Subgroups — #91529

Order 480.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic
2	#6719	C₂	1	1	yes	maximal, characteristic
3	C₃×C₂²⋊Dic₅	C₂	1	1	yes	maximal, characteristic
4	Dic₃⋊Dic₅	C₂	1	2	yes	maximal, complemented
5	Dic₃×Dic₅	C₂	1	2	yes	maximal, complemented
6	#4072	—	3	1	no	maximal, Hall
7	C₂³×C₃×C₅	C₂²	1	1	yes	characteristic
8	C₂×Dic₁₅	C₂²	1	2	yes	—
9	C₂×C₅×Dic₃	C₄	1	4	yes	complemented
10	C₂×C₅×Dic₃	C₂²	1	1	yes	characteristic
11	C₂×C₅×Dic₃	C₂²	1	1	yes	characteristic
12	C₂×C₃×Dic₅	C₂²	1	2	yes	—
13	#877	—	5	1	no	maximal, Hall
14	C₂²×C₄×C₅	—	3	1	no	—
15	C₂²⋊Dic₅	—	3	1	no	—
16	C₂²⋊Dic₅	D₃	1	1	yes	characteristic
17	C₄⋊Dic₅	—	3	2	no	—
18	C₄×Dic₅	—	3	2	no	—
19	C₂²×C₃×C₅	—	2	2	no	—
20	C₂²×C₃×C₅	C₂×C₄	1	2	yes	—
21	C₂²×C₃×C₅	C₂³	1	1	yes	characteristic
22	Dic₁₅	—	2	2	no	—
23	C₃×Dic₅	—	2	2	no	—
24	C₅×Dic₃	C₂×C₄	1	2	yes	—
25	C₅×Dic₃	C₂×C₄	1	2	yes	—
26	C₂²×Dic₃	D₅	1	1	yes	characteristic
27	C₃×C₂²⋊C₄	—	5	1	no	—
28	#217	—	5	1	no	—
29	Dic₃⋊C₄	—	5	2	no	—
30	C₄×Dic₃	—	5	2	no	—
31	C₂³×C₅	C₂×D₃	1	1	yes	characteristic
32	C₂×C₄×C₅	—	3	4	no	—
33	C₂×C₄×C₅	—	3	1	no	—
34	C₂×C₄×C₅	—	3	1	no	—
35	C₂×Dic₅	—	3	2	no	—
36	C₂×Dic₅	C₂×D₃	1	2	yes	—
37	#117	—	15	1	no	Sylow
38	C₂×C₃×C₅	—	2	2	no	—
39	C₂×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₃×C₅	#41	1	2	yes	—
41	C₂³×C₃	C₂×D₅	1	1	yes	characteristic
42	C₂×C₃×C₄	—	5	2	no	—
43	C₂×Dic₃	—	5	2	no	—
44	C₂×Dic₃	Dic₅	1	4	yes	complemented
45	C₂×Dic₃	C₂×D₅	1	1	yes	characteristic
46	C₂×Dic₃	C₂×D₅	1	1	yes	characteristic
47	C₂²×C₅	—	2	2	no	—
48	C₂²×C₅	C₄×D₃	1	2	yes	—
49	C₂²×C₅	C₂²×D₃	1	1	yes	characteristic
50	C₄×C₅	—	3	2	no	—
51	C₄×C₅	—	3	2	no	—
52	Dic₅	—	2	2	no	—
53	Dic₅	—	6	2	no	—
54	C₂²×C₄	—	3	1	no	—
55	C₄⋊C₄	—	15	2	no	—
56	C₂²⋊C₄	—	5	1	no	—
57	C₂²⋊C₄	—	15	1	no	—
58	C₄²	—	15	2	no	—
59	C₃×C₅	#117	1	1	yes	characteristic, complemented, Hall
60	C₂²×C₃	—	2	2	no	—
61	C₂²×C₃	C₂×Dic₅	1	2	yes	—
62	C₂²×C₃	C₂²×D₅	1	1	yes	characteristic
63	C₃×C₄	—	10	2	no	—
64	Dic₃	—	10	2	no	—
65	Dic₃	C₂×Dic₅	1	2	yes	—
66	Dic₃	C₂×Dic₅	1	2	yes	—
67	C₂×C₅	—	2	2	no	—
68	C₂×C₅	C₂×C₄×D₃	1	1	yes	characteristic
69	C₂×C₅	#237	1	2	yes	—
70	C₂³	D₃×D₅	1	1	yes	characteristic
71	C₂×C₄	—	3	4	no	—
72	C₂×C₄	—	3	1	no	—
73	C₂×C₄	—	3	1	no	—
74	C₂×C₄	—	5	2	no	—
75	C₂×C₄	—	15	2	no	—
76	C₂×C₃	—	2	2	no	—
77	C₂×C₃	#709	1	2	yes	—
78	C₂×C₃	C₂²×Dic₅	1	1	yes	characteristic
79	C₅	#877	1	1	yes	characteristic, complemented, Sylow
80	C₂²	—	2	2	no	—
81	C₂²	D₃×Dic₅	1	2	yes	—
82	C₂²	C₂×D₃×D₅	1	1	yes	characteristic
83	C₄	—	3	2	no	—
84	C₄	—	3	2	no	—
85	C₄	—	10	2	no	—
86	C₄	—	30	2	no	—
87	C₃	#4072	1	1	yes	characteristic, complemented, Sylow
88	C₂	—	2	2	no	—
89	C₂	#6779	1	2	yes	—
90	C₂	C₂×D₃×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×D₃×D₅
commutator	C₂×C₃×C₅	C₂²×C₄
Fitting, prime radical	C₂³×C₃×C₅	C₂²
socle, cosocle	C₂²×C₃×C₅	C₂³
hypercenter	C₂³	D₃×D₅
hypocenter	C₃×C₅	#117
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂³

Complements

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

N	H ≅ G/N	type
4. Dic₃⋊Dic₅	C₂	semidirect
5. Dic₃×Dic₅	C₂	semidirect
9. C₂×C₅×Dic₃	C₄	semidirect
44. C₂×Dic₃	Dic₅	semidirect
59. C₃×C₅	#117	semidirect
79. C₅	#877	semidirect
87. C₃	#4072	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×Dic₃	2
#6719	2
C₃×C₂²⋊Dic₅	2
Dic₃⋊Dic₅	2
Dic₃×Dic₅	2
#4072	3
#877	5