Subgroups — #20291695

Order 6784.

1540 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811692	C₂	1	8	yes	maximal, complemented
2	#16811691	C₂	1	2	yes	maximal, complemented
3	#16811690	C₂	1	6	yes	maximal, complemented
4	#41×#6157	C₂	1	8	yes	maximal, complemented
5	C₂×#13133296	C₂	1	6	yes	maximal, complemented
6	C₂×#41×D₅₃	C₂	1	1	yes	maximal, characteristic
7	Q₈×#6157	C₂²	1	16	yes	complemented
8	D₄×#6157	C₂²	1	48	yes	complemented
9	#13133296	C₂²	1	4	yes	—
10	#13133296	C₂²	1	12	yes	—
11	#13133296	C₂²	1	12	yes	complemented
12	#13133296	C₂²	1	12	yes	complemented
13	C₂×C₄⋊#6157	C₂²	1	12	yes	complemented
14	C₂×C₂²⋊#6157	C₂²	1	12	yes	—
15	C₂×C₄×#6157	C₂²	1	12	yes	complemented
16	C₂×#11719949	C₄	1	1	yes	characteristic, complemented
17	C₂×#11719945	C₄	1	3	yes	complemented
18	C₂×#11719947	C₄	1	3	yes	complemented
19	#41×D₅₃	C₄	1	4	yes	complemented
20	#41×D₅₃	C₄	1	4	yes	complemented
21	#41×D₅₃	C₂²	1	4	yes	—
22	#41×D₅₃	C₂²	1	4	yes	—
23	C₂×Q₈×D₅₃	C₂²	1	1	yes	characteristic
24	C₂×D₄×D₅₃	C₂²	1	3	yes	—
25	C₂²×C₄×D₅₃	C₂²	1	3	yes	—
26	C₂×#41×C₅₃	C₄	1	1	yes	characteristic, complemented
27	C₂³×D₅₃	C₂³	1	3	yes	—
28	C₂²×#6157	C₂³	1	24	yes	—
29	#41×C₅₃	C₂×C₄	1	4	yes	complemented
30	#41×C₅₃	C₂×C₄	1	4	yes	complemented
31	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic, complemented
32	C₂×D₄×C₅₃	C₂×C₄	1	3	yes	complemented
33	C₂²×C₄×C₅₃	C₂×C₄	1	3	yes	complemented
34	C₂×C₂²⋊D₅₃	C₂×C₄	1	6	yes	complemented
35	C₂²×Dic₅₃	C₂×C₄	1	3	yes	complemented
36	#11719949	C₂×C₄	1	4	yes	complemented
37	#11719949	C₂×C₄	1	4	yes	complemented
38	Q₈×D₅₃	C₂×C₄	1	4	yes	complemented
39	Q₈×D₅₃	C₂³	1	4	yes	—
40	#11719947	C₂×C₄	1	12	yes	complemented
41	#11719947	C₂×C₄	1	12	yes	complemented
42	D₄×D₅₃	C₂×C₄	1	12	yes	complemented
43	D₄×D₅₃	C₂³	1	12	yes	—
44	#11719945	C₂×C₄	1	12	yes	complemented
45	#11719945	C₂×C₄	1	12	yes	complemented
46	C₂×D₂₁₂	C₂×C₄	1	3	yes	complemented
47	C₂×C₄×D₅₃	C₂×C₄	1	6	yes	complemented
48	C₂×C₄×D₅₃	C₂×C₄	1	6	yes	complemented
49	C₂×C₄×D₅₃	C₂×C₄	1	3	yes	complemented
50	C₂×C₄×D₅₃	C₂×C₄	1	3	yes	complemented
51	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
52	C₂×C₄×D₅₃	C₂³	1	3	yes	—
53	C₂×C₄×D₅₃	C₂³	1	6	yes	—
