Subgroups — #27353075

Order 13568.

778 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	C₂×#16810676	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11719917	C₂	1	1	yes	maximal, characteristic
3	#20273468	C₂	1	4	yes	maximal, complemented
4	#20273465	C₂	1	4	yes	maximal, complemented
5	#20273461	C₂	1	4	yes	maximal, complemented
6	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719917	C₄	1	1	yes	characteristic
8	C₂²×#11719917	C₄	1	1	yes	characteristic
9	#16810676	—	2	1	no	—
10	#16810676	—	2	1	no	—
11	#16810676	C₂²	1	2	yes	—
12	#16810676	C₂²	1	2	yes	—
13	C₂×C₄⋊#88322	C₂²	1	2	yes	—
14	C₂×C₂²⋊#88322	—	2	1	no	—
15	C₄×#11719917	C₄	1	2	yes	—
16	C₄×#11719917	C₄	1	2	yes	—
17	C₄×#11719917	C₂²	1	2	yes	—
18	C₄×#11719917	C₂²	1	2	yes	—
19	C₂×#13133128	C₄	1	2	yes	—
20	C₂×C₄×#88322	—	2	1	no	—
21	C₂×C₄×#88322	C₂²	1	2	yes	—
22	#16810375	C₂²	1	8	yes	complemented
23	#16810373	C₂²	1	8	yes	complemented
24	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
25	C₅₃×#431	—	2	1	no	—
26	C₅₃×#431	—	2	1	no	—
27	C₅₃×#431	C₂²	1	2	yes	—
28	C₅₃×#431	C₂²	1	2	yes	—
29	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
30	C₂×C₂²⋊C₈×C₅₃	—	2	1	no	—
31	C₂×C₄×C₈×C₅₃	—	2	1	no	—
32	C₂×#11719917	—	2	4	no	—
33	C₂×#11719917	C₈	1	8	yes	complemented
34	C₂×#11719917	C₂×C₄	1	2	yes	—
35	C₂×#11719917	C₂×C₄	1	2	yes	—
36	C₂×#11719917	C₂×C₄	1	2	yes	—
37	C₂×#11719917	C₂×C₄	1	2	yes	—
38	C₂²×#88322	—	2	2	no	—
39	C₂²×#88322	C₂×C₄	1	2	yes	—
40	C₂²×#88322	C₂×C₄	1	2	yes	—
41	C₄⋊#88322	—	2	2	no	—
42	C₄⋊#88322	C₂³	1	4	yes	—
43	C₂²⋊#88322	—	2	2	no	—
44	C₂²⋊#88322	—	2	2	no	—
45	#13133128	C₂×C₄	1	4	yes	—
46	#13133128	C₂×C₄	1	4	yes	—
47	C₄×#88322	—	2	2	no	—
48	C₄×#88322	—	2	2	no	—
49	C₄×#88322	C₂×C₄	1	4	yes	—
50	C₄×#88322	C₂³	1	4	yes	—
51	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
53	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	—	2	2	no	—
55	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₅₃	C₂×C₄	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	D₄	1	1	yes	characteristic
61	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
62	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
63	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
64	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₅₃	—	2	2	no	—
68	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
69	C₂²⋊C₈×C₅₃	—	2	2	no	—
70	C₂²⋊C₈×C₅₃	—	2	2	no	—
71	C₄×C₈×C₅₃	—	2	2	no	—
72	C₄×C₈×C₅₃	—	2	2	no	—
73	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	—	4	2	no	—
75	C₂²×C₄×C₅₃	—	4	2	no	—
76	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
86	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
87	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
96	C₂×C₈×C₅₃	—	2	4	no	—
97	C₂×C₈×C₅₃	—	4	4	no	—
98	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
99	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
100	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
101	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
102	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
103	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
104	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
111	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
112	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
113	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
114	#11719917	—	2	4	no	—
115	#11719917	—	2	4	no	—
116	#11719917	C₂×C₈	1	8	yes	complemented
117	#11719917	C₂×C₈	1	8	yes	complemented
118	C₂×#88322	—	2	4	no	—
119	C₂×#88322	—	2	4	no	—
120	C₂×#88322	—	4	4	no	—
121	C₂×#88322	C₂×C₈	1	8	yes	complemented
122	C₂×#88322	C₂²×C₄	1	4	yes	—
123	C₂×#88322	C₂²×C₄	1	4	yes	—
124	C₂³×C₅₃	—	4	1	no	—
125	C₂³×C₅₃	—	4	1	no	—
126	C₂³×C₅₃	C₄⋊C₈	1	4	yes	—
127	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	—	2	2	no	—
132	C₂×C₄×C₅₃	—	2	2	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	—	4	1	no	—
137	C₂×C₄×C₅₃	—	4	1	no	—
138	C₂×C₄×C₅₃	—	4	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	C₄⋊C₈	1	4	yes	—
143	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
144	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
145	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
155	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
156	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
