Subgroups — C₇×#472 #90118

Order 448.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
2	C₂×C₇×D₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#131	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#108	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#100	C₂	1	2	yes	maximal, complemented
6	C₇×#41	C₂²	1	1	yes	characteristic
7	C₂×C₇×D₄	C₂²	1	2	yes	—
8	C₇×SD₁₆	—	2	2	no	—
9	C₇×D₈	—	2	2	no	—
10	C₇×D₈	—	2	1	no	—
11	C₇×D₈	—	2	1	no	—
12	C₇×M₁₆	—	2	1	no	—
13	C₇×M₁₆	C₂²	1	1	yes	characteristic
14	C₇×M₁₆	C₂²	1	2	yes	—
15	C₂×C₇×C₈	—	2	1	no	—
16	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
17	#472	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₇	—	2	2	no	—
19	C₇×Q₈	D₄	1	1	yes	characteristic
20	C₇×D₄	—	2	1	no	—
21	C₇×D₄	—	2	2	no	—
22	C₇×D₄	—	2	2	no	—
23	C₇×D₄	D₄	1	1	yes	characteristic
24	C₂×C₄×C₇	—	2	1	no	—
25	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
26	C₇×C₈	—	2	1	no	—
27	C₇×C₈	—	2	2	no	—
28	C₇×C₈	D₄	1	1	yes	characteristic
29	C₇×C₈	D₄	1	1	yes	characteristic
30	AGL(1,ℤ/8)	C₂×C₇	1	2	yes	complemented
31	C₂×D₈	C₂×C₇	1	1	yes	characteristic, complemented
32	#131	C₂×C₇	1	1	yes	characteristic, complemented
33	#108	C₂×C₇	1	1	yes	characteristic, complemented
34	#100	C₂×C₇	1	2	yes	complemented
35	C₂²×C₇	—	2	1	no	—
36	C₂²×C₇	—	4	2	no	—
37	C₂²×C₇	—	8	2	no	—
38	C₂²×C₇	#41	1	1	yes	characteristic
39	C₄×C₇	—	2	1	no	—
40	C₄×C₇	C₂×D₄	1	1	yes	characteristic
41	C₄×C₇	C₂×D₄	1	1	yes	characteristic
42	#41	C₂²×C₇	1	1	yes	characteristic
43	C₂×D₄	C₂²×C₇	1	2	yes	—
44	SD₁₆	—	2	2	no	—
45	D₈	—	2	2	no	—
46	D₈	—	2	1	no	—
47	D₈	—	2	1	no	—
48	M₁₆	—	2	1	no	—
49	M₁₆	C₂²×C₇	1	1	yes	characteristic
50	M₁₆	C₂²×C₇	1	2	yes	—
51	C₂×C₈	—	2	1	no	—
52	C₂×C₈	C₂²×C₇	1	1	yes	characteristic
53	C₂×C₇	—	2	1	no	—
54	C₂×C₇	—	4	1	no	—
55	C₂×C₇	—	8	2	no	—
56	C₂×C₇	#121	1	1	yes	characteristic
57	C₂³	—	2	2	no	—
58	Q₈	C₇×D₄	1	1	yes	characteristic
59	D₄	—	2	1	no	—
60	D₄	—	2	2	no	—
61	D₄	—	2	2	no	—
62	D₄	C₇×D₄	1	1	yes	characteristic
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
65	C₈	—	2	1	no	—
66	C₈	—	2	2	no	—
67	C₈	C₇×D₄	1	1	yes	characteristic
68	C₈	C₇×D₄	1	1	yes	characteristic
69	C₇	#472	1	1	yes	characteristic, direct product factor, Sylow
70	C₂²	—	2	1	no	—
71	C₂²	—	4	2	no	—
72	C₂²	—	8	2	no	—
73	C₂²	C₇×#41	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	C₂×C₇×D₄	1	1	yes	characteristic
76	C₄	C₂×C₇×D₄	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	4	1	no	—
79	C₂	—	8	2	no	—
80	C₂	C₇×#121	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇	#121
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₇×AGL(1,ℤ/8)	C₂	semidirect
2. C₂×C₇×D₈	C₂	semidirect
3. C₇×#131	C₂	semidirect
4. C₇×#108	C₂	semidirect
5. C₇×#100	C₂	semidirect
17. #472	C₇	direct (N × H)
30. AGL(1,ℤ/8)	C₂×C₇	semidirect
31. C₂×D₈	C₂×C₇	semidirect
32. #131	C₂×C₇	semidirect
33. #108	C₂×C₇	semidirect
34. #100	C₂×C₇	semidirect
69. C₇	#472	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×AGL(1,ℤ/8)	2
C₂×C₇×D₈	2
C₇×#131	2
C₇×#108	2
C₇×#100	2
#472	7

Subgroups — C7×#472 #90118

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#472 #90118