Subgroups — C₂²×C₅⋊((C₃⋊D₃)⋊C₃) #11762008

Order 1080.

244 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅⋊((C₃⋊D₃)⋊C₃)	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃⋊D₁₅	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₃×D₁₅	—	3	1	no	maximal
5	C₂×C₅×H(3,𝔽₃)	C₂²	1	3	yes	complemented
6	C₅⋊((C₃⋊D₃)⋊C₃)	C₂²	1	4	yes	direct product factor
7	C₂²×(C₃⋊D₃)⋊C₃	—	5	1	no	maximal, Hall
8	C₂²×C₃²×C₅	—	2	1	no	—
9	C₂²×C₃²×C₅	D₃	1	1	yes	characteristic, complemented
10	C₂²×C₃²×C₅	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂×C₃⋊D₁₅	C₂×C₃	1	6	yes	complemented
12	C₂×C₃×D₁₅	—	3	6	no	—
13	C₅×H(3,𝔽₃)	C₂³	1	1	yes	characteristic, complemented, Hall
14	C₂²×D₁₅	—	3	1	no	—
15	C₂²×D₁₅	—	9	1	no	—
16	C₂²×C₃×D₅	—	9	1	no	—
17	C₂²×H(3,𝔽₃)	D₅	1	1	yes	characteristic, complemented
18	C₂×(C₃⋊D₃)⋊C₃	—	5	6	no	—
19	C₂×C₃²×C₅	—	2	3	no	—
20	C₂×C₃²×C₅	C₂×D₃	1	3	yes	complemented
21	C₂×C₃²×C₅	C₂²×C₃	1	3	yes	complemented
22	C₃⋊D₁₅	C₂²×C₃	1	4	yes	complemented
23	C₃×D₁₅	—	3	4	no	—
24	C₂²×C₃⋊D₃	—	5	1	no	—
25	C₂²×C₃×D₃	—	15	1	no	—
26	C₂²×C₃×C₅	—	3	1	no	—
27	C₂²×C₃×C₅	—	3	1	no	—
28	C₂²×C₃×C₅	—	6	1	no	—
29	C₂²×C₃×C₅	C₃×D₃	1	1	yes	characteristic
30	C₂×D₁₅	—	3	6	no	—
31	C₂×D₁₅	—	9	6	no	—
32	C₂×C₃×D₅	—	9	6	no	—
33	C₂×H(3,𝔽₃)	C₂×D₅	1	3	yes	complemented
34	(C₃⋊D₃)⋊C₃	—	5	4	no	—
35	C₃²×C₅	—	2	1	no	—
36	C₃²×C₅	C₂²×D₃	1	1	yes	characteristic, complemented
37	C₃²×C₅	C₂³×C₃	1	1	yes	characteristic, complemented
38	C₂²×D₅	—	9	1	no	Hall
39	C₂²×C₃²	—	2	1	no	—
40	C₂²×C₃²	C₃×D₅	1	1	yes	characteristic, complemented
41	C₂²×C₃²	D₁₅	1	1	yes	characteristic, complemented
42	C₂×C₃⋊D₃	—	5	6	no	—
43	C₂×C₃×D₃	—	15	6	no	—
44	C₂×C₃×C₅	—	3	3	no	—
45	C₂×C₃×C₅	—	3	3	no	—
46	C₂×C₃×C₅	—	6	3	no	—
47	C₂×C₃×C₅	C₂×C₃×D₃	1	3	yes	—
48	D₁₅	—	3	4	no	—
49	D₁₅	—	9	4	no	—
50	C₃×D₅	—	9	4	no	—
51	H(3,𝔽₃)	C₂²×D₅	1	1	yes	characteristic, complemented, Sylow
52	C₂³×C₃	—	45	1	no	—
53	C₂²×D₃	—	15	1	no	—
54	C₂²×D₃	—	45	1	no	—
55	C₂²×C₅	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic, complemented
56	C₂×D₅	—	9	6	no	—
57	C₂×C₃²	—	2	3	no	—
58	C₂×C₃²	C₂×C₃×D₅	1	3	yes	complemented
59	C₂×C₃²	C₂×D₁₅	1	3	yes	complemented
60	C₃⋊D₃	—	5	4	no	—
61	C₃×D₃	—	15	4	no	—
62	C₃×C₅	—	3	1	no	—
63	C₃×C₅	—	3	1	no	—
64	C₃×C₅	—	6	1	no	—
65	C₃×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
66	C₂²×C₃	—	3	1	no	—
67	C₂²×C₃	—	3	1	no	—
68	C₂²×C₃	—	6	1	no	—
69	C₂²×C₃	—	45	6	no	—
70	C₂²×C₃	C₃×D₁₅	1	1	yes	characteristic
71	C₂×D₃	—	15	6	no	—
72	C₂×D₃	—	45	6	no	—
73	C₂×C₅	C₂×(C₃⋊D₃)⋊C₃	1	3	yes	complemented
74	D₅	—	9	4	no	—
75	C₃²	—	2	1	no	—
76	C₃²	C₂²×C₃×D₅	1	1	yes	characteristic, complemented
77	C₃²	C₂²×D₁₅	1	1	yes	characteristic, complemented
78	C₂³	—	45	1	no	Sylow
79	C₂×C₃	—	3	3	no	—
80	C₂×C₃	—	3	3	no	—
81	C₂×C₃	—	6	3	no	—
82	C₂×C₃	—	45	4	no	—
83	C₂×C₃	C₂×C₃×D₁₅	1	3	yes	—
84	D₃	—	15	4	no	—
85	D₃	—	45	4	no	—
86	C₅	C₂²×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic, complemented, Sylow
87	C₂²	—	45	6	no	—
88	C₂²	C₅⋊((C₃⋊D₃)⋊C₃)	1	1	yes	characteristic, direct product factor
89	C₃	—	3	1	no	—
90	C₃	—	3	1	no	—
91	C₃	—	6	1	no	—
92	C₃	C₂²×C₃×D₁₅	1	1	yes	characteristic
93	C₂	—	45	4	no	—
94	C₂	C₂×C₅⋊((C₃⋊D₃)⋊C₃)	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²	C₅⋊((C₃⋊D₃)⋊C₃)
commutator, cosocle, hypocenter	C₃²×C₅	C₂³×C₃
Frattini	C₃	C₂²×C₃×D₁₅
Fitting	C₂²×C₅×H(3,𝔽₃)	C₂
socle	C₂²×C₃×C₅	C₃×D₃
LCS	G ▷ C₃²×C₅
UCS	C₁ ◁ C₂²

