Subgroups — #16840510

Order 3904.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14519765	C₂	1	2	yes	maximal, complemented
2	#14519762	C₂	1	4	yes	maximal, complemented
3	#14519755	C₂	1	4	yes	maximal, complemented
4	#14519752	C₂	1	4	yes	maximal, complemented
5	#125×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₂₄₄	C₂²	1	4	yes	—
7	C₂×D₂₄₄	C₂²	1	2	yes	—
8	C₂×C₄×D₆₁	C₂²	1	4	yes	—
9	C₂×Dic₁₂₂	C₂²	1	2	yes	—
10	C₄⋊C₄×C₆₁	C₂²	1	4	yes	—
11	C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
12	C₄²×C₆₁	C₂²	1	1	yes	characteristic
13	#11758198	C₂²	1	4	yes	—
14	#11758198	C₂²	1	4	yes	—
15	#11758198	C₂²	1	4	yes	—
16	#11758196	C₂²	1	4	yes	—
17	C₂²×D₆₁	C₂³	1	4	yes	—
18	C₂×C₄×C₆₁	C₂³	1	4	yes	—
19	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₆₁	C₂³	1	2	yes	—
21	C₂×Dic₆₁	C₂³	1	4	yes	—
22	D₂₄₄	—	4	2	no	—
23	D₂₄₄	—	4	4	no	—
24	C₄×D₆₁	—	4	4	no	—
25	Dic₁₂₂	—	4	2	no	—
26	C₂²×C₆₁	C₂⁴	1	1	yes	characteristic
27	C₂×D₆₁	—	2	4	no	—
28	C₂×D₆₁	—	2	4	no	—
29	C₂×D₆₁	—	2	4	no	—
30	C₄×C₆₁	—	2	2	no	—
31	C₄×C₆₁	—	2	4	no	—
32	C₄×C₆₁	—	2	1	no	—
33	Dic₆₁	—	2	4	no	—
34	C₂×C₆₁	H(5,𝔽₂)	1	2	yes	—
35	C₂×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
36	D₆₁	—	4	4	no	—
37	#562	—	61	1	no	maximal, Sylow
38	C₆₁	#562	1	1	yes	characteristic, complemented, Sylow
39	#125	D₆₁	1	1	yes	characteristic, complemented
40	#124	—	61	2	no	—
41	#123	—	61	4	no	—
42	C₂²⋊Q₈	—	61	4	no	—
43	#121	—	61	4	no	—
44	C₂×Q₈	—	61	2	no	—
45	C₂×D₄	—	61	4	no	—
46	C₂×D₄	—	61	2	no	—
47	C₂²×C₄	—	61	4	no	—
48	C₄⋊C₄	—	61	4	no	—
49	C₄⋊C₄	C₂×D₆₁	1	4	yes	—
50	C₄⋊C₄	C₂×D₆₁	1	2	yes	—
51	C₂²⋊C₄	—	61	4	no	—
52	C₂²⋊C₄	—	61	4	no	—
53	C₂²⋊C₄	—	61	4	no	—
54	C₄²	C₂×D₆₁	1	1	yes	characteristic
55	C₂³	—	61	4	no	—
56	Q₈	—	244	2	no	—
57	D₄	—	244	4	no	—
58	D₄	—	244	2	no	—
59	C₂×C₄	—	61	4	no	—
60	C₂×C₄	—	244	4	no	—
61	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₆₁	1	2	yes	—
63	C₂×C₄	C₂²×D₆₁	1	4	yes	—
64	C₂²	—	122	4	no	—
65	C₂²	—	122	4	no	—
66	C₂²	—	122	4	no	—
67	C₂²	C₂³×D₆₁	1	1	yes	characteristic
68	C₄	—	2	4	no	—
69	C₄	—	2	2	no	—
70	C₄	—	2	1	no	—
71	C₄	—	122	4	no	—
72	C₂	—	244	4	no	—
73	C₂	#14519823	1	2	yes	—
74	C₂	#14519825	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₆₁
commutator, socle, cosocle	C₂²×C₆₁	C₂⁴
Fitting	#125×C₆₁	C₂
hypercenter	#125	D₆₁
hypocenter	C₆₁	#562
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. #14519765	C₂	semidirect
2. #14519762	C₂	semidirect
3. #14519755	C₂	semidirect
4. #14519752	C₂	semidirect
5. #125×C₆₁	C₂	semidirect
38. C₆₁	#562	semidirect
39. #125	D₆₁	semidirect

Maximal subgroups

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

subgroup	index
#14519765	2
#14519762	2
#14519755	2
#14519752	2
#125×C₆₁	2
#562	61