Subgroups — #91990

Order 480.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#711	C₂	1	1	yes	maximal, characteristic
2	#6762	C₂	1	2	yes	maximal, complemented
3	C₂×C₃⋊#170	C₂	1	2	yes	maximal, complemented
4	#6758	C₂	1	1	yes	maximal, characteristic, complemented
5	D₅⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#4131	—	3	1	no	maximal, Hall
7	C₃×C₅×D₄	C₄	1	1	yes	characteristic
8	C₃×C₂²⋊D₅	C₄	1	2	yes	—
9	C₂×C₃×Dic₅	C₂²	1	2	yes	—
10	C₃×C₄×D₅	C₂²	1	1	yes	characteristic
11	C₃×Dic₁₀	C₄	1	1	yes	characteristic
12	C₃⋊#170	C₂²	1	1	yes	characteristic
13	C₃⋊#170	C₂²	1	1	yes	characteristic
14	C₃⋊#170	C₂²	1	2	yes	—
15	#948	—	5	1	no	maximal, Hall
16	#711	D₃	1	1	yes	characteristic
17	#704	—	3	2	no	—
18	C₂×#170	—	3	2	no	—
19	#700	—	3	1	no	—
20	D₅⋊C₈	—	3	1	no	—
21	C₂²×C₃×C₅	C₂×C₄	1	2	yes	—
22	C₂×C₃×D₅	C₂×C₄	1	1	yes	characteristic
23	C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₃×Dic₅	C₂×C₄	1	2	yes	—
25	C₃×Dic₅	C₂³	1	1	yes	characteristic
26	C₃×#41	—	5	1	no	—
27	#208	—	5	1	no	—
28	#208	—	5	2	no	—
29	C₂×C₃⋊C₈	—	5	1	no	—
30	C₂×C₃⋊C₈	—	5	2	no	—
31	C₅×D₄	Dic₃	1	1	yes	characteristic
32	C₂²⋊D₅	Dic₃	1	2	yes	—
33	C₂×Dic₅	C₂×D₃	1	2	yes	—
34	C₄×D₅	C₂×D₃	1	1	yes	characteristic
35	Dic₁₀	Dic₃	1	1	yes	characteristic
36	#170	—	3	1	no	—
37	#170	—	3	1	no	—
38	#170	—	3	2	no	—
39	#131	—	15	1	no	Sylow
40	C₂×C₃×C₅	—	2	2	no	—
41	C₂×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
42	C₃×D₅	—	2	1	no	—
43	C₃×Q₈	—	5	1	no	—
44	C₃×D₄	—	5	2	no	—
45	C₃×D₄	AGL(1,𝔽₅)	1	1	yes	characteristic
46	C₂×C₃×C₄	—	5	2	no	—
47	C₂×C₃×C₄	—	5	1	no	—
48	C₃⋊C₈	—	5	1	no	—
49	C₃⋊C₈	—	5	1	no	—
50	C₃⋊C₈	—	5	2	no	—
51	C₂²×C₅	C₂×Dic₃	1	2	yes	—
52	C₂×D₅	C₂×Dic₃	1	1	yes	characteristic
53	C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
54	Dic₅	C₂×Dic₃	1	2	yes	—
55	Dic₅	C₂²×D₃	1	1	yes	characteristic
56	#41	—	5	1	no	—
57	M₁₆	—	15	1	no	—
58	M₁₆	—	15	2	no	—
59	C₂×C₈	—	15	1	no	—
60	C₂×C₈	—	15	2	no	—
61	C₃×C₅	#131	1	1	yes	characteristic, complemented, Hall
62	C₂²×C₃	—	5	1	no	—
63	C₂²×C₃	C₂×AGL(1,𝔽₅)	1	2	yes	—
64	C₃×C₄	—	5	2	no	—
65	C₃×C₄	—	5	1	no	—
66	C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
67	C₂×C₅	—	2	2	no	—
68	C₂×C₅	C₂²×Dic₃	1	1	yes	characteristic
69	D₅	—	2	1	no	—
70	Q₈	—	5	1	no	—
71	D₄	—	5	2	no	—
72	D₄	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
73	C₂×C₄	—	5	2	no	—
74	C₂×C₄	—	5	1	no	—
75	C₈	—	15	1	no	—
76	C₈	—	15	1	no	—
77	C₈	—	15	2	no	—
78	C₂×C₃	—	2	2	no	—
79	C₂×C₃	—	10	1	no	—
80	C₂×C₃	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
81	C₅	#948	1	1	yes	characteristic, complemented, Sylow
82	C₂²	—	5	1	no	—
83	C₂²	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
84	C₄	—	5	1	no	—
85	C₄	—	5	2	no	—
86	C₄	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
87	C₃	#4131	1	1	yes	characteristic, complemented, Sylow
88	C₂	—	2	2	no	—
89	C₂	—	10	1	no	—
90	C₂	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×ΓL(1,𝔽₁₆)
commutator, socle	C₂×C₃×C₅	C₂²×C₄
Fitting	C₃×C₅×D₄	C₄
prime radical	C₃×#711	C₂
cosocle	C₃×Dic₅	C₂³
hypercenter	D₄	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#131
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ D₄

Complements

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

N	H ≅ G/N	type
2. #6762	C₂	semidirect
3. C₂×C₃⋊#170	C₂	semidirect
4. #6758	C₂	semidirect
5. D₅⋊(C₃⋊C₈)	C₂	semidirect
61. C₃×C₅	#131	semidirect
81. C₅	#948	semidirect
87. C₃	#4131	semidirect

Maximal subgroups

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

subgroup	index
C₃×#711	2
#6762	2
C₂×C₃⋊#170	2
#6758	2
D₅⋊(C₃⋊C₈)	2
#4131	3
#948	5