157	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
158	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
159	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
160	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
163	C₈×C₅₃	—	4	4	no	—
164	#88322	—	2	4	no	—
165	#88322	—	4	4	no	—
166	#88322	C₂²×C₈	1	8	yes	—
167	#17787	—	53	1	no	maximal, Sylow
168	C₂²×C₅₃	—	2	1	no	—
169	C₂²×C₅₃	—	2	1	no	—
170	C₂²×C₅₃	—	2	1	no	—
171	C₂²×C₅₃	—	2	1	no	—
172	C₂²×C₅₃	—	2	1	no	—
173	C₂²×C₅₃	—	2	1	no	—
174	C₂²×C₅₃	—	4	4	no	—
175	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
176	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
177	C₂²×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
178	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
179	C₄×C₅₃	—	2	1	no	—
180	C₄×C₅₃	—	2	1	no	—
181	C₄×C₅₃	—	2	1	no	—
182	C₄×C₅₃	—	2	1	no	—
183	C₄×C₅₃	—	2	1	no	—
184	C₄×C₅₃	—	2	1	no	—
185	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
186	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
187	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
188	C₄×C₅₃	#430	1	2	yes	—
189	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
190	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
191	C₄×C₅₃	#575	1	2	yes	—
192	C₂×#431	—	53	1	no	—
193	C₂×#431	D₅₃	1	1	yes	characteristic
194	C₂×C₄×M₁₆	—	53	1	no	—
195	#1565	—	53	4	no	—
196	#1565	—	53	4	no	—
197	#1564	—	53	4	no	—
198	C₂×C₅₃	—	2	2	no	—
199	C₂×C₅₃	—	2	2	no	—
200	C₂×C₅₃	—	2	2	no	—
201	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
202	C₂×C₅₃	#2911	1	1	yes	characteristic
203	C₂×C₅₃	C₂×#430	1	1	yes	characteristic
204	C₂²×M₁₆	—	53	1	no	—
205	C₂²×M₁₆	—	53	1	no	—
206	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
207	#431	—	2	1	no	—
208	#431	—	2	1	no	—
209	#431	—	53	2	no	—
210	#431	—	53	2	no	—
211	#431	—	106	1	no	—
212	#431	—	106	1	no	—
213	#431	C₂×D₅₃	1	2	yes	—
214	#431	C₂×D₅₃	1	2	yes	—
215	C₂×C₄⋊C₈	—	53	2	no	—
216	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
217	C₂×C₂²⋊C₈	—	2	1	no	—
218	C₂×C₂²⋊C₈	—	106	1	no	—
219	C₄×M₁₆	—	53	2	no	—
220	C₄×M₁₆	—	53	2	no	—
221	C₄×M₁₆	—	53	2	no	—
222	C₄×M₁₆	—	53	2	no	—
223	C₂×#97	—	53	2	no	—
224	C₂×C₄×C₈	—	2	1	no	—
225	C₂×C₄×C₈	—	53	2	no	—
226	C₂×C₄×C₈	—	106	1	no	—
227	#335	—	53	8	no	—
228	#334	—	53	8	no	—
229	C₅₃	#17787	1	1	yes	characteristic, complemented, Sylow
230	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
231	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
232	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
233	C₂×M₁₆	—	53	2	no	—
234	C₂×M₁₆	—	53	2	no	—
235	C₂×M₁₆	—	53	8	no	—
236	C₂×M₁₆	—	53	2	no	—
237	C₂×M₁₆	—	53	2	no	—
238	C₂×M₁₆	—	106	4	no	—
239	C₂²×C₈	—	2	2	no	—
240	C₂²×C₈	—	53	2	no	—
241	C₂²×C₈	—	53	2	no	—
242	C₂²×C₈	—	106	2	no	—
243	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
244	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
245	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
246	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
247	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
248	C₂×C₄²	C₄×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	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
254	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
255	C₄⋊C₈	—	2	2	no	—
256	C₄⋊C₈	—	53	4	no	—
257	C₄⋊C₈	—	106	2	no	—
258	C₄⋊C₈	C₂²×D₅₃	1	4	yes	—
259	C₂²⋊C₈	—	2	2	no	—
260	C₂²⋊C₈	—	2	2	no	—
261	C₂²⋊C₈	—	106	2	no	—
262	C₂²⋊C₈	—	106	2	no	—
263	#97	—	53	4	no	—
264	#97	—	53	4	no	—
265	C₄×C₈	—	2	2	no	—
266	C₄×C₈	—	2	2	no	—
267	C₄×C₈	—	53	4	no	—
268	C₄×C₈	—	53	4	no	—
269	C₄×C₈	—	106	2	no	—
270	C₄×C₈	—	106	2	no	—
271	C₂⁴	#11719919	1	1	yes	characteristic
272	C₂²×C₄	—	4	2	no	—
273	C₂²×C₄	—	4	2	no	—
274	C₂²×C₄	C₈×D₅₃	1	4	yes	—
275	C₂²×C₄	#11719912	1	4	yes	—
276	C₂²×C₄	#11719919	1	1	yes	characteristic
277	C₂²×C₄	#11719919	1	1	yes	characteristic
278	C₂²×C₄	#11719919	1	1	yes	characteristic
279	C₂²×C₄	#11719919	1	1	yes	characteristic
280	C₂²×C₄	#11719919	1	1	yes	characteristic
281	C₂²×C₄	#11719919	1	1	yes	characteristic
282	C₂²×C₄	#11719919	1	1	yes	characteristic
283	C₂²×C₄	#11719919	1	1	yes	characteristic
284	C₂²×C₄	#11719919	1	1	yes	characteristic
285	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
286	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
