Subgroups — C₈×#13133248 #27353517

Order 13568.

714 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11719945	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133248	C₂	1	1	yes	maximal, characteristic
3	#20273532	C₂	1	2	yes	maximal, complemented
4	#20273525	C₂	1	2	yes	maximal, complemented
5	C₈×#11719925	C₂	1	2	yes	maximal, complemented
6	C₈×#11719924	C₂	1	2	yes	maximal, complemented
7	#20273465	C₂	1	1	yes	maximal, characteristic, complemented
8	#20273110	C₂	1	1	yes	maximal, characteristic, complemented
9	#20273100	C₂	1	1	yes	maximal, characteristic
10	C₈×#11719917	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₈×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13133248	C₄	1	1	yes	characteristic
13	C₄×#11719945	C₂²	1	1	yes	characteristic
14	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
15	C₈×Dic₁₀₆	C₂²	1	1	yes	characteristic
16	#16810728	C₄	1	2	yes	—
17	#16810721	C₄	1	2	yes	—
18	C₄×#11719925	C₂²	1	2	yes	—
19	C₄×#11719924	C₂²	1	2	yes	—
20	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
21	C₈×D₂₁₂	C₂²	1	1	yes	characteristic
22	C₈×C₂²⋊D₅₃	—	2	1	no	—
23	C₈×C₂²⋊D₅₃	—	2	1	no	—
24	#16810676	C₂²	1	1	yes	characteristic
25	#16810676	C₂²	1	1	yes	characteristic
26	#16810674	—	2	1	no	—
27	#16810663	C₄	1	1	yes	characteristic
28	#16810632	C₄	1	1	yes	characteristic
29	#16810622	C₄	1	1	yes	characteristic
30	C₄×#11719917	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₅₃	—	2	1	no	—
32	#16810375	C₂²	1	2	yes	—
33	#16810373	C₂²	1	2	yes	—
34	Q₈⋊#88322	C₂²	1	2	yes	—
35	Q₈⋊#88322	C₂²	1	2	yes	—
36	#16810353	C₂²	1	2	yes	—
37	#16810350	C₂²	1	2	yes	—
38	#16810345	C₂²	1	2	yes	—
39	C₈×#88322	C₂²	1	2	yes	—
40	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
41	C₈×Q₈×C₅₃	—	2	1	no	—
42	C₈×Q₈×C₅₃	—	2	1	no	—
43	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
44	C₈×Q₈×C₅₃	C₂²	1	2	yes	—
45	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
46	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
47	C₂×C₄×C₈×C₅₃	—	2	1	no	—
48	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
49	C₂×#11719945	C₂×C₄	1	1	yes	characteristic
50	#13133248	C₈	1	8	yes	direct product factor
51	C₂×#11719925	C₂×C₄	1	2	yes	—
52	C₂×#11719924	C₂×C₄	1	2	yes	—
53	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
54	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
55	C₂×C₈×D₅₃	—	2	1	no	—
56	C₂×C₈×D₅₃	—	2	1	no	—
57	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
58	C₄×Dic₁₀₆	C₂³	1	1	yes	characteristic
59	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
60	C₄×D₂₁₂	C₂³	1	1	yes	characteristic
61	C₄×C₂²⋊D₅₃	—	2	1	no	—
62	C₄×C₂²⋊D₅₃	—	2	1	no	—
