Subgroups — #27387402

Order 13568.

1002 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810648	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810648	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810645	C₂	1	1	yes	maximal, characteristic
4	C₂×#16810580	C₂	1	1	yes	maximal, characteristic
5	#20277195	C₂	1	1	yes	maximal, characteristic
6	#20275438	C₂	1	1	yes	maximal, characteristic, complemented
7	#20275437	C₂	1	1	yes	maximal, characteristic, complemented
8	#20275437	C₂	1	1	yes	maximal, characteristic, complemented
9	#20275434	C₂	1	1	yes	maximal, characteristic, complemented
10	#20275431	C₂	1	1	yes	maximal, characteristic, complemented
11	#20275430	C₂	1	1	yes	maximal, characteristic, complemented
12	#20275430	C₂	1	1	yes	maximal, characteristic, complemented
13	#20274410	C₂	1	1	yes	maximal, characteristic
14	#336×D₅₃	C₂	1	1	yes	maximal, characteristic
15	C₅₃×#1940	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×#11719912	C₂²	1	1	yes	characteristic
17	C₂²×#11719912	C₂²	1	1	yes	characteristic
18	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
19	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
20	C₂×#13133174	C₂²	1	1	yes	characteristic
21	C₂×#13133174	C₂²	1	1	yes	characteristic
22	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
23	#16810648	—	2	4	no	—
24	#16810648	—	2	4	no	—
25	#16810645	—	2	4	no	—
26	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
27	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
29	C₂×#13133146	C₂²	1	1	yes	characteristic
30	C₂×#13133146	C₂²	1	1	yes	characteristic
31	C₂×#13133146	C₂²	1	1	yes	characteristic
32	C₂×#13133146	C₂²	1	1	yes	characteristic
33	C₂×#13133146	C₂²	1	1	yes	characteristic
34	C₂×#13133146	C₂²	1	1	yes	characteristic
35	#16810580	—	2	4	no	—
36	C₂×#13133127	C₂²	1	1	yes	characteristic
37	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
38	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
39	#16810520	C₄	1	1	yes	characteristic
40	#16810520	C₄	1	1	yes	characteristic
41	#16810518	C₄	1	1	yes	characteristic
42	#16810518	C₄	1	1	yes	characteristic
43	#16810497	C₄	1	1	yes	characteristic
44	#16810497	C₄	1	1	yes	characteristic
45	#16810494	C₄	1	1	yes	characteristic
46	#16810494	C₄	1	1	yes	characteristic
47	C₄×#11719919	C₂²	1	1	yes	characteristic
48	C₄×#11719919	C₂²	1	1	yes	characteristic
49	C₄×C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
50	#16810378	C₂²	1	1	yes	characteristic
51	#16810378	C₂²	1	1	yes	characteristic
52	#16810377	C₂²	1	1	yes	characteristic
53	#16810377	C₂²	1	1	yes	characteristic
54	#16810377	C₂²	1	1	yes	characteristic
55	#16810377	C₂²	1	1	yes	characteristic
56	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
57	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
