Subgroups — C₄⋊#66472 #13049600

Order 1296.

956 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#66472	C₂	1	2	yes	maximal
2	C₄×C₃⋊(C₃⋊(C₃⋊D₃))	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃⋊(C₃⋊(C₃⋊D₃))	C₂²	1	1	yes	characteristic
4	#66472	—	2	2	no	—
5	C₃⁴×C₄	C₄	1	1	yes	characteristic, complemented
6	C₃⋊(C₃⋊(C₃⋊Dic₃))	C₄	1	1	yes	characteristic, complemented
7	C₄×C₃⋊(C₃⋊D₃)	—	6	20	no	—
8	C₂×C₃⁴	C₂×C₄	1	1	yes	characteristic
9	C₃⋊(C₃⋊(C₃⋊D₃))	D₄	1	1	yes	characteristic
10	C₃⋊(C₃⋊(C₃⋊D₃))	Q₈	1	1	yes	characteristic
11	C₄⋊#157	—	9	10	no	maximal
12	C₂×C₃⋊(C₃⋊D₃)	—	6	20	no	—
13	C₃³×C₄	—	2	20	no	—
14	C₃⋊(C₃⋊Dic₃)	—	6	20	no	—
15	C₃⁴	C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
16	C₂×#157	—	9	20	no	—
17	C₄×C₃⋊D₃	—	9	10	no	—
18	C₄×C₃⋊D₃	—	18	60	no	—
19	C₂×C₃³	—	2	20	no	—
20	C₃⋊(C₃⋊D₃)	—	6	20	no	—
21	C₃⋊(C₃⋊D₃)	—	6	20	no	—
22	C₂×C₃⋊D₃	—	9	10	no	—
23	C₂×C₃⋊D₃	—	18	60	no	—
24	#157	—	18	20	no	—
25	C₃²×C₄	—	2	60	no	—
26	C₃²×C₄	#157	1	10	yes	complemented
27	C₃⋊Dic₃	—	9	10	no	—
28	C₃⋊Dic₃	—	18	60	no	—
29	C₃³	—	2	20	no	—
30	C₄×D₃	—	54	20	no	—
31	C₂×C₃²	—	2	60	no	—
32	C₂×C₃²	C₂×#157	1	10	yes	—
33	C₃⋊D₃	—	9	10	no	—
34	C₃⋊D₃	—	9	10	no	—
35	C₃⋊D₃	—	18	60	no	—
36	C₃⋊D₃	—	18	60	no	—
37	C₄⋊C₄	—	81	1	no	Sylow
38	C₂×D₃	—	54	20	no	—
39	C₃×C₄	—	2	20	no	—
40	Dic₃	—	54	20	no	—
41	C₃²	—	2	60	no	—
42	C₃²	C₄⋊#157	1	10	yes	complemented
43	C₂×C₄	—	81	1	no	—
44	C₂×C₄	—	81	2	no	—
45	C₂×C₃	—	2	20	no	—
46	D₃	—	54	20	no	—
47	D₃	—	54	20	no	—
48	C₂²	—	81	1	no	—
49	C₄	—	81	1	no	—
50	C₄	—	162	2	no	—
51	C₄	#66472	1	1	yes	characteristic, complemented
52	C₃	—	2	20	no	—
53	C₂	—	81	1	no	—
54	C₂	—	81	1	no	—
55	C₂	C₂×#66472	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×#66472
commutator, socle	C₂×C₃⁴	C₂×C₄
Fitting	C₃⁴×C₄	C₄
prime radical, cosocle	C₂×C₃⋊(C₃⋊(C₃⋊D₃))	C₂²
hypercenter	C₄	#66472
hypocenter	C₃⁴	C₄⋊C₄
LCS	G ▷ C₂×C₃⁴ ▷ C₃⁴
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
5. C₃⁴×C₄	C₄	semidirect
6. C₃⋊(C₃⋊(C₃⋊Dic₃))	C₄	semidirect
15. C₃⁴	C₄⋊C₄	semidirect
26. C₃²×C₄	#157	semidirect
42. C₃²	C₄⋊#157	semidirect
51. C₄	#66472	semidirect

Maximal subgroups

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

subgroup	index
C₂×#66472	2
C₄×C₃⋊(C₃⋊(C₃⋊D₃))	2
C₄⋊#157	9

Subgroups — C4⋊#66472 #13049600

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊#66472 #13049600