Subgroups — #27349700

Order 13568.

834 conjugacy classes of subgroups.

The subgroup lattice has 366 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	#20273008	C₂	1	4	yes	maximal, complemented
4	#20273005	C₂	1	4	yes	maximal, complemented
5	#20273004	C₂	1	4	yes	maximal, complemented
6	C₂×C₅₃×#431	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719917	C₄	1	2	yes	—
8	#16810676	C₄	1	2	yes	—
9	#16810676	C₄	1	2	yes	—
10	#16810676	C₂²	1	2	yes	—
11	#16810676	C₂²	1	2	yes	—
12	C₂×C₄⋊#88322	C₄	1	2	yes	—
13	C₂×C₂²⋊#88322	C₄	1	2	yes	—
14	C₄×#11719917	C₄	1	2	yes	—
15	C₄×#11719917	C₄	1	2	yes	—
16	C₄×#11719917	C₂²	1	2	yes	—
17	C₄×#11719917	C₂²	1	2	yes	—
18	C₂×#13133128	C₄	1	2	yes	—
19	C₂×C₄×#88322	C₂²	1	2	yes	—
20	C₂×C₄×#88322	C₂²	1	2	yes	—
21	#16810345	C₂²	1	8	yes	complemented
22	#16810344	C₂²	1	8	yes	complemented
23	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
24	C₅₃×#431	C₄	1	2	yes	—
25	C₅₃×#431	C₄	1	2	yes	—
26	C₅₃×#431	C₂²	1	2	yes	—
27	C₅₃×#431	C₂²	1	2	yes	—
28	C₂×C₄⋊C₈×C₅₃	C₄	1	2	yes	—
29	C₂×C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
30	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
31	C₂×#11719917	—	2	8	no	—
32	C₂×#11719917	C₂×C₄	1	4	yes	—
33	C₂×#11719917	C₂×C₄	1	4	yes	—
34	C₂²×#88322	C₂×C₄	1	4	yes	—
35	C₂²×#88322	C₂×C₄	1	4	yes	—
36	C₄⋊#88322	C₂×C₄	1	4	yes	—
37	C₄⋊#88322	C₂×C₄	1	4	yes	—
38	C₂²⋊#88322	C₂×C₄	1	4	yes	—
39	C₂²⋊#88322	C₂×C₄	1	4	yes	—
40	#13133128	C₂×C₄	1	4	yes	—
41	#13133128	C₂×C₄	1	4	yes	—
42	C₄×#88322	C₂×C₄	1	4	yes	—
43	C₄×#88322	C₂×C₄	1	4	yes	—
44	C₄×#88322	C₂³	1	4	yes	—
45	C₄×#88322	C₂³	1	4	yes	—
46	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
62	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
63	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
64	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
65	C₂²⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
66	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
67	C₄×C₈×C₅₃	C₂³	1	4	yes	—
68	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	—	4	2	no	—
70	C₂²×C₄×C₅₃	—	4	2	no	—
71	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
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	M₁₆	1	8	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	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂²×C₄	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	C₂×C₈×C₅₃	—	4	4	no	—
91	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
92	C₄²×C₅₃	C₄²	1	1	yes	characteristic
93	C₄²×C₅₃	C₄²	1	1	yes	characteristic
94	C₄²×C₅₃	C₄²	1	1	yes	characteristic
95	C₄²×C₅₃	C₄²	1	1	yes	characteristic
96	C₄²×C₅₃	C₄²	1	1	yes	characteristic
97	C₄²×C₅₃	C₄²	1	1	yes	characteristic
98	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₅₃	C₂²×C₄	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	C₂⁴	1	1	yes	characteristic
108	#11719917	—	2	8	no	—
109	#11719917	—	2	8	no	—
110	C₂×#88322	—	2	8	no	—
111	C₂×#88322	—	4	4	no	—
112	C₂×#88322	C₂²×C₄	1	8	yes	—
113	C₂×#88322	C₂²×C₄	1	8	yes	—
114	C₂³×C₅₃	—	4	1	no	—
115	C₂³×C₅₃	—	4	1	no	—
116	C₂³×C₅₃	#97	1	4	yes	—
117	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
118	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
119	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₄×C₅₃	—	2	2	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	2	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	4	1	no	—
127	C₂×C₄×C₅₃	—	4	1	no	—
128	C₂×C₄×C₅₃	—	4	1	no	—
129	C₂×C₄×C₅₃	—	4	1	no	—
130	C₂×C₄×C₅₃	—	4	1	no	—
131	C₂×C₄×C₅₃	—	4	1	no	—
132	C₂×C₄×C₅₃	#97	1	4	yes	—
133	C₂×C₄×C₅₃	#97	1	4	yes	—
134	C₂×C₄×C₅₃	#97	1	4	yes	—
135	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
141	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
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×M₁₆	1	4	yes	—
145	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
