Subgroups — #21448072

Order 7808.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₄₈₈	C₂	1	4	yes	maximal
2	#16840142	C₂	1	2	yes	maximal
3	C₄×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₂₄₄	—	2	2	no	—
5	C₂×Dic₂₄₄	C₂²	1	4	yes	—
6	#14519777	—	2	2	no	—
7	Dic₄₈₈	—	2	8	no	—
8	C₂×C₁₆×C₆₁	C₂²	1	2	yes	—
9	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
10	C₂×Dic₁₂₂	—	2	4	no	—
11	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
12	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
13	C₄²×C₆₁	D₄	1	1	yes	characteristic
14	C₄⋊Dic₆₁	—	4	4	no	—
15	Dic₂₄₄	—	2	8	no	—
16	Dic₂₄₄	—	4	4	no	—
17	C₁₆×C₆₁	D₄	1	4	yes	—
18	C₂×C₄×C₆₁	D₈	1	2	yes	—
19	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
20	C₂×Dic₆₁	—	4	4	no	—
21	Dic₁₂₂	—	4	4	no	—
22	Dic₁₂₂	—	4	4	no	—
23	C₈×C₆₁	D₈	1	2	yes	—
24	C₈×C₆₁	C₂×D₄	1	2	yes	—
25	C₂²×C₆₁	C₂×D₈	1	1	yes	characteristic
26	C₄×C₆₁	Q₃₂	1	4	yes	complemented
27	C₄×C₆₁	#127	1	1	yes	characteristic
28	C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
29	Dic₆₁	—	8	4	no	—
30	#2247	—	61	1	no	maximal, Sylow
31	C₂×C₆₁	#493	1	1	yes	characteristic
32	C₂×C₆₁	C₂×Q₃₂	1	2	yes	—
33	C₂×Q₃₂	—	61	4	no	—
34	#494	—	61	2	no	—
35	C₄×C₁₆	D₆₁	1	1	yes	characteristic
36	C₆₁	#2247	1	1	yes	characteristic, complemented, Sylow
37	C₂×Q₁₆	—	61	4	no	—
38	C₂×Q₁₆	—	122	2	no	—
39	C₄⋊Q₈	—	122	2	no	—
40	Q₃₂	—	122	8	no	—
41	C₂×C₁₆	C₂×D₆₁	1	2	yes	—
42	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
43	C₂×Q₈	—	122	4	no	—
44	Q₁₆	—	122	8	no	—
45	Q₁₆	—	244	4	no	—
46	C₂×C₈	D₂₄₄	1	1	yes	characteristic
47	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
48	C₄⋊C₄	—	244	4	no	—
49	C₄²	D₂₄₄	1	1	yes	characteristic
50	C₁₆	D₂₄₄	1	4	yes	—
51	Q₈	—	244	4	no	—
52	Q₈	—	244	4	no	—
53	C₂×C₄	—	244	4	no	—
54	C₂×C₄	D₄₈₈	1	2	yes	—
55	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
56	C₈	D₄₈₈	1	2	yes	—
57	C₈	C₂×D₂₄₄	1	2	yes	—
58	C₂²	C₂×D₄₈₈	1	1	yes	characteristic
59	C₄	—	488	4	no	—
60	C₄	Dic₄₈₈	1	4	yes	complemented
61	C₄	#14519774	1	1	yes	characteristic
62	C₄	C₂×D₄₈₈	1	1	yes	characteristic
63	C₂	#16840139	1	1	yes	characteristic
64	C₂	C₂×Dic₄₈₈	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄₈₈
commutator	C₂×C₈×C₆₁	C₂³
Frattini	C₂×C₈	C₂²×D₆₁
Fitting	C₄×C₁₆×C₆₁	C₂
socle	C₂²×C₆₁	C₂×D₈
hypercenter	C₄×C₁₆	D₆₁
hypocenter	C₆₁	#2247
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
26. C₄×C₆₁	Q₃₂	semidirect
36. C₆₁	#2247	semidirect
60. C₄	Dic₄₈₈	semidirect