Subgroups — Q₁₆×C₇⋊D₇ #13096557

Order 1568.

300 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₇⋊D₇	C₂	1	2	yes	maximal
2	C₇²×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#11716074	C₂	1	2	yes	maximal, complemented
4	#11716064	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₇⋊D₇	C₂	1	1	yes	maximal, characteristic
6	C₇²×Q₈	C₂²	1	2	yes	—
7	C₄×C₇⋊D₇	—	2	2	no	—
8	C₄×C₇⋊D₇	C₂²	1	1	yes	characteristic
9	#87648	—	2	2	no	—
10	#87648	C₂²	1	2	yes	—
11	C₇²×C₈	C₂²	1	1	yes	characteristic
12	C₇⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
13	D₇×Q₁₆	—	7	8	no	maximal
14	C₂×C₇⋊D₇	D₄	1	1	yes	characteristic
15	C₄×C₇²	—	2	2	no	—
16	C₄×C₇²	C₂³	1	1	yes	characteristic
17	C₇⋊Dic₇	—	2	2	no	—
18	C₇⋊Dic₇	D₄	1	1	yes	characteristic
19	Q₈×D₇	—	7	16	no	—
20	C₇×Q₁₆	D₇	1	8	yes	complemented
21	#1144	—	7	16	no	—
22	Dic₂₈	—	7	8	no	—
23	C₈×D₇	—	7	8	no	—
24	C₂×C₇²	C₂×D₄	1	1	yes	characteristic
25	C₇⋊D₇	Q₁₆	1	2	yes	direct product factor
26	C₇×Q₈	C₂×D₇	1	16	yes	—
27	C₄×D₇	—	7	8	no	—
28	C₄×D₇	—	14	16	no	—
29	Dic₁₄	—	7	16	no	—
30	Dic₁₄	—	14	16	no	—
31	C₇×C₈	C₂×D₇	1	8	yes	—
32	C₇⋊C₈	—	7	8	no	—
33	C₇²	C₂×Q₁₆	1	1	yes	characteristic, complemented, Sylow
34	C₂×Q₁₆	—	49	1	no	Sylow
35	C₂×D₇	—	7	8	no	—
36	C₄×C₇	—	2	16	no	—
37	C₄×C₇	C₂²×D₇	1	8	yes	—
38	Dic₇	—	7	8	no	—
39	Dic₇	—	14	16	no	—
40	C₂×Q₈	—	49	2	no	—
41	Q₁₆	—	49	2	no	—
42	Q₁₆	—	49	1	no	—
43	Q₁₆	C₇⋊D₇	1	1	yes	characteristic, direct product factor
44	C₂×C₈	—	49	1	no	—
45	C₂×C₇	D₄×D₇	1	8	yes	—
46	D₇	—	7	16	no	—
47	Q₈	—	49	2	no	—
48	Q₈	—	98	2	no	—
49	Q₈	C₂×C₇⋊D₇	1	2	yes	—
50	C₂×C₄	—	49	1	no	—
51	C₂×C₄	—	98	2	no	—
52	C₈	—	49	1	no	—
53	C₈	C₂×C₇⋊D₇	1	1	yes	characteristic
54	C₇	D₇×Q₁₆	1	8	yes	complemented
55	C₂²	—	49	1	no	—
56	C₄	—	2	2	no	—
57	C₄	—	49	1	no	—
58	C₄	—	98	2	no	—
59	C₄	C₂²×C₇⋊D₇	1	1	yes	characteristic
60	C₂	—	49	2	no	—
61	C₂	D₄×C₇⋊D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×C₇⋊D₇
commutator, cosocle	C₄×C₇²	C₂³
Frattini	C₄	C₂²×C₇⋊D₇
Fitting	C₇²×Q₁₆	C₂
socle	C₂×C₇²	C₂×D₄
prime radical	C₂×C₇⋊D₇	D₄
hypercenter	Q₁₆	C₇⋊D₇
hypocenter	C₇²	C₂×Q₁₆
LCS	G ▷ C₄×C₇² ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
2. C₇²×Q₁₆	C₂	semidirect
3. #11716074	C₂	semidirect
4. #11716064	C₂	semidirect
20. C₇×Q₁₆	D₇	semidirect
25. C₇⋊D₇	Q₁₆	direct (N × H)
33. C₇²	C₂×Q₁₆	semidirect
43. Q₁₆	C₇⋊D₇	direct (N × H)
54. C₇	D₇×Q₁₆	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
Q₈×C₇⋊D₇	2
C₇²×Q₁₆	2
#11716074	2
#11716064	2
C₈×C₇⋊D₇	2
D₇×Q₁₆	7

Subgroups — Q16×C7⋊D7 #13096557

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₁₆×C₇⋊D₇ #13096557