63	#13133176	C₂×C₄	1	1	yes	characteristic
64	#13133174	—	2	1	no	—
65	C₄²×D₅₃	—	2	1	no	—
66	#13133150	C₂×C₄	1	2	yes	—
67	#13133148	C₂×C₄	1	2	yes	—
68	C₄⋊#88322	—	2	1	no	—
69	C₄⋊#88322	C₂³	1	1	yes	characteristic
70	C₄⋊#88322	C₂³	1	1	yes	characteristic
71	#13133146	—	2	1	no	—
72	#13133146	—	2	1	no	—
73	#13133141	C₂×C₄	1	2	yes	—
74	#13133141	C₂×C₄	1	2	yes	—
75	#13133140	C₂×C₄	1	2	yes	—
76	#13133137	C₂×C₄	1	2	yes	—
77	C₂²⋊#88322	—	2	1	no	—
78	C₂²⋊#88322	—	2	1	no	—
79	#13133129	—	2	1	no	—
80	#13133129	—	2	1	no	—
81	#13133128	C₂×C₄	1	2	yes	—
82	C₄×#88322	—	2	1	no	—
83	C₄×#88322	C₂³	1	2	yes	—
84	C₄×#88322	C₂³	1	2	yes	—
85	C₈×Dic₅₃	—	2	1	no	—
86	C₈×Dic₅₃	—	2	1	no	—
87	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
88	C₂²×C₈×C₅₃	—	2	1	no	—
89	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
90	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
91	C₄×Q₈×C₅₃	—	2	1	no	—
92	C₄×Q₈×C₅₃	—	2	1	no	—
93	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
94	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
95	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
96	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
97	C₂×C₄²×C₅₃	—	2	1	no	—
98	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
99	C₄⋊C₈×C₅₃	—	2	2	no	—
100	C₄⋊C₈×C₅₃	—	2	2	no	—
101	C₄⋊C₈×C₅₃	—	2	1	no	—
102	C₄⋊C₈×C₅₃	C₂³	1	2	yes	—
103	C₄×C₈×C₅₃	—	2	2	no	—
104	C₄×C₈×C₅₃	—	2	2	no	—
105	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
106	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
107	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
108	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
109	C₂×Q₈×C₅₃	—	2	1	no	—
110	C₂×Q₈×C₅₃	—	2	2	no	—
111	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	complemented
112	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
113	C₂²×C₄×C₅₃	—	2	1	no	—
114	C₂²×C₄×C₅₃	—	2	1	no	—
115	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
118	C₂×C₂²⋊D₅₃	—	2	1	no	—
119	#11719945	—	2	2	no	—
120	#11719945	C₂×C₈	1	4	yes	—
121	C₂×D₂₁₂	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₄×D₅₃	—	2	1	no	—
123	C₂×C₄×D₅₃	—	2	1	no	—
124	C₂×C₄×D₅₃	—	2	1	no	—
125	C₂×Dic₁₀₆	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₈×C₅₃	—	2	2	no	—
127	C₂×C₈×C₅₃	—	2	2	no	—
128	C₂×C₈×C₅₃	—	4	1	no	—
129	C₂×C₈×C₅₃	—	4	2	no	—
130	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
132	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
133	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
134	C₄⋊C₄×C₅₃	—	2	2	no	—
135	C₄⋊C₄×C₅₃	—	2	1	no	—
136	C₄⋊C₄×C₅₃	—	2	2	no	—
137	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
138	C₄²×C₅₃	—	2	2	no	—
139	C₄²×C₅₃	—	2	2	no	—
