Subgroups — #14558370

Order 2368.

810 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11924584	C₂	1	5	yes	maximal, complemented
2	#11924582	C₂	1	10	yes	maximal, complemented
3	C₂×#10596966	C₂	1	5	yes	maximal, complemented
4	#41×D₃₇	C₂	1	10	yes	maximal, complemented
5	2₋¹⁺⁴×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#41×C₃₇	C₂²	1	10	yes	complemented
7	C₂×Q₈×C₃₇	C₂²	1	5	yes	complemented
8	#10596966	C₂²	1	10	yes	complemented
9	#10596966	C₂²	1	20	yes	complemented
10	Q₈×D₃₇	C₂²	1	10	yes	complemented
11	#10596964	C₂²	1	10	yes	complemented
12	D₄×D₃₇	C₂²	1	30	yes	complemented
13	#10596962	C₂²	1	30	yes	complemented
14	C₂×D₁₄₈	C₂²	1	15	yes	complemented
15	C₂×C₄×D₃₇	C₂²	1	15	yes	complemented
16	C₂²×D₃₇	C₂³	1	15	yes	—
17	Q₈×C₃₇	C₂³	1	10	yes	—
18	D₄×C₃₇	C₂³	1	10	yes	—
19	C₂×C₄×C₃₇	C₂³	1	15	yes	—
20	C₂²⋊D₃₇	C₂³	1	20	yes	—
21	C₂×Dic₃₇	C₂³	1	5	yes	—
22	D₁₄₈	C₂³	1	30	yes	—
23	C₄×D₃₇	C₂³	1	10	yes	—
24	C₄×D₃₇	C₂³	1	30	yes	—
25	Dic₇₄	C₂³	1	10	yes	—
26	C₂²×C₃₇	C₂⁴	1	5	yes	—
27	C₂×D₃₇	—	4	15	no	—
28	C₂×D₃₇	C₂⁴	1	10	yes	—
29	C₄×C₃₇	C₂⁴	1	10	yes	—
30	Dic₃₇	C₂⁴	1	5	yes	—
31	Dic₃₇	C₂⁴	1	1	yes	characteristic
32	C₂×C₃₇	—	2	5	no	—
33	C₂×C₃₇	C₂⁵	1	1	yes	characteristic
34	D₃₇	—	2	10	no	—
35	#585	—	37	1	no	maximal, Sylow
36	C₃₇	#585	1	1	yes	characteristic, complemented, Sylow
37	2₋¹⁺⁴	—	37	5	no	—
38	2₋¹⁺⁴	D₃₇	1	1	yes	characteristic, complemented
39	H(5,𝔽₂)	—	37	10	no	—
40	C₂×#41	—	37	10	no	—
41	C₂×#41	—	37	5	no	—
42	#41	—	37	10	no	—
43	#41	—	37	20	no	—
44	#41	—	37	10	no	—
45	#41	—	37	30	no	—
46	#41	C₂×D₃₇	1	10	yes	complemented
47	C₂×Q₈	—	37	10	no	—
48	C₂×Q₈	C₂×D₃₇	1	5	yes	complemented
49	C₂×D₄	—	37	30	no	—
50	C₂×D₄	—	37	15	no	—
51	C₂²×C₄	—	37	15	no	—
52	C₂³	—	37	15	no	—
53	Q₈	—	37	10	no	—
54	Q₈	C₂²×D₃₇	1	10	yes	—
55	D₄	—	37	20	no	—
56	D₄	—	37	30	no	—
57	D₄	C₂²×D₃₇	1	10	yes	—
58	C₂×C₄	—	37	5	no	—
59	C₂×C₄	—	37	10	no	—
60	C₂×C₄	—	37	30	no	—
61	C₂×C₄	C₂²×D₃₇	1	15	yes	—
62	C₂²	—	37	10	no	—
63	C₂²	—	148	15	no	—
64	C₂²	C₂³×D₃₇	1	5	yes	—
65	C₄	—	37	1	no	—
66	C₄	—	37	5	no	—
67	C₄	C₂³×D₃₇	1	10	yes	—
68	C₂	—	2	5	no	—
69	C₂	—	74	10	no	—
70	C₂	C₂⁴×D₃₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×D₃₇
commutator, socle, cosocle	C₂×C₃₇	C₂⁵
Fitting	2₋¹⁺⁴×C₃₇	C₂
hypercenter	2₋¹⁺⁴	D₃₇
hypocenter	C₃₇	#585
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ 2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. #11924584	C₂	semidirect
2. #11924582	C₂	semidirect
3. C₂×#10596966	C₂	semidirect
4. #41×D₃₇	C₂	semidirect
5. 2₋¹⁺⁴×C₃₇	C₂	semidirect
6. #41×C₃₇	C₂²	semidirect
7. C₂×Q₈×C₃₇	C₂²	semidirect
8. #10596966	C₂²	semidirect
9. #10596966	C₂²	semidirect
10. Q₈×D₃₇	C₂²	semidirect
11. #10596964	C₂²	semidirect
12. D₄×D₃₇	C₂²	semidirect
13. #10596962	C₂²	semidirect
14. C₂×D₁₄₈	C₂²	semidirect
15. C₂×C₄×D₃₇	C₂²	semidirect
36. C₃₇	#585	semidirect
38. 2₋¹⁺⁴	D₃₇	semidirect
46. #41	C₂×D₃₇	semidirect
48. C₂×Q₈	C₂×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
#11924584	2
#11924582	2
C₂×#10596966	2
#41×D₃₇	2
2₋¹⁺⁴×C₃₇	2
#585	37