58	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
59	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
60	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
61	C₅₃×#336	C₂²	1	1	yes	characteristic
62	C₅₃×#336	C₂²	1	1	yes	characteristic
63	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
66	C₂×#11719912	—	2	2	no	—
67	C₂×#11719912	—	2	2	no	—
68	C₂×#11719912	—	2	2	no	—
69	C₂×#11719912	—	2	2	no	—
70	C₂×#11719912	—	2	2	no	—
71	C₂×#11719912	D₄	1	4	yes	—
72	C₂²×#88322	C₂³	1	1	yes	characteristic
73	C₂²×#88322	C₂³	1	1	yes	characteristic
74	C₂²×#88322	C₂³	1	1	yes	characteristic
75	C₂²×#88322	C₂³	1	1	yes	characteristic
76	C₂×C₈×D₅₃	—	2	2	no	—
77	C₂×C₈×D₅₃	—	2	2	no	—
78	C₂×C₈×D₅₃	—	2	2	no	—
79	C₂×C₈×D₅₃	—	2	2	no	—
80	C₂×C₈×D₅₃	—	2	2	no	—
81	C₂×C₈×D₅₃	Q₈	1	4	yes	—
82	#13133174	—	2	2	no	—
83	#13133174	—	2	2	no	—
84	#13133174	D₄	1	4	yes	—
85	#13133174	Q₈	1	4	yes	—
86	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
87	C₂×C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
88	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
89	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
90	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
91	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
92	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
93	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
94	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
95	C₂×#11719919	C₂×C₄	1	1	yes	characteristic
96	C₂×#11719921	C₂×C₄	1	1	yes	characteristic
97	C₂×#11719921	C₂×C₄	1	1	yes	characteristic
98	C₂×#11719921	C₂×C₄	1	1	yes	characteristic
99	C₂×#11719921	C₂×C₄	1	1	yes	characteristic
100	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
101	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
102	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
103	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
104	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
105	C₄²×D₅₃	—	2	2	no	—
106	C₄²×D₅₃	—	2	2	no	—
107	C₄⋊#88322	—	2	2	no	—
108	C₄⋊#88322	—	2	2	no	—
109	C₄⋊#88322	—	2	2	no	—
110	#13133146	—	2	2	no	—
111	#13133146	—	2	2	no	—
112	#13133146	—	2	2	no	—
113	#13133146	—	2	2	no	—
114	#13133146	—	2	2	no	—
115	#13133146	—	2	2	no	—
116	#13133127	—	2	2	no	—
117	C₄×#88322	—	2	2	no	—
118	C₈×Dic₅₃	—	2	2	no	—
119	#13133124	C₂×C₄	1	1	yes	characteristic
120	#13133124	C₂×C₄	1	1	yes	characteristic
121	#13133124	C₂×C₄	1	1	yes	characteristic
122	#13133124	C₂×C₄	1	1	yes	characteristic
123	#13133124	C₂×C₄	1	1	yes	characteristic
124	#13133124	C₂×C₄	1	1	yes	characteristic
125	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
126	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
127	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
