Subgroups — C₂×C₇×M₃₂ #90149

Order 448.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₇×M₃₂	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₇×C₁₆	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄×C₇	C₄	1	1	yes	characteristic
5	C₂×C₇×C₈	C₄	1	1	yes	characteristic
6	C₂×C₇×C₈	C₄	1	2	yes	—
7	C₂×C₇×C₈	C₂²	1	2	yes	—
8	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₇×C₁₆	C₂²	1	4	yes	complemented
10	C₂×M₃₂	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂³×C₇	C₈	1	1	yes	characteristic
12	C₂×C₄×C₇	C₈	1	2	yes	—
13	C₂×C₄×C₇	C₈	1	1	yes	characteristic
14	C₂×C₄×C₇	C₂×C₄	1	2	yes	—
15	C₂×C₄×C₇	C₂×C₄	1	1	yes	characteristic
16	C₇×C₈	C₂×C₄	1	2	yes	—
17	C₇×C₈	C₂×C₄	1	1	yes	characteristic
18	C₇×C₈	C₂³	1	1	yes	characteristic
19	C₂²×C₈	C₂×C₇	1	1	yes	characteristic
20	M₃₂	C₂×C₇	1	4	yes	direct product factor
21	C₂×C₁₆	C₂×C₇	1	2	yes	complemented
22	C₂²×C₇	—	2	2	no	—
23	C₂²×C₇	C₂×C₈	1	2	yes	—
24	C₂²×C₇	C₂×C₈	1	1	yes	characteristic
25	C₄×C₇	C₂×C₈	1	2	yes	—
26	C₄×C₇	C₂×C₈	1	1	yes	characteristic
27	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₄	C₄×C₇	1	1	yes	characteristic
29	C₂×C₈	C₄×C₇	1	1	yes	characteristic
30	C₂×C₈	C₄×C₇	1	2	yes	—
31	C₂×C₈	C₂²×C₇	1	1	yes	characteristic
32	C₂×C₈	C₂²×C₇	1	2	yes	—
33	C₁₆	C₂²×C₇	1	4	yes	complemented
34	C₂×C₇	—	2	2	no	—
35	C₂×C₇	M₃₂	1	2	yes	direct product factor
36	C₂×C₇	C₂²×C₈	1	1	yes	characteristic
37	C₂³	C₇×C₈	1	1	yes	characteristic
38	C₂×C₄	C₇×C₈	1	2	yes	—
39	C₂×C₄	C₇×C₈	1	1	yes	characteristic
40	C₂×C₄	C₂×C₄×C₇	1	2	yes	—
41	C₂×C₄	C₂×C₄×C₇	1	1	yes	characteristic
42	C₈	C₂×C₄×C₇	1	2	yes	—
43	C₈	C₂×C₄×C₇	1	1	yes	characteristic
44	C₈	C₂³×C₇	1	1	yes	characteristic
45	C₇	C₂×M₃₂	1	1	yes	characteristic, direct product factor, Sylow
46	C₂²	—	2	2	no	—
47	C₂²	C₂×C₇×C₈	1	2	yes	—
48	C₂²	C₂×C₇×C₈	1	1	yes	characteristic
49	C₄	C₂×C₇×C₈	1	1	yes	characteristic
50	C₄	C₂×C₇×C₈	1	2	yes	—
51	C₄	C₂²×C₄×C₇	1	1	yes	characteristic
52	C₂	—	2	2	no	—
53	C₂	C₇×M₃₂	1	2	yes	direct product factor
54	C₂	C₂²×C₇×C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₇×C₈	C₂²
commutator	C₂	C₂²×C₇×C₈
Frattini, cosocle	C₈	C₂³×C₇
socle	C₂²×C₇	C₂×C₈
prime radical	C₂³×C₇	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×M₃₂	C₂	direct (N × H)
3. C₂×C₇×C₁₆	C₂	semidirect
9. C₇×C₁₆	C₂²	semidirect
10. C₂×M₃₂	C₇	direct (N × H)
20. M₃₂	C₂×C₇	direct (N × H)
21. C₂×C₁₆	C₂×C₇	semidirect
33. C₁₆	C₂²×C₇	semidirect
35. C₂×C₇	M₃₂	direct (N × H)
45. C₇	C₂×M₃₂	direct (N × H)
53. C₂	C₇×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
C₂²×C₇×C₈	2
C₇×M₃₂	2
C₂×C₇×C₁₆	2
C₂×M₃₂	7

Subgroups — C2×C7×M32 #90149

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×M₃₂ #90149