Subgroups — C₇×#6396 #13096099

Order 1568.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇²×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₇×#1136	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₇²	C₂²	1	1	yes	characteristic
5	C₇²×C₈	—	2	1	no	—
6	C₇×C₇⋊C₈	—	2	1	no	—
7	C₇×C₇⋊C₈	C₄	1	2	yes	—
8	C₇×#108	—	7	1	no	maximal
9	#6396	C₇	1	1	yes	maximal, characteristic, direct product factor
10	C₂²×C₇²	Q₈	1	1	yes	characteristic
11	C₄×C₇²	C₂×C₄	1	1	yes	characteristic
12	C₄×C₇²	D₄	1	1	yes	characteristic
13	C₇×M₁₆	—	2	3	no	—
14	C₇×M₁₆	—	7	1	no	—
15	C₇×M₁₆	D₇	1	1	yes	characteristic
16	C₇×M₁₆	C₂×C₇	1	1	yes	characteristic
17	C₂×C₇×C₈	—	7	1	no	—
18	#1136	C₂×C₇	1	1	yes	characteristic
19	C₂×C₇⋊C₈	C₂×C₇	1	1	yes	characteristic
20	C₂×C₇²	—	2	1	no	—
21	C₂×C₇²	C₄⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₇	—	2	3	no	—
23	C₂×C₄×C₇	C₂×D₇	1	1	yes	characteristic
24	C₂×C₄×C₇	C₂²×C₇	1	1	yes	characteristic
25	C₇×C₈	—	2	6	no	—
26	C₇×C₈	—	2	1	no	—
27	C₇×C₈	—	2	1	no	—
28	C₇×C₈	—	7	2	no	—
29	C₇×C₈	—	14	1	no	—
30	C₇⋊C₈	—	2	1	no	—
31	C₇⋊C₈	C₄×C₇	1	2	yes	—
32	C₇²	#108	1	1	yes	characteristic, complemented, Sylow
33	#108	—	7	1	no	Sylow
34	C₂²×C₇	—	2	3	no	—
35	C₂²×C₇	Dic₁₄	1	1	yes	characteristic
36	C₂²×C₇	C₇×Q₈	1	1	yes	characteristic
37	C₄×C₇	—	2	3	no	—
38	C₄×C₇	—	2	3	no	—
39	C₄×C₇	C₄×D₇	1	1	yes	characteristic
40	C₄×C₇	C₂²⋊D₇	1	1	yes	characteristic
41	C₄×C₇	C₂×C₄×C₇	1	1	yes	characteristic
42	C₄×C₇	C₇×D₄	1	1	yes	characteristic
43	M₁₆	—	7	1	no	—
44	M₁₆	C₇×D₇	1	1	yes	characteristic
45	C₂×C₈	—	7	1	no	—
46	C₂×C₇	—	2	1	no	—
47	C₂×C₇	—	2	1	no	—
48	C₂×C₇	—	2	3	no	—
49	C₂×C₇	—	4	3	no	—
50	C₂×C₇	Dic₇⋊C₄	1	1	yes	characteristic
51	C₂×C₇	C₇×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄	C₂×C₇×D₇	1	1	yes	characteristic
53	C₈	—	2	1	no	—
54	C₈	—	7	2	no	—
55	C₈	—	14	1	no	—
56	C₇	—	2	3	no	—
57	C₇	#6396	1	1	yes	characteristic, direct product factor
58	C₇	C₇×#108	1	1	yes	characteristic, complemented
59	C₂²	C₇×Dic₁₄	1	1	yes	characteristic
60	C₄	C₄×C₇×D₇	1	1	yes	characteristic
61	C₄	C₇×C₂²⋊D₇	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	C₇×Dic₇⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇	C₂²⋊D₇
commutator	C₄×C₇	C₂×C₄×C₇
Frattini	C₂×C₄	C₂×C₇×D₇
Fitting	C₇²×M₁₆	C₂
socle	C₂×C₇²	C₄⋊C₄
prime radical	C₂²×C₇²	Q₈
cosocle	C₂×C₄×C₇	C₂²×C₇
hypercenter	C₇×M₁₆	D₇
hypocenter	C₇	C₇×#108
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄×C₇ ◁ C₂×C₄×C₇ ◁ C₇×M₁₆

Complements

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

N	H ≅ G/N	type
9. #6396	C₇	direct (N × H)
32. C₇²	#108	semidirect
57. C₇	#6396	direct (N × H)
58. C₇	C₇×#108	semidirect

Maximal subgroups

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

subgroup	index
C₇²×M₁₆	2
C₇×#1136	2
C₂×C₇×C₇⋊C₈	2
C₇×#108	7
#6396	7

Subgroups — C7×#6396 #13096099

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#6396 #13096099