128	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
129	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
130	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
131	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
132	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
133	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
134	C₄⋊C₈×C₅₃	—	2	2	no	—
135	C₄⋊C₈×C₅₃	—	2	2	no	—
136	C₄⋊C₈×C₅₃	—	2	2	no	—
137	#97×C₅₃	—	2	2	no	—
138	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
139	#95×C₅₃	C₂×C₄	1	1	yes	characteristic
140	C₂³×D₅₃	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
143	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
145	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
146	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
147	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
148	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
149	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
150	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
151	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
152	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
153	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
154	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
155	C₂×C₄×D₅₃	—	2	2	no	—
156	C₂×C₄×D₅₃	—	2	2	no	—
157	C₂×C₄×D₅₃	—	2	2	no	—
158	C₂×C₄×D₅₃	—	2	2	no	—
159	C₂×C₄×D₅₃	—	4	1	no	—
160	C₂×C₄×D₅₃	—	4	1	no	—
161	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
162	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
163	C₂×C₄×D₅₃	C₂×Q₈	1	2	yes	—
164	C₂×C₄×D₅₃	C₂×Q₈	1	2	yes	—
165	C₂×C₄×D₅₃	#41	1	2	yes	—
166	C₂×C₄×D₅₃	#41	1	2	yes	—
167	C₂×C₈×C₅₃	—	2	1	no	—
168	C₂×C₈×C₅₃	—	2	1	no	—
169	C₂×C₈×C₅₃	—	2	1	no	—
170	C₂×C₈×C₅₃	—	2	1	no	—
171	C₂×C₈×C₅₃	—	2	1	no	—
172	C₂×C₈×C₅₃	—	2	1	no	—
173	C₂×C₈×C₅₃	—	2	1	no	—
174	C₂×C₈×C₅₃	—	2	1	no	—
175	C₂×C₈×C₅₃	—	2	1	no	—
176	C₂×C₈×C₅₃	—	2	1	no	—
177	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
178	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
179	C₄⋊C₄×C₅₃	—	2	2	no	—
180	C₄⋊C₄×C₅₃	—	4	1	no	—
181	C₄²×C₅₃	—	2	1	no	—
182	C₄²×C₅₃	—	2	1	no	—
183	C₄²×C₅₃	—	2	1	no	—
184	C₄²×C₅₃	—	2	1	no	—
185	C₄²×C₅₃	C₂×D₄	1	2	yes	—
186	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
187	#11719921	—	2	2	no	—
188	#11719921	—	2	2	no	—
189	#11719921	C₄⋊C₄	1	4	yes	—
190	#11719921	C₄⋊C₄	1	4	yes	—
191	C₄⋊Dic₅₃	—	2	2	no	—
192	C₄⋊Dic₅₃	—	4	1	no	—
193	#11719919	—	2	2	no	—
194	#11719919	—	2	2	no	—
195	#11719919	—	2	2	no	—
196	#11719919	—	2	2	no	—
197	#11719919	—	4	1	no	—
198	#11719919	—	4	1	no	—
199	#11719919	C₄⋊C₄	1	4	yes	—
200	#11719919	C₄⋊C₄	1	4	yes	—
201	C₄×Dic₅₃	—	2	1	no	—
202	C₄×Dic₅₃	—	2	1	no	—
203	C₄×Dic₅₃	—	2	1	no	—
204	C₄×Dic₅₃	—	2	1	no	—
205	C₄×Dic₅₃	—	2	2	no	—