140	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
141	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
142	C₄²×C₅₃	#41	1	1	yes	characteristic
143	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
144	C₂²⋊Dic₅₃	—	2	1	no	—
145	#11719925	C₂×C₈	1	8	yes	complemented
146	#11719924	C₂×C₈	1	8	yes	complemented
147	#11719921	—	2	1	no	—
148	#11719921	—	2	1	no	—
149	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
150	C₄⋊Dic₅₃	C₂²×C₄	1	1	yes	characteristic
151	#11719919	—	2	1	no	—
152	#11719919	—	2	1	no	—
153	C₄×Dic₅₃	—	2	1	no	—
154	C₄×Dic₅₃	—	2	1	no	—
155	C₄×Dic₅₃	—	2	1	no	—
156	#11719917	—	2	2	no	—
157	#11719917	C₂×C₈	1	4	yes	complemented
158	C₂×#88322	—	2	1	no	—
159	C₂×#88322	—	2	1	no	—
160	C₂×#88322	—	2	2	no	—
161	C₂×#88322	C₂²×C₄	1	2	yes	—
162	C₂×#88322	C₂²×C₄	1	2	yes	—
163	C₈×D₅₃	—	4	2	no	—
164	C₈×D₅₃	—	4	2	no	—
165	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
166	C₂²×D₅₃	—	2	1	no	—
167	Q₈×C₅₃	—	2	2	no	—
168	Q₈×C₅₃	—	2	4	no	—
169	Q₈×C₅₃	C₂²×C₈	1	4	yes	—
170	C₂×C₄×C₅₃	—	2	2	no	—
171	C₂×C₄×C₅₃	—	2	4	no	—
172	C₂×C₄×C₅₃	—	2	1	no	—
173	C₂×C₄×C₅₃	—	2	1	no	—
174	C₂×C₄×C₅₃	—	2	2	no	—
175	C₂×C₄×C₅₃	—	4	1	no	—
176	C₂×C₄×C₅₃	—	4	2	no	—
177	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
178	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
179	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
180	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
181	C₂×C₄×C₅₃	#131	1	2	yes	—
182	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
183	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
185	C₂²⋊D₅₃	—	2	4	no	—
186	C₂×Dic₅₃	—	2	1	no	—
187	C₂×Dic₅₃	—	2	1	no	—
188	C₂×Dic₅₃	—	2	1	no	—
189	D₂₁₂	—	2	1	no	—
190	D₂₁₂	C₂²×C₈	1	2	yes	—
191	C₄×D₅₃	—	2	4	no	—
192	C₄×D₅₃	—	4	2	no	—
193	C₄×D₅₃	—	4	1	no	—
194	C₄×D₅₃	—	4	1	no	—
195	Dic₁₀₆	—	2	1	no	—
196	Dic₁₀₆	C₂²×C₈	1	2	yes	—
197	C₈×C₅₃	—	2	1	no	—
198	C₈×C₅₃	—	2	1	no	—
199	C₈×C₅₃	—	2	1	no	—
200	C₈×C₅₃	—	4	2	no	—
201	C₈×C₅₃	#137	1	2	yes	complemented
202	#88322	—	2	2	no	—
203	#88322	—	4	2	no	—
204	#88322	—	4	1	no	—
205	#88322	—	4	1	no	—
206	#88322	C₂²×C₈	1	4	yes	—
207	C₈×#137	—	53	1	no	maximal, Sylow
208	C₂²×C₅₃	—	2	1	no	—
209	C₂²×C₅₃	—	2	1	no	—
210	C₂²×C₅₃	C₈×D₄	1	2	yes	—
211	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
212	C₂×D₅₃	—	2	2	no	—
213	C₂×D₅₃	—	4	1	no	—
214	C₂×D₅₃	—	4	1	no	—
215	C₄×C₅₃	—	2	4	no	—
216	C₄×C₅₃	—	2	1	no	—
217	C₄×C₅₃	—	2	1	no	—
218	C₄×C₅₃	—	2	1	no	—
219	C₄×C₅₃	—	4	2	no	—
220	C₄×C₅₃	C₈×D₄	1	2	yes	—
221	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
222	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
223	C₄×C₅₃	C₂×#137	1	1	yes	characteristic
