Subgroups — #13096566

Order 1568.

380 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₇²	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊(C₇⋊Dic₇)	C₂	1	2	yes	maximal
3	C₄⋊(C₇⋊Dic₇)	C₂	1	2	yes	maximal, complemented
4	C₄×C₇⋊Dic₇	C₂	1	2	yes	maximal, complemented
5	C₂³×C₇²	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₇²	C₄	1	4	yes	complemented
7	C₂×C₄×C₇²	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₇²	C₂²	1	1	yes	characteristic
9	C₂×C₇⋊Dic₇	C₂²	1	4	yes	—
10	#6489	—	7	8	no	maximal
11	C₂²×C₇²	—	2	2	no	—
12	C₂²×C₇²	C₂×C₄	1	2	yes	—
13	C₂²×C₇²	C₂³	1	1	yes	characteristic
14	C₄×C₇²	C₂×C₄	1	2	yes	—
15	C₄×C₇²	C₂×C₄	1	2	yes	—
16	C₇⋊Dic₇	—	2	4	no	—
17	C₂²×C₄×C₇	D₇	1	8	yes	—
18	C₂²⋊Dic₇	—	7	16	no	—
19	C₄⋊Dic₇	—	7	16	no	—
20	C₄×Dic₇	—	7	16	no	—
21	C₂×C₇²	—	2	2	no	—
22	C₂×C₇²	C₂²×C₄	1	1	yes	characteristic
23	C₂×C₇²	#41	1	2	yes	—
24	C₂³×C₇	C₂×D₇	1	8	yes	—
25	C₂×C₄×C₇	Dic₇	1	32	yes	complemented
26	C₂×C₄×C₇	C₂×D₇	1	8	yes	—
27	C₂×C₄×C₇	C₂×D₇	1	8	yes	—
28	C₂×Dic₇	—	7	32	no	—
29	C₇²	#117	1	1	yes	characteristic, complemented, Sylow
30	#117	—	49	1	no	Sylow
31	C₂²×C₇	—	2	16	no	—
32	C₂²×C₇	C₂×Dic₇	1	16	yes	—
33	C₂²×C₇	C₂²×D₇	1	8	yes	—
34	C₄×C₇	C₂×Dic₇	1	16	yes	—
35	C₄×C₇	C₂×Dic₇	1	16	yes	—
36	Dic₇	—	14	32	no	—
37	C₂²×C₄	C₇⋊D₇	1	1	yes	characteristic
38	C₄⋊C₄	—	49	2	no	—
39	C₂²⋊C₄	—	49	2	no	—
40	C₄²	—	49	2	no	—
41	C₂×C₇	—	2	16	no	—
42	C₂×C₇	#1157	1	16	yes	—
43	C₂×C₇	C₂²×Dic₇	1	8	yes	—
44	C₂³	C₂×C₇⋊D₇	1	1	yes	characteristic
45	C₂×C₄	—	49	4	no	—
46	C₂×C₄	C₇⋊Dic₇	1	4	yes	complemented
47	C₂×C₄	C₂×C₇⋊D₇	1	1	yes	characteristic
48	C₂×C₄	C₂×C₇⋊D₇	1	1	yes	characteristic
49	C₇	#6489	1	8	yes	complemented
50	C₂²	—	2	2	no	—
51	C₂²	C₂×C₇⋊Dic₇	1	2	yes	—
52	C₂²	C₂²×C₇⋊D₇	1	1	yes	characteristic
53	C₄	—	98	4	no	—
54	C₄	C₂×C₇⋊Dic₇	1	2	yes	—
55	C₄	C₂×C₇⋊Dic₇	1	2	yes	—
56	C₂	—	2	2	no	—
57	C₂	#11716129	1	2	yes	—
58	C₂	C₂²×C₇⋊Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₇⋊D₇
commutator	C₂×C₇²	C₂²×C₄
Frattini	C₂²	C₂²×C₇⋊D₇
Fitting	C₂²×C₄×C₇²	C₂
socle, cosocle	C₂²×C₇²	C₂³
prime radical	C₂³×C₇²	C₂²
hypercenter	C₂²×C₄	C₇⋊D₇
hypocenter	C₇²	#117
LCS	G ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₇⋊Dic₇)	C₂	semidirect
4. C₄×C₇⋊Dic₇	C₂	semidirect
6. C₂×C₄×C₇²	C₄	semidirect
25. C₂×C₄×C₇	Dic₇	semidirect
29. C₇²	#117	semidirect
46. C₂×C₄	C₇⋊Dic₇	semidirect
49. C₇	#6489	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₇²	2
C₂²⋊(C₇⋊Dic₇)	2
C₄⋊(C₇⋊Dic₇)	2
C₄×C₇⋊Dic₇	2
#6489	7