206	C₄×Dic₅₃	—	2	2	no	—
207	C₄×Dic₅₃	C₂×D₄	1	2	yes	—
208	C₄×Dic₅₃	C₂×Q₈	1	2	yes	—
209	C₂×#88322	—	2	1	no	—
210	C₂×#88322	—	2	1	no	—
211	C₂×#88322	—	2	1	no	—
212	C₂×#88322	—	2	1	no	—
213	C₂×#88322	—	2	1	no	—
214	C₂×#88322	—	2	1	no	—
215	C₂×#88322	—	2	1	no	—
216	C₂×#88322	—	2	1	no	—
217	C₂×#88322	—	2	1	no	—
218	C₂×#88322	—	2	1	no	—
219	C₂×#88322	C₂×D₄	1	2	yes	—
220	C₂×#88322	C₂×Q₈	1	2	yes	—
221	#11719912	—	2	8	no	—
222	#11719912	—	4	4	no	—
223	C₈×D₅₃	—	2	8	no	—
224	C₈×D₅₃	—	4	4	no	—
225	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
226	C₂²×D₅₃	—	2	1	no	—
227	C₂²×D₅₃	—	2	1	no	—
228	C₂²×D₅₃	—	2	1	no	—
229	C₂²×D₅₃	—	2	1	no	—
230	C₂²×D₅₃	C₂×C₄⋊C₄	1	2	yes	—
231	C₂²×D₅₃	C₄×D₄	1	2	yes	—
232	C₂²×D₅₃	C₄×Q₈	1	2	yes	—
233	C₂×C₄×C₅₃	—	2	1	no	—
234	C₂×C₄×C₅₃	—	2	1	no	—
235	C₂×C₄×C₅₃	—	2	1	no	—
236	C₂×C₄×C₅₃	—	2	1	no	—
237	C₂×C₄×C₅₃	—	2	1	no	—
238	C₂×C₄×C₅₃	—	2	1	no	—
239	C₂×C₄×C₅₃	—	2	1	no	—
240	C₂×C₄×C₅₃	—	2	1	no	—
241	C₂×C₄×C₅₃	—	2	1	no	—
242	C₂×C₄×C₅₃	—	2	1	no	—
243	C₂×C₄×C₅₃	—	2	1	no	—
244	C₂×C₄×C₅₃	—	2	1	no	—
245	C₂×C₄×C₅₃	—	2	1	no	—
246	C₂×C₄×C₅₃	—	2	1	no	—
247	C₂×C₄×C₅₃	—	2	1	no	—
248	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
249	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
250	C₂×C₄×C₅₃	#131	1	2	yes	—
251	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
252	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
253	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
254	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
255	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
256	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
257	C₂×Dic₅₃	—	2	1	no	—
258	C₂×Dic₅₃	—	2	1	no	—
259	C₂×Dic₅₃	—	2	1	no	—
260	C₂×Dic₅₃	—	2	1	no	—
261	C₂×Dic₅₃	—	2	1	no	—
262	C₂×Dic₅₃	—	2	1	no	—
263	C₂×Dic₅₃	—	2	1	no	—
264	C₂×Dic₅₃	—	2	1	no	—
265	C₂×Dic₅₃	—	2	1	no	—
266	C₂×Dic₅₃	—	2	1	no	—
267	C₂×Dic₅₃	—	2	1	no	—
268	C₂×Dic₅₃	—	2	1	no	—
269	C₂×Dic₅₃	—	2	1	no	—
270	C₂×Dic₅₃	—	2	1	no	—
271	C₂×Dic₅₃	—	2	1	no	—
272	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
273	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
274	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
275	C₂×Dic₅₃	C₄×D₄	1	2	yes	—
276	C₂×Dic₅₃	C₄×Q₈	1	2	yes	—
277	C₂×Dic₅₃	#131	1	2	yes	—
278	C₄×D₅₃	—	4	4	no	—
279	C₄×D₅₃	—	4	4	no	—
280	C₄×D₅₃	#121	1	4	yes	—
281	C₄×D₅₃	C₂²⋊Q₈	1	4	yes	—
282	C₄×D₅₃	#125	1	4	yes	—
283	C₄×D₅₃	C₄⋊Q₈	1	4	yes	—
284	C₈×C₅₃	—	2	2	no	—
285	C₈×C₅₃	—	2	2	no	—
286	C₈×C₅₃	—	4	1	no	—
287	C₈×C₅₃	—	4	1	no	—
288	#88322	—	2	2	no	—
289	#88322	—	2	2	no	—
290	#88322	—	4	1	no	—
291	#88322	—	4	1	no	—
292	#21148	—	53	1	no	maximal, Sylow
293	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
294	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
