Subgroups — #27353509

Order 13568.

802 conjugacy classes of subgroups.

The subgroup lattice has 626 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, complemented
2	C₄×#13133241	C₂	1	1	yes	maximal, characteristic
3	C₈×D₄×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#20273537	C₂	1	1	yes	maximal, characteristic, complemented
5	#20273529	C₂	1	1	yes	maximal, characteristic, complemented
6	#20273527	C₂	1	1	yes	maximal, characteristic, complemented
7	#20273518	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273514	C₂	1	1	yes	maximal, characteristic, complemented
9	D₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
10	#20273507	C₂	1	1	yes	maximal, characteristic, complemented
11	#20273464	C₂	1	1	yes	maximal, characteristic, complemented
12	#20273104	C₂	1	1	yes	maximal, characteristic, complemented
13	#20273099	C₂	1	1	yes	maximal, characteristic, complemented
14	#20272994	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₃×#1580	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13133241	C₄	1	1	yes	characteristic
17	C₂²×C₈×D₅₃	—	2	1	no	—
18	C₄×#11719947	C₂²	1	1	yes	characteristic
19	C₄×D₄×D₅₃	C₂²	1	1	yes	characteristic
20	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic, complemented
21	#16810747	C₂²	1	1	yes	characteristic, complemented
22	#16810735	C₄	1	1	yes	characteristic
23	#16810732	C₄	1	1	yes	characteristic
24	#16810722	C₄	1	1	yes	characteristic
25	#16810720	C₄	1	1	yes	characteristic
26	C₄×#11719913	C₂²	1	1	yes	characteristic
27	C₄×#11719923	C₂²	1	1	yes	characteristic
28	C₄×#11719922	C₂²	1	1	yes	characteristic
29	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
30	D₄×#88322	C₂²	1	1	yes	characteristic
31	D₄×#88322	C₂²	1	1	yes	characteristic, complemented
32	#16810688	C₂²	1	1	yes	characteristic
33	C₈×C₂²⋊D₅₃	—	2	1	no	—
34	C₈×C₂²⋊D₅₃	—	2	1	no	—
35	#16810686	—	2	1	no	—
36	#16810686	—	2	1	no	—
37	#16810675	C₂²	1	1	yes	characteristic
38	#16810674	C₂²	1	1	yes	characteristic
39	#16810673	—	2	1	no	—
40	#16810664	C₄	1	1	yes	characteristic
41	C₄⋊C₈×D₅₃	C₂²	1	1	yes	characteristic
42	#16810626	C₄	1	1	yes	characteristic
43	#16810621	C₄	1	1	yes	characteristic
44	C₂²⋊C₈×D₅₃	—	2	1	no	—
45	C₄×#11719912	C₂²	1	1	yes	characteristic
46	C₂×C₈×Dic₅₃	—	2	1	no	—
47	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
48	#16810376	C₂²	1	1	yes	characteristic, complemented
49	#16810375	C₂²	1	1	yes	characteristic, complemented
50	#16810373	C₂²	1	1	yes	characteristic, complemented
51	#16810355	C₂²	1	1	yes	characteristic, complemented
52	#16810353	C₂²	1	1	yes	characteristic, complemented
53	#16810352	C₂²	1	1	yes	characteristic
54	D₄⋊#88322	C₂²	1	1	yes	characteristic
55	D₄⋊#88322	C₂²	1	1	yes	characteristic, complemented
56	#16810350	C₂²	1	1	yes	characteristic
