Subgroups — #16840498

Order 3904.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14519764	C₂	1	4	yes	maximal, complemented
2	#14519758	C₂	1	4	yes	maximal, complemented
3	C₄×Dic₁₂₂	C₂	1	2	yes	maximal, complemented
4	D₄×Dic₆₁	C₂	1	4	yes	maximal, complemented
5	#127×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×D₄×C₆₁	C₂²	1	4	yes	—
7	C₂×D₄×C₆₁	C₂²	1	2	yes	—
8	C₂²×Dic₆₁	C₂²	1	4	yes	—
9	C₂×Dic₁₂₂	C₂²	1	2	yes	—
10	C₄²×C₆₁	C₂²	1	1	yes	characteristic
11	C₂²⋊Dic₆₁	C₂²	1	8	yes	—
12	C₂²⋊Dic₆₁	C₂²	1	4	yes	—
13	C₄⋊Dic₆₁	C₂²	1	2	yes	—
14	#11758196	C₂²	1	4	yes	—
15	C₄×Dic₆₁	C₂²	1	4	yes	—
16	C₂³×C₆₁	C₂³	1	4	yes	—
17	D₄×C₆₁	—	2	8	no	—
18	D₄×C₆₁	—	4	2	no	—
19	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₆₁	C₂³	1	2	yes	—
21	C₂×Dic₆₁	—	4	4	no	—
22	C₂×Dic₆₁	C₂³	1	4	yes	—
23	C₂×Dic₆₁	C₂³	1	4	yes	—
24	Dic₁₂₂	—	2	4	no	—
25	C₂²×C₆₁	—	2	8	no	—
26	C₂²×C₆₁	—	2	4	no	—
27	C₂²×C₆₁	C₂⁴	1	1	yes	characteristic
28	C₄×C₆₁	—	2	1	no	—
29	C₄×C₆₁	#41	1	4	yes	—
30	Dic₆₁	—	2	4	no	—
31	Dic₆₁	—	2	4	no	—
32	C₂×C₆₁	—	4	4	no	—
33	C₂×C₆₁	C₂×#41	1	2	yes	—
34	C₂×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
35	#559	—	61	1	no	maximal, Sylow
36	C₆₁	#559	1	1	yes	characteristic, complemented, Sylow
37	#127	D₆₁	1	1	yes	characteristic
38	#124	—	61	4	no	—
39	#123	—	61	4	no	—
40	C₄×Q₈	—	61	2	no	—
41	C₄×D₄	—	61	4	no	—
42	C₂×Q₈	—	61	2	no	—
43	C₂×D₄	C₂×D₆₁	1	4	yes	—
44	C₂×D₄	C₂×D₆₁	1	2	yes	—
45	C₂²×C₄	—	61	4	no	—
46	C₄⋊C₄	—	61	4	no	—
47	C₄⋊C₄	—	61	2	no	—
48	C₂²⋊C₄	—	61	8	no	—
49	C₂²⋊C₄	—	61	4	no	—
50	C₄²	—	61	4	no	—
51	C₄²	C₂×D₆₁	1	1	yes	characteristic
52	C₂³	C₂²×D₆₁	1	4	yes	—
53	Q₈	—	122	4	no	—
54	D₄	—	2	8	no	—
55	D₄	—	4	2	no	—
56	C₂×C₄	—	61	4	no	—
57	C₂×C₄	—	61	4	no	—
58	C₂×C₄	—	244	4	no	—
59	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₆₁	1	2	yes	—
61	C₂²	—	2	8	no	—
62	C₂²	—	2	4	no	—
63	C₂²	C₂³×D₆₁	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	122	4	no	—
66	C₄	—	122	4	no	—
67	C₄	#11758224	1	4	yes	—
68	C₂	—	4	4	no	—
69	C₂	C₂×#11758224	1	2	yes	—
70	C₂	#14519822	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₆₁
commutator, socle, cosocle	C₂²×C₆₁	C₂⁴
Fitting, prime radical	#127×C₆₁	C₂
hypercenter	#127	D₆₁
hypocenter	C₆₁	#559
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #14519764	C₂	semidirect
2. #14519758	C₂	semidirect
3. C₄×Dic₁₂₂	C₂	semidirect
4. D₄×Dic₆₁	C₂	semidirect
36. C₆₁	#559	semidirect

Maximal subgroups

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

subgroup	index
#14519764	2
#14519758	2
C₄×Dic₁₂₂	2
D₄×Dic₆₁	2
#127×C₆₁	2
#559	61