Subgroups — #27353520

Order 13568.

714 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11719947	C₂	1	1	yes	maximal, characteristic
2	C₈×Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#13133246	C₂	1	1	yes	maximal, characteristic
4	#20273535	C₂	1	1	yes	maximal, characteristic, complemented
5	#20273532	C₂	1	1	yes	maximal, characteristic, complemented
6	#20273527	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273521	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273519	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273518	C₂	1	1	yes	maximal, characteristic, complemented
10	SD₁₆×#88322	C₂	1	1	yes	maximal, characteristic, complemented
11	#20273466	C₂	1	1	yes	maximal, characteristic, complemented
12	#20273108	C₂	1	1	yes	maximal, characteristic, complemented
13	#20273099	C₂	1	1	yes	maximal, characteristic
14	#20272994	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×SD₁₆⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13133246	C₄	1	1	yes	characteristic
17	C₄×#11719947	C₂²	1	1	yes	characteristic
18	C₄×Q₈×D₅₃	C₂²	1	1	yes	characteristic
19	Q₈×#88322	C₂²	1	1	yes	characteristic
20	C₈×Dic₁₀₆	—	2	1	no	—
21	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
22	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
23	#16810747	—	2	1	no	—
24	#16810747	C₂²	1	1	yes	characteristic
25	#16810747	C₂²	1	1	yes	characteristic
26	#16810729	C₄	1	1	yes	characteristic
27	#16810727	C₄	1	1	yes	characteristic
28	#16810724	C₄	1	1	yes	characteristic
29	#16810720	C₄	1	1	yes	characteristic
30	C₄×Dic₂₁₂	C₂²	1	1	yes	characteristic
31	C₄×#11719925	C₂²	1	1	yes	characteristic
32	C₄×#11719923	C₂²	1	1	yes	characteristic
33	D₄×#88322	C₂²	1	1	yes	characteristic
34	C₈×C₂²⋊D₅₃	—	2	1	no	—
35	#16810686	—	2	1	no	—
36	#16810675	C₂²	1	1	yes	characteristic
37	#16810674	C₂²	1	1	yes	characteristic
38	#16810673	—	2	1	no	—
39	#16810662	C₄	1	1	yes	characteristic
40	C₄⋊C₈×D₅₃	—	2	1	no	—
41	C₄⋊C₈×D₅₃	C₂²	1	1	yes	characteristic
42	#16810630	C₄	1	1	yes	characteristic
43	#16810621	C₄	1	1	yes	characteristic
44	C₄×#11719912	C₂²	1	1	yes	characteristic
45	C₂×C₈×Dic₅₃	—	2	1	no	—
46	C₄×C₈×D₅₃	—	2	1	no	—
47	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
48	#16810375	C₂²	1	1	yes	characteristic
49	#16810374	C₂²	1	1	yes	characteristic
50	#16810373	C₂²	1	1	yes	characteristic
51	#16810355	C₂²	1	1	yes	characteristic
52	#16810355	C₂²	1	1	yes	characteristic
53	Q₈⋊#88322	C₂²	1	1	yes	characteristic
54	#16810353	C₂²	1	1	yes	characteristic
55	#16810353	C₂²	1	1	yes	characteristic
56	D₄⋊#88322	C₂²	1	1	yes	characteristic
57	#16810345	C₂²	1	1	yes	characteristic
58	C₈×#88322	C₂²	1	1	yes	characteristic
59	C₈×#88322	C₂²	1	1	yes	characteristic
60	C₈×Q₈×C₅₃	C₂²	1	1	yes	characteristic
61	C₄×SD₁₆×C₅₃	C₂²	1	1	yes	characteristic
