Subgroups — #27349739

Order 13568.

772 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20283348	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810676	C₂	1	1	yes	maximal, characteristic
3	#20273015	C₂	1	4	yes	maximal, complemented
4	#20273007	C₂	1	2	yes	maximal, complemented
5	#20273005	C₂	1	2	yes	maximal, complemented
6	#20272998	C₂	1	2	yes	maximal, complemented
7	#20272996	C₂	1	2	yes	maximal, complemented
8	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
9	#16810680	C₄	1	2	yes	—
10	#16810680	C₄	1	2	yes	—
11	#16810676	C₄	1	2	yes	—
12	#16810676	C₄	1	2	yes	—
13	#16810676	C₂²	1	1	yes	characteristic
14	#16810676	C₂²	1	2	yes	—
15	#16810676	C₂²	1	1	yes	characteristic
16	#16810676	C₂²	1	2	yes	—
17	#16810676	C₂²	1	2	yes	—
18	#16810647	C₄	1	2	yes	—
19	C₂×C₄⋊#88322	C₄	1	2	yes	—
20	#16810587	C₄	1	2	yes	—
21	C₂×C₂²⋊#88322	C₄	1	2	yes	—
22	C₄×#11719917	C₂²	1	2	yes	—
23	C₂×C₄×#88322	C₂²	1	2	yes	—
24	#16810345	C₂²	1	4	yes	complemented
25	#16810345	C₂²	1	4	yes	complemented
26	#16810344	C₂²	1	4	yes	complemented
27	C₈×#88322	C₂²	1	4	yes	complemented
28	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
29	C₅₃×#431	C₄	1	2	yes	—
30	C₅₃×#431	C₄	1	2	yes	—
31	C₅₃×#431	C₂²	1	1	yes	characteristic
32	C₅₃×#431	C₂²	1	1	yes	characteristic
33	C₅₃×#431	C₂²	1	2	yes	—
34	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
35	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
36	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
37	C₂×#11719917	C₂×C₄	1	4	yes	—
38	C₂²×#88322	C₂×C₄	1	4	yes	—
39	C₄⋊#88322	C₂×C₄	1	2	yes	—
40	C₄⋊#88322	C₂×C₄	1	2	yes	—
41	C₄⋊#88322	C₂×C₄	1	4	yes	—
42	C₄⋊#88322	C₂×C₄	1	4	yes	—
43	C₄⋊#88322	C₂×C₄	1	4	yes	—
44	C₂²⋊#88322	C₂×C₄	1	2	yes	—
45	C₂²⋊#88322	C₂×C₄	1	2	yes	—
46	C₂²⋊#88322	C₂×C₄	1	4	yes	—
47	C₂²⋊#88322	C₂×C₄	1	4	yes	—
48	C₂²⋊#88322	C₂×C₄	1	4	yes	—
49	#13133128	C₂×C₄	1	4	yes	—
50	C₄×#88322	C₂×C₄	1	4	yes	—
51	C₄×#88322	C₂³	1	2	yes	—
52	C₄×#88322	C₂³	1	2	yes	—
53	C₄×#88322	C₂³	1	4	yes	—
54	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
58	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
70	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
71	C₄⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
72	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
73	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
74	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
75	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
76	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
77	C₄×C₈×C₅₃	C₂³	1	2	yes	—
78	C₄×C₈×C₅₃	C₂³	1	2	yes	—
79	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	—	2	4	no	—
81	C₂²×C₄×C₅₃	—	2	4	no	—
82	C₂²×C₄×C₅₃	—	4	2	no	—
83	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
88	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
89	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
90	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂²×C₄	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	1	yes	characteristic
96	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₅₃	—	2	8	no	—
102	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
103	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
104	C₄²×C₅₃	C₄²	1	1	yes	characteristic
105	C₄²×C₅₃	C₄²	1	1	yes	characteristic
106	C₄²×C₅₃	C₄²	1	1	yes	characteristic
107	C₄²×C₅₃	C₄²	1	1	yes	characteristic
108	C₄²×C₅₃	C₄²	1	1	yes	characteristic
109	C₄²×C₅₃	C₄²	1	1	yes	characteristic
110	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
115	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
117	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
118	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
119	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
120	#11719917	—	2	8	no	—
121	C₂×#88322	—	4	4	no	—
122	C₂×#88322	C₂²×C₄	1	4	yes	—
123	C₂×#88322	C₂²×C₄	1	4	yes	—
124	C₂×#88322	C₂²×C₄	1	8	yes	—
125	C₂³×C₅₃	—	2	2	no	—
126	C₂³×C₅₃	—	2	2	no	—
127	C₂³×C₅₃	—	4	1	no	—
128	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
129	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
130	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
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	—	2	2	no	—
