318	#405	C₂²×C₅₃	1	2	yes	—
319	#405	C₂²×C₅₃	1	2	yes	—
320	#405	C₂²×C₅₃	1	2	yes	—
321	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
322	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
323	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
324	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
325	C₅₃	#60682	1	1	yes	characteristic, direct product factor, Sylow
326	C₂×#41	C₂³×C₅₃	1	4	yes	—
327	C₂³×C₄	C₂³×C₅₃	1	1	yes	characteristic
328	#135	—	2	8	no	—
329	#135	—	2	8	no	—
330	C₂×Q₁₆	C₂³×C₅₃	1	2	yes	—
331	C₂×Q₁₆	C₂³×C₅₃	1	2	yes	—
332	C₂×SD₁₆	C₂³×C₅₃	1	4	yes	—
333	C₂×SD₁₆	C₂³×C₅₃	1	4	yes	—
334	C₂×D₈	C₂³×C₅₃	1	2	yes	—
335	C₂×D₈	C₂³×C₅₃	1	2	yes	—
336	C₂×M₁₆	—	2	4	no	—
337	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
338	C₂×M₁₆	C₂³×C₅₃	1	2	yes	—
339	C₂²×C₈	—	2	4	no	—
340	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
341	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
342	C₂²×C₈	C₂³×C₅₃	1	2	yes	—
343	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
344	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
345	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
346	C₄⋊Q₈	C₂³×C₅₃	1	2	yes	—
347	C₄⋊Q₈	C₂³×C₅₃	1	4	yes	—
348	#127	C₂³×C₅₃	1	2	yes	—
349	#126	—	2	8	no	—
350	#126	—	2	8	no	—
351	#125	—	2	8	no	—
352	#125	C₂³×C₅₃	1	4	yes	—
353	#124	—	2	8	no	—
354	#124	C₂³×C₅₃	1	4	yes	—
355	#123	—	2	4	no	—
356	#123	—	2	8	no	—
357	#123	—	2	8	no	—
358	C₂²⋊Q₈	—	2	4	no	—
359	C₂²⋊Q₈	—	2	8	no	—
360	C₂²⋊Q₈	C₂³×C₅₃	1	8	yes	—
361	#121	—	2	4	no	—
362	#121	—	2	8	no	—
363	#121	C₂³×C₅₃	1	8	yes	—
364	#120	—	2	4	no	—
365	C₄×Q₈	—	2	4	no	—
366	C₄×Q₈	C₂³×C₅₃	1	8	yes	—
367	C₄×D₄	—	2	8	no	—
368	C₄×D₄	—	2	8	no	—
369	C₄×D₄	—	2	4	no	—
370	C₄×D₄	C₂³×C₅₃	1	8	yes	—
371	#117	—	2	4	no	—
372	#117	D₄×C₅₃	1	4	yes	—
373	#117	C₂³×C₅₃	1	4	yes	—
374	#117	C₂³×C₅₃	1	2	yes	—
375	#117	C₂³×C₅₃	1	2	yes	—
376	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
377	C₂×C₂²⋊C₄	D₄×C₅₃	1	2	yes	—
378	C₂×C₄²	—	2	4	no	—
379	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
380	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
381	#107	C₂³×C₅₃	1	4	yes	—
382	#107	C₂³×C₅₃	1	4	yes	—
383	#106	C₂³×C₅₃	1	4	yes	—
384	#106	C₂³×C₅₃	1	4	yes	—
385	#103	C₂³×C₅₃	1	8	yes	—
386	#103	C₂³×C₅₃	1	8	yes	—
387	#102	C₂³×C₅₃	1	8	yes	—
388	#102	C₂³×C₅₃	1	8	yes	—
389	#97	C₂³×C₅₃	1	2	yes	—
390	#97	C₂³×C₅₃	1	2	yes	—
391	#97	C₂³×C₅₃	1	2	yes	—
392	#97	C₂³×C₅₃	1	2	yes	—
393	C₄×C₈	C₂³×C₅₃	1	2	yes	—
394	C₄×C₈	C₂³×C₅₃	1	2	yes	—
395	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
396	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
397	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
398	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
399	C₂⁴	C₂×D₄×C₅₃	1	1	yes	characteristic
400	#41	—	2	8	no	—
401	#41	—	2	8	no	—
402	C₂×Q₈	—	2	4	no	—
403	C₂×Q₈	C₂⁴×C₅₃	1	4	yes	—
404	C₂×D₄	—	2	4	no	—
