Subgroups — C₅×Dic₃×#10597244 #49137263

Order 36300.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#10597244	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×#10597244	—	3	1	no	maximal, Hall
3	C₃×C₅×#10597244	C₄	1	1	yes	characteristic, complemented, Hall
4	C₅×C₁₁²×Dic₃	C₅	1	1	yes	maximal, characteristic, complemented
5	Dic₃×#10597244	C₅	1	5	yes	maximal, direct product factor
6	C₂×C₅×#10597244	D₃	1	1	yes	characteristic
7	C₂×C₃×#10597244	C₂×C₅	1	5	yes	—
8	C₂×C₃×C₅×C₁₁²	C₂×C₅	1	1	yes	characteristic
9	C₅×Dic₃×C₁₁⋊C₅	—	11	2	no	maximal
10	C₅×#10597244	Dic₃	1	1	yes	characteristic, direct product factor, Hall
11	C₄×#10597244	—	3	5	no	—
12	C₄×C₅×C₁₁²	—	3	1	no	—
13	C₃×C₅×C₁₁²	C₄×C₅	1	1	yes	characteristic, complemented
14	C₃×#10597244	C₄×C₅	1	5	yes	complemented
15	C₂×C₃×C₅×C₁₁⋊C₅	—	11	2	no	—
16	C₁₁²×Dic₃	C₅²	1	1	yes	characteristic, complemented, Hall
17	C₂×C₅×C₁₁²	C₅×D₃	1	1	yes	characteristic
18	C₂×#10597244	C₅×D₃	1	5	yes	—
19	C₄×C₅×C₁₁⋊C₅	—	33	2	no	—
20	C₃×C₅×C₁₁⋊C₅	—	11	2	no	—
21	C₂×C₃×C₁₁²	C₂×C₅²	1	1	yes	characteristic
22	C₅×C₁₁×Dic₃	—	5	2	no	—
23	C₅×C₁₁×Dic₃	C₁₁⋊C₅	1	2	yes	complemented
24	Dic₃×C₁₁⋊C₅	—	11	10	no	—
25	C₅×C₁₁²	C₅×Dic₃	1	1	yes	characteristic, complemented
26	#10597244	C₅×Dic₃	1	5	yes	direct product factor
27	C₂×C₅×C₁₁⋊C₅	—	11	2	no	—
28	C₄×C₁₁²	—	3	1	no	Hall
29	C₃×C₁₁²	C₄×C₅²	1	1	yes	characteristic, complemented, Hall
30	C₂×C₃×C₅×C₁₁	—	5	2	no	—
31	C₂×C₃×C₅×C₁₁	C₂×C₁₁⋊C₅	1	2	yes	—
32	C₂×C₃×C₁₁⋊C₅	—	11	10	no	—
33	C₅²×Dic₃	—	121	1	no	Hall
34	C₅×C₁₁⋊C₅	—	11	2	no	—
35	C₂×C₁₁²	C₅²×D₃	1	1	yes	characteristic
36	C₄×C₅×C₁₁	—	3	2	no	—
37	C₄×C₅×C₁₁	—	15	2	no	—
38	C₄×C₁₁⋊C₅	—	33	10	no	—
39	C₃×C₅×C₁₁	—	5	2	no	—
40	C₃×C₅×C₁₁	C₄×C₁₁⋊C₅	1	2	yes	complemented
41	C₃×C₁₁⋊C₅	—	11	10	no	—
42	C₂×C₃×C₅²	—	121	1	no	—
43	C₁₁×Dic₃	—	5	2	no	—
44	C₁₁×Dic₃	C₅×C₁₁⋊C₅	1	2	yes	complemented
45	C₁₁²	C₅²×Dic₃	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₅×C₁₁	—	5	2	no	—
47	C₂×C₅×C₁₁	D₃×C₁₁⋊C₅	1	2	yes	—
48	C₂×C₁₁⋊C₅	—	11	10	no	—
49	C₄×C₅²	—	363	1	no	Hall
50	C₃×C₅²	—	121	1	no	Hall
51	C₂×C₃×C₁₁	—	5	2	no	—
52	C₂×C₃×C₁₁	C₂×C₅×C₁₁⋊C₅	1	2	yes	—
53	C₅×Dic₃	—	121	5	no	—
54	C₅×Dic₃	#10597244	1	1	yes	characteristic, direct product factor
55	C₅×C₁₁	—	5	2	no	—
56	C₅×C₁₁	Dic₃×C₁₁⋊C₅	1	2	yes	complemented
57	C₁₁⋊C₅	—	11	10	no	—
58	C₂×C₅²	—	121	1	no	—
59	C₄×C₁₁	—	3	2	no	—
60	C₄×C₁₁	—	15	2	no	—
61	C₃×C₁₁	—	5	2	no	—
62	C₃×C₁₁	C₄×C₅×C₁₁⋊C₅	1	2	yes	complemented
63	C₂×C₃×C₅	—	121	5	no	—
64	C₂×C₃×C₅	C₂×#10597244	1	1	yes	characteristic
65	C₅²	—	121	1	no	Sylow
66	C₂×C₁₁	—	5	2	no	—
67	C₂×C₁₁	C₅×D₃×C₁₁⋊C₅	1	2	yes	—
68	C₄×C₅	—	3	1	no	—
69	C₄×C₅	—	363	5	no	—
70	C₃×C₅	—	121	5	no	—
71	C₃×C₅	C₄×#10597244	1	1	yes	characteristic, complemented
72	Dic₃	C₅×#10597244	1	1	yes	characteristic, direct product factor, Hall
73	C₁₁	—	5	2	no	—
74	C₁₁	C₅×Dic₃×C₁₁⋊C₅	1	2	yes	complemented
75	C₂×C₅	—	121	5	no	—
76	C₂×C₅	D₃×#10597244	1	1	yes	characteristic
77	C₂×C₃	C₂×C₅×#10597244	1	1	yes	characteristic
78	C₅	—	121	5	no	—
79	C₅	Dic₃×#10597244	1	1	yes	characteristic, direct product factor
80	C₄	—	3	1	no	Sylow
81	C₃	C₄×C₅×#10597244	1	1	yes	characteristic, complemented, Sylow
82	C₂	C₅×D₃×#10597244	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₅	D₃×#10597244
commutator, hypocenter	C₃×C₁₁²	C₄×C₅²
Frattini	C₂	C₅×D₃×#10597244
Fitting, socle	C₂×C₃×C₅×C₁₁²	C₂×C₅
prime radical	C₂×C₃×C₅×#10597244	C₂
cosocle	C₂×C₃×C₁₁²	C₂×C₅²
LCS	G ▷ C₃×C₁₁²
UCS	C₁ ◁ C₂×C₅

