Subgroups — C₂×C₃×ΓL(1,𝔽₂₅) #64023

Order 288.

147 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×D₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×ΓL(1,𝔽₂₅)	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₃×M₁₆	—	3	1	no	maximal
6	C₂×ΓL(1,𝔽₂₅)	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃×D₃	C₄	1	1	yes	characteristic
8	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
9	C₂×C₃×Dic₃	C₄	1	1	yes	characteristic
10	C₃×C₄×D₃	C₄	1	2	yes	—
11	C₃×C₄×D₃	C₂²	1	2	yes	—
12	C₃²×C₈	C₂²	1	2	yes	complemented
13	C₃×C₃⋊C₈	C₂²	1	2	yes	complemented
14	C₂²×C₃×C₄	—	3	1	no	—
15	C₂×C₄×D₃	C₂×C₃	1	1	yes	characteristic
16	C₃×M₁₆	—	3	4	no	—
17	C₂×C₃×C₈	—	2	1	no	—
18	C₂×C₃×C₈	—	3	1	no	—
19	C₂×C₃×C₈	D₃	1	1	yes	characteristic, complemented
20	C₂×C₃×C₈	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
22	ΓL(1,𝔽₂₅)	C₂×C₃	1	4	yes	direct product factor
23	C₂²×C₃²	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×D₃	—	2	2	no	—
25	C₂×C₃×D₃	C₂×C₄	1	2	yes	—
26	C₃²×C₄	C₂×C₄	1	1	yes	characteristic
27	C₃²×C₄	C₂³	1	1	yes	characteristic
28	C₃×Dic₃	C₂×C₄	1	2	yes	—
29	C₂×M₁₆	—	3	1	no	Sylow
30	C₂³×C₃	—	3	1	no	—
31	C₂²×D₃	C₃×C₄	1	1	yes	characteristic
32	C₂×C₃×C₄	—	2	1	no	—
33	C₂×C₃×C₄	—	3	1	no	—
34	C₂×C₃×C₄	—	3	2	no	—
35	C₂×C₃×C₄	—	3	2	no	—
36	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
37	C₂×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
38	C₂×Dic₃	C₃×C₄	1	1	yes	characteristic
39	C₄×D₃	C₃×C₄	1	2	yes	—
40	C₄×D₃	C₂²×C₃	1	2	yes	—
41	C₃×C₈	—	2	2	no	—
42	C₃×C₈	—	3	2	no	—
43	C₃×C₈	C₂×D₃	1	2	yes	complemented
44	C₃×C₈	C₂²×C₃	1	2	yes	complemented
45	C₃⋊C₈	C₂²×C₃	1	2	yes	complemented
46	C₂×C₃²	M₁₆	1	2	yes	complemented
47	C₂×C₃²	C₂²×C₄	1	1	yes	characteristic
48	C₃×D₃	—	2	2	no	—
49	C₂²×C₄	—	3	1	no	—
50	M₁₆	—	3	4	no	—
51	C₂×C₈	—	3	1	no	—
52	C₂×C₈	C₃×D₃	1	1	yes	characteristic, complemented
53	C₂²×C₃	—	2	1	no	—
54	C₂²×C₃	—	3	2	no	—
55	C₂²×C₃	—	6	2	no	—
56	C₂²×C₃	C₄×D₃	1	1	yes	characteristic
57	C₂²×C₃	C₂×C₃×C₄	1	1	yes	characteristic
58	C₂×D₃	—	2	2	no	—
59	C₂×D₃	C₂×C₃×C₄	1	2	yes	—
60	C₃×C₄	—	2	1	no	—
61	C₃×C₄	—	2	1	no	—
62	C₃×C₄	—	3	2	no	—
63	C₃×C₄	C₄×D₃	1	1	yes	characteristic
64	C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
65	C₃×C₄	C₂²×D₃	1	1	yes	characteristic
