Subgroups — C₂×C₃⋊(C₃⋊(D₄₁⋊C₂₅)) #49146708

Order 36900.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₄₁⋊C₂₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃⋊(C₃⋊(D₄₁⋊C₂₅))	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃⋊(D₄₁⋊C₂₅)	—	3	4	no	maximal
4	C₃²×C₄₁⋊C₂₅	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₅×C₃⋊D₁₂₃	C₅	1	1	yes	maximal, characteristic
6	C₂×C₃×C₄₁⋊C₂₅	D₃	1	4	yes	complemented
7	C₃⋊(D₄₁⋊C₂₅)	—	3	8	no	—
8	C₂×D₄₁⋊C₂₅	—	9	1	no	Hall
9	C₅×C₃⋊D₁₂₃	C₂×C₅	1	2	yes	—
10	C₂×C₃²×C₅×C₄₁	C₂×C₅	1	1	yes	characteristic
11	C₃×C₄₁⋊C₂₅	C₂×D₃	1	4	yes	complemented
12	C₂×C₅×D₁₂₃	—	3	4	no	—
13	C₂×C₄₁⋊C₂₅	C₃⋊D₃	1	1	yes	characteristic, complemented
14	D₄₁⋊C₂₅	—	9	2	no	—
15	C₃²×C₅×C₄₁	C₂²×C₅	1	1	yes	characteristic
16	C₂×C₃⋊D₁₂₃	C₂₅	1	1	yes	characteristic, complemented, Hall
17	C₂×C₃×C₅×C₄₁	C₅×D₃	1	4	yes	—
18	C₅×D₁₂₃	—	3	8	no	—
19	C₄₁⋊C₂₅	C₂×C₃⋊D₃	1	1	yes	characteristic, complemented, Hall
20	C₂×C₃⋊D₃×C₂₅	—	41	1	no	maximal, Hall
21	C₂×C₅×D₄₁	—	9	1	no	—
22	C₂×C₃²×C₄₁	C₂×C₂₅	1	1	yes	characteristic, complemented
23	C₃⋊D₁₂₃	C₂×C₂₅	1	2	yes	complemented
24	C₃×C₅×C₄₁	C₂×C₅×D₃	1	4	yes	—
25	C₂×D₁₂₃	—	3	4	no	—
26	C₂×C₃²×C₂₅	—	41	1	no	—
27	C₃⋊D₃×C₂₅	—	41	2	no	—
28	C₂×C₅×C₄₁	C₅×C₃⋊D₃	1	1	yes	characteristic
29	C₅×D₄₁	—	9	2	no	—
30	C₃²×C₄₁	C₂²×C₂₅	1	1	yes	characteristic, complemented, Hall
31	C₂×D₃×C₂₅	—	123	4	no	—
32	C₂×C₃×C₄₁	D₃×C₂₅	1	4	yes	complemented
33	D₁₂₃	—	3	8	no	—
34	C₃²×C₂₅	—	41	1	no	Hall
35	C₅×C₄₁	C₂×C₅×C₃⋊D₃	1	1	yes	characteristic
36	C₂×C₅×C₃⋊D₃	—	41	1	no	—
37	C₂×D₄₁	—	9	1	no	Hall
38	C₂×C₃×C₂₅	—	41	4	no	—
39	D₃×C₂₅	—	123	8	no	—
40	C₃×C₄₁	C₂×D₃×C₂₅	1	4	yes	complemented
41	C₂²×C₂₅	—	369	1	no	Hall
42	C₂×C₃²×C₅	D₄₁⋊C₅	1	1	yes	characteristic
43	C₅×C₃⋊D₃	—	41	2	no	—
44	C₂×C₄₁	C₃⋊D₃×C₂₅	1	1	yes	characteristic, complemented
45	D₄₁	—	9	2	no	—
46	C₃×C₂₅	—	41	4	no	—
47	C₂×C₅×D₃	—	123	4	no	—
48	C₂×C₂₅	—	41	1	no	—
49	C₂×C₂₅	—	369	2	no	—
50	C₃²×C₅	C₂×D₄₁⋊C₅	1	1	yes	characteristic
51	C₄₁	C₂×C₃⋊D₃×C₂₅	1	1	yes	characteristic, complemented, Sylow
52	C₂×C₃⋊D₃	—	41	1	no	Hall
53	C₂×C₃×C₅	C₃⋊(D₄₁⋊C₅)	1	4	yes	—
54	C₅×D₃	—	123	8	no	—
55	C₂₅	—	41	1	no	Sylow
56	C₂²×C₅	—	369	1	no	—
57	C₂×C₃²	D₄₁⋊C₂₅	1	1	yes	characteristic, complemented
58	C₃⋊D₃	—	41	2	no	—
59	C₃×C₅	C₂×C₃⋊(D₄₁⋊C₅)	1	4	yes	—
60	C₂×D₃	—	123	4	no	—
61	C₂×C₅	—	369	2	no	—
62	C₂×C₅	C₃⋊(C₃⋊(D₄₁⋊C₅))	1	1	yes	characteristic
63	C₃²	C₂×D₄₁⋊C₂₅	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₃	C₃⋊(D₄₁⋊C₂₅)	1	4	yes	complemented
65	D₃	—	123	8	no	—
66	C₅	C₂×C₃⋊(C₃⋊(D₄₁⋊C₅))	1	1	yes	characteristic
67	C₂²	—	369	1	no	Sylow
68	C₃	C₂×C₃⋊(D₄₁⋊C₂₅)	1	4	yes	complemented
69	C₂	—	369	2	no	—
70	C₂	C₃⋊(C₃⋊(D₄₁⋊C₂₅))	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₅	C₃⋊(C₃⋊(D₄₁⋊C₅))
commutator, hypocenter	C₃²×C₄₁	C₂²×C₂₅
Frattini	C₅	C₂×C₃⋊(C₃⋊(D₄₁⋊C₅))
Fitting, socle	C₂×C₃²×C₅×C₄₁	C₂×C₅
prime radical	C₂×C₅×C₃⋊D₁₂₃	C₅
cosocle	C₃²×C₅×C₄₁	C₂²×C₅
LCS	G ▷ C₃²×C₄₁
UCS	C₁ ◁ C₂×C₅

