Subgroups — #20288118

Order 6784.

996 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811253	C₂	1	4	yes	maximal, complemented
2	#16811252	C₂	1	4	yes	maximal, complemented
3	#16811250	C₂	1	4	yes	maximal, complemented
4	#16811248	C₂	1	4	yes	maximal
5	#16811180	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811177	C₂	1	8	yes	maximal, complemented
7	#16811176	C₂	1	2	yes	maximal, complemented
8	C₂×#13133174	C₂	1	1	yes	maximal, characteristic
9	#117×D₅₃	C₂	1	2	yes	maximal
10	C₅₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
12	#13133217	C₂²	1	2	yes	complemented
13	#13133216	C₂²	1	4	yes	complemented
14	#13133216	C₂²	1	8	yes	complemented
15	#13133215	C₂²	1	8	yes	complemented
16	#13133209	C₂²	1	2	yes	complemented
17	#13133208	C₂²	1	4	yes	complemented
18	#13133207	C₂²	1	8	yes	complemented
19	#13133207	C₂²	1	8	yes	complemented
20	#13133197	C₂²	1	8	yes	—
21	#13133196	C₂²	1	4	yes	—
22	#13133195	C₂²	1	8	yes	—
23	#13133195	C₂²	1	8	yes	—
24	#13133194	C₂²	1	8	yes	—
25	#13133193	C₂²	1	8	yes	—
26	C₄×Dic₁₀₆	C₂²	1	8	yes	complemented
27	C₄×Dic₁₀₆	C₂²	1	4	yes	complemented
28	#13133176	C₂²	1	2	yes	—
29	#13133175	C₂²	1	4	yes	—
30	#13133174	C₂²	1	4	yes	—
31	#13133174	C₂²	1	4	yes	—
32	#13133174	C₂²	1	2	yes	—
33	#13133174	C₂²	1	2	yes	—
34	#13133173	C₂²	1	4	yes	—
35	C₂×#11719919	C₂²	1	2	yes	—
36	C₄⋊C₄×D₅₃	C₂²	1	4	yes	—
37	C₂×#11719921	C₂²	1	2	yes	—
38	C₂²⋊C₄×D₅₃	C₂²	1	4	yes	—
39	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
40	C₄²×D₅₃	C₂²	1	4	yes	—
41	C₄⋊Q₈×C₅₃	C₂²	1	2	yes	complemented
42	#125×C₅₃	C₂²	1	2	yes	complemented
43	C₂²⋊Q₈×C₅₃	C₂²	1	4	yes	—
44	C₄×Q₈×C₅₃	C₂²	1	4	yes	complemented
45	#117×C₅₃	C₂²	1	2	yes	—
46	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
47	C₂³×D₅₃	C₂³	1	1	yes	characteristic
48	C₂×Q₈×C₅₃	C₂³	1	4	yes	—
49	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
50	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
51	C₂²×Dic₅₃	C₂³	1	2	yes	—
52	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₅₃	Q₈	1	8	yes	complemented
54	C₂×C₄×D₅₃	C₂³	1	8	yes	—
55	C₂×C₄×D₅₃	C₂³	1	2	yes	—
56	C₂×C₄×D₅₃	C₂³	1	2	yes	—
57	C₂×Dic₁₀₆	C₂³	1	8	yes	—
58	C₂×Dic₁₀₆	C₂³	1	4	yes	—
59	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
60	C₄⋊C₄×C₅₃	C₂³	1	8	yes	—
61	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
62	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
63	C₄²×C₅₃	C₂³	1	2	yes	—
64	C₄²×C₅₃	C₂³	1	1	yes	characteristic
65	C₄²×C₅₃	C₂³	1	1	yes	characteristic
66	C₄²×C₅₃	C₂³	1	4	yes	—
67	C₂²⋊Dic₅₃	C₂³	1	4	yes	—
68	#11719921	C₂³	1	8	yes	—
69	#11719921	C₂³	1	4	yes	—
70	#11719921	C₂³	1	4	yes	—
71	C₄⋊Dic₅₃	C₂³	1	4	yes	—
72	C₄⋊Dic₅₃	C₂³	1	8	yes	—
73	C₄⋊Dic₅₃	C₂³	1	4	yes	—
74	#11719919	C₂³	1	8	yes	—
75	#11719919	C₂³	1	4	yes	—
76	#11719919	C₂³	1	4	yes	—
77	#11719919	C₂³	1	8	yes	—
