Subgroups — #27276780

Order 13568.

588 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274867	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊#88322	C₂	1	2	yes	maximal
4	#20272556	C₂	1	2	yes	maximal
5	C₄²×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	—	2	1	no	—
7	C₂×C₄⋊#88322	—	2	1	no	—
8	C₂×C₄⋊#88322	—	2	1	no	—
9	C₂×C₄⋊#88322	C₂²	1	2	yes	—
10	C₂×C₄×#88322	C₂²	1	2	yes	—
11	#16810378	—	2	1	no	—
12	Q₈⋊#88322	—	2	4	no	—
13	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
17	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
18	C₄×C₄⋊C₄×C₅₃	—	2	1	no	—
19	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
20	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
22	C₄³×C₅₃	—	2	1	no	—
23	C₄³×C₅₃	C₂²	1	1	yes	characteristic
24	C₂²×#88322	—	2	2	no	—
25	C₂²×#88322	—	2	2	no	—
26	C₄⋊#88322	—	2	4	no	—
27	C₄⋊#88322	—	2	2	no	—
28	C₄⋊#88322	—	2	2	no	—
29	C₄⋊#88322	—	4	2	no	—
30	C₄⋊#88322	—	4	2	no	—
31	C₄×#88322	—	2	2	no	—
32	C₄×#88322	—	2	2	no	—
33	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₄×Q₈×C₅₃	—	2	2	no	—
35	C₄×Q₈×C₅₃	—	2	2	no	—
36	C₄×Q₈×C₅₃	—	2	1	no	—
37	C₄×Q₈×C₅₃	—	2	1	no	—
38	C₄×Q₈×C₅₃	—	2	2	no	—
39	C₄×Q₈×C₅₃	C₈	1	4	yes	complemented
40	C₄×Q₈×C₅₃	D₄	1	2	yes	—
41	C₄×Q₈×C₅₃	Q₈	1	2	yes	—
42	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
43	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
44	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
45	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₅₃	—	2	1	no	—
49	C₂×C₄²×C₅₃	—	2	1	no	—
50	C₂×C₄²×C₅₃	—	2	1	no	—
51	C₂×C₄²×C₅₃	—	2	1	no	—
52	C₂×C₄²×C₅₃	—	2	1	no	—
53	C₂×C₄²×C₅₃	—	2	1	no	—
54	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
61	#95×C₅₃	—	2	1	no	—
62	#95×C₅₃	—	2	1	no	—
63	#95×C₅₃	—	2	1	no	—
64	C₂×Q₈×C₅₃	—	2	1	no	—
65	C₂×Q₈×C₅₃	—	2	1	no	—
66	C₂×Q₈×C₅₃	—	2	1	no	—
67	C₂×Q₈×C₅₃	C₄⋊C₄	1	2	yes	—
68	C₂×Q₈×C₅₃	C₂×C₈	1	2	yes	—
69	C₂×Q₈×C₅₃	M₁₆	1	2	yes	—
70	C₂²×C₄×C₅₃	—	2	1	no	—
71	C₂²×C₄×C₅₃	—	2	1	no	—
72	C₂²×C₄×C₅₃	—	2	1	no	—
73	C₂²×C₄×C₅₃	—	2	1	no	—
74	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
81	C₄⋊C₄×C₅₃	—	2	4	no	—
82	C₄⋊C₄×C₅₃	—	2	4	no	—
83	C₄⋊C₄×C₅₃	—	2	1	no	—
84	C₄⋊C₄×C₅₃	—	2	2	no	—
85	C₄⋊C₄×C₅₃	—	2	2	no	—
86	C₄⋊C₄×C₅₃	—	2	1	no	—
87	C₄⋊C₄×C₅₃	—	2	1	no	—
88	C₄⋊C₄×C₅₃	C₄⋊C₄	1	2	yes	—
89	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
90	C₄⋊C₄×C₅₃	M₁₆	1	2	yes	—
91	C₄²×C₅₃	—	2	4	no	—
92	C₄²×C₅₃	—	2	4	no	—
93	C₄²×C₅₃	—	2	2	no	—
94	C₄²×C₅₃	—	2	2	no	—
95	C₄²×C₅₃	—	2	2	no	—
96	C₄²×C₅₃	—	2	1	no	—
97	C₄²×C₅₃	—	2	1	no	—
98	C₄²×C₅₃	—	2	1	no	—
99	C₄²×C₅₃	—	2	1	no	—
100	C₄²×C₅₃	—	4	1	no	—
101	C₄²×C₅₃	—	4	2	no	—
102	C₄²×C₅₃	—	4	2	no	—
103	C₄²×C₅₃	—	4	1	no	—
104	C₄²×C₅₃	—	4	1	no	—
105	C₄²×C₅₃	C₂×C₈	1	2	yes	—
106	C₄²×C₅₃	M₁₆	1	2	yes	—
107	C₄²×C₅₃	SD₁₆	1	2	yes	—
108	C₄²×C₅₃	Q₁₆	1	2	yes	—
109	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
111	C₄²×C₅₃	#41	1	1	yes	characteristic
112	C₄²×C₅₃	#41	1	1	yes	characteristic
113	C₂×#88322	—	2	4	no	—
114	C₂×#88322	—	4	2	no	—
115	C₂×#88322	—	4	2	no	—
116	C₂×#88322	—	4	2	no	—
