Subgroups — C₃²×#3998 #13086250

Order 1440.

156 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#168	C₂	1	1	yes	maximal, characteristic
2	C₃²×#674	C₂	1	2	yes	maximal, complemented
3	C₃×#3998	C₃	1	4	yes	maximal, direct product factor
4	C₂×C₃²×C₄×C₅	C₄	1	1	yes	characteristic
5	C₃²×#168	C₄	1	1	yes	characteristic
6	C₃²×#168	C₂²	1	1	yes	characteristic
7	C₃²×M₃₂	—	5	1	no	maximal, Hall
8	C₂×C₃×#168	C₂×C₃	1	4	yes	—
9	C₃×#674	C₂×C₃	1	8	yes	complemented
10	C₂²×C₃²×C₅	C₈	1	1	yes	characteristic
11	C₃²×C₄×C₅	C₈	1	1	yes	characteristic
12	C₃²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
13	#3998	C₃²	1	1	yes	characteristic, direct product factor, Hall
14	C₂×C₃²×C₈	—	5	1	no	—
15	C₃²×C₁₆	—	5	2	no	—
16	C₂×C₃×C₄×C₅	C₃×C₄	1	4	yes	—
17	C₃×#168	C₃×C₄	1	4	yes	—
18	C₃×#168	C₂²×C₃	1	4	yes	—
19	C₃×M₃₂	—	5	4	no	—
20	C₂×C₃²×C₅	—	2	1	no	—
21	C₂×C₃²×C₅	C₂×C₈	1	1	yes	characteristic
22	C₂×#168	C₂×C₃²	1	1	yes	characteristic
23	#674	C₂×C₃²	1	2	yes	complemented
24	C₂×C₃²×C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
25	C₃²×C₈	—	5	1	no	—
26	C₃²×C₈	—	5	1	no	—
27	C₂²×C₃×C₅	C₃×C₈	1	4	yes	—
28	C₃×C₄×C₅	C₃×C₈	1	4	yes	—
29	C₃×C₄×C₅	C₂×C₃×C₄	1	4	yes	—
30	C₂×C₃×C₈	—	5	4	no	—
31	C₃×C₁₆	—	5	8	no	—
32	C₃²×C₅	M₃₂	1	1	yes	characteristic, complemented, Hall
33	C₂×C₄×C₅	C₃²×C₄	1	1	yes	characteristic
34	#168	C₃²×C₄	1	1	yes	characteristic
35	#168	C₂²×C₃²	1	1	yes	characteristic
36	C₂²×C₃²	#170	1	1	yes	characteristic
37	C₃²×C₄	#170	1	1	yes	characteristic
38	C₃²×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
39	M₃₂	—	5	1	no	Sylow
40	C₂×C₃×C₅	—	2	4	no	—
41	C₂×C₃×C₅	C₂×C₃×C₈	1	4	yes	—
42	C₂×C₃×C₄	C₃×AGL(1,𝔽₅)	1	4	yes	—
43	C₃×C₈	—	5	4	no	—
44	C₃×C₈	—	5	4	no	—
45	C₂²×C₅	C₃²×C₈	1	1	yes	characteristic
46	C₄×C₅	C₃²×C₈	1	1	yes	characteristic
47	C₄×C₅	C₂×C₃²×C₄	1	1	yes	characteristic
48	C₂×C₃²	—	2	1	no	—
49	C₂×C₃²	C₂×#170	1	1	yes	characteristic
50	C₂×C₈	—	5	1	no	—
51	C₁₆	—	5	2	no	—
52	C₃×C₅	C₃×M₃₂	1	4	yes	complemented
53	C₂²×C₃	C₃×#170	1	4	yes	—
54	C₃×C₄	C₃×#170	1	4	yes	—
55	C₃×C₄	C₂×C₃×AGL(1,𝔽₅)	1	4	yes	—
56	C₂×C₅	—	2	1	no	—
57	C₂×C₅	C₂×C₃²×C₈	1	1	yes	characteristic
58	C₃²	#3998	1	1	yes	characteristic, direct product factor, Sylow
59	C₂×C₄	C₃²×AGL(1,𝔽₅)	1	1	yes	characteristic
60	C₈	—	5	1	no	—
61	C₈	—	5	1	no	—
62	C₂×C₃	—	2	4	no	—
63	C₂×C₃	C₂×C₃×#170	1	4	yes	—
64	C₅	C₃²×M₃₂	1	1	yes	characteristic, complemented, Sylow
65	C₂²	C₃²×#170	1	1	yes	characteristic
66	C₄	C₃²×#170	1	1	yes	characteristic
67	C₄	C₂×C₃²×AGL(1,𝔽₅)	1	1	yes	characteristic
68	C₃	C₃×#3998	1	4	yes	direct product factor
69	C₂	—	2	1	no	—
70	C₂	C₂×C₃²×#170	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₄	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₃²×C₈
Frattini	C₄	C₂×C₃²×AGL(1,𝔽₅)
Fitting	C₂×C₃²×C₄×C₅	C₄
socle	C₂×C₃²×C₅	C₂×C₈
prime radical	C₂²×C₃²×C₅	C₈
cosocle	#168	C₂²×C₃²
hypercenter	C₂×C₃²×C₄	AGL(1,𝔽₅)
hypocenter	C₅	C₃²×M₃₂
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃²×C₄ ◁ C₂×C₃²×C₄

Complements

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

N	H ≅ G/N	type
2. C₃²×#674	C₂	semidirect
3. C₃×#3998	C₃	direct (N × H)
9. C₃×#674	C₂×C₃	semidirect
13. #3998	C₃²	direct (N × H)
23. #674	C₂×C₃²	semidirect
32. C₃²×C₅	M₃₂	semidirect
52. C₃×C₅	C₃×M₃₂	semidirect
58. C₃²	#3998	direct (N × H)
64. C₅	C₃²×M₃₂	semidirect
68. C₃	C₃×#3998	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×#168	2
C₃²×#674	2
C₃×#3998	3
C₃²×M₃₂	5

Subgroups — C32×#3998 #13086250

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#3998 #13086250