224	C₄×C₅₃	#575	1	1	yes	characteristic
225	Dic₅₃	—	2	2	no	—
226	Dic₅₃	—	4	1	no	—
227	Dic₅₃	—	4	1	no	—
228	C₈×#41	—	53	1	no	—
229	C₂×C₈×Q₈	D₅₃	1	1	yes	characteristic, complemented
230	C₄×#137	—	53	1	no	—
231	#1579	—	53	2	no	—
232	SD₁₆⋊C₈	—	53	2	no	—
233	C₈×Q₁₆	—	53	2	no	—
234	C₈×SD₁₆	—	53	2	no	—
235	#1565	—	53	1	no	—
236	#1479	—	53	1	no	—
237	#1477	—	53	1	no	—
238	C₈×M₁₆	—	53	1	no	—
239	C₂×C₅₃	—	2	2	no	—
240	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
241	C₂×C₅₃	C₄×#137	1	1	yes	characteristic
242	C₂×C₅₃	#2955	1	1	yes	characteristic
243	D₅₃	—	4	2	no	—
244	C₂×#137	—	53	1	no	—
245	C₄×#41	—	53	1	no	—
246	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
247	C₈×Q₈	—	2	1	no	—
248	C₈×Q₈	—	2	1	no	—
249	C₈×Q₈	—	53	1	no	—
250	C₈×Q₈	—	53	1	no	—
251	C₈×Q₈	C₂×D₅₃	1	2	yes	—
252	C₈×Q₈	C₂×D₅₃	1	2	yes	—
253	#441	—	53	2	no	—
254	#440	—	53	2	no	—
255	C₄×Q₁₆	—	53	2	no	—
256	C₄×SD₁₆	—	53	2	no	—
257	C₈×D₄	—	53	1	no	—
258	C₈×D₄	—	53	1	no	—
259	C₈×D₄	—	106	1	no	—
260	C₈×D₄	—	106	1	no	—
261	#431	—	53	1	no	—
262	#431	—	53	1	no	—
263	#431	—	106	1	no	—
264	#428	—	53	1	no	—
265	C₂×C₄⋊C₈	—	2	1	no	—
266	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
267	#419	—	53	1	no	—
268	#417	—	53	1	no	—
269	C₄×M₁₆	—	53	1	no	—
270	C₂×C₄×C₈	—	2	1	no	—
271	C₂×C₄×C₈	—	106	1	no	—
272	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
273	#335	—	53	2	no	—
274	#334	—	53	2	no	—
275	Q₈⋊C₈	—	53	2	no	—
276	Q₈⋊C₈	—	53	2	no	—
277	Q₈⋊C₈	—	53	2	no	—
278	D₄⋊C₈	—	53	2	no	—
279	#322	—	53	2	no	—
280	C₈²	—	53	2	no	—
281	C₅₃	C₈×#137	1	1	yes	characteristic, complemented, Sylow
282	C₂×#41	—	53	1	no	—
283	C₂²×Q₈	C₄×D₅₃	1	1	yes	characteristic
284	#137	—	53	8	no	—
285	C₂×Q₁₆	—	53	2	no	—
286	C₂×SD₁₆	—	53	2	no	—
287	C₂×M₁₆	—	53	1	no	—
288	C₂×M₁₆	—	53	1	no	—
289	C₂²×C₈	—	2	1	no	—
290	C₂²×C₈	—	106	1	no	—
291	C₂²×C₈	—	106	1	no	—
292	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
293	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
294	C₄×Q₈	—	2	1	no	—
295	C₄×Q₈	—	2	1	no	—
296	C₄×Q₈	—	53	1	no	—
297	C₄×Q₈	—	53	1	no	—
298	C₄×Q₈	C₄×D₅₃	1	2	yes	—
299	C₄×Q₈	C₂²×D₅₃	1	2	yes	—
300	C₄×D₄	—	53	1	no	—
301	C₄×D₄	—	53	1	no	—
302	C₄×D₄	—	106	1	no	—
303	C₄×D₄	—	106	1	no	—
304	#117	—	53	1	no	—
305	#117	—	106	1	no	—
306	C₂×C₄⋊C₄	—	2	1	no	—
307	C₂×C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
308	C₂×C₄²	—	2	1	no	—
309	C₂×C₄²	—	106	1	no	—
310	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
311	#107	—	53	2	no	—
312	#106	—	53	2	no	—
313	C₄⋊C₈	—	2	2	no	—
314	C₄⋊C₈	—	2	2	no	—
315	C₄⋊C₈	—	2	1	no	—
316	C₄⋊C₈	—	53	1	no	—
317	C₄⋊C₈	—	53	1	no	—
318	C₄⋊C₈	—	106	1	no	—
319	C₄⋊C₈	—	106	1	no	—
320	C₄⋊C₈	—	106	1	no	—