54	C₂×C₄×D₅₃	C₂³	1	6	yes	—
55	C₂×Dic₁₀₆	C₂×C₄	1	3	yes	complemented
56	C₂²⋊#6157	C₂³	1	8	yes	—
57	C₂²⋊#6157	C₂³	1	24	yes	—
58	C₄⋊#6157	C₂³	1	8	yes	—
59	C₄⋊#6157	C₂³	1	24	yes	—
60	C₄×#6157	C₂³	1	8	yes	—
61	C₄×#6157	C₂³	1	24	yes	—
62	C₂³×C₅₃	C₂²×C₄	1	3	yes	—
63	C₂²×D₅₃	—	4	6	no	—
64	C₂²×D₅₃	C₂²×C₄	1	3	yes	—
65	C₂²×D₅₃	C₂²×C₄	1	6	yes	—
66	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
67	C₂²×D₅₃	C₂⁴	1	6	yes	—
68	C₂×#6157	—	4	24	no	—
69	C₂×#6157	C₂⁴	1	16	yes	—
70	Q₈×C₅₃	C₂²×C₄	1	4	yes	—
71	D₄×C₅₃	C₂²×C₄	1	12	yes	—
72	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
73	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
75	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
76	C₂²⋊D₅₃	C₂²×C₄	1	24	yes	—
77	C₂×Dic₅₃	C₂²×C₄	1	3	yes	—
78	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
79	C₂×Dic₅₃	C₂²×C₄	1	6	yes	—
80	C₂×Dic₅₃	C₂²×C₄	1	6	yes	—
81	D₂₁₂	C₂²×C₄	1	12	yes	—
82	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
83	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
84	C₄×D₅₃	C₂²×C₄	1	6	yes	—
85	C₄×D₅₃	C₂²×C₄	1	6	yes	—
86	C₄×D₅₃	C₂²×C₄	1	6	yes	—
87	C₄×D₅₃	C₂²×C₄	1	6	yes	—
88	C₄×D₅₃	C₂²×C₄	1	6	yes	—
89	C₄×D₅₃	C₂⁴	1	1	yes	characteristic
90	C₄×D₅₃	C₂⁴	1	6	yes	—
91	C₄×D₅₃	C₂⁴	1	1	yes	characteristic
92	Dic₁₀₆	C₂²×C₄	1	12	yes	—
93	C₂²×C₅₃	—	4	3	no	—
94	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
95	C₂²×C₅₃	C₂³×C₄	1	6	yes	—
96	C₂×D₅₃	—	2	12	no	—
97	C₂×D₅₃	—	2	2	no	—
98	C₂×D₅₃	—	4	6	no	—
99	C₂×D₅₃	—	4	3	no	—
100	C₂×D₅₃	C₂³×C₄	1	1	yes	characteristic
101	C₂×D₅₃	C₂³×C₄	1	6	yes	—
102	C₂×D₅₃	C₂⁵	1	1	yes	characteristic
103	#6157	—	2	16	no	—
104	C₄×C₅₃	C₂³×C₄	1	6	yes	—
105	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
106	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
107	Dic₅₃	C₂³×C₄	1	6	yes	—
108	Dic₅₃	C₂³×C₄	1	1	yes	characteristic
109	Dic₅₃	C₂³×C₄	1	1	yes	characteristic
110	#3428	—	53	1	no	maximal, Sylow
111	C₂×C₅₃	—	2	1	no	—
112	C₂×C₅₃	—	2	6	no	—
113	C₂×C₅₃	C₂⁴×C₄	1	1	yes	characteristic
114	D₅₃	—	2	1	no	—
115	D₅₃	—	2	6	no	—
116	D₅₃	#585	1	2	yes	—
117	C₂²×#41	—	53	1	no	—
118	#520	—	53	8	no	—
119	#519	—	53	2	no	—
120	#518	—	53	6	no	—
121	C₄×#41	—	53	8	no	—
122	C₂×#117	—	53	6	no	—