287	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
288	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
289	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
290	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
291	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
292	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
293	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
294	M₁₆	—	53	8	no	—
295	M₁₆	—	53	8	no	—
296	M₁₆	—	106	4	no	—
297	M₁₆	—	106	4	no	—
298	C₂×C₈	—	2	4	no	—
299	C₂×C₈	—	4	4	no	—
300	C₂×C₈	—	53	8	no	—
301	C₂×C₈	—	53	4	no	—
302	C₂×C₈	—	53	4	no	—
303	C₂×C₈	—	106	4	no	—
304	C₂×C₈	—	106	4	no	—
305	C₂×C₈	—	212	4	no	—
306	C₄²	#11719919	1	1	yes	characteristic
307	C₄²	#11719919	1	1	yes	characteristic
308	C₄²	#11719919	1	1	yes	characteristic
309	C₄²	#11719919	1	1	yes	characteristic
310	C₄²	#11719919	1	1	yes	characteristic
311	C₄²	#11719919	1	1	yes	characteristic
312	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
313	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
314	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
315	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
316	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
317	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
318	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
319	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
320	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
321	C₄²	C₂³×D₅₃	1	1	yes	characteristic
322	C₂³	—	4	1	no	—
323	C₂³	—	4	1	no	—
324	C₂³	#13133146	1	4	yes	—
325	C₂³	C₂×#11719919	1	1	yes	characteristic
326	C₂³	C₂×#11719919	1	1	yes	characteristic
327	C₂³	C₂×#11719919	1	1	yes	characteristic
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	#13133146	1	4	yes	—
341	C₂×C₄	#13133146	1	4	yes	—
342	C₂×C₄	#13133146	1	4	yes	—
343	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
344	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
345	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
346	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
347	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
348	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
349	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
350	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
351	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
352	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
353	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
354	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
355	C₂×C₄	C₂×#11719912	1	2	yes	—
356	C₂×C₄	C₂×#11719912	1	2	yes	—
357	C₂×C₄	C₂×#11719912	1	2	yes	—
358	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
359	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
360	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
361	C₈	—	4	4	no	—
362	C₈	—	53	8	no	—
363	C₈	—	106	4	no	—
364	C₈	—	212	4	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	2	1	no	—
369	C₂²	—	2	1	no	—
370	C₂²	—	2	1	no	—
371	C₂²	—	4	4	no	—
372	C₂²	C₂×#13133146	1	2	yes	—
373	C₂²	C₂×#13133146	1	2	yes	—
374	C₂²	C₂×#13133146	1	2	yes	—
375	C₂²	C₂²×#11719919	1	1	yes	characteristic
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	2	1	no	—
382	C₄	C₂×#13133146	1	2	yes	—
383	C₄	C₂×#13133146	1	2	yes	—
384	C₄	C₂×#13133146	1	2	yes	—
385	C₄	#16810671	1	2	yes	—
386	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
387	C₄	C₂²×#11719912	1	1	yes	characteristic
388	C₄	#16811421	1	2	yes	—
389	C₂	—	2	2	no	—
390	C₂	—	2	2	no	—
391	C₂	—	2	2	no	—
392	C₂	C₂²×#13133146	1	1	yes	characteristic
393	C₂	#20283304	1	1	yes	characteristic
394	C₂	C₂×#16810671	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#431	C₂
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×#431	D₅₃
hypocenter	C₅₃	#17787
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #20273468	C₂	semidirect
4. #20273465	C₂	semidirect
5. #20273461	C₂	semidirect
22. #16810375	C₂²	semidirect
23. #16810373	C₂²	semidirect
33. C₂×#11719917	C₈	semidirect
116. #11719917	C₂×C₈	semidirect
117. #11719917	C₂×C₈	semidirect
121. C₂×#88322	C₂×C₈	semidirect
229. C₅₃	#17787	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810676	2
C₂×C₄×#11719917	2
#20273468	2
#20273465	2
#20273461	2
C₂×C₅₃×#431	2
#17787	53