Subgroups — C₂×C₂²⋊#15690200 #37566977

Order 24352.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15690200	C₂	1	2	yes	maximal, complemented
2	C₂²⋊#15690200	C₂	1	4	yes	maximal, direct product factor
3	C₂³×D₇₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#15690200	—	2	4	no	—
5	C₂×#15690200	C₂²	1	4	yes	complemented
6	C₂²×D₇₆₁	—	2	4	no	—
7	C₂²×D₇₆₁	C₄	1	2	yes	complemented
8	C₂²×D₇₆₁	C₄	1	1	yes	characteristic, complemented
9	C₂²×D₇₆₁	C₂²	1	2	yes	—
10	C₂²×D₇₆₁	C₂²	1	1	yes	characteristic
11	C₂³×C₇₆₁	C₄	1	1	yes	characteristic, complemented
12	C₂²×C₇₆₁	—	2	2	no	—
13	C₂²×C₇₆₁	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₇₆₁	C₂×C₄	1	2	yes	complemented
15	C₂×D₇₆₁	—	2	2	no	—
16	C₂×D₇₆₁	—	2	4	no	—
17	C₂×D₇₆₁	—	2	4	no	—
18	C₂×D₇₆₁	C₂×C₄	1	1	yes	characteristic
19	C₂×D₇₆₁	C₂×C₄	1	2	yes	complemented
20	C₂×D₇₆₁	D₄	1	4	yes	—
21	C₂×D₇₆₁	C₂³	1	1	yes	characteristic
22	#15690200	—	2	4	no	—
23	C₂×C₇₆₁	—	2	2	no	—
24	C₂×C₇₆₁	C₂²⋊C₄	1	2	yes	complemented
25	C₂×C₇₆₁	C₂²×C₄	1	1	yes	characteristic
26	D₇₆₁	—	2	2	no	—
27	D₇₆₁	C₂²⋊C₄	1	2	yes	complemented
28	D₇₆₁	C₂×D₄	1	2	yes	—
29	C₇₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
30	C₂×C₂²⋊C₄	—	761	1	no	maximal, Sylow
31	C₂⁴	—	761	1	no	—
32	C₂²×C₄	—	761	2	no	—
33	C₂²⋊C₄	—	761	4	no	—
34	C₂³	—	1522	4	no	—
35	C₂³	—	761	1	no	—
36	C₂³	—	761	2	no	—
37	C₂³	—	761	2	no	—
38	C₂³	—	761	1	no	—
39	C₂³	#15690200	1	1	yes	characteristic, complemented
40	C₂×C₄	—	1522	4	no	—
41	C₂×C₄	—	761	4	no	—
42	C₂²	—	2	2	no	—
43	C₂²	—	1522	4	no	—
44	C₂²	—	1522	4	no	—
45	C₂²	—	1522	2	no	—
46	C₂²	—	761	2	no	—
47	C₂²	—	761	4	no	—
48	C₂²	—	761	1	no	—
49	C₂²	—	761	1	no	—
50	C₂²	C₂×#15690200	1	1	yes	characteristic
51	C₂²	C₂×#15690200	1	2	yes	complemented
52	C₄	—	1522	4	no	—
53	C₂	—	2	2	no	—
54	C₂	—	1522	2	no	—
55	C₂	—	761	2	no	—
56	C₂	—	761	2	no	—
57	C₂	C₂²⋊#15690200	1	2	yes	direct product factor
58	C₂	C₂²×#15690200	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15690200
commutator	C₂×C₇₆₁	C₂²×C₄
Frattini	C₂	C₂²×#15690200
Fitting	C₂³×C₇₆₁	C₄
socle	C₂²×C₇₆₁	C₂×C₄
prime radical	C₂³×D₇₆₁	C₂
cosocle	C₂×D₇₆₁	C₂³
hypercenter	C₂³	#15690200
hypocenter	C₇₆₁	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₇₆₁ ▷ C₇₆₁
UCS	C₁ ◁ C₂² ◁ C₂³

Complements

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

N	H ≅ G/N	type
1. C₂²×#15690200	C₂	semidirect
2. C₂²⋊#15690200	C₂	direct (N × H)
5. C₂×#15690200	C₂²	semidirect
7. C₂²×D₇₆₁	C₄	semidirect
8. C₂²×D₇₆₁	C₄	semidirect
11. C₂³×C₇₆₁	C₄	semidirect
14. C₂²×C₇₆₁	C₂×C₄	semidirect
19. C₂×D₇₆₁	C₂×C₄	semidirect
24. C₂×C₇₆₁	C₂²⋊C₄	semidirect
27. D₇₆₁	C₂²⋊C₄	semidirect
29. C₇₆₁	C₂×C₂²⋊C₄	semidirect
39. C₂³	#15690200	semidirect
51. C₂²	C₂×#15690200	semidirect
57. C₂	C₂²⋊#15690200	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#15690200	2
C₂²⋊#15690200	2
C₂³×D₇₆₁	2
C₂×C₂²⋊C₄	761

Subgroups — C2×C22⋊#15690200 #37566977

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂²⋊#15690200 #37566977