62	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
63	C₅₃×#335	C₂²	1	1	yes	characteristic
64	C₅₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
65	C₅₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
66	C₅₃×#322	C₂²	1	1	yes	characteristic
67	C₂×#11719947	C₂×C₄	1	1	yes	characteristic
68	C₂×Q₈×D₅₃	C₂×C₄	1	1	yes	characteristic
69	#13133246	C₈	1	8	yes	complemented
70	C₂×Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
71	C₂×#11719925	C₂×C₄	1	1	yes	characteristic
72	C₂×#11719923	C₂×C₄	1	1	yes	characteristic
73	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
74	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
75	C₂²×#88322	—	2	1	no	—
76	C₂×C₈×D₅₃	—	2	1	no	—
77	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
78	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
79	Q₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
80	C₄×Dic₁₀₆	—	2	1	no	—
81	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
82	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
83	#13133182	—	2	1	no	—
84	#13133182	C₂×C₄	1	1	yes	characteristic
85	#13133182	C₂×C₄	1	1	yes	characteristic
86	D₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
87	C₄×C₂²⋊D₅₃	—	2	1	no	—
88	#13133177	—	2	1	no	—
89	#13133175	C₂×C₄	1	1	yes	characteristic
90	#13133173	—	2	1	no	—
91	C₄⋊C₄×D₅₃	—	2	1	no	—
92	C₄⋊C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄×Dic₅₃	—	2	1	no	—
94	C₄²×D₅₃	—	2	1	no	—
95	C₄²×D₅₃	C₂³	1	1	yes	characteristic
96	#13133151	C₂×C₄	1	1	yes	characteristic
97	#13133150	C₂×C₄	1	1	yes	characteristic
98	#13133148	C₂×C₄	1	1	yes	characteristic
99	C₄⋊#88322	—	2	1	no	—
100	C₄⋊#88322	C₂³	1	1	yes	characteristic
101	C₄⋊#88322	C₂³	1	1	yes	characteristic
102	#13133146	—	2	1	no	—
103	#13133146	—	2	1	no	—
104	#13133146	—	2	1	no	—
105	#13133146	—	2	1	no	—
106	#13133146	—	2	1	no	—
107	#13133146	—	2	1	no	—
108	#13133142	C₂×C₄	1	1	yes	characteristic
109	#13133142	C₂×C₄	1	1	yes	characteristic
110	#13133141	C₂×C₄	1	1	yes	characteristic
111	#13133140	C₂×C₄	1	1	yes	characteristic
112	#13133140	C₂×C₄	1	1	yes	characteristic
113	#13133138	C₂×C₄	1	1	yes	characteristic
114	C₂²⋊#88322	—	2	1	no	—
115	#13133129	—	2	1	no	—
116	#13133129	—	2	1	no	—
117	#13133127	C₂×C₄	1	1	yes	characteristic
118	#13133127	C₂×C₄	1	1	yes	characteristic
119	C₄×#88322	—	2	1	no	—
120	C₄×#88322	C₂³	1	1	yes	characteristic
121	C₄×#88322	C₂³	1	1	yes	characteristic
122	C₄×#88322	C₂³	1	1	yes	characteristic
123	C₈×Dic₅₃	—	2	1	no	—
124	C₈×Dic₅₃	—	2	1	no	—
125	C₈×Dic₅₃	—	2	1	no	—
126	C₈×Dic₅₃	—	2	1	no	—
127	C₈×Dic₅₃	—	2	1	no	—
128	C₈×Dic₅₃	D₄	1	1	yes	characteristic
129	C₈×Dic₅₃	D₄	1	1	yes	characteristic
130	C₂×SD₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
131	C₂²×C₈×C₅₃	—	2	1	no	—
132	C₄×Q₈×C₅₃	C₂³	1	1	yes	characteristic
133	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
