Subgroups — #14558159

Order 2368.

242 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11924515	C₂	1	4	yes	maximal, complemented
2	#11924513	C₂	1	8	yes	maximal, complemented
3	C₂×C₂²⋊Dic₃₇	C₂	1	2	yes	maximal
4	C₂×C₂²⋊C₄×C₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂⁴×C₃₇	C₂²	1	1	yes	characteristic
6	C₂²×C₄×C₃₇	C₂²	1	2	yes	—
7	C₂²×Dic₃₇	C₂²	1	4	yes	—
8	C₂×Dic₇₄	C₂²	1	4	yes	complemented
9	C₂²⋊C₄×C₃₇	C₂²	1	4	yes	—
10	C₂²⋊Dic₃₇	C₂²	1	4	yes	—
11	C₂²⋊Dic₃₇	C₂²	1	4	yes	—
12	C₄⋊Dic₃₇	C₂²	1	4	yes	complemented
13	#10596936	C₂²	1	8	yes	complemented
14	C₂³×C₃₇	—	4	2	no	—
15	C₂³×C₃₇	Q₈	1	4	yes	complemented
16	C₂³×C₃₇	C₂³	1	1	yes	characteristic
17	C₂³×C₃₇	C₂³	1	2	yes	—
18	C₂×C₄×C₃₇	—	4	2	no	—
19	C₂×C₄×C₃₇	C₂³	1	4	yes	—
20	C₂×Dic₃₇	—	4	4	no	—
21	C₂×Dic₃₇	C₂³	1	8	yes	—
22	Dic₇₄	—	4	4	no	—
23	C₂²×C₃₇	—	2	4	no	—
24	C₂²×C₃₇	—	2	2	no	—
25	C₂²×C₃₇	—	4	4	no	—
26	C₂²×C₃₇	C₂×Q₈	1	2	yes	—
27	C₂²×C₃₇	C₂×Q₈	1	4	yes	—
28	C₂²×C₃₇	C₂⁴	1	1	yes	characteristic
29	C₄×C₃₇	—	2	4	no	—
30	Dic₃₇	—	2	8	no	—
31	C₂×C₃₇	—	2	4	no	—
32	C₂×C₃₇	—	2	2	no	—
33	C₂×C₃₇	C₂²×Q₈	1	1	yes	characteristic
34	C₂×C₃₇	H(5,𝔽₂)	1	2	yes	—
35	#543	—	37	1	no	maximal, Sylow
36	C₃₇	#543	1	1	yes	characteristic, complemented, Sylow
37	C₂²⋊Q₈	—	37	4	no	—
38	C₂²⋊Q₈	—	37	8	no	—
39	C₂×C₂²⋊C₄	—	37	2	no	—
40	C₂×C₂²⋊C₄	D₃₇	1	1	yes	characteristic
41	C₂⁴	C₂×D₃₇	1	1	yes	characteristic
42	C₂×Q₈	—	37	4	no	—
43	C₂²×C₄	—	37	4	no	—
44	C₂²×C₄	C₂×D₃₇	1	2	yes	—
45	C₄⋊C₄	—	37	8	no	—
46	C₄⋊C₄	—	37	4	no	—
47	C₂²⋊C₄	—	37	4	no	—
48	C₂²⋊C₄	—	37	4	no	—
49	C₂²⋊C₄	C₂×D₃₇	1	4	yes	—
50	C₂³	—	4	2	no	—
51	C₂³	Dic₇₄	1	4	yes	complemented
52	C₂³	C₂²×D₃₇	1	1	yes	characteristic
53	C₂³	C₂²×D₃₇	1	2	yes	—
54	Q₈	—	148	4	no	—
55	C₂×C₄	—	4	2	no	—
56	C₂×C₄	—	37	8	no	—
57	C₂×C₄	—	148	4	no	—
58	C₂×C₄	C₂²×D₃₇	1	4	yes	—
59	C₂²	—	2	2	no	—
60	C₂²	—	2	4	no	—
61	C₂²	—	4	4	no	—
62	C₂²	C₂×Dic₇₄	1	2	yes	—
63	C₂²	C₂×Dic₇₄	1	4	yes	—
64	C₂²	C₂³×D₃₇	1	1	yes	characteristic
65	C₄	—	2	4	no	—
66	C₄	—	74	8	no	—
67	C₂	—	2	2	no	—
68	C₂	—	2	4	no	—
69	C₂	C₂²×Dic₇₄	1	1	yes	characteristic
70	C₂	#11924581	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₇
commutator, socle, cosocle	C₂²×C₃₇	C₂⁴
Fitting	C₂×C₂²⋊C₄×C₃₇	C₂
prime radical	C₂⁴×C₃₇	C₂²
hypercenter	C₂×C₂²⋊C₄	D₃₇
hypocenter	C₃₇	#543
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. #11924515	C₂	semidirect
2. #11924513	C₂	semidirect
8. C₂×Dic₇₄	C₂²	semidirect
12. C₄⋊Dic₃₇	C₂²	semidirect
13. #10596936	C₂²	semidirect
15. C₂³×C₃₇	Q₈	semidirect
36. C₃₇	#543	semidirect
51. C₂³	Dic₇₄	semidirect

Maximal subgroups

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

subgroup	index
#11924515	2
#11924513	2
C₂×C₂²⋊Dic₃₇	2
C₂×C₂²⋊C₄×C₃₇	2
#543	37