Subgroups — #16840197

Order 3904.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄₈₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#14519720	C₂	1	2	yes	maximal, complemented
3	#14519718	C₂	1	2	yes	maximal, complemented
4	#14519717	C₂	1	1	yes	maximal, characteristic, complemented
5	#135×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#41×C₆₁	—	2	1	no	—
7	C₂×D₂₄₄	—	2	1	no	—
8	Q₁₆×C₆₁	C₂²	1	1	yes	characteristic, complemented
9	SD₁₆×C₆₁	—	2	1	no	—
10	D₈×C₆₁	C₂²	1	1	yes	characteristic, complemented
11	C₂×C₈×C₆₁	C₂²	1	1	yes	characteristic
12	D₄₈₈	—	2	1	no	—
13	D₄₈₈	C₂²	1	2	yes	—
14	#11758185	C₂²	1	2	yes	complemented
15	C₂²×D₆₁	—	4	1	no	—
16	Q₈×C₆₁	—	2	1	no	—
17	D₄×C₆₁	—	2	1	no	—
18	D₄×C₆₁	—	4	1	no	—
19	C₂×C₄×C₆₁	—	4	1	no	—
20	C₂×C₄×C₆₁	D₄	1	1	yes	characteristic
21	D₂₄₄	—	2	2	no	—
22	D₂₄₄	—	4	1	no	—
23	C₈×C₆₁	D₄	1	1	yes	characteristic
24	C₈×C₆₁	C₂³	1	1	yes	characteristic
25	C₂²×C₆₁	—	4	1	no	—
26	C₂²×C₆₁	D₈	1	1	yes	characteristic
27	C₂×D₆₁	—	4	2	no	—
28	C₂×D₆₁	—	8	2	no	—
29	C₄×C₆₁	—	4	1	no	—
30	C₄×C₆₁	D₈	1	1	yes	characteristic
31	C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₆₁	—	2	1	no	—
33	C₂×C₆₁	—	8	1	no	—
34	C₂×C₆₁	C₂×D₈	1	1	yes	characteristic
35	D₆₁	—	8	2	no	—
36	#509	—	61	1	no	maximal, Sylow
37	C₆₁	#509	1	1	yes	characteristic, complemented, Sylow
38	#135	D₆₁	1	1	yes	characteristic, complemented
39	C₂×D₈	—	61	1	no	—
40	SD₃₂	—	61	2	no	—
41	D₁₆	—	61	2	no	—
42	M₃₂	—	61	1	no	—
43	#41	—	2	1	no	—
44	C₂×D₄	—	122	1	no	—
45	Q₁₆	C₂×D₆₁	1	1	yes	characteristic, complemented
46	SD₁₆	—	2	1	no	—
47	D₈	—	61	2	no	—
48	D₈	—	122	1	no	—
49	D₈	C₂×D₆₁	1	1	yes	characteristic, complemented
50	C₂×C₈	C₂×D₆₁	1	1	yes	characteristic
51	C₁₆	—	61	2	no	—
52	C₂³	—	244	1	no	—
53	Q₈	—	2	1	no	—
54	D₄	—	2	1	no	—
55	D₄	—	4	1	no	—
56	D₄	—	122	2	no	—
57	D₄	—	244	1	no	—
58	C₂×C₄	—	4	1	no	—
59	C₂×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
60	C₈	C₂²⋊D₆₁	1	1	yes	characteristic
61	C₈	C₂²×D₆₁	1	1	yes	characteristic
62	C₂²	—	4	1	no	—
63	C₂²	—	488	2	no	—
64	C₂²	—	244	2	no	—
65	C₂²	#11758199	1	1	yes	characteristic
66	C₄	—	4	1	no	—
67	C₄	#11758199	1	1	yes	characteristic
68	C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	—	8	1	no	—
71	C₂	—	488	2	no	—
72	C₂	C₂×#11758199	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#11758199
commutator, cosocle	C₈×C₆₁	C₂³
Frattini	C₈	C₂²×D₆₁
Fitting	#135×C₆₁	C₂
socle	C₂×C₆₁	C₂×D₈
hypercenter	#135	D₆₁
hypocenter	C₆₁	#509
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
1. C₂×D₄₈₈	C₂	semidirect
2. #14519720	C₂	semidirect
3. #14519718	C₂	semidirect
4. #14519717	C₂	semidirect
5. #135×C₆₁	C₂	semidirect
8. Q₁₆×C₆₁	C₂²	semidirect
10. D₈×C₆₁	C₂²	semidirect
14. #11758185	C₂²	semidirect
37. C₆₁	#509	semidirect
38. #135	D₆₁	semidirect
45. Q₁₆	C₂×D₆₁	semidirect
49. D₈	C₂×D₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄₈₈	2
#14519720	2
#14519718	2
#14519717	2
#135×C₆₁	2
#509	61