Subgroups — C₃²×#3992 #13086244

Order 1440.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
2	C₃²×D₅⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₃²×C₈×D₅	C₂	1	1	yes	maximal, characteristic
4	C₃×#3992	C₃	1	4	yes	maximal, direct product factor
5	C₂×C₃²×AGL(1,𝔽₅)	C₄	1	2	yes	—
6	C₃²×C₄×D₅	C₂²	1	1	yes	characteristic
7	C₃²×#170	C₄	1	2	yes	complemented
8	C₃²×C₅×C₈	C₄	1	1	yes	characteristic, complemented
9	C₃²×#168	C₄	1	1	yes	characteristic, complemented
10	C₃²×#97	—	5	1	no	maximal, Hall
11	C₃×C₄×AGL(1,𝔽₅)	C₂×C₃	1	4	yes	—
12	C₃×D₅⋊C₈	C₂×C₃	1	4	yes	—
13	C₃×C₈×D₅	C₂×C₃	1	4	yes	—
14	C₂×C₃²×D₅	C₂×C₄	1	1	yes	characteristic
15	C₃²×AGL(1,𝔽₅)	—	2	2	no	—
16	C₃²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
17	C₃²×Dic₅	C₂×C₄	1	1	yes	characteristic
18	#3992	C₃²	1	1	yes	characteristic, direct product factor, Hall
19	C₂×C₃²×C₈	—	5	1	no	—
20	C₂×C₃²×C₈	—	5	1	no	—
21	C₃²×C₄²	—	5	1	no	—
22	C₂×C₃×AGL(1,𝔽₅)	C₃×C₄	1	8	yes	—
23	C₃×C₄×D₅	C₂²×C₃	1	4	yes	—
24	C₃×#170	C₃×C₄	1	8	yes	complemented
25	C₃×C₅×C₈	C₃×C₄	1	4	yes	complemented
26	C₃×#168	C₃×C₄	1	4	yes	complemented
27	C₃×#97	—	5	4	no	—
28	C₂×C₃²×C₅	C₄²	1	1	yes	characteristic
29	C₃²×D₅	M₁₆	1	2	yes	—
30	C₄×AGL(1,𝔽₅)	C₂×C₃²	1	1	yes	characteristic
31	D₅⋊C₈	C₂×C₃²	1	1	yes	characteristic
32	C₈×D₅	C₂×C₃²	1	1	yes	characteristic
33	C₂×C₃²×C₄	—	5	1	no	—
34	C₂×C₃²×C₄	—	5	2	no	—
35	C₃²×C₈	—	5	2	no	—
36	C₃²×C₈	—	5	1	no	—
37	C₃²×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
38	C₂×C₃×D₅	C₂×C₃×C₄	1	4	yes	—
39	C₃×AGL(1,𝔽₅)	—	2	8	no	—
40	C₃×C₄×C₅	C₂×C₃×C₄	1	4	yes	—
41	C₃×Dic₅	C₂×C₃×C₄	1	4	yes	—
42	C₂×C₃×C₈	—	5	4	no	—
43	C₂×C₃×C₈	—	5	4	no	—
44	C₃×C₄²	—	5	4	no	—
45	C₃²×C₅	#97	1	1	yes	characteristic, complemented, Hall
46	C₂×AGL(1,𝔽₅)	C₃²×C₄	1	2	yes	—
47	C₄×D₅	C₂²×C₃²	1	1	yes	characteristic
48	#170	C₃²×C₄	1	2	yes	complemented
49	C₅×C₈	C₃²×C₄	1	1	yes	characteristic, complemented
50	#168	C₃²×C₄	1	1	yes	characteristic, complemented
51	C₂²×C₃²	—	5	1	no	—
52	C₃²×C₄	—	5	1	no	—
53	C₃²×C₄	—	10	2	no	—
54	C₃²×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
55	#97	—	5	1	no	Sylow
56	C₂×C₃×C₅	C₃×C₄²	1	4	yes	—
57	C₃×D₅	C₃×M₁₆	1	8	yes	—
58	C₂×C₃×C₄	—	5	4	no	—
59	C₂×C₃×C₄	—	5	8	no	—
60	C₃×C₈	—	5	8	no	—
61	C₃×C₈	—	5	4	no	—
62	C₃×C₈	C₃×AGL(1,𝔽₅)	1	4	yes	complemented
63	C₂×D₅	C₂×C₃²×C₄	1	1	yes	characteristic
64	AGL(1,𝔽₅)	—	2	2	no	—
65	C₄×C₅	C₂×C₃²×C₄	1	1	yes	characteristic
66	Dic₅	C₂×C₃²×C₄	1	1	yes	characteristic
67	C₂×C₃²	—	5	2	no	—
68	C₂×C₃²	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
69	C₂×C₈	—	5	1	no	—
70	C₂×C₈	—	5	1	no	—
71	C₄²	—	5	1	no	—
72	C₃×C₅	C₃×#97	1	4	yes	complemented
73	C₂²×C₃	—	5	4	no	—
74	C₃×C₄	—	5	4	no	—
75	C₃×C₄	—	10	8	no	—
76	C₃×C₄	C₂×C₃×AGL(1,𝔽₅)	1	4	yes	—
77	C₂×C₅	C₃²×C₄²	1	1	yes	characteristic
78	D₅	C₃²×M₁₆	1	2	yes	—
79	C₃²	#3992	1	1	yes	characteristic, direct product factor, Sylow
80	C₂×C₄	—	5	1	no	—
81	C₂×C₄	—	5	2	no	—
82	C₈	—	5	2	no	—
83	C₈	—	5	1	no	—
84	C₈	C₃²×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
85	C₂×C₃	—	5	8	no	—
86	C₂×C₃	C₃×C₄×AGL(1,𝔽₅)	1	4	yes	—
87	C₅	C₃²×#97	1	1	yes	characteristic, complemented, Sylow
88	C₂²	—	5	1	no	—
89	C₄	—	5	1	no	—
90	C₄	—	10	2	no	—
91	C₄	C₂×C₃²×AGL(1,𝔽₅)	1	1	yes	characteristic
92	C₃	C₃×#3992	1	4	yes	direct product factor
93	C₂	—	5	2	no	—
94	C₂	C₃²×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₄	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₃²×C₄²
Frattini	C₄	C₂×C₃²×AGL(1,𝔽₅)
Fitting	C₃²×C₅×C₈	C₄
socle	C₂×C₃²×C₅	C₄²
prime radical	C₂×C₃²×D₅	C₂×C₄
cosocle	C₄×D₅	C₂²×C₃²
hypercenter	C₃²×C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₃²×#97
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₃²×C₄ ◁ C₃²×C₈

