Subgroups — #27402904

Order 13568.

832 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810754	C₂	1	2	yes	maximal
2	#20275439	C₂	1	4	yes	maximal
3	#20275434	C₂	1	2	yes	maximal
4	#20274867	C₂	1	1	yes	maximal, characteristic
5	#20274867	C₂	1	2	yes	maximal
6	#20274568	C₂	1	2	yes	maximal
7	C₄×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
8	C₄²×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
9	#16810754	—	2	8	no	—
10	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊#88322	C₂²	1	2	yes	—
12	C₂×C₄⋊#88322	C₂²	1	2	yes	—
13	C₂×C₄⋊#88322	C₂²	1	4	yes	—
14	C₂×C₄⋊#88322	C₂²	1	4	yes	—
15	C₂×C₄⋊#88322	C₂²	1	2	yes	—
16	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
17	C₂×#13133128	C₂²	1	4	yes	—
18	C₂×C₄×#88322	C₂²	1	2	yes	—
19	#16810378	C₂²	1	2	yes	—
20	#16810378	C₂²	1	4	yes	—
21	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
25	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
27	C₄×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
30	C₄³×C₅₃	C₂²	1	1	yes	characteristic
31	C₄³×C₅₃	C₂²	1	2	yes	—
32	C₂²×#88322	C₂³	1	2	yes	—
33	C₂²×#88322	C₂³	1	4	yes	—
34	C₂²×#88322	C₂³	1	2	yes	—
35	C₄⋊#88322	—	2	8	no	—
36	C₄⋊#88322	—	2	4	no	—
37	C₄⋊#88322	—	2	2	no	—
38	C₄⋊#88322	—	2	4	no	—
39	C₄⋊#88322	—	4	4	no	—
40	C₄⋊#88322	—	4	2	no	—
41	C₄⋊#88322	Q₈	1	4	yes	complemented
42	#13133128	—	2	8	no	—
43	C₄×#88322	—	2	4	no	—
44	C₂²×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₄×Q₈×C₅₃	—	2	4	no	—
46	C₄×Q₈×C₅₃	—	2	4	no	—
47	C₄×Q₈×C₅₃	D₄	1	4	yes	—
48	C₄×Q₈×C₅₃	Q₈	1	4	yes	—
49	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
58	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
60	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
61	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
65	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
67	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
68	#95×C₅₃	C₂×C₄	1	2	yes	—
69	#95×C₅₃	C₂×C₄	1	2	yes	—
70	#95×C₅₃	C₂×C₄	1	2	yes	—
71	C₂×Q₈×C₅₃	—	2	2	no	—
72	C₂×Q₈×C₅₃	—	2	2	no	—
73	C₂×Q₈×C₅₃	C₄⋊C₄	1	4	yes	—
74	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
85	C₄⋊C₄×C₅₃	—	2	4	no	—
86	C₄⋊C₄×C₅₃	—	2	4	no	—
87	C₄⋊C₄×C₅₃	—	2	2	no	—
88	C₄⋊C₄×C₅₃	C₄⋊C₄	1	8	yes	—
89	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
90	C₄⋊C₄×C₅₃	M₁₆	1	4	yes	—
91	C₄²×C₅₃	—	2	4	no	—
92	C₄²×C₅₃	—	2	4	no	—
93	C₄²×C₅₃	—	2	2	no	—
94	C₄²×C₅₃	—	2	4	no	—
95	C₄²×C₅₃	—	2	2	no	—
96	C₄²×C₅₃	—	2	1	no	—
97	C₄²×C₅₃	—	2	2	no	—
98	C₄²×C₅₃	—	2	1	no	—
99	C₄²×C₅₃	—	2	2	no	—
100	C₄²×C₅₃	—	4	1	no	—
101	C₄²×C₅₃	—	4	2	no	—
102	C₄²×C₅₃	M₁₆	1	4	yes	—
103	C₄²×C₅₃	C₂×D₄	1	2	yes	—
104	C₄²×C₅₃	C₂×D₄	1	4	yes	—
105	C₄²×C₅₃	C₂×Q₈	1	4	yes	—
106	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
107	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
108	C₄²×C₅₃	#41	1	2	yes	—
109	C₂×#88322	—	2	2	no	—
110	C₂×#88322	—	2	2	no	—
111	C₂×#88322	—	2	2	no	—
112	C₂×#88322	—	2	4	no	—
113	C₂×#88322	—	2	2	no	—
114	C₂×#88322	—	2	4	no	—
115	C₂×#88322	—	2	2	no	—
116	C₂×#88322	—	2	4	no	—
117	C₂×#88322	C₂×Q₈	1	4	yes	—
118	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
119	Q₈×C₅₃	—	2	8	no	—
120	C₂×C₄×C₅₃	—	2	1	no	—
121	C₂×C₄×C₅₃	—	2	2	no	—
122	C₂×C₄×C₅₃	—	2	2	no	—
123	C₂×C₄×C₅₃	—	2	2	no	—
124	C₂×C₄×C₅₃	—	2	1	no	—
125	C₂×C₄×C₅₃	—	2	1	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	—	2	1	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	1	no	—
131	C₂×C₄×C₅₃	—	2	1	no	—
132	C₂×C₄×C₅₃	—	2	2	no	—
133	C₂×C₄×C₅₃	—	2	1	no	—
134	C₂×C₄×C₅₃	—	2	2	no	—
135	C₂×C₄×C₅₃	—	2	1	no	—
136	C₂×C₄×C₅₃	—	2	1	no	—
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
139	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
141	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
142	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
143	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
144	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
145	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
146	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
147	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