57	#16810345	C₂²	1	1	yes	characteristic, complemented
58	C₈×#88322	C₂²	1	1	yes	characteristic, complemented
59	C₈×#88322	C₂²	1	1	yes	characteristic, complemented
60	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
61	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic
62	C₈×D₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
63	C₅₃×#334	C₂²	1	1	yes	characteristic, complemented
64	C₅₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
65	C₅₃×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
66	C₅₃×#322	C₂²	1	1	yes	characteristic, complemented
67	C₂×#11719947	C₂×C₄	1	1	yes	characteristic
68	C₂×D₄×D₅₃	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₄×D₅₃	—	2	1	no	—
70	#13133241	C₈	1	8	yes	complemented
71	C₂×#11719913	C₂×C₄	1	1	yes	characteristic
72	C₂×#11719923	C₂×C₄	1	1	yes	characteristic
73	C₂×#11719922	C₂×C₄	1	1	yes	characteristic
74	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
75	C₂×#11719912	C₂×C₄	1	1	yes	characteristic
76	C₂²×#88322	—	2	1	no	—
77	C₂²×#88322	—	2	1	no	—
78	C₂×C₈×D₅₃	—	2	1	no	—
79	C₂×C₈×D₅₃	—	2	1	no	—
80	C₂×C₈×D₅₃	—	2	1	no	—
81	C₂×C₈×D₅₃	—	4	4	no	—
82	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
83	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
84	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
85	#13133182	C₂×C₄	1	1	yes	characteristic
86	C₄×D₂₁₂	C₂³	1	1	yes	characteristic
87	D₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
88	D₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
89	#13133179	C₂×C₄	1	1	yes	characteristic
90	C₄×C₂²⋊D₅₃	—	2	1	no	—
91	C₄×C₂²⋊D₅₃	—	2	1	no	—
92	#13133177	—	2	1	no	—
93	#13133177	—	2	1	no	—
94	#13133175	C₂×C₄	1	1	yes	characteristic
95	#13133173	—	2	1	no	—
96	C₄⋊C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
97	C₂²⋊C₄×D₅₃	—	2	1	no	—
98	C₂×C₄×Dic₅₃	—	2	1	no	—
99	C₄²×D₅₃	C₂³	1	1	yes	characteristic
100	#13133150	C₂×C₄	1	1	yes	characteristic
101	#13133149	C₂×C₄	1	1	yes	characteristic
102	#13133148	C₂×C₄	1	1	yes	characteristic
103	C₄⋊#88322	C₂³	1	1	yes	characteristic
104	C₄⋊#88322	C₂³	1	1	yes	characteristic
105	#13133146	—	2	1	no	—
106	#13133146	—	2	1	no	—
107	#13133146	—	2	1	no	—
108	#13133146	—	2	1	no	—
109	#13133142	C₂×C₄	1	1	yes	characteristic
110	#13133140	C₂×C₄	1	1	yes	characteristic
111	#13133139	C₂×C₄	1	1	yes	characteristic
112	#13133138	C₂×C₄	1	1	yes	characteristic
113	#13133138	C₂×C₄	1	1	yes	characteristic
114	#13133137	C₂×C₄	1	1	yes	characteristic
115	C₂²⋊#88322	—	2	1	no	—
116	C₂²⋊#88322	—	2	1	no	—
117	#13133129	—	2	1	no	—
118	#13133129	—	2	1	no	—
119	#13133129	—	2	1	no	—
120	#13133129	—	2	1	no	—
121	#13133127	C₂×C₄	1	1	yes	characteristic
122	#13133127	C₂×C₄	1	1	yes	characteristic
123	C₄×#88322	C₂³	1	1	yes	characteristic
