Subgroups — C₂×C₃×M₃₂ #5377

Order 192.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₈	C₂	1	1	yes	maximal, characteristic
2	C₃×M₃₂	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₃×C₁₆	C₂	1	2	yes	maximal, complemented
4	C₂×M₃₂	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂²×C₃×C₄	C₄	1	1	yes	characteristic
6	C₂×C₃×C₈	C₄	1	1	yes	characteristic
7	C₂×C₃×C₈	C₄	1	2	yes	—
8	C₂×C₃×C₈	C₂²	1	2	yes	—
9	C₂×C₃×C₈	C₂²	1	1	yes	characteristic
10	C₃×C₁₆	C₂²	1	4	yes	complemented
11	C₂²×C₈	C₂×C₃	1	1	yes	characteristic
12	M₃₂	C₂×C₃	1	4	yes	direct product factor
13	C₂×C₁₆	C₂×C₃	1	2	yes	complemented
14	C₂³×C₃	C₈	1	1	yes	characteristic
15	C₂×C₃×C₄	C₈	1	2	yes	—
16	C₂×C₃×C₄	C₈	1	1	yes	characteristic
17	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
18	C₂×C₃×C₄	C₂×C₄	1	1	yes	characteristic
19	C₃×C₈	C₂×C₄	1	2	yes	—
20	C₃×C₈	C₂×C₄	1	1	yes	characteristic
21	C₃×C₈	C₂³	1	1	yes	characteristic
22	C₂²×C₄	C₃×C₄	1	1	yes	characteristic
23	C₂×C₈	C₃×C₄	1	2	yes	—
24	C₂×C₈	C₃×C₄	1	1	yes	characteristic
25	C₂×C₈	C₂²×C₃	1	1	yes	characteristic
26	C₂×C₈	C₂²×C₃	1	2	yes	—
27	C₁₆	C₂²×C₃	1	4	yes	complemented
28	C₂²×C₃	—	2	2	no	—
29	C₂²×C₃	C₂×C₈	1	2	yes	—
30	C₂²×C₃	C₂×C₈	1	1	yes	characteristic
31	C₃×C₄	C₂×C₈	1	2	yes	—
32	C₃×C₄	C₂×C₈	1	1	yes	characteristic
33	C₃×C₄	C₂²×C₄	1	1	yes	characteristic
34	C₂³	C₃×C₈	1	1	yes	characteristic
35	C₂×C₄	C₃×C₈	1	2	yes	—
36	C₂×C₄	C₃×C₈	1	1	yes	characteristic
37	C₂×C₄	C₂×C₃×C₄	1	2	yes	—
38	C₂×C₄	C₂×C₃×C₄	1	1	yes	characteristic
39	C₈	C₂×C₃×C₄	1	1	yes	characteristic
40	C₈	C₂×C₃×C₄	1	2	yes	—
41	C₈	C₂³×C₃	1	1	yes	characteristic
42	C₂×C₃	—	2	2	no	—
43	C₂×C₃	M₃₂	1	2	yes	direct product factor
44	C₂×C₃	C₂²×C₈	1	1	yes	characteristic
45	C₂²	—	2	2	no	—
46	C₂²	C₂×C₃×C₈	1	2	yes	—
47	C₂²	C₂×C₃×C₈	1	1	yes	characteristic
48	C₄	C₂×C₃×C₈	1	1	yes	characteristic
49	C₄	C₂×C₃×C₈	1	2	yes	—
50	C₄	C₂²×C₃×C₄	1	1	yes	characteristic
51	C₃	C₂×M₃₂	1	1	yes	characteristic, direct product factor, Sylow
52	C₂	—	2	2	no	—
53	C₂	C₃×M₃₂	1	2	yes	direct product factor
54	C₂	C₂²×C₃×C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₂²
commutator	C₂	C₂²×C₃×C₈
Frattini, cosocle	C₈	C₂³×C₃
socle	C₂²×C₃	C₂×C₈
prime radical	C₂³×C₃	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×M₃₂	C₂	direct (N × H)
3. C₂×C₃×C₁₆	C₂	semidirect
4. C₂×M₃₂	C₃	direct (N × H)
10. C₃×C₁₆	C₂²	semidirect
12. M₃₂	C₂×C₃	direct (N × H)
13. C₂×C₁₆	C₂×C₃	semidirect
27. C₁₆	C₂²×C₃	semidirect
43. C₂×C₃	M₃₂	direct (N × H)
51. C₃	C₂×M₃₂	direct (N × H)
53. C₂	C₃×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×C₃×C₈	2
C₃×M₃₂	2
C₂×C₃×C₁₆	2
C₂×M₃₂	3

Subgroups — C2×C3×M32 #5377

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×M₃₂ #5377