134	#106×C₅₃	C₂×C₄	1	1	yes	characteristic
135	C₄⋊C₈×C₅₃	—	2	1	no	—
136	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
137	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
138	#103×C₅₃	C₂×C₄	1	1	yes	characteristic
139	#102×C₅₃	C₂×C₄	1	1	yes	characteristic
140	C₂²⋊C₈×C₅₃	—	2	1	no	—
141	C₄×C₈×C₅₃	—	2	1	no	—
142	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
144	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
145	C₂×Q₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
146	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
147	C₂²×C₄×C₅₃	—	2	1	no	—
148	C₂×C₂²⋊D₅₃	—	2	1	no	—
149	C₂²×Dic₅₃	—	2	1	no	—
150	C₂²×Dic₅₃	—	2	1	no	—
151	Q₈×D₅₃	—	2	2	no	—
152	Q₈×D₅₃	C₂×C₈	1	4	yes	complemented
153	#11719947	—	2	2	no	—
154	#11719947	C₂×C₈	1	4	yes	—
155	C₂×C₄×D₅₃	—	2	1	no	—
156	C₂×C₄×D₅₃	—	2	1	no	—
157	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
158	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
159	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
160	C₂×Dic₁₀₆	—	2	1	no	—
161	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
162	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
163	SD₁₆×C₅₃	C₂×C₈	1	4	yes	complemented
164	C₂×C₈×C₅₃	—	2	1	no	—
165	C₂×C₈×C₅₃	—	2	1	no	—
166	C₂×C₈×C₅₃	—	2	1	no	—
167	C₂×C₈×C₅₃	—	4	2	no	—
168	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
169	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
170	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
171	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
172	C₄⋊C₄×C₅₃	—	2	1	no	—
173	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
174	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
175	C₂²⋊C₄×C₅₃	—	2	1	no	—
176	C₄²×C₅₃	—	2	1	no	—
177	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
178	C₂²⋊Dic₅₃	—	2	1	no	—
179	#11719925	C₂×C₈	1	4	yes	complemented
180	#11719923	C₂×C₈	1	4	yes	complemented
181	#11719921	—	2	1	no	—
182	C₄⋊Dic₅₃	—	2	1	no	—
183	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
184	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
185	#11719919	—	2	1	no	—
186	#11719919	—	2	1	no	—
187	#11719919	—	2	1	no	—
188	#11719919	—	2	1	no	—
189	#11719919	—	2	1	no	—
190	C₄×Dic₅₃	—	2	1	no	—
191	C₄×Dic₅₃	—	2	1	no	—
192	C₄×Dic₅₃	—	2	1	no	—
193	C₄×Dic₅₃	—	2	1	no	—
194	C₄×Dic₅₃	—	2	1	no	—
195	C₄×Dic₅₃	—	2	1	no	—
196	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
197	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
198	C₄×Dic₅₃	#41	1	1	yes	characteristic
199	C₂×#88322	—	2	1	no	—
200	C₂×#88322	—	2	1	no	—
201	C₂×#88322	—	2	1	no	—
202	C₂×#88322	—	4	2	no	—
203	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
204	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
205	C₂×#88322	C₂×D₄	1	1	yes	characteristic
206	C₂×#88322	C₂×D₄	1	1	yes	characteristic