124	C₄×#88322	C₂³	1	1	yes	characteristic
125	C₄×#88322	C₂³	1	1	yes	characteristic
126	C₈×Dic₅₃	—	2	1	no	—
127	C₈×Dic₅₃	—	2	1	no	—
128	C₈×Dic₅₃	—	2	1	no	—
129	C₈×Dic₅₃	D₄	1	1	yes	characteristic
130	C₈×Dic₅₃	D₄	1	1	yes	characteristic
131	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
132	C₂²×C₈×C₅₃	—	2	1	no	—
133	C₂²×C₈×C₅₃	—	2	1	no	—
134	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
135	C₄×D₄×C₅₃	C₂³	1	1	yes	characteristic
136	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
137	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
138	C₄⋊C₈×C₅₃	C₂³	1	1	yes	characteristic
139	#102×C₅₃	C₂×C₄	1	1	yes	characteristic
140	#102×C₅₃	C₂×C₄	1	1	yes	characteristic
141	C₂²⋊C₈×C₅₃	—	2	1	no	—
142	C₂²⋊C₈×C₅₃	—	2	1	no	—
143	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
144	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
145	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
146	C₂³×D₅₃	—	2	1	no	—
147	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
148	C₂×D₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
149	C₂²×C₄×C₅₃	—	2	1	no	—
150	C₂²×C₄×C₅₃	—	2	1	no	—
151	C₂×C₂²⋊D₅₃	—	2	1	no	—
152	C₂×C₂²⋊D₅₃	—	2	1	no	—
153	C₂²×Dic₅₃	—	2	1	no	—
154	C₂²×Dic₅₃	—	2	1	no	—
155	C₂²×Dic₅₃	—	2	1	no	—
156	#11719947	—	2	2	no	—
157	#11719947	C₂×C₈	1	4	yes	complemented
158	D₄×D₅₃	—	2	2	no	—
159	D₄×D₅₃	C₂×C₈	1	4	yes	complemented
160	C₂×D₂₁₂	C₂²×C₄	1	1	yes	characteristic
161	C₂×C₄×D₅₃	—	2	1	no	—
162	C₂×C₄×D₅₃	—	2	1	no	—
163	C₂×C₄×D₅₃	—	2	1	no	—
164	C₂×C₄×D₅₃	—	4	2	no	—
165	C₂×C₄×D₅₃	—	4	2	no	—
166	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
167	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
168	C₂×C₄×D₅₃	#41	1	1	yes	characteristic
169	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
170	D₈×C₅₃	C₂×C₈	1	4	yes	complemented
171	C₂×C₈×C₅₃	—	2	1	no	—
172	C₂×C₈×C₅₃	—	2	1	no	—
173	C₂×C₈×C₅₃	—	2	1	no	—
174	C₂×C₈×C₅₃	—	4	2	no	—
175	C₂×C₈×C₅₃	—	4	2	no	—
176	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
177	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
178	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
179	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
180	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
181	C₄⋊C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
182	C₂²⋊C₄×C₅₃	—	2	1	no	—
183	C₂²⋊C₄×C₅₃	—	2	1	no	—
184	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
185	C₂²⋊Dic₅₃	—	2	1	no	—
186	C₂²⋊Dic₅₃	—	2	1	no	—
187	#11719923	C₂×C₈	1	4	yes	complemented
188	#11719922	C₂×C₈	1	4	yes	complemented
189	#11719921	—	2	1	no	—
190	#11719921	—	2	1	no	—
191	#11719921	—	2	1	no	—
192	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
193	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
194	#11719919	—	2	1	no	—
