Subgroups — C₇×B(4,𝔽₂) #90103

Order 448.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#120	C₂	1	3	yes	maximal, complemented
3	C₇×#99	C₂	1	3	yes	maximal, complemented
4	C₂⁴×C₇	C₂²	1	1	yes	characteristic, complemented
5	C₇×#41	—	2	3	no	—
6	C₂×C₇×D₄	—	2	3	no	—
7	C₂×C₇×D₄	—	2	3	no	—
8	C₂×C₇×D₄	C₂²	1	3	yes	—
9	C₇×C₂²⋊C₄	—	2	3	no	—
10	C₇×C₂²⋊C₄	C₂²	1	3	yes	complemented
11	B(4,𝔽₂)	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₂³×C₇	—	2	1	no	—
13	C₂³×C₇	—	2	3	no	—
14	C₂³×C₇	—	4	2	no	—
15	C₂³×C₇	D₄	1	3	yes	complemented
16	C₂³×C₇	C₂³	1	1	yes	characteristic
17	C₇×Q₈	—	2	1	no	—
18	C₇×D₄	—	2	3	no	—
19	C₇×D₄	—	2	6	no	—
20	C₇×D₄	—	4	6	no	—
21	C₂×C₄×C₇	—	2	3	no	—
22	C₂×C₄×C₇	—	2	3	no	—
23	C₂×C₄×C₇	D₄	1	3	yes	complemented
24	H(5,𝔽₂)	C₂×C₇	1	1	yes	characteristic, complemented
25	#120	C₂×C₇	1	3	yes	complemented
26	#99	C₂×C₇	1	3	yes	complemented
27	C₂²×C₇	—	2	6	no	—
28	C₂²×C₇	—	2	3	no	—
29	C₂²×C₇	—	4	1	no	—
30	C₂²×C₇	—	4	3	no	—
31	C₂²×C₇	—	4	3	no	—
32	C₂²×C₇	—	4	1	no	—
33	C₂²×C₇	—	8	1	no	—
34	C₂²×C₇	C₂×D₄	1	3	yes	—
35	C₄×C₇	—	2	3	no	—
36	C₄×C₇	—	4	3	no	—
37	C₂⁴	C₂²×C₇	1	1	yes	characteristic, complemented
38	#41	—	2	3	no	—
39	C₂×D₄	—	2	3	no	—
40	C₂×D₄	—	2	3	no	—
41	C₂×D₄	C₂²×C₇	1	3	yes	—
42	C₂²⋊C₄	—	2	3	no	—
43	C₂²⋊C₄	C₂²×C₇	1	3	yes	complemented
44	C₂×C₇	—	2	3	no	—
45	C₂×C₇	—	4	2	no	—
46	C₂×C₇	—	4	3	no	—
47	C₂×C₇	#120	1	1	yes	characteristic
48	C₂³	—	2	1	no	—
49	C₂³	—	2	3	no	—
50	C₂³	—	4	2	no	—
51	C₂³	C₇×D₄	1	3	yes	complemented
52	C₂³	C₂³×C₇	1	1	yes	characteristic
53	Q₈	—	2	1	no	—
54	D₄	—	2	3	no	—
55	D₄	—	2	6	no	—
56	D₄	—	4	6	no	—
57	C₂×C₄	—	2	3	no	—
58	C₂×C₄	—	2	3	no	—
59	C₂×C₄	C₇×D₄	1	3	yes	complemented
60	C₇	B(4,𝔽₂)	1	1	yes	characteristic, direct product factor, Sylow
61	C₂²	—	2	6	no	—
62	C₂²	—	2	3	no	—
63	C₂²	—	4	1	no	—
64	C₂²	—	4	3	no	—
65	C₂²	—	4	3	no	—
66	C₂²	—	4	1	no	—
67	C₂²	—	8	1	no	—
68	C₂²	C₂×C₇×D₄	1	3	yes	—
69	C₄	—	2	3	no	—
70	C₄	—	4	3	no	—
71	C₂	—	2	3	no	—
72	C₂	—	4	2	no	—
73	C₂	—	4	3	no	—
74	C₂	C₇×#120	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇	#120
commutator, Frattini, cosocle	C₂³	C₂³×C₇
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×H(5,𝔽₂)	C₂	semidirect
2. C₇×#120	C₂	semidirect
3. C₇×#99	C₂	semidirect
4. C₂⁴×C₇	C₂²	semidirect
10. C₇×C₂²⋊C₄	C₂²	semidirect
11. B(4,𝔽₂)	C₇	direct (N × H)
15. C₂³×C₇	D₄	semidirect
23. C₂×C₄×C₇	D₄	semidirect
24. H(5,𝔽₂)	C₂×C₇	semidirect
25. #120	C₂×C₇	semidirect
26. #99	C₂×C₇	semidirect
37. C₂⁴	C₂²×C₇	semidirect
43. C₂²⋊C₄	C₂²×C₇	semidirect
51. C₂³	C₇×D₄	semidirect
59. C₂×C₄	C₇×D₄	semidirect
60. C₇	B(4,𝔽₂)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×H(5,𝔽₂)	2
C₇×#120	2
C₇×#99	2
B(4,𝔽₂)	7

Subgroups — C7×B(4,𝔽2) #90103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×B(4,𝔽₂) #90103