Subgroups — C₂×#11716129 #13096731

Order 1568.

820 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, complemented
2	C₂×C₂²⋊(C₇⋊D₇)	C₂	1	2	yes	maximal, complemented
3	#11716129	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊(C₇⋊D₇)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×C₇⋊D₇	C₂	1	2	yes	maximal, complemented
6	C₂×#87648	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₇²	C₂²	1	1	yes	characteristic
8	C₂²×C₇⋊D₇	C₂²	1	2	yes	—
9	C₂×C₄×C₇²	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇²	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇²	C₂²	1	4	yes	complemented
12	C₂²⋊(C₇⋊D₇)	C₂²	1	8	yes	complemented
13	C₂×C₇⋊Dic₇	C₂²	1	2	yes	—
14	C₄⋊(C₇⋊D₇)	C₂²	1	4	yes	complemented
15	C₄×C₇⋊D₇	C₂²	1	8	yes	complemented
16	#87648	C₂²	1	4	yes	complemented
17	C₂×#1157	—	7	8	no	maximal
18	C₂²×C₇²	—	2	2	no	—
19	C₂²×C₇²	C₂³	1	2	yes	—
20	C₂²×C₇²	C₂³	1	1	yes	characteristic
21	C₂×C₇⋊D₇	—	2	4	no	—
22	C₂×C₇⋊D₇	C₂³	1	4	yes	—
23	C₄×C₇²	C₂³	1	2	yes	—
24	C₄×C₇²	C₂³	1	2	yes	—
25	C₇⋊Dic₇	C₂³	1	4	yes	—
26	C₂²×C₄×C₇	D₇	1	8	yes	complemented
27	C₂×C₂²⋊D₇	—	7	16	no	—
28	#1157	—	7	64	no	—
29	C₂×D₂₈	—	7	8	no	—
30	C₂×C₄×D₇	—	7	16	no	—
31	C₂×Dic₁₄	—	7	8	no	—
32	C₂×C₇²	—	2	2	no	—
33	C₂×C₇²	#41	1	2	yes	complemented
34	C₂×C₇²	C₂⁴	1	1	yes	characteristic
35	C₇⋊D₇	—	2	4	no	—
36	C₂³×C₇	C₂×D₇	1	8	yes	—
37	C₂²×D₇	—	7	16	no	—
38	C₂×C₄×C₇	C₂×D₇	1	8	yes	—
39	C₂×C₄×C₇	C₂×D₇	1	8	yes	—
40	C₂×C₄×C₇	C₂×D₇	1	32	yes	complemented
41	C₂²⋊D₇	—	7	64	no	—
42	C₂×Dic₇	—	7	16	no	—
43	D₂₈	—	7	32	no	—
44	C₄×D₇	—	7	64	no	—
45	Dic₁₄	—	7	32	no	—
46	C₇²	C₂×#41	1	1	yes	characteristic, complemented, Sylow
47	C₂×#41	—	49	1	no	Sylow
48	C₂²×C₇	—	2	16	no	—
49	C₂²×C₇	C₂²×D₇	1	16	yes	—
50	C₂²×C₇	C₂²×D₇	1	8	yes	—
51	C₂×D₇	—	7	32	no	—
52	C₂×D₇	—	14	32	no	—
53	C₄×C₇	C₂²×D₇	1	16	yes	—
54	C₄×C₇	C₂²×D₇	1	16	yes	—
55	Dic₇	—	7	32	no	—
56	#41	—	49	8	no	—
57	C₂×Q₈	—	49	1	no	—
58	C₂×D₄	—	49	2	no	—
59	C₂×D₄	—	49	1	no	—
60	C₂²×C₄	—	49	2	no	—
61	C₂²×C₄	C₇⋊D₇	1	1	yes	characteristic, complemented
62	C₂×C₇	—	2	16	no	—
63	C₂×C₇	#1157	1	16	yes	complemented
64	C₂×C₇	C₂³×D₇	1	8	yes	—
65	D₇	—	14	32	no	—
66	C₂³	—	49	2	no	—
67	C₂³	C₂×C₇⋊D₇	1	1	yes	characteristic
68	Q₈	—	49	4	no	—
69	D₄	—	49	8	no	—
70	D₄	—	49	4	no	—
71	C₂×C₄	—	49	2	no	—
72	C₂×C₄	—	49	8	no	—
73	C₂×C₄	C₂×C₇⋊D₇	1	1	yes	characteristic
74	C₂×C₄	C₂×C₇⋊D₇	1	1	yes	characteristic
75	C₂×C₄	C₂×C₇⋊D₇	1	4	yes	complemented
76	C₇	C₂×#1157	1	8	yes	complemented
77	C₂²	—	2	2	no	—
78	C₂²	—	49	4	no	—
79	C₂²	—	98	4	no	—
80	C₂²	C₂²×C₇⋊D₇	1	2	yes	—
81	C₂²	C₂²×C₇⋊D₇	1	1	yes	characteristic
82	C₄	—	49	4	no	—
83	C₄	C₂²×C₇⋊D₇	1	2	yes	—
84	C₄	C₂²×C₇⋊D₇	1	2	yes	—
85	C₂	—	2	2	no	—
86	C₂	—	98	4	no	—
87	C₂	#11716129	1	2	yes	direct product factor
88	C₂	C₂³×C₇⋊D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₇⋊D₇
commutator, cosocle	C₂×C₇²	C₂⁴
Frattini	C₂	C₂³×C₇⋊D₇
Fitting	C₂²×C₄×C₇²	C₂
socle	C₂²×C₇²	C₂³
hypercenter	C₂²×C₄	C₇⋊D₇
hypocenter	C₇²	C₂×#41
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
1. C₂²×C₄×C₇²	C₂	semidirect
2. C₂×C₂²⋊(C₇⋊D₇)	C₂	semidirect
3. #11716129	C₂	direct (N × H)
4. C₂×C₄⋊(C₇⋊D₇)	C₂	semidirect
5. C₂×C₄×C₇⋊D₇	C₂	semidirect
6. C₂×#87648	C₂	semidirect
11. C₂×C₄×C₇²	C₂²	semidirect
12. C₂²⋊(C₇⋊D₇)	C₂²	semidirect
14. C₄⋊(C₇⋊D₇)	C₂²	semidirect
15. C₄×C₇⋊D₇	C₂²	semidirect
16. #87648	C₂²	semidirect
26. C₂²×C₄×C₇	D₇	semidirect
33. C₂×C₇²	#41	semidirect
40. C₂×C₄×C₇	C₂×D₇	semidirect
46. C₇²	C₂×#41	semidirect
61. C₂²×C₄	C₇⋊D₇	semidirect
63. C₂×C₇	#1157	semidirect
75. C₂×C₄	C₂×C₇⋊D₇	semidirect
76. C₇	C₂×#1157	semidirect
87. C₂	#11716129	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
C₂×C₂²⋊(C₇⋊D₇)	2
#11716129	2
C₂×C₄⋊(C₇⋊D₇)	2
C₂×C₄×C₇⋊D₇	2
C₂×#87648	2
C₂×#1157	7

Subgroups — C2×#11716129 #13096731

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11716129 #13096731