195	#11719919	—	2	1	no	—
196	#11719919	—	2	1	no	—
197	C₄×Dic₅₃	—	2	1	no	—
198	C₄×Dic₅₃	—	2	1	no	—
199	C₄×Dic₅₃	—	2	1	no	—
200	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
201	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
202	C₄×Dic₅₃	#41	1	1	yes	characteristic
203	C₂×#88322	—	2	1	no	—
204	C₂×#88322	—	2	1	no	—
205	C₂×#88322	—	2	1	no	—
206	C₂×#88322	—	4	2	no	—
207	C₂×#88322	—	4	2	no	—
208	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
209	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
210	C₂×#88322	C₂×D₄	1	1	yes	characteristic
211	C₂×#88322	C₂×D₄	1	1	yes	characteristic
212	#11719913	C₂×C₈	1	4	yes	complemented
213	#11719912	—	2	2	no	—
214	#11719912	C₂×C₈	1	4	yes	complemented
215	C₈×D₅₃	—	2	2	no	—
216	C₈×D₅₃	—	4	1	no	—
217	C₈×D₅₃	—	4	2	no	—
218	C₈×D₅₃	—	4	2	no	—
219	C₈×D₅₃	—	4	2	no	—
220	C₂³×C₅₃	—	2	1	no	—
221	C₂³×C₅₃	—	2	1	no	—
222	C₂²×D₅₃	—	2	1	no	—
223	C₂²×D₅₃	—	2	4	no	—
224	C₂²×D₅₃	—	4	2	no	—
225	C₂²×D₅₃	—	4	2	no	—
226	C₂²×D₅₃	C₄×D₄	1	1	yes	characteristic
227	D₄×C₅₃	—	2	1	no	—
228	D₄×C₅₃	—	2	1	no	—
229	D₄×C₅₃	C₂²×C₈	1	2	yes	—
230	D₄×C₅₃	C₂²×C₈	1	2	yes	complemented
231	C₂×C₄×C₅₃	—	2	1	no	—
232	C₂×C₄×C₅₃	—	2	1	no	—
233	C₂×C₄×C₅₃	—	4	1	no	—
234	C₂×C₄×C₅₃	—	4	1	no	—
235	C₂×C₄×C₅₃	—	4	1	no	—
236	C₂×C₄×C₅₃	—	4	1	no	—
237	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
238	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
239	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
240	C₂²⋊D₅₃	—	2	4	no	—
241	C₂²⋊D₅₃	—	2	4	no	—
242	C₂×Dic₅₃	—	2	1	no	—
243	C₂×Dic₅₃	—	2	4	no	—
244	C₂×Dic₅₃	—	2	1	no	—
245	C₂×Dic₅₃	—	2	1	no	—
246	C₂×Dic₅₃	—	4	2	no	—
247	C₂×Dic₅₃	—	4	2	no	—
248	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
249	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
250	C₂×Dic₅₃	C₄×D₄	1	1	yes	characteristic
251	D₂₁₂	—	2	1	no	—
252	D₂₁₂	C₂²×C₈	1	2	yes	—
253	C₄×D₅₃	—	2	1	no	—
254	C₄×D₅₃	—	2	1	no	—
255	C₄×D₅₃	—	4	1	no	—
256	C₄×D₅₃	—	4	2	no	—
257	C₄×D₅₃	—	4	2	no	—
258	C₄×D₅₃	—	4	1	no	—
259	C₄×D₅₃	—	4	1	no	—
260	C₄×D₅₃	C₂²×C₈	1	2	yes	—
261	C₄×D₅₃	#131	1	2	yes	—
262	Dic₁₀₆	—	2	1	no	—
263	Dic₁₀₆	C₂²×C₈	1	2	yes	complemented
264	C₈×C₅₃	—	2	1	no	—
265	C₈×C₅₃	—	2	1	no	—
266	C₈×C₅₃	—	2	1	no	—
267	C₈×C₅₃	—	4	1	no	—
268	C₈×C₅₃	—	4	1	no	—
269	C₈×C₅₃	—	4	1	no	—
270	C₈×C₅₃	C₂²×C₈	1	2	yes	complemented
271	#88322	—	2	1	no	—
272	#88322	—	2	1	no	—
273	#88322	—	4	1	no	—
274	#88322	—	4	1	no	—
275	#88322	—	4	1	no	—
276	#88322	C₂²×C₈	1	2	yes	complemented
277	#88322	AGL(1,ℤ/8)	1	2	yes	complemented
278	C₈×AGL(1,ℤ/8)	—	53	1	no	maximal, Sylow
279	C₂²×C₅₃	—	2	2	no	—
280	C₂²×C₅₃	—	2	2	no	—
281	C₂²×C₅₃	—	4	1	no	—
282	C₂²×C₅₃	—	4	1	no	—
283	C₂²×C₅₃	—	4	1	no	—
284	C₂²×C₅₃	—	4	1	no	—
285	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
