Subgroups — C₂⁵×(C₃⋊D₃)⋊C₃ #13180120

Order 1728.

20523 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×(C₃⋊D₃)⋊C₃	C₂	1	62	yes	maximal, direct product factor
3	C₂⁵×C₃⋊D₃	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂⁵×C₃×D₃	—	3	1	no	maximal
5	C₂⁴×H(3,𝔽₃)	C₂²	1	31	yes	complemented
6	C₂³×(C₃⋊D₃)⋊C₃	C₂²	1	620	yes	direct product factor
7	C₂⁵×C₃²	—	2	1	no	—
8	C₂⁵×C₃²	D₃	1	1	yes	characteristic, complemented
9	C₂⁵×C₃²	C₂×C₃	1	1	yes	characteristic, complemented
10	C₂⁴×C₃⋊D₃	C₂×C₃	1	62	yes	complemented
11	C₂⁴×C₃×D₃	—	3	62	no	—
12	C₂³×H(3,𝔽₃)	C₂³	1	155	yes	complemented
13	C₂²×(C₃⋊D₃)⋊C₃	C₂³	1	1240	yes	direct product factor
14	C₂⁶×C₃	—	9	1	no	—
15	C₂⁵×D₃	—	3	1	no	—
16	C₂⁵×D₃	—	9	1	no	—
17	C₂⁴×C₃²	—	2	31	no	—
18	C₂⁴×C₃²	C₂×D₃	1	31	yes	complemented
19	C₂⁴×C₃²	C₂²×C₃	1	31	yes	complemented
20	C₂³×C₃⋊D₃	C₂²×C₃	1	620	yes	complemented
21	C₂³×C₃×D₃	—	3	620	no	—
22	C₂²×H(3,𝔽₃)	C₂⁴	1	155	yes	complemented
23	C₂×(C₃⋊D₃)⋊C₃	C₂⁴	1	496	yes	direct product factor
24	C₂⁵×C₃	—	3	1	no	—
25	C₂⁵×C₃	—	3	1	no	—
26	C₂⁵×C₃	—	6	1	no	—
27	C₂⁵×C₃	—	9	62	no	—
28	C₂⁵×C₃	C₃×D₃	1	1	yes	characteristic
29	C₂⁴×D₃	—	3	62	no	—
30	C₂⁴×D₃	—	9	62	no	—
31	C₂³×C₃²	—	2	155	no	—
32	C₂³×C₃²	C₂²×D₃	1	155	yes	complemented
33	C₂³×C₃²	C₂³×C₃	1	155	yes	complemented
34	C₂²×C₃⋊D₃	C₂³×C₃	1	1240	yes	complemented
35	C₂²×C₃×D₃	—	3	1240	no	—
36	C₂⁶	—	9	1	no	Sylow
37	C₂×H(3,𝔽₃)	C₂⁵	1	31	yes	complemented
38	(C₃⋊D₃)⋊C₃	C₂⁵	1	32	yes	direct product factor
39	C₂⁴×C₃	—	3	31	no	—
40	C₂⁴×C₃	—	3	31	no	—
41	C₂⁴×C₃	—	6	31	no	—
42	C₂⁴×C₃	—	9	620	no	—
43	C₂⁴×C₃	C₂×C₃×D₃	1	31	yes	—
44	C₂³×D₃	—	3	620	no	—
45	C₂³×D₃	—	9	620	no	—
46	C₂²×C₃²	—	2	155	no	—
47	C₂²×C₃²	C₂³×D₃	1	155	yes	complemented
48	C₂²×C₃²	C₂⁴×C₃	1	155	yes	complemented
49	C₂×C₃⋊D₃	C₂⁴×C₃	1	496	yes	complemented
50	C₂×C₃×D₃	—	3	496	no	—
51	C₂⁵	—	9	62	no	—
52	C₂⁵	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic, direct product factor
53	H(3,𝔽₃)	C₂⁶	1	1	yes	characteristic, complemented, Sylow
54	C₂³×C₃	—	3	155	no	—
55	C₂³×C₃	—	3	155	no	—
56	C₂³×C₃	—	6	155	no	—
57	C₂³×C₃	—	9	1240	no	—
58	C₂³×C₃	C₂²×C₃×D₃	1	155	yes	—
59	C₂²×D₃	—	3	1240	no	—
60	C₂²×D₃	—	9	1240	no	—
61	C₂×C₃²	—	2	31	no	—
62	C₂×C₃²	C₂⁴×D₃	1	31	yes	complemented
63	C₂×C₃²	C₂⁵×C₃	1	31	yes	complemented
64	C₃⋊D₃	C₂⁵×C₃	1	32	yes	complemented
65	C₃×D₃	—	3	32	no	—
66	C₂⁴	—	9	620	no	—
67	C₂⁴	C₂×(C₃⋊D₃)⋊C₃	1	31	yes	direct product factor
68	C₂²×C₃	—	3	155	no	—
69	C₂²×C₃	—	3	155	no	—
70	C₂²×C₃	—	6	155	no	—
71	C₂²×C₃	—	9	496	no	—
72	C₂²×C₃	C₂³×C₃×D₃	1	155	yes	—
73	C₂×D₃	—	3	496	no	—
74	C₂×D₃	—	9	496	no	—
75	C₃²	—	2	1	no	—
76	C₃²	C₂⁵×D₃	1	1	yes	characteristic, complemented
77	C₃²	C₂⁶×C₃	1	1	yes	characteristic, complemented
78	C₂³	—	9	1240	no	—
79	C₂³	C₂²×(C₃⋊D₃)⋊C₃	1	155	yes	direct product factor
80	C₂×C₃	—	3	31	no	—
81	C₂×C₃	—	3	31	no	—
82	C₂×C₃	—	6	31	no	—
83	C₂×C₃	—	9	32	no	—
84	C₂×C₃	C₂⁴×C₃×D₃	1	31	yes	—
85	D₃	—	3	32	no	—
86	D₃	—	9	32	no	—
87	C₂²	—	9	496	no	—
88	C₂²	C₂³×(C₃⋊D₃)⋊C₃	1	155	yes	direct product factor
89	C₃	—	3	1	no	—
90	C₃	—	3	1	no	—
91	C₃	—	6	1	no	—
92	C₃	C₂⁵×C₃×D₃	1	1	yes	characteristic
93	C₂	—	9	32	no	—
94	C₂	C₂⁴×(C₃⋊D₃)⋊C₃	1	31	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂⁵	(C₃⋊D₃)⋊C₃
commutator, cosocle, hypocenter	C₃²	C₂⁶×C₃
Frattini	C₃	C₂⁵×C₃×D₃
Fitting	C₂⁵×H(3,𝔽₃)	C₂
socle	C₂⁵×C₃	C₃×D₃
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂⁵

