Subgroups — C₂×#11716068 #13096563

Order 1568.

460 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	1	yes	maximal, characteristic
3	C₂²×C₇⋊Dic₇	C₂	1	1	yes	maximal, characteristic
4	#11716068	C₂	1	4	yes	maximal, direct product factor
5	C₂³×C₇²	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₇²	—	2	2	no	—
7	C₂×C₄×C₇²	C₂²	1	2	yes	—
8	C₂×C₇⋊Dic₇	—	2	2	no	—
9	C₂×C₇⋊Dic₇	C₄	1	4	yes	complemented
10	C₂×C₇⋊Dic₇	C₂²	1	2	yes	—
11	C₂×C₇⋊Dic₇	C₂²	1	2	yes	—
12	C₂×Dic₇⋊C₄	—	7	8	no	maximal
13	C₂²×C₇²	C₂×C₄	1	2	yes	—
14	C₂²×C₇²	D₄	1	2	yes	—
15	C₂²×C₇²	Q₈	1	2	yes	—
16	C₂²×C₇²	C₂³	1	1	yes	characteristic
17	C₄×C₇²	—	2	2	no	—
18	C₇⋊Dic₇	—	2	2	no	—
19	C₇⋊Dic₇	C₂×C₄	1	4	yes	complemented
20	C₂²×C₄×C₇	D₇	1	8	yes	—
21	C₂²×Dic₇	—	7	8	no	—
22	C₂²×Dic₇	—	7	8	no	—
23	Dic₇⋊C₄	—	7	32	no	—
24	C₂×C₇²	C₄⋊C₄	1	4	yes	complemented
25	C₂×C₇²	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₇²	C₂×D₄	1	1	yes	characteristic
27	C₂×C₇²	C₂×Q₈	1	1	yes	characteristic
28	C₂³×C₇	C₂×D₇	1	8	yes	—
29	C₂×C₄×C₇	—	2	16	no	—
30	C₂×C₄×C₇	C₂×D₇	1	16	yes	—
31	C₂×Dic₇	—	7	32	no	—
32	C₂×Dic₇	—	7	16	no	—
33	C₂×Dic₇	—	7	16	no	—
34	C₂×Dic₇	—	14	16	no	—
35	C₇²	C₂×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
36	C₂×C₄⋊C₄	—	49	1	no	Sylow
37	C₂²×C₇	Dic₁₄	1	16	yes	—
38	C₂²×C₇	C₄×D₇	1	16	yes	—
39	C₂²×C₇	C₂²⋊D₇	1	16	yes	—
40	C₂²×C₇	C₂²×D₇	1	8	yes	—
41	C₄×C₇	—	2	16	no	—
42	Dic₇	—	7	32	no	—
43	Dic₇	—	14	16	no	—
44	C₂²×C₄	—	49	1	no	—
45	C₂²×C₄	—	49	1	no	—
46	C₂²×C₄	C₇⋊D₇	1	1	yes	characteristic
47	C₄⋊C₄	—	49	4	no	—
48	C₂×C₇	Dic₇⋊C₄	1	32	yes	complemented
49	C₂×C₇	C₂×Dic₁₄	1	8	yes	—
50	C₂×C₇	C₂×C₄×D₇	1	8	yes	—
51	C₂×C₇	C₂×C₂²⋊D₇	1	8	yes	—
52	C₂³	C₂×C₇⋊D₇	1	1	yes	characteristic
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	—	49	4	no	—
55	C₂×C₄	—	49	2	no	—
56	C₂×C₄	—	49	2	no	—
57	C₂×C₄	—	98	2	no	—
58	C₂×C₄	C₂×C₇⋊D₇	1	2	yes	—
59	C₇	C₂×Dic₇⋊C₄	1	8	yes	complemented
60	C₂²	#87648	1	2	yes	—
61	C₂²	C₄×C₇⋊D₇	1	2	yes	—
62	C₂²	C₂²⋊(C₇⋊D₇)	1	2	yes	—
63	C₂²	C₂²×C₇⋊D₇	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	49	4	no	—
66	C₄	—	98	2	no	—
67	C₂	#11716068	1	4	yes	direct product factor
68	C₂	C₂×#87648	1	1	yes	characteristic
69	C₂	C₂×C₄×C₇⋊D₇	1	1	yes	characteristic
70	C₂	C₂×C₂²⋊(C₇⋊D₇)	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. #11716068	C₂	direct (N × H)
9. C₂×C₇⋊Dic₇	C₄	semidirect
19. C₇⋊Dic₇	C₂×C₄	semidirect
24. C₂×C₇²	C₄⋊C₄	semidirect
35. C₇²	C₂×C₄⋊C₄	semidirect
48. C₂×C₇	Dic₇⋊C₄	semidirect
59. C₇	C₂×Dic₇⋊C₄	semidirect
67. C₂	#11716068	direct (N × H)

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
#11716068	2
C₂×Dic₇⋊C₄	7

Subgroups — C2×#11716068 #13096563

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11716068 #13096563