78	#11719919	C₂³	1	8	yes	—
79	#11719919	C₂³	1	8	yes	—
80	C₄×Dic₅₃	C₂³	1	8	yes	—
81	C₄×Dic₅₃	C₂³	1	4	yes	—
82	C₄×Dic₅₃	C₂³	1	2	yes	—
83	C₄×Dic₅₃	C₂³	1	2	yes	—
84	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
85	C₂²×D₅₃	—	4	2	no	—
86	C₂²×D₅₃	C₂×Q₈	1	4	yes	—
87	C₂²×D₅₃	C₂⁴	1	2	yes	—
88	Q₈×C₅₃	—	4	4	no	—
89	C₂×C₄×C₅₃	—	4	2	no	—
90	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
91	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
92	C₂×C₄×C₅₃	C₂⁴	1	8	yes	—
93	C₂×C₄×C₅₃	C₂⁴	1	2	yes	—
94	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
96	C₂×Dic₅₃	—	4	2	no	—
97	C₂×Dic₅₃	C₂×Q₈	1	4	yes	—
98	C₂×Dic₅₃	C₂⁴	1	8	yes	—
99	C₂×Dic₅₃	C₂⁴	1	4	yes	—
100	C₂×Dic₅₃	C₂⁴	1	2	yes	—
101	C₄×D₅₃	—	4	8	no	—
102	C₄×D₅₃	C₂×Q₈	1	8	yes	—
103	C₄×D₅₃	C₂×Q₈	1	8	yes	—
104	Dic₁₀₆	—	4	4	no	—
105	Dic₁₀₆	—	4	8	no	—
106	C₂²×C₅₃	—	2	2	no	—
107	C₂²×C₅₃	C₂²×Q₈	1	2	yes	—
108	C₂²×C₅₃	C₂⁵	1	1	yes	characteristic
109	C₂×D₅₃	—	2	4	no	—
110	C₂×D₅₃	—	4	4	no	—
111	C₂×D₅₃	C₂²×Q₈	1	4	yes	—
112	C₄×C₅₃	—	2	2	no	—
113	C₄×C₅₃	—	2	8	no	—
114	C₄×C₅₃	—	2	2	no	—
115	C₄×C₅₃	C₂²×Q₈	1	2	yes	—
116	C₄×C₅₃	C₂²×Q₈	1	2	yes	—
117	Dic₅₃	—	2	8	no	—
118	Dic₅₃	—	2	4	no	—
119	Dic₅₃	C₂²×Q₈	1	4	yes	—
120	#3458	—	53	1	no	maximal, Sylow
121	C₂×C₅₃	—	2	2	no	—
122	C₂×C₅₃	C₂³×Q₈	1	1	yes	characteristic
123	C₂×C₅₃	#585	1	2	yes	—
124	D₅₃	—	2	4	no	—
125	#544	—	53	4	no	—
126	#544	—	53	4	no	—
127	#544	—	53	4	no	—
128	#543	—	53	4	no	—
129	#533	—	53	8	no	—
130	#533	—	53	2	no	—
131	#533	—	53	1	no	—
132	#533	D₅₃	1	1	yes	characteristic, complemented
133	C₂×#117	—	53	2	no	—
134	C₂×#117	—	53	1	no	—
135	C₅₃	#3458	1	1	yes	characteristic, complemented, Sylow
136	C₂³×C₄	—	53	1	no	—
137	C₄⋊Q₈	—	53	8	no	—
138	C₄⋊Q₈	—	53	8	no	—
139	C₄⋊Q₈	—	53	4	no	—
140	C₄⋊Q₈	—	53	2	no	—
141	C₄⋊Q₈	C₂×D₅₃	1	2	yes	complemented
142	#125	—	53	8	no	—
143	#125	—	53	8	no	—
144	#125	—	53	4	no	—
145	#125	—	53	2	no	—
146	#125	C₂×D₅₃	1	2	yes	complemented
147	C₂²⋊Q₈	—	53	8	no	—
148	C₂²⋊Q₈	—	53	4	no	—
149	C₂²⋊Q₈	—	53	8	no	—
150	C₂²⋊Q₈	—	53	8	no	—
151	C₂²⋊Q₈	—	53	8	no	—
152	C₂²⋊Q₈	—	53	8	no	—
153	C₂²⋊Q₈	C₂×D₅₃	1	4	yes	—
154	C₄×Q₈	—	53	8	no	—
155	C₄×Q₈	—	53	4	no	—
156	C₄×Q₈	C₂×D₅₃	1	4	yes	complemented
157	#117	—	53	2	no	—
158	#117	—	53	4	no	—
159	#117	—	53	4	no	—
160	#117	—	53	4	no	—
161	#117	—	53	4	no	—
162	#117	—	53	2	no	—
163	#117	—	53	2	no	—
164	#117	C₂×D₅₃	1	2	yes	—
165	C₂×C₄⋊C₄	—	53	2	no	—
166	C₂×C₄⋊C₄	—	53	4	no	—
167	C₂×C₂²⋊C₄	—	53	2	no	—
168	C₂×C₂²⋊C₄	—	53	4	no	—
169	C₂×C₄²	—	53	1	no	—
170	C₂×C₄²	—	53	4	no	—
171	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
172	C₂⁴	—	53	1	no	—
173	C₂×Q₈	—	53	8	no	—
174	C₂×Q₈	—	53	4	no	—
175	C₂×Q₈	C₂²×D₅₃	1	4	yes	—
176	C₂²×C₄	—	53	1	no	—
177	C₂²×C₄	—	53	8	no	—
178	C₂²×C₄	—	53	2	no	—
179	C₂²×C₄	—	53	2	no	—
180	C₂²×C₄	—	53	2	no	—
