Subgroups — H(5,𝔽₂)×#3862 #72246194

Order 4704.

830 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#3862	C₂	1	6	yes	maximal, complemented
2	C₂×D₄×#3862	C₂	1	9	yes	maximal, complemented
3	C₇²×H(5,𝔽₂)	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂³×#3862	C₂²	1	6	yes	complemented
5	Q₈×#3862	C₂²	1	2	yes	complemented
6	D₄×#3862	C₂²	1	18	yes	complemented
7	C₂×C₄×#3862	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	2	no	maximal
11	C₂²×#3862	—	4	6	no	—
12	C₂²×#3862	C₂³	1	9	yes	—
13	C₄×#3862	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	12	no	—
19	C₂×D₄×ΓL(1,𝔽₈)	—	7	18	no	—
20	C₂×#3862	—	2	9	no	—
21	C₂×#3862	C₂⁴	1	1	yes	characteristic
22	C₇×H(5,𝔽₂)	—	3	2	no	—
23	C₇×H(5,𝔽₂)	ΓL(1,𝔽₈)	1	2	yes	complemented
24	C₂²×C₇²	—	4	6	no	—
25	C₂²×C₇²	C₂³×C₃	1	9	yes	—
26	C₄×C₇²	C₂³×C₃	1	6	yes	—
27	C₂³×ΓL(1,𝔽₈)	—	7	12	no	—
28	Q₈×ΓL(1,𝔽₈)	—	7	4	no	—
29	D₄×ΓL(1,𝔽₈)	—	7	36	no	—
30	C₂×C₄×ΓL(1,𝔽₈)	—	7	18	no	—
31	#3862	H(5,𝔽₂)	1	1	yes	characteristic, direct product factor, Hall
32	C₇×#41	—	3	12	no	—
33	C₇×#41	C₂×ΓL(1,𝔽₈)	1	12	yes	complemented
34	C₂×C₇×D₄	—	3	18	no	—
35	C₂×C₇×D₄	C₂×ΓL(1,𝔽₈)	1	18	yes	complemented
36	C₂×C₇²	—	2	9	no	—
37	C₂×C₇²	C₂⁴×C₃	1	1	yes	characteristic
38	C₃×H(5,𝔽₂)	—	49	1	no	Hall
39	C₂²×ΓL(1,𝔽₈)	—	7	18	no	—
40	C₂²×ΓL(1,𝔽₈)	—	28	12	no	—
41	C₄×ΓL(1,𝔽₈)	—	7	12	no	—
42	C₂³×C₇	—	3	12	no	—
43	C₂³×C₇	C₂²×ΓL(1,𝔽₈)	1	12	yes	complemented
44	C₇×Q₈	—	3	4	no	—
45	C₇×Q₈	C₂²×ΓL(1,𝔽₈)	1	4	yes	complemented
46	C₇×D₄	—	3	36	no	—
47	C₇×D₄	C₂²×ΓL(1,𝔽₈)	1	36	yes	complemented
48	C₂×C₄×C₇	—	3	18	no	—
49	C₂×C₄×C₇	C₂²×ΓL(1,𝔽₈)	1	18	yes	complemented
50	C₇²	C₃×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
51	C₃×#41	—	49	6	no	—
52	C₂×C₃×D₄	—	49	9	no	—
53	C₂×ΓL(1,𝔽₈)	—	7	2	no	—
54	C₂×ΓL(1,𝔽₈)	—	14	18	no	—
55	H(5,𝔽₂)	#3862	1	1	yes	characteristic, direct product factor, Sylow
56	C₂²×C₇	—	3	18	no	—
57	C₂²×C₇	—	4	12	no	—
58	C₂²×C₇	—	12	12	no	—
59	C₂²×C₇	C₂³×ΓL(1,𝔽₈)	1	18	yes	—
60	C₄×C₇	—	3	12	no	—
61	C₄×C₇	C₂³×ΓL(1,𝔽₈)	1	12	yes	—
62	C₂³×C₃	—	49	6	no	—
63	C₃×Q₈	—	49	2	no	—
64	C₃×D₄	—	49	18	no	—
65	C₂×C₃×C₄	—	49	9	no	—
66	ΓL(1,𝔽₈)	—	7	2	no	—
67	#41	C₂×#3862	1	6	yes	complemented
68	C₂×D₄	C₂×#3862	1	9	yes	complemented
69	C₂×C₇	—	2	18	no	—
70	C₂×C₇	—	3	2	no	—
71	C₂×C₇	—	6	18	no	—
72	C₂×C₇	C₂⁴×ΓL(1,𝔽₈)	1	2	yes	—
73	C₂²×C₃	—	49	9	no	—
74	C₂²×C₃	—	196	6	no	—
75	C₃×C₄	—	49	6	no	—
76	C₂³	C₂²×#3862	1	6	yes	complemented
77	Q₈	C₂²×#3862	1	2	yes	complemented
78	D₄	C₂²×#3862	1	18	yes	complemented
79	C₂×C₄	C₂²×#3862	1	9	yes	complemented
80	C₇	—	3	2	no	—
81	C₇	ΓL(1,𝔽₈)×H(5,𝔽₂)	1	2	yes	complemented
82	C₂×C₃	—	49	1	no	—
83	C₂×C₃	—	98	9	no	—
84	C₂²	—	4	6	no	—
85	C₂²	C₂³×#3862	1	9	yes	—
86	C₄	C₂³×#3862	1	6	yes	—
87	C₃	—	49	1	no	Sylow
88	C₂	—	2	9	no	—
89	C₂	C₂⁴×#3862	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×#3862
commutator, socle, cosocle	C₂×C₇²	C₂⁴×C₃
Fitting	C₇²×H(5,𝔽₂)	C₃
hypercenter	H(5,𝔽₂)	#3862
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×#3862	C₂	semidirect
2. C₂×D₄×#3862	C₂	semidirect
3. C₇²×H(5,𝔽₂)	C₃	semidirect
4. C₂³×#3862	C₂²	semidirect
5. Q₈×#3862	C₂²	semidirect
6. D₄×#3862	C₂²	semidirect
7. C₂×C₄×#3862	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
23. C₇×H(5,𝔽₂)	ΓL(1,𝔽₈)	semidirect
31. #3862	H(5,𝔽₂)	direct (N × H)
33. C₇×#41	C₂×ΓL(1,𝔽₈)	semidirect
35. C₂×C₇×D₄	C₂×ΓL(1,𝔽₈)	semidirect
43. C₂³×C₇	C₂²×ΓL(1,𝔽₈)	semidirect
45. C₇×Q₈	C₂²×ΓL(1,𝔽₈)	semidirect
47. C₇×D₄	C₂²×ΓL(1,𝔽₈)	semidirect
49. C₂×C₄×C₇	C₂²×ΓL(1,𝔽₈)	semidirect
50. C₇²	C₃×H(5,𝔽₂)	semidirect
55. H(5,𝔽₂)	#3862	direct (N × H)
67. #41	C₂×#3862	semidirect
68. C₂×D₄	C₂×#3862	semidirect
76. C₂³	C₂²×#3862	semidirect
77. Q₈	C₂²×#3862	semidirect
78. D₄	C₂²×#3862	semidirect
79. C₂×C₄	C₂²×#3862	semidirect
81. 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×#3862	2
C₂×D₄×#3862	2
C₇²×H(5,𝔽₂)	3
ΓL(1,𝔽₈)×H(5,𝔽₂)	7

Subgroups — H(5,𝔽2)×#3862 #72246194

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(5,𝔽₂)×#3862 #72246194