Subgroups — C₇×#470 #90116

Order 448.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₇×C₂²⋊Q₈	C₂	1	2	yes	maximal
4	C₇×#106	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#103	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×Q₈	C₂²	1	2	yes	—
7	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₇×Q₁₆	—	2	2	no	—
9	C₇×M₁₆	—	2	2	no	—
10	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₇×C₄⋊C₄	—	2	2	no	—
13	C₇×C₄⋊C₄	C₂²	1	2	yes	—
14	C₇×C₂²⋊C₄	—	2	2	no	—
15	#470	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂³×C₇	D₄	1	1	yes	characteristic
17	C₇×Q₈	—	2	2	no	—
18	C₇×Q₈	—	2	2	no	—
19	C₂×C₄×C₇	—	2	1	no	—
20	C₂×C₄×C₇	—	2	1	no	—
21	C₂×C₄×C₇	—	2	2	no	—
22	C₂×C₄×C₇	—	2	2	no	—
23	C₂×C₄×C₇	D₄	1	1	yes	characteristic
24	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₇×C₈	—	2	1	no	—
26	C₇×C₈	D₄	1	2	yes	—
27	C₂×Q₁₆	C₂×C₇	1	1	yes	characteristic, complemented
28	C₂×M₁₆	C₂×C₇	1	1	yes	characteristic
29	C₂²⋊Q₈	C₂×C₇	1	2	yes	—
30	#106	C₂×C₇	1	1	yes	characteristic, complemented
31	#103	C₂×C₇	1	2	yes	complemented
32	C₂²×C₇	—	2	1	no	—
33	C₂²×C₇	—	2	2	no	—
34	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
35	C₄×C₇	—	2	1	no	—
36	C₄×C₇	—	4	2	no	—
37	C₄×C₇	—	4	2	no	—
38	C₄×C₇	C₂×D₄	1	1	yes	characteristic
39	C₄×C₇	#41	1	1	yes	characteristic
40	C₂×Q₈	C₂²×C₇	1	2	yes	—
41	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
42	Q₁₆	—	2	2	no	—
43	M₁₆	—	2	2	no	—
44	C₂×C₈	C₂²×C₇	1	1	yes	characteristic
45	C₂×C₈	C₂²×C₇	1	1	yes	characteristic
46	C₄⋊C₄	—	2	2	no	—
47	C₄⋊C₄	C₂²×C₇	1	2	yes	—
48	C₂²⋊C₄	—	2	2	no	—
49	C₂×C₇	—	4	1	no	—
50	C₂×C₇	#121	1	1	yes	characteristic
51	C₂×C₇	#137	1	2	yes	—
52	C₂³	C₇×D₄	1	1	yes	characteristic
53	Q₈	—	2	2	no	—
54	Q₈	—	2	2	no	—
55	C₂×C₄	—	2	1	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	C₇×D₄	1	1	yes	characteristic
60	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
61	C₈	—	2	1	no	—
62	C₈	C₇×D₄	1	2	yes	—
63	C₇	#470	1	1	yes	characteristic, direct product factor, Sylow
64	C₂²	—	2	1	no	—
65	C₂²	—	2	2	no	—
66	C₂²	C₂×C₇×D₄	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	4	2	no	—
69	C₄	—	4	2	no	—
70	C₄	C₂×C₇×D₄	1	1	yes	characteristic
71	C₄	C₇×#41	1	1	yes	characteristic
72	C₂	—	4	1	no	—
73	C₂	C₇×#121	1	1	yes	characteristic
74	C₂	C₇×#137	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₇×Q₁₆	C₂	semidirect
4. C₇×#106	C₂	semidirect
5. C₇×#103	C₂	semidirect
15. #470	C₇	direct (N × H)
27. C₂×Q₁₆	C₂×C₇	semidirect
30. #106	C₂×C₇	semidirect
31. #103	C₂×C₇	semidirect
63. C₇	#470	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×Q₁₆	2
C₂×C₇×M₁₆	2
C₇×C₂²⋊Q₈	2
C₇×#106	2
C₇×#103	2
#470	7

Subgroups — C7×#470 #90116

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#470 #90116