Subgroups — #13086306

Order 1440.

228 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃⋊(C₃⋊#170)	C₂	1	2	yes	maximal
2	C₃²×C₄×Dic₅	C₂	1	1	yes	maximal, characteristic
3	#91233	—	3	4	no	maximal
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	C₃⋊(C₃⋊#170)	—	2	2	no	—
8	C₄⋊(C₃⋊(C₃⋊C₈))	—	5	1	no	maximal, Hall
9	C₂×C₃⋊#170	—	3	8	no	—
10	C₃×C₄×Dic₅	D₃	1	4	yes	—
11	C₂²×C₃²×C₅	C₂×C₄	1	1	yes	characteristic
12	C₃²×C₄×C₅	—	2	1	no	—
13	C₃²×Dic₅	C₈	1	2	yes	complemented
14	C₃²×Dic₅	D₄	1	1	yes	characteristic
15	C₃²×Dic₅	Q₈	1	1	yes	characteristic
16	#4004	—	9	1	no	Hall
17	C₃²×C₄²	—	5	1	no	—
18	C₂×C₃⋊(C₃⋊C₈)	—	5	2	no	—
19	C₂×C₃×C₄×C₅	Dic₃	1	4	yes	—
20	C₂×C₃×Dic₅	Dic₃	1	4	yes	—
21	C₂×C₃×Dic₅	C₂×D₃	1	4	yes	—
22	C₃⋊#170	—	6	8	no	—
23	C₄⋊(C₃⋊C₈)	—	15	4	no	—
24	C₂×C₃²×C₅	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₃²×C₅	C₂×C₈	1	1	yes	characteristic
26	C₂×C₃²×C₅	M₁₆	1	1	yes	characteristic
27	C₂×#170	—	9	2	no	—
28	C₄×Dic₅	C₃⋊D₃	1	1	yes	characteristic
29	C₂×C₃²×C₄	—	5	1	no	—
30	C₂×C₃²×C₄	—	5	1	no	—
31	C₂×C₃²×C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
32	C₃⋊(C₃⋊C₈)	—	10	2	no	—
33	C₂²×C₃×C₅	C₂×Dic₃	1	4	yes	—
34	C₃×C₄×C₅	—	2	4	no	—
35	C₃×Dic₅	C₃⋊C₈	1	8	yes	complemented
36	C₃×Dic₅	Dic₆	1	4	yes	—
37	C₃×Dic₅	D₁₂	1	4	yes	—
38	C₃×C₄²	—	5	4	no	—
39	C₂×C₃⋊C₈	—	15	8	no	—
40	C₃²×C₅	C₄⋊C₈	1	1	yes	characteristic, complemented, Hall
41	C₂×C₄×C₅	C₃⋊Dic₃	1	1	yes	characteristic
42	C₂×Dic₅	C₃⋊Dic₃	1	1	yes	characteristic
43	C₂×Dic₅	C₂×C₃⋊D₃	1	1	yes	characteristic
44	#170	—	18	2	no	—
45	C₂²×C₃²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
46	C₃²×C₄	—	2	1	no	—
47	C₃²×C₄	—	5	2	no	—
48	C₃²×C₄	—	5	1	no	—
49	C₃²×C₄	—	5	1	no	—
50	C₄⋊C₈	—	45	1	no	Sylow
51	C₂×C₃×C₅	C₂×C₃⋊C₈	1	4	yes	—
52	C₂×C₃×C₅	#208	1	4	yes	—
53	C₂×C₃×C₅	C₄⋊Dic₃	1	4	yes	—
54	C₂×C₃×C₄	—	5	4	no	—
55	C₂×C₃×C₄	—	5	4	no	—
56	C₂×C₃×C₄	ΓL(1,𝔽₁₆)	1	4	yes	—
57	C₃⋊C₈	—	30	8	no	—
58	C₂²×C₅	C₂×C₃⋊Dic₃	1	1	yes	characteristic
59	C₄×C₅	—	2	1	no	—
60	Dic₅	C₃⋊(C₃⋊C₈)	1	2	yes	complemented
61	Dic₅	#634	1	1	yes	characteristic
62	Dic₅	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
63	C₂×C₃²	D₅⋊C₈	1	1	yes	characteristic
64	C₂×C₃²	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
65	C₂×C₃²	#704	1	1	yes	characteristic
66	C₂×C₈	—	45	2	no	—
67	C₄²	—	5	1	no	—
68	C₃×C₅	C₄⋊(C₃⋊C₈)	1	4	yes	complemented
69	C₂²×C₃	C₂×ΓL(1,𝔽₁₆)	1	4	yes	—
70	C₃×C₄	—	2	4	no	—
71	C₃×C₄	—	5	8	no	—
72	C₃×C₄	—	5	4	no	—
73	C₃×C₄	—	5	4	no	—
74	C₂×C₅	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
75	C₂×C₅	#3748	1	1	yes	characteristic
76	C₂×C₅	C₄⋊(C₃⋊Dic₃)	1	1	yes	characteristic
77	C₃²	#4004	1	1	yes	characteristic, complemented, Sylow
78	C₂×C₄	—	5	1	no	—
79	C₂×C₄	—	5	1	no	—
80	C₂×C₄	C₃⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
81	C₈	—	90	2	no	—
82	C₂×C₃	D₅⋊(C₃⋊C₈)	1	4	yes	—
83	C₂×C₃	C₄⋊ΓL(1,𝔽₁₆)	1	4	yes	—
84	C₂×C₃	#6762	1	4	yes	—
85	C₅	C₄⋊(C₃⋊(C₃⋊C₈))	1	1	yes	characteristic, complemented, Sylow
86	C₂²	C₂×C₃⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	5	2	no	—
89	C₄	—	5	1	no	—
90	C₄	—	5	1	no	—
91	C₃	#91233	1	4	yes	complemented
92	C₂	D₅⋊(C₃⋊(C₃⋊C₈))	1	1	yes	characteristic
93	C₂	C₄⋊(C₃⋊ΓL(1,𝔽₁₆))	1	1	yes	characteristic
94	C₂	#10624097	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×C₃⋊ΓL(1,𝔽₁₆)
commutator	C₂×C₃²×C₅	C₂×C₈
Fitting	C₂×C₃²×C₄×C₅	C₄
socle, prime radical	C₂²×C₃²×C₅	C₂×C₄
cosocle	C₂×C₃²×Dic₅	C₂²
hypercenter	C₂×C₄	C₃⋊ΓL(1,𝔽₁₆)
hypocenter	C₃²×C₅	C₄⋊C₈
LCS	G ▷ C₂×C₃²×C₅ ▷ C₃²×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
13. C₃²×Dic₅	C₈	semidirect
35. C₃×Dic₅	C₃⋊C₈	semidirect
40. C₃²×C₅	C₄⋊C₈	semidirect
60. Dic₅	C₃⋊(C₃⋊C₈)	semidirect
68. C₃×C₅	C₄⋊(C₃⋊C₈)	semidirect
77. C₃²	#4004	semidirect
85. C₅	C₄⋊(C₃⋊(C₃⋊C₈))	semidirect
91. C₃	#91233	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃⋊(C₃⋊#170)	2
C₃²×C₄×Dic₅	2
#91233	3
C₄⋊(C₃⋊(C₃⋊C₈))	5