295	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
296	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
297	C₂²×C₅₃	C₂×#131	1	1	yes	characteristic
298	C₂²×C₅₃	#568	1	1	yes	characteristic
299	C₂²×C₅₃	#568	1	1	yes	characteristic
300	C₂×D₅₃	—	2	2	no	—
301	C₂×D₅₃	—	2	2	no	—
302	C₂×D₅₃	—	2	2	no	—
303	C₂×D₅₃	—	2	2	no	—
304	C₂×D₅₃	—	2	2	no	—
305	C₂×D₅₃	—	2	2	no	—
306	C₂×D₅₃	#389	1	4	yes	—
307	C₄×C₅₃	—	2	2	no	—
308	C₄×C₅₃	—	2	2	no	—
309	C₄×C₅₃	—	2	2	no	—
310	C₄×C₅₃	—	4	1	no	—
311	C₄×C₅₃	—	4	1	no	—
312	C₄×C₅₃	—	4	1	no	—
313	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
314	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
315	C₄×C₅₃	C₂×#125	1	1	yes	characteristic
316	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
317	Dic₅₃	—	2	2	no	—
318	Dic₅₃	—	2	2	no	—
319	Dic₅₃	—	2	2	no	—
320	Dic₅₃	—	4	1	no	—
321	Dic₅₃	—	4	1	no	—
322	Dic₅₃	—	4	1	no	—
323	Dic₅₃	#389	1	4	yes	—
324	C₂×#424	—	53	1	no	—
325	C₂×#424	—	53	1	no	—
326	C₂×#423	—	53	1	no	—
327	C₂×#405	—	53	1	no	—
328	#2296	—	53	1	no	—
329	#1940	—	53	1	no	—
330	#1940	—	53	1	no	—
331	#1940	—	53	1	no	—
332	#1940	D₅₃	1	1	yes	characteristic, complemented
333	#1939	—	53	1	no	—
334	#1939	—	53	1	no	—
335	#1939	—	53	1	no	—
336	#1939	—	53	1	no	—
337	#1729	—	53	1	no	—
338	C₂×#336	—	53	1	no	—
339	C₂×C₅₃	C₂×#389	1	1	yes	characteristic
340	C₂×C₅₃	#2906	1	1	yes	characteristic
341	C₂×C₅₃	#2906	1	1	yes	characteristic
342	C₂×C₅₃	#2933	1	1	yes	characteristic
343	C₂×C₅₃	#2933	1	1	yes	characteristic
344	C₂×C₅₃	#2960	1	1	yes	characteristic
345	C₂×C₅₃	#2961	1	1	yes	characteristic
346	D₅₃	—	2	4	no	—
347	C₂²×M₁₆	—	53	1	no	—
348	C₂²×M₁₆	—	53	1	no	—
349	C₂³×C₈	—	53	1	no	—
350	C₂³×C₈	—	53	1	no	—
351	C₂×#117	—	53	1	no	—
352	C₂×#117	—	53	1	no	—
353	C₂²×C₄²	—	53	1	no	—
354	#424	—	106	4	no	—
355	#424	—	106	4	no	—
356	#423	—	106	4	no	—
357	C₂×C₄⋊C₈	—	53	1	no	—
358	C₂×C₄⋊C₈	—	53	1	no	—
359	C₂×C₄⋊C₈	—	53	1	no	—
360	C₂×C₄⋊C₈	—	53	1	no	—
361	C₂×C₄⋊C₈	—	53	1	no	—
362	C₂×C₄⋊C₈	—	53	1	no	—
363	C₂×C₄⋊C₈	—	53	1	no	—
364	C₂×C₄⋊C₈	—	53	1	no	—
365	C₂×C₄⋊C₈	—	53	1	no	—
366	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
367	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
368	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
369	#405	—	106	4	no	—
370	C₂×#97	—	53	1	no	—
371	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
372	C₂×C₄×C₈	—	53	1	no	—
373	C₂×C₄×C₈	—	53	1	no	—
374	#389	—	53	1	no	—
375	#389	—	53	1	no	—
376	#389	—	53	1	no	—
377	#389	—	53	1	no	—
378	#385	—	53	1	no	—
379	#385	—	53	1	no	—
380	#385	—	53	1	no	—
381	#385	—	53	1	no	—
382	C₄×C₄⋊C₄	—	53	1	no	—
383	C₄×C₄⋊C₄	—	53	1	no	—
384	C₄×C₄⋊C₄	—	53	1	no	—
385	C₄×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
386	#336	—	53	1	no	—
387	#336	—	53	1	no	—
388	#336	—	53	1	no	—
