Subgroups — #14558368

Order 2368.

810 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11924583	C₂	1	6	yes	maximal, complemented
2	#11924581	C₂	1	9	yes	maximal, complemented
3	C₂×#10596964	C₂	1	9	yes	maximal, complemented
4	#41×D₃₇	C₂	1	6	yes	maximal, complemented
5	H(5,𝔽₂)×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#41×C₃₇	C₂²	1	6	yes	complemented
7	C₂×D₄×C₃₇	C₂²	1	9	yes	complemented
8	C₂×C₂²⋊D₃₇	C₂²	1	18	yes	complemented
9	C₂²×Dic₃₇	C₂²	1	6	yes	complemented
10	#10596966	C₂²	1	2	yes	complemented
11	Q₈×D₃₇	C₂²	1	6	yes	complemented
12	#10596964	C₂²	1	18	yes	complemented
13	#10596964	C₂²	1	36	yes	complemented
14	D₄×D₃₇	C₂²	1	18	yes	complemented
15	#10596962	C₂²	1	18	yes	complemented
16	C₂×C₄×D₃₇	C₂²	1	9	yes	complemented
17	C₂×Dic₇₄	C₂²	1	9	yes	complemented
18	C₂³×C₃₇	C₂³	1	6	yes	—
19	C₂²×D₃₇	C₂³	1	9	yes	—
20	Q₈×C₃₇	C₂³	1	2	yes	—
21	D₄×C₃₇	C₂³	1	18	yes	—
22	C₂×C₄×C₃₇	C₂³	1	9	yes	—
23	C₂²⋊D₃₇	C₂³	1	36	yes	—
24	C₂×Dic₃₇	C₂³	1	18	yes	—
25	C₂×Dic₃₇	C₂³	1	9	yes	—
26	D₁₄₈	C₂³	1	6	yes	—
27	C₄×D₃₇	C₂³	1	6	yes	—
28	C₄×D₃₇	C₂³	1	18	yes	—
29	Dic₇₄	C₂³	1	18	yes	—
30	C₂²×C₃₇	—	4	6	no	—
31	C₂²×C₃₇	C₂⁴	1	9	yes	—
32	C₂×D₃₇	—	4	9	no	—
33	C₂×D₃₇	C₂⁴	1	6	yes	—
34	C₄×C₃₇	C₂⁴	1	6	yes	—
35	Dic₃₇	C₂⁴	1	9	yes	—
36	Dic₃₇	C₂⁴	1	1	yes	characteristic
37	C₂×C₃₇	—	2	9	no	—
38	C₂×C₃₇	C₂⁵	1	1	yes	characteristic
39	D₃₇	—	2	6	no	—
40	#585	—	37	1	no	maximal, Sylow
41	C₃₇	#585	1	1	yes	characteristic, complemented, Sylow
42	2₋¹⁺⁴	—	37	6	no	—
43	H(5,𝔽₂)	—	37	9	no	—
44	H(5,𝔽₂)	D₃₇	1	1	yes	characteristic, complemented
45	C₂×#41	—	37	6	no	—
46	C₂×#41	—	37	9	no	—
47	#41	—	37	2	no	—
48	#41	—	37	18	no	—
49	#41	—	37	36	no	—
50	#41	—	37	18	no	—
51	#41	C₂×D₃₇	1	6	yes	complemented
52	C₂×Q₈	—	37	6	no	—
53	C₂×Q₈	—	37	9	no	—
54	C₂×D₄	—	37	18	no	—
55	C₂×D₄	—	37	18	no	—
56	C₂×D₄	C₂×D₃₇	1	9	yes	complemented
57	C₂²×C₄	—	37	6	no	—
58	C₂²×C₄	—	37	9	no	—
59	C₂³	—	37	9	no	—
60	C₂³	C₂²×D₃₇	1	6	yes	—
61	Q₈	—	37	18	no	—
62	Q₈	C₂²×D₃₇	1	2	yes	—
63	D₄	—	37	36	no	—
64	D₄	—	37	6	no	—
65	D₄	C₂²×D₃₇	1	18	yes	—
66	C₂×C₄	—	37	9	no	—
67	C₂×C₄	—	37	18	no	—
68	C₂×C₄	—	37	6	no	—
69	C₂×C₄	—	37	18	no	—
70	C₂×C₄	C₂²×D₃₇	1	9	yes	—
71	C₂²	—	4	6	no	—
72	C₂²	—	37	6	no	—
73	C₂²	—	148	9	no	—
74	C₂²	C₂³×D₃₇	1	9	yes	—
75	C₄	—	37	1	no	—
76	C₄	—	37	9	no	—
77	C₄	C₂³×D₃₇	1	6	yes	—
78	C₂	—	2	9	no	—
79	C₂	—	74	6	no	—
80	C₂	C₂⁴×D₃₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×D₃₇
commutator, socle, cosocle	C₂×C₃₇	C₂⁵
Fitting	H(5,𝔽₂)×C₃₇	C₂
hypercenter	H(5,𝔽₂)	D₃₇
hypocenter	C₃₇	#585
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. #11924583	C₂	semidirect
2. #11924581	C₂	semidirect
3. C₂×#10596964	C₂	semidirect
4. #41×D₃₇	C₂	semidirect
5. H(5,𝔽₂)×C₃₇	C₂	semidirect
6. #41×C₃₇	C₂²	semidirect
7. C₂×D₄×C₃₇	C₂²	semidirect
8. C₂×C₂²⋊D₃₇	C₂²	semidirect
9. C₂²×Dic₃₇	C₂²	semidirect
10. #10596966	C₂²	semidirect
11. Q₈×D₃₇	C₂²	semidirect
12. #10596964	C₂²	semidirect
13. #10596964	C₂²	semidirect
14. D₄×D₃₇	C₂²	semidirect
15. #10596962	C₂²	semidirect
16. C₂×C₄×D₃₇	C₂²	semidirect
17. C₂×Dic₇₄	C₂²	semidirect
41. C₃₇	#585	semidirect
44. H(5,𝔽₂)	D₃₇	semidirect
51. #41	C₂×D₃₇	semidirect
56. C₂×D₄	C₂×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
#11924583	2
#11924581	2
C₂×#10596964	2
#41×D₃₇	2
H(5,𝔽₂)×C₃₇	2
#585	37