207	Dic₂₁₂	C₂×C₈	1	4	yes	complemented
208	#11719912	—	2	2	no	—
209	#11719912	C₂×C₈	1	4	yes	complemented
210	C₈×D₅₃	—	2	2	no	—
211	C₈×D₅₃	—	4	1	no	—
212	C₈×D₅₃	—	4	2	no	—
213	C₂³×C₅₃	—	2	1	no	—
214	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
215	Q₈×C₅₃	—	2	1	no	—
216	Q₈×C₅₃	C₂²×C₈	1	2	yes	—
217	D₄×C₅₃	—	2	1	no	—
218	D₄×C₅₃	C₂²×C₈	1	2	yes	—
219	C₂×C₄×C₅₃	—	2	1	no	—
220	C₂×C₄×C₅₃	—	2	1	no	—
221	C₂×C₄×C₅₃	—	2	1	no	—
222	C₂×C₄×C₅₃	—	4	1	no	—
223	C₂×C₄×C₅₃	—	4	1	no	—
224	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
225	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
227	C₂²⋊D₅₃	—	2	4	no	—
228	C₂×Dic₅₃	—	2	1	no	—
229	C₂×Dic₅₃	—	2	1	no	—
230	C₂×Dic₅₃	—	2	4	no	—
231	C₂×Dic₅₃	—	2	1	no	—
232	C₂×Dic₅₃	—	2	1	no	—
233	C₂×Dic₅₃	—	4	2	no	—
234	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
235	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
236	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
237	C₄×D₅₃	—	2	4	no	—
238	C₄×D₅₃	—	2	1	no	—
239	C₄×D₅₃	—	2	1	no	—
240	C₄×D₅₃	—	4	1	no	—
241	C₄×D₅₃	—	4	2	no	—
242	C₄×D₅₃	C₂²×C₈	1	2	yes	—
243	C₄×D₅₃	#131	1	2	yes	—
244	Dic₁₀₆	—	2	1	no	—
245	Dic₁₀₆	—	2	1	no	—
246	Dic₁₀₆	—	2	4	no	—
247	Dic₁₀₆	C₂²×C₈	1	2	yes	—
248	Dic₁₀₆	C₂²×C₈	1	2	yes	—
249	C₈×C₅₃	—	2	1	no	—
250	C₈×C₅₃	—	2	1	no	—
251	C₈×C₅₃	—	2	1	no	—
252	C₈×C₅₃	—	4	1	no	—
253	C₈×C₅₃	—	4	1	no	—
254	C₈×C₅₃	—	4	1	no	—
255	C₈×C₅₃	C₂²×C₈	1	2	yes	—
256	#88322	—	2	1	no	—
257	#88322	—	2	1	no	—
258	#88322	—	4	1	no	—
259	#88322	—	4	1	no	—
260	#88322	—	4	1	no	—
261	#88322	C₂²×C₈	1	2	yes	—
262	#88322	#137	1	2	yes	complemented
263	C₈×#137	—	53	1	no	maximal, Sylow
264	C₂²×C₅₃	—	2	2	no	—
265	C₂²×C₅₃	—	4	1	no	—
266	C₂²×C₅₃	—	4	1	no	—
267	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
268	C₂×D₅₃	—	2	1	no	—
269	C₂×D₅₃	—	2	1	no	—
270	C₂×D₅₃	C₈×D₄	1	2	yes	—
271	C₄×C₅₃	—	2	2	no	—
272	C₄×C₅₃	—	2	1	no	—
273	C₄×C₅₃	—	4	1	no	—
274	C₄×C₅₃	—	4	1	no	—
275	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
276	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
277	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
278	C₄×C₅₃	#575	1	1	yes	characteristic
279	Dic₅₃	—	2	2	no	—
280	Dic₅₃	—	2	2	no	—
281	Dic₅₃	—	2	1	no	—
282	Dic₅₃	—	2	1	no	—
283	Dic₅₃	—	4	1	no	—
284	Dic₅₃	—	4	1	no	—
285	Dic₅₃	C₈×D₄	1	2	yes	—
286	C₈×#41	—	53	1	no	—
287	C₂×C₈×Q₈	—	53	1	no	—
288	C₄×#137	—	53	1	no	—
289	#1579	—	53	1	no	—
290	#1579	—	53	1	no	—
291	SD₁₆⋊C₈	—	53	1	no	—
292	SD₁₆⋊C₈	D₅₃	1	1	yes	characteristic, complemented
293	C₈×Q₁₆	—	53	1	no	—
294	C₈×Q₁₆	—	53	1	no	—
295	C₈×SD₁₆	—	53	1	no	—
296	C₈×SD₁₆	—	53	1	no	—
297	#1565	—	53	1	no	—
298	#1479	—	53	1	no	—
299	#1477	—	53	1	no	—
300	C₈×M₁₆	—	53	1	no	—
301	C₂×C₅₃	—	4	2	no	—
302	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
303	C₂×C₅₃	C₄×#137	1	1	yes	characteristic