66	C₃×C₄	C₂³×C₃	1	1	yes	characteristic
67	Dic₃	C₂×C₃×C₄	1	2	yes	—
68	C₃²	C₂×M₁₆	1	1	yes	characteristic, complemented, Sylow
69	C₂³	—	3	1	no	—
70	C₂×C₄	—	3	1	no	—
71	C₂×C₄	—	3	2	no	—
72	C₂×C₄	—	3	2	no	—
73	C₂×C₄	C₂×C₃×D₃	1	1	yes	characteristic
74	C₈	—	3	2	no	—
75	C₈	C₂×C₃×D₃	1	2	yes	complemented
76	C₂×C₃	—	2	2	no	—
77	C₂×C₃	—	2	1	no	—
78	C₂×C₃	—	6	2	no	—
79	C₂×C₃	ΓL(1,𝔽₂₅)	1	2	yes	direct product factor
80	C₂×C₃	C₃×M₁₆	1	2	yes	complemented
81	C₂×C₃	C₂×C₄×D₃	1	1	yes	characteristic
82	C₂×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
83	D₃	—	2	2	no	—
84	C₂²	—	3	2	no	—
85	C₂²	—	6	2	no	—
86	C₂²	C₃×C₄×D₃	1	1	yes	characteristic
87	C₄	—	3	2	no	—
88	C₄	C₃×C₄×D₃	1	1	yes	characteristic
89	C₄	C₂²×C₃×D₃	1	1	yes	characteristic
90	C₃	—	2	1	no	—
91	C₃	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic, direct product factor
92	C₃	C₂×C₃×M₁₆	1	1	yes	characteristic, complemented
93	C₂	—	6	2	no	—
94	C₂	C₃×ΓL(1,𝔽₂₅)	1	2	yes	direct product factor
95	C₂	C₂×C₃×C₄×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₃
commutator	C₂×C₃	C₂²×C₃×C₄
Frattini	C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×C₈	C₂
socle	C₂²×C₃²	C₂×C₄
prime radical	C₂²×C₃×D₃	C₄
cosocle	C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×C₈	D₃
hypocenter	C₃	C₂×C₃×M₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃²×C₈	C₂	semidirect
3. C₂×C₃×C₃⋊C₈	C₂	semidirect
4. C₃×ΓL(1,𝔽₂₅)	C₂	direct (N × H)
6. C₂×ΓL(1,𝔽₂₅)	C₃	direct (N × H)
12. C₃²×C₈	C₂²	semidirect
13. C₃×C₃⋊C₈	C₂²	semidirect
19. C₂×C₃×C₈	D₃	semidirect
20. C₂×C₃×C₈	C₂×C₃	semidirect
21. C₂×C₃⋊C₈	C₂×C₃	semidirect
22. ΓL(1,𝔽₂₅)	C₂×C₃	direct (N × H)
43. C₃×C₈	C₂×D₃	semidirect
44. C₃×C₈	C₂²×C₃	semidirect
45. C₃⋊C₈	C₂²×C₃	semidirect
46. C₂×C₃²	M₁₆	semidirect
52. C₂×C₈	C₃×D₃	semidirect
68. C₃²	C₂×M₁₆	semidirect
75. C₈	C₂×C₃×D₃	semidirect
79. C₂×C₃	ΓL(1,𝔽₂₅)	direct (N × H)
80. C₂×C₃	C₃×M₁₆	semidirect
91. C₃	C₂×ΓL(1,𝔽₂₅)	direct (N × H)
92. C₃	C₂×C₃×M₁₆	semidirect
94. C₂	C₃×ΓL(1,𝔽₂₅)	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×C₃×C₄×D₃	2
C₂×C₃²×C₈	2
C₂×C₃×C₃⋊C₈	2
C₃×ΓL(1,𝔽₂₅)	2
C₂×C₃×M₁₆	3
C₂×ΓL(1,𝔽₂₅)	3

Subgroups — C2×C3×ΓL(1,𝔽25) #64023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×ΓL(1,𝔽₂₅) #64023