Subgroups — #1044×C₁₁²⋊C₃ #49137621

Order 36300.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅⋊D₅×C₁₁²⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₁₁²×#1044	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₅²×C₁₁²⋊C₃	C₄	1	1	yes	characteristic, complemented, Hall
4	AGL(1,𝔽₅)×C₁₁²⋊C₃	—	5	2	no	maximal
5	C₁₁²×C₅⋊D₅	C₂×C₃	1	1	yes	characteristic
6	D₅×C₁₁²⋊C₃	—	5	2	no	—
7	D₅×C₁₁²⋊C₃	—	10	2	no	—
8	C₅²×C₁₁²	C₃×C₄	1	1	yes	characteristic, complemented, Hall
9	C₁₁²×AGL(1,𝔽₅)	—	5	2	no	—
10	C₅×C₁₁²⋊C₃	—	2	2	no	—
11	C₅×C₁₁²⋊C₃	AGL(1,𝔽₅)	1	2	yes	complemented
12	C₄×C₁₁²⋊C₃	—	25	1	no	Hall
13	D₅×C₁₁²	—	5	2	no	—
14	D₅×C₁₁²	—	10	2	no	—
15	C₁₁×#1044	—	3	4	no	—
16	C₂×C₁₁²⋊C₃	—	25	1	no	—
17	C₅×C₁₁²	—	2	2	no	—
18	C₅×C₁₁²	C₃×AGL(1,𝔽₅)	1	2	yes	complemented
19	C₁₁×C₅⋊D₅	—	3	4	no	—
20	C₄×C₁₁²	—	25	1	no	Hall
21	C₁₁²⋊C₃	#1044	1	1	yes	characteristic, direct product factor, Hall
22	C₃×#1044	—	121	1	no	maximal, Hall
23	C₅²×C₁₁	—	3	4	no	—
24	C₂×C₁₁²	—	25	1	no	—
25	C₁₁×AGL(1,𝔽₅)	—	15	8	no	—
26	C₃×C₅⋊D₅	—	121	1	no	—
27	C₁₁²	C₃×#1044	1	1	yes	characteristic, complemented, Sylow
28	D₅×C₁₁	—	15	8	no	—
29	D₅×C₁₁	—	30	8	no	—
30	#1044	C₁₁²⋊C₃	1	1	yes	characteristic, direct product factor, Hall
31	C₃×C₅²	—	121	1	no	Hall
32	C₃×AGL(1,𝔽₅)	—	605	2	no	—
33	C₅×C₁₁	—	3	8	no	—
34	C₅×C₁₁	—	6	8	no	—
35	C₅⋊D₅	C₂×C₁₁²⋊C₃	1	1	yes	characteristic
36	C₄×C₁₁	—	75	4	no	—
37	C₃×D₅	—	605	2	no	—
38	C₃×D₅	—	1210	2	no	—
39	C₅²	C₄×C₁₁²⋊C₃	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₁₁	—	75	4	no	—
41	AGL(1,𝔽₅)	—	5	2	no	—
42	C₃×C₅	—	121	2	no	—
43	C₃×C₅	—	242	2	no	—
44	C₃×C₄	—	3025	1	no	Hall
45	C₁₁	—	3	4	no	—
46	D₅	—	5	2	no	—
47	D₅	—	10	2	no	—
48	C₂×C₃	—	3025	1	no	—
49	C₅	—	2	2	no	—
50	C₅	AGL(1,𝔽₅)×C₁₁²⋊C₃	1	2	yes	complemented
51	C₄	—	25	1	no	Sylow
52	C₃	—	121	1	no	Sylow
53	C₂	—	25	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅²×C₁₁²	C₃×C₄
prime radical	C₅⋊D₅×C₁₁²⋊C₃	C₂
cosocle	C₁₁²×C₅⋊D₅	C₂×C₃
LCS	G ▷ C₅²×C₁₁²
UCS	C₁

Complements

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

N	H ≅ G/N	type
2. C₁₁²×#1044	C₃	semidirect
3. C₅²×C₁₁²⋊C₃	C₄	semidirect
8. C₅²×C₁₁²	C₃×C₄	semidirect
11. C₅×C₁₁²⋊C₃	AGL(1,𝔽₅)	semidirect
18. C₅×C₁₁²	C₃×AGL(1,𝔽₅)	semidirect
21. C₁₁²⋊C₃	#1044	direct (N × H)
27. C₁₁²	C₃×#1044	semidirect
30. #1044	C₁₁²⋊C₃	direct (N × H)
39. C₅²	C₄×C₁₁²⋊C₃	semidirect
50. C₅	AGL(1,𝔽₅)×C₁₁²⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₅⋊D₅×C₁₁²⋊C₃	2
C₁₁²×#1044	3
AGL(1,𝔽₅)×C₁₁²⋊C₃	5
C₃×#1044	121

Subgroups — #1044×C112⋊C3 #49137621

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1044×C₁₁²⋊C₃ #49137621