117	C₂×#88322	—	4	2	no	—
118	C₂×#88322	—	4	2	no	—
119	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	Q₈×C₅₃	—	2	2	no	—
121	Q₈×C₅₃	—	2	2	no	—
122	Q₈×C₅₃	—	2	2	no	—
123	Q₈×C₅₃	C₄⋊C₈	1	4	yes	complemented
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	1	no	—
128	C₂×C₄×C₅₃	—	2	1	no	—
129	C₂×C₄×C₅₃	—	2	1	no	—
130	C₂×C₄×C₅₃	—	2	1	no	—
131	C₂×C₄×C₅₃	—	2	1	no	—
132	C₂×C₄×C₅₃	—	2	1	no	—
133	C₂×C₄×C₅₃	—	2	2	no	—
134	C₂×C₄×C₅₃	—	2	2	no	—
135	C₂×C₄×C₅₃	—	2	2	no	—
136	C₂×C₄×C₅₃	—	2	1	no	—
137	C₂×C₄×C₅₃	—	2	1	no	—
138	C₂×C₄×C₅₃	—	2	1	no	—
139	C₂×C₄×C₅₃	—	4	1	no	—
140	C₂×C₄×C₅₃	—	4	1	no	—
141	C₂×C₄×C₅₃	—	4	1	no	—
142	C₂×C₄×C₅₃	—	4	1	no	—
143	C₂×C₄×C₅₃	—	4	1	no	—
144	C₂×C₄×C₅₃	—	4	1	no	—
145	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
146	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
147	C₂×C₄×C₅₃	#103	1	2	yes	—
148	C₂×C₄×C₅₃	#103	1	2	yes	—
149	C₂×C₄×C₅₃	#104	1	2	yes	—
150	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
152	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
153	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
162	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
163	C₂×C₄×C₅₃	#137	1	1	yes	characteristic
164	#88322	—	4	4	no	—
165	#88322	—	8	2	no	—
166	#8564	—	53	1	no	maximal, Sylow
167	C₂²×C₅₃	#380	1	1	yes	characteristic
168	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
169	C₂²×C₅₃	#407	1	1	yes	characteristic
170	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
171	C₂²×C₅₃	#417	1	1	yes	characteristic
172	C₂²×C₅₃	C₂×#104	1	1	yes	characteristic
173	C₂²×C₅₃	#421	1	1	yes	characteristic
174	C₄×C₅₃	—	2	4	no	—
175	C₄×C₅₃	—	2	1	no	—
176	C₄×C₅₃	—	2	1	no	—
177	C₄×C₅₃	—	2	2	no	—
178	C₄×C₅₃	—	2	2	no	—
179	C₄×C₅₃	—	4	2	no	—
180	C₄×C₅₃	—	4	2	no	—
181	C₄×C₅₃	—	4	1	no	—
182	C₄×C₅₃	—	4	2	no	—
183	C₄×C₅₃	Q₈⋊C₈	1	4	yes	complemented
184	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
185	C₄×C₅₃	#423	1	1	yes	characteristic
186	C₄×C₅₃	#431	1	1	yes	characteristic
187	C₄×C₅₃	#432	1	1	yes	characteristic
188	C₄×C₅₃	#460	1	1	yes	characteristic
189	C₄×C₅₃	#462	1	1	yes	characteristic
190	C₄×C₅₃	#475	1	1	yes	characteristic
191	C₄×C₅₃	#477	1	1	yes	characteristic
192	C₄²×Q₈	D₅₃	1	1	yes	characteristic
193	#1842	—	53	1	no	—
194	C₄×C₄⋊C₈	—	53	1	no	—
195	C₂×Q₈⋊C₈	—	53	2	no	—
196	#1279	—	53	2	no	—
197	C₂×C₅₃	C₂×Q₈⋊C₈	1	1	yes	characteristic
198	C₂×C₅₃	#1477	1	1	yes	characteristic
199	C₂×C₅₃	#1487	1	1	yes	characteristic
200	C₂×C₅₃	#1489	1	1	yes	characteristic
201	C₂×C₅₃	#1797	1	1	yes	characteristic
202	C₂×C₅₃	#1803	1	1	yes	characteristic
203	C₂×C₅₃	#1806	1	1	yes	characteristic
204	C₂×C₄×Q₈	Dic₅₃	1	1	yes	characteristic
205	C₂×C₄×Q₈	Dic₅₃	1	1	yes	characteristic
206	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
207	C₂×C₄⋊C₈	—	53	2	no	—
208	C₂×C₄⋊C₈	—	106	1	no	—
209	C₂×C₄⋊C₈	—	106	1	no	—
210	C₂×C₄⋊C₈	—	106	1	no	—
211	C₂×C₄×C₈	—	53	2	no	—
212	C₄×C₄⋊C₄	—	2	1	no	—
213	C₄×C₄⋊C₄	—	2	1	no	—
214	C₄×C₄⋊C₄	—	2	1	no	—
215	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
