Subgroups — #27537700

Order 13568.

574 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276547	C₂	1	2	yes	maximal, complemented
2	#20276541	C₂	1	2	yes	maximal, complemented
3	#20276541	C₂	1	2	yes	maximal, complemented
4	D₄×#11719908	C₂	1	2	yes	maximal, complemented
5	#20276522	C₂	1	2	yes	maximal, complemented
6	#20276329	C₂	1	2	yes	maximal, complemented
7	#20276326	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂²⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#2929	C₂	1	1	yes	maximal, characteristic
10	C₂×#13133157	C₂²	1	1	yes	characteristic
11	C₂×#13133157	C₂²	1	1	yes	characteristic
12	C₂×#13133157	C₂²	1	2	yes	complemented
13	C₂²×#11719908	C₂²	1	1	yes	characteristic
14	C₂²×#11719908	C₂²	1	1	yes	characteristic
15	C₂²×#11719908	C₂²	1	2	yes	complemented
16	C₄⋊#11719908	C₂²	1	2	yes	complemented
17	C₄⋊#11719908	C₂²	1	2	yes	complemented
18	C₂²⋊#11719908	C₂²	1	2	yes	—
19	C₂²⋊#11719908	C₂²	1	2	yes	—
20	C₂²⋊#11719908	C₂²	1	2	yes	—
21	C₂²⋊#11719908	C₂²	1	2	yes	—
22	C₂²⋊#11719908	C₂²	1	2	yes	complemented
23	C₂²⋊#11719908	C₂²	1	2	yes	complemented
24	#16810399	C₂²	1	2	yes	complemented
25	C₄×#11719908	C₂²	1	2	yes	complemented
26	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
27	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#525	C₄	1	1	yes	characteristic
29	C₅₃×#435	C₄	1	2	yes	—
30	C₅₃×#435	C₄	1	2	yes	—
31	C₈×D₄×C₅₃	C₂²	1	2	yes	—
32	C₈×D₄×C₅₃	C₂²	1	2	yes	—
33	C₅₃×#424	C₄	1	1	yes	characteristic
34	C₅₃×#408	C₄	1	1	yes	characteristic
35	C₅₃×#407	C₄	1	1	yes	characteristic
36	C₅₃×#405	C₂²	1	1	yes	characteristic
37	#13133157	—	4	1	no	—
38	#13133157	—	4	1	no	—
39	#13133157	—	4	2	no	—
40	C₂×#11719908	—	4	1	no	—
41	C₂×#11719908	—	4	1	no	—
42	C₂×#11719908	—	4	2	no	—
43	C₂×#11719908	C₂³	1	2	yes	—
44	C₂×#11719908	C₂³	1	2	yes	—
45	C₂×#11719908	C₂³	1	2	yes	—
46	C₂×#11719908	C₂³	1	2	yes	—
47	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
49	#131×C₅₃	—	4	1	no	—
50	#131×C₅₃	—	4	1	no	—
51	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
53	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	—	4	1	no	—
55	C₂²×C₈×C₅₃	—	4	1	no	—
56	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
61	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
63	#123×C₅₃	C₈	1	2	yes	—
64	C₂²⋊Q₈×C₅₃	C₈	1	1	yes	characteristic
65	C₂²⋊Q₈×C₅₃	C₈	1	1	yes	characteristic
66	#121×C₅₃	C₈	1	1	yes	characteristic
67	#121×C₅₃	C₈	1	1	yes	characteristic
68	#120×C₅₃	C₈	1	2	yes	—
69	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
70	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
71	#117×C₅₃	C₂×C₄	1	1	yes	characteristic
72	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
73	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
74	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
75	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
76	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
77	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
78	#97×C₅₃	C₂×C₄	1	2	yes	—
79	C₄×C₈×C₅₃	C₂³	1	2	yes	—
80	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
81	#41×C₅₃	—	4	1	no	—
82	#41×C₅₃	—	4	1	no	—
83	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
84	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
85	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
86	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
87	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	—	4	1	no	—
89	C₂²×C₄×C₅₃	—	4	1	no	—