389	#336	—	53	1	no	—
390	#336	—	53	1	no	—
391	#336	—	53	1	no	—
392	#336	C₂×D₅₃	1	1	yes	characteristic
393	#336	C₂×D₅₃	1	1	yes	characteristic
394	C₅₃	#21148	1	1	yes	characteristic, complemented, Sylow
395	C₂³×C₄	—	53	1	no	—
396	C₂³×C₄	—	53	1	no	—
397	C₂³×C₄	—	53	1	no	—
398	C₂×M₁₆	—	53	4	no	—
399	C₂×M₁₆	—	106	2	no	—
400	C₂×M₁₆	—	106	2	no	—
401	C₂×M₁₆	—	106	2	no	—
402	C₂×M₁₆	—	106	2	no	—
403	C₂×M₁₆	—	106	2	no	—
404	C₂²×C₈	—	53	1	no	—
405	C₂²×C₈	—	53	1	no	—
406	C₂²×C₈	—	53	4	no	—
407	C₂²×C₈	—	53	1	no	—
408	C₂²×C₈	—	53	1	no	—
409	C₂²×C₈	—	106	2	no	—
410	C₂²×C₈	—	106	2	no	—
411	C₂²×C₈	—	106	2	no	—
412	C₂²×C₈	—	106	2	no	—
413	C₂²×C₈	—	106	2	no	—
414	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
415	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
416	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
417	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
418	#117	—	53	4	no	—
419	#117	—	53	4	no	—
420	#117	—	106	2	no	—
421	#117	—	106	2	no	—
422	C₂×C₄⋊C₄	—	53	1	no	—
423	C₂×C₄⋊C₄	—	53	1	no	—
424	C₂×C₄⋊C₄	—	53	1	no	—
425	C₂×C₄⋊C₄	—	53	1	no	—
426	C₂×C₄⋊C₄	—	53	1	no	—
427	C₂×C₄⋊C₄	—	53	1	no	—
428	C₂×C₄⋊C₄	—	53	1	no	—
429	C₂×C₄⋊C₄	—	53	1	no	—
430	C₂×C₄⋊C₄	—	53	1	no	—
431	C₂×C₄⋊C₄	—	53	1	no	—
432	C₂×C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
433	C₂×C₄⋊C₄	C₄×D₅₃	1	1	yes	characteristic
434	C₂×C₂²⋊C₄	—	53	1	no	—
435	C₂×C₂²⋊C₄	—	53	1	no	—
436	C₂×C₂²⋊C₄	—	53	1	no	—
437	C₂×C₂²⋊C₄	—	53	1	no	—
438	C₂×C₄²	—	53	1	no	—
439	C₂×C₄²	—	53	1	no	—
440	C₂×C₄²	—	53	1	no	—
441	C₂×C₄²	—	53	1	no	—
442	C₂×C₄²	—	53	1	no	—
443	C₂×C₄²	—	106	2	no	—
444	C₂×C₄²	—	106	2	no	—
445	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
446	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
447	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
448	C₄⋊C₈	—	2	2	no	—
449	C₄⋊C₈	—	2	2	no	—
450	C₄⋊C₈	—	2	2	no	—
451	C₄⋊C₈	—	106	2	no	—
452	C₄⋊C₈	—	106	2	no	—
453	C₄⋊C₈	—	106	2	no	—
454	C₄⋊C₈	—	106	2	no	—
455	C₄⋊C₈	—	106	2	no	—
456	C₄⋊C₈	—	106	2	no	—
457	C₄⋊C₈	—	106	2	no	—
458	C₄⋊C₈	—	106	2	no	—
459	C₄⋊C₈	—	106	2	no	—
460	#97	—	2	2	no	—
461	#97	—	106	2	no	—
462	C₄×C₈	—	106	2	no	—
463	C₄×C₈	—	106	2	no	—
464	#95	—	53	1	no	—
465	#95	—	53	1	no	—
466	#95	—	53	1	no	—
467	#95	—	53	1	no	—
468	#95	—	53	1	no	—
469	#95	—	53	1	no	—
470	#95	C₄×D₅₃	1	1	yes	characteristic
471	#95	C₄×D₅₃	1	1	yes	characteristic
472	C₂⁴	—	53	1	no	—
473	C₂²×C₄	—	53	1	no	—
474	C₂²×C₄	—	53	2	no	—
475	C₂²×C₄	—	53	2	no	—
476	C₂²×C₄	—	53	2	no	—
477	C₂²×C₄	—	53	2	no	—
478	C₂²×C₄	—	53	2	no	—
479	C₂²×C₄	—	53	2	no	—
480	C₂²×C₄	—	53	1	no	—
481	C₂²×C₄	—	53	1	no	—
482	C₂²×C₄	—	53	1	no	—
483	C₂²×C₄	—	53	1	no	—
484	C₂²×C₄	—	53	1	no	—
485	C₂²×C₄	—	53	1	no	—
486	C₂²×C₄	—	106	2	no	—
487	C₂²×C₄	—	106	2	no	—
488	C₂²×C₄	—	106	2	no	—
