Subgroups — Dic₇×Q₃₂ #11729249

Order 896.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×Dic₇	C₂	1	2	yes	maximal
3	#89359	C₂	1	2	yes	maximal
4	#89299	C₂	1	1	yes	maximal, characteristic
5	C₁₆×Dic₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇×Q₁₆	C₂²	1	2	yes	—
7	Q₈×Dic₇	—	2	2	no	—
8	C₇×Q₃₂	C₄	1	4	yes	complemented
9	C₂×C₇×C₁₆	C₂²	1	1	yes	characteristic
10	#6409	—	2	2	no	—
11	#6392	C₂²	1	2	yes	—
12	C₈×Dic₇	C₂²	1	1	yes	characteristic
13	C₂×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₄×Q₃₂	—	7	1	no	maximal, Sylow
15	C₂×C₇×Q₈	—	2	2	no	—
16	C₇×Q₁₆	—	2	2	no	—
17	C₇×Q₁₆	C₂×C₄	1	4	yes	—
18	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
19	C₄⋊Dic₇	—	2	2	no	—
20	C₄⋊Dic₇	—	4	2	no	—
21	C₄×Dic₇	—	4	2	no	—
22	C₄×Dic₇	D₄	1	1	yes	characteristic
23	C₂×C₇⋊C₈	D₄	1	1	yes	characteristic
24	C₇×C₁₆	C₂×C₄	1	2	yes	—
25	C₇⋊C₁₆	—	2	1	no	—
26	C₂×Q₃₂	D₇	1	1	yes	characteristic
27	C₄×Q₁₆	—	7	2	no	—
28	#366	—	7	1	no	—
29	#358	—	7	2	no	—
30	C₄×C₁₆	—	7	1	no	—
31	C₇×Q₈	—	2	4	no	—
32	C₇×Q₈	—	4	2	no	—
33	C₂×C₄×C₇	—	4	2	no	—
34	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₇	—	4	2	no	—
36	C₂×Dic₇	D₈	1	1	yes	characteristic
37	C₂×Dic₇	D₈	1	1	yes	characteristic
38	C₇×C₈	C₂²×C₄	1	1	yes	characteristic
39	C₇×C₈	#41	1	1	yes	characteristic
40	C₇⋊C₈	—	2	1	no	—
41	C₂×Q₁₆	C₂×D₇	1	2	yes	—
42	C₄×Q₈	—	14	2	no	—
43	Q₃₂	Dic₇	1	4	yes	direct product factor
44	C₂×C₁₆	—	7	1	no	—
45	C₂×C₁₆	C₂×D₇	1	1	yes	characteristic
46	#107	—	7	2	no	—
47	#103	—	14	2	no	—
48	C₄×C₈	—	7	1	no	—
49	C₂²×C₇	C₂×D₈	1	1	yes	characteristic
50	C₄×C₇	—	4	4	no	—
51	C₄×C₇	C₄×D₄	1	1	yes	characteristic
52	C₄×C₇	#135	1	1	yes	characteristic
53	Dic₇	—	2	1	no	—
54	Dic₇	—	8	2	no	—
55	Dic₇	Q₃₂	1	2	yes	direct product factor
56	C₂×Q₈	—	2	2	no	—
57	Q₁₆	—	2	2	no	—
58	Q₁₆	C₂×Dic₇	1	4	yes	—
59	C₂×C₈	—	7	1	no	—
60	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
61	C₄⋊C₄	—	14	2	no	—
62	C₄⋊C₄	—	28	2	no	—
63	C₄²	—	7	1	no	—
64	C₄²	—	28	2	no	—
65	C₁₆	—	14	1	no	—
66	C₁₆	C₂×Dic₇	1	2	yes	—
67	C₂×C₇	C₄×D₈	1	1	yes	characteristic
68	C₂×C₇	C₂×Q₃₂	1	1	yes	characteristic
69	C₂×C₇	#508	1	1	yes	characteristic
70	Q₈	—	2	4	no	—
71	Q₈	—	4	2	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	7	1	no	—
74	C₂×C₄	—	7	1	no	—
75	C₂×C₄	—	28	2	no	—
76	C₂×C₄	D₄×D₇	1	1	yes	characteristic
77	C₈	—	14	1	no	—
78	C₈	#1159	1	1	yes	characteristic
79	C₈	C₂²×Dic₇	1	1	yes	characteristic
80	C₇	C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
81	C₂²	D₇×D₈	1	1	yes	characteristic
82	C₄	—	4	4	no	—
83	C₄	—	7	2	no	—
84	C₄	—	14	1	no	—
85	C₄	—	56	2	no	—
86	C₄	#6475	1	1	yes	characteristic
87	C₄	D₄×Dic₇	1	1	yes	characteristic
88	C₂	D₇×Q₃₂	1	1	yes	characteristic
89	C₂	#89691	1	1	yes	characteristic
90	C₂	D₈×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₇×D₈
commutator	C₇×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₇
Fitting	C₂×C₇×Q₃₂	C₂
socle, prime radical	C₂²×C₇	C₂×D₈
cosocle	C₂×C₇×C₈	C₂³
hypercenter	C₂×Q₃₂	D₇
hypocenter	C₇	C₄×Q₃₂
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
8. C₇×Q₃₂	C₄	semidirect
43. Q₃₂	Dic₇	direct (N × H)
55. Dic₇	Q₃₂	direct (N × H)
80. C₇	C₄×Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×Q₃₂	2
Q₁₆×Dic₇	2
#89359	2
#89299	2
C₁₆×Dic₇	2
C₄×Q₃₂	7

Subgroups — Dic7×Q32 #11729249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₇×Q₃₂ #11729249