Subgroups — C₂×#11716066 #13096561

Order 1568.

340 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	#11716066	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₇⋊(C₇⋊C₈)	C₂	1	2	yes	maximal, complemented
4	C₂³×C₇²	C₄	1	1	yes	characteristic
5	C₂×C₄×C₇²	C₄	1	2	yes	—
6	C₂×C₄×C₇²	C₄	1	1	yes	characteristic
7	C₂×C₄×C₇²	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₇²	C₂²	1	2	yes	—
9	C₇⋊(C₇⋊C₈)	C₂²	1	4	yes	complemented
10	C₂×#1136	—	7	8	no	maximal
11	C₂²×C₇²	—	2	2	no	—
12	C₂²×C₇²	C₂×C₄	1	2	yes	—
13	C₂²×C₇²	C₂×C₄	1	1	yes	characteristic
14	C₄×C₇²	C₂×C₄	1	1	yes	characteristic
15	C₄×C₇²	C₂×C₄	1	2	yes	—
16	C₄×C₇²	C₂³	1	1	yes	characteristic
17	C₂²×C₄×C₇	D₇	1	8	yes	—
18	#1136	—	7	32	no	—
19	C₂×C₇⋊C₈	—	7	16	no	—
20	C₂×C₇²	—	2	2	no	—
21	C₂×C₇²	M₁₆	1	2	yes	complemented
22	C₂×C₇²	C₂²×C₄	1	1	yes	characteristic
23	C₂³×C₇	Dic₇	1	8	yes	—
24	C₂×C₄×C₇	Dic₇	1	16	yes	—
25	C₂×C₄×C₇	Dic₇	1	8	yes	—
26	C₂×C₄×C₇	C₂×D₇	1	16	yes	—
27	C₂×C₄×C₇	C₂×D₇	1	8	yes	—
28	C₇⋊C₈	—	7	32	no	—
29	C₇²	C₂×M₁₆	1	1	yes	characteristic, complemented, Sylow
30	C₂×M₁₆	—	49	1	no	Sylow
31	C₂²×C₇	—	2	16	no	—
32	C₂²×C₇	C₂×Dic₇	1	16	yes	—
33	C₂²×C₇	C₂×Dic₇	1	8	yes	—
34	C₄×C₇	C₂×Dic₇	1	16	yes	—
35	C₄×C₇	C₂×Dic₇	1	8	yes	—
36	C₄×C₇	C₂²×D₇	1	8	yes	—
37	C₂²×C₄	C₇⋊D₇	1	1	yes	characteristic
38	M₁₆	—	49	4	no	—
39	C₂×C₈	—	49	2	no	—
40	C₂×C₇	—	2	16	no	—
41	C₂×C₇	#1136	1	16	yes	complemented
42	C₂×C₇	C₂²×Dic₇	1	8	yes	—
43	C₂³	C₇⋊Dic₇	1	1	yes	characteristic
44	C₂×C₄	C₇⋊Dic₇	1	1	yes	characteristic
45	C₂×C₄	C₇⋊Dic₇	1	2	yes	—
46	C₂×C₄	C₂×C₇⋊D₇	1	2	yes	—
47	C₂×C₄	C₂×C₇⋊D₇	1	1	yes	characteristic
48	C₈	—	49	4	no	—
49	C₇	C₂×#1136	1	8	yes	complemented
50	C₂²	—	2	2	no	—
51	C₂²	C₂×C₇⋊Dic₇	1	2	yes	—
52	C₂²	C₂×C₇⋊Dic₇	1	1	yes	characteristic
53	C₄	C₂×C₇⋊Dic₇	1	2	yes	—
54	C₄	C₂×C₇⋊Dic₇	1	1	yes	characteristic
55	C₄	C₂²×C₇⋊D₇	1	1	yes	characteristic
56	C₂	—	2	2	no	—
57	C₂	#11716066	1	2	yes	direct product factor
58	C₂	C₂²×C₇⋊Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₇⋊D₇
commutator	C₂×C₇²	C₂²×C₄
Frattini	C₄	C₂²×C₇⋊D₇
Fitting	C₂²×C₄×C₇²	C₂
socle	C₂²×C₇²	C₂×C₄
prime radical	C₂³×C₇²	C₄
cosocle	C₄×C₇²	C₂³
hypercenter	C₂²×C₄	C₇⋊D₇
hypocenter	C₇²	C₂×M₁₆
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
2. #11716066	C₂	direct (N × H)
3. C₂×C₇⋊(C₇⋊C₈)	C₂	semidirect
9. C₇⋊(C₇⋊C₈)	C₂²	semidirect
21. C₂×C₇²	M₁₆	semidirect
29. C₇²	C₂×M₁₆	semidirect
41. C₂×C₇	#1136	semidirect
49. C₇	C₂×#1136	semidirect
57. C₂	#11716066	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
#11716066	2
C₂×C₇⋊(C₇⋊C₈)	2
C₂×#1136	7

Subgroups — C2×#11716066 #13096561

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11716066 #13096561