Subgroups — C₇×#469 #90115

Order 448.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×D₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#121	C₂	1	2	yes	maximal, complemented
4	C₇×#106	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#102	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×D₄	—	2	2	no	—
7	C₂×C₇×D₄	C₂²	1	2	yes	—
8	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₇×D₈	—	2	2	no	—
10	C₇×M₁₆	—	2	2	no	—
11	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
13	C₇×C₄⋊C₄	C₂²	1	2	yes	—
14	C₇×C₂²⋊C₄	—	2	2	no	—
15	#469	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂³×C₇	—	2	2	no	—
17	C₂³×C₇	D₄	1	1	yes	characteristic
18	C₇×D₄	—	2	2	no	—
19	C₇×D₄	—	2	2	no	—
20	C₇×D₄	—	4	4	no	—
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	C₂³	1	1	yes	characteristic
26	C₇×C₈	—	2	1	no	—
27	C₇×C₈	D₄	1	2	yes	complemented
28	C₂×D₈	C₂×C₇	1	1	yes	characteristic, complemented
29	C₂×M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
30	#121	C₂×C₇	1	2	yes	complemented
31	#106	C₂×C₇	1	1	yes	characteristic, complemented
32	#102	C₂×C₇	1	2	yes	complemented
33	C₂²×C₇	—	2	1	no	—
34	C₂²×C₇	—	2	2	no	—
35	C₂²×C₇	—	4	2	no	—
36	C₂²×C₇	—	4	2	no	—
37	C₂²×C₇	—	4	2	no	—
38	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
39	C₄×C₇	—	2	1	no	—
40	C₄×C₇	—	4	2	no	—
41	C₄×C₇	C₂×D₄	1	1	yes	characteristic
42	C₄×C₇	#41	1	1	yes	characteristic
43	C₂×D₄	—	2	2	no	—
44	C₂×D₄	C₂²×C₇	1	2	yes	—
45	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
46	D₈	—	2	2	no	—
47	M₁₆	—	2	2	no	—
48	C₂×C₈	C₂²×C₇	1	1	yes	characteristic
49	C₂×C₈	C₂²×C₇	1	1	yes	characteristic
50	C₄⋊C₄	C₂²×C₇	1	2	yes	—
51	C₂²⋊C₄	—	2	2	no	—
52	C₂×C₇	—	4	1	no	—
53	C₂×C₇	—	8	2	no	—
54	C₂×C₇	#121	1	1	yes	characteristic
55	C₂×C₇	AGL(1,ℤ/8)	1	2	yes	—
56	C₂³	—	2	2	no	—
57	C₂³	C₇×D₄	1	1	yes	characteristic
58	D₄	—	2	2	no	—
59	D₄	—	2	2	no	—
60	D₄	—	4	4	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	C₇×D₄	1	1	yes	characteristic
65	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
66	C₈	—	2	1	no	—
67	C₈	C₇×D₄	1	2	yes	complemented
68	C₇	#469	1	1	yes	characteristic, direct product factor, Sylow
69	C₂²	—	2	1	no	—
70	C₂²	—	2	2	no	—
71	C₂²	—	4	2	no	—
72	C₂²	—	4	2	no	—
73	C₂²	—	4	2	no	—
74	C₂²	C₂×C₇×D₄	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	2	no	—
77	C₄	C₂×C₇×D₄	1	1	yes	characteristic
78	C₄	C₇×#41	1	1	yes	characteristic
79	C₂	—	4	1	no	—
80	C₂	—	8	2	no	—
81	C₂	C₇×#121	1	1	yes	characteristic
82	C₂	C₇×AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₇
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×D₈	C₂	semidirect
2. C₂×C₇×M₁₆	C₂	semidirect
3. C₇×#121	C₂	semidirect
4. C₇×#106	C₂	semidirect
5. C₇×#102	C₂	semidirect
15. #469	C₇	direct (N × H)
27. C₇×C₈	D₄	semidirect
28. C₂×D₈	C₂×C₇	semidirect
29. C₂×M₁₆	C₂×C₇	semidirect
30. #121	C₂×C₇	semidirect
31. #106	C₂×C₇	semidirect
32. #102	C₂×C₇	semidirect
67. C₈	C₇×D₄	semidirect
68. C₇	#469	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×D₈	2
C₂×C₇×M₁₆	2
C₇×#121	2
C₇×#106	2
C₇×#102	2
#469	7

Subgroups — C7×#469 #90115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#469 #90115