405	C₂×D₄	—	2	8	no	—
406	C₂×D₄	—	2	4	no	—
407	C₂×D₄	C₂⁴×C₅₃	1	4	yes	—
408	C₂²×C₄	—	2	2	no	—
409	C₂²×C₄	—	2	2	no	—
410	C₂²×C₄	—	2	4	no	—
411	C₂²×C₄	—	2	8	no	—
412	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
413	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
414	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
415	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
416	C₂²×C₄	C₂⁴×C₅₃	1	2	yes	—
417	C₂²×C₄	C₂⁴×C₅₃	1	1	yes	characteristic
418	Q₁₆	—	2	4	no	—
419	Q₁₆	—	2	4	no	—
420	SD₁₆	—	2	8	no	—
421	SD₁₆	—	2	8	no	—
422	D₈	—	2	4	no	—
423	D₈	—	2	4	no	—
424	M₁₆	—	2	4	no	—
425	M₁₆	—	2	4	no	—
426	C₂×C₈	—	2	2	no	—
427	C₂×C₈	—	2	2	no	—
428	C₂×C₈	—	2	4	no	—
429	C₂×C₈	—	2	4	no	—
430	C₂×C₈	#41×C₅₃	1	4	yes	—
431	C₂×C₈	#41×C₅₃	1	4	yes	—
432	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
433	C₂×C₈	C₂⁴×C₅₃	1	2	yes	—
434	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
435	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
436	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
437	C₂×C₈	C₂⁴×C₅₃	1	1	yes	characteristic
438	C₄⋊C₄	—	2	8	no	—
439	C₄⋊C₄	—	2	4	no	—
440	C₄⋊C₄	—	2	8	no	—
441	C₄⋊C₄	—	2	8	no	—
442	C₄⋊C₄	—	2	4	no	—
443	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
444	C₄⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
445	C₄⋊C₄	C₂⁴×C₅₃	1	8	yes	—
446	C₂²⋊C₄	—	2	4	no	—
447	C₂²⋊C₄	—	2	8	no	—
448	C₂²⋊C₄	—	2	8	no	—
449	C₂²⋊C₄	—	2	8	no	—
450	C₂²⋊C₄	—	2	8	no	—
451	C₂²⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
452	C₂²⋊C₄	C₂×D₄×C₅₃	1	4	yes	—
453	C₄²	—	2	4	no	—
454	C₄²	—	2	8	no	—
455	C₄²	C₂×D₄×C₅₃	1	2	yes	—
456	C₄²	C₂×D₄×C₅₃	1	2	yes	—
457	C₄²	C₂⁴×C₅₃	1	2	yes	—
458	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
459	C₄²	C₂⁴×C₅₃	1	1	yes	characteristic
460	C₂³	—	2	2	no	—
461	C₂³	—	2	4	no	—
462	C₂³	—	4	2	no	—
463	C₂³	C₂²×D₄×C₅₃	1	2	yes	—
464	C₂³	C₂²×D₄×C₅₃	1	1	yes	characteristic
465	Q₈	—	2	4	no	—
466	Q₈	—	2	4	no	—
467	Q₈	—	8	4	no	—
468	D₄	—	2	4	no	—
469	D₄	—	2	4	no	—
470	D₄	—	4	8	no	—
471	D₄	—	8	4	no	—
472	D₄	—	8	8	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	2	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	4	no	—
478	C₂×C₄	—	2	8	no	—
479	C₂×C₄	—	4	2	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	4	8	no	—
482	C₂×C₄	—	8	8	no	—
483	C₂×C₄	C₂²×D₄×C₅₃	1	4	yes	—
484	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
485	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
486	C₂×C₄	C₂²×D₄×C₅₃	1	2	yes	—
487	C₂×C₄	C₂²×D₄×C₅₃	1	1	yes	characteristic
488	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
489	C₂×C₄	C₂×#41×C₅₃	1	2	yes	—
490	C₂×C₄	C₂⁵×C₅₃	1	1	yes	characteristic
491	C₈	—	2	2	no	—
492	C₈	—	2	2	no	—
493	C₈	C₂×#41×C₅₃	1	2	yes	—
494	C₈	C₂×#41×C₅₃	1	2	yes	—
495	C₈	C₂×#41×C₅₃	1	2	yes	—
496	C₈	C₂×#41×C₅₃	1	2	yes	—
497	C₂²	—	2	2	no	—
498	C₂²	—	2	4	no	—
499	C₂²	—	2	2	no	—
