Subgroups — H(5,𝔽₂)×#3861 #72246152

Order 4704.

1660 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#3861	C₂	1	6	yes	maximal, complemented
2	C₂×D₄×#3861	C₂	1	9	yes	maximal, complemented
3	C₇²×H(5,𝔽₂)	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂³×#3861	C₂²	1	6	yes	complemented
5	Q₈×#3861	C₂²	1	2	yes	complemented
6	D₄×#3861	C₂²	1	18	yes	complemented
7	C₂×C₄×#3861	C₂²	1	9	yes	complemented
8	C₇²×#41	C₂×C₃	1	6	yes	complemented
9	C₂×C₇²×D₄	C₂×C₃	1	9	yes	complemented
10	ΓL(1,𝔽₈)×H(5,𝔽₂)	—	7	8	no	maximal
11	C₂²×#3861	—	4	6	no	—
12	C₂²×#3861	C₂³	1	9	yes	—
13	C₄×#3861	C₂³	1	6	yes	—
14	C₂³×C₇²	C₂²×C₃	1	6	yes	complemented
15	C₇²×Q₈	C₂²×C₃	1	2	yes	complemented
16	C₇²×D₄	C₂²×C₃	1	18	yes	complemented
17	C₂×C₄×C₇²	C₂²×C₃	1	9	yes	complemented
18	#41×ΓL(1,𝔽₈)	—	7	48	no	—
19	C₂×D₄×ΓL(1,𝔽₈)	—	7	72	no	—
20	C₂×#3861	—	2	9	no	—
21	C₂×#3861	C₂⁴	1	1	yes	characteristic
22	C₇×H(5,𝔽₂)	ΓL(1,𝔽₈)	1	8	yes	complemented
23	C₂²×C₇²	—	4	6	no	—
24	C₂²×C₇²	C₂³×C₃	1	9	yes	—
25	C₄×C₇²	C₂³×C₃	1	6	yes	—
26	C₂³×ΓL(1,𝔽₈)	—	7	48	no	—
27	Q₈×ΓL(1,𝔽₈)	—	7	16	no	—
28	D₄×ΓL(1,𝔽₈)	—	7	144	no	—
29	C₂×C₄×ΓL(1,𝔽₈)	—	7	72	no	—
30	#3861	H(5,𝔽₂)	1	1	yes	characteristic, direct product factor, Hall
31	C₇×#41	C₂×ΓL(1,𝔽₈)	1	48	yes	complemented
32	C₂×C₇×D₄	C₂×ΓL(1,𝔽₈)	1	72	yes	complemented
33	C₂×C₇²	—	2	9	no	—
34	C₂×C₇²	C₂⁴×C₃	1	1	yes	characteristic
35	C₃×H(5,𝔽₂)	—	49	1	no	Hall
36	C₂²×ΓL(1,𝔽₈)	—	7	72	no	—
37	C₂²×ΓL(1,𝔽₈)	—	28	48	no	—
38	C₄×ΓL(1,𝔽₈)	—	7	48	no	—
39	C₂³×C₇	C₂²×ΓL(1,𝔽₈)	1	48	yes	complemented
40	C₇×Q₈	C₂²×ΓL(1,𝔽₈)	1	16	yes	complemented
41	C₇×D₄	C₂²×ΓL(1,𝔽₈)	1	144	yes	complemented
42	C₂×C₄×C₇	C₂²×ΓL(1,𝔽₈)	1	72	yes	complemented
43	C₇²	C₃×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
44	C₃×#41	—	49	6	no	—
45	C₂×C₃×D₄	—	49	9	no	—
46	C₂×ΓL(1,𝔽₈)	—	7	8	no	—
47	C₂×ΓL(1,𝔽₈)	—	14	72	no	—
48	H(5,𝔽₂)	#3861	1	1	yes	characteristic, direct product factor, Sylow
49	C₂²×C₇	—	4	48	no	—
50	C₂²×C₇	C₂³×ΓL(1,𝔽₈)	1	72	yes	—
51	C₄×C₇	C₂³×ΓL(1,𝔽₈)	1	48	yes	—
52	C₂³×C₃	—	49	6	no	—
53	C₃×Q₈	—	49	2	no	—
