Subgroups — #91119

Order 480.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊Dic₁₅	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃⋊#168	C₂	1	1	yes	maximal, characteristic
4	#3967	—	3	1	no	maximal, Hall
5	C₃×C₅×Q₈	—	2	1	no	—
6	C₃×C₅×Q₈	C₄	1	2	yes	complemented
7	C₂×C₃×C₄×C₅	—	2	1	no	—
8	C₂×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₅	—	2	1	no	—
10	C₃⋊#168	—	2	1	no	—
11	#835	—	5	1	no	maximal, Hall
12	C₂×C₅×Q₈	D₃	1	1	yes	characteristic
13	C₄⋊Dic₅	—	3	1	no	—
14	C₂×#168	—	3	1	no	—
15	C₂²×C₃×C₅	D₄	1	1	yes	characteristic
16	C₃×C₄×C₅	—	2	2	no	—
17	C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₃×C₄×C₅	D₄	1	1	yes	characteristic
19	Dic₁₅	—	4	1	no	—
20	C₂×C₃×Q₈	D₅	1	1	yes	characteristic
21	C₄⋊Dic₃	—	5	1	no	—
22	C₂×C₃⋊C₈	—	5	1	no	—
23	C₅×Q₈	—	2	1	no	—
24	C₅×Q₈	Dic₃	1	2	yes	complemented
25	C₂×C₄×C₅	—	2	1	no	—
26	C₂×C₄×C₅	C₂×D₃	1	1	yes	characteristic
27	C₂×Dic₅	—	6	1	no	—
28	#168	—	6	1	no	—
29	#103	—	15	1	no	Sylow
30	C₂×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₃×C₅	SD₁₆	1	1	yes	characteristic
32	C₂×C₃×C₅	Q₁₆	1	1	yes	characteristic
33	C₃×Q₈	—	2	1	no	—
34	C₃×Q₈	Dic₅	1	2	yes	complemented
35	C₂×C₃×C₄	—	2	1	no	—
36	C₂×C₃×C₄	C₂×D₅	1	1	yes	characteristic
37	C₂×Dic₃	—	10	1	no	—
38	C₃⋊C₈	—	10	1	no	—
39	C₂²×C₅	#67	1	1	yes	characteristic
40	C₄×C₅	—	2	2	no	—
41	C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
42	C₄×C₅	#67	1	1	yes	characteristic
43	Dic₅	—	12	1	no	—
44	C₂×Q₈	D₁₅	1	1	yes	characteristic
45	C₂×C₈	—	15	1	no	—
46	C₄⋊C₄	—	15	1	no	—
47	C₃×C₅	#103	1	1	yes	characteristic, complemented, Hall
48	C₂²×C₃	C₂²⋊D₅	1	1	yes	characteristic
49	C₃×C₄	—	2	2	no	—
50	C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
51	C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
52	Dic₃	—	20	1	no	—
53	C₂×C₅	#215	1	1	yes	characteristic
54	C₂×C₅	#216	1	1	yes	characteristic
55	C₂×C₅	#217	1	1	yes	characteristic
56	Q₈	—	2	1	no	—
57	Q₈	Dic₁₅	1	2	yes	complemented
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	30	1	no	—
60	C₂×C₄	C₂×D₁₅	1	1	yes	characteristic
61	C₈	—	30	1	no	—
62	C₂×C₃	#688	1	1	yes	characteristic
63	C₂×C₃	#689	1	1	yes	characteristic
64	C₂×C₃	C₂²⋊Dic₅	1	1	yes	characteristic
65	C₅	#835	1	1	yes	characteristic, complemented, Sylow
66	C₂²	#1220	1	1	yes	characteristic
67	C₄	—	2	2	no	—
68	C₄	—	60	1	no	—
69	C₄	C₂×Dic₁₅	1	1	yes	characteristic
70	C₄	#1220	1	1	yes	characteristic
71	C₃	#3967	1	1	yes	characteristic, complemented, Sylow
72	C₂	#6717	1	1	yes	characteristic
73	C₂	#6718	1	1	yes	characteristic
74	C₂	#6719	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#1220
commutator	C₃×C₄×C₅	C₂×C₄
Frattini	C₂×C₄	C₂×D₁₅
Fitting	C₂×C₃×C₅×Q₈	C₂
socle, prime radical	C₂²×C₃×C₅	D₄
cosocle	C₂×C₃×C₄×C₅	C₂²
hypercenter	C₂×Q₈	D₁₅
hypocenter	C₃×C₅	#103
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
6. C₃×C₅×Q₈	C₄	semidirect
24. C₅×Q₈	Dic₃	semidirect
34. C₃×Q₈	Dic₅	semidirect
47. C₃×C₅	#103	semidirect
57. Q₈	Dic₁₅	semidirect
65. C₅	#835	semidirect
71. C₃	#3967	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×Q₈	2
C₄⋊Dic₁₅	2
C₂×C₃⋊#168	2
#3967	3
#835	5