489	C₂²×C₄	—	106	2	no	—
490	C₂²×C₄	—	212	1	no	—
491	C₂²×C₄	—	212	1	no	—
492	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
493	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
494	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
495	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
496	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
497	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
498	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
499	M₁₆	—	106	8	no	—
500	M₁₆	—	212	4	no	—
501	C₂×C₈	—	2	1	no	—
502	C₂×C₈	—	2	1	no	—
503	C₂×C₈	—	2	1	no	—
504	C₂×C₈	—	2	1	no	—
505	C₂×C₈	—	2	1	no	—
506	C₂×C₈	—	2	1	no	—
507	C₂×C₈	—	2	1	no	—
508	C₂×C₈	—	2	1	no	—
509	C₂×C₈	—	2	1	no	—
510	C₂×C₈	—	2	1	no	—
511	C₂×C₈	—	53	2	no	—
512	C₂×C₈	—	53	2	no	—
513	C₂×C₈	—	106	8	no	—
514	C₂×C₈	—	106	1	no	—
515	C₂×C₈	—	106	1	no	—
516	C₂×C₈	—	106	1	no	—
517	C₂×C₈	—	106	1	no	—
518	C₂×C₈	—	106	1	no	—
519	C₂×C₈	—	106	1	no	—
520	C₂×C₈	—	106	1	no	—
521	C₂×C₈	—	106	1	no	—
522	C₂×C₈	—	106	1	no	—
523	C₂×C₈	—	106	1	no	—
524	C₂×C₈	—	212	4	no	—
525	C₂×C₈	D₄×D₅₃	1	2	yes	—
526	C₂×C₈	Q₈×D₅₃	1	2	yes	—
527	C₄⋊C₄	—	2	2	no	—
528	C₄⋊C₄	—	4	1	no	—
529	C₄⋊C₄	—	53	4	no	—
530	C₄⋊C₄	—	53	4	no	—
531	C₄⋊C₄	—	106	2	no	—
532	C₄⋊C₄	—	106	2	no	—
533	C₄⋊C₄	—	106	2	no	—
534	C₄⋊C₄	—	106	2	no	—
535	C₄⋊C₄	—	106	2	no	—
536	C₄⋊C₄	—	212	1	no	—
537	C₄⋊C₄	—	212	1	no	—
538	C₄⋊C₄	—	212	1	no	—
539	C₂²⋊C₄	—	53	4	no	—
540	C₂²⋊C₄	—	53	4	no	—
541	C₂²⋊C₄	—	106	2	no	—
542	C₂²⋊C₄	—	106	2	no	—
543	C₄²	—	2	1	no	—
544	C₄²	—	2	1	no	—
545	C₄²	—	2	1	no	—
546	C₄²	—	2	1	no	—
547	C₄²	—	53	2	no	—
548	C₄²	—	53	2	no	—
549	C₄²	—	106	2	no	—
550	C₄²	—	106	2	no	—
551	C₄²	—	106	1	no	—
552	C₄²	—	106	1	no	—
553	C₄²	—	106	1	no	—
554	C₄²	—	106	1	no	—
555	C₄²	D₄×D₅₃	1	2	yes	—
556	C₄²	Q₈×D₅₃	1	2	yes	—
557	C₂³	—	53	2	no	—
558	C₂³	—	53	2	no	—
559	C₂³	—	53	2	no	—
560	C₂³	—	106	1	no	—
561	C₂³	—	106	1	no	—
562	C₂³	—	106	1	no	—
563	C₂³	—	106	1	no	—
564	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
565	C₂×C₄	—	2	1	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	1	no	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	1	no	—
572	C₂×C₄	—	2	1	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	2	1	no	—
578	C₂×C₄	—	2	1	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	53	2	no	—
581	C₂×C₄	—	53	2	no	—
582	C₂×C₄	—	53	2	no	—
583	C₂×C₄	—	53	4	no	—
584	C₂×C₄	—	53	4	no	—
585	C₂×C₄	—	53	4	no	—
586	C₂×C₄	—	53	4	no	—
587	C₂×C₄	—	53	2	no	—
588	C₂×C₄	—	53	2	no	—
589	C₂×C₄	—	53	2	no	—
590	C₂×C₄	—	106	1	no	—
591	C₂×C₄	—	106	1	no	—
592	C₂×C₄	—	106	1	no	—
593	C₂×C₄	—	106	1	no	—
594	C₂×C₄	—	106	1	no	—
595	C₂×C₄	—	106	1	no	—
596	C₂×C₄	—	106	1	no	—
597	C₂×C₄	—	106	1	no	—
598	C₂×C₄	—	106	1	no	—
599	C₂×C₄	—	106	1	no	—
600	C₂×C₄	—	106	1	no	—