54	C₃×D₄	—	49	18	no	—
55	C₂×C₃×C₄	—	49	9	no	—
56	ΓL(1,𝔽₈)	—	7	8	no	—
57	#41	C₂×#3861	1	6	yes	complemented
58	C₂×D₄	C₂×#3861	1	9	yes	complemented
59	C₂×C₇	—	2	72	no	—
60	C₂×C₇	C₂⁴×ΓL(1,𝔽₈)	1	8	yes	—
61	C₂²×C₃	—	49	9	no	—
62	C₂²×C₃	—	196	6	no	—
63	C₃×C₄	—	49	6	no	—
64	C₂³	C₂²×#3861	1	6	yes	complemented
65	Q₈	C₂²×#3861	1	2	yes	complemented
66	D₄	C₂²×#3861	1	18	yes	complemented
67	C₂×C₄	C₂²×#3861	1	9	yes	complemented
68	C₇	ΓL(1,𝔽₈)×H(5,𝔽₂)	1	8	yes	complemented
69	C₂×C₃	—	49	1	no	—
70	C₂×C₃	—	98	9	no	—
71	C₂²	—	4	6	no	—
72	C₂²	C₂³×#3861	1	9	yes	—
73	C₄	C₂³×#3861	1	6	yes	—
74	C₃	—	49	1	no	Sylow
75	C₂	—	2	9	no	—
76	C₂	C₂⁴×#3861	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×#3861
commutator, socle, cosocle	C₂×C₇²	C₂⁴×C₃
Fitting	C₇²×H(5,𝔽₂)	C₃
hypercenter	H(5,𝔽₂)	#3861
hypocenter	C₇²	C₃×H(5,𝔽₂)
LCS	G ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. #41×#3861	C₂	semidirect
2. C₂×D₄×#3861	C₂	semidirect
3. C₇²×H(5,𝔽₂)	C₃	semidirect
4. C₂³×#3861	C₂²	semidirect
5. Q₈×#3861	C₂²	semidirect
6. D₄×#3861	C₂²	semidirect
7. C₂×C₄×#3861	C₂²	semidirect
8. C₇²×#41	C₂×C₃	semidirect
9. C₂×C₇²×D₄	C₂×C₃	semidirect
14. C₂³×C₇²	C₂²×C₃	semidirect
15. C₇²×Q₈	C₂²×C₃	semidirect
16. C₇²×D₄	C₂²×C₃	semidirect
17. C₂×C₄×C₇²	C₂²×C₃	semidirect
22. C₇×H(5,𝔽₂)	ΓL(1,𝔽₈)	semidirect
30. #3861	H(5,𝔽₂)	direct (N × H)
31. C₇×#41	C₂×ΓL(1,𝔽₈)	semidirect
32. C₂×C₇×D₄	C₂×ΓL(1,𝔽₈)	semidirect
39. C₂³×C₇	C₂²×ΓL(1,𝔽₈)	semidirect
40. C₇×Q₈	C₂²×ΓL(1,𝔽₈)	semidirect
41. C₇×D₄	C₂²×ΓL(1,𝔽₈)	semidirect
42. C₂×C₄×C₇	C₂²×ΓL(1,𝔽₈)	semidirect
43. C₇²	C₃×H(5,𝔽₂)	semidirect
48. H(5,𝔽₂)	#3861	direct (N × H)
57. #41	C₂×#3861	semidirect
58. C₂×D₄	C₂×#3861	semidirect
64. C₂³	C₂²×#3861	semidirect
65. Q₈	C₂²×#3861	semidirect
66. D₄	C₂²×#3861	semidirect
67. C₂×C₄	C₂²×#3861	semidirect
68. C₇	ΓL(1,𝔽₈)×H(5,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
#41×#3861	2
C₂×D₄×#3861	2
C₇²×H(5,𝔽₂)	3
ΓL(1,𝔽₈)×H(5,𝔽₂)	7

Subgroups — H(5,𝔽2)×#3861 #72246152

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(5,𝔽₂)×#3861 #72246152