90	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	M₁₆×C₅₃	—	4	1	no	—
98	M₁₆×C₅₃	—	4	2	no	—
99	M₁₆×C₅₃	—	4	1	no	—
100	C₂×C₈×C₅₃	—	2	1	no	—
101	C₂×C₈×C₅₃	—	2	1	no	—
102	C₂×C₈×C₅₃	—	2	1	no	—
103	C₂×C₈×C₅₃	—	2	1	no	—
104	C₂×C₈×C₅₃	—	2	1	no	—
105	C₂×C₈×C₅₃	—	2	1	no	—
106	C₂×C₈×C₅₃	—	2	2	no	—
107	C₂×C₈×C₅₃	—	2	2	no	—
108	C₂×C₈×C₅₃	—	4	1	no	—
109	C₂×C₈×C₅₃	—	4	1	no	—
110	C₂×C₈×C₅₃	—	4	1	no	—
111	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
112	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
116	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
117	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
118	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
119	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
120	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
121	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
122	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
123	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
124	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
125	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
126	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
127	#11719908	—	2	2	no	—
128	#11719908	—	2	2	no	—
129	#11719908	—	2	2	no	—
130	#11719908	—	2	2	no	—
131	C₂³×C₅₃	—	2	2	no	—
132	C₂³×C₅₃	—	4	1	no	—
133	C₂³×C₅₃	—	4	1	no	—
134	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
135	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
136	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
137	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
138	Q₈×C₅₃	—	4	1	no	—
139	D₄×C₅₃	—	2	4	no	—
140	D₄×C₅₃	—	4	2	no	—
141	D₄×C₅₃	—	4	1	no	—
142	D₄×C₅₃	—	4	2	no	—
143	C₂×C₄×C₅₃	—	2	1	no	—
144	C₂×C₄×C₅₃	—	2	1	no	—
145	C₂×C₄×C₅₃	—	2	1	no	—
146	C₂×C₄×C₅₃	—	2	2	no	—
147	C₂×C₄×C₅₃	—	2	1	no	—
148	C₂×C₄×C₅₃	—	2	1	no	—
149	C₂×C₄×C₅₃	—	2	1	no	—
150	C₂×C₄×C₅₃	—	2	2	no	—
151	C₂×C₄×C₅₃	—	4	1	no	—
152	C₂×C₄×C₅₃	—	4	1	no	—
153	C₂×C₄×C₅₃	—	4	1	no	—
154	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
158	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
159	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
160	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
161	C₈×C₅₃	—	2	1	no	—
162	C₈×C₅₃	—	2	1	no	—
163	C₈×C₅₃	—	2	1	no	—
164	C₈×C₅₃	—	2	2	no	—
165	C₈×C₅₃	—	2	2	no	—
166	C₈×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
167	C₈×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
168	#33490	—	53	1	no	maximal, Sylow
169	C₂²×C₅₃	—	2	1	no	—
170	C₂²×C₅₃	—	2	1	no	—
171	C₂²×C₅₃	—	2	2	no	—
172	C₂²×C₅₃	—	2	1	no	—
173	C₂²×C₅₃	—	2	1	no	—
174	C₂²×C₅₃	—	2	1	no	—
175	C₂²×C₅₃	—	2	2	no	—
176	C₂²×C₅₃	—	2	1	no	—
177	C₂²×C₅₃	—	2	1	no	—
178	C₂²×C₅₃	—	2	1	no	—
179	C₂²×C₅₃	—	2	2	no	—
180	C₂²×C₅₃	—	4	4	no	—
181	C₂²×C₅₃	#504	1	2	yes	—
182	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
183	C₄×C₅₃	—	2	2	no	—
184	C₄×C₅₃	—	2	2	no	—
185	C₄×C₅₃	—	2	1	no	—
186	C₄×C₅₃	—	2	1	no	—
187	C₄×C₅₃	—	2	1	no	—
188	C₄×C₅₃	—	2	2	no	—
189	C₄×C₅₃	#518	1	1	yes	characteristic
190	C₄×C₅₃	#568	1	1	yes	characteristic
191	#2929	D₅₃	1	1	yes	characteristic
192	#2169	—	53	2	no	—
193	#2168	—	53	2	no	—
194	#2168	—	53	2	no	—
195	D₄×C₁₆	—	53	2	no	—
196	#2164	—	53	2	no	—
197	#2113	—	53	2	no	—
198	#2112	—	53	1	no	—
199	C₂×C₂²⋊C₁₆	—	53	1	no	—
200	C₂×C₅₃	—	2	2	no	—
201	C₂×C₅₃	—	4	2	no	—
202	C₂×C₅₃	—	4	1	no	—
203	C₂×C₅₃	—	4	1	no	—
204	C₂×C₅₃	#2972	1	1	yes	characteristic
205	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
