Subgroups — #14558270

Order 2368.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11924530	C₂	1	2	yes	maximal, complemented
2	#11924517	C₂	1	3	yes	maximal, complemented
3	#11924508	C₂	1	6	yes	maximal, complemented
4	#11924504	C₂	1	3	yes	maximal, complemented
5	#127×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₃₇	C₂²	1	1	yes	characteristic
7	C₂×D₄×C₃₇	C₂²	1	3	yes	—
8	C₂×D₄×C₃₇	C₂²	1	3	yes	complemented
9	C₂×C₂²⋊D₃₇	C₂²	1	6	yes	—
10	C₂²×Dic₃₇	C₂²	1	3	yes	complemented
11	C₄²×C₃₇	C₂²	1	1	yes	characteristic, complemented
12	C₂²⋊Dic₃₇	C₂²	1	3	yes	—
13	C₂²⋊Dic₃₇	C₂²	1	3	yes	complemented
14	#10596938	C₂²	1	6	yes	—
15	#10596936	C₂²	1	6	yes	complemented
16	C₂³×C₃₇	C₂³	1	1	yes	characteristic
17	C₂³×C₃₇	C₂³	1	3	yes	—
18	C₂²×D₃₇	—	4	3	no	—
19	C₂²×D₃₇	C₂³	1	2	yes	—
20	D₄×C₃₇	—	4	3	no	—
21	D₄×C₃₇	—	4	3	no	—
22	C₂×C₄×C₃₇	C₂³	1	3	yes	—
23	C₂²⋊D₃₇	—	4	6	no	—
24	C₂×Dic₃₇	—	4	3	no	—
25	C₂×Dic₃₇	C₂³	1	6	yes	—
26	C₂²×C₃₇	—	2	3	no	—
27	C₂²×C₃₇	—	2	6	no	—
28	C₂²×C₃₇	—	2	3	no	—
29	C₂²×C₃₇	C₂⁴	1	1	yes	characteristic
30	C₂×D₃₇	—	2	6	no	—
31	C₂×D₃₇	—	4	4	no	—
32	C₄×C₃₇	—	2	3	no	—
33	Dic₃₇	—	2	6	no	—
34	C₂×C₃₇	—	4	1	no	—
35	C₂×C₃₇	—	4	3	no	—
36	C₂×C₃₇	H(5,𝔽₂)	1	3	yes	—
37	D₃₇	—	4	2	no	—
38	#560	—	37	1	no	maximal, Sylow
39	C₃₇	#560	1	1	yes	characteristic, complemented, Sylow
40	#127	D₃₇	1	1	yes	characteristic, complemented
41	#126	—	37	2	no	—
42	#123	—	37	3	no	—
43	#121	—	37	6	no	—
44	#120	—	37	3	no	—
45	C₂⁴	—	37	1	no	—
46	C₂×D₄	—	37	6	no	—
47	C₂×D₄	C₂×D₃₇	1	3	yes	—
48	C₂×D₄	C₂×D₃₇	1	3	yes	complemented
49	C₂²×C₄	—	37	3	no	—
50	C₄⋊C₄	—	37	6	no	—
51	C₂²⋊C₄	—	37	3	no	—
52	C₂²⋊C₄	—	37	3	no	—
53	C₂²⋊C₄	—	37	6	no	—
54	C₄²	C₂×D₃₇	1	1	yes	characteristic, complemented
55	C₂³	—	37	2	no	—
56	C₂³	—	148	3	no	—
57	C₂³	C₂²×D₃₇	1	1	yes	characteristic
58	C₂³	C₂²×D₃₇	1	3	yes	—
59	D₄	—	4	3	no	—
60	D₄	—	4	3	no	—
61	D₄	—	148	6	no	—
62	C₂×C₄	—	37	6	no	—
63	C₂×C₄	—	148	3	no	—
64	C₂×C₄	C₂²×D₃₇	1	3	yes	—
65	C₂²	—	2	3	no	—
66	C₂²	—	2	6	no	—
67	C₂²	—	2	3	no	—
68	C₂²	—	74	6	no	—
69	C₂²	—	148	4	no	—
70	C₂²	C₂³×D₃₇	1	1	yes	characteristic
71	C₄	—	2	3	no	—
72	C₄	—	74	6	no	—
73	C₂	—	4	1	no	—
74	C₂	—	4	3	no	—
75	C₂	—	148	2	no	—
76	C₂	#11924581	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₇
commutator, socle, cosocle	C₂²×C₃₇	C₂⁴
Fitting	#127×C₃₇	C₂
hypercenter	#127	D₃₇
hypocenter	C₃₇	#560
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #11924530	C₂	semidirect
2. #11924517	C₂	semidirect
3. #11924508	C₂	semidirect
4. #11924504	C₂	semidirect
5. #127×C₃₇	C₂	semidirect
8. C₂×D₄×C₃₇	C₂²	semidirect
10. C₂²×Dic₃₇	C₂²	semidirect
11. C₄²×C₃₇	C₂²	semidirect
13. C₂²⋊Dic₃₇	C₂²	semidirect
15. #10596936	C₂²	semidirect
39. C₃₇	#560	semidirect
40. #127	D₃₇	semidirect
48. C₂×D₄	C₂×D₃₇	semidirect
54. C₄²	C₂×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
#11924530	2
#11924517	2
#11924508	2
#11924504	2
#127×C₃₇	2
#560	37