Subgroups — C₂×C₃⋊(C₃⋊#672) #13085094

Order 1440.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₃⋊(C₃⋊#672)	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃⋊#672	—	3	4	no	maximal
4	C₂×C₃²×C₄×C₅	C₄	1	1	yes	characteristic
5	C₃²×C₅×C₈	C₄	1	1	yes	characteristic
6	C₃²×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₃⋊(C₃⋊C₁₆)	—	5	1	no	maximal, Hall
8	C₂×C₃×C₅×C₈	D₃	1	4	yes	—
9	C₃⋊#672	—	3	8	no	—
10	C₂²×C₃²×C₅	C₈	1	1	yes	characteristic
11	C₃²×C₄×C₅	C₈	1	1	yes	characteristic
12	C₃²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
13	C₂×#672	—	9	1	no	Hall
14	C₂×C₃²×C₈	D₅	1	1	yes	characteristic
15	C₃⋊(C₃⋊C₁₆)	—	5	2	no	—
16	C₂×C₃×C₄×C₅	Dic₃	1	4	yes	—
17	C₃×C₅×C₈	Dic₃	1	4	yes	—
18	C₃×C₅×C₈	C₂×D₃	1	4	yes	—
19	C₂×C₃⋊C₁₆	—	15	4	no	—
20	C₂×C₃²×C₅	C₁₆	1	2	yes	complemented
21	C₂×C₃²×C₅	C₂×C₈	1	1	yes	characteristic
22	C₂×C₅×C₈	C₃⋊D₃	1	1	yes	characteristic
23	#672	—	9	2	no	—
24	C₂×C₃²×C₄	Dic₅	1	1	yes	characteristic
25	C₃²×C₈	Dic₅	1	1	yes	characteristic
26	C₃²×C₈	C₂×D₅	1	1	yes	characteristic
27	C₂²×C₃×C₅	C₃⋊C₈	1	4	yes	—
28	C₃×C₄×C₅	C₃⋊C₈	1	4	yes	—
29	C₃×C₄×C₅	C₂×Dic₃	1	4	yes	—
30	C₂×C₃×C₈	D₁₅	1	4	yes	—
31	C₃⋊C₁₆	—	15	8	no	—
32	C₃²×C₅	C₂×C₁₆	1	1	yes	characteristic, complemented, Hall
33	C₂×C₄×C₅	C₃⋊Dic₃	1	1	yes	characteristic
34	C₅×C₈	C₃⋊Dic₃	1	1	yes	characteristic
35	C₅×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
36	C₂²×C₃²	#168	1	1	yes	characteristic
37	C₃²×C₄	#168	1	1	yes	characteristic
38	C₃²×C₄	C₂×Dic₅	1	1	yes	characteristic
39	C₂×C₁₆	—	45	1	no	Sylow
40	C₂×C₃×C₅	C₃⋊C₁₆	1	8	yes	complemented
41	C₂×C₃×C₅	C₂×C₃⋊C₈	1	4	yes	—
42	C₂×C₃×C₄	Dic₁₅	1	4	yes	—
43	C₃×C₈	Dic₁₅	1	4	yes	—
44	C₃×C₈	C₂×D₁₅	1	4	yes	—
45	C₂²×C₅	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
46	C₄×C₅	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
47	C₄×C₅	C₂×C₃⋊Dic₃	1	1	yes	characteristic
48	C₂×C₃²	#672	1	2	yes	complemented
49	C₂×C₃²	C₂×#168	1	1	yes	characteristic
50	C₂×C₈	C₃⋊D₁₅	1	1	yes	characteristic
51	C₁₆	—	45	2	no	—
52	C₃×C₅	C₂×C₃⋊C₁₆	1	4	yes	complemented
53	C₂²×C₃	C₃⋊#168	1	4	yes	—
54	C₃×C₄	C₃⋊#168	1	4	yes	—
55	C₃×C₄	C₂×Dic₁₅	1	4	yes	—
56	C₂×C₅	C₃⋊(C₃⋊C₁₆)	1	2	yes	complemented
57	C₂×C₅	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
58	C₃²	C₂×#672	1	1	yes	characteristic, complemented, Sylow
59	C₂×C₄	C₃⋊Dic₁₅	1	1	yes	characteristic
60	C₈	C₃⋊Dic₁₅	1	1	yes	characteristic
61	C₈	C₂×C₃⋊D₁₅	1	1	yes	characteristic
62	C₂×C₃	C₃⋊#672	1	8	yes	complemented
63	C₂×C₃	C₂×C₃⋊#168	1	4	yes	—
64	C₅	C₂×C₃⋊(C₃⋊C₁₆)	1	1	yes	characteristic, complemented, Sylow
65	C₂²	C₃⋊(C₃⋊#168)	1	1	yes	characteristic
66	C₄	C₃⋊(C₃⋊#168)	1	1	yes	characteristic
67	C₄	C₂×C₃⋊Dic₁₅	1	1	yes	characteristic
68	C₃	C₂×C₃⋊#672	1	4	yes	complemented
69	C₂	C₃⋊(C₃⋊#672)	1	2	yes	direct product factor
70	C₂	C₂×C₃⋊(C₃⋊#168)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₈	C₃⋊D₁₅
commutator, hypocenter	C₃²×C₅	C₂×C₁₆
Frattini	C₈	C₂×C₃⋊D₁₅
Fitting	C₂×C₃²×C₅×C₈	C₂
socle, prime radical	C₂²×C₃²×C₅	C₈
cosocle	C₃²×C₅×C₈	C₂²
LCS	G ▷ C₃²×C₅
UCS	C₁ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
2. C₃⋊(C₃⋊#672)	C₂	direct (N × H)
20. C₂×C₃²×C₅	C₁₆	semidirect
32. C₃²×C₅	C₂×C₁₆	semidirect
40. C₂×C₃×C₅	C₃⋊C₁₆	semidirect
48. C₂×C₃²	#672	semidirect
52. C₃×C₅	C₂×C₃⋊C₁₆	semidirect
56. C₂×C₅	C₃⋊(C₃⋊C₁₆)	semidirect
58. C₃²	C₂×#672	semidirect
62. C₂×C₃	C₃⋊#672	semidirect
64. C₅	C₂×C₃⋊(C₃⋊C₁₆)	semidirect
68. C₃	C₂×C₃⋊#672	semidirect
69. C₂	C₃⋊(C₃⋊#672)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₅×C₈	2
C₃⋊(C₃⋊#672)	2
C₂×C₃⋊#672	3
C₂×C₃⋊(C₃⋊C₁₆)	5

Subgroups — C2×C3⋊(C3⋊#672) #13085094

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃⋊(C₃⋊#672) #13085094