148	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
149	C₂×C₄×C₅₃	#131	1	2	yes	—
150	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
154	C₂×C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂×C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
156	#88322	—	2	4	no	—
157	#88322	—	4	2	no	—
158	#88322	—	4	4	no	—
159	#22722	—	53	1	no	maximal, Sylow
160	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
161	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
162	C₂²×C₅₃	#520	1	1	yes	characteristic
163	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
164	C₂²×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
165	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
166	C₂²×C₅₃	#568	1	1	yes	characteristic
167	C₄×C₅₃	—	2	4	no	—
168	C₄×C₅₃	—	2	2	no	—
169	C₄×C₅₃	—	2	4	no	—
170	C₄×C₅₃	—	2	2	no	—
171	C₄×C₅₃	—	2	2	no	—
172	C₄×C₅₃	—	2	4	no	—
173	C₄×C₅₃	—	4	2	no	—
174	C₄×C₅₃	—	4	1	no	—
175	C₄×C₅₃	#423	1	4	yes	—
176	C₄×C₅₃	#446	1	4	yes	complemented
177	C₄×C₅₃	D₄×Q₈	1	1	yes	characteristic
178	C₄×C₅₃	#550	1	1	yes	characteristic
179	C₄×C₅₃	#557	1	1	yes	characteristic
180	C₄×C₅₃	Q₈²	1	1	yes	characteristic
181	C₂×#446	—	53	2	no	—
182	C₄²×Q₈	D₅₃	1	1	yes	characteristic
183	#1940	—	53	4	no	—
184	#1939	—	53	2	no	—
185	#1842	—	53	1	no	—
186	#1842	—	53	2	no	—
187	#1767	—	53	2	no	—
188	C₄×C₄⋊C₈	—	53	1	no	—
189	C₂×C₅₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
190	C₂×C₅₃	C₂×#423	1	1	yes	characteristic
191	C₂×C₅₃	#2906	1	1	yes	characteristic
192	C₂×C₅₃	C₂×#446	1	1	yes	characteristic
193	C₂×C₅₃	Q₈×M₁₆	1	1	yes	characteristic
194	C₂×C₅₃	#2989	1	1	yes	characteristic
195	C₂×C₅₃	#2990	1	1	yes	characteristic
196	C₂×C₄×Q₈	Dic₅₃	1	1	yes	characteristic
197	C₂×C₄×Q₈	Dic₅₃	1	1	yes	characteristic
198	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
199	#446	—	106	8	no	—
200	C₂×C₄⋊C₈	—	53	1	no	—
201	C₂×C₄⋊C₈	—	53	1	no	—
202	C₂×C₄⋊C₈	—	53	2	no	—
203	C₂×C₄⋊C₈	—	53	2	no	—
204	C₂×C₄⋊C₈	—	53	4	no	—
205	C₂×C₄⋊C₈	—	53	4	no	—
206	C₂×C₄⋊C₈	—	53	2	no	—
207	C₂×#97	—	53	4	no	—
208	C₂×C₄×C₈	—	53	2	no	—
209	C₄×C₄⋊C₄	Dic₅₃	1	2	yes	—
210	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
211	C₄×C₄⋊C₄	Dic₅₃	1	2	yes	—
212	C₄×C₄⋊C₄	Dic₅₃	1	1	yes	characteristic
213	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
214	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
215	C₄³	C₂×D₅₃	1	1	yes	characteristic
216	C₄³	C₂×D₅₃	1	2	yes	—
217	#336	—	53	4	no	—
218	#336	—	53	2	no	—
219	C₅₃	#22722	1	1	yes	characteristic, complemented, Sylow
220	C₂²×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
221	C₂²×C₈	—	53	2	no	—
222	C₂²×C₈	—	53	2	no	—
223	C₂²×C₈	—	53	4	no	—
224	C₄×Q₈	—	2	4	no	—
225	C₄×Q₈	—	2	4	no	—
226	C₄×Q₈	Dic₁₀₆	1	4	yes	—
227	C₄×Q₈	D₂₁₂	1	4	yes	—
228	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
229	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
230	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
231	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
232	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
233	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
234	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
235	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
236	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
237	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
238	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
239	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
240	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
241	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
242	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
243	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
244	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
245	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
246	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
247	C₄⋊C₈	—	53	4	no	—
248	C₄⋊C₈	—	106	4	no	—
249	C₄⋊C₈	—	106	8	no	—
250	C₄⋊C₈	—	106	2	no	—
251	C₄⋊C₈	—	106	4	no	—
252	C₄⋊C₈	—	212	4	no	—
253	C₄⋊C₈	—	212	2	no	—
254	#97	—	106	8	no	—
255	C₄×C₈	—	106	4	no	—
256	#95	C₂×Dic₅₃	1	2	yes	—
257	#95	C₂×Dic₅₃	1	2	yes	—