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×H(3,𝔽₃)	C₂	semidirect
2. C₂×C₅⋊((C₃⋊D₃)⋊C₃)	C₂	direct (N × H)
3. C₂²×C₃⋊D₁₅	C₃	semidirect
5. C₂×C₅×H(3,𝔽₃)	C₂²	semidirect
6. C₅⋊((C₃⋊D₃)⋊C₃)	C₂²	direct (N × H)
9. C₂²×C₃²×C₅	D₃	semidirect
10. C₂²×C₃²×C₅	C₂×C₃	semidirect
11. C₂×C₃⋊D₁₅	C₂×C₃	semidirect
13. C₅×H(3,𝔽₃)	C₂³	semidirect
17. C₂²×H(3,𝔽₃)	D₅	semidirect
20. C₂×C₃²×C₅	C₂×D₃	semidirect
21. C₂×C₃²×C₅	C₂²×C₃	semidirect
22. C₃⋊D₁₅	C₂²×C₃	semidirect
33. C₂×H(3,𝔽₃)	C₂×D₅	semidirect
36. C₃²×C₅	C₂²×D₃	semidirect
37. C₃²×C₅	C₂³×C₃	semidirect
40. C₂²×C₃²	C₃×D₅	semidirect
41. C₂²×C₃²	D₁₅	semidirect
51. H(3,𝔽₃)	C₂²×D₅	semidirect
55. C₂²×C₅	(C₃⋊D₃)⋊C₃	semidirect
58. C₂×C₃²	C₂×C₃×D₅	semidirect
59. C₂×C₃²	C₂×D₁₅	semidirect
73. C₂×C₅	C₂×(C₃⋊D₃)⋊C₃	semidirect
76. C₃²	C₂²×C₃×D₅	semidirect
77. C₃²	C₂²×D₁₅	semidirect
86. C₅	C₂²×(C₃⋊D₃)⋊C₃	semidirect
88. C₂²	C₅⋊((C₃⋊D₃)⋊C₃)	direct (N × H)
94. C₂	C₂×C₅⋊((C₃⋊D₃)⋊C₃)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×H(3,𝔽₃)	2
C₂×C₅⋊((C₃⋊D₃)⋊C₃)	2
C₂²×C₃⋊D₁₅	3
C₂²×C₃×D₁₅	3
C₂²×(C₃⋊D₃)⋊C₃	5

Subgroups — C22×C5⋊((C3⋊D3)⋊C3) #11762008

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅⋊((C₃⋊D₃)⋊C₃) #11762008