Subgroups — C₇×#496 #90142

Order 448.

144 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	C₂×C₇×D₈	C₂	1	2	yes	maximal, complemented
3	C₇×#127	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#124	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#97	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₇×D₄	—	2	2	no	—
8	C₂×C₇×D₄	C₂²	1	1	yes	characteristic
9	C₂×C₇×D₄	C₂²	1	2	yes	—
10	C₇×SD₁₆	—	2	4	no	—
11	C₇×D₈	—	2	4	no	—
12	C₂×C₇×C₈	C₂²	1	2	yes	—
13	C₇×C₂²⋊C₄	—	2	2	no	—
14	C₄²×C₇	C₂²	1	1	yes	characteristic
15	#496	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂³×C₇	—	2	1	no	—
17	C₂³×C₇	—	2	2	no	—
18	C₇×Q₈	—	2	2	no	—
19	C₇×D₄	—	2	2	no	—
20	C₇×D₄	—	2	2	no	—
21	C₇×D₄	—	2	2	no	—
22	C₇×D₄	—	4	4	no	—
23	C₂×C₄×C₇	—	2	1	no	—
24	C₂×C₄×C₇	D₄	1	2	yes	—
25	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
26	C₇×C₈	D₄	1	4	yes	complemented
27	C₂×SD₁₆	C₂×C₇	1	2	yes	complemented
28	C₂×D₈	C₂×C₇	1	2	yes	complemented
29	#127	C₂×C₇	1	1	yes	characteristic, complemented
30	#124	C₂×C₇	1	1	yes	characteristic, complemented
31	#97	C₂×C₇	1	1	yes	characteristic, complemented
32	C₂²×C₇	—	4	1	no	—
33	C₂²×C₇	—	4	2	no	—
34	C₂²×C₇	—	4	4	no	—
35	C₂²×C₇	—	4	2	no	—
36	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
37	C₄×C₇	—	2	2	no	—
38	C₄×C₇	—	4	1	no	—
39	C₄×C₇	C₂×D₄	1	2	yes	—
40	C₂×Q₈	C₂²×C₇	1	1	yes	characteristic
41	C₂×D₄	—	2	2	no	—
42	C₂×D₄	C₂²×C₇	1	1	yes	characteristic
43	C₂×D₄	C₂²×C₇	1	2	yes	—
44	SD₁₆	—	2	4	no	—
45	D₈	—	2	4	no	—
46	C₂×C₈	C₂²×C₇	1	2	yes	—
47	C₂²⋊C₄	—	2	2	no	—
48	C₄²	C₂²×C₇	1	1	yes	characteristic
49	C₂×C₇	—	8	2	no	—
50	C₂×C₇	—	8	1	no	—
51	C₂×C₇	#127	1	1	yes	characteristic
52	C₂×C₇	AGL(1,ℤ/8)	1	2	yes	—
53	C₂³	—	2	1	no	—
54	C₂³	—	2	2	no	—
55	Q₈	—	2	2	no	—
56	D₄	—	2	2	no	—
57	D₄	—	2	2	no	—
58	D₄	—	2	2	no	—
59	D₄	—	4	4	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	C₇×D₄	1	2	yes	—
62	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
63	C₈	C₇×D₄	1	4	yes	complemented
64	C₇	#496	1	1	yes	characteristic, direct product factor, Sylow
65	C₂²	—	4	1	no	—
66	C₂²	—	4	2	no	—
67	C₂²	—	4	4	no	—
68	C₂²	—	4	2	no	—
69	C₂²	C₂×C₇×D₄	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	4	1	no	—
72	C₄	C₂×C₇×D₄	1	2	yes	—
73	C₂	—	8	2	no	—
74	C₂	—	8	1	no	—
75	C₂	C₇×#127	1	1	yes	characteristic
76	C₂	C₇×AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₇
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
2. C₂×C₇×D₈	C₂	semidirect
3. C₇×#127	C₂	semidirect
4. C₇×#124	C₂	semidirect
5. C₇×#97	C₂	semidirect
15. #496	C₇	direct (N × H)
26. C₇×C₈	D₄	semidirect
27. C₂×SD₁₆	C₂×C₇	semidirect
28. C₂×D₈	C₂×C₇	semidirect
29. #127	C₂×C₇	semidirect
30. #124	C₂×C₇	semidirect
31. #97	C₂×C₇	semidirect
63. C₈	C₇×D₄	semidirect
64. C₇	#496	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×SD₁₆	2
C₂×C₇×D₈	2
C₇×#127	2
C₇×#124	2
C₇×#97	2
#496	7

Subgroups — C7×#496 #90142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#496 #90142