Subgroups — C₂×#3743 #64109

Order 288.

204 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₃⋊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	#3743	C₂	1	4	yes	maximal, direct product factor
5	C₂×ΓL(1,𝔽₂₅)	—	3	4	no	maximal
6	C₂²×C₃⋊D₃	C₄	1	1	yes	characteristic
7	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
8	C₂×C₃⋊Dic₃	C₄	1	1	yes	characteristic
9	C₄×C₃⋊D₃	C₄	1	2	yes	—
10	C₄×C₃⋊D₃	C₂²	1	2	yes	—
11	C₃²×C₈	C₂²	1	2	yes	complemented
12	C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	complemented
13	C₂×C₄×D₃	—	3	4	no	—
14	C₂×C₃×C₈	D₃	1	4	yes	complemented
15	C₂×C₃⋊C₈	—	3	4	no	—
16	ΓL(1,𝔽₂₅)	—	3	16	no	—
17	C₂²×C₃²	C₂×C₄	1	1	yes	characteristic
18	C₂×C₃⋊D₃	—	2	2	no	—
19	C₂×C₃⋊D₃	C₂×C₄	1	2	yes	—
20	C₃²×C₄	C₂×C₄	1	1	yes	characteristic
21	C₃²×C₄	C₂³	1	1	yes	characteristic
22	C₃⋊Dic₃	C₂×C₄	1	2	yes	—
23	C₂×M₁₆	—	9	1	no	Sylow
24	C₂²×D₃	—	3	4	no	—
25	C₂×C₃×C₄	C₂×D₃	1	4	yes	—
26	C₂×Dic₃	—	3	4	no	—
27	C₄×D₃	—	3	8	no	—
28	C₄×D₃	—	3	8	no	—
29	C₃×C₈	C₂×D₃	1	8	yes	complemented
30	C₃⋊C₈	—	3	8	no	—
31	C₂×C₃²	M₁₆	1	2	yes	complemented
32	C₂×C₃²	C₂²×C₄	1	1	yes	characteristic
33	C₃⋊D₃	—	2	2	no	—
34	C₂²×C₄	—	9	1	no	—
35	M₁₆	—	9	4	no	—
36	C₂×C₈	—	9	1	no	—
37	C₂×C₈	C₃⋊D₃	1	1	yes	characteristic, complemented
38	C₂²×C₃	C₄×D₃	1	4	yes	—
39	C₂×D₃	—	3	8	no	—
40	C₂×D₃	—	6	8	no	—
41	C₃×C₄	C₄×D₃	1	4	yes	—
42	C₃×C₄	C₂²×D₃	1	4	yes	—
43	Dic₃	—	3	8	no	—
44	C₃²	C₂×M₁₆	1	1	yes	characteristic, complemented, Sylow
45	C₂³	—	9	1	no	—
46	C₂×C₄	—	9	1	no	—
47	C₂×C₄	—	9	2	no	—
48	C₂×C₄	—	9	2	no	—
49	C₂×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
50	C₈	—	9	2	no	—
51	C₈	C₂×C₃⋊D₃	1	2	yes	complemented
52	C₂×C₃	ΓL(1,𝔽₂₅)	1	8	yes	complemented
53	C₂×C₃	C₂×C₄×D₃	1	4	yes	—
54	D₃	—	6	8	no	—
55	C₂²	—	9	2	no	—
56	C₂²	—	18	2	no	—
57	C₂²	C₄×C₃⋊D₃	1	1	yes	characteristic
58	C₄	—	9	2	no	—
59	C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
60	C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
61	C₃	C₂×ΓL(1,𝔽₂₅)	1	4	yes	complemented
62	C₂	—	18	2	no	—
63	C₂	#3743	1	2	yes	direct product factor
64	C₂	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₃⋊D₃
commutator	C₂×C₃²	C₂²×C₄
Frattini	C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×C₈	C₂
socle	C₂²×C₃²	C₂×C₄
prime radical	C₂²×C₃⋊D₃	C₄
cosocle	C₃²×C₄	C₂³
hypercenter	C₂×C₈	C₃⋊D₃
hypocenter	C₃²	C₂×M₁₆
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×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. #3743	C₂	direct (N × H)
11. C₃²×C₈	C₂²	semidirect
12. C₃⋊(C₃⋊C₈)	C₂²	semidirect
14. C₂×C₃×C₈	D₃	semidirect
29. C₃×C₈	C₂×D₃	semidirect
31. C₂×C₃²	M₁₆	semidirect
37. C₂×C₈	C₃⋊D₃	semidirect
44. C₃²	C₂×M₁₆	semidirect
51. C₈	C₂×C₃⋊D₃	semidirect
52. C₂×C₃	ΓL(1,𝔽₂₅)	semidirect
61. C₃	C₂×ΓL(1,𝔽₂₅)	semidirect
63. C₂	#3743	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×C₄×C₃⋊D₃	2
C₂×C₃²×C₈	2
C₂×C₃⋊(C₃⋊C₈)	2
#3743	2
C₂×ΓL(1,𝔽₂₅)	3

Subgroups — C2×#3743 #64109

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#3743 #64109