304	C₂×C₅₃	#2955	1	1	yes	characteristic
305	D₅₃	—	2	2	no	—
306	C₂×#137	—	53	1	no	—
307	C₄×#41	—	53	1	no	—
308	C₂×C₄×Q₈	—	53	1	no	—
309	C₈×Q₈	—	53	1	no	—
310	C₈×Q₈	—	53	1	no	—
311	C₈×Q₈	—	53	1	no	—
312	C₈×Q₈	—	53	1	no	—
313	C₈×Q₈	—	53	1	no	—
314	C₈×Q₈	—	106	1	no	—
315	C₈×Q₈	—	106	1	no	—
316	C₈×Q₈	C₂×D₅₃	1	1	yes	characteristic
317	#441	—	53	1	no	—
318	#441	—	53	1	no	—
319	#440	—	53	1	no	—
320	#440	—	53	1	no	—
321	C₄×Q₁₆	—	53	1	no	—
322	C₄×Q₁₆	—	53	1	no	—
323	C₄×SD₁₆	—	53	1	no	—
324	C₄×SD₁₆	C₂×D₅₃	1	1	yes	characteristic
325	C₈×D₄	—	53	1	no	—
326	C₈×D₄	—	106	1	no	—
327	C₈×D₄	—	106	1	no	—
328	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
329	#431	—	53	1	no	—
330	#431	—	53	1	no	—
331	#431	—	106	1	no	—
332	#428	—	53	1	no	—
333	C₂×C₄⋊C₈	—	53	1	no	—
334	C₂×C₄⋊C₈	—	106	1	no	—
335	#419	—	53	1	no	—
336	#417	—	53	1	no	—
337	C₄×M₁₆	—	53	1	no	—
338	C₂×C₄×C₈	—	53	1	no	—
339	C₂×C₄×C₈	—	106	1	no	—
340	C₂×C₄×C₈	—	106	1	no	—
341	#335	—	53	1	no	—
342	#335	C₂×D₅₃	1	1	yes	characteristic
343	#334	—	53	1	no	—
344	#334	—	53	1	no	—
345	Q₈⋊C₈	—	53	1	no	—
346	Q₈⋊C₈	—	53	1	no	—
347	Q₈⋊C₈	—	53	1	no	—
348	Q₈⋊C₈	—	53	1	no	—
349	Q₈⋊C₈	—	53	1	no	—
350	Q₈⋊C₈	C₂×D₅₃	1	1	yes	characteristic
351	D₄⋊C₈	—	53	1	no	—
352	D₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
353	#322	—	53	1	no	—
354	#322	C₂×D₅₃	1	1	yes	characteristic
355	C₈²	—	53	1	no	—
356	C₈²	—	53	1	no	—
357	C₅₃	C₈×#137	1	1	yes	characteristic, complemented, Sylow
358	C₂×#41	—	53	1	no	—
359	C₂²×Q₈	—	53	1	no	—
360	#137	—	53	8	no	—
361	C₂×Q₁₆	—	53	1	no	—
362	C₂×Q₁₆	—	53	1	no	—
363	C₂×SD₁₆	—	53	1	no	—
364	C₂×SD₁₆	C₄×D₅₃	1	1	yes	characteristic
365	C₂×M₁₆	—	53	1	no	—
366	C₂×M₁₆	—	53	1	no	—
367	C₂²×C₈	—	2	1	no	—
368	C₂²×C₈	—	53	1	no	—
369	C₂²×C₈	—	53	1	no	—
370	C₂²×C₈	—	106	1	no	—
371	C₂²×C₈	—	106	1	no	—
372	C₄×Q₈	—	53	1	no	—
373	C₄×Q₈	—	53	1	no	—
374	C₄×Q₈	—	53	1	no	—
375	C₄×Q₈	—	53	1	no	—
376	C₄×Q₈	—	53	1	no	—
377	C₄×Q₈	—	106	1	no	—
378	C₄×Q₈	—	106	1	no	—
379	C₄×Q₈	C₂²×D₅₃	1	1	yes	characteristic
380	C₄×D₄	—	53	1	no	—
381	C₄×D₄	—	106	1	no	—
382	C₄×D₄	—	106	1	no	—
383	C₄×D₄	C₂²×D₅₃	1	1	yes	characteristic
384	#117	—	53	1	no	—
385	#117	—	106	1	no	—
386	C₂×C₄⋊C₄	—	53	1	no	—
387	C₂×C₄⋊C₄	—	106	1	no	—
388	C₂×C₄²	—	53	1	no	—
389	C₂×C₄²	—	106	1	no	—
390	C₂×C₄²	—	106	1	no	—
391	#107	—	53	1	no	—
392	#107	—	53	1	no	—
393	#106	—	53	1	no	—
394	#106	C₄×D₅₃	1	1	yes	characteristic
395	C₄⋊C₈	—	2	1	no	—
396	C₄⋊C₈	—	53	1	no	—
397	C₄⋊C₈	—	53	1	no	—
398	C₄⋊C₈	—	106	1	no	—
399	C₄⋊C₈	—	106	1	no	—
400	C₄⋊C₈	—	106	1	no	—
401	C₄⋊C₈	—	106	1	no	—
402	C₄⋊C₈	—	106	1	no	—
403	C₄⋊C₈	—	106	1	no	—
404	C₄⋊C₈	—	106	1	no	—
405	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
406	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
407	#103	—	53	1	no	—
408	#103	—	53	1	no	—
409	#103	—	53	1	no	—
410	#103	—	53	1	no	—
