Subgroups — C₇×#6395 #13096097

Order 1568.

133 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	C₂×C₇²×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₇×D₇	—	2	1	no	—
5	C₂×C₄×C₇²	C₂²	1	1	yes	characteristic
6	C₂×C₇×Dic₇	—	2	1	no	—
7	C₇×D₂₈	—	2	1	no	—
8	C₇×D₂₈	C₄	1	2	yes	complemented
9	C₇²×C₈	—	2	1	no	—
10	C₇×#102	—	7	1	no	maximal
11	#6395	C₇	1	1	yes	maximal, characteristic, direct product factor
12	C₂²×C₇²	D₄	1	1	yes	characteristic
13	C₂×C₇×D₇	—	2	2	no	—
14	C₂×C₇×D₇	—	4	1	no	—
15	C₂×C₇×D₇	—	4	1	no	—
16	C₄×C₇²	C₂×C₄	1	1	yes	characteristic
17	C₄×C₇²	D₄	1	1	yes	characteristic
18	C₇×Dic₇	—	4	1	no	—
19	C₂×C₇×D₄	—	7	1	no	—
20	C₂×D₂₈	C₂×C₇	1	1	yes	characteristic
21	C₂×C₇×C₈	—	2	3	no	—
22	C₂×C₇×C₈	D₇	1	1	yes	characteristic, complemented
23	C₂×C₇×C₈	C₂×C₇	1	1	yes	characteristic, complemented
24	C₇×C₄⋊C₄	—	7	1	no	—
25	C₄⋊Dic₇	C₂×C₇	1	1	yes	characteristic, complemented
26	C₂×C₇²	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₇²	D₈	1	1	yes	characteristic
28	C₂×C₇²	SD₁₆	1	1	yes	characteristic
29	C₇×D₇	—	4	2	no	—
30	C₂³×C₇	—	14	1	no	—
31	C₂²×D₇	—	2	1	no	—
32	C₇×D₄	—	7	2	no	—
33	C₇×D₄	—	14	1	no	—
34	C₂×C₄×C₇	—	2	3	no	—
35	C₂×C₄×C₇	—	14	1	no	—
36	C₂×C₄×C₇	C₂×D₇	1	1	yes	characteristic
37	C₂×C₄×C₇	C₂²×C₇	1	1	yes	characteristic
38	C₂×Dic₇	—	2	1	no	—
39	D₂₈	—	2	1	no	—
40	D₂₈	C₄×C₇	1	2	yes	complemented
41	C₇×C₈	—	2	6	no	—
42	C₇×C₈	—	2	1	no	—
43	C₇×C₈	—	2	1	no	—
44	C₇²	#102	1	1	yes	characteristic, complemented, Sylow
45	#102	—	7	1	no	Sylow
46	C₂²×C₇	—	2	3	no	—
47	C₂²×C₇	—	14	2	no	—
48	C₂²×C₇	—	28	1	no	—
49	C₂²×C₇	—	28	1	no	—
50	C₂²×C₇	D₂₈	1	1	yes	characteristic
51	C₂²×C₇	C₇×D₄	1	1	yes	characteristic
52	C₂×D₇	—	2	2	no	—
53	C₂×D₇	—	4	1	no	—
54	C₂×D₇	—	4	1	no	—
55	C₄×C₇	—	2	3	no	—
56	C₄×C₇	—	2	3	no	—
57	C₄×C₇	—	28	1	no	—
58	C₄×C₇	C₄×D₇	1	1	yes	characteristic
59	C₄×C₇	C₂²⋊D₇	1	1	yes	characteristic
60	C₄×C₇	C₂×C₄×C₇	1	1	yes	characteristic
61	C₄×C₇	C₇×D₄	1	1	yes	characteristic
62	Dic₇	—	4	1	no	—
63	C₂×D₄	—	7	1	no	—
64	C₂×C₈	C₇×D₇	1	1	yes	characteristic, complemented
65	C₄⋊C₄	—	7	1	no	—
66	C₂×C₇	—	2	3	no	—
67	C₂×C₇	—	2	3	no	—
68	C₂×C₇	—	2	3	no	—
69	C₂×C₇	—	28	2	no	—
70	C₂×C₇	#1132	1	1	yes	characteristic
71	C₂×C₇	D₅₆	1	1	yes	characteristic
72	C₂×C₇	(C₂×D₇)⋊C₄	1	1	yes	characteristic
73	C₂×C₇	C₇×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₇	C₇×D₈	1	1	yes	characteristic
75	C₂×C₇	C₇×SD₁₆	1	1	yes	characteristic
76	D₇	—	4	2	no	—
77	C₂³	—	14	1	no	—
78	D₄	—	7	2	no	—
79	D₄	—	14	1	no	—
80	C₂×C₄	—	14	1	no	—
81	C₂×C₄	C₂×C₇×D₇	1	1	yes	characteristic
82	C₈	—	2	1	no	—
83	C₇	—	2	3	no	—
84	C₇	#6395	1	1	yes	characteristic, direct product factor
85	C₇	C₇×#102	1	1	yes	characteristic, complemented
86	C₂²	—	14	2	no	—
87	C₂²	—	28	1	no	—
88	C₂²	—	28	1	no	—
89	C₂²	C₇×D₂₈	1	1	yes	characteristic
90	C₄	—	28	1	no	—
91	C₄	C₄×C₇×D₇	1	1	yes	characteristic
92	C₄	C₇×C₂²⋊D₇	1	1	yes	characteristic
93	C₂	—	28	2	no	—
94	C₂	C₇×#1132	1	1	yes	characteristic
95	C₂	C₇×D₅₆	1	1	yes	characteristic
96	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	C₂²×C₇²	D₄
prime radical	C₂×C₇×D₂₈	C₂
cosocle	C₂×C₄×C₇	C₂²×C₇
hypercenter	C₂×C₇×C₈	D₇
hypocenter	C₇	C₇×#102
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
2. C₂×C₇²×C₈	C₂	semidirect
3. C₇×C₄⋊Dic₇	C₂	semidirect
8. C₇×D₂₈	C₄	semidirect
11. #6395	C₇	direct (N × H)
22. C₂×C₇×C₈	D₇	semidirect
23. C₂×C₇×C₈	C₂×C₇	semidirect
25. C₄⋊Dic₇	C₂×C₇	semidirect
40. D₂₈	C₄×C₇	semidirect
44. C₇²	#102	semidirect
64. C₂×C₈	C₇×D₇	semidirect
84. C₇	#6395	direct (N × H)
85. C₇	C₇×#102	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
C₂×C₇²×C₈	2
C₇×C₄⋊Dic₇	2
C₇×#102	7
#6395	7

Subgroups — C7×#6395 #13096097

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#6395 #13096097