Subgroups — Q₈×C₇₁⋊C₃₂ #31918083

Order 18176.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₇₁⋊C₃₂)	C₂	1	3	yes	maximal
2	C₄×C₇₁⋊C₃₂	C₂	1	3	yes	maximal
3	Q₈×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇₁⋊C₃₂	C₂²	1	3	yes	—
5	C₂×C₇₁⋊C₃₂	C₂²	1	1	yes	characteristic
6	C₈×Q₈×C₇₁	C₄	1	1	yes	characteristic
7	C₄⋊C₁₆×C₇₁	C₄	1	3	yes	—
8	C₄×C₁₆×C₇₁	C₂²	1	3	yes	—
9	C₇₁⋊C₃₂	—	2	3	no	—
10	C₇₁⋊C₃₂	Q₈	1	2	yes	direct product factor
11	C₄×Q₈×C₇₁	C₈	1	1	yes	characteristic
12	C₂×C₁₆×C₇₁	C₂×C₄	1	3	yes	—
13	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
14	C₄⋊C₈×C₇₁	C₈	1	3	yes	—
15	C₄×C₈×C₇₁	C₂×C₄	1	3	yes	—
16	C₂×Q₈×C₇₁	C₁₆	1	1	yes	characteristic
17	C₂×C₈×C₇₁	C₂×C₈	1	3	yes	—
18	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
19	C₄⋊C₄×C₇₁	C₁₆	1	3	yes	—
20	C₄²×C₇₁	C₂×C₈	1	3	yes	—
21	C₁₆×C₇₁	—	2	3	no	—
22	C₁₆×C₇₁	C₂×Q₈	1	1	yes	characteristic
23	C₁₆×C₇₁	#41	1	1	yes	characteristic
24	Q₈×C₇₁	C₃₂	1	4	yes	complemented
25	C₂×C₄×C₇₁	C₂×C₁₆	1	3	yes	—
26	C₂×C₄×C₇₁	C₂×C₁₆	1	3	yes	—
27	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
28	C₈×C₇₁	—	2	3	no	—
29	C₈×C₇₁	C₄×Q₈	1	1	yes	characteristic
30	C₈×C₇₁	#131	1	1	yes	characteristic
31	C₂²×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
32	C₄×C₇₁	—	2	3	no	—
33	C₄×C₇₁	C₂×C₃₂	1	6	yes	—
34	C₄×C₇₁	C₈×Q₈	1	1	yes	characteristic
35	C₄×C₇₁	#504	1	1	yes	characteristic
36	Q₈×C₃₂	—	71	1	no	maximal, Sylow
37	C₂×C₇₁	Q₈×C₁₆	1	1	yes	characteristic
38	C₂×C₇₁	C₂²×C₃₂	1	1	yes	characteristic
39	C₂×C₇₁	#2258	1	1	yes	characteristic
40	Q₈×C₁₆	D₇₁	1	1	yes	characteristic
41	C₄⋊C₃₂	—	71	3	no	—
42	C₄×C₃₂	—	71	3	no	—
43	C₇₁	Q₈×C₃₂	1	1	yes	characteristic, complemented, Sylow
44	C₈×Q₈	Dic₇₁	1	1	yes	characteristic
45	C₂×C₃₂	—	71	1	no	—
46	C₂×C₃₂	—	71	3	no	—
47	C₄⋊C₁₆	Dic₇₁	1	3	yes	—
48	C₄×C₁₆	C₂×D₇₁	1	3	yes	—
49	C₄×Q₈	C₇₁⋊C₈	1	1	yes	characteristic
50	C₂×C₁₆	C₂×Dic₇₁	1	3	yes	—
51	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
52	C₄⋊C₈	C₇₁⋊C₈	1	3	yes	—
53	C₄×C₈	C₂×Dic₇₁	1	3	yes	—
54	C₃₂	—	71	2	no	—
55	C₃₂	—	142	3	no	—
56	C₂×Q₈	C₇₁⋊C₁₆	1	1	yes	characteristic
57	C₂×C₈	C₂×C₇₁⋊C₈	1	3	yes	—
58	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
59	C₄⋊C₄	C₇₁⋊C₁₆	1	3	yes	—
60	C₄²	C₂×C₇₁⋊C₈	1	3	yes	—
61	C₁₆	—	2	3	no	—
62	C₁₆	Q₈×D₇₁	1	1	yes	characteristic
63	C₁₆	#11765866	1	1	yes	characteristic
64	Q₈	C₇₁⋊C₃₂	1	4	yes	direct product factor
65	C₂×C₄	C₂×C₇₁⋊C₁₆	1	3	yes	—
66	C₂×C₄	C₂×C₇₁⋊C₁₆	1	3	yes	—
67	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
68	C₈	—	2	3	no	—
69	C₈	Q₈×Dic₇₁	1	1	yes	characteristic
70	C₈	#14556135	1	1	yes	characteristic
71	C₂²	C₂²×C₇₁⋊C₁₆	1	1	yes	characteristic
72	C₄	—	2	3	no	—
73	C₄	C₂×C₇₁⋊C₃₂	1	6	yes	—
74	C₄	Q₈×C₇₁⋊C₈	1	1	yes	characteristic
75	C₄	#17986416	1	1	yes	characteristic
76	C₂	Q₈×C₇₁⋊C₁₆	1	1	yes	characteristic
77	C₂	C₂²×C₇₁⋊C₃₂	1	1	yes	characteristic
78	C₂	#22590634	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₇₁
commutator	C₂×C₇₁	C₂²×C₃₂
Fitting	Q₈×C₁₆×C₇₁	C₂
socle, prime radical	C₂²×C₇₁	C₂²×C₁₆
cosocle	C₂×C₁₆×C₇₁	C₂³
hypercenter	Q₈×C₁₆	D₇₁
hypocenter	C₇₁	Q₈×C₃₂
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₁₆ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
10. C₇₁⋊C₃₂	Q₈	direct (N × H)
24. Q₈×C₇₁	C₃₂	semidirect
43. C₇₁	Q₈×C₃₂	semidirect
64. Q₈	C₇₁⋊C₃₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄⋊(C₇₁⋊C₃₂)	2
C₄×C₇₁⋊C₃₂	2
Q₈×C₁₆×C₇₁	2
Q₈×C₃₂	71

Subgroups — Q8×C71⋊C32 #31918083

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₇₁⋊C₃₂ #31918083