Subgroups — C₇×#467 #90113

Order 448.

114 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	C₂²×C₇×C₈	C₂	1	1	yes	maximal, characteristic
3	C₇×C₂²⋊Q₈	C₂	1	2	yes	maximal
4	C₇×#107	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	C₂×C₇×C₈	—	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	#467	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	C₂²×C₈	C₂×C₇	1	1	yes	characteristic
29	C₂²⋊Q₈	C₂×C₇	1	2	yes	—
30	#107	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	1	no	—
34	C₂²×C₇	Q₁₆	1	2	yes	complemented
35	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
36	C₄×C₇	—	2	1	no	—
37	C₄×C₇	—	4	2	no	—
38	C₄×C₇	—	4	2	no	—
39	C₄×C₇	C₂×D₄	1	1	yes	characteristic
40	C₄×C₇	#41	1	1	yes	characteristic
41	C₂×Q₈	C₂²×C₇	1	2	yes	—
42	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
43	Q₁₆	—	2	2	no	—
44	C₂×C₈	—	2	2	no	—
45	C₂×C₈	C₂²×C₇	1	1	yes	characteristic
46	C₂×C₈	C₂²×C₇	1	1	yes	characteristic
47	C₄⋊C₄	—	2	2	no	—
48	C₄⋊C₄	C₂²×C₇	1	2	yes	—
49	C₂²⋊C₄	—	2	2	no	—
50	C₂×C₇	—	2	2	no	—
51	C₂×C₇	#121	1	1	yes	characteristic
52	C₂×C₇	C₂×Q₁₆	1	1	yes	characteristic
53	C₂×C₇	#135	1	1	yes	characteristic
54	C₂³	C₇×D₄	1	1	yes	characteristic
55	Q₈	—	2	2	no	—
56	Q₈	—	2	2	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	C₇×D₄	1	1	yes	characteristic
62	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	C₇×D₄	1	2	yes	—
65	C₇	#467	1	1	yes	characteristic, direct product factor, Sylow
66	C₂²	—	2	1	no	—
67	C₂²	—	2	1	no	—
68	C₂²	C₇×Q₁₆	1	2	yes	complemented
69	C₂²	C₂×C₇×D₄	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	4	2	no	—
72	C₄	—	4	2	no	—
73	C₄	C₂×C₇×D₄	1	1	yes	characteristic
74	C₄	C₇×#41	1	1	yes	characteristic
75	C₂	—	2	2	no	—
76	C₂	C₇×#121	1	1	yes	characteristic
77	C₂	C₂×C₇×Q₁₆	1	1	yes	characteristic
78	C₂	C₇×#135	1	1	yes	characteristic

	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×#107	C₂	semidirect
5. C₇×#103	C₂	semidirect
15. #467	C₇	direct (N × H)
27. C₂×Q₁₆	C₂×C₇	semidirect
30. #107	C₂×C₇	semidirect
31. #103	C₂×C₇	semidirect
34. C₂²×C₇	Q₁₆	semidirect
65. C₇	#467	direct (N × H)
68. C₂²	C₇×Q₁₆	semidirect

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
C₂²×C₇×C₈	2
C₇×C₂²⋊Q₈	2
C₇×#107	2
C₇×#103	2
#467	7

Subgroups — C7×#467 #90113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#467 #90113