216	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
217	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
218	C₄³	—	2	1	no	—
219	C₄³	C₂×D₅₃	1	1	yes	characteristic
220	#336	—	106	1	no	—
221	Q₈⋊C₈	—	106	4	no	—
222	C₅₃	#8564	1	1	yes	characteristic, complemented, Sylow
223	C₂²×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
224	C₂²×C₈	—	106	2	no	—
225	C₂²×C₈	—	106	2	no	—
226	C₄×Q₈	—	2	2	no	—
227	C₄×Q₈	—	2	2	no	—
228	C₄×Q₈	—	2	1	no	—
229	C₄×Q₈	—	2	1	no	—
230	C₄×Q₈	—	2	2	no	—
231	C₄×Q₈	#88322	1	4	yes	complemented
232	C₄×Q₈	Dic₁₀₆	1	2	yes	—
233	C₄×Q₈	D₂₁₂	1	2	yes	—
234	C₂×C₄⋊C₄	—	2	1	no	—
235	C₂×C₄⋊C₄	—	2	1	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
238	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
239	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
240	C₂×C₄²	—	2	1	no	—
241	C₂×C₄²	—	2	1	no	—
242	C₂×C₄²	—	2	1	no	—
243	C₂×C₄²	—	2	1	no	—
244	C₂×C₄²	—	2	1	no	—
245	C₂×C₄²	—	2	1	no	—
246	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
247	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
248	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
249	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
250	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
251	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
252	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
253	C₄⋊C₈	—	106	4	no	—
254	C₄⋊C₈	—	106	2	no	—
255	C₄⋊C₈	—	106	2	no	—
256	C₄⋊C₈	—	212	2	no	—
257	C₄⋊C₈	—	212	2	no	—
258	C₄×C₈	—	106	2	no	—
259	C₄×C₈	—	106	2	no	—
260	#95	—	2	1	no	—
261	#95	—	2	1	no	—
262	#95	—	2	1	no	—
263	C₂×Q₈	—	2	1	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	2	1	no	—
266	C₂×Q₈	C₂×#88322	1	2	yes	—
267	C₂×Q₈	#11719917	1	2	yes	—
268	C₂×Q₈	C₄⋊Dic₅₃	1	2	yes	—
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	2	1	no	—
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	—	2	1	no	—
273	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
274	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
275	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
276	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
277	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
278	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
279	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
280	C₂×C₈	—	106	4	no	—
281	C₂×C₈	—	212	2	no	—
282	C₂×C₈	—	212	2	no	—
283	C₂×C₈	—	212	2	no	—
284	C₂×C₈	—	212	2	no	—
285	C₂×C₈	—	212	2	no	—
286	C₄⋊C₄	—	2	4	no	—
287	C₄⋊C₄	—	2	4	no	—
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	2	1	no	—
293	C₄⋊C₄	C₂×#88322	1	2	yes	—
294	C₄⋊C₄	#11719917	1	2	yes	—
295	C₄⋊C₄	C₄⋊Dic₅₃	1	2	yes	—
296	C₄²	—	2	4	no	—
297	C₄²	—	2	4	no	—
298	C₄²	—	2	2	no	—
299	C₄²	—	2	2	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	2	1	no	—
302	C₄²	—	2	1	no	—
303	C₄²	—	2	1	no	—
304	C₄²	—	2	1	no	—
305	C₄²	—	4	1	no	—
306	C₄²	—	4	2	no	—
307	C₄²	—	4	2	no	—
308	C₄²	—	4	1	no	—
309	C₄²	—	4	1	no	—
310	C₄²	C₂×#88322	1	2	yes	—
311	C₄²	#11719917	1	2	yes	—
312	C₄²	#11719924	1	2	yes	—
313	C₄²	#11719925	1	2	yes	—
