Subgroups — C₃²×#4002 #13086253

Order 1440.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×#170	C₂	1	2	yes	maximal
2	C₃²×C₄×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₃×#4002	C₃	1	4	yes	maximal, direct product factor
4	C₂×C₃²×C₄×C₅	C₄	1	1	yes	characteristic
5	C₂×C₃²×Dic₅	C₄	1	1	yes	characteristic
6	C₂×C₃²×Dic₅	C₂²	1	1	yes	characteristic
7	C₃²×#170	C₄	1	4	yes	complemented
8	C₃²×#97	—	5	1	no	maximal, Hall
9	C₂×C₃×#170	C₂×C₃	1	8	yes	—
10	C₃×C₄×Dic₅	C₂×C₃	1	4	yes	—
11	C₂²×C₃²×C₅	C₂×C₄	1	1	yes	characteristic
12	C₃²×C₄×C₅	—	2	1	no	—
13	C₃²×Dic₅	—	2	1	no	—
14	C₃²×Dic₅	C₂×C₄	1	2	yes	—
15	#4002	C₃²	1	1	yes	characteristic, direct product factor, Hall
16	C₂×C₃²×C₈	—	5	2	no	—
17	C₃²×C₄²	—	5	1	no	—
18	C₂×C₃×C₄×C₅	C₃×C₄	1	4	yes	—
19	C₂×C₃×Dic₅	C₃×C₄	1	4	yes	—
20	C₂×C₃×Dic₅	C₂²×C₃	1	4	yes	—
21	C₃×#170	C₃×C₄	1	16	yes	complemented
22	C₃×#97	—	5	4	no	—
23	C₂×C₃²×C₅	C₄²	1	1	yes	characteristic
24	C₂×C₃²×C₅	M₁₆	1	1	yes	characteristic
25	C₂×C₃²×C₅	M₁₆	1	1	yes	characteristic
26	C₂×#170	C₂×C₃²	1	2	yes	—
27	C₄×Dic₅	C₂×C₃²	1	1	yes	characteristic
28	C₂×C₃²×C₄	—	5	1	no	—
29	C₂×C₃²×C₄	—	5	1	no	—
30	C₂×C₃²×C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
31	C₃²×C₈	—	5	4	no	—
32	C₂²×C₃×C₅	C₂×C₃×C₄	1	4	yes	—
33	C₃×C₄×C₅	—	2	4	no	—
34	C₃×Dic₅	—	2	4	no	—
35	C₃×Dic₅	C₂×C₃×C₄	1	8	yes	—
36	C₂×C₃×C₈	—	5	8	no	—
37	C₃×C₄²	—	5	4	no	—
38	C₃²×C₅	#97	1	1	yes	characteristic, complemented, Hall
39	C₂×C₄×C₅	C₃²×C₄	1	1	yes	characteristic
40	C₂×Dic₅	C₃²×C₄	1	1	yes	characteristic
41	C₂×Dic₅	C₂²×C₃²	1	1	yes	characteristic
42	#170	C₃²×C₄	1	4	yes	complemented
43	C₂²×C₃²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
44	C₃²×C₄	—	2	1	no	—
45	C₃²×C₄	—	5	2	no	—
46	C₃²×C₄	—	10	1	no	—
47	#97	—	5	1	no	Sylow
48	C₂×C₃×C₅	C₃×C₄²	1	4	yes	—
49	C₂×C₃×C₅	C₃×M₁₆	1	4	yes	—
50	C₂×C₃×C₅	C₃×M₁₆	1	4	yes	—
51	C₂×C₃×C₄	—	5	4	no	—
52	C₂×C₃×C₄	—	5	4	no	—
53	C₂×C₃×C₄	C₃×AGL(1,𝔽₅)	1	4	yes	—
54	C₃×C₈	—	5	16	no	—
55	C₂²×C₅	C₂×C₃²×C₄	1	1	yes	characteristic
56	C₄×C₅	—	2	1	no	—
57	Dic₅	—	2	1	no	—
58	Dic₅	C₂×C₃²×C₄	1	2	yes	—
59	C₂×C₃²	#700	1	1	yes	characteristic
60	C₂×C₃²	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
61	C₂×C₃²	#704	1	1	yes	characteristic
62	C₂×C₈	—	5	2	no	—
63	C₄²	—	5	1	no	—
64	C₃×C₅	C₃×#97	1	4	yes	complemented
65	C₂²×C₃	C₂×C₃×AGL(1,𝔽₅)	1	4	yes	—
66	C₃×C₄	—	2	4	no	—
67	C₃×C₄	—	5	8	no	—
68	C₃×C₄	—	10	4	no	—
69	C₂×C₅	C₃²×C₄²	1	1	yes	characteristic
70	C₂×C₅	C₃²×M₁₆	1	1	yes	characteristic
71	C₂×C₅	C₃²×M₁₆	1	1	yes	characteristic
72	C₃²	#4002	1	1	yes	characteristic, direct product factor, Sylow
73	C₂×C₄	—	5	1	no	—
74	C₂×C₄	—	5	1	no	—
75	C₂×C₄	C₃²×AGL(1,𝔽₅)	1	1	yes	characteristic
76	C₈	—	5	4	no	—
77	C₂×C₃	C₃×#700	1	4	yes	—
78	C₂×C₃	C₃×C₄×AGL(1,𝔽₅)	1	4	yes	—
79	C₂×C₃	C₃×#704	1	4	yes	—
80	C₅	C₃²×#97	1	1	yes	characteristic, complemented, Sylow
81	C₂²	C₂×C₃²×AGL(1,𝔽₅)	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	5	2	no	—
84	C₄	—	10	1	no	—
85	C₃	C₃×#4002	1	4	yes	direct product factor
86	C₂	C₃²×#700	1	1	yes	characteristic
87	C₂	C₃²×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
88	C₂	C₃²×#704	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₃²×C₄²
Frattini	C₂²	C₂×C₃²×AGL(1,𝔽₅)
Fitting	C₂×C₃²×C₄×C₅	C₄
socle, prime radical	C₂²×C₃²×C₅	C₂×C₄
cosocle	C₂×Dic₅	C₂²×C₃²
hypercenter	C₂×C₃²×C₄	AGL(1,𝔽₅)
hypocenter	C₅	C₃²×#97
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃² ◁ C₂×C₃²×C₄

Complements

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

N	H ≅ G/N	type
3. C₃×#4002	C₃	direct (N × H)
7. C₃²×#170	C₄	semidirect
15. #4002	C₃²	direct (N × H)
21. C₃×#170	C₃×C₄	semidirect
38. C₃²×C₅	#97	semidirect
42. #170	C₃²×C₄	semidirect
64. C₃×C₅	C₃×#97	semidirect
72. C₃²	#4002	direct (N × H)
80. C₅	C₃²×#97	semidirect
85. C₃	C₃×#4002	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×#170	2
C₃²×C₄×Dic₅	2
C₃×#4002	3
C₃²×#97	5

Subgroups — C32×#4002 #13086253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#4002 #13086253