Subgroups — C₇×#460 #90106

Order 448.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×SD₁₆	C₂	1	2	yes	maximal, complemented
2	C₇×#127	C₂	1	1	yes	maximal, characteristic
3	C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#102	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₇×D₄	—	2	1	no	—
8	C₂×C₇×D₄	—	2	1	no	—
9	C₂×C₇×D₄	C₂²	1	2	yes	—
10	C₇×SD₁₆	—	2	4	no	—
11	C₂×C₇×C₈	C₂²	1	2	yes	—
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	#460	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂³×C₇	—	2	2	no	—
18	C₇×Q₈	—	2	1	no	—
19	C₇×Q₈	D₄	1	2	yes	complemented
20	C₇×D₄	—	2	2	no	—
21	C₇×D₄	—	2	2	no	—
22	C₇×D₄	—	4	2	no	—
23	C₇×D₄	—	4	2	no	—
24	C₂×C₄×C₇	—	2	1	no	—
25	C₂×C₄×C₇	—	2	1	no	—
26	C₂×C₄×C₇	D₄	1	1	yes	characteristic
27	C₂×C₄×C₇	D₄	1	1	yes	characteristic
28	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
29	C₇×C₈	—	2	2	no	—
30	C₂×SD₁₆	C₂×C₇	1	2	yes	complemented
31	#127	C₂×C₇	1	1	yes	characteristic
32	C₄×Q₈	C₂×C₇	1	1	yes	characteristic, complemented
33	C₄⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
34	#102	C₂×C₇	1	2	yes	complemented
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	2	no	—
40	C₄×C₇	—	2	1	no	—
41	C₄×C₇	—	4	1	no	—
42	C₄×C₇	SD₁₆	1	2	yes	complemented
43	C₄×C₇	C₂×D₄	1	1	yes	characteristic
44	C₄×C₇	#41	1	1	yes	characteristic
45	C₂×Q₈	C₂²×C₇	1	1	yes	characteristic
46	C₂×D₄	—	2	1	no	—
47	C₂×D₄	—	2	1	no	—
48	C₂×D₄	C₂²×C₇	1	2	yes	—
49	SD₁₆	—	2	4	no	—
50	C₂×C₈	C₂²×C₇	1	2	yes	—
51	C₄⋊C₄	—	2	1	no	—
52	C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
53	C₄²	—	2	1	no	—
54	C₄²	C₂²×C₇	1	1	yes	characteristic
55	C₂×C₇	—	8	2	no	—
56	C₂×C₇	#121	1	1	yes	characteristic
57	C₂×C₇	C₂×SD₁₆	1	1	yes	characteristic
58	C₂×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
59	C₂³	—	2	2	no	—
60	Q₈	—	2	1	no	—
61	Q₈	C₇×D₄	1	2	yes	complemented
62	D₄	—	2	2	no	—
63	D₄	—	2	2	no	—
64	D₄	—	4	2	no	—
65	D₄	—	4	2	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	C₇×D₄	1	1	yes	characteristic
69	C₂×C₄	C₇×D₄	1	1	yes	characteristic
70	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
71	C₈	—	2	2	no	—
72	C₇	#460	1	1	yes	characteristic, direct product factor, Sylow
73	C₂²	—	4	2	no	—
74	C₂²	—	4	2	no	—
75	C₂²	—	4	2	no	—
76	C₂²	C₂×C₇×D₄	1	1	yes	characteristic
77	C₄	—	2	2	no	—
78	C₄	—	2	1	no	—
79	C₄	—	4	1	no	—
80	C₄	C₇×SD₁₆	1	2	yes	complemented
81	C₄	C₂×C₇×D₄	1	1	yes	characteristic
82	C₄	C₇×#41	1	1	yes	characteristic
83	C₂	—	8	2	no	—
84	C₂	C₇×#121	1	1	yes	characteristic
85	C₂	C₂×C₇×SD₁₆	1	1	yes	characteristic
86	C₂	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₇
prime radical	C₇×#127	C₂
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. C₂×C₇×SD₁₆	C₂	semidirect
3. C₄×C₇×Q₈	C₂	semidirect
4. C₇×C₄⋊C₈	C₂	semidirect
5. C₇×#102	C₂	semidirect
16. #460	C₇	direct (N × H)
19. C₇×Q₈	D₄	semidirect
30. C₂×SD₁₆	C₂×C₇	semidirect
32. C₄×Q₈	C₂×C₇	semidirect
33. C₄⋊C₈	C₂×C₇	semidirect
34. #102	C₂×C₇	semidirect
42. C₄×C₇	SD₁₆	semidirect
61. Q₈	C₇×D₄	semidirect
72. C₇	#460	direct (N × H)
80. C₄	C₇×SD₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×SD₁₆	2
C₇×#127	2
C₄×C₇×Q₈	2
C₇×C₄⋊C₈	2
C₇×#102	2
#460	7

Subgroups — C7×#460 #90106

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#460 #90106