137	C₂×C₄×C₅₃	—	2	2	no	—
138	C₂×C₄×C₅₃	—	2	2	no	—
139	C₂×C₄×C₅₃	—	2	2	no	—
140	C₂×C₄×C₅₃	—	2	2	no	—
141	C₂×C₄×C₅₃	—	2	2	no	—
142	C₂×C₄×C₅₃	—	2	2	no	—
143	C₂×C₄×C₅₃	—	2	2	no	—
144	C₂×C₄×C₅₃	—	4	1	no	—
145	C₂×C₄×C₅₃	—	4	1	no	—
146	C₂×C₄×C₅₃	—	4	1	no	—
147	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
150	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
156	C₂×C₄×C₅₃	#131	1	4	yes	—
157	C₂×C₄×C₅₃	#131	1	4	yes	—
158	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
161	C₈×C₅₃	—	2	4	no	—
162	C₈×C₅₃	—	2	4	no	—
163	#88322	—	2	8	no	—
164	#88322	—	2	4	no	—
165	#88322	—	2	4	no	—
166	#17326	—	53	1	no	maximal, Sylow
167	C₂²×C₅₃	—	2	1	no	—
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	#405	1	2	yes	—
176	C₂²×C₅₃	#405	1	2	yes	—
177	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
178	C₄×C₅₃	—	2	1	no	—
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	#405	1	2	yes	—
186	C₄×C₅₃	#405	1	2	yes	—
187	C₄×C₅₃	C₂×#131	1	2	yes	—
188	C₄×C₅₃	C₂×#131	1	2	yes	—
189	C₄×C₅₃	#568	1	2	yes	—
190	#2920	—	53	1	no	—
191	C₂×#431	—	53	1	no	—
192	C₂×#431	D₅₃	1	1	yes	characteristic
193	#1454	—	53	4	no	—
194	#1452	—	53	2	no	—
195	#1451	—	53	2	no	—
196	#1450	—	53	2	no	—
197	C₈×M₁₆	—	53	2	no	—
198	C₂×C₅₃	—	2	2	no	—
199	C₂×C₅₃	—	2	2	no	—
200	C₂×C₅₃	—	4	1	no	—
201	C₂×C₅₃	C₂×#405	1	1	yes	characteristic
202	C₂×C₅₃	C₂×#405	1	1	yes	characteristic
203	C₂×C₅₃	#2873	1	1	yes	characteristic
204	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
205	#432	—	53	2	no	—
206	#432	—	53	2	no	—
207	#431	—	53	2	no	—
208	#431	—	53	2	no	—
209	#431	—	53	1	no	—
210	#431	—	53	1	no	—
211	#431	—	53	2	no	—
212	#431	—	53	2	no	—
213	#431	—	53	2	no	—
214	#431	Dic₅₃	1	2	yes	—
215	#431	Dic₅₃	1	2	yes	—
216	#431	C₂×D₅₃	1	1	yes	characteristic
217	#431	C₂×D₅₃	1	1	yes	characteristic
218	#431	C₂×D₅₃	1	2	yes	—
219	#423	—	53	2	no	—
220	C₂×C₄⋊C₈	—	53	2	no	—
221	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	—
222	#407	—	53	2	no	—
223	C₂×C₂²⋊C₈	—	53	2	no	—
224	C₂×C₂²⋊C₈	Dic₅₃	1	2	yes	—
225	C₄×M₁₆	—	53	2	no	—
226	C₂×C₄×C₈	—	53	2	no	—
227	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
228	#322	—	53	4	no	—
229	#322	—	53	4	no	—
230	#322	—	53	4	no	—
231	C₈²	—	53	4	no	—
232	C₅₃	#17326	1	1	yes	characteristic, complemented, Sylow
233	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
234	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
235	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
236	C₂×M₁₆	—	53	4	no	—
237	C₂²×C₈	—	53	4	no	—
238	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
239	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
240	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
241	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
242	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
243	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
244	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
245	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
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	C₄⋊C₈	—	53	2	no	—
252	C₄⋊C₈	—	53	2	no	—
253	C₄⋊C₈	—	53	4	no	—
254	C₄⋊C₈	—	53	4	no	—
255	C₄⋊C₈	—	53	4	no	—
256	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
257	C₄⋊C₈	C₂×Dic₅₃	1	2	yes	—
258	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
259	C₂²⋊C₈	—	53	2	no	—
260	C₂²⋊C₈	—	53	2	no	—
261	C₂²⋊C₈	—	53	4	no	—
262	C₂²⋊C₈	—	53	4	no	—
263	C₂²⋊C₈	—	53	4	no	—
264	C₂²⋊C₈	C₂×Dic₅₃	1	2	yes	—
265	C₂²⋊C₈	C₂×Dic₅₃	1	2	yes	—
266	C₂²⋊C₈	C₂×Dic₅₃	1	4	yes	—
267	#97	—	53	4	no	—
268	C₄×C₈	—	53	4	no	—
269	C₄×C₈	—	53	2	no	—
270	C₄×C₈	—	53	4	no	—
271	C₄×C₈	—	53	2	no	—
272	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
273	C₄×C₈	C₂²×D₅₃	1	2	yes	—
274	C₄×C₈	C₂²×D₅₃	1	2	yes	—
275	C₂⁴	C₄×Dic₅₃	1	1	yes	characteristic
276	C₂²×C₄	—	2	4	no	—
277	C₂²×C₄	—	2	4	no	—
278	C₂²×C₄	—	4	2	no	—
279	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
280	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
281	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
282	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
283	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
284	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