Complements

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

N	H ≅ G/N	type
1. C₂⁵×H(3,𝔽₃)	C₂	semidirect
2. C₂⁴×(C₃⋊D₃)⋊C₃	C₂	direct (N × H)
3. C₂⁵×C₃⋊D₃	C₃	semidirect
5. C₂⁴×H(3,𝔽₃)	C₂²	semidirect
6. C₂³×(C₃⋊D₃)⋊C₃	C₂²	direct (N × H)
8. C₂⁵×C₃²	D₃	semidirect
9. C₂⁵×C₃²	C₂×C₃	semidirect
10. C₂⁴×C₃⋊D₃	C₂×C₃	semidirect
12. C₂³×H(3,𝔽₃)	C₂³	semidirect
13. C₂²×(C₃⋊D₃)⋊C₃	C₂³	direct (N × H)
18. C₂⁴×C₃²	C₂×D₃	semidirect
19. C₂⁴×C₃²	C₂²×C₃	semidirect
20. C₂³×C₃⋊D₃	C₂²×C₃	semidirect
22. C₂²×H(3,𝔽₃)	C₂⁴	semidirect
23. C₂×(C₃⋊D₃)⋊C₃	C₂⁴	direct (N × H)
32. C₂³×C₃²	C₂²×D₃	semidirect
33. C₂³×C₃²	C₂³×C₃	semidirect
34. C₂²×C₃⋊D₃	C₂³×C₃	semidirect
37. C₂×H(3,𝔽₃)	C₂⁵	semidirect
38. (C₃⋊D₃)⋊C₃	C₂⁵	direct (N × H)
47. C₂²×C₃²	C₂³×D₃	semidirect
48. C₂²×C₃²	C₂⁴×C₃	semidirect
49. C₂×C₃⋊D₃	C₂⁴×C₃	semidirect
52. C₂⁵	(C₃⋊D₃)⋊C₃	direct (N × H)
53. H(3,𝔽₃)	C₂⁶	semidirect
62. C₂×C₃²	C₂⁴×D₃	semidirect
63. C₂×C₃²	C₂⁵×C₃	semidirect
64. C₃⋊D₃	C₂⁵×C₃	semidirect
67. C₂⁴	C₂×(C₃⋊D₃)⋊C₃	direct (N × H)
76. C₃²	C₂⁵×D₃	semidirect
77. C₃²	C₂⁶×C₃	semidirect
79. C₂³	C₂²×(C₃⋊D₃)⋊C₃	direct (N × H)
88. C₂²	C₂³×(C₃⋊D₃)⋊C₃	direct (N × H)
94. C₂	C₂⁴×(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
C₂⁵×H(3,𝔽₃)	2
C₂⁴×(C₃⋊D₃)⋊C₃	2
C₂⁵×C₃⋊D₃	3
C₂⁵×C₃×D₃	3

Subgroups — C25×(C3⋊D3)⋊C3 #13180120

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁵×(C₃⋊D₃)⋊C₃ #13180120