601	C₂×C₄	—	106	1	no	—
602	C₂×C₄	—	106	1	no	—
603	C₂×C₄	—	106	1	no	—
604	C₂×C₄	—	106	1	no	—
605	C₂×C₄	—	212	4	no	—
606	C₂×C₄	—	212	4	no	—
607	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
608	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
609	C₂×C₄	#13133225	1	2	yes	—
610	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
611	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
612	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
613	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
614	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
615	C₂×C₄	#41×D₅₃	1	1	yes	characteristic
616	C₈	—	2	2	no	—
617	C₈	—	2	2	no	—
618	C₈	—	4	1	no	—
619	C₈	—	4	1	no	—
620	C₈	—	106	2	no	—
621	C₈	—	106	2	no	—
622	C₈	—	212	1	no	—
623	C₈	—	212	1	no	—
624	C₂²	—	53	4	no	—
625	C₂²	—	106	2	no	—
626	C₂²	—	106	2	no	—
627	C₂²	—	106	2	no	—
628	C₂²	—	106	2	no	—
629	C₂²	—	106	2	no	—
630	C₂²	—	106	2	no	—
631	C₂²	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
632	C₂²	C₄×D₄×D₅₃	1	1	yes	characteristic
633	C₂²	C₄×Q₈×D₅₃	1	1	yes	characteristic
634	C₂²	#131×D₅₃	1	1	yes	characteristic
635	C₂²	C₂×#13133225	1	1	yes	characteristic
636	C₂²	#16811388	1	1	yes	characteristic
637	C₂²	#16811389	1	1	yes	characteristic
638	C₄	—	2	2	no	—
639	C₄	—	2	2	no	—
640	C₄	—	2	2	no	—
641	C₄	—	4	1	no	—
642	C₄	—	4	1	no	—
643	C₄	—	4	1	no	—
644	C₄	—	53	4	no	—
645	C₄	—	106	2	no	—
646	C₄	—	106	2	no	—
647	C₄	—	106	2	no	—
648	C₄	—	212	1	no	—
649	C₄	—	212	1	no	—
650	C₄	—	212	1	no	—
651	C₄	#121×D₅₃	1	1	yes	characteristic
652	C₄	C₂²⋊Q₈×D₅₃	1	1	yes	characteristic
653	C₄	#125×D₅₃	1	1	yes	characteristic
654	C₄	C₄⋊Q₈×D₅₃	1	1	yes	characteristic
655	C₂	—	106	4	no	—
656	C₂	#389×D₅₃	1	1	yes	characteristic
657	C₂	#20283282	1	1	yes	characteristic
658	C₂	#20283284	1	1	yes	characteristic
659	C₂	#20283438	1	1	yes	characteristic
660	C₂	#20283441	1	1	yes	characteristic
661	C₂	#20283646	1	1	yes	characteristic
662	C₂	#20283651	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂³×C₄
Fitting	C₅₃×#1940	C₂
prime radical	C₂³×D₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1940	D₅₃
hypocenter	C₅₃	#21148
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1940

Complements

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

N	H ≅ G/N	type
6. #20275438	C₂	semidirect
7. #20275437	C₂	semidirect
8. #20275437	C₂	semidirect
9. #20275434	C₂	semidirect
10. #20275431	C₂	semidirect
11. #20275430	C₂	semidirect
12. #20275430	C₂	semidirect
15. C₅₃×#1940	C₂	semidirect
332. #1940	D₅₃	semidirect
394. C₅₃	#21148	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16810648	2
C₂×#16810645	2
C₂×#16810580	2
#20277195	2
#20275438	2
#20275437	2
#20275434	2
#20275431	2
#20275430	2
#20274410	2
#336×D₅₃	2
C₅₃×#1940	2
#21148	53