Subgroups — C₇×#488 #90134

Order 448.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#124	C₂	1	1	yes	maximal, characteristic
3	C₇×#103	C₂	1	2	yes	maximal, complemented
4	C₇×#102	C₂	1	2	yes	maximal
5	C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
8	C₂×C₇×C₈	C₂²	1	2	yes	—
9	C₇×C₄⋊C₄	—	2	2	no	—
10	C₇×C₄⋊C₄	C₂²	1	2	yes	—
11	C₇×C₂²⋊C₄	—	2	2	no	—
12	C₄²×C₇	C₂²	1	1	yes	characteristic
13	#488	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₂³×C₇	—	2	1	no	—
15	C₇×Q₈	—	2	2	no	—
16	C₇×D₄	—	2	2	no	—
17	C₂×C₄×C₇	—	2	1	no	—
18	C₂×C₄×C₇	—	2	2	no	—
19	C₂×C₄×C₇	D₄	1	2	yes	—
20	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
21	C₇×C₈	—	2	2	no	—
22	#125	C₂×C₇	1	1	yes	characteristic, complemented
23	#124	C₂×C₇	1	1	yes	characteristic
24	#103	C₂×C₇	1	2	yes	complemented
25	#102	C₂×C₇	1	2	yes	—
26	C₄×C₈	C₂×C₇	1	1	yes	characteristic, complemented
27	C₂²×C₇	—	4	1	no	—
28	C₂²×C₇	—	4	2	no	—
29	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
30	C₄×C₇	—	2	2	no	—
31	C₄×C₇	—	4	1	no	—
32	C₄×C₇	—	4	2	no	—
33	C₄×C₇	#41	1	2	yes	—
34	C₂×Q₈	C₂²×C₇	1	1	yes	characteristic
35	C₂×D₄	C₂²×C₇	1	1	yes	characteristic
36	C₂×C₈	C₂²×C₇	1	2	yes	—
37	C₄⋊C₄	—	2	2	no	—
38	C₄⋊C₄	C₂²×C₇	1	2	yes	—
39	C₂²⋊C₄	—	2	2	no	—
40	C₄²	C₂²×C₇	1	1	yes	characteristic
41	C₂×C₇	—	8	1	no	—
42	C₂×C₇	#124	1	1	yes	characteristic
43	C₂×C₇	#135	1	2	yes	—
44	C₂³	—	2	1	no	—
45	Q₈	—	2	2	no	—
46	D₄	—	2	2	no	—
47	C₂×C₄	—	2	1	no	—
48	C₂×C₄	—	2	2	no	—
49	C₂×C₄	C₇×D₄	1	2	yes	—
50	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
51	C₈	—	2	2	no	—
52	C₇	#488	1	1	yes	characteristic, direct product factor, Sylow
53	C₂²	—	4	1	no	—
54	C₂²	—	4	2	no	—
55	C₂²	C₂×C₇×D₄	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	4	1	no	—
58	C₄	—	4	2	no	—
59	C₄	C₇×#41	1	2	yes	—
60	C₂	—	8	1	no	—
61	C₂	C₇×#124	1	1	yes	characteristic
62	C₂	C₇×#135	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₇
prime radical	C₂×C₇×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#125	C₂	semidirect
3. C₇×#103	C₂	semidirect
5. C₄×C₇×C₈	C₂	semidirect
13. #488	C₇	direct (N × H)
22. #125	C₂×C₇	semidirect
24. #103	C₂×C₇	semidirect
26. C₄×C₈	C₂×C₇	semidirect
52. C₇	#488	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×#125	2
C₇×#124	2
C₇×#103	2
C₇×#102	2
C₄×C₇×C₈	2
#488	7

Subgroups — C7×#488 #90134

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#488 #90134