Subgroups — #120⋊(C₁₃⋊C₃) #11928752

Order 1248.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊(C₂²⋊(C₁₃⋊C₃))	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#120	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₂×C₂²⋊(C₁₃⋊C₃)	—	4	1	no	maximal
4	C₂⁴×C₁₃	C₂×C₃	1	1	yes	characteristic, complemented
5	C₂×D₄×C₁₃	—	3	1	no	—
6	C₁₃×C₂²⋊C₄	—	3	1	no	—
7	C₂²⋊(C₁₃⋊C₃)	—	4	1	no	—
8	C₂²⋊(C₁₃⋊C₃)	—	8	2	no	—
9	C₂³×C₁₃	—	3	1	no	—
10	C₂³×C₁₃	—	6	2	no	—
11	C₂³×C₁₃	A₄	1	1	yes	characteristic, complemented
12	D₄×C₁₃	—	6	2	no	—
13	C₂×C₄×C₁₃	—	3	1	no	—
14	#120⋊C₃	—	13	1	no	maximal, Hall
15	C₂×C₁₃⋊C₃	—	16	1	no	—
16	C₂²×C₁₃	—	2	2	no	—
17	C₂²×C₁₃	—	3	2	no	—
18	C₂²×C₁₃	—	6	1	no	—
19	C₂²×C₁₃	—	6	2	no	—
20	C₂²×C₁₃	—	6	2	no	—
21	C₂²×C₁₃	C₂×A₄	1	1	yes	characteristic
22	C₄×C₁₃	—	6	1	no	—
23	C₂²⋊A₄	—	13	1	no	—
24	C₁₃⋊C₃	—	16	1	no	Hall
25	#120	C₁₃⋊C₃	1	1	yes	characteristic, complemented, Sylow
26	C₂×C₁₃	—	3	1	no	—
27	C₂×C₁₃	—	4	1	no	—
28	C₂×C₁₃	—	6	2	no	—
29	C₂×A₄	—	52	1	no	—
30	C₂⁴	C₂×C₁₃⋊C₃	1	1	yes	characteristic, complemented
31	C₂×D₄	—	3	1	no	—
32	C₂²⋊C₄	—	3	1	no	—
33	C₁₃	#120⋊C₃	1	1	yes	characteristic, complemented, Sylow
34	A₄	—	52	1	no	—
35	A₄	—	104	2	no	—
36	C₂³	—	3	1	no	—
37	C₂³	—	6	2	no	—
38	C₂³	C₂²⋊(C₁₃⋊C₃)	1	1	yes	characteristic, complemented
39	D₄	—	6	2	no	—
40	C₂×C₄	—	3	1	no	—
41	C₂×C₃	—	208	1	no	—
42	C₂²	—	2	2	no	—
43	C₂²	—	3	2	no	—
44	C₂²	—	6	1	no	—
45	C₂²	—	6	2	no	—
46	C₂²	—	6	2	no	—
47	C₂²	C₂×C₂²⋊(C₁₃⋊C₃)	1	1	yes	characteristic
48	C₄	—	6	1	no	—
49	C₃	—	208	1	no	Sylow
50	C₂	—	3	1	no	—
51	C₂	—	4	1	no	—
52	C₂	—	6	2	no	—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₂⁴×C₁₃	C₂×C₃
Frattini	C₂²	C₂×C₂²⋊(C₁₃⋊C₃)
Fitting	C₁₃×#120	C₃
socle	C₂²×C₁₃	C₂×A₄
LCS	G ▷ C₂⁴×C₁₃
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₂²⋊(C₂²⋊(C₁₃⋊C₃))	C₂	semidirect
2. C₁₃×#120	C₃	semidirect
4. C₂⁴×C₁₃	C₂×C₃	semidirect
11. C₂³×C₁₃	A₄	semidirect
25. #120	C₁₃⋊C₃	semidirect
30. C₂⁴	C₂×C₁₃⋊C₃	semidirect
33. C₁₃	#120⋊C₃	semidirect
38. C₂³	C₂²⋊(C₁₃⋊C₃)	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊(C₂²⋊(C₁₃⋊C₃))	2
C₁₃×#120	3
C₂×C₂²⋊(C₁₃⋊C₃)	4
#120⋊C₃	13

Subgroups — #120⋊(C13⋊C3) #11928752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #120⋊(C₁₃⋊C₃) #11928752