Subgroups — C₂×C₄⋊(C₇⋊Dic₇) #13096565

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	2	yes	maximal
3	C₄⋊(C₇⋊Dic₇)	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₇²	C₂²	1	1	yes	characteristic
5	C₂×C₄×C₇²	C₄	1	4	yes	complemented
6	C₂×C₄×C₇²	C₂²	1	2	yes	—
7	C₂×C₇⋊Dic₇	—	2	4	no	—
8	C₂×C₇⋊Dic₇	C₂²	1	4	yes	—
9	C₂×C₄⋊Dic₇	—	7	8	no	maximal
10	C₂²×C₇²	C₂×C₄	1	2	yes	—
11	C₂²×C₇²	D₄	1	2	yes	—
12	C₂²×C₇²	Q₈	1	2	yes	—
13	C₂²×C₇²	C₂³	1	1	yes	characteristic
14	C₄×C₇²	C₂×C₄	1	4	yes	complemented
15	C₇⋊Dic₇	—	2	4	no	—
16	C₂²×C₄×C₇	D₇	1	8	yes	—
17	C₂²×Dic₇	—	7	16	no	—
18	C₄⋊Dic₇	—	7	32	no	—
19	C₂×C₇²	C₄⋊C₄	1	4	yes	complemented
20	C₂×C₇²	C₂²×C₄	1	1	yes	characteristic
21	C₂×C₇²	C₂×D₄	1	1	yes	characteristic
22	C₂×C₇²	C₂×Q₈	1	1	yes	characteristic
23	C₂³×C₇	C₂×D₇	1	8	yes	—
24	C₂×C₄×C₇	Dic₇	1	32	yes	complemented
25	C₂×C₄×C₇	C₂×D₇	1	16	yes	—
26	C₂×Dic₇	—	7	32	no	—
27	C₂×Dic₇	—	14	32	no	—
28	C₇²	C₂×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
29	C₂×C₄⋊C₄	—	49	1	no	Sylow
30	C₂²×C₇	Dic₁₄	1	16	yes	—
31	C₂²×C₇	D₂₈	1	16	yes	—
32	C₂²×C₇	C₂×Dic₇	1	16	yes	—
33	C₂²×C₇	C₂²×D₇	1	8	yes	—
34	C₄×C₇	C₂×Dic₇	1	32	yes	complemented
35	Dic₇	—	14	32	no	—
36	C₂²×C₄	—	49	2	no	—
37	C₂²×C₄	C₇⋊D₇	1	1	yes	characteristic
38	C₄⋊C₄	—	49	4	no	—
39	C₂×C₇	C₄⋊Dic₇	1	32	yes	complemented
40	C₂×C₇	C₂×Dic₁₄	1	8	yes	—
41	C₂×C₇	C₂×D₂₈	1	8	yes	—
42	C₂×C₇	C₂²×Dic₇	1	8	yes	—
43	C₂³	C₂×C₇⋊D₇	1	1	yes	characteristic
44	C₂×C₄	—	49	4	no	—
45	C₂×C₄	—	98	4	no	—
46	C₂×C₄	C₇⋊Dic₇	1	4	yes	complemented
47	C₂×C₄	C₂×C₇⋊D₇	1	2	yes	—
48	C₇	C₂×C₄⋊Dic₇	1	8	yes	complemented
49	C₂²	#87648	1	2	yes	—
50	C₂²	C₄⋊(C₇⋊D₇)	1	2	yes	—
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	4	yes	complemented
55	C₂	C₄⋊(C₇⋊Dic₇)	1	4	yes	direct product factor
56	C₂	C₂×#87648	1	1	yes	characteristic
57	C₂	C₂×C₄⋊(C₇⋊D₇)	1	1	yes	characteristic
58	C₂	C₂²×C₇⋊Dic₇	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
3. C₄⋊(C₇⋊Dic₇)	C₂	direct (N × H)
5. C₂×C₄×C₇²	C₄	semidirect
14. C₄×C₇²	C₂×C₄	semidirect
19. C₂×C₇²	C₄⋊C₄	semidirect
24. C₂×C₄×C₇	Dic₇	semidirect
28. C₇²	C₂×C₄⋊C₄	semidirect
34. C₄×C₇	C₂×Dic₇	semidirect
39. C₂×C₇	C₄⋊Dic₇	semidirect
46. C₂×C₄	C₇⋊Dic₇	semidirect
48. C₇	C₂×C₄⋊Dic₇	semidirect
54. C₄	C₂×C₇⋊Dic₇	semidirect
55. C₂	C₄⋊(C₇⋊Dic₇)	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
C₄⋊(C₇⋊Dic₇)	2
C₂×C₄⋊Dic₇	7

Subgroups — C2×C4⋊(C7⋊Dic7) #13096565

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊(C₇⋊Dic₇) #13096565