Subgroups — C₇×#428 #90074

Order 448.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₇×#117	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₇×#107	C₂	1	2	yes	maximal, complemented
5	C₇×#106	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₇	—	2	1	no	—
7	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₇×M₁₆	C₄	1	4	yes	complemented
9	C₂×C₇×C₈	C₂²	1	2	yes	—
10	C₇×C₄⋊C₄	—	2	1	no	—
11	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₇×C₄⋊C₄	C₂²	1	2	yes	—
13	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₇×C₂²⋊C₄	—	2	1	no	—
15	C₄²×C₇	—	2	1	no	—
16	#428	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂³×C₇	D₄	1	1	yes	characteristic
18	C₂×C₄×C₇	—	2	1	no	—
19	C₂×C₄×C₇	—	2	1	no	—
20	C₂×C₄×C₇	—	2	2	no	—
21	C₂×C₄×C₇	—	4	2	no	—
22	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₇	D₄	1	1	yes	characteristic
24	C₂×C₄×C₇	Q₈	1	2	yes	—
25	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
26	C₇×C₈	C₂×C₄	1	4	yes	complemented
27	C₂×M₁₆	C₂×C₇	1	1	yes	characteristic
28	#117	C₂×C₇	1	1	yes	characteristic
29	C₂×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic
30	#107	C₂×C₇	1	2	yes	complemented
31	#106	C₂×C₇	1	2	yes	complemented
32	C₂²×C₇	—	2	1	no	—
33	C₂²×C₇	—	2	1	no	—
34	C₂²×C₇	C₄⋊C₄	1	2	yes	—
35	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
36	C₄×C₇	—	4	1	no	—
37	C₄×C₇	—	4	2	no	—
38	C₄×C₇	—	4	1	no	—
39	C₄×C₇	C₄⋊C₄	1	2	yes	—
40	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
41	C₄×C₇	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄	—	2	1	no	—
43	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
44	M₁₆	C₄×C₇	1	4	yes	complemented
45	C₂×C₈	C₂²×C₇	1	2	yes	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
48	C₄⋊C₄	C₂²×C₇	1	2	yes	—
49	C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
50	C₂²⋊C₄	—	2	1	no	—
51	C₄²	—	2	1	no	—
52	C₂×C₇	—	2	2	no	—
53	C₂×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₂×C₇	#137	1	1	yes	characteristic
56	C₂³	C₇×D₄	1	1	yes	characteristic
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	4	2	no	—
61	C₂×C₄	C₂×C₄×C₇	1	2	yes	—
62	C₂×C₄	C₇×D₄	1	1	yes	characteristic
63	C₂×C₄	C₇×Q₈	1	2	yes	—
64	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
65	C₈	C₂×C₄×C₇	1	4	yes	complemented
66	C₇	#428	1	1	yes	characteristic, direct product factor, Sylow
67	C₂²	—	2	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	C₇×C₄⋊C₄	1	2	yes	—
70	C₂²	C₂×C₇×D₄	1	1	yes	characteristic
71	C₄	—	4	2	no	—
72	C₄	—	4	1	no	—
73	C₄	—	4	1	no	—
74	C₄	C₇×C₄⋊C₄	1	2	yes	—
75	C₄	C₂²×C₄×C₇	1	1	yes	characteristic
76	C₄	C₂×C₇×Q₈	1	1	yes	characteristic
77	C₂	—	2	2	no	—
78	C₂	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
79	C₂	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₂	C₇×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×D₄
commutator	C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂³×C₇
prime radical	C₂³×C₇	D₄
LCS	G ▷ 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
4. C₇×#107	C₂	semidirect
5. C₇×#106	C₂	semidirect
8. C₇×M₁₆	C₄	semidirect
16. #428	C₇	direct (N × H)
26. C₇×C₈	C₂×C₄	semidirect
30. #107	C₂×C₇	semidirect
31. #106	C₂×C₇	semidirect
44. M₁₆	C₄×C₇	semidirect
65. C₈	C₂×C₄×C₇	semidirect
66. C₇	#428	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×M₁₆	2
C₇×#117	2
C₂×C₇×C₄⋊C₄	2
C₇×#107	2
C₇×#106	2
#428	7

Subgroups — C7×#428 #90074

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#428 #90074