Subgroups — #27349727

Order 13568.

742 conjugacy classes of subgroups.

The subgroup lattice has 378 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	#20273010	C₂	1	4	yes	maximal, complemented
5	#20273002	C₂	1	2	yes	maximal, complemented
6	M₁₆×#88322	C₂	1	2	yes	maximal, complemented
7	C₅₃×#2920	C₂	1	1	yes	maximal, characteristic
8	#16810680	C₄	1	2	yes	—
9	#16810680	C₄	1	2	yes	—
10	#16810676	C₄	1	2	yes	—
11	#16810676	C₄	1	2	yes	—
12	#16810676	C₂²	1	2	yes	—
13	#16810676	C₂²	1	2	yes	—
14	#16810676	C₂²	1	2	yes	—
15	#16810676	C₂²	1	2	yes	—
16	#16810647	C₄	1	2	yes	—
17	C₂×C₄⋊#88322	C₄	1	2	yes	—
18	#16810587	C₄	1	2	yes	—
19	C₂×C₂²⋊#88322	C₄	1	2	yes	—
20	C₄×#11719917	C₂²	1	2	yes	—
21	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
22	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
23	#16810345	C₂²	1	8	yes	complemented
24	#16810344	C₂²	1	4	yes	complemented
25	C₈×#88322	C₂²	1	4	yes	complemented
26	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
27	C₅₃×#432	C₄	1	2	yes	—
28	C₅₃×#432	C₄	1	2	yes	—
29	C₅₃×#431	C₂²	1	2	yes	—
30	C₅₃×#431	C₂²	1	2	yes	—
31	C₅₃×#423	C₄	1	2	yes	—
32	C₅₃×#407	C₄	1	2	yes	—
33	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
34	C₂×#11719917	C₂×C₄	1	4	yes	—
35	C₂²×#88322	C₂×C₄	1	2	yes	—
36	C₂²×#88322	C₂×C₄	1	2	yes	—
37	C₄⋊#88322	C₂×C₄	1	4	yes	—
38	C₄⋊#88322	C₂×C₄	1	4	yes	—
39	C₄⋊#88322	C₂×C₄	1	4	yes	—
40	C₄⋊#88322	C₂×C₄	1	4	yes	—
41	C₂²⋊#88322	C₂×C₄	1	4	yes	—
42	C₂²⋊#88322	C₂×C₄	1	4	yes	—
43	C₂²⋊#88322	C₂×C₄	1	4	yes	—
44	C₂²⋊#88322	C₂×C₄	1	4	yes	—
45	#13133128	C₂×C₄	1	4	yes	—
46	C₄×#88322	C₂×C₄	1	2	yes	—
47	C₄×#88322	C₂×C₄	1	2	yes	—
48	C₄×#88322	C₂³	1	2	yes	—
49	C₄×#88322	C₂³	1	2	yes	—
50	C₄×#88322	C₂³	1	4	yes	—
51	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
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	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	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	C₂×C₄	1	4	yes	—
68	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
69	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
70	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
71	#97×C₅₃	C₂×C₄	1	4	yes	—
72	C₄×C₈×C₅₃	C₂³	1	4	yes	—
73	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	—	2	4	no	—
75	C₂²×C₄×C₅₃	—	4	2	no	—
76	C₂²×C₄×C₅₃	—	4	2	no	—
77	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
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	C₂²×C₄	1	1	yes	characteristic
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	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₅₃	C₂²×C₄	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	M₁₆×C₅₃	—	2	8	no	—
96	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
97	C₄²×C₅₃	C₄²	1	1	yes	characteristic
98	C₄²×C₅₃	C₄²	1	1	yes	characteristic
99	C₄²×C₅₃	C₄²	1	1	yes	characteristic
100	C₄²×C₅₃	C₄²	1	1	yes	characteristic
101	C₄²×C₅₃	C₄²	1	1	yes	characteristic
102	C₄²×C₅₃	C₄²	1	1	yes	characteristic
103	C₄²×C₅₃	C₂²×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	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₅₃	C₂²×C₄	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	C₂⁴	1	1	yes	characteristic
113	#11719917	—	4	4	no	—
114	C₂×#88322	—	4	2	no	—
115	C₂×#88322	—	4	2	no	—
116	C₂×#88322	C₂²×C₄	1	4	yes	—
117	C₂×#88322	C₂²×C₄	1	4	yes	—
118	C₂×#88322	C₂²×C₄	1	8	yes	—
119	C₂³×C₅₃	—	2	2	no	—
120	C₂³×C₅₃	—	4	1	no	—
121	C₂³×C₅₃	—	4	1	no	—
122	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
123	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
124	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	2	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
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	C₂×C₄²	1	1	yes	characteristic
143	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
144	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
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	#131	1	4	yes	—