181	C₂²×C₄	—	53	8	no	—
182	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
183	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
184	C₄⋊C₄	—	53	8	no	—
185	C₄⋊C₄	—	53	4	no	—
186	C₄⋊C₄	—	53	4	no	—
187	C₄⋊C₄	—	53	4	no	—
188	C₄⋊C₄	—	53	8	no	—
189	C₄⋊C₄	—	53	8	no	—
190	C₄⋊C₄	—	53	8	no	—
191	C₄⋊C₄	—	53	8	no	—
192	C₄⋊C₄	—	53	4	no	—
193	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
194	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
195	C₄⋊C₄	C₂²×D₅₃	1	8	yes	—
196	C₂²⋊C₄	—	53	4	no	—
197	C₂²⋊C₄	—	53	8	no	—
198	C₂²⋊C₄	—	53	4	no	—
199	C₂²⋊C₄	—	53	4	no	—
200	C₂²⋊C₄	C₂²×D₅₃	1	4	yes	—
201	C₄²	—	53	8	no	—
202	C₄²	—	53	2	no	—
203	C₄²	—	53	2	no	—
204	C₄²	—	53	4	no	—
205	C₄²	C₂²×D₅₃	1	1	yes	characteristic
206	C₄²	C₂²×D₅₃	1	1	yes	characteristic
207	C₄²	C₂²×D₅₃	1	4	yes	—
208	C₄²	C₂²×D₅₃	1	2	yes	—
209	C₂³	—	53	2	no	—
210	C₂³	—	53	4	no	—
211	C₂³	—	212	2	no	—
212	C₂³	C₂³×D₅₃	1	1	yes	characteristic
213	Q₈	—	4	4	no	—
214	Q₈	—	212	4	no	—
215	Q₈	—	212	8	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	53	4	no	—
218	C₂×C₄	—	53	2	no	—
219	C₂×C₄	—	53	8	no	—
220	C₂×C₄	—	53	8	no	—
221	C₂×C₄	—	53	8	no	—
222	C₂×C₄	—	53	4	no	—
223	C₂×C₄	—	212	8	no	—
224	C₂×C₄	—	212	2	no	—
225	C₂×C₄	Q₈×D₅₃	1	4	yes	—
226	C₂×C₄	C₂³×D₅₃	1	8	yes	—
227	C₂×C₄	C₂³×D₅₃	1	2	yes	—
228	C₂×C₄	C₂³×D₅₃	1	2	yes	—
229	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
230	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
231	C₂²	—	2	2	no	—
232	C₂²	—	53	4	no	—
233	C₂²	—	106	4	no	—
234	C₂²	—	212	4	no	—
235	C₂²	C₂×Q₈×D₅₃	1	2	yes	—
236	C₂²	C₂⁴×D₅₃	1	1	yes	characteristic
237	C₄	—	2	2	no	—
238	C₄	—	2	2	no	—
239	C₄	—	2	8	no	—
240	C₄	—	53	4	no	—
241	C₄	—	106	8	no	—
242	C₄	—	106	4	no	—
243	C₄	C₂×Q₈×D₅₃	1	2	yes	—
244	C₄	C₂×Q₈×D₅₃	1	2	yes	—
245	C₂	—	2	2	no	—
246	C₂	—	106	4	no	—
247	C₂	C₂²×Q₈×D₅₃	1	1	yes	characteristic
248	C₂	#16811459	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂⁵
Frattini	C₂²	C₂⁴×D₅₃
Fitting	C₅₃×#533	C₂
hypercenter	#533	D₅₃
hypocenter	C₅₃	#3458
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #16811253	C₂	semidirect
2. #16811252	C₂	semidirect
3. #16811250	C₂	semidirect
5. #16811180	C₂	semidirect
6. #16811177	C₂	semidirect
7. #16811176	C₂	semidirect
10. C₅₃×#533	C₂	semidirect
12. #13133217	C₂²	semidirect
13. #13133216	C₂²	semidirect
14. #13133216	C₂²	semidirect
15. #13133215	C₂²	semidirect
16. #13133209	C₂²	semidirect
17. #13133208	C₂²	semidirect
18. #13133207	C₂²	semidirect
19. #13133207	C₂²	semidirect
26. C₄×Dic₁₀₆	C₂²	semidirect
27. C₄×Dic₁₀₆	C₂²	semidirect
41. C₄⋊Q₈×C₅₃	C₂²	semidirect
42. #125×C₅₃	C₂²	semidirect
44. C₄×Q₈×C₅₃	C₂²	semidirect
53. C₂×C₄×D₅₃	Q₈	semidirect
132. #533	D₅₃	semidirect
135. C₅₃	#3458	semidirect
141. C₄⋊Q₈	C₂×D₅₃	semidirect
146. #125	C₂×D₅₃	semidirect
156. C₄×Q₈	C₂×D₅₃	semidirect