Subgroups — C₇×#451 #90097

Order 448.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×Q₁₆	C₂	1	2	yes	maximal, complemented
3	C₇×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
4	C₇×#103	C₂	1	2	yes	maximal, complemented
5	C₇×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇×Q₈	—	2	4	no	—
7	C₂×C₇×Q₈	—	2	1	no	—
8	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₇×Q₈	C₂²	1	2	yes	—
10	C₂²×C₄×C₇	—	2	1	no	—
11	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
12	C₇×Q₁₆	—	2	4	no	—
13	C₂×C₇×C₈	C₂²	1	2	yes	—
14	C₇×C₄⋊C₄	—	2	1	no	—
15	C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₇×C₂²⋊C₄	—	2	1	no	—
17	#451	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂³×C₇	D₄	1	1	yes	characteristic
19	C₇×Q₈	—	2	4	no	—
20	C₇×Q₈	—	2	2	no	—
21	C₇×Q₈	—	2	2	no	—
22	C₇×Q₈	D₄	1	4	yes	—
23	C₂×C₄×C₇	—	2	4	no	—
24	C₂×C₄×C₇	—	2	2	no	—
25	C₂×C₄×C₇	—	2	2	no	—
26	C₂×C₄×C₇	—	2	1	no	—
27	C₂×C₄×C₇	—	2	1	no	—
28	C₂×C₄×C₇	D₄	1	1	yes	characteristic
29	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
30	C₇×C₈	—	2	2	no	—
31	C₂²×Q₈	C₂×C₇	1	1	yes	characteristic
32	C₂×Q₁₆	C₂×C₇	1	2	yes	complemented
33	C₂²⋊Q₈	C₂×C₇	1	1	yes	characteristic
34	#103	C₂×C₇	1	2	yes	complemented
35	C₂²⋊C₈	C₂×C₇	1	1	yes	characteristic
36	C₂²×C₇	—	2	1	no	—
37	C₂²×C₇	—	2	1	no	—
38	C₂²×C₇	Q₁₆	1	2	yes	complemented
39	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
40	C₄×C₇	—	2	4	no	—
41	C₄×C₇	—	2	1	no	—
42	C₄×C₇	—	4	1	no	—
43	C₄×C₇	—	4	1	no	—
44	C₄×C₇	C₂×D₄	1	2	yes	—
45	C₂×Q₈	—	2	4	no	—
46	C₂×Q₈	—	2	1	no	—
47	C₂×Q₈	C₂²×C₇	1	1	yes	characteristic
48	C₂×Q₈	C₂²×C₇	1	2	yes	—
49	C₂²×C₄	—	2	1	no	—
50	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
51	Q₁₆	—	2	4	no	—
52	C₂×C₈	C₂²×C₇	1	2	yes	—
53	C₄⋊C₄	—	2	1	no	—
54	C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
55	C₂²⋊C₄	—	2	1	no	—
56	C₂×C₇	—	2	2	no	—
57	C₂×C₇	#120	1	1	yes	characteristic
58	C₂×C₇	C₂×Q₁₆	1	1	yes	characteristic
59	C₂×C₇	#137	1	1	yes	characteristic
60	C₂³	C₇×D₄	1	1	yes	characteristic
61	Q₈	—	2	4	no	—
62	Q₈	—	2	2	no	—
63	Q₈	—	2	2	no	—
64	Q₈	C₇×D₄	1	4	yes	—
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	C₇×D₄	1	1	yes	characteristic
71	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
72	C₈	—	2	2	no	—
73	C₇	#451	1	1	yes	characteristic, direct product factor, Sylow
74	C₂²	—	2	1	no	—
75	C₂²	—	2	1	no	—
76	C₂²	C₇×Q₁₆	1	2	yes	complemented
77	C₂²	C₂×C₇×D₄	1	1	yes	characteristic
78	C₄	—	2	4	no	—
79	C₄	—	2	1	no	—
80	C₄	—	4	1	no	—
81	C₄	—	4	1	no	—
82	C₄	C₂×C₇×D₄	1	2	yes	—
83	C₂	—	2	2	no	—
84	C₂	C₇×#120	1	1	yes	characteristic
85	C₂	C₂×C₇×Q₁₆	1	1	yes	characteristic
86	C₂	C₇×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₇
prime radical	C₂³×C₇	D₄
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
2. C₂×C₇×Q₁₆	C₂	semidirect
4. C₇×#103	C₂	semidirect
17. #451	C₇	direct (N × H)
32. C₂×Q₁₆	C₂×C₇	semidirect
34. #103	C₂×C₇	semidirect
38. C₂²×C₇	Q₁₆	semidirect
73. C₇	#451	direct (N × H)
76. C₂²	C₇×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₇×Q₈	2
C₂×C₇×Q₁₆	2
C₇×C₂²⋊Q₈	2
C₇×#103	2
C₇×C₂²⋊C₈	2
#451	7

Subgroups — C7×#451 #90097

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#451 #90097