123	C₅₃	#3428	1	1	yes	characteristic, complemented, Sylow
124	C₂×#41	—	53	4	no	—
125	C₂×#41	—	53	4	no	—
126	C₂×#41	—	53	4	no	—
127	C₂×#41	—	53	4	no	—
128	C₂×#41	—	53	1	no	—
129	C₂×#41	—	53	3	no	—
130	C₂×#41	—	53	3	no	—
131	C₂×#41	#6157	1	1	yes	characteristic, complemented
132	C₂²×Q₈	—	53	1	no	—
133	C₂²×D₄	—	53	3	no	—
134	C₂³×C₄	—	53	3	no	—
135	C₄×Q₈	—	53	16	no	—
136	C₄×D₄	—	53	48	no	—
137	#117	—	53	4	no	—
138	#117	—	53	12	no	—
139	#117	—	53	12	no	—
140	#117	—	53	12	no	—
141	C₂×C₄⋊C₄	—	53	12	no	—
142	C₂×C₂²⋊C₄	—	53	12	no	—
143	C₂×C₄²	—	53	12	no	—
144	C₂⁴	—	53	3	no	—
145	#41	—	53	4	no	—
146	#41	—	53	4	no	—
147	#41	—	53	12	no	—
148	#41	—	53	12	no	—
149	#41	—	53	12	no	—
150	#41	—	53	12	no	—
151	#41	C₂×#6157	1	4	yes	complemented
152	#41	C₂×#6157	1	4	yes	complemented
153	C₂×Q₈	—	53	4	no	—
154	C₂×Q₈	—	53	4	no	—
155	C₂×Q₈	—	53	3	no	—
156	C₂×Q₈	C₂×#6157	1	1	yes	characteristic, complemented
157	C₂×D₄	—	53	6	no	—
158	C₂×D₄	—	53	12	no	—
159	C₂×D₄	—	53	12	no	—
160	C₂×D₄	—	53	3	no	—
161	C₂×D₄	C₂×#6157	1	3	yes	complemented
162	C₂²×C₄	—	53	3	no	—
163	C₂²×C₄	—	53	3	no	—
164	C₂²×C₄	—	53	3	no	—
165	C₂²×C₄	—	53	6	no	—
166	C₂²×C₄	—	53	6	no	—
167	C₂²×C₄	—	53	6	no	—
168	C₂²×C₄	—	53	6	no	—
169	C₂²×C₄	—	53	1	no	—
170	C₂²×C₄	—	53	3	no	—
171	C₂²×C₄	—	53	24	no	—
172	C₂²×C₄	C₂×#6157	1	3	yes	complemented
173	C₄⋊C₄	—	53	8	no	—
174	C₄⋊C₄	—	53	24	no	—
175	C₂²⋊C₄	—	53	8	no	—
176	C₂²⋊C₄	—	53	24	no	—
177	C₄²	—	53	8	no	—
178	C₄²	—	53	24	no	—
179	C₂³	—	53	1	no	—
180	C₂³	—	53	6	no	—
181	C₂³	—	53	6	no	—
182	C₂³	—	53	3	no	—
183	C₂³	—	212	6	no	—
184	C₂³	C₂²×#6157	1	3	yes	—
185	Q₈	—	53	12	no	—
186	Q₈	C₂²×#6157	1	4	yes	—
187	D₄	—	53	24	no	—
188	D₄	—	53	12	no	—
189	D₄	C₂²×#6157	1	12	yes	—
190	C₂×C₄	—	53	1	no	—
191	C₂×C₄	—	53	3	no	—
192	C₂×C₄	—	53	6	no	—
193	C₂×C₄	—	53	6	no	—
194	C₂×C₄	—	53	1	no	—
195	C₂×C₄	—	53	6	no	—
196	C₂×C₄	—	53	6	no	—
197	C₂×C₄	—	53	6	no	—
198	C₂×C₄	—	53	6	no	—
199	C₂×C₄	—	53	6	no	—
200	C₂×C₄	—	53	1	no	—
201	C₂×C₄	—	53	1	no	—
202	C₂×C₄	—	53	6	no	—
203	C₂×C₄	—	53	1	no	—
204	C₂×C₄	—	53	16	no	—
205	C₂×C₄	—	212	24	no	—
206	C₂×C₄	C₂²×#6157	1	3	yes	—
207	C₂×C₄	C₂²×#6157	1	6	yes	—
208	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