411	#103	—	53	1	no	—
412	#103	C₄×D₅₃	1	1	yes	characteristic
413	#102	—	53	1	no	—
414	#102	C₄×D₅₃	1	1	yes	characteristic
415	C₂²⋊C₈	—	2	1	no	—
416	C₂²⋊C₈	—	106	1	no	—
417	C₂²⋊C₈	—	106	1	no	—
418	C₂²⋊C₈	—	106	1	no	—
419	#97	—	53	1	no	—
420	#97	—	53	1	no	—
421	C₄×C₈	—	2	1	no	—
422	C₄×C₈	—	53	1	no	—
423	C₄×C₈	—	53	1	no	—
424	C₄×C₈	—	53	1	no	—
425	C₄×C₈	—	53	1	no	—
426	C₄×C₈	—	53	1	no	—
427	C₄×C₈	—	106	1	no	—
428	C₄×C₈	—	106	1	no	—
429	C₄×C₈	—	106	1	no	—
430	C₄×C₈	—	106	1	no	—
431	C₄×C₈	—	106	1	no	—
432	C₄×C₈	—	106	1	no	—
433	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
434	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
435	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
436	#41	—	53	4	no	—
437	#41	—	106	2	no	—
438	C₂×Q₈	—	53	4	no	—
439	C₂×Q₈	—	53	1	no	—
440	C₂×Q₈	—	53	1	no	—
441	C₂×Q₈	—	106	1	no	—
442	C₂×Q₈	—	106	2	no	—
443	C₂×Q₈	C₂×C₄×D₅₃	1	1	yes	characteristic
444	C₂×D₄	—	106	1	no	—
445	C₂×D₄	C₂×C₄×D₅₃	1	1	yes	characteristic
446	C₂²×C₄	—	2	1	no	—
447	C₂²×C₄	—	53	1	no	—
448	C₂²×C₄	—	53	1	no	—
449	C₂²×C₄	—	53	1	no	—
450	C₂²×C₄	—	106	1	no	—
451	C₂²×C₄	—	106	1	no	—
452	C₂²×C₄	—	106	1	no	—
453	C₂²×C₄	—	106	1	no	—
454	Q₁₆	—	53	4	no	—
455	Q₁₆	—	53	4	no	—
456	SD₁₆	—	53	4	no	—
457	SD₁₆	C₈×D₅₃	1	4	yes	complemented
458	M₁₆	—	53	4	no	—
459	M₁₆	—	106	2	no	—
460	C₂×C₈	—	2	1	no	—
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	4	2	no	—
464	C₂×C₈	—	53	1	no	—
465	C₂×C₈	—	53	1	no	—
466	C₂×C₈	—	53	1	no	—
467	C₂×C₈	—	53	1	no	—
468	C₂×C₈	—	106	2	no	—
469	C₂×C₈	—	106	1	no	—
470	C₂×C₈	—	106	1	no	—
471	C₂×C₈	—	106	1	no	—
472	C₂×C₈	—	212	1	no	—
473	C₂×C₈	—	212	2	no	—
474	C₂×C₈	—	212	2	no	—
475	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
476	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
477	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
478	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	53	1	no	—
481	C₄⋊C₄	—	53	1	no	—
482	C₄⋊C₄	—	106	1	no	—
483	C₄⋊C₄	—	106	1	no	—
484	C₄⋊C₄	—	106	1	no	—
485	C₄⋊C₄	—	106	1	no	—
486	C₄⋊C₄	—	106	1	no	—
487	C₄⋊C₄	—	106	1	no	—
488	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
489	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
490	C₂²⋊C₄	—	2	1	no	—
491	C₂²⋊C₄	—	106	1	no	—
492	C₂²⋊C₄	—	106	1	no	—
493	C₄²	—	2	1	no	—
494	C₄²	—	53	1	no	—
495	C₄²	—	53	1	no	—
496	C₄²	—	53	1	no	—
497	C₄²	—	106	1	no	—
498	C₄²	—	106	1	no	—
499	C₄²	—	106	1	no	—
500	C₄²	—	106	1	no	—
501	C₄²	—	106	1	no	—
502	C₄²	—	106	1	no	—
503	C₄²	C₂³×D₅₃	1	1	yes	characteristic
504	C₂³	—	2	1	no	—
505	C₂³	—	53	1	no	—
506	Q₈	—	2	1	no	—
507	Q₈	—	53	2	no	—
508	Q₈	—	53	2	no	—
509	Q₈	—	106	4	no	—
510	Q₈	—	106	1	no	—
511	Q₈	—	106	1	no	—
512	Q₈	C₂×C₈×D₅₃	1	2	yes	—
513	D₄	—	2	1	no	—
514	D₄	—	106	4	no	—
515	D₄	C₂×C₈×D₅₃	1	2	yes	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	1	no	—
