Subgroups — #91528

Order 480.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Dic₅	C₂	1	1	yes	maximal, characteristic
2	#6719	C₂	1	1	yes	maximal, characteristic
3	C₅×#217	C₂	1	1	yes	maximal, characteristic
4	#6668	C₂	1	2	yes	maximal, complemented
5	Dic₃×Dic₅	C₂	1	2	yes	maximal, complemented
6	#4023	—	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	C₂²	1	2	yes	—
10	C₂×C₃×Dic₅	C₄	1	4	yes	complemented
11	C₂×C₃×Dic₅	C₂²	1	1	yes	characteristic
12	C₂×C₃×Dic₅	C₂²	1	1	yes	characteristic
13	#926	—	5	1	no	maximal, Hall
14	C₂²×Dic₅	D₃	1	1	yes	characteristic
15	C₅×C₂²⋊C₄	—	3	1	no	—
16	C₂²⋊Dic₅	—	3	1	no	—
17	#683	—	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	C₂×C₄	1	2	yes	—
24	C₃×Dic₅	C₂×C₄	1	2	yes	—
25	C₅×Dic₃	—	2	2	no	—
26	C₂²×C₃×C₄	—	5	1	no	—
27	#217	—	5	1	no	—
28	#217	D₅	1	1	yes	characteristic
29	C₄⋊Dic₃	—	5	2	no	—
30	C₄×Dic₃	—	5	2	no	—
31	C₂³×C₅	C₂×D₃	1	1	yes	characteristic
32	C₂×C₄×C₅	—	3	2	no	—
33	C₂×Dic₅	—	3	2	no	—
34	C₂×Dic₅	Dic₃	1	4	yes	complemented
35	C₂×Dic₅	C₂×D₃	1	1	yes	characteristic
36	C₂×Dic₅	C₂×D₃	1	1	yes	characteristic
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	4	no	—
43	C₂×C₃×C₄	—	5	1	no	—
44	C₂×C₃×C₄	—	5	1	no	—
45	C₂×Dic₃	—	5	2	no	—
46	C₂×Dic₃	C₂×D₅	1	2	yes	—
47	C₂²×C₅	—	2	2	no	—
48	C₂²×C₅	C₂×Dic₃	1	2	yes	—
49	C₂²×C₅	C₂²×D₃	1	1	yes	characteristic
50	C₄×C₅	—	6	2	no	—
51	Dic₅	—	6	2	no	—
52	Dic₅	C₂×Dic₃	1	2	yes	—
53	Dic₅	C₂×Dic₃	1	2	yes	—
54	C₂²×C₄	—	5	1	no	—
55	C₄⋊C₄	—	15	2	no	—
56	C₂²⋊C₄	—	3	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	C₄×D₅	1	2	yes	—
62	C₂²×C₃	C₂²×D₅	1	1	yes	characteristic
63	C₃×C₄	—	5	2	no	—
64	C₃×C₄	—	5	2	no	—
65	Dic₃	—	2	2	no	—
66	Dic₃	—	10	2	no	—
67	C₂×C₅	—	2	2	no	—
68	C₂×C₅	#235	1	2	yes	—
69	C₂×C₅	C₂²×Dic₃	1	1	yes	characteristic
70	C₂³	D₃×D₅	1	1	yes	characteristic
71	C₂×C₄	—	3	2	no	—
72	C₂×C₄	—	5	4	no	—
73	C₂×C₄	—	5	1	no	—
74	C₂×C₄	—	5	1	no	—
75	C₂×C₄	—	15	2	no	—
76	C₂×C₃	—	2	2	no	—
77	C₂×C₃	C₂×C₄×D₅	1	1	yes	characteristic
78	C₂×C₃	#711	1	2	yes	—
79	C₅	#926	1	1	yes	characteristic, complemented, Sylow
80	C₂²	—	2	2	no	—
81	C₂²	D₅×Dic₃	1	2	yes	—
82	C₂²	C₂×D₃×D₅	1	1	yes	characteristic
83	C₄	—	5	2	no	—
84	C₄	—	5	2	no	—
85	C₄	—	6	2	no	—
86	C₄	—	30	2	no	—
87	C₃	#4023	1	1	yes	characteristic, complemented, Sylow
88	C₂	—	2	2	no	—
89	C₂	C₂×D₅×Dic₃	1	1	yes	characteristic
90	C₂	#6782	1	2	yes	—

	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. #6668	C₂	semidirect
5. Dic₃×Dic₅	C₂	semidirect
10. C₂×C₃×Dic₅	C₄	semidirect
34. C₂×Dic₅	Dic₃	semidirect
59. C₃×C₅	#117	semidirect
79. C₅	#926	semidirect
87. C₃	#4023	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×Dic₅	2
#6719	2
C₅×#217	2
#6668	2
Dic₃×Dic₅	2
#4023	3
#926	5