Complements

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

N	H ≅ G/N	type
4. C₃×#3992	C₃	direct (N × H)
7. C₃²×#170	C₄	semidirect
8. C₃²×C₅×C₈	C₄	semidirect
9. C₃²×#168	C₄	semidirect
18. #3992	C₃²	direct (N × H)
24. C₃×#170	C₃×C₄	semidirect
25. C₃×C₅×C₈	C₃×C₄	semidirect
26. C₃×#168	C₃×C₄	semidirect
37. C₃²×C₈	AGL(1,𝔽₅)	semidirect
45. C₃²×C₅	#97	semidirect
48. #170	C₃²×C₄	semidirect
49. C₅×C₈	C₃²×C₄	semidirect
50. #168	C₃²×C₄	semidirect
62. C₃×C₈	C₃×AGL(1,𝔽₅)	semidirect
72. C₃×C₅	C₃×#97	semidirect
79. C₃²	#3992	direct (N × H)
84. C₈	C₃²×AGL(1,𝔽₅)	semidirect
87. C₅	C₃²×#97	semidirect
92. C₃	C₃×#3992	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₄×AGL(1,𝔽₅)	2
C₃²×D₅⋊C₈	2
C₃²×C₈×D₅	2
C₃×#3992	3
C₃²×#97	5

Subgroups — C32×#3992 #13086244

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#3992 #13086244