206	C₂×C₅₃	#3407	1	1	yes	characteristic
207	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
208	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
209	#525	Dic₅₃	1	1	yes	characteristic
210	C₂×M₃₂	—	53	1	no	—
211	C₂×M₃₂	—	53	1	no	—
212	C₂×M₃₂	—	53	2	no	—
213	C₂²×C₁₆	—	53	1	no	—
214	C₂²×C₁₆	—	53	1	no	—
215	C₂²×C₁₆	—	53	2	no	—
216	#435	Dic₅₃	1	2	yes	—
217	#435	Dic₅₃	1	2	yes	—
218	C₈×D₄	C₂×D₅₃	1	2	yes	—
219	C₈×D₄	C₂×D₅₃	1	2	yes	—
220	#424	Dic₅₃	1	1	yes	characteristic
221	#408	Dic₅₃	1	1	yes	characteristic
222	#407	Dic₅₃	1	1	yes	characteristic
223	#405	C₂×D₅₃	1	1	yes	characteristic
224	C₄⋊C₁₆	—	53	2	no	—
225	C₄⋊C₁₆	—	53	2	no	—
226	C₂²⋊C₁₆	—	53	2	no	—
227	C₂²⋊C₁₆	—	53	2	no	—
228	C₂²⋊C₁₆	—	53	2	no	—
229	C₂²⋊C₁₆	—	53	2	no	—
230	C₂²⋊C₁₆	—	53	2	no	—
231	C₂²⋊C₁₆	—	53	2	no	—
232	#346	—	53	2	no	—
233	C₄×C₁₆	—	53	2	no	—
234	C₅₃	#33490	1	1	yes	characteristic, complemented, Sylow
235	C₂×#41	C₂×Dic₅₃	1	1	yes	characteristic
236	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
237	#131	—	4	1	no	—
238	#131	—	4	1	no	—
239	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
240	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
241	C₂×M₁₆	C₂×Dic₅₃	1	1	yes	characteristic
242	C₂²×C₈	—	4	1	no	—
243	C₂²×C₈	—	4	1	no	—
244	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
245	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
246	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
247	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
248	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
249	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
250	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
251	#123	#88322	1	2	yes	—
252	C₂²⋊Q₈	#88322	1	1	yes	characteristic
253	C₂²⋊Q₈	#88322	1	1	yes	characteristic
254	#121	#88322	1	1	yes	characteristic
255	#121	#88322	1	1	yes	characteristic
256	#120	#88322	1	2	yes	—
257	C₄×D₄	C₂×Dic₅₃	1	2	yes	—
258	C₄×D₄	C₂×Dic₅₃	1	2	yes	—
259	#117	C₂×Dic₅₃	1	1	yes	characteristic
260	M₃₂	—	212	1	no	—
261	M₃₂	—	212	2	no	—
262	M₃₂	—	212	1	no	—
263	C₂×C₁₆	—	53	2	no	—
264	C₂×C₁₆	—	53	2	no	—
265	C₂×C₁₆	—	53	2	no	—
266	C₂×C₁₆	—	53	2	no	—
267	C₂×C₁₆	—	212	1	no	—
268	C₂×C₁₆	—	212	2	no	—
269	C₂×C₁₆	—	212	1	no	—
270	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
271	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
272	C₂²⋊C₈	C₂×Dic₅₃	1	2	yes	—
273	C₂²⋊C₈	C₂×Dic₅₃	1	2	yes	—
274	C₂²⋊C₈	C₂×Dic₅₃	1	2	yes	—
275	C₂²⋊C₈	C₂×Dic₅₃	1	2	yes	—
276	#97	C₂×Dic₅₃	1	2	yes	—
277	C₄×C₈	C₂²×D₅₃	1	2	yes	—
278	C₂⁴	C₂×#88322	1	1	yes	characteristic
279	#41	—	4	1	no	—
280	#41	—	4	1	no	—
281	C₂×Q₈	C₂×#88322	1	1	yes	characteristic
282	C₂×D₄	C₂×#88322	1	2	yes	—
283	C₂×D₄	C₂×#88322	1	2	yes	—
284	C₂×D₄	C₂×#88322	1	2	yes	—
285	C₂×D₄	C₂×#88322	1	1	yes	characteristic
286	C₂²×C₄	—	4	1	no	—
287	C₂²×C₄	—	4	1	no	—
288	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
289	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
290	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
291	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
292	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
293	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
294	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
295	M₁₆	—	4	2	no	—
296	M₁₆	—	4	1	no	—
297	M₁₆	—	4	1	no	—
298	C₂×C₈	—	2	1	no	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	2	1	no	—
302	C₂×C₈	—	2	1	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	4	1	no	—
