Subgroups — #14558276

Order 2368.

260 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11924523	C₂	1	3	yes	maximal, complemented
2	#11924522	C₂	1	6	yes	maximal, complemented
3	#11924505	C₂	1	2	yes	maximal, complemented
4	#11924504	C₂	1	3	yes	maximal, complemented
5	#120×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×C₃₇	C₂²	1	1	yes	characteristic, complemented
7	C₂³×D₃₇	C₂²	1	1	yes	characteristic, complemented
8	C₂×D₄×C₃₇	C₂²	1	3	yes	—
9	C₂×C₂²⋊D₃₇	C₂²	1	6	yes	—
10	C₂×Dic₇₄	C₂²	1	3	yes	complemented
11	C₂²⋊C₄×C₃₇	C₂²	1	3	yes	complemented
12	C₂²⋊Dic₃₇	C₂²	1	3	yes	complemented
13	C₂²⋊Dic₃₇	C₂²	1	6	yes	complemented
14	#10596938	C₂²	1	6	yes	complemented
15	C₄×Dic₃₇	C₂²	1	3	yes	complemented
16	C₂³×C₃₇	—	4	3	no	—
17	C₂³×C₃₇	C₂³	1	1	yes	characteristic
18	C₂³×C₃₇	C₂³	1	3	yes	—
19	C₂²×D₃₇	—	4	3	no	—
20	C₂²×D₃₇	C₂³	1	2	yes	—
21	D₄×C₃₇	—	4	3	no	—
22	C₂×C₄×C₃₇	C₂³	1	3	yes	—
23	C₂²⋊D₃₇	—	4	6	no	—
24	C₂×Dic₃₇	C₂³	1	6	yes	—
25	Dic₇₄	—	4	3	no	—
26	C₂²×C₃₇	—	2	6	no	—
27	C₂²×C₃₇	—	2	3	no	—
28	C₂²×C₃₇	—	2	3	no	—
29	C₂²×C₃₇	—	4	4	no	—
30	C₂²×C₃₇	C₂⁴	1	1	yes	characteristic
31	C₂×D₃₇	—	2	6	no	—
32	C₂×D₃₇	—	4	4	no	—
33	C₄×C₃₇	—	2	3	no	—
34	Dic₃₇	—	2	6	no	—
35	C₂×C₃₇	—	4	1	no	—
36	C₂×C₃₇	—	4	3	no	—
37	C₂×C₃₇	H(5,𝔽₂)	1	3	yes	—
38	D₃₇	—	4	2	no	—
39	H(3,𝔽₄)	—	37	1	no	maximal, Sylow
40	C₃₇	H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
41	#124	—	37	3	no	—
42	#124	—	37	6	no	—
43	#120	—	37	2	no	—
44	#120	—	37	3	no	—
45	#120	D₃₇	1	1	yes	characteristic, complemented
46	C₂⁴	—	37	1	no	—
47	C₂⁴	C₂×D₃₇	1	1	yes	characteristic, complemented
48	C₂×Q₈	—	37	3	no	—
49	C₂×D₄	—	37	6	no	—
50	C₂×D₄	C₂×D₃₇	1	3	yes	—
51	C₂²⋊C₄	—	37	3	no	—
52	C₂²⋊C₄	—	37	6	no	—
53	C₂²⋊C₄	—	37	6	no	—
54	C₂²⋊C₄	C₂×D₃₇	1	3	yes	complemented
55	C₄²	—	37	3	no	—
56	C₂³	—	4	3	no	—
57	C₂³	—	37	2	no	—
58	C₂³	—	148	3	no	—
59	C₂³	C₂²×D₃₇	1	1	yes	characteristic
60	C₂³	C₂²×D₃₇	1	3	yes	—
61	Q₈	—	148	3	no	—
62	D₄	—	4	3	no	—
63	D₄	—	148	6	no	—
64	C₂×C₄	—	37	6	no	—
65	C₂×C₄	C₂²×D₃₇	1	3	yes	—
66	C₂²	—	2	3	no	—
67	C₂²	—	2	3	no	—
68	C₂²	—	2	6	no	—
69	C₂²	—	4	4	no	—
70	C₂²	—	74	6	no	—
71	C₂²	—	148	4	no	—
72	C₂²	C₂³×D₃₇	1	1	yes	characteristic
73	C₄	—	2	3	no	—
74	C₄	—	74	6	no	—
75	C₂	—	4	3	no	—
76	C₂	—	4	1	no	—
77	C₂	—	148	2	no	—
78	C₂	#11924581	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₇
commutator, socle, cosocle	C₂²×C₃₇	C₂⁴
Fitting	#120×C₃₇	C₂
hypercenter	#120	D₃₇
hypocenter	C₃₇	H(3,𝔽₄)
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ #120

Complements

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

N	H ≅ G/N	type
1. #11924523	C₂	semidirect
2. #11924522	C₂	semidirect
3. #11924505	C₂	semidirect
4. #11924504	C₂	semidirect
5. #120×C₃₇	C₂	semidirect
6. C₂⁴×C₃₇	C₂²	semidirect
7. C₂³×D₃₇	C₂²	semidirect
10. C₂×Dic₇₄	C₂²	semidirect
11. C₂²⋊C₄×C₃₇	C₂²	semidirect
12. C₂²⋊Dic₃₇	C₂²	semidirect
13. C₂²⋊Dic₃₇	C₂²	semidirect
14. #10596938	C₂²	semidirect
15. C₄×Dic₃₇	C₂²	semidirect
40. C₃₇	H(3,𝔽₄)	semidirect
45. #120	D₃₇	semidirect
47. C₂⁴	C₂×D₃₇	semidirect
54. C₂²⋊C₄	C₂×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
#11924523	2
#11924522	2
#11924505	2
#11924504	2
#120×C₃₇	2
H(3,𝔽₄)	37