Subgroups — C₇×#462 #90108

Order 448.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×Q₁₆	C₂	1	2	yes	maximal
2	C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
4	C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₇×#103	C₂	1	2	yes	maximal
6	C₂×C₇×Q₈	C₂²	1	2	yes	—
7	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
8	C₇×Q₁₆	—	2	4	no	—
9	C₂×C₇×C₈	C₂²	1	2	yes	—
10	C₇×C₄⋊C₄	—	2	1	no	—
11	C₇×C₄⋊C₄	—	2	1	no	—
12	C₇×C₄⋊C₄	—	2	1	no	—
13	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₄²×C₇	—	2	1	no	—
15	C₄²×C₇	C₂²	1	1	yes	characteristic
16	#462	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₇×Q₈	—	2	1	no	—
18	C₇×Q₈	—	2	2	no	—
19	C₇×Q₈	—	2	2	no	—
20	C₇×Q₈	D₄	1	2	yes	—
21	C₂×C₄×C₇	—	2	1	no	—
22	C₂×C₄×C₇	—	2	1	no	—
23	C₂×C₄×C₇	—	2	2	no	—
24	C₂×C₄×C₇	D₄	1	1	yes	characteristic
25	C₂×C₄×C₇	D₄	1	1	yes	characteristic
26	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
27	C₇×C₈	—	2	2	no	—
28	C₂×Q₁₆	C₂×C₇	1	2	yes	—
29	C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic
30	C₄×Q₈	C₂×C₇	1	1	yes	characteristic
31	C₄⋊C₈	C₂×C₇	1	1	yes	characteristic
32	#103	C₂×C₇	1	2	yes	—
33	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
34	C₄×C₇	—	2	2	no	—
35	C₄×C₇	—	2	1	no	—
36	C₄×C₇	—	4	2	no	—
37	C₄×C₇	—	4	1	no	—
38	C₄×C₇	Q₁₆	1	2	yes	complemented
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	C₂×Q₈	C₂²×C₇	1	1	yes	characteristic
43	Q₁₆	—	2	4	no	—
44	C₂×C₈	C₂²×C₇	1	2	yes	—
45	C₄⋊C₄	—	2	1	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	—	2	1	no	—
48	C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
49	C₄²	—	2	1	no	—
50	C₄²	C₂²×C₇	1	1	yes	characteristic
51	C₂×C₇	#121	1	1	yes	characteristic
52	C₂×C₇	C₂×Q₁₆	1	1	yes	characteristic
53	C₂×C₇	#137	1	1	yes	characteristic
54	Q₈	—	2	1	no	—
55	Q₈	—	2	2	no	—
56	Q₈	—	2	2	no	—
57	Q₈	C₇×D₄	1	2	yes	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	C₇×D₄	1	1	yes	characteristic
62	C₂×C₄	C₇×D₄	1	1	yes	characteristic
63	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
64	C₈	—	2	2	no	—
65	C₇	#462	1	1	yes	characteristic, direct product factor, Sylow
66	C₂²	C₂×C₇×D₄	1	1	yes	characteristic
67	C₄	—	2	2	no	—
68	C₄	—	2	1	no	—
69	C₄	—	4	2	no	—
70	C₄	—	4	1	no	—
71	C₄	C₇×Q₁₆	1	2	yes	complemented
72	C₄	C₂×C₇×D₄	1	1	yes	characteristic
73	C₄	C₇×#41	1	1	yes	characteristic
74	C₂	C₇×#121	1	1	yes	characteristic
75	C₂	C₂×C₇×Q₁₆	1	1	yes	characteristic
76	C₂	C₇×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₇
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
16. #462	C₇	direct (N × H)
38. C₄×C₇	Q₁₆	semidirect
65. C₇	#462	direct (N × H)
71. C₄	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
C₇×C₄⋊Q₈	2
C₄×C₇×Q₈	2
C₇×C₄⋊C₈	2
C₇×#103	2
#462	7

Subgroups — C7×#462 #90108

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#462 #90108