307	C₂×C₈	—	4	1	no	—
308	C₂×C₈	—	4	1	no	—
309	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
310	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
311	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
312	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
313	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
314	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
315	C₄⋊C₄	C₂×#88322	1	1	yes	characteristic
316	C₄⋊C₄	C₂×#88322	1	1	yes	characteristic
317	C₄⋊C₄	C₂×#88322	1	2	yes	—
318	C₄⋊C₄	C₂×#88322	1	2	yes	—
319	C₂²⋊C₄	C₂×#88322	1	2	yes	—
320	C₂²⋊C₄	C₂×#88322	1	2	yes	—
321	C₂²⋊C₄	C₂×#88322	1	2	yes	—
322	C₂²⋊C₄	C₂×#88322	1	2	yes	—
323	C₂²⋊C₄	C₂×#88322	1	2	yes	—
324	C₄²	C₂²×Dic₅₃	1	2	yes	—
325	C₁₆	—	106	2	no	—
326	C₁₆	—	106	2	no	—
327	C₁₆	—	106	2	no	—
328	C₁₆	—	106	2	no	—
329	C₂³	—	2	2	no	—
330	C₂³	—	4	1	no	—
331	C₂³	—	4	1	no	—
332	C₂³	C₂²×#88322	1	1	yes	characteristic
333	C₂³	C₂²×#88322	1	1	yes	characteristic
334	C₂³	C₂²×#88322	1	2	yes	—
335	C₂³	C₂²×#88322	1	1	yes	characteristic
336	Q₈	—	4	1	no	—
337	D₄	—	2	4	no	—
338	D₄	—	4	2	no	—
339	D₄	—	4	1	no	—
340	D₄	—	4	2	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	4	1	no	—
350	C₂×C₄	—	4	1	no	—
351	C₂×C₄	—	4	1	no	—
352	C₂×C₄	C₂²×#88322	1	2	yes	—
353	C₂×C₄	C₂²×#88322	1	2	yes	—
354	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
355	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
356	C₂×C₄	C₂²×#88322	1	2	yes	—
357	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
358	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	2	1	no	—
362	C₈	—	2	2	no	—
363	C₈	—	2	2	no	—
364	C₈	#13133262	1	1	yes	characteristic
365	C₈	#13133262	1	1	yes	characteristic
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	2	2	no	—
369	C₂²	—	2	1	no	—
370	C₂²	—	2	1	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	2	2	no	—
373	C₂²	—	2	1	no	—
374	C₂²	—	2	1	no	—
375	C₂²	—	2	1	no	—
376	C₂²	—	2	2	no	—
377	C₂²	—	4	4	no	—
378	C₂²	#16811040	1	2	yes	—
379	C₂²	C₂³×#88322	1	1	yes	characteristic
380	C₄	—	2	2	no	—
381	C₄	—	2	2	no	—
382	C₄	—	2	1	no	—
383	C₄	—	2	1	no	—
384	C₄	—	2	1	no	—
385	C₄	—	2	2	no	—
386	C₄	#16811098	1	1	yes	characteristic
387	C₄	#16811390	1	1	yes	characteristic
388	C₂	—	2	2	no	—
389	C₂	—	4	1	no	—
390	C₂	—	4	1	no	—
391	C₂	—	4	2	no	—
392	C₂	#20283739	1	1	yes	characteristic
393	C₂	C₂×#16811040	1	1	yes	characteristic
394	C₂	#20287818	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#2929	C₂
prime radical	C₅₃×#525	C₄
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	#2929	D₅₃
hypocenter	C₅₃	#33490
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ #2929

Complements

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

N	H ≅ G/N	type
1. #20276547	C₂	semidirect
2. #20276541	C₂	semidirect
3. #20276541	C₂	semidirect
4. D₄×#11719908	C₂	semidirect
5. #20276522	C₂	semidirect
6. #20276329	C₂	semidirect
7. #20276326	C₂	semidirect
8. C₂×C₂²⋊#11719908	C₂	semidirect
12. C₂×#13133157	C₂²	semidirect
15. C₂²×#11719908	C₂²	semidirect
16. C₄⋊#11719908	C₂²	semidirect
17. C₄⋊#11719908	C₂²	semidirect
22. C₂²⋊#11719908	C₂²	semidirect
23. C₂²⋊#11719908	C₂²	semidirect
24. #16810399	C₂²	semidirect
25. C₄×#11719908	C₂²	semidirect
234. C₅₃	#33490	semidirect

Maximal subgroups

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

subgroup	index
#20276547	2
#20276541	2
D₄×#11719908	2
#20276522	2
#20276329	2
#20276326	2
C₂×C₂²⋊#11719908	2
C₅₃×#2929	2
#33490	53