286	C₂×D₅₃	—	2	1	no	—
287	C₂×D₅₃	—	2	1	no	—
288	C₂×D₅₃	—	2	2	no	—
289	C₂×D₅₃	—	4	8	no	—
290	C₂×D₅₃	—	4	1	no	—
291	C₂×D₅₃	—	4	1	no	—
292	C₂×D₅₃	C₈×D₄	1	2	yes	—
293	C₄×C₅₃	—	2	1	no	—
294	C₄×C₅₃	—	4	1	no	—
295	C₄×C₅₃	—	4	1	no	—
296	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
297	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
298	C₄×C₅₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
299	C₄×C₅₃	#575	1	1	yes	characteristic
300	Dic₅₃	—	2	2	no	—
301	Dic₅₃	—	2	1	no	—
302	Dic₅₃	—	2	1	no	—
303	Dic₅₃	—	4	1	no	—
304	Dic₅₃	—	4	1	no	—
305	Dic₅₃	C₈×D₄	1	2	yes	—
306	C₈×#41	—	53	1	no	—
307	C₄×AGL(1,ℤ/8)	—	53	1	no	—
308	C₂×C₈×D₄	—	53	1	no	—
309	#1580	—	53	1	no	—
310	#1580	D₅₃	1	1	yes	characteristic, complemented
311	SD₁₆⋊C₈	—	53	1	no	—
312	SD₁₆⋊C₈	—	53	1	no	—
313	C₈×SD₁₆	—	53	1	no	—
314	C₈×SD₁₆	—	53	1	no	—
315	C₈×D₈	—	53	1	no	—
316	C₈×D₈	—	53	1	no	—
317	#1565	—	53	1	no	—
318	#1478	—	53	1	no	—
319	#1477	—	53	1	no	—
320	C₈×M₁₆	—	53	1	no	—
321	C₂×C₅₃	—	4	2	no	—
322	C₂×C₅₃	—	4	2	no	—
323	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
324	C₂×C₅₃	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
325	C₂×C₅₃	#2955	1	1	yes	characteristic
326	D₅₃	—	2	2	no	—
327	D₅₃	—	4	2	no	—
328	C₂×AGL(1,ℤ/8)	—	53	1	no	—
329	C₂³×C₈	—	106	1	no	—
330	C₄×#41	—	53	1	no	—
331	C₂×C₄×D₄	—	53	1	no	—
332	C₈×Q₈	—	53	1	no	—
333	C₈×Q₈	—	53	1	no	—
334	#442	—	53	1	no	—
335	#442	—	53	1	no	—
336	#440	—	53	1	no	—
337	#440	—	53	1	no	—
338	C₄×SD₁₆	—	53	1	no	—
339	C₄×SD₁₆	—	53	1	no	—
340	C₄×D₈	—	53	1	no	—
341	C₄×D₈	C₂×D₅₃	1	1	yes	characteristic
342	C₈×D₄	—	53	1	no	—
343	C₈×D₄	—	53	1	no	—
344	C₈×D₄	—	53	1	no	—
345	C₈×D₄	—	53	1	no	—
346	C₈×D₄	—	106	1	no	—
347	C₈×D₄	—	106	1	no	—
348	C₈×D₄	—	106	1	no	—
349	C₈×D₄	—	106	1	no	—
350	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic
351	C₈×D₄	C₂×D₅₃	1	1	yes	characteristic, complemented
352	#431	—	53	1	no	—
353	#431	—	53	1	no	—
354	#431	—	106	1	no	—
355	#428	—	53	1	no	—
356	C₂×C₄⋊C₈	—	53	1	no	—
357	#418	—	53	1	no	—
358	#417	—	53	1	no	—
359	C₂×C₂²⋊C₈	—	106	1	no	—
360	C₄×M₁₆	—	53	1	no	—
361	C₂×C₄×C₈	—	53	1	no	—
362	C₂×C₄×C₈	—	106	1	no	—
363	#335	—	53	1	no	—
364	#335	—	53	1	no	—
365	#334	—	53	1	no	—
366	#334	C₂×D₅₃	1	1	yes	characteristic, complemented
367	Q₈⋊C₈	—	53	1	no	—
368	Q₈⋊C₈	—	53	1	no	—
369	D₄⋊C₈	—	53	1	no	—
370	D₄⋊C₈	—	53	1	no	—
371	D₄⋊C₈	—	53	1	no	—
372	D₄⋊C₈	—	53	1	no	—
373	D₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
374	D₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic, complemented
375	#322	—	53	1	no	—
376	#322	C₂×D₅₃	1	1	yes	characteristic, complemented
377	C₈²	—	53	1	no	—
378	C₈²	—	53	1	no	—
379	C₅₃	C₈×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
380	C₂×#41	—	53	1	no	—
381	C₂²×D₄	—	53	1	no	—
