Subgroups — C₇×#456 #90102

Order 448.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic
2	C₇×#137	C₂	1	2	yes	maximal, complemented
3	C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#104	C₂	1	2	yes	maximal, complemented
5	C₇×#101	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#41	—	2	2	no	—
7	C₇×#41	—	2	2	no	—
8	C₇×#41	C₂²	1	2	yes	—
9	C₂×C₇×Q₈	—	2	2	no	—
10	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
12	C₇×Q₁₆	—	2	2	no	—
13	C₇×SD₁₆	—	2	2	no	—
14	C₇×M₁₆	C₂²	1	2	yes	—
15	C₇×C₄⋊C₄	—	2	2	no	—
16	C₄²×C₇	C₂²	1	1	yes	characteristic
17	#456	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₇×Q₈	—	2	2	no	—
19	C₇×Q₈	—	2	2	no	—
20	C₇×Q₈	—	2	2	no	—
21	C₇×Q₈	D₄	1	2	yes	—
22	C₇×D₄	—	2	2	no	—
23	C₇×D₄	—	2	2	no	—
24	C₇×D₄	D₄	1	2	yes	—
25	C₂×C₄×C₇	—	2	4	no	—
26	C₂×C₄×C₇	—	2	1	no	—
27	C₂×C₄×C₇	—	2	2	no	—
28	C₂×C₄×C₇	—	2	1	no	—
29	C₂×C₄×C₇	D₄	1	2	yes	—
30	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
31	C₇×C₈	—	2	2	no	—
32	2₋¹⁺⁴	C₂×C₇	1	1	yes	characteristic
33	#137	C₂×C₇	1	2	yes	complemented
34	C₄⋊Q₈	C₂×C₇	1	1	yes	characteristic, complemented
35	#104	C₂×C₇	1	2	yes	complemented
36	#101	C₂×C₇	1	1	yes	characteristic, complemented
37	C₂²×C₇	—	2	2	no	—
38	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
39	C₄×C₇	—	2	2	no	—
40	C₄×C₇	—	2	2	no	—
41	C₄×C₇	—	2	2	no	—
42	C₄×C₇	—	4	1	no	—
43	C₄×C₇	C₂×D₄	1	2	yes	—
44	#41	—	2	2	no	—
45	#41	—	2	2	no	—
46	#41	C₂²×C₇	1	2	yes	—
47	C₂×Q₈	—	2	2	no	—
48	C₂×Q₈	C₂²×C₇	1	1	yes	characteristic
49	C₂×Q₈	C₂²×C₇	1	1	yes	characteristic
50	Q₁₆	—	2	2	no	—
51	SD₁₆	—	2	2	no	—
52	M₁₆	C₂²×C₇	1	2	yes	—
53	C₄⋊C₄	—	2	2	no	—
54	C₄²	C₂²×C₇	1	1	yes	characteristic
55	C₂×C₇	—	2	1	no	—
56	C₂×C₇	—	4	2	no	—
57	C₂×C₇	#120	1	1	yes	characteristic
58	Q₈	—	2	2	no	—
59	Q₈	—	2	2	no	—
60	Q₈	—	2	2	no	—
61	Q₈	C₇×D₄	1	2	yes	—
62	D₄	—	2	2	no	—
63	D₄	—	2	2	no	—
64	D₄	C₇×D₄	1	2	yes	—
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	C₇×D₄	1	2	yes	—
70	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
71	C₈	—	2	2	no	—
72	C₇	#456	1	1	yes	characteristic, direct product factor, Sylow
73	C₂²	—	2	2	no	—
74	C₂²	C₂×C₇×D₄	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	2	2	no	—
77	C₄	—	2	2	no	—
78	C₄	—	4	1	no	—
79	C₄	C₂×C₇×D₄	1	2	yes	—
80	C₂	—	2	1	no	—
81	C₂	—	4	2	no	—
82	C₂	C₇×#120	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇	#120
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₇
prime radical	C₇×2₋¹⁺⁴	C₂
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
2. C₇×#137	C₂	semidirect
3. C₇×C₄⋊Q₈	C₂	semidirect
4. C₇×#104	C₂	semidirect
5. C₇×#101	C₂	semidirect
17. #456	C₇	direct (N × H)
33. #137	C₂×C₇	semidirect
34. C₄⋊Q₈	C₂×C₇	semidirect
35. #104	C₂×C₇	semidirect
36. #101	C₂×C₇	semidirect
72. C₇	#456	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×2₋¹⁺⁴	2
C₇×#137	2
C₇×C₄⋊Q₈	2
C₇×#104	2
C₇×#101	2
#456	7

Subgroups — C7×#456 #90102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#456 #90102