Subgroups — #13096564

Order 1568.

370 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	C₂×#87648	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊(C₇⋊Dic₇)	C₂	1	2	yes	maximal
4	C₄⋊(C₇⋊Dic₇)	C₂	1	1	yes	maximal, characteristic, complemented
5	#11716068	C₂	1	2	yes	maximal, complemented
6	C₂³×C₇²	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₇²	—	2	2	no	—
8	C₂×C₄×C₇²	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₇²	C₂²	1	1	yes	characteristic
10	C₂×C₇⋊Dic₇	C₂²	1	2	yes	—
11	C₂×C₇⋊Dic₇	C₂²	1	2	yes	—
12	#87648	—	2	2	no	—
13	#6487	—	7	8	no	maximal
14	C₂²×C₇²	—	2	1	no	—
15	C₂²×C₇²	—	2	1	no	—
16	C₂²×C₇²	Q₈	1	2	yes	complemented
17	C₂²×C₇²	C₂³	1	1	yes	characteristic
18	C₄×C₇²	—	2	1	no	—
19	C₄×C₇²	D₄	1	2	yes	—
20	C₇⋊Dic₇	—	2	2	no	—
21	C₇⋊Dic₇	—	2	2	no	—
22	C₂²×C₄×C₇	D₇	1	8	yes	—
23	C₂×Dic₁₄	—	7	8	no	—
24	C₂²⋊Dic₇	—	7	16	no	—
25	C₄⋊Dic₇	—	7	8	no	—
26	Dic₇⋊C₄	—	7	16	no	—
27	C₂×C₇²	—	2	2	no	—
28	C₂×C₇²	C₂×D₄	1	1	yes	characteristic
29	C₂×C₇²	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₇²	#41	1	1	yes	characteristic
31	C₂³×C₇	C₂×D₇	1	8	yes	—
32	C₂×C₄×C₇	—	2	16	no	—
33	C₂×C₄×C₇	C₂×D₇	1	8	yes	—
34	C₂×C₄×C₇	C₂×D₇	1	8	yes	—
35	C₂×Dic₇	—	7	16	no	—
36	C₂×Dic₇	—	7	16	no	—
37	Dic₁₄	—	14	16	no	—
38	C₇²	C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
39	C₂²⋊Q₈	—	49	1	no	Sylow
40	C₂²×C₇	—	2	8	no	—
41	C₂²×C₇	—	2	8	no	—
42	C₂²×C₇	Dic₁₄	1	16	yes	complemented
43	C₂²×C₇	C₂²×D₇	1	8	yes	—
44	C₄×C₇	—	2	8	no	—
45	C₄×C₇	C₂²⋊D₇	1	16	yes	—
46	Dic₇	—	14	16	no	—
47	Dic₇	—	14	16	no	—
48	C₂×Q₈	—	49	1	no	—
49	C₂²×C₄	C₇⋊D₇	1	1	yes	characteristic
50	C₄⋊C₄	—	49	2	no	—
51	C₄⋊C₄	—	49	1	no	—
52	C₂²⋊C₄	—	49	2	no	—
53	C₂×C₇	—	2	16	no	—
54	C₂×C₇	C₂×Dic₁₄	1	8	yes	—
55	C₂×C₇	#1157	1	8	yes	—
56	C₂×C₇	C₂×C₂²⋊D₇	1	8	yes	—
57	C₂³	C₂×C₇⋊D₇	1	1	yes	characteristic
58	Q₈	—	98	2	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	49	2	no	—
61	C₂×C₄	—	49	2	no	—
62	C₂×C₄	C₂×C₇⋊D₇	1	1	yes	characteristic
63	C₂×C₄	C₂×C₇⋊D₇	1	1	yes	characteristic
64	C₇	#6487	1	8	yes	complemented
65	C₂²	—	2	1	no	—
66	C₂²	—	2	1	no	—
67	C₂²	#87648	1	2	yes	complemented
68	C₂²	C₂²×C₇⋊D₇	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	98	2	no	—
71	C₄	—	98	2	no	—
72	C₄	C₂²⋊(C₇⋊D₇)	1	2	yes	—
73	C₂	—	2	2	no	—
74	C₂	C₂×#87648	1	1	yes	characteristic
75	C₂	#11716129	1	1	yes	characteristic
76	C₂	C₂×C₂²⋊(C₇⋊D₇)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×C₇⋊D₇
commutator, socle, cosocle	C₂²×C₇²	C₂³
Fitting	C₂²×C₄×C₇²	C₂
prime radical	C₂³×C₇²	C₂²
hypercenter	C₂²×C₄	C₇⋊D₇
hypocenter	C₇²	C₂²⋊Q₈
LCS	G ▷ C₂²×C₇² ▷ C₇²
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#87648	C₂	semidirect
4. C₄⋊(C₇⋊Dic₇)	C₂	semidirect
5. #11716068	C₂	semidirect
16. C₂²×C₇²	Q₈	semidirect
38. C₇²	C₂²⋊Q₈	semidirect
42. C₂²×C₇	Dic₁₄	semidirect
64. C₇	#6487	semidirect
67. C₂²	#87648	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
C₂×#87648	2
C₂²⋊(C₇⋊Dic₇)	2
C₄⋊(C₇⋊Dic₇)	2
#11716068	2
#6487	7