152	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
155	C₈×C₅₃	—	2	8	no	—
156	#88322	—	2	4	no	—
157	#88322	—	2	4	no	—
158	#88322	—	2	8	no	—
159	#17324	—	53	1	no	maximal, Sylow
160	C₂²×C₅₃	—	2	1	no	—
161	C₂²×C₅₃	—	2	1	no	—
162	C₂²×C₅₃	—	2	1	no	—
163	C₂²×C₅₃	—	2	1	no	—
164	C₂²×C₅₃	—	2	1	no	—
165	C₂²×C₅₃	—	2	1	no	—
166	C₂²×C₅₃	—	2	1	no	—
167	C₂²×C₅₃	—	2	1	no	—
168	C₂²×C₅₃	—	4	4	no	—
169	C₂²×C₅₃	#405	1	2	yes	—
170	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
171	C₄×C₅₃	—	2	1	no	—
172	C₄×C₅₃	—	2	1	no	—
173	C₄×C₅₃	—	2	1	no	—
174	C₄×C₅₃	—	2	1	no	—
175	C₄×C₅₃	—	2	1	no	—
176	C₄×C₅₃	—	2	1	no	—
177	C₄×C₅₃	—	2	1	no	—
178	C₄×C₅₃	—	2	1	no	—
179	C₄×C₅₃	#405	1	2	yes	—
180	C₄×C₅₃	C₂×#131	1	2	yes	—
181	C₄×C₅₃	#568	1	2	yes	—
182	C₄×C₅₃	#568	1	2	yes	—
183	#2920	—	53	1	no	—
184	#2920	D₅₃	1	1	yes	characteristic
185	C₂×#431	—	53	1	no	—
186	#1454	—	53	4	no	—
187	#1453	—	53	4	no	—
188	#1451	—	53	2	no	—
189	C₈×M₁₆	—	53	2	no	—
190	C₂×C₅₃	—	2	2	no	—
191	C₂×C₅₃	—	4	1	no	—
192	C₂×C₅₃	—	4	1	no	—
193	C₂×C₅₃	C₂×#405	1	1	yes	characteristic
194	C₂×C₅₃	#2873	1	1	yes	characteristic
195	C₂×C₅₃	#2873	1	1	yes	characteristic
196	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
197	#432	—	53	2	no	—
198	#432	—	53	2	no	—
199	#432	Dic₅₃	1	2	yes	—
200	#432	Dic₅₃	1	2	yes	—
201	#431	—	53	2	no	—
202	#431	—	53	2	no	—
203	#431	—	53	2	no	—
204	#431	—	53	2	no	—
205	#431	—	53	2	no	—
206	#431	—	53	2	no	—
207	#431	C₂×D₅₃	1	2	yes	—
208	#431	C₂×D₅₃	1	2	yes	—
209	#423	—	53	2	no	—
210	#423	Dic₅₃	1	2	yes	—
211	C₂×C₄⋊C₈	—	53	2	no	—
212	#407	—	53	2	no	—
213	#407	Dic₅₃	1	2	yes	—
214	C₂×C₂²⋊C₈	—	53	2	no	—
215	C₄×M₁₆	—	53	2	no	—
216	C₄×M₁₆	C₂×D₅₃	1	2	yes	—
217	C₂×C₄×C₈	—	53	1	no	—
218	C₂×C₄×C₈	—	53	1	no	—
219	#322	—	53	4	no	—
220	#322	—	53	8	no	—
221	C₈²	—	53	4	no	—
222	C₅₃	#17324	1	1	yes	characteristic, complemented, Sylow
223	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
224	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
225	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
226	C₂×M₁₆	—	53	4	no	—
227	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
228	C₂²×C₈	—	53	2	no	—
229	C₂²×C₈	—	53	2	no	—
230	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
231	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
232	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
233	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
234	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
235	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
236	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
237	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
238	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
239	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
240	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
241	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
242	C₄⋊C₈	—	53	4	no	—
243	C₄⋊C₈	—	53	4	no	—
244	C₄⋊C₈	—	53	4	no	—
245	C₄⋊C₈	—	53	4	no	—
246	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
247	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
248	C₂²⋊C₈	—	53	4	no	—
249	C₂²⋊C₈	—	53	4	no	—
250	C₂²⋊C₈	—	53	4	no	—
251	C₂²⋊C₈	—	53	4	no	—
252	C₂²⋊C₈	C₂×Dic₅₃	1	4	yes	—
253	C₂²⋊C₈	C₂×Dic₅₃	1	4	yes	—
254	#97	—	53	4	no	—
255	#97	C₂×Dic₅₃	1	4	yes	—
256	C₄×C₈	—	53	2	no	—
257	C₄×C₈	—	53	2	no	—
258	C₄×C₈	—	53	2	no	—
259	C₄×C₈	—	53	4	no	—
260	C₄×C₈	—	53	2	no	—
261	C₄×C₈	C₂²×D₅₃	1	4	yes	—
262	C₂⁴	C₄×Dic₅₃	1	1	yes	characteristic
263	C₂²×C₄	—	2	4	no	—
264	C₂²×C₄	—	4	2	no	—
265	C₂²×C₄	—	4	2	no	—
266	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
267	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
268	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
269	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
270	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
271	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