382	C₂³×C₄	—	106	1	no	—
383	AGL(1,ℤ/8)	—	53	8	no	—
384	C₂×SD₁₆	—	53	1	no	—
385	C₂×SD₁₆	—	53	1	no	—
386	C₂×D₈	—	53	1	no	—
387	C₂×D₈	C₄×D₅₃	1	1	yes	characteristic
388	C₂×M₁₆	—	53	1	no	—
389	C₂×M₁₆	—	53	1	no	—
390	C₂²×C₈	—	2	1	no	—
391	C₂²×C₈	—	2	1	no	—
392	C₂²×C₈	—	53	1	no	—
393	C₂²×C₈	—	53	1	no	—
394	C₂²×C₈	—	106	1	no	—
395	C₂²×C₈	—	106	1	no	—
396	C₂²×C₈	—	106	1	no	—
397	C₂²×C₈	—	106	1	no	—
398	C₂²×C₈	—	106	1	no	—
399	C₂²×C₈	—	212	4	no	—
400	C₄×Q₈	—	53	1	no	—
401	C₄×Q₈	—	53	1	no	—
402	C₄×D₄	—	53	1	no	—
403	C₄×D₄	—	53	1	no	—
404	C₄×D₄	—	53	1	no	—
405	C₄×D₄	—	53	1	no	—
406	C₄×D₄	—	106	1	no	—
407	C₄×D₄	—	106	1	no	—
408	C₄×D₄	—	106	1	no	—
409	C₄×D₄	—	106	1	no	—
410	C₄×D₄	C₂²×D₅₃	1	1	yes	characteristic
411	C₄×D₄	C₂²×D₅₃	1	1	yes	characteristic
412	#117	—	53	1	no	—
413	#117	—	106	1	no	—
414	C₂×C₄⋊C₄	—	53	1	no	—
415	C₂×C₂²⋊C₄	—	106	1	no	—
416	C₂×C₄²	—	53	1	no	—
417	C₂×C₄²	—	106	1	no	—
418	#107	—	53	1	no	—
419	#107	C₄×D₅₃	1	1	yes	characteristic
420	#106	—	53	1	no	—
421	#106	—	53	1	no	—
422	C₄⋊C₈	—	53	1	no	—
423	C₄⋊C₈	—	53	1	no	—
424	C₄⋊C₈	—	106	1	no	—
425	C₄⋊C₈	—	106	1	no	—
426	C₄⋊C₈	—	106	1	no	—
427	C₄⋊C₈	—	106	1	no	—
428	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
429	C₄⋊C₈	C₂²×D₅₃	1	1	yes	characteristic
430	#103	—	53	1	no	—
431	#103	—	53	1	no	—
432	#102	—	53	1	no	—
433	#102	—	53	1	no	—
434	#102	—	53	1	no	—
435	#102	—	53	1	no	—
436	#102	C₄×D₅₃	1	1	yes	characteristic
437	#102	C₄×D₅₃	1	1	yes	characteristic
438	C₂²⋊C₈	—	2	1	no	—
439	C₂²⋊C₈	—	2	1	no	—
440	C₂²⋊C₈	—	106	1	no	—
441	C₂²⋊C₈	—	106	1	no	—
442	C₂²⋊C₈	—	106	1	no	—
443	C₂²⋊C₈	—	106	1	no	—
444	C₂²⋊C₈	—	106	1	no	—
445	C₂²⋊C₈	—	106	1	no	—
446	#97	—	53	1	no	—
447	#97	—	53	1	no	—
448	C₄×C₈	—	53	1	no	—
449	C₄×C₈	—	53	1	no	—
450	C₄×C₈	—	53	1	no	—
451	C₄×C₈	—	53	1	no	—
452	C₄×C₈	—	53	1	no	—
453	C₄×C₈	—	106	1	no	—
454	C₄×C₈	—	106	1	no	—
455	C₄×C₈	—	106	1	no	—
456	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
457	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
458	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
459	C₂⁴	—	106	1	no	—
460	#41	—	53	4	no	—
461	#41	—	106	2	no	—
462	C₂×Q₈	—	53	1	no	—
463	C₂×D₄	—	53	4	no	—
464	C₂×D₄	—	53	1	no	—
465	C₂×D₄	—	106	2	no	—
466	C₂×D₄	—	106	1	no	—
467	C₂×D₄	—	106	1	no	—
468	C₂×D₄	C₂×C₄×D₅₃	1	1	yes	characteristic
469	C₂×D₄	C₂×C₄×D₅₃	1	1	yes	characteristic
470	C₂²×C₄	—	2	1	no	—
471	C₂²×C₄	—	2	1	no	—
472	C₂²×C₄	—	53	1	no	—
473	C₂²×C₄	—	53	1	no	—
474	C₂²×C₄	—	53	1	no	—
475	C₂²×C₄	—	106	1	no	—
476	C₂²×C₄	—	106	1	no	—
477	C₂²×C₄	—	106	1	no	—
478	C₂²×C₄	—	106	1	no	—
479	C₂²×C₄	—	106	1	no	—
480	C₂²×C₄	—	106	1	no	—
481	C₂²×C₄	—	212	2	no	—
482	C₂²×C₄	—	212	2	no	—
483	SD₁₆	—	53	4	no	—
484	SD₁₆	—	53	4	no	—
485	D₈	—	53	4	no	—
486	D₈	C₈×D₅₃	1	4	yes	complemented
487	M₁₆	—	53	4	no	—
