Subgroups — C₇×#6397 #13096100

Order 1568.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×D₂₈	C₂	1	1	yes	maximal, characteristic
2	C₇²×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#1136	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₇×D₇	C₄	1	2	yes	—
5	C₂×C₄×C₇²	C₂²	1	1	yes	characteristic
6	C₇×D₂₈	—	2	1	no	—
7	C₇×D₂₈	—	2	1	no	—
8	C₇²×C₈	—	2	1	no	—
9	C₇×C₇⋊C₈	—	2	1	no	—
10	C₇×#100	—	7	1	no	maximal
11	#6397	C₇	1	1	yes	maximal, characteristic, direct product factor
12	C₂²×C₇²	C₂×C₄	1	1	yes	characteristic
13	C₂×C₇×D₇	—	2	2	no	—
14	C₂×C₇×D₇	—	4	2	no	—
15	C₄×C₇²	D₄	1	1	yes	characteristic
16	C₄×C₇²	D₄	1	1	yes	characteristic
17	C₂×C₇×D₄	—	7	1	no	—
18	C₂×D₂₈	C₂×C₇	1	1	yes	characteristic
19	C₇×M₁₆	—	2	3	no	—
20	C₇×M₁₆	—	7	1	no	—
21	C₇×M₁₆	D₇	1	1	yes	characteristic, complemented
22	C₇×M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
23	#1136	C₂×C₇	1	1	yes	characteristic, complemented
24	C₂×C₇²	—	2	1	no	—
25	C₂×C₇²	C₂²⋊C₄	1	1	yes	characteristic
26	C₇×D₇	—	4	2	no	—
27	C₂³×C₇	—	7	2	no	—
28	C₂²×D₇	C₄×C₇	1	2	yes	—
29	C₇×D₄	—	14	1	no	—
30	C₇×D₄	—	14	1	no	—
31	C₂×C₄×C₇	—	2	3	no	—
32	C₂×C₄×C₇	C₂×D₇	1	1	yes	characteristic
33	C₂×C₄×C₇	C₂²×C₇	1	1	yes	characteristic
34	D₂₈	—	2	1	no	—
35	D₂₈	—	2	1	no	—
36	C₇×C₈	—	2	6	no	—
37	C₇×C₈	—	2	1	no	—
38	C₇×C₈	—	2	1	no	—
39	C₇×C₈	—	14	1	no	—
40	C₇⋊C₈	—	2	1	no	—
41	C₇²	#100	1	1	yes	characteristic, complemented, Sylow
42	#100	—	7	1	no	Sylow
43	C₂²×C₇	—	2	3	no	—
44	C₂²×C₇	—	14	2	no	—
45	C₂²×C₇	—	28	2	no	—
46	C₂²×C₇	C₄×D₇	1	1	yes	characteristic
47	C₂²×C₇	C₂×C₄×C₇	1	1	yes	characteristic
48	C₂×D₇	—	2	2	no	—
49	C₂×D₇	—	4	2	no	—
50	C₄×C₇	—	2	3	no	—
51	C₄×C₇	—	2	3	no	—
52	C₄×C₇	D₂₈	1	1	yes	characteristic
53	C₄×C₇	C₂²⋊D₇	1	1	yes	characteristic
54	C₄×C₇	C₇×D₄	1	1	yes	characteristic
55	C₄×C₇	C₇×D₄	1	1	yes	characteristic
56	C₂×D₄	—	7	1	no	—
57	M₁₆	—	7	1	no	—
58	M₁₆	C₇×D₇	1	1	yes	characteristic, complemented
59	C₂×C₇	—	2	1	no	—
60	C₂×C₇	—	2	1	no	—
61	C₂×C₇	—	2	3	no	—
62	C₂×C₇	—	4	3	no	—
63	C₂×C₇	—	28	2	no	—
64	C₂×C₇	(C₂×D₇)⋊C₄	1	1	yes	characteristic
65	C₂×C₇	C₇×C₂²⋊C₄	1	1	yes	characteristic
66	D₇	—	4	2	no	—
67	C₂³	—	7	2	no	—
68	D₄	—	14	1	no	—
69	D₄	—	14	1	no	—
70	C₂×C₄	C₂×C₇×D₇	1	1	yes	characteristic
71	C₈	—	2	1	no	—
72	C₈	—	14	1	no	—
73	C₇	—	2	3	no	—
74	C₇	#6397	1	1	yes	characteristic, direct product factor
75	C₇	C₇×#100	1	1	yes	characteristic, complemented
76	C₂²	—	14	2	no	—
77	C₂²	—	28	2	no	—
78	C₂²	C₄×C₇×D₇	1	1	yes	characteristic
79	C₄	C₇×D₂₈	1	1	yes	characteristic
80	C₄	C₇×C₂²⋊D₇	1	1	yes	characteristic
81	C₂	—	2	1	no	—
82	C₂	—	28	2	no	—
83	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	C₂×C₇×D₂₈	C₂
cosocle	C₂×C₄×C₇	C₂²×C₇
hypercenter	C₇×M₁₆	D₇
hypocenter	C₇	C₇×#100
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
2. C₇²×M₁₆	C₂	semidirect
3. C₇×#1136	C₂	semidirect
11. #6397	C₇	direct (N × H)
21. C₇×M₁₆	D₇	semidirect
22. C₇×M₁₆	C₂×C₇	semidirect
23. #1136	C₂×C₇	semidirect
41. C₇²	#100	semidirect
58. M₁₆	C₇×D₇	semidirect
74. C₇	#6397	direct (N × H)
75. C₇	C₇×#100	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×D₂₈	2
C₇²×M₁₆	2
C₇×#1136	2
C₇×#100	7
#6397	7

Subgroups — C7×#6397 #13096100

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#6397 #13096100