Subgroups — #91815

Order 480.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×C₅	C₂	1	1	yes	maximal, characteristic
2	#6719	C₂	1	2	yes	maximal
3	C₄⋊Dic₁₅	C₂	1	2	yes	maximal, complemented
4	C₄×Dic₁₅	C₂	1	2	yes	maximal, complemented
5	#4072	—	3	1	no	maximal, Hall
6	C₂³×C₃×C₅	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₄×C₅	C₄	1	4	yes	complemented
8	C₂×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₂×Dic₁₅	C₂²	1	4	yes	—
11	#926	—	5	1	no	maximal, Hall
12	C₂²×C₄×C₅	D₃	1	1	yes	characteristic
13	C₂²⋊Dic₅	—	3	2	no	—
14	C₄⋊Dic₅	—	3	2	no	—
15	C₄×Dic₅	—	3	2	no	—
16	C₂²×C₃×C₅	—	2	2	no	—
17	C₂²×C₃×C₅	C₂×C₄	1	2	yes	—
18	C₂²×C₃×C₅	C₂³	1	1	yes	characteristic
19	C₃×C₄×C₅	C₂×C₄	1	2	yes	—
20	C₃×C₄×C₅	C₂×C₄	1	2	yes	—
21	Dic₁₅	—	2	4	no	—
22	C₂²×C₃×C₄	D₅	1	1	yes	characteristic
23	#217	—	5	2	no	—
24	C₄⋊Dic₃	—	5	2	no	—
25	C₄×Dic₃	—	5	2	no	—
26	C₂³×C₅	C₂×D₃	1	1	yes	characteristic
27	C₂×C₄×C₅	Dic₃	1	4	yes	complemented
28	C₂×C₄×C₅	C₂×D₃	1	1	yes	characteristic
29	C₂×C₄×C₅	C₂×D₃	1	1	yes	characteristic
30	C₂×Dic₅	—	3	4	no	—
31	#117	—	15	1	no	Sylow
32	C₂×C₃×C₅	—	2	2	no	—
33	C₂×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₃×C₅	#41	1	2	yes	—
35	C₂³×C₃	C₂×D₅	1	1	yes	characteristic
36	C₂×C₃×C₄	Dic₅	1	4	yes	complemented
37	C₂×C₃×C₄	C₂×D₅	1	1	yes	characteristic
38	C₂×C₃×C₄	C₂×D₅	1	1	yes	characteristic
39	C₂×Dic₃	—	5	4	no	—
40	C₂²×C₅	—	2	2	no	—
41	C₂²×C₅	C₂×Dic₃	1	2	yes	—
42	C₂²×C₅	C₂²×D₃	1	1	yes	characteristic
43	C₄×C₅	C₂×Dic₃	1	2	yes	—
44	C₄×C₅	C₂×Dic₃	1	2	yes	—
45	Dic₅	—	6	4	no	—
46	C₂²×C₄	D₁₅	1	1	yes	characteristic
47	C₄⋊C₄	—	15	2	no	—
48	C₂²⋊C₄	—	15	2	no	—
49	C₄²	—	15	2	no	—
50	C₃×C₅	#117	1	1	yes	characteristic, complemented, Hall
51	C₂²×C₃	—	2	2	no	—
52	C₂²×C₃	C₂×Dic₅	1	2	yes	—
53	C₂²×C₃	C₂²×D₅	1	1	yes	characteristic
54	C₃×C₄	C₂×Dic₅	1	2	yes	—
55	C₃×C₄	C₂×Dic₅	1	2	yes	—
56	Dic₃	—	10	4	no	—
57	C₂×C₅	—	2	2	no	—
58	C₂×C₅	#235	1	2	yes	—
59	C₂×C₅	C₂²×Dic₃	1	1	yes	characteristic
60	C₂³	C₂×D₁₅	1	1	yes	characteristic
61	C₂×C₄	—	15	4	no	—
62	C₂×C₄	Dic₁₅	1	4	yes	complemented
63	C₂×C₄	C₂×D₁₅	1	1	yes	characteristic
64	C₂×C₄	C₂×D₁₅	1	1	yes	characteristic
65	C₂×C₃	—	2	2	no	—
66	C₂×C₃	#709	1	2	yes	—
67	C₂×C₃	C₂²×Dic₅	1	1	yes	characteristic
68	C₅	#926	1	1	yes	characteristic, complemented, Sylow
69	C₂²	—	2	2	no	—
70	C₂²	C₂×Dic₁₅	1	2	yes	—
71	C₂²	C₂²×D₁₅	1	1	yes	characteristic
72	C₄	—	30	4	no	—
73	C₄	C₂×Dic₁₅	1	2	yes	—
74	C₄	C₂×Dic₁₅	1	2	yes	—
75	C₃	#4072	1	1	yes	characteristic, complemented, Sylow
76	C₂	—	2	2	no	—
77	C₂	#6817	1	2	yes	—
78	C₂	C₂²×Dic₁₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₅
commutator	C₂×C₃×C₅	C₂²×C₄
Frattini	C₂²	C₂²×D₁₅
Fitting	C₂²×C₃×C₄×C₅	C₂
socle, cosocle	C₂²×C₃×C₅	C₂³
prime radical	C₂³×C₃×C₅	C₂²
hypercenter	C₂²×C₄	D₁₅
hypocenter	C₃×C₅	#117
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
3. C₄⋊Dic₁₅	C₂	semidirect
4. C₄×Dic₁₅	C₂	semidirect
7. C₂×C₃×C₄×C₅	C₄	semidirect
27. C₂×C₄×C₅	Dic₃	semidirect
36. C₂×C₃×C₄	Dic₅	semidirect
50. C₃×C₅	#117	semidirect
62. C₂×C₄	Dic₁₅	semidirect
68. C₅	#926	semidirect
75. 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×C₃×C₄×C₅	2
#6719	2
C₄⋊Dic₁₅	2
C₄×Dic₁₅	2
#4072	3
#926	5