488	M₁₆	—	106	2	no	—
489	C₂×C₈	—	2	1	no	—
490	C₂×C₈	—	2	1	no	—
491	C₂×C₈	—	2	1	no	—
492	C₂×C₈	—	4	2	no	—
493	C₂×C₈	—	4	2	no	—
494	C₂×C₈	—	53	1	no	—
495	C₂×C₈	—	53	1	no	—
496	C₂×C₈	—	53	1	no	—
497	C₂×C₈	—	53	1	no	—
498	C₂×C₈	—	106	2	no	—
499	C₂×C₈	—	106	1	no	—
500	C₂×C₈	—	106	1	no	—
501	C₂×C₈	—	106	1	no	—
502	C₂×C₈	—	212	1	no	—
503	C₂×C₈	—	212	2	no	—
504	C₂×C₈	—	212	2	no	—
505	C₂×C₈	—	212	2	no	—
506	C₂×C₈	—	212	2	no	—
507	C₂×C₈	—	212	2	no	—
508	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
509	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
510	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
511	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
512	C₄⋊C₄	—	53	1	no	—
513	C₄⋊C₄	—	53	1	no	—
514	C₄⋊C₄	—	106	1	no	—
515	C₄⋊C₄	—	106	1	no	—
516	C₄⋊C₄	—	106	1	no	—
517	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
518	C₄⋊C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
519	C₂²⋊C₄	—	2	1	no	—
520	C₂²⋊C₄	—	2	1	no	—
521	C₂²⋊C₄	—	106	1	no	—
522	C₂²⋊C₄	—	106	1	no	—
523	C₂²⋊C₄	—	106	1	no	—
524	C₂²⋊C₄	—	106	1	no	—
525	C₂²⋊C₄	—	106	1	no	—
526	C₄²	—	53	1	no	—
527	C₄²	—	53	1	no	—
528	C₄²	—	53	1	no	—
529	C₄²	—	106	1	no	—
530	C₄²	—	106	1	no	—
531	C₄²	—	106	1	no	—
532	C₄²	C₂³×D₅₃	1	1	yes	characteristic
533	C₂³	—	2	1	no	—
534	C₂³	—	2	1	no	—
535	C₂³	—	53	1	no	—
536	C₂³	—	106	4	no	—
537	C₂³	—	106	1	no	—
538	C₂³	—	212	2	no	—
539	C₂³	—	212	2	no	—
540	Q₈	—	53	2	no	—
541	Q₈	—	106	1	no	—
542	D₄	—	2	1	no	—
543	D₄	—	2	1	no	—
544	D₄	—	53	2	no	—
545	D₄	—	106	4	no	—
546	D₄	—	106	4	no	—
547	D₄	—	106	1	no	—
548	D₄	C₂×C₈×D₅₃	1	2	yes	—
549	D₄	C₂×C₈×D₅₃	1	2	yes	complemented
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	53	1	no	—
557	C₂×C₄	—	53	2	no	—
558	C₂×C₄	—	53	2	no	—
559	C₂×C₄	—	53	1	no	—
560	C₂×C₄	—	53	1	no	—
561	C₂×C₄	—	106	4	no	—
562	C₂×C₄	—	106	1	no	—
563	C₂×C₄	—	106	1	no	—
564	C₂×C₄	—	106	1	no	—
565	C₂×C₄	—	106	1	no	—
566	C₂×C₄	—	106	1	no	—
567	C₂×C₄	—	212	1	no	—
568	C₂×C₄	—	212	2	no	—
569	C₂×C₄	—	212	2	no	—
570	C₂×C₄	—	212	2	no	—
571	C₂×C₄	—	212	2	no	—
572	C₂×C₄	—	212	1	no	—
573	C₂×C₄	—	212	1	no	—
574	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
575	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
576	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
577	C₈	—	2	1	no	—
578	C₈	—	2	1	no	—
579	C₈	—	2	1	no	—
580	C₈	—	4	1	no	—
581	C₈	—	4	1	no	—
582	C₈	—	4	1	no	—
583	C₈	—	53	2	no	—
584	C₈	—	53	2	no	—
585	C₈	—	106	1	no	—
586	C₈	—	106	1	no	—
587	C₈	—	212	1	no	—
588	C₈	—	212	1	no	—
589	C₈	—	212	1	no	—
590	C₈	C₂×C₈×D₅₃	1	2	yes	complemented
591	C₂²	—	2	2	no	—
592	C₂²	—	2	2	no	—
593	C₂²	—	4	1	no	—
594	C₂²	—	4	1	no	—
595	C₂²	—	4	1	no	—
596	C₂²	—	4	1	no	—
597	C₂²	—	53	2	no	—
598	C₂²	—	106	1	no	—
599	C₂²	—	106	1	no	—
600	C₂²	—	106	2	no	—
