Subgroups — #16839840

Order 3904.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11758194	C₂	1	2	yes	maximal, complemented
2	C₄⋊#92418	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#92418	C₂	1	2	yes	maximal, complemented
4	C₄×#92418	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₄×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×D₄×C₆₁	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₆₁	C₂²	1	2	yes	—
8	C₄⋊C₄×C₆₁	C₄	1	1	yes	characteristic
9	C₂²⋊C₄×C₆₁	C₄	1	2	yes	—
10	C₄²×C₆₁	C₂²	1	1	yes	characteristic
11	#11758194	—	2	4	no	—
12	C₂×#92418	C₂²	1	1	yes	characteristic
13	C₂×#92418	C₂²	1	2	yes	—
14	C₂×#92418	C₂²	1	1	yes	characteristic
15	C₂³×C₆₁	C₂×C₄	1	2	yes	—
16	D₄×C₆₁	—	2	2	no	—
17	C₂×C₄×C₆₁	—	2	2	no	—
18	C₂×C₄×C₆₁	—	2	2	no	—
19	C₂×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₆₁	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
23	#92418	—	2	2	no	—
24	#92418	—	2	1	no	—
25	#92418	D₄	1	2	yes	complemented
26	C₂²×C₆₁	—	2	2	no	—
27	C₂²×C₆₁	—	2	2	no	—
28	C₂²×C₆₁	—	2	2	no	—
29	C₂²×C₆₁	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₆₁	—	2	2	no	—
31	C₄×C₆₁	—	2	1	no	—
32	C₄×C₆₁	M₁₆	1	2	yes	complemented
33	C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
34	C₄×C₆₁	#41	1	1	yes	characteristic
35	C₂×C₆₁	—	4	2	no	—
36	C₂×C₆₁	C₄×D₄	1	1	yes	characteristic
37	C₂×C₆₁	C₂×M₁₆	1	1	yes	characteristic
38	C₂×C₆₁	#131	1	1	yes	characteristic
39	#436	—	61	1	no	maximal, Sylow
40	C₆₁	#436	1	1	yes	characteristic, complemented, Sylow
41	C₂×M₁₆	—	61	2	no	—
42	C₄×D₄	D₆₁	1	1	yes	characteristic
43	C₄⋊C₈	—	61	1	no	—
44	C₂²⋊C₈	—	61	2	no	—
45	C₄×C₈	—	61	1	no	—
46	C₂×D₄	Dic₆₁	1	1	yes	characteristic
47	C₂²×C₄	C₂×D₆₁	1	2	yes	—
48	M₁₆	—	122	4	no	—
49	C₂×C₈	—	61	1	no	—
50	C₂×C₈	—	61	1	no	—
51	C₂×C₈	—	61	2	no	—
52	C₄⋊C₄	Dic₆₁	1	1	yes	characteristic
53	C₂²⋊C₄	Dic₆₁	1	2	yes	—
54	C₄²	C₂×D₆₁	1	1	yes	characteristic
55	C₂³	C₂×Dic₆₁	1	2	yes	—
56	D₄	—	2	2	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	C₂×Dic₆₁	1	1	yes	characteristic
60	C₂×C₄	C₂×Dic₆₁	1	2	yes	—
61	C₂×C₄	C₂×Dic₆₁	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
63	C₈	—	61	2	no	—
64	C₈	—	122	2	no	—
65	C₈	—	122	1	no	—
66	C₂²	—	2	2	no	—
67	C₂²	—	2	2	no	—
68	C₂²	—	2	2	no	—
69	C₂²	C₂²×Dic₆₁	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	2	1	no	—
72	C₄	#11758194	1	2	yes	complemented
73	C₄	D₄×D₆₁	1	1	yes	characteristic
74	C₄	#11758224	1	1	yes	characteristic
75	C₂	—	4	2	no	—
76	C₂	D₄×Dic₆₁	1	1	yes	characteristic
77	C₂	C₂×#11758194	1	1	yes	characteristic
78	C₂	#14519784	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₆₁
commutator, socle	C₂²×C₆₁	C₂²×C₄
Fitting	C₄×D₄×C₆₁	C₂
prime radical	C₂×D₄×C₆₁	C₄
cosocle	C₂×C₄×C₆₁	C₂³
hypercenter	C₄×D₄	D₆₁
hypocenter	C₆₁	#436
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#11758194	C₂	semidirect
2. C₄⋊#92418	C₂	semidirect
3. C₂²⋊#92418	C₂	semidirect
4. C₄×#92418	C₂	semidirect
25. #92418	D₄	semidirect
32. C₄×C₆₁	M₁₆	semidirect
40. C₆₁	#436	semidirect
72. C₄	#11758194	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11758194	2
C₄⋊#92418	2
C₂²⋊#92418	2
C₄×#92418	2
C₄×D₄×C₆₁	2
#436	61