Subgroups — C₂×#30654824 #45658298

Order 32700.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²⋊C₃×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#30654824	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₅⋊D₅₄₅	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₅²⋊C₃×C₁₀₉	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₅⋊D₅₄₅	C₂×C₃	1	2	yes	complemented
6	C₂×C₅²×C₁₀₉	C₂×C₃	1	1	yes	characteristic, complemented
7	C₅²×C₁₀₉	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
8	C₂×D₅₄₅	—	15	2	no	—
9	C₂×C₃×D₁₀₉	—	25	1	no	maximal, Hall
10	C₂×C₅×C₁₀₉	—	3	2	no	—
11	D₅₄₅	—	15	4	no	—
12	C₂×C₃×C₁₀₉	—	25	1	no	—
13	C₃×D₁₀₉	—	25	2	no	—
14	C₅×C₁₀₉	—	3	2	no	—
15	C₂×D₁₀₉	—	25	1	no	Hall
16	C₃×C₁₀₉	—	25	1	no	Hall
17	C₂×(C₅⋊D₅)⋊C₃	—	109	1	no	maximal, Hall
18	C₂×C₁₀₉	(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented
19	D₁₀₉	—	25	2	no	—
20	C₂×C₅²⋊C₃	D₁₀₉	1	1	yes	characteristic, complemented
21	(C₅⋊D₅)⋊C₃	—	109	2	no	—
22	C₁₀₉	C₂×(C₅⋊D₅)⋊C₃	1	1	yes	characteristic, complemented, Sylow
23	C₂×C₅⋊D₅	—	109	1	no	Hall
24	C₅²⋊C₃	C₂×D₁₀₉	1	1	yes	characteristic, complemented, Hall
25	C₂×C₅²	C₃×D₁₀₉	1	1	yes	characteristic, complemented
26	C₅⋊D₅	—	109	2	no	—
27	C₅²	C₂×C₃×D₁₀₉	1	1	yes	characteristic, complemented, Sylow
28	C₂×D₅	—	1635	2	no	—
29	C₂²×C₃	—	2725	1	no	Hall
30	C₂×C₅	—	3	2	no	—
31	D₅	—	1635	4	no	—
32	C₂×C₃	—	25	1	no	—
33	C₂×C₃	—	2725	2	no	—
34	C₅	—	3	2	no	—
35	C₂²	—	2725	1	no	Sylow
36	C₃	—	25	1	no	Sylow
37	C₂	—	2725	2	no	—
38	C₂	#30654824	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	#30654824
commutator, cosocle, hypocenter	C₅²×C₁₀₉	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₅²×C₁₀₉	C₂×C₃
LCS	G ▷ C₅²×C₁₀₉
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×C₅²⋊C₃×C₁₀₉	C₂	semidirect
2. #30654824	C₂	direct (N × H)
3. C₂×C₅⋊D₅₄₅	C₃	semidirect
4. C₅²⋊C₃×C₁₀₉	C₂²	semidirect
5. C₅⋊D₅₄₅	C₂×C₃	semidirect
6. C₂×C₅²×C₁₀₉	C₂×C₃	semidirect
7. C₅²×C₁₀₉	C₂²×C₃	semidirect
18. C₂×C₁₀₉	(C₅⋊D₅)⋊C₃	semidirect
20. C₂×C₅²⋊C₃	D₁₀₉	semidirect
22. C₁₀₉	C₂×(C₅⋊D₅)⋊C₃	semidirect
24. C₅²⋊C₃	C₂×D₁₀₉	semidirect
25. C₂×C₅²	C₃×D₁₀₉	semidirect
27. C₅²	C₂×C₃×D₁₀₉	semidirect
38. C₂	#30654824	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅²⋊C₃×C₁₀₉	2
#30654824	2
C₂×C₅⋊D₅₄₅	3
C₂×C₃×D₁₀₉	25
C₂×(C₅⋊D₅)⋊C₃	109

Subgroups — C2×#30654824 #45658298

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#30654824 #45658298