Subgroups — C₇×#6390 #13096098

Order 1568.

117 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×Dic₁₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇²×C₈	C₂	1	1	yes	maximal, characteristic
3	C₇×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₇²	C₂²	1	1	yes	characteristic
5	C₂×C₇×Dic₇	—	2	1	no	—
6	C₂×C₇×Dic₇	—	2	1	no	—
7	C₇×Dic₁₄	—	2	1	no	—
8	C₇×Dic₁₄	C₄	1	2	yes	complemented
9	C₇²×C₈	—	2	1	no	—
10	C₇×#103	—	7	1	no	maximal
11	#6390	C₇	1	1	yes	maximal, characteristic, direct product factor
12	C₂²×C₇²	D₄	1	1	yes	characteristic
13	C₄×C₇²	C₂×C₄	1	1	yes	characteristic
14	C₄×C₇²	D₄	1	1	yes	characteristic
15	C₇×Dic₇	—	2	2	no	—
16	C₇×Dic₇	—	4	1	no	—
17	C₂×C₇×Q₈	—	7	1	no	—
18	C₂×Dic₁₄	C₂×C₇	1	1	yes	characteristic
19	C₂×C₇×C₈	—	2	3	no	—
20	C₂×C₇×C₈	D₇	1	1	yes	characteristic
21	C₂×C₇×C₈	C₂×C₇	1	1	yes	characteristic
22	C₇×C₄⋊C₄	—	7	1	no	—
23	C₄⋊Dic₇	C₂×C₇	1	1	yes	characteristic
24	C₂×C₇²	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₇²	SD₁₆	1	1	yes	characteristic
26	C₂×C₇²	Q₁₆	1	1	yes	characteristic
27	C₇×Q₈	—	7	2	no	—
28	C₇×Q₈	—	14	1	no	—
29	C₂×C₄×C₇	—	2	3	no	—
30	C₂×C₄×C₇	—	14	1	no	—
31	C₂×C₄×C₇	—	14	1	no	—
32	C₂×C₄×C₇	C₂×D₇	1	1	yes	characteristic
33	C₂×C₄×C₇	C₂²×C₇	1	1	yes	characteristic
34	C₂×Dic₇	—	2	1	no	—
35	C₂×Dic₇	—	2	1	no	—
36	Dic₁₄	—	2	1	no	—
37	Dic₁₄	C₄×C₇	1	2	yes	complemented
38	C₇×C₈	—	2	6	no	—
39	C₇×C₈	—	2	1	no	—
40	C₇×C₈	—	2	1	no	—
41	C₇²	#103	1	1	yes	characteristic, complemented, Sylow
42	#103	—	7	1	no	Sylow
43	C₂²×C₇	—	2	3	no	—
44	C₂²×C₇	D₂₈	1	1	yes	characteristic
45	C₂²×C₇	C₇×D₄	1	1	yes	characteristic
46	C₄×C₇	—	2	3	no	—
47	C₄×C₇	—	2	3	no	—
48	C₄×C₇	—	14	2	no	—
49	C₄×C₇	—	28	1	no	—
50	C₄×C₇	C₄×D₇	1	1	yes	characteristic
51	C₄×C₇	C₂²⋊D₇	1	1	yes	characteristic
52	C₄×C₇	C₂×C₄×C₇	1	1	yes	characteristic
53	C₄×C₇	C₇×D₄	1	1	yes	characteristic
54	Dic₇	—	2	2	no	—
55	Dic₇	—	4	1	no	—
56	C₂×Q₈	—	7	1	no	—
57	C₂×C₈	C₇×D₇	1	1	yes	characteristic
58	C₄⋊C₄	—	7	1	no	—
59	C₂×C₇	—	2	3	no	—
60	C₂×C₇	—	2	3	no	—
61	C₂×C₇	—	2	3	no	—
62	C₂×C₇	#1132	1	1	yes	characteristic
63	C₂×C₇	Dic₂₈	1	1	yes	characteristic
64	C₂×C₇	(C₂×D₇)⋊C₄	1	1	yes	characteristic
65	C₂×C₇	C₇×C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₇	C₇×SD₁₆	1	1	yes	characteristic
67	C₂×C₇	C₇×Q₁₆	1	1	yes	characteristic
68	Q₈	—	7	2	no	—
69	Q₈	—	14	1	no	—
70	C₂×C₄	—	14	1	no	—
71	C₂×C₄	—	14	1	no	—
72	C₂×C₄	C₂×C₇×D₇	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₇	—	2	3	no	—
75	C₇	#6390	1	1	yes	characteristic, direct product factor
76	C₇	C₇×#103	1	1	yes	characteristic, complemented
77	C₂²	C₇×D₂₈	1	1	yes	characteristic
78	C₄	—	14	2	no	—
79	C₄	—	28	1	no	—
80	C₄	C₄×C₇×D₇	1	1	yes	characteristic
81	C₄	C₇×C₂²⋊D₇	1	1	yes	characteristic
82	C₂	C₇×#1132	1	1	yes	characteristic
83	C₂	C₇×Dic₂₈	1	1	yes	characteristic
84	C₂	C₇×(C₂×D₇)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₇	D₂₈
commutator	C₄×C₇	C₂×C₄×C₇
Frattini	C₂×C₄	C₂×C₇×D₇
Fitting	C₂×C₇²×C₈	C₂
socle, prime radical	C₂²×C₇²	D₄
cosocle	C₂×C₄×C₇	C₂²×C₇
hypercenter	C₂×C₇×C₈	D₇
hypocenter	C₇	C₇×#103
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄×C₇ ◁ C₂×C₇×C₈

Complements

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

N	H ≅ G/N	type
8. C₇×Dic₁₄	C₄	semidirect
11. #6390	C₇	direct (N × H)
37. Dic₁₄	C₄×C₇	semidirect
41. C₇²	#103	semidirect
75. C₇	#6390	direct (N × H)
76. C₇	C₇×#103	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×Dic₁₄	2
C₂×C₇²×C₈	2
C₇×C₄⋊Dic₇	2
C₇×#103	7
#6390	7

Subgroups — C7×#6390 #13096098

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#6390 #13096098