519	C₂×C₄	—	4	1	no	—
520	C₂×C₄	—	4	1	no	—
521	C₂×C₄	—	53	1	no	—
522	C₂×C₄	—	53	2	no	—
523	C₂×C₄	—	53	2	no	—
524	C₂×C₄	—	53	1	no	—
525	C₂×C₄	—	53	1	no	—
526	C₂×C₄	—	106	4	no	—
527	C₂×C₄	—	106	4	no	—
528	C₂×C₄	—	106	1	no	—
529	C₂×C₄	—	106	1	no	—
530	C₂×C₄	—	106	1	no	—
531	C₂×C₄	—	106	1	no	—
532	C₂×C₄	—	106	1	no	—
533	C₂×C₄	—	106	1	no	—
534	C₂×C₄	—	212	1	no	—
535	C₂×C₄	—	212	2	no	—
536	C₂×C₄	—	212	2	no	—
537	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
538	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
539	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
540	C₈	—	2	1	no	—
541	C₈	—	2	1	no	—
542	C₈	—	2	1	no	—
543	C₈	—	4	1	no	—
544	C₈	—	4	1	no	—
545	C₈	—	4	1	no	—
546	C₈	—	53	2	no	—
547	C₈	—	53	2	no	—
548	C₈	—	106	1	no	—
549	C₈	—	106	1	no	—
550	C₈	—	212	1	no	—
551	C₈	—	212	1	no	—
552	C₈	—	212	1	no	—
553	C₈	C₂×C₈×D₅₃	1	2	yes	—
554	C₂²	—	2	2	no	—
555	C₂²	—	4	1	no	—
556	C₂²	—	4	1	no	—
557	C₂²	—	53	2	no	—
558	C₂²	—	106	1	no	—
559	C₂²	—	106	1	no	—
560	C₂²	C₄×D₄×D₅₃	1	1	yes	characteristic
561	C₄	—	2	1	no	—
562	C₄	—	2	2	no	—
563	C₄	—	4	1	no	—
564	C₄	—	4	1	no	—
565	C₄	—	53	2	no	—
566	C₄	—	106	1	no	—
567	C₄	—	106	1	no	—
568	C₄	—	106	2	no	—
569	C₄	—	106	2	no	—
570	C₄	—	212	1	no	—
571	C₄	—	212	1	no	—
572	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
573	C₄	#131×D₅₃	1	1	yes	characteristic
574	C₄	#137×D₅₃	1	1	yes	characteristic
575	C₄	#16811419	1	1	yes	characteristic
576	C₂	—	4	2	no	—
577	C₂	—	106	2	no	—
578	C₂	C₈×D₄×D₅₃	1	1	yes	characteristic
579	C₂	#20283537	1	1	yes	characteristic
580	C₂	#20283615	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×SD₁₆⋊C₈	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#11719947	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	SD₁₆⋊C₈	D₅₃
hypocenter	C₅₃	C₈×#137
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ SD₁₆⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₈×Q₈×D₅₃	C₂	semidirect
4. #20273535	C₂	semidirect
5. #20273532	C₂	semidirect
6. #20273527	C₂	semidirect
7. #20273521	C₂	semidirect
8. #20273519	C₂	semidirect
9. #20273518	C₂	semidirect
10. SD₁₆×#88322	C₂	semidirect
11. #20273466	C₂	semidirect
12. #20273108	C₂	semidirect
14. #20272994	C₂	semidirect
15. C₅₃×SD₁₆⋊C₈	C₂	semidirect
69. #13133246	C₈	semidirect
152. Q₈×D₅₃	C₂×C₈	semidirect
163. SD₁₆×C₅₃	C₂×C₈	semidirect
179. #11719925	C₂×C₈	semidirect
180. #11719923	C₂×C₈	semidirect
207. Dic₂₁₂	C₂×C₈	semidirect
209. #11719912	C₂×C₈	semidirect
262. #88322	#137	semidirect
292. SD₁₆⋊C₈	D₅₃	semidirect
357. C₅₃	C₈×#137	semidirect
457. SD₁₆	C₈×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
C₈×#11719947	2
C₈×Q₈×D₅₃	2
C₄×#13133246	2
#20273535	2
#20273532	2
#20273527	2
#20273521	2
#20273519	2
#20273518	2
SD₁₆×#88322	2
#20273466	2
#20273108	2
#20273099	2
#20272994	2
C₅₃×SD₁₆⋊C₈	2
C₈×#137	53