Subgroups — C₂³×M₃₂ #62621

Order 256.

1167 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×M₃₂	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₁₆	C₂	1	2	yes	maximal, complemented
4	C₂⁴×C₄	C₄	1	1	yes	characteristic
5	C₂³×C₈	C₄	1	1	yes	characteristic
6	C₂³×C₈	C₄	1	14	yes	—
7	C₂³×C₈	C₂²	1	14	yes	—
8	C₂³×C₈	C₂²	1	1	yes	characteristic
9	C₂×M₃₂	C₂²	1	112	yes	direct product factor
10	C₂²×C₁₆	C₂²	1	28	yes	complemented
11	C₂⁵	C₈	1	1	yes	characteristic
12	C₂³×C₄	C₈	1	14	yes	—
13	C₂³×C₄	C₈	1	1	yes	characteristic
14	C₂³×C₄	C₂×C₄	1	14	yes	—
15	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈	C₂×C₄	1	14	yes	—
17	C₂²×C₈	C₂×C₄	1	84	yes	—
18	C₂²×C₈	C₂×C₄	1	7	yes	—
19	C₂²×C₈	C₂³	1	28	yes	—
20	C₂²×C₈	C₂³	1	7	yes	—
21	M₃₂	C₂³	1	64	yes	direct product factor
22	C₂×C₁₆	C₂³	1	56	yes	complemented
23	C₂⁴	—	2	8	no	—
24	C₂⁴	C₂×C₈	1	14	yes	—
25	C₂⁴	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₄	C₂×C₈	1	84	yes	—
27	C₂²×C₄	C₂×C₈	1	14	yes	—
28	C₂²×C₄	C₂×C₈	1	7	yes	—
29	C₂²×C₄	C₂²×C₄	1	28	yes	—
30	C₂²×C₄	C₂²×C₄	1	7	yes	—
31	C₂×C₈	C₂²×C₄	1	56	yes	—
32	C₂×C₈	C₂²×C₄	1	28	yes	—
33	C₂×C₈	C₂²×C₄	1	21	yes	—
34	C₂×C₈	C₂⁴	1	7	yes	—
35	C₂×C₈	C₂⁴	1	8	yes	—
36	C₁₆	C₂⁴	1	16	yes	complemented
37	C₂³	—	2	56	no	—
38	C₂³	M₃₂	1	8	yes	direct product factor
39	C₂³	C₂²×C₈	1	28	yes	—
40	C₂³	C₂²×C₈	1	7	yes	—
41	C₂×C₄	C₂²×C₈	1	56	yes	—
42	C₂×C₄	C₂²×C₈	1	21	yes	—
43	C₂×C₄	C₂²×C₈	1	28	yes	—
44	C₂×C₄	C₂³×C₄	1	8	yes	—
45	C₂×C₄	C₂³×C₄	1	7	yes	—
46	C₈	C₂³×C₄	1	7	yes	—
47	C₈	C₂³×C₄	1	8	yes	—
48	C₈	C₂⁵	1	1	yes	characteristic
49	C₂²	—	2	56	no	—
50	C₂²	C₂×M₃₂	1	28	yes	direct product factor
51	C₂²	C₂³×C₈	1	8	yes	—
52	C₂²	C₂³×C₈	1	7	yes	—
53	C₄	C₂³×C₈	1	7	yes	—
54	C₄	C₂³×C₈	1	8	yes	—
55	C₄	C₂⁴×C₄	1	1	yes	characteristic
56	C₂	—	2	8	no	—
57	C₂	C₂²×M₃₂	1	14	yes	direct product factor
58	C₂	C₂⁴×C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈	C₂²
commutator	C₂	C₂⁴×C₈
Frattini, cosocle	C₈	C₂⁵
socle	C₂⁴	C₂×C₈
prime radical	C₂⁵	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂²×M₃₂	C₂	direct (N × H)
3. C₂³×C₁₆	C₂	semidirect
9. C₂×M₃₂	C₂²	direct (N × H)
10. C₂²×C₁₆	C₂²	semidirect
21. M₃₂	C₂³	direct (N × H)
22. C₂×C₁₆	C₂³	semidirect
36. C₁₆	C₂⁴	semidirect
38. C₂³	M₃₂	direct (N × H)
50. C₂²	C₂×M₃₂	direct (N × H)
57. C₂	C₂²×M₃₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₈	2
C₂²×M₃₂	2
C₂³×C₁₆	2

Subgroups — C23×M32 #62621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×M₃₂ #62621