285	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
286	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
287	C₂²×C₄	C₄×Dic₅₃	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×C₄×D₅₃	1	1	yes	characteristic
292	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
293	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
294	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
295	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
296	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
297	M₁₆	—	106	8	no	—
298	C₂×C₈	—	2	8	no	—
299	C₂×C₈	—	53	4	no	—
300	C₂×C₈	—	53	4	no	—
301	C₂×C₈	—	53	8	no	—
302	C₂×C₈	—	212	4	no	—
303	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
304	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
305	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
306	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
307	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
308	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
309	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
310	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
311	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
312	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
313	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
314	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
315	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
316	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
317	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
318	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
319	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
320	C₄²	C₂³×D₅₃	1	1	yes	characteristic
321	C₂³	—	2	2	no	—
322	C₂³	—	2	2	no	—
323	C₂³	—	4	1	no	—
324	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
325	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
326	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
327	C₂×C₄	—	2	2	no	—
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	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	4	1	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
344	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
345	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
346	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
347	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
348	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
349	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
350	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
351	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
352	C₂×C₄	#13133223	1	4	yes	—
353	C₂×C₄	#13133224	1	4	yes	—
354	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
355	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
356	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
357	C₈	—	2	4	no	—
358	C₈	—	2	4	no	—
359	C₈	—	106	8	no	—
360	C₈	—	106	4	no	—
361	C₈	—	106	4	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	2	1	no	—
369	C₂²	—	4	4	no	—
370	C₂²	#16810578	1	2	yes	—
371	C₂²	#16810579	1	2	yes	—
372	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	#16810578	1	2	yes	—
381	C₄	#16810579	1	2	yes	—
382	C₄	C₂×#13133223	1	2	yes	—
383	C₄	C₂×#13133224	1	2	yes	—
384	C₄	#16811389	1	2	yes	—
385	C₂	—	2	2	no	—
386	C₂	—	2	2	no	—
387	C₂	—	4	1	no	—
388	C₂	C₂×#16810578	1	1	yes	characteristic
389	C₂	C₂×#16810579	1	1	yes	characteristic
390	C₂	#20283133	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₄²
Fitting	C₂×C₅₃×#431	C₂
prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂×#431	D₅₃
hypocenter	C₅₃	#17326
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #20273015	C₂	semidirect
4. #20273007	C₂	semidirect
5. #20273005	C₂	semidirect
6. #20272998	C₂	semidirect
7. #20272996	C₂	semidirect
24. #16810345	C₂²	semidirect
25. #16810345	C₂²	semidirect
26. #16810344	C₂²	semidirect
27. C₈×#88322	C₂²	semidirect
232. C₅₃	#17326	semidirect

Maximal subgroups

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

subgroup	index
#20283348	2
C₂×#16810676	2
#20273015	2
#20273007	2
#20273005	2
#20272998	2
#20272996	2
C₂×C₅₃×#431	2
#17326	53