500	C₂²	—	4	4	no	—
501	C₂²	—	4	8	no	—
502	C₂²	—	4	4	no	—
503	C₂²	C₅₃×#532	1	2	yes	—
504	C₂²	C₂³×D₄×C₅₃	1	1	yes	characteristic
505	C₄	—	2	4	no	—
506	C₄	—	2	2	no	—
507	C₄	—	2	2	no	—
508	C₄	—	2	2	no	—
509	C₄	—	4	8	no	—
510	C₄	—	4	4	no	—
511	C₄	C₅₃×#532	1	2	yes	—
512	C₄	C₂²×#41×C₅₃	1	1	yes	characteristic
513	C₄	C₂²×#41×C₅₃	1	1	yes	characteristic
514	C₂	—	2	2	no	—
515	C₂	—	4	2	no	—
516	C₂	—	8	4	no	—
517	C₂	C₂×C₅₃×#532	1	1	yes	characteristic
518	C₂	C₅₃×#3584	1	2	yes	—

	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×#117×C₅₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅₃×#3455	C₂	semidirect
2. C₅₃×#3177	C₂	semidirect
3. C₅₃×#3176	C₂	semidirect
4. C₅₃×#3175	C₂	semidirect
5. C₅₃×#3174	C₂	semidirect
6. C₅₃×#3173	C₂	semidirect
7. C₅₃×#3160	C₂	semidirect
8. C₅₃×#3147	C₂	semidirect
9. C₅₃×#3135	C₂	semidirect
10. C₅₃×#3054	C₂	semidirect
11. C₅₃×#3050	C₂	semidirect
12. C₅₃×#2943	C₂	semidirect
13. C₄×#135×C₅₃	C₂	semidirect
14. C₂×C₅₃×#405	C₂	semidirect
30. C₅₃×#500	C₂²	semidirect
31. C₅₃×#499	C₂²	semidirect
32. C₅₃×#499	C₂²	semidirect
33. C₅₃×#498	C₂²	semidirect
34. C₅₃×#495	C₂²	semidirect
35. C₅₃×#495	C₂²	semidirect
36. C₅₃×#494	C₂²	semidirect
37. C₅₃×#493	C₂²	semidirect
38. C₅₃×#492	C₂²	semidirect
39. C₅₃×#491	C₂²	semidirect
40. C₅₃×#490	C₂²	semidirect
41. C₅₃×#489	C₂²	semidirect
48. C₅₃×#442	C₂²	semidirect
49. C₅₃×#441	C₂²	semidirect
50. C₅₃×#440	C₂²	semidirect
51. C₄×Q₁₆×C₅₃	C₂²	semidirect
52. C₄×SD₁₆×C₅₃	C₂²	semidirect
53. C₄×D₈×C₅₃	C₂²	semidirect
237. #60682	C₅₃	direct (N × H)
255. #3455	C₂×C₅₃	semidirect
256. #3177	C₂×C₅₃	semidirect
257. #3176	C₂×C₅₃	semidirect
258. #3175	C₂×C₅₃	semidirect
259. #3174	C₂×C₅₃	semidirect
260. #3173	C₂×C₅₃	semidirect
261. #3160	C₂×C₅₃	semidirect
262. #3147	C₂×C₅₃	semidirect
263. #3135	C₂×C₅₃	semidirect
264. #3054	C₂×C₅₃	semidirect
265. #3050	C₂×C₅₃	semidirect
266. #2943	C₂×C₅₃	semidirect
267. C₄×#135	C₂×C₅₃	semidirect
268. C₂×#405	C₂×C₅₃	semidirect
289. #500	C₂²×C₅₃	semidirect
290. #499	C₂²×C₅₃	semidirect
291. #499	C₂²×C₅₃	semidirect
292. #498	C₂²×C₅₃	semidirect
293. #495	C₂²×C₅₃	semidirect
294. #495	C₂²×C₅₃	semidirect
295. #494	C₂²×C₅₃	semidirect
296. #493	C₂²×C₅₃	semidirect
297. #492	C₂²×C₅₃	semidirect
298. #491	C₂²×C₅₃	semidirect
299. #490	C₂²×C₅₃	semidirect
300. #489	C₂²×C₅₃	semidirect
307. #442	C₂²×C₅₃	semidirect
308. #441	C₂²×C₅₃	semidirect
309. #440	C₂²×C₅₃	semidirect
310. C₄×Q₁₆	C₂²×C₅₃	semidirect
311. C₄×SD₁₆	C₂²×C₅₃	semidirect
312. C₄×D₈	C₂²×C₅₃	semidirect
325. C₅₃	#60682	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×#3455	2
C₅₃×#3177	2
C₅₃×#3176	2
C₅₃×#3175	2
C₅₃×#3174	2
C₅₃×#3173	2
C₅₃×#3160	2
C₅₃×#3147	2
C₅₃×#3135	2
C₅₃×#3054	2
C₅₃×#3050	2
C₅₃×#2943	2
C₄×#135×C₅₃	2
C₂×C₅₃×#405	2
#60682	53

Subgroups — C53×#60682 #27268194

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#60682 #27268194