Subgroups — Q₁₆×Dic₇ #89955

Order 448.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₇	C₂	1	2	yes	maximal
3	#6409	C₂	1	2	yes	maximal
4	#6392	C₂	1	1	yes	maximal, characteristic
5	C₈×Dic₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇×Q₈	C₂²	1	2	yes	—
7	C₇×Q₁₆	C₄	1	4	yes	complemented
8	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₇	—	2	2	no	—
10	C₄⋊Dic₇	C₂²	1	2	yes	—
11	C₄×Dic₇	—	2	2	no	—
12	C₄×Dic₇	C₂²	1	1	yes	characteristic
13	C₂×C₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₄×Q₁₆	—	7	1	no	maximal, Sylow
15	C₇×Q₈	—	2	2	no	—
16	C₇×Q₈	C₂×C₄	1	4	yes	—
17	C₂×C₄×C₇	—	2	2	no	—
18	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
19	C₂×Dic₇	—	2	2	no	—
20	C₂×Dic₇	D₄	1	1	yes	characteristic
21	C₂×Dic₇	D₄	1	1	yes	characteristic
22	C₇×C₈	C₂×C₄	1	2	yes	—
23	C₇⋊C₈	—	2	1	no	—
24	C₂×Q₁₆	D₇	1	1	yes	characteristic
25	C₄×Q₈	—	7	2	no	—
26	#107	—	7	1	no	—
27	#103	—	7	2	no	—
28	C₄×C₈	—	7	1	no	—
29	C₂²×C₇	C₂×D₄	1	1	yes	characteristic
30	C₄×C₇	—	2	4	no	—
31	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₇	#41	1	1	yes	characteristic
33	Dic₇	—	2	1	no	—
34	Dic₇	—	4	2	no	—
35	Dic₇	Q₁₆	1	2	yes	direct product factor
36	C₂×Q₈	C₂×D₇	1	2	yes	—
37	Q₁₆	Dic₇	1	4	yes	direct product factor
38	C₂×C₈	—	7	1	no	—
39	C₂×C₈	C₂×D₇	1	1	yes	characteristic
40	C₄⋊C₄	—	7	2	no	—
41	C₄⋊C₄	—	14	2	no	—
42	C₄²	—	7	1	no	—
43	C₄²	—	14	2	no	—
44	C₂×C₇	C₄×D₄	1	1	yes	characteristic
45	C₂×C₇	C₂×Q₁₆	1	1	yes	characteristic
46	C₂×C₇	#135	1	1	yes	characteristic
47	Q₈	—	2	2	no	—
48	Q₈	C₂×Dic₇	1	4	yes	—
49	C₂×C₄	—	2	2	no	—
50	C₂×C₄	—	7	1	no	—
51	C₂×C₄	—	7	1	no	—
52	C₂×C₄	—	14	2	no	—
53	C₂×C₄	C₂²×D₇	1	1	yes	characteristic
54	C₈	—	14	1	no	—
55	C₈	C₂×Dic₇	1	2	yes	—
56	C₇	C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
57	C₂²	D₄×D₇	1	1	yes	characteristic
58	C₄	—	2	4	no	—
59	C₄	—	7	2	no	—
60	C₄	—	14	1	no	—
61	C₄	—	28	2	no	—
62	C₄	#1159	1	1	yes	characteristic
63	C₄	C₂²×Dic₇	1	1	yes	characteristic
64	C₂	D₇×Q₁₆	1	1	yes	characteristic
65	C₂	#6482	1	1	yes	characteristic
66	C₂	D₄×Dic₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
7. C₇×Q₁₆	C₄	semidirect
35. Dic₇	Q₁₆	direct (N × H)
37. Q₁₆	Dic₇	direct (N × H)
56. C₇	C₄×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×Q₁₆	2
Q₈×Dic₇	2
#6409	2
#6392	2
C₈×Dic₇	2
C₄×Q₁₆	7

Subgroups — Q16×Dic7 #89955

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₁₆×Dic₇ #89955