314	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
315	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
316	C₄²	#11719945	1	1	yes	characteristic
317	C₄²	#11719945	1	1	yes	characteristic
318	C₂³	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
319	Q₈	—	2	2	no	—
320	Q₈	—	2	2	no	—
321	Q₈	—	2	2	no	—
322	Q₈	C₄⋊#88322	1	4	yes	complemented
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	4	1	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	C₂²⋊#88322	1	2	yes	—
345	C₂×C₄	C₂²⋊#88322	1	2	yes	—
346	C₂×C₄	#13133141	1	2	yes	—
347	C₂×C₄	#13133141	1	2	yes	—
348	C₂×C₄	#13133144	1	2	yes	—
349	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
350	C₂×C₄	#13133176	1	1	yes	characteristic
351	C₂×C₄	#13133191	1	1	yes	characteristic
352	C₂×C₄	#13133191	1	1	yes	characteristic
353	C₂×C₄	#13133196	1	1	yes	characteristic
354	C₂×C₄	#13133196	1	1	yes	characteristic
355	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
356	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
357	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
358	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
359	C₂×C₄	C₂×#11719924	1	1	yes	characteristic
360	C₂×C₄	C₂×#11719925	1	1	yes	characteristic
361	C₂×C₄	#13133242	1	1	yes	characteristic
362	C₂×C₄	#13133247	1	1	yes	characteristic
363	C₈	—	424	2	no	—
364	C₈	—	212	4	no	—
365	C₂²	#16810483	1	1	yes	characteristic
366	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
367	C₂²	#16810587	1	1	yes	characteristic
368	C₂²	C₂×#13133141	1	1	yes	characteristic
369	C₂²	#16810624	1	1	yes	characteristic
370	C₂²	C₂×#13133144	1	1	yes	characteristic
371	C₂²	#16810640	1	1	yes	characteristic
372	C₄	—	2	4	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	2	no	—
376	C₄	—	2	2	no	—
377	C₄	—	4	1	no	—
378	C₄	—	4	2	no	—
379	C₄	—	4	2	no	—
380	C₄	—	4	2	no	—
381	C₄	Q₈⋊#88322	1	4	yes	complemented
382	C₄	C₂×C₄⋊#88322	1	1	yes	characteristic
383	C₄	#16810647	1	1	yes	characteristic
384	C₄	#16810676	1	1	yes	characteristic
385	C₄	#16810680	1	1	yes	characteristic
386	C₄	#16810823	1	1	yes	characteristic
387	C₄	#16810833	1	1	yes	characteristic
388	C₄	#16810908	1	1	yes	characteristic
389	C₄	#16810918	1	1	yes	characteristic
390	C₂	C₂×Q₈⋊#88322	1	1	yes	characteristic
391	C₂	#20273102	1	1	yes	characteristic
392	C₂	#20273144	1	1	yes	characteristic
393	C₂	#20273154	1	1	yes	characteristic
394	C₂	#20274695	1	1	yes	characteristic
395	C₂	#20274720	1	1	yes	characteristic
396	C₂	#20274735	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄²×Q₈×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄²×Q₈	D₅₃
hypocenter	C₅₃	#8564
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
39. C₄×Q₈×C₅₃	C₈	semidirect
123. Q₈×C₅₃	C₄⋊C₈	semidirect
183. C₄×C₅₃	Q₈⋊C₈	semidirect
222. C₅₃	#8564	semidirect
231. C₄×Q₈	#88322	semidirect
322. Q₈	C₄⋊#88322	semidirect
381. C₄	Q₈⋊#88322	semidirect

Maximal subgroups

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

subgroup	index
#20274867	2
C₄×C₄⋊#88322	2
C₂×Q₈⋊#88322	2
#20272556	2
C₄²×Q₈×C₅₃	2
#8564	53