146	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
147	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
150	C₈×C₅₃	—	2	8	no	—
151	#88322	—	2	8	no	—
152	#88322	—	2	8	no	—
153	#17319	—	53	1	no	maximal, Sylow
154	C₂²×C₅₃	—	2	1	no	—
155	C₂²×C₅₃	—	2	1	no	—
156	C₂²×C₅₃	—	2	1	no	—
157	C₂²×C₅₃	—	2	1	no	—
158	C₂²×C₅₃	—	2	1	no	—
159	C₂²×C₅₃	—	2	1	no	—
160	C₂²×C₅₃	—	4	4	no	—
161	C₂²×C₅₃	C₂×#97	1	2	yes	—
162	C₂²×C₅₃	C₂×#97	1	2	yes	—
163	C₂²×C₅₃	C₂×#97	1	2	yes	—
164	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
165	C₄×C₅₃	—	2	1	no	—
166	C₄×C₅₃	—	2	1	no	—
167	C₄×C₅₃	—	2	1	no	—
168	C₄×C₅₃	—	2	1	no	—
169	C₄×C₅₃	—	2	1	no	—
170	C₄×C₅₃	—	2	1	no	—
171	C₄×C₅₃	C₂×#97	1	2	yes	—
172	C₄×C₅₃	C₂×#97	1	2	yes	—
173	C₄×C₅₃	C₂×#97	1	2	yes	—
174	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
175	C₄×C₅₃	#568	1	2	yes	—
176	C₄×C₅₃	#568	1	2	yes	—
177	C₂×#431	—	53	1	no	—
178	C₂×#431	D₅₃	1	1	yes	characteristic
179	C₂×C₄×M₁₆	—	53	1	no	—
180	#1452	—	53	4	no	—
181	#1451	—	53	4	no	—
182	#1451	—	53	4	no	—
183	C₂×C₅₃	—	2	2	no	—
184	C₂×C₅₃	—	2	2	no	—
185	C₂×C₅₃	—	2	2	no	—
186	C₂×C₅₃	C₂²×#97	1	1	yes	characteristic
187	C₂×C₅₃	#2873	1	1	yes	characteristic
188	C₂×C₅₃	#2873	1	1	yes	characteristic
189	C₂²×M₁₆	—	53	2	no	—
190	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
191	#431	—	53	2	no	—
192	#431	—	53	2	no	—
193	#431	—	53	2	no	—
194	#431	—	53	2	no	—
195	#431	Dic₅₃	1	2	yes	—
196	#431	Dic₅₃	1	2	yes	—
197	#431	C₂×D₅₃	1	2	yes	—
198	#431	C₂×D₅₃	1	2	yes	—
199	C₂×C₄⋊C₈	—	53	2	no	—
200	C₂×C₄⋊C₈	Dic₅₃	1	2	yes	—
201	C₂×C₂²⋊C₈	—	53	2	no	—
202	C₂×C₂²⋊C₈	Dic₅₃	1	2	yes	—
203	C₄×M₁₆	—	53	2	no	—
204	C₄×M₁₆	—	53	2	no	—
205	C₄×M₁₆	—	53	2	no	—
206	C₄×M₁₆	—	53	2	no	—
207	C₂×#97	—	53	2	no	—
208	C₂×C₄×C₈	—	53	2	no	—
209	C₂×C₄×C₈	—	53	2	no	—
210	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
211	#322	—	53	8	no	—
212	#322	—	53	8	no	—
213	C₅₃	#17319	1	1	yes	characteristic, complemented, Sylow
214	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
215	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
216	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
217	C₂×M₁₆	—	53	4	no	—
218	C₂×M₁₆	—	53	4	no	—
219	C₂×M₁₆	—	106	8	no	—
220	C₂²×C₈	—	53	4	no	—
221	C₂²×C₈	—	53	4	no	—
222	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
223	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
224	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
225	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
226	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
227	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
228	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
229	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
230	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
231	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
232	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
233	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
234	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
235	C₄⋊C₈	—	53	4	no	—
236	C₄⋊C₈	—	53	4	no	—
237	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
238	C₄⋊C₈	C₂×Dic₅₃	1	4	yes	—
239	C₂²⋊C₈	—	53	4	no	—
240	C₂²⋊C₈	—	53	4	no	—
241	C₂²⋊C₈	C₂×Dic₅₃	1	4	yes	—
242	C₂²⋊C₈	C₂×Dic₅₃	1	4	yes	—
243	#97	—	53	4	no	—
244	#97	—	53	4	no	—
245	C₄×C₈	—	53	4	no	—
246	C₄×C₈	—	53	4	no	—
247	C₄×C₈	—	53	4	no	—
248	C₄×C₈	—	53	4	no	—
249	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
250	C₄×C₈	C₂²×D₅₃	1	4	yes	—
251	C₂⁴	C₄×Dic₅₃	1	1	yes	characteristic
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	#11719912	1	8	yes	—
255	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
256	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
257	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
258	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