601	C₂²	—	212	8	no	—
602	C₂²	—	212	1	no	—
603	C₂²	—	212	1	no	—
604	C₂²	C₄×D₄×D₅₃	1	1	yes	characteristic
605	C₄	—	2	1	no	—
606	C₄	—	4	1	no	—
607	C₄	—	4	1	no	—
608	C₄	—	53	2	no	—
609	C₄	—	106	1	no	—
610	C₄	—	106	1	no	—
611	C₄	—	106	2	no	—
612	C₄	—	212	1	no	—
613	C₄	—	212	1	no	—
614	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
615	C₄	#131×D₅₃	1	1	yes	characteristic
616	C₄	AGL(1,ℤ/8)×D₅₃	1	1	yes	characteristic
617	C₄	#16811419	1	1	yes	characteristic
618	C₂	—	4	2	no	—
619	C₂	—	4	2	no	—
620	C₂	—	106	2	no	—
621	C₂	—	212	2	no	—
622	C₂	C₈×D₄×D₅₃	1	1	yes	characteristic
623	C₂	#20283526	1	1	yes	characteristic
624	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×#1580	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#13133241	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#1580	D₅₃
hypocenter	C₅₃	C₈×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1580

Complements

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

N	H ≅ G/N	type
1. C₈×#11719947	C₂	semidirect
3. C₈×D₄×D₅₃	C₂	semidirect
4. #20273537	C₂	semidirect
5. #20273529	C₂	semidirect
6. #20273527	C₂	semidirect
7. #20273518	C₂	semidirect
8. #20273514	C₂	semidirect
9. D₈×#88322	C₂	semidirect
10. #20273507	C₂	semidirect
11. #20273464	C₂	semidirect
12. #20273104	C₂	semidirect
13. #20273099	C₂	semidirect
14. #20272994	C₂	semidirect
15. C₅₃×#1580	C₂	semidirect
20. C₈×Dic₁₀₆	C₂²	semidirect
21. #16810747	C₂²	semidirect
31. D₄×#88322	C₂²	semidirect
48. #16810376	C₂²	semidirect
49. #16810375	C₂²	semidirect
50. #16810373	C₂²	semidirect
51. #16810355	C₂²	semidirect
52. #16810353	C₂²	semidirect
55. D₄⋊#88322	C₂²	semidirect
57. #16810345	C₂²	semidirect
58. C₈×#88322	C₂²	semidirect
59. C₈×#88322	C₂²	semidirect
62. C₈×D₄×C₅₃	C₂²	semidirect
63. C₅₃×#334	C₂²	semidirect
65. C₅₃×D₄⋊C₈	C₂²	semidirect
66. C₅₃×#322	C₂²	semidirect
70. #13133241	C₈	semidirect
157. #11719947	C₂×C₈	semidirect
159. D₄×D₅₃	C₂×C₈	semidirect
170. D₈×C₅₃	C₂×C₈	semidirect
187. #11719923	C₂×C₈	semidirect
188. #11719922	C₂×C₈	semidirect
212. #11719913	C₂×C₈	semidirect
214. #11719912	C₂×C₈	semidirect
230. D₄×C₅₃	C₂²×C₈	semidirect
263. Dic₁₀₆	C₂²×C₈	semidirect
270. C₈×C₅₃	C₂²×C₈	semidirect
276. #88322	C₂²×C₈	semidirect
277. #88322	AGL(1,ℤ/8)	semidirect
310. #1580	D₅₃	semidirect
351. C₈×D₄	C₂×D₅₃	semidirect
366. #334	C₂×D₅₃	semidirect
374. D₄⋊C₈	C₂×D₅₃	semidirect
376. #322	C₂×D₅₃	semidirect
379. C₅₃	C₈×AGL(1,ℤ/8)	semidirect
486. D₈	C₈×D₅₃	semidirect
549. D₄	C₂×C₈×D₅₃	semidirect
590. C₈	C₂×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
C₄×#13133241	2
C₈×D₄×D₅₃	2
#20273537	2
#20273529	2
#20273527	2
#20273518	2
#20273514	2
D₈×#88322	2
#20273507	2
#20273464	2
#20273104	2
#20273099	2
#20272994	2
C₅₃×#1580	2
C₈×AGL(1,ℤ/8)	53