321	C₄⋊C₈	C₂²×D₅₃	1	2	yes	—
322	#103	—	53	2	no	—
323	#103	—	53	2	no	—
324	#103	—	53	2	no	—
325	#102	—	53	2	no	—
326	C₂²⋊C₈	—	106	1	no	—
327	C₂²⋊C₈	—	106	1	no	—
328	C₂²⋊C₈	—	106	1	no	—
329	C₂²⋊C₈	—	106	1	no	—
330	#97	—	53	2	no	—
331	C₄×C₈	—	2	2	no	—
332	C₄×C₈	—	2	2	no	—
333	C₄×C₈	—	53	2	no	—
334	C₄×C₈	—	53	2	no	—
335	C₄×C₈	—	106	1	no	—
336	C₄×C₈	—	106	1	no	—
337	C₄×C₈	—	106	1	no	—
338	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
339	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
340	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
341	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
342	#41	—	53	4	no	—
343	#41	—	106	2	no	—
344	C₂×Q₈	—	2	1	no	—
345	C₂×Q₈	—	2	2	no	—
346	C₂×Q₈	—	53	1	no	—
347	C₂×Q₈	C₈×D₅₃	1	4	yes	complemented
348	C₂×Q₈	C₂×C₄×D₅₃	1	2	yes	—
349	C₂×D₄	—	53	1	no	—
350	C₂×D₄	—	106	1	no	—
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	106	1	no	—
354	C₂²×C₄	—	106	1	no	—
355	C₂²×C₄	—	106	1	no	—
356	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
357	C₂²×C₄	#11719945	1	1	yes	characteristic
358	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
359	Q₁₆	—	53	8	no	—
360	SD₁₆	—	53	8	no	—
361	M₁₆	—	53	4	no	—
362	M₁₆	—	106	2	no	—
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	—	2	2	no	—
365	C₂×C₈	—	4	1	no	—
366	C₂×C₈	—	4	2	no	—
367	C₂×C₈	—	53	2	no	—
368	C₂×C₈	—	53	2	no	—
369	C₂×C₈	—	106	1	no	—
370	C₂×C₈	—	106	1	no	—
371	C₂×C₈	—	106	2	no	—
372	C₂×C₈	—	212	2	no	—
373	C₂×C₈	—	212	2	no	—
374	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
375	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
376	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
377	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
378	C₄⋊C₄	—	2	2	no	—
379	C₄⋊C₄	—	2	1	no	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	—	53	1	no	—
382	C₄⋊C₄	—	53	1	no	—
383	C₄⋊C₄	—	106	1	no	—
384	C₄⋊C₄	—	106	1	no	—
385	C₄⋊C₄	C₂×C₄×D₅₃	1	2	yes	—
386	C₂²⋊C₄	—	106	1	no	—
387	C₂²⋊C₄	—	106	1	no	—
388	C₂²⋊C₄	—	106	1	no	—
389	C₄²	—	2	2	no	—
390	C₄²	—	2	2	no	—
391	C₄²	—	106	1	no	—
392	C₄²	—	106	1	no	—
393	C₄²	—	106	1	no	—
394	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
395	C₄²	#11719945	1	1	yes	characteristic
396	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
397	C₄²	C₂³×D₅₃	1	1	yes	characteristic
398	C₂³	—	106	1	no	—
399	C₂³	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
400	Q₈	—	2	2	no	—
401	Q₈	—	2	4	no	—
402	Q₈	—	53	2	no	—
403	Q₈	—	106	1	no	—
404	Q₈	C₂×C₈×D₅₃	1	4	yes	—
405	D₄	—	53	2	no	—
406	D₄	—	106	4	no	—
407	D₄	—	106	1	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	4	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	—	106	1	no	—