259	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
260	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
261	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
262	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
263	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
266	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
267	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
268	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
269	C₂²×C₄	C₂×C₄×D₅₃	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	C₂²×Dic₅₃	1	1	yes	characteristic
273	M₁₆	—	106	8	no	—
274	M₁₆	—	106	8	no	—
275	C₂×C₈	—	4	4	no	—
276	C₂×C₈	—	53	8	no	—
277	C₂×C₈	—	53	8	no	—
278	C₂×C₈	—	106	8	no	—
279	C₂×C₈	—	212	4	no	—
280	C₂×C₈	C₂²×Dic₅₃	1	8	yes	—
281	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
282	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
283	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
284	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
285	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
286	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
287	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
288	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
289	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
290	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
291	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
292	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
293	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
294	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
295	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
296	C₄²	C₂³×D₅₃	1	1	yes	characteristic
297	C₂³	—	4	1	no	—
298	C₂³	—	4	1	no	—
299	C₂³	#13133127	1	4	yes	—
300	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
301	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
302	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	#13133127	1	4	yes	—
316	C₂×C₄	#13133127	1	4	yes	—
317	C₂×C₄	#13133127	1	4	yes	—
318	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
319	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
320	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
321	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
322	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
323	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
324	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
325	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
326	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
327	C₂×C₄	C₂×#11719912	1	4	yes	—
328	C₂×C₄	C₂×#11719912	1	4	yes	—
329	C₂×C₄	C₂×#11719912	1	4	yes	—
330	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
331	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
332	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
333	C₈	—	2	8	no	—
334	C₈	—	106	8	no	—
335	C₈	—	106	8	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	4	4	no	—
343	C₂²	C₂×#13133127	1	2	yes	—
344	C₂²	C₂×#13133127	1	2	yes	—
345	C₂²	C₂×#13133127	1	2	yes	—
346	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	C₂×#13133127	1	2	yes	—
354	C₄	C₂×#13133127	1	2	yes	—
355	C₄	C₂×#13133127	1	2	yes	—
356	C₄	C₂²×#11719912	1	2	yes	—
357	C₄	#16811389	1	2	yes	—
358	C₄	#16811390	1	2	yes	—
359	C₂	—	2	2	no	—
360	C₂	—	2	2	no	—
361	C₂	—	2	2	no	—
362	C₂	C₂²×#13133127	1	1	yes	characteristic
363	C₂	#20283133	1	1	yes	characteristic
364	C₂	#20283134	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	#17319
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. #20273008	C₂	semidirect
4. #20273005	C₂	semidirect
5. #20273004	C₂	semidirect
21. #16810345	C₂²	semidirect
22. #16810344	C₂²	semidirect
213. C₅₃	#17319	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
#20273008	2
#20273005	2
#20273004	2
C₂×C₅₃×#431	2
#17319	53