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×C₄₁⋊C₂₅	C₂	semidirect
2. C₃⋊(C₃⋊(D₄₁⋊C₂₅))	C₂	direct (N × H)
4. C₃²×C₄₁⋊C₂₅	C₂²	semidirect
6. C₂×C₃×C₄₁⋊C₂₅	D₃	semidirect
11. C₃×C₄₁⋊C₂₅	C₂×D₃	semidirect
13. C₂×C₄₁⋊C₂₅	C₃⋊D₃	semidirect
16. C₂×C₃⋊D₁₂₃	C₂₅	semidirect
19. C₄₁⋊C₂₅	C₂×C₃⋊D₃	semidirect
22. C₂×C₃²×C₄₁	C₂×C₂₅	semidirect
23. C₃⋊D₁₂₃	C₂×C₂₅	semidirect
30. C₃²×C₄₁	C₂²×C₂₅	semidirect
32. C₂×C₃×C₄₁	D₃×C₂₅	semidirect
40. C₃×C₄₁	C₂×D₃×C₂₅	semidirect
44. C₂×C₄₁	C₃⋊D₃×C₂₅	semidirect
51. C₄₁	C₂×C₃⋊D₃×C₂₅	semidirect
57. C₂×C₃²	D₄₁⋊C₂₅	semidirect
63. C₃²	C₂×D₄₁⋊C₂₅	semidirect
64. C₂×C₃	C₃⋊(D₄₁⋊C₂₅)	semidirect
68. C₃	C₂×C₃⋊(D₄₁⋊C₂₅)	semidirect
70. C₂	C₃⋊(C₃⋊(D₄₁⋊C₂₅))	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×C₃²×C₄₁⋊C₂₅	2
C₃⋊(C₃⋊(D₄₁⋊C₂₅))	2
C₂×C₃⋊(D₄₁⋊C₂₅)	3
C₂×C₅×C₃⋊D₁₂₃	5
C₂×C₃⋊D₃×C₂₅	41

Subgroups — C2×C3⋊(C3⋊(D41⋊C25)) #49146708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃⋊(C₃⋊(D₄₁⋊C₂₅)) #49146708