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#10597244	C₄	semidirect
4. C₅×C₁₁²×Dic₃	C₅	semidirect
5. Dic₃×#10597244	C₅	direct (N × H)
10. C₅×#10597244	Dic₃	direct (N × H)
13. C₃×C₅×C₁₁²	C₄×C₅	semidirect
14. C₃×#10597244	C₄×C₅	semidirect
16. C₁₁²×Dic₃	C₅²	semidirect
23. C₅×C₁₁×Dic₃	C₁₁⋊C₅	semidirect
25. C₅×C₁₁²	C₅×Dic₃	semidirect
26. #10597244	C₅×Dic₃	direct (N × H)
29. C₃×C₁₁²	C₄×C₅²	semidirect
40. C₃×C₅×C₁₁	C₄×C₁₁⋊C₅	semidirect
44. C₁₁×Dic₃	C₅×C₁₁⋊C₅	semidirect
45. C₁₁²	C₅²×Dic₃	semidirect
54. C₅×Dic₃	#10597244	direct (N × H)
56. C₅×C₁₁	Dic₃×C₁₁⋊C₅	semidirect
62. C₃×C₁₁	C₄×C₅×C₁₁⋊C₅	semidirect
71. C₃×C₅	C₄×#10597244	semidirect
72. Dic₃	C₅×#10597244	direct (N × H)
74. C₁₁	C₅×Dic₃×C₁₁⋊C₅	semidirect
79. C₅	Dic₃×#10597244	direct (N × H)
81. C₃	C₄×C₅×#10597244	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#10597244	2
C₄×C₅×#10597244	3
C₅×C₁₁²×Dic₃	5
Dic₃×#10597244	5
C₅×Dic₃×C₁₁⋊C₅	11

Subgroups — C5×Dic3×#10597244 #49137263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×Dic₃×#10597244 #49137263