Subgroups — #49137622

Order 36300.

112 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	C₁₁×C₁₁⋊#1044	—	3	1	no	maximal, Hall
3	C₅²×C₁₁²⋊C₃	C₄	1	1	yes	characteristic, complemented, Hall
4	#20300037	—	5	1	no	maximal
5	#20300037	—	5	1	no	maximal
6	C₁₁²×C₅⋊D₅	D₃	1	1	yes	characteristic
7	D₅×C₁₁²⋊C₃	—	5	1	no	—
8	D₅×C₁₁²⋊C₃	—	5	1	no	—
9	D₅×C₁₁²⋊C₃	—	10	2	no	—
10	C₅²×C₁₁²	Dic₃	1	1	yes	characteristic, complemented, Hall
11	C₁₁×C₁₁⋊AGL(1,𝔽₅)	—	15	1	no	—
12	C₁₁×C₁₁⋊AGL(1,𝔽₅)	—	15	1	no	—
13	C₅×C₁₁²⋊C₃	—	2	2	no	—
14	C₅×C₁₁²⋊C₃	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
15	C₅×C₁₁²⋊C₃	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
16	#13089712	—	25	1	no	Hall
17	D₅×C₁₁²	—	5	1	no	—
18	D₅×C₁₁²	—	5	1	no	—
19	D₅×C₁₁²	—	10	2	no	—
20	C₁₁⋊#1044	—	3	1	no	—
21	C₁₁×#1044	—	33	1	no	—
22	C₂×C₁₁²⋊C₃	—	25	1	no	—
23	C₅×C₁₁²	—	2	2	no	—
24	C₅×C₁₁²	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
25	C₅×C₁₁²	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
26	C₁₁×C₅⋊D₅	—	3	1	no	—
27	C₁₁×C₅⋊D₅	—	6	1	no	—
28	C₁₁×C₅⋊D₅	—	6	1	no	—
29	C₁₁×Dic₁₁	—	75	1	no	Hall
30	C₁₁²⋊C₃	#1044	1	1	yes	characteristic, complemented, Hall
31	C₃⋊#1044	—	121	1	no	maximal, Hall
32	C₅²×C₁₁	—	3	1	no	—
33	C₅²×C₁₁	—	6	1	no	—
34	C₅²×C₁₁	—	6	1	no	—
35	C₂×C₁₁²	—	25	1	no	—
36	C₁₁⋊AGL(1,𝔽₅)	—	15	1	no	—
37	C₁₁⋊AGL(1,𝔽₅)	—	15	1	no	—
38	C₁₁×AGL(1,𝔽₅)	—	165	1	no	—
39	C₁₁×AGL(1,𝔽₅)	—	165	1	no	—
40	C₃×C₅⋊D₅	—	242	1	no	—
41	C₁₁²	C₃⋊#1044	1	1	yes	characteristic, complemented, Sylow
42	D₅×C₁₁	—	15	1	no	—
43	D₅×C₁₁	—	15	1	no	—
44	D₅×C₁₁	—	15	1	no	—
45	D₅×C₁₁	—	30	1	no	—
46	D₅×C₁₁	—	30	1	no	—
47	D₅×C₁₁	—	30	1	no	—
48	D₅×C₁₁	—	30	2	no	—
49	D₅×C₁₁	—	30	2	no	—
50	D₅×C₁₁	—	30	4	no	—
51	#1044	—	33	1	no	Hall
52	C₃×C₅²	—	242	1	no	Hall
53	ΓL(1,𝔽₁₆)	—	605	1	no	—
54	ΓL(1,𝔽₁₆)	—	605	1	no	—
55	C₅×C₁₁	—	3	1	no	—
56	C₅×C₁₁	—	3	1	no	—
57	C₅×C₁₁	—	3	1	no	—
58	C₅×C₁₁	—	6	2	no	—
59	C₅×C₁₁	—	6	2	no	—
60	C₅×C₁₁	—	6	4	no	—
61	C₅×C₁₁	—	6	1	no	—
62	C₅×C₁₁	—	6	1	no	—
63	C₅×C₁₁	—	6	1	no	—
64	C₅⋊D₅	#10624391	1	1	yes	characteristic
65	C₄×C₁₁	—	825	1	no	—
66	Dic₁₁	—	75	1	no	—
67	C₃×D₅	—	1210	1	no	—
68	C₃×D₅	—	1210	1	no	—
69	C₃×D₅	—	1210	2	no	—
70	C₅²	#13089712	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₁₁	—	75	1	no	—
72	C₂×C₁₁	—	150	1	no	—
73	C₂×C₁₁	—	150	1	no	—
74	AGL(1,𝔽₅)	—	165	1	no	—
75	AGL(1,𝔽₅)	—	165	1	no	—
76	C₃×C₅	—	242	2	no	—
77	C₃×C₅	—	242	1	no	—
78	C₃×C₅	—	242	1	no	—
79	Dic₃	—	3025	1	no	Hall
80	C₁₁	—	3	1	no	—
81	C₁₁	—	6	1	no	—
82	C₁₁	—	6	1	no	—
83	D₅	—	5	1	no	—
84	D₅	—	5	1	no	—
85	D₅	—	10	2	no	—
86	C₂×C₃	—	6050	1	no	—
87	C₅	—	2	2	no	—
88	C₅	#20300037	1	1	yes	characteristic, complemented
89	C₅	#20300037	1	1	yes	characteristic, complemented
90	C₄	—	825	1	no	Sylow
91	C₃	—	242	1	no	Sylow
92	C₂	—	25	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, hypocenter	C₅²×C₁₁²⋊C₃	C₄
Fitting, socle	C₅²×C₁₁²	Dic₃
prime radical, cosocle	C₅⋊D₅×C₁₁²⋊C₃	C₂
LCS	G ▷ C₅²×C₁₁²⋊C₃
UCS	C₁

Complements

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

N	H ≅ G/N	type
3. C₅²×C₁₁²⋊C₃	C₄	semidirect
10. C₅²×C₁₁²	Dic₃	semidirect
14. C₅×C₁₁²⋊C₃	AGL(1,𝔽₅)	semidirect
15. C₅×C₁₁²⋊C₃	AGL(1,𝔽₅)	semidirect
24. C₅×C₁₁²	ΓL(1,𝔽₁₆)	semidirect
25. C₅×C₁₁²	ΓL(1,𝔽₁₆)	semidirect
30. C₁₁²⋊C₃	#1044	semidirect
41. C₁₁²	C₃⋊#1044	semidirect
70. C₅²	#13089712	semidirect
88. C₅	#20300037	semidirect
89. C₅	#20300037	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
C₁₁×C₁₁⋊#1044	3
#20300037	5
C₃⋊#1044	121