Subgroups — H(5,𝔽₂)×D₃₇ #14558362

Order 2368.

898 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11924582	C₂	1	6	yes	maximal, complemented
2	#11924581	C₂	1	9	yes	maximal, complemented
3	#41×D₃₇	C₂	1	6	yes	maximal, complemented
4	C₂×D₄×D₃₇	C₂	1	9	yes	maximal, complemented
5	H(5,𝔽₂)×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₃₇	C₂²	1	6	yes	complemented
7	#41×C₃₇	C₂²	1	6	yes	complemented
8	C₂×D₄×C₃₇	C₂²	1	9	yes	complemented
9	C₂×C₂²⋊D₃₇	C₂²	1	18	yes	complemented
10	#10596966	C₂²	1	6	yes	complemented
11	Q₈×D₃₇	C₂²	1	2	yes	complemented
12	#10596964	C₂²	1	18	yes	complemented
13	D₄×D₃₇	C₂²	1	18	yes	complemented
14	D₄×D₃₇	C₂²	1	36	yes	complemented
15	#10596962	C₂²	1	18	yes	complemented
16	C₂×D₁₄₈	C₂²	1	9	yes	complemented
17	C₂×C₄×D₃₇	C₂²	1	9	yes	complemented
18	C₂³×C₃₇	C₂³	1	6	yes	complemented
19	C₂²×D₃₇	—	4	12	no	—
20	C₂²×D₃₇	C₂³	1	9	yes	—
21	C₂²×D₃₇	C₂³	1	18	yes	complemented
22	Q₈×C₃₇	C₂³	1	2	yes	complemented
23	D₄×C₃₇	C₂³	1	18	yes	complemented
24	C₂×C₄×C₃₇	C₂³	1	9	yes	complemented
25	C₂²⋊D₃₇	C₂³	1	36	yes	complemented
26	C₂×Dic₃₇	C₂³	1	9	yes	complemented
27	D₁₄₈	C₂³	1	18	yes	complemented
28	C₄×D₃₇	C₂³	1	6	yes	—
29	C₄×D₃₇	C₂³	1	18	yes	complemented
30	Dic₇₄	C₂³	1	6	yes	complemented
31	C₂²×C₃₇	—	4	6	no	—
32	C₂²×C₃₇	C₂⁴	1	9	yes	—
33	C₂×D₃₇	—	2	18	no	—
34	C₂×D₃₇	—	4	18	no	—
35	C₂×D₃₇	C₂⁴	1	9	yes	—
36	C₂×D₃₇	C₂⁴	1	1	yes	characteristic
37	C₄×C₃₇	C₂⁴	1	6	yes	—
38	Dic₃₇	C₂⁴	1	6	yes	—
39	C₂×C₃₇	—	2	9	no	—
40	C₂×C₃₇	C₂⁵	1	1	yes	characteristic
41	D₃₇	—	2	9	no	—
42	D₃₇	H(5,𝔽₂)	1	2	yes	direct product factor
43	C₂×H(5,𝔽₂)	—	37	1	no	maximal, Sylow
44	C₃₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
45	H(5,𝔽₂)	—	37	6	no	—
46	H(5,𝔽₂)	—	37	9	no	—
47	H(5,𝔽₂)	D₃₇	1	1	yes	characteristic, direct product factor
48	C₂×#41	—	37	6	no	—
49	C₂²×D₄	—	37	9	no	—
50	C₂⁴	—	37	6	no	—
51	#41	—	37	6	no	—
52	#41	—	37	18	no	—
53	#41	—	37	18	no	—
54	#41	C₂×D₃₇	1	6	yes	complemented
55	C₂×Q₈	—	37	2	no	—
56	C₂×D₄	—	37	18	no	—
57	C₂×D₄	—	37	18	no	—
58	C₂×D₄	—	37	36	no	—
59	C₂×D₄	—	37	9	no	—
60	C₂×D₄	C₂×D₃₇	1	9	yes	complemented
61	C₂²×C₄	—	37	9	no	—
62	C₂³	—	37	9	no	—
63	C₂³	—	37	18	no	—
64	C₂³	—	148	12	no	—
65	C₂³	C₂²×D₃₇	1	6	yes	complemented
66	Q₈	—	37	6	no	—
67	Q₈	C₂²×D₃₇	1	2	yes	complemented
68	D₄	—	37	36	no	—
69	D₄	—	37	18	no	—
70	D₄	C₂²×D₃₇	1	18	yes	complemented
71	C₂×C₄	—	37	9	no	—
72	C₂×C₄	—	37	6	no	—
73	C₂×C₄	—	37	18	no	—
74	C₂×C₄	C₂²×D₃₇	1	9	yes	complemented
75	C₂²	—	4	6	no	—
76	C₂²	—	37	1	no	—
77	C₂²	—	37	9	no	—
78	C₂²	—	74	18	no	—
79	C₂²	—	148	18	no	—
80	C₂²	C₂³×D₃₇	1	9	yes	—
81	C₄	—	37	6	no	—
82	C₄	C₂³×D₃₇	1	6	yes	—
83	C₂	—	2	9	no	—
84	C₂	—	37	2	no	—
85	C₂	—	74	9	no	—
86	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	C₂×H(5,𝔽₂)
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. #11924582	C₂	semidirect
2. #11924581	C₂	semidirect
3. #41×D₃₇	C₂	semidirect
4. C₂×D₄×D₃₇	C₂	semidirect
5. H(5,𝔽₂)×C₃₇	C₂	semidirect
6. C₂³×D₃₇	C₂²	semidirect
7. #41×C₃₇	C₂²	semidirect
8. C₂×D₄×C₃₇	C₂²	semidirect
9. C₂×C₂²⋊D₃₇	C₂²	semidirect
10. #10596966	C₂²	semidirect
11. Q₈×D₃₇	C₂²	semidirect
12. #10596964	C₂²	semidirect
13. D₄×D₃₇	C₂²	semidirect
14. D₄×D₃₇	C₂²	semidirect
15. #10596962	C₂²	semidirect
16. C₂×D₁₄₈	C₂²	semidirect
17. C₂×C₄×D₃₇	C₂²	semidirect
18. C₂³×C₃₇	C₂³	semidirect
21. C₂²×D₃₇	C₂³	semidirect
22. Q₈×C₃₇	C₂³	semidirect
23. D₄×C₃₇	C₂³	semidirect
24. C₂×C₄×C₃₇	C₂³	semidirect
25. C₂²⋊D₃₇	C₂³	semidirect
26. C₂×Dic₃₇	C₂³	semidirect
27. D₁₄₈	C₂³	semidirect
29. C₄×D₃₇	C₂³	semidirect
30. Dic₇₄	C₂³	semidirect
42. D₃₇	H(5,𝔽₂)	direct (N × H)
44. C₃₇	C₂×H(5,𝔽₂)	semidirect
47. H(5,𝔽₂)	D₃₇	direct (N × H)
54. #41	C₂×D₃₇	semidirect
60. C₂×D₄	C₂×D₃₇	semidirect
65. C₂³	C₂²×D₃₇	semidirect
67. Q₈	C₂²×D₃₇	semidirect
70. D₄	C₂²×D₃₇	semidirect
74. C₂×C₄	C₂²×D₃₇	semidirect

Maximal subgroups

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

subgroup	index
#11924582	2
#11924581	2
#41×D₃₇	2
C₂×D₄×D₃₇	2
H(5,𝔽₂)×C₃₇	2
C₂×H(5,𝔽₂)	37

Subgroups — H(5,𝔽2)×D37 #14558362

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(5,𝔽₂)×D₃₇ #14558362