272	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
273	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
274	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
275	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
276	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
277	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
278	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
279	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
280	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
281	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
282	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
283	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
284	M₁₆	—	2	8	no	—
285	M₁₆	—	212	4	no	—
286	C₂×C₈	—	53	4	no	—
287	C₂×C₈	—	53	4	no	—
288	C₂×C₈	—	53	8	no	—
289	C₂×C₈	—	212	2	no	—
290	C₂×C₈	—	212	2	no	—
291	C₂×C₈	C₂²×Dic₅₃	1	8	yes	—
292	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
293	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
294	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
295	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
296	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
297	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
298	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
299	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
300	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
301	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
302	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
303	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
304	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
305	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
306	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
307	C₄²	C₂³×D₅₃	1	1	yes	characteristic
308	C₂³	—	2	2	no	—
309	C₂³	—	4	1	no	—
310	C₂³	—	4	1	no	—
311	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
312	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
313	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
332	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
333	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
334	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
335	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
336	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
337	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
338	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
339	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
340	C₂×C₄	#13133224	1	4	yes	—
341	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
342	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
343	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
344	C₈	—	2	8	no	—
345	C₈	—	106	4	no	—
346	C₈	—	106	4	no	—
347	C₈	—	106	8	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	4	4	no	—
357	C₂²	#16810579	1	2	yes	—
358	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	#16810579	1	2	yes	—
368	C₄	C₂×#13133224	1	2	yes	—
369	C₄	#16811389	1	2	yes	—
370	C₄	#16811389	1	2	yes	—
371	C₂	—	2	2	no	—
372	C₂	—	4	1	no	—
373	C₂	—	4	1	no	—
374	C₂	C₂×#16810579	1	1	yes	characteristic
375	C₂	#20283133	1	1	yes	characteristic
376	C₂	#20283133	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₄²
Fitting	C₅₃×#2920	C₂
prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	#2920	D₅₃
hypocenter	C₅₃	#17324
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ #2920

Complements

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

N	H ≅ G/N	type
3. #20273015	C₂	semidirect
4. #20273010	C₂	semidirect
5. #20273002	C₂	semidirect
6. M₁₆×#88322	C₂	semidirect
23. #16810345	C₂²	semidirect
24. #16810344	C₂²	semidirect
25. C₈×#88322	C₂²	semidirect
222. C₅₃	#17324	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
#20273010	2
#20273002	2
M₁₆×#88322	2
C₅₃×#2920	2
#17324	53