416	C₂×C₄	—	106	4	no	—
417	C₂×C₄	—	106	1	no	—
418	C₂×C₄	—	106	1	no	—
419	C₂×C₄	—	212	2	no	—
420	C₂×C₄	—	212	1	no	—
421	C₂×C₄	—	212	1	no	—
422	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
423	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
424	C₂×C₄	C₄×C₂²⋊D₅₃	1	1	yes	characteristic
425	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
426	C₂×C₄	#13133223	1	2	yes	—
427	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
428	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
429	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
430	C₈	—	2	1	no	—
431	C₈	—	2	1	no	—
432	C₈	—	2	1	no	—
433	C₈	—	4	2	no	—
434	C₈	—	53	4	no	—
435	C₈	—	106	2	no	—
436	C₈	—	212	2	no	—
437	C₈	—	212	1	no	—
438	C₈	—	212	1	no	—
439	C₈	#13133248	1	2	yes	direct product factor
440	C₂²	—	2	1	no	—
441	C₂²	—	2	1	no	—
442	C₂²	—	106	2	no	—
443	C₂²	—	212	1	no	—
444	C₂²	—	212	1	no	—
445	C₂²	C₈×C₂²⋊D₅₃	1	2	yes	—
446	C₂²	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
447	C₄	—	2	1	no	—
448	C₄	—	2	1	no	—
449	C₄	—	2	1	no	—
450	C₄	—	2	4	no	—
451	C₄	—	4	2	no	—
452	C₄	—	106	2	no	—
453	C₄	—	212	1	no	—
454	C₄	—	212	1	no	—
455	C₄	C₈×C₂²⋊D₅₃	1	2	yes	—
456	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
457	C₄	C₂×#13133223	1	1	yes	characteristic
458	C₄	C₂×#13133248	1	1	yes	characteristic
459	C₄	#16811421	1	1	yes	characteristic
460	C₂	—	2	2	no	—
461	C₂	—	212	2	no	—
462	C₂	C₂×C₈×C₂²⋊D₅₃	1	1	yes	characteristic
463	C₂	C₄×#13133248	1	1	yes	characteristic
464	C₂	#20283616	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₈×Q₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#11719945	C₂×C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×C₈×Q₈	D₅₃
hypocenter	C₅₃	C₈×#137
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. #20273532	C₂	semidirect
4. #20273525	C₂	semidirect
5. C₈×#11719925	C₂	semidirect
6. C₈×#11719924	C₂	semidirect
7. #20273465	C₂	semidirect
8. #20273110	C₂	semidirect
10. C₈×#11719917	C₂	semidirect
11. C₂×C₈×Q₈×C₅₃	C₂	semidirect
50. #13133248	C₈	direct (N × H)
111. C₂×Q₈×C₅₃	C₂×C₈	semidirect
145. #11719925	C₂×C₈	semidirect
146. #11719924	C₂×C₈	semidirect
157. #11719917	C₂×C₈	semidirect
201. C₈×C₅₃	#137	semidirect
229. C₂×C₈×Q₈	D₅₃	semidirect
281. C₅₃	C₈×#137	semidirect
347. C₂×Q₈	C₈×D₅₃	semidirect
439. C₈	#13133248	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈×#11719945	2
C₄×#13133248	2
#20273532	2
#20273525	2
C₈×#11719925	2
C₈×#11719924	2
#20273465	2
#20273110	2
#20273100	2
C₈×#11719917	2
C₂×C₈×Q₈×C₅₃	2
C₈×#137	53

Subgroups — C8×#13133248 #27353517

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#13133248 #27353517