Subgroups — C₂³×C₇₁⋊C₃₂ #32155555

Order 18176.

542 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇₁⋊C₃₂	C₂	1	14	yes	maximal, direct product factor
2	C₂³×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇₁⋊C₃₂	C₂²	1	28	yes	direct product factor
4	C₂³×C₈×C₇₁	C₄	1	1	yes	characteristic
5	C₂²×C₁₆×C₇₁	C₄	1	7	yes	—
6	C₂²×C₁₆×C₇₁	C₂²	1	7	yes	—
7	C₇₁⋊C₃₂	C₂³	1	8	yes	direct product factor
8	C₂³×C₄×C₇₁	C₈	1	1	yes	characteristic
9	C₂²×C₈×C₇₁	C₈	1	7	yes	—
10	C₂²×C₈×C₇₁	C₂×C₄	1	7	yes	—
11	C₂×C₁₆×C₇₁	C₂×C₄	1	21	yes	—
12	C₂×C₁₆×C₇₁	C₂³	1	7	yes	—
13	C₂⁴×C₇₁	C₁₆	1	1	yes	characteristic
14	C₂²×C₄×C₇₁	C₁₆	1	7	yes	—
15	C₂²×C₄×C₇₁	C₂×C₈	1	7	yes	—
16	C₂×C₈×C₇₁	C₂×C₈	1	21	yes	—
17	C₂×C₈×C₇₁	C₂²×C₄	1	7	yes	—
18	C₁₆×C₇₁	C₂²×C₄	1	7	yes	—
19	C₁₆×C₇₁	C₂⁴	1	1	yes	characteristic
20	C₂³×C₇₁	C₃₂	1	8	yes	complemented
21	C₂³×C₇₁	C₂×C₁₆	1	7	yes	—
22	C₂×C₄×C₇₁	C₂×C₁₆	1	21	yes	—
23	C₂×C₄×C₇₁	C₂²×C₈	1	7	yes	—
24	C₈×C₇₁	C₂²×C₈	1	7	yes	—
25	C₈×C₇₁	C₂³×C₄	1	1	yes	characteristic
26	C₂²×C₇₁	C₂×C₃₂	1	28	yes	complemented
27	C₂²×C₇₁	C₂²×C₁₆	1	7	yes	—
28	C₄×C₇₁	C₂²×C₁₆	1	7	yes	—
29	C₄×C₇₁	C₂³×C₈	1	1	yes	characteristic
30	C₂³×C₃₂	—	71	1	no	maximal, Sylow
31	C₂×C₇₁	C₂²×C₃₂	1	14	yes	complemented
32	C₂×C₇₁	C₂³×C₁₆	1	1	yes	characteristic
33	C₂³×C₁₆	D₇₁	1	1	yes	characteristic
34	C₂²×C₃₂	—	71	14	no	—
35	C₇₁	C₂³×C₃₂	1	1	yes	characteristic, complemented, Sylow
36	C₂³×C₈	Dic₇₁	1	1	yes	characteristic
37	C₂²×C₁₆	Dic₇₁	1	7	yes	—
38	C₂²×C₁₆	C₂×D₇₁	1	7	yes	—
39	C₂×C₃₂	—	71	28	no	—
40	C₂³×C₄	C₇₁⋊C₈	1	1	yes	characteristic
41	C₂²×C₈	C₇₁⋊C₈	1	7	yes	—
42	C₂²×C₈	C₂×Dic₇₁	1	7	yes	—
43	C₂×C₁₆	C₂×Dic₇₁	1	21	yes	—
44	C₂×C₁₆	C₂²×D₇₁	1	7	yes	—
45	C₃₂	—	71	8	no	—
46	C₂⁴	C₇₁⋊C₁₆	1	1	yes	characteristic
47	C₂²×C₄	C₇₁⋊C₁₆	1	7	yes	—
48	C₂²×C₄	C₂×C₇₁⋊C₈	1	7	yes	—
49	C₂×C₈	C₂×C₇₁⋊C₈	1	21	yes	—
50	C₂×C₈	C₂²×Dic₇₁	1	7	yes	—
51	C₁₆	C₂²×Dic₇₁	1	7	yes	—
52	C₁₆	C₂³×D₇₁	1	1	yes	characteristic
53	C₂³	C₇₁⋊C₃₂	1	8	yes	direct product factor
54	C₂³	C₂×C₇₁⋊C₁₆	1	7	yes	—
55	C₂×C₄	C₂×C₇₁⋊C₁₆	1	21	yes	—
56	C₂×C₄	C₂²×C₇₁⋊C₈	1	7	yes	—
57	C₈	C₂²×C₇₁⋊C₈	1	7	yes	—
58	C₈	C₂³×Dic₇₁	1	1	yes	characteristic
59	C₂²	C₂×C₇₁⋊C₃₂	1	28	yes	direct product factor
60	C₂²	C₂²×C₇₁⋊C₁₆	1	7	yes	—
61	C₄	C₂²×C₇₁⋊C₁₆	1	7	yes	—
62	C₄	C₂³×C₇₁⋊C₈	1	1	yes	characteristic
63	C₂	C₂²×C₇₁⋊C₃₂	1	14	yes	direct product factor
64	C₂	C₂³×C₇₁⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₁₆	D₇₁
commutator, hypocenter	C₇₁	C₂³×C₃₂
Frattini	C₁₆	C₂³×D₇₁
Fitting	C₂³×C₁₆×C₇₁	C₂
socle, prime radical	C₂⁴×C₇₁	C₁₆
cosocle	C₁₆×C₇₁	C₂⁴
LCS	G ▷ C₇₁
UCS	C₁ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×C₇₁⋊C₃₂	C₂	direct (N × H)
3. C₂×C₇₁⋊C₃₂	C₂²	direct (N × H)
7. C₇₁⋊C₃₂	C₂³	direct (N × H)
20. C₂³×C₇₁	C₃₂	semidirect
26. C₂²×C₇₁	C₂×C₃₂	semidirect
31. C₂×C₇₁	C₂²×C₃₂	semidirect
35. C₇₁	C₂³×C₃₂	semidirect
53. C₂³	C₇₁⋊C₃₂	direct (N × H)
59. C₂²	C₂×C₇₁⋊C₃₂	direct (N × H)
63. C₂	C₂²×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
C₂²×C₇₁⋊C₃₂	2
C₂³×C₁₆×C₇₁	2
C₂³×C₃₂	71

Subgroups — C23×C71⋊C32 #32155555

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₇₁⋊C₃₂ #32155555