Subgroups — C₂²×ΓL(1,𝔽₂₅) #5737

Order 192.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×ΓL(1,𝔽₂₅)	C₂	1	12	yes	maximal, direct product factor
5	C₂²×M₁₆	—	3	1	no	maximal, Sylow
6	C₂³×D₃	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
8	C₂²×Dic₃	C₄	1	1	yes	characteristic
9	C₂×C₄×D₃	C₄	1	6	yes	—
10	C₂×C₄×D₃	C₂²	1	6	yes	—
11	C₂×C₃×C₈	C₂²	1	6	yes	complemented
12	C₂×C₃⋊C₈	C₂²	1	6	yes	complemented
13	ΓL(1,𝔽₂₅)	C₂²	1	16	yes	direct product factor
14	C₂³×C₄	—	3	1	no	—
15	C₂×M₁₆	—	3	12	no	—
16	C₂²×C₈	—	3	1	no	—
17	C₂²×C₈	D₃	1	1	yes	characteristic, complemented
18	C₂³×C₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×D₃	—	2	4	no	—
20	C₂²×D₃	C₂×C₄	1	6	yes	—
21	C₂×C₃×C₄	C₂×C₄	1	3	yes	—
22	C₂×C₃×C₄	C₂³	1	3	yes	—
23	C₂×Dic₃	C₂×C₄	1	6	yes	—
24	C₄×D₃	C₂×C₄	1	12	yes	—
25	C₄×D₃	C₂³	1	4	yes	—
26	C₃×C₈	C₂³	1	4	yes	complemented
27	C₃⋊C₈	C₂³	1	4	yes	complemented
28	C₂⁴	—	3	1	no	—
29	C₂²×C₄	—	3	1	no	—
30	C₂²×C₄	—	3	6	no	—
31	C₂²×C₄	—	3	6	no	—
32	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
33	M₁₆	—	3	16	no	—
34	C₂×C₈	—	3	6	no	—
35	C₂×C₈	C₂×D₃	1	6	yes	complemented
36	C₂²×C₃	M₁₆	1	4	yes	complemented
37	C₂²×C₃	C₂²×C₄	1	3	yes	—
38	C₂×D₃	—	2	12	no	—
39	C₂×D₃	C₂²×C₄	1	4	yes	—
40	C₃×C₄	C₂²×C₄	1	3	yes	—
41	C₃×C₄	C₂⁴	1	1	yes	characteristic
42	Dic₃	C₂²×C₄	1	4	yes	—
43	C₂³	—	3	6	no	—
44	C₂³	—	6	4	no	—
45	C₂³	C₄×D₃	1	1	yes	characteristic
46	C₂×C₄	—	3	6	no	—
47	C₂×C₄	—	3	12	no	—
48	C₂×C₄	—	3	4	no	—
49	C₂×C₄	C₄×D₃	1	3	yes	—
50	C₂×C₄	C₂²×D₃	1	3	yes	—
51	C₈	—	3	4	no	—
52	C₈	C₂²×D₃	1	4	yes	complemented
53	C₂×C₃	C₂×M₁₆	1	6	yes	complemented
54	C₂×C₃	C₂³×C₄	1	1	yes	characteristic
55	D₃	—	2	4	no	—
56	C₂²	—	3	4	no	—
57	C₂²	—	6	12	no	—
58	C₂²	ΓL(1,𝔽₂₅)	1	4	yes	direct product factor
59	C₂²	C₂×C₄×D₃	1	3	yes	—
60	C₄	—	3	4	no	—
61	C₄	C₂×C₄×D₃	1	3	yes	—
62	C₄	C₂³×D₃	1	1	yes	characteristic
63	C₃	C₂²×M₁₆	1	1	yes	characteristic, complemented, Sylow
64	C₂	—	6	4	no	—
65	C₂	C₂×ΓL(1,𝔽₂₅)	1	6	yes	direct product factor
66	C₂	C₂²×C₄×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃
commutator	C₂×C₃	C₂³×C₄
Frattini	C₄	C₂³×D₃
Fitting	C₂²×C₃×C₈	C₂
socle	C₂³×C₃	C₂×C₄
prime radical	C₂³×D₃	C₄
cosocle	C₃×C₄	C₂⁴
hypercenter	C₂²×C₈	D₃
hypocenter	C₃	C₂²×M₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×C₈	C₂	semidirect
3. C₂²×C₃⋊C₈	C₂	semidirect
4. C₂×ΓL(1,𝔽₂₅)	C₂	direct (N × H)
11. C₂×C₃×C₈	C₂²	semidirect
12. C₂×C₃⋊C₈	C₂²	semidirect
13. ΓL(1,𝔽₂₅)	C₂²	direct (N × H)
17. C₂²×C₈	D₃	semidirect
26. C₃×C₈	C₂³	semidirect
27. C₃⋊C₈	C₂³	semidirect
35. C₂×C₈	C₂×D₃	semidirect
36. C₂²×C₃	M₁₆	semidirect
52. C₈	C₂²×D₃	semidirect
53. C₂×C₃	C₂×M₁₆	semidirect
58. C₂²	ΓL(1,𝔽₂₅)	direct (N × H)
63. C₃	C₂²×M₁₆	semidirect
65. C₂	C₂×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
C₂²×C₄×D₃	2
C₂²×C₃×C₈	2
C₂²×C₃⋊C₈	2
C₂×ΓL(1,𝔽₂₅)	2
C₂²×M₁₆	3

Subgroups — C22×ΓL(1,𝔽25) #5737

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×ΓL(1,𝔽₂₅) #5737