209	C₂×C₄	C₂²×#6157	1	6	yes	—
210	C₂²	—	4	3	no	—
211	C₂²	—	53	1	no	—
212	C₂²	—	53	1	no	—
213	C₂²	—	53	6	no	—
214	C₂²	—	106	12	no	—
215	C₂²	—	106	2	no	—
216	C₂²	—	212	3	no	—
217	C₂²	—	212	6	no	—
218	C₂²	C₂³×#6157	1	1	yes	characteristic
219	C₂²	C₂³×#6157	1	6	yes	—
220	C₄	—	53	1	no	—
221	C₄	—	53	1	no	—
222	C₄	—	53	6	no	—
223	C₄	—	106	16	no	—
224	C₄	C₂³×#6157	1	1	yes	characteristic
225	C₄	C₂³×#6157	1	6	yes	—
226	C₄	C₂³×#6157	1	1	yes	characteristic
227	C₂	—	2	1	no	—
228	C₂	—	2	6	no	—
229	C₂	—	53	2	no	—
230	C₂	—	106	1	no	—
231	C₂	—	106	6	no	—
232	C₂	C₂⁴×#6157	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂³×#6157
commutator, socle	C₂×C₅₃	C₂⁴×C₄
Frattini	C₂	C₂⁴×#6157
Fitting	C₂×#41×C₅₃	C₄
hypercenter	C₂×#41	#6157
hypocenter	C₅₃	#3428
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #16811692	C₂	semidirect
2. #16811691	C₂	semidirect
3. #16811690	C₂	semidirect
4. #41×#6157	C₂	semidirect
5. C₂×#13133296	C₂	semidirect
7. Q₈×#6157	C₂²	semidirect
8. D₄×#6157	C₂²	semidirect
11. #13133296	C₂²	semidirect
12. #13133296	C₂²	semidirect
13. C₂×C₄⋊#6157	C₂²	semidirect
15. C₂×C₄×#6157	C₂²	semidirect
16. C₂×#11719949	C₄	semidirect
17. C₂×#11719945	C₄	semidirect
18. C₂×#11719947	C₄	semidirect
19. #41×D₅₃	C₄	semidirect
20. #41×D₅₃	C₄	semidirect
26. C₂×#41×C₅₃	C₄	semidirect
29. #41×C₅₃	C₂×C₄	semidirect
30. #41×C₅₃	C₂×C₄	semidirect
31. C₂×Q₈×C₅₃	C₂×C₄	semidirect
32. C₂×D₄×C₅₃	C₂×C₄	semidirect
33. C₂²×C₄×C₅₃	C₂×C₄	semidirect
34. C₂×C₂²⋊D₅₃	C₂×C₄	semidirect
35. C₂²×Dic₅₃	C₂×C₄	semidirect
36. #11719949	C₂×C₄	semidirect
37. #11719949	C₂×C₄	semidirect
38. Q₈×D₅₃	C₂×C₄	semidirect
40. #11719947	C₂×C₄	semidirect
41. #11719947	C₂×C₄	semidirect
42. D₄×D₅₃	C₂×C₄	semidirect
44. #11719945	C₂×C₄	semidirect
45. #11719945	C₂×C₄	semidirect
46. C₂×D₂₁₂	C₂×C₄	semidirect
47. C₂×C₄×D₅₃	C₂×C₄	semidirect
48. C₂×C₄×D₅₃	C₂×C₄	semidirect
49. C₂×C₄×D₅₃	C₂×C₄	semidirect
50. C₂×C₄×D₅₃	C₂×C₄	semidirect
55. C₂×Dic₁₀₆	C₂×C₄	semidirect
123. C₅₃	#3428	semidirect
131. C₂×#41	#6157	semidirect
151. #41	C₂×#6157	semidirect
152. #41	C₂×#6157	semidirect
156. C₂×Q₈	C₂×#6157	semidirect
161. C₂×D₄	C₂×#6157	semidirect
172. C₂²×C₄	C₂×#6157	semidirect