258	#95	C₂×Dic₅₃	1	2	yes	—
259	C₂×Q₈	—	2	2	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	C₄⋊Dic₅₃	1	4	yes	—
262	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
263	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
264	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
265	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
266	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
267	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
268	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
269	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
270	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
271	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
272	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
273	C₂×C₈	—	53	4	no	—
274	C₂×C₈	—	106	2	no	—
275	C₂×C₈	—	106	2	no	—
276	C₂×C₈	—	106	2	no	—
277	C₂×C₈	—	106	4	no	—
278	C₂×C₈	—	106	2	no	—
279	C₂×C₈	—	106	4	no	—
280	C₂×C₈	—	106	2	no	—
281	C₂×C₈	—	106	4	no	—
282	C₄⋊C₄	—	2	4	no	—
283	C₄⋊C₄	—	2	4	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	#11719917	1	4	yes	—
286	C₄⋊C₄	C₄⋊Dic₅₃	1	8	yes	—
287	C₄⋊C₄	C₄⋊Dic₅₃	1	4	yes	—
288	C₄²	—	2	4	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	4	no	—
291	C₄²	—	2	4	no	—
292	C₄²	—	2	2	no	—
293	C₄²	—	2	1	no	—
294	C₄²	—	2	2	no	—
295	C₄²	—	2	1	no	—
296	C₄²	—	2	2	no	—
297	C₄²	—	4	1	no	—
298	C₄²	—	4	2	no	—
299	C₄²	#11719917	1	4	yes	—
300	C₄²	C₂×Dic₁₀₆	1	4	yes	—
301	C₄²	C₂×Dic₁₀₆	1	2	yes	—
302	C₄²	C₂×D₂₁₂	1	2	yes	—
303	C₄²	C₂×D₂₁₂	1	4	yes	—
304	C₄²	Q₈×D₅₃	1	2	yes	—
305	C₄²	#11719949	1	2	yes	—
306	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
307	Q₈	—	2	8	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
327	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
328	C₂×C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
329	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
330	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
331	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
332	C₂×C₄	C₂×#11719917	1	4	yes	—
333	C₂×C₄	C₂×#11719917	1	2	yes	—
334	C₂×C₄	C₂×#11719917	1	2	yes	—
335	C₂×C₄	C₂×#11719917	1	2	yes	—
336	C₂×C₄	C₂×#11719917	1	2	yes	—
337	C₂×C₄	#13133224	1	2	yes	—
338	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
339	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
340	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
341	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
342	C₂×C₄	#13133263	1	1	yes	characteristic
343	C₂×C₄	#13133264	1	1	yes	characteristic
344	C₈	—	106	4	no	—
345	C₈	—	212	2	no	—
346	C₈	—	212	4	no	—
347	C₂²	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
348	C₂²	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
349	C₂²	#16811106	1	1	yes	characteristic
350	C₂²	C₂²×#11719917	1	1	yes	characteristic
351	C₂²	C₂²×#11719917	1	1	yes	characteristic
352	C₂²	C₂×#13133224	1	1	yes	characteristic
353	C₂²	#16811390	1	1	yes	characteristic
354	C₄	—	2	2	no	—
355	C₄	—	2	4	no	—
356	C₄	—	2	2	no	—
357	C₄	—	2	4	no	—
358	C₄	—	2	2	no	—
359	C₄	—	2	4	no	—
360	C₄	—	4	2	no	—
361	C₄	—	4	1	no	—
362	C₄	#16810647	1	4	yes	—
363	C₄	#16810754	1	4	yes	complemented
364	C₄	Q₈×D₂₁₂	1	1	yes	characteristic
365	C₄	#16811295	1	1	yes	characteristic
366	C₄	#16811352	1	1	yes	characteristic
367	C₄	Q₈×Dic₁₀₆	1	1	yes	characteristic
368	C₂	Q₈×C₄⋊Dic₅₃	1	1	yes	characteristic
369	C₂	C₂×#16810647	1	1	yes	characteristic
370	C₂	#20283283	1	1	yes	characteristic
371	C₂	C₂×#16810754	1	1	yes	characteristic
372	C₂	Q₈×#11719917	1	1	yes	characteristic
373	C₂	#20283832	1	1	yes	characteristic
374	C₂	#20283840	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle, prime radical	C₂³×C₅₃	C₂³×C₄
Fitting	C₄²×Q₈×C₅₃	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₄²×Q₈	D₅₃
hypocenter	C₅₃	#22722
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
41. C₄⋊#88322	Q₈	semidirect
176. C₄×C₅₃	#446	semidirect
219. C₅₃	#22722	semidirect
363. C₄	#16810754	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810754	2
#20275439	2
#20275434	2
#20274867	2
#20274568	2
C₄×C₄⋊#88322	2
C₄²×Q₈×C₅₃	2
#22722	53