Subgroups — #11924513

Order 1184.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₇₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊C₄×C₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂²⋊Dic₃₇	C₂	1	1	yes	maximal, characteristic
5	C₄⋊Dic₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#10596936	C₂	1	1	yes	maximal, characteristic, complemented
7	#10596936	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₃₇	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₃₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₃₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₇	—	2	2	no	—
12	C₂×Dic₃₇	C₂²	1	1	yes	characteristic
13	C₂×Dic₃₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₇	C₂²	1	1	yes	characteristic
16	Dic₇₄	—	2	2	no	—
17	C₂²×C₃₇	—	2	1	no	—
18	C₂²×C₃₇	—	2	1	no	—
19	C₂²×C₃₇	Q₈	1	2	yes	complemented
20	C₂²×C₃₇	C₂³	1	1	yes	characteristic
21	C₄×C₃₇	—	2	1	no	—
22	C₄×C₃₇	—	2	1	no	—
23	Dic₃₇	—	2	1	no	—
24	Dic₃₇	—	2	1	no	—
25	Dic₃₇	—	2	1	no	—
26	Dic₃₇	D₄	1	2	yes	—
27	C₂×C₃₇	—	2	2	no	—
28	C₂×C₃₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₃₇	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₃₇	#41	1	1	yes	characteristic
31	C₃₇	C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
32	C₂²⋊Q₈	—	37	1	no	maximal, Sylow
33	C₂×Q₈	—	37	1	no	—
34	C₂²×C₄	—	37	1	no	—
35	C₄⋊C₄	—	37	1	no	—
36	C₄⋊C₄	—	37	1	no	—
37	C₄⋊C₄	—	37	1	no	—
38	C₂²⋊C₄	—	37	1	no	—
39	C₂²⋊C₄	D₃₇	1	1	yes	characteristic
40	C₂³	C₂×D₃₇	1	1	yes	characteristic
41	Q₈	—	74	2	no	—
42	C₂×C₄	—	37	1	no	—
43	C₂×C₄	—	37	1	no	—
44	C₂×C₄	—	37	1	no	—
45	C₂×C₄	—	37	1	no	—
46	C₂×C₄	—	74	2	no	—
47	C₂×C₄	C₂×D₃₇	1	1	yes	characteristic
48	C₂×C₄	C₂×D₃₇	1	1	yes	characteristic
49	C₂²	—	2	1	no	—
50	C₂²	—	2	1	no	—
51	C₂²	Dic₇₄	1	2	yes	complemented
52	C₂²	C₂²×D₃₇	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	2	1	no	—
55	C₄	—	37	2	no	—
56	C₄	—	74	1	no	—
57	C₄	—	74	1	no	—
58	C₄	—	74	1	no	—
59	C₂	—	2	2	no	—
60	C₂	C₂×Dic₇₄	1	1	yes	characteristic
61	C₂	D₄×D₃₇	1	1	yes	characteristic
62	C₂	#10596964	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃₇
commutator, socle, cosocle	C₂²×C₃₇	C₂³
Fitting	C₂²⋊C₄×C₃₇	C₂
prime radical	C₂³×C₃₇	C₂²
hypercenter	C₂²⋊C₄	D₃₇
hypocenter	C₃₇	C₂²⋊Q₈
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₇₄	C₂	semidirect
5. C₄⋊Dic₃₇	C₂	semidirect
6. #10596936	C₂	semidirect
7. #10596936	C₂	semidirect
19. C₂²×C₃₇	Q₈	semidirect
31. C₃₇	C₂²⋊Q₈	semidirect
51. C₂²	Dic₇₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₃₇	2
C₂×Dic₇₄	2
C₂²⋊C₄×C₃₇	2
C₂²⋊Dic₃₇	2
C₄⋊Dic₃₇	2
#10596936	2
C₂²⋊Q₈	37