Subgroups — C₇×#6398 #13096101

Order 1568.

106 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	C₇²×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₇×#1136	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₇²	C₂²	1	1	yes	characteristic
5	C₂×C₇×Dic₇	C₄	1	2	yes	—
6	C₇×Dic₁₄	—	2	1	no	—
7	C₇×Dic₁₄	—	2	1	no	—
8	C₇²×C₈	—	2	1	no	—
9	C₇×C₇⋊C₈	—	2	1	no	—
10	C₇×#101	—	7	1	no	maximal
11	#6398	C₇	1	1	yes	maximal, characteristic, direct product factor
12	C₂²×C₇²	C₂×C₄	1	1	yes	characteristic
13	C₄×C₇²	D₄	1	1	yes	characteristic
14	C₄×C₇²	D₄	1	1	yes	characteristic
15	C₇×Dic₇	—	2	2	no	—
16	C₂×C₇×Q₈	—	7	1	no	—
17	C₂×Dic₁₄	C₂×C₇	1	1	yes	characteristic
18	C₇×M₁₆	—	2	3	no	—
19	C₇×M₁₆	—	7	1	no	—
20	C₇×M₁₆	D₇	1	1	yes	characteristic
21	C₇×M₁₆	C₂×C₇	1	1	yes	characteristic
22	#1136	C₂×C₇	1	1	yes	characteristic
23	C₂×C₇²	—	2	1	no	—
24	C₂×C₇²	C₂²⋊C₄	1	1	yes	characteristic
25	C₇×Q₈	—	14	1	no	—
26	C₇×Q₈	—	14	1	no	—
27	C₂×C₄×C₇	—	2	3	no	—
28	C₂×C₄×C₇	—	7	2	no	—
29	C₂×C₄×C₇	C₂×D₇	1	1	yes	characteristic
30	C₂×C₄×C₇	C₂²×C₇	1	1	yes	characteristic
31	C₂×Dic₇	C₄×C₇	1	2	yes	—
32	Dic₁₄	—	2	1	no	—
33	Dic₁₄	—	2	1	no	—
34	C₇×C₈	—	2	6	no	—
35	C₇×C₈	—	2	1	no	—
36	C₇×C₈	—	2	1	no	—
37	C₇×C₈	—	14	1	no	—
38	C₇⋊C₈	—	2	1	no	—
39	C₇²	#101	1	1	yes	characteristic, complemented, Sylow
40	#101	—	7	1	no	Sylow
41	C₂²×C₇	—	2	3	no	—
42	C₂²×C₇	C₄×D₇	1	1	yes	characteristic
43	C₂²×C₇	C₂×C₄×C₇	1	1	yes	characteristic
44	C₄×C₇	—	2	3	no	—
45	C₄×C₇	—	2	3	no	—
46	C₄×C₇	—	14	2	no	—
47	C₄×C₇	D₂₈	1	1	yes	characteristic
48	C₄×C₇	C₂²⋊D₇	1	1	yes	characteristic
49	C₄×C₇	C₇×D₄	1	1	yes	characteristic
50	C₄×C₇	C₇×D₄	1	1	yes	characteristic
51	Dic₇	—	2	2	no	—
52	C₂×Q₈	—	7	1	no	—
53	M₁₆	—	7	1	no	—
54	M₁₆	C₇×D₇	1	1	yes	characteristic
55	C₂×C₇	—	2	1	no	—
56	C₂×C₇	—	2	1	no	—
57	C₂×C₇	—	2	3	no	—
58	C₂×C₇	—	4	3	no	—
59	C₂×C₇	(C₂×D₇)⋊C₄	1	1	yes	characteristic
60	C₂×C₇	C₇×C₂²⋊C₄	1	1	yes	characteristic
61	Q₈	—	14	1	no	—
62	Q₈	—	14	1	no	—
63	C₂×C₄	—	7	2	no	—
64	C₂×C₄	C₂×C₇×D₇	1	1	yes	characteristic
65	C₈	—	2	1	no	—
66	C₈	—	14	1	no	—
67	C₇	—	2	3	no	—
68	C₇	#6398	1	1	yes	characteristic, direct product factor
69	C₇	C₇×#101	1	1	yes	characteristic, complemented
70	C₂²	C₄×C₇×D₇	1	1	yes	characteristic
71	C₄	—	14	2	no	—
72	C₄	C₇×D₂₈	1	1	yes	characteristic
73	C₄	C₇×C₂²⋊D₇	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	C₇×(C₂×D₇)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₇	(C₂×D₇)⋊C₄
commutator	C₂²×C₇	C₂×C₄×C₇
Frattini	C₂×C₄	C₂×C₇×D₇
Fitting	C₇²×M₁₆	C₂
socle	C₂×C₇²	C₂²⋊C₄
prime radical	C₂²×C₇²	C₂×C₄
cosocle	C₂×C₄×C₇	C₂²×C₇
hypercenter	C₇×M₁₆	D₇
hypocenter	C₇	C₇×#101
LCS	G ▷ C₂²×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ C₇×M₁₆

Complements

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

N	H ≅ G/N	type
11. #6398	C₇	direct (N × H)
39. C₇²	#101	semidirect
68. C₇	#6398	direct (N × H)
69. C₇	C₇×#101	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
C₇²×M₁₆	2
C₇×#1136	2
C₇×#101	7
#6398	7

Subgroups — C7×#6398 #13096101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#6398 #13096101