Subgroups — C₃×C₂²⋊#168 #91032

Order 480.

100 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	C₂×C₃×#168	C₂	1	2	yes	maximal, complemented
3	C₂²⋊#168	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂³×C₃×C₅	C₄	1	1	yes	characteristic
5	C₂×C₃×C₄×C₅	—	2	2	no	—
6	C₂×C₃×C₄×C₅	C₄	1	1	yes	characteristic
7	C₂×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₃×#168	—	2	2	no	—
9	C₃×C₂²⋊C₈	—	5	1	no	maximal, Hall
10	C₂²×C₄×C₅	C₂×C₃	1	1	yes	characteristic
11	C₂×#168	C₂×C₃	1	2	yes	complemented
12	C₂²×C₃×C₅	—	2	1	no	—
13	C₂²×C₃×C₅	—	2	1	no	—
14	C₂²×C₃×C₅	C₈	1	2	yes	complemented
15	C₂²×C₃×C₅	C₂×C₄	1	1	yes	characteristic
16	C₃×C₄×C₅	—	2	1	no	—
17	C₃×C₄×C₅	D₄	1	2	yes	—
18	C₂²×C₃×C₄	D₅	1	1	yes	characteristic
19	C₂×C₃×C₈	—	5	2	no	—
20	C₂³×C₅	C₃×C₄	1	1	yes	characteristic
21	C₂×C₄×C₅	—	2	2	no	—
22	C₂×C₄×C₅	C₃×C₄	1	1	yes	characteristic
23	C₂×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
24	#168	—	2	2	no	—
25	C₂²⋊C₈	—	5	1	no	Sylow
26	C₂×C₃×C₅	—	2	2	no	—
27	C₂×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₃×C₅	C₂×C₈	1	1	yes	characteristic
29	C₂×C₃×C₅	M₁₆	1	1	yes	characteristic
30	C₂³×C₃	Dic₅	1	1	yes	characteristic
31	C₂×C₃×C₄	—	2	2	no	—
32	C₂×C₃×C₄	Dic₅	1	1	yes	characteristic
33	C₂×C₃×C₄	C₂×D₅	1	1	yes	characteristic
34	C₃×C₈	—	10	2	no	—
35	C₂²×C₅	—	2	1	no	—
36	C₂²×C₅	—	2	1	no	—
37	C₂²×C₅	C₃×C₈	1	2	yes	complemented
38	C₂²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
39	C₄×C₅	—	2	1	no	—
40	C₄×C₅	C₃×D₄	1	2	yes	—
41	C₂²×C₄	C₃×D₅	1	1	yes	characteristic
42	C₂×C₈	—	5	2	no	—
43	C₃×C₅	C₂²⋊C₈	1	1	yes	characteristic, complemented, Hall
44	C₂²×C₃	—	2	1	no	—
45	C₂²×C₃	—	2	1	no	—
46	C₂²×C₃	#168	1	2	yes	complemented
47	C₂²×C₃	C₂×Dic₅	1	1	yes	characteristic
48	C₃×C₄	—	2	1	no	—
49	C₃×C₄	C₂²⋊D₅	1	2	yes	—
50	C₂×C₅	—	2	2	no	—
51	C₂×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₅	C₂×C₃×C₈	1	1	yes	characteristic
53	C₂×C₅	C₃×M₁₆	1	1	yes	characteristic
54	C₂³	C₃×Dic₅	1	1	yes	characteristic
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	C₃×Dic₅	1	1	yes	characteristic
57	C₂×C₄	C₂×C₃×D₅	1	1	yes	characteristic
58	C₈	—	10	2	no	—
59	C₂×C₃	—	2	2	no	—
60	C₂×C₃	C₂×#168	1	1	yes	characteristic
61	C₂×C₃	#681	1	1	yes	characteristic
62	C₂×C₃	C₂²⋊Dic₅	1	1	yes	characteristic
63	C₅	C₃×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
64	C₂²	—	2	1	no	—
65	C₂²	—	2	1	no	—
66	C₂²	C₃×#168	1	2	yes	complemented
67	C₂²	C₂×C₃×Dic₅	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	C₃×C₂²⋊D₅	1	2	yes	—
70	C₃	C₂²⋊#168	1	1	yes	characteristic, direct product factor, Sylow
71	C₂	—	2	2	no	—
72	C₂	C₂×C₃×#168	1	1	yes	characteristic
73	C₂	C₃×#681	1	1	yes	characteristic
74	C₂	C₃×C₂²⋊Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₅
commutator	C₂×C₅	C₂×C₃×C₈
Frattini	C₂×C₄	C₂×C₃×D₅
Fitting	C₂²×C₃×C₄×C₅	C₂
socle	C₂²×C₃×C₅	C₂×C₄
prime radical	C₂³×C₃×C₅	C₄
cosocle	C₂×C₄×C₅	C₂²×C₃
hypercenter	C₂²×C₃×C₄	D₅
hypocenter	C₅	C₃×C₂²⋊C₈
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×#168	C₂	semidirect
3. C₂²⋊#168	C₃	direct (N × H)
11. C₂×#168	C₂×C₃	semidirect
14. C₂²×C₃×C₅	C₈	semidirect
37. C₂²×C₅	C₃×C₈	semidirect
43. C₃×C₅	C₂²⋊C₈	semidirect
46. C₂²×C₃	#168	semidirect
63. C₅	C₃×C₂²⋊C₈	semidirect
66. C₂²	C₃×#168	semidirect
70. C₃	C₂²⋊#168	direct (N × H)

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
C₂×C₃×#168	2
C₂²⋊#168	3
C₃×C₂²⋊C₈	5

Subgroups — C3×C22⋊#168 #91032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₂²⋊#168 #91032