Subgroups — SL(2,𝔽₃)×D₂₇ #13047029

Order 1296.

81 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	SL(2,𝔽₃)×C₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	D₉×SL(2,𝔽₃)	—	3	1	no	maximal
3	Q₈×D₂₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×D₂₇	—	4	1	no	maximal
5	C₉×SL(2,𝔽₃)	D₃	1	1	yes	characteristic
6	Q₈×C₂₇	C₂×C₃	1	1	yes	characteristic, complemented
7	C₄×D₂₇	—	3	1	no	—
8	Dic₅₄	—	3	1	no	—
9	Q₈⋊C₂₇	—	2	1	no	—
10	C₂×C₃×C₂₇	—	4	1	no	—
11	C₃×D₂₇	—	4	2	no	—
12	D₃×SL(2,𝔽₃)	—	9	1	no	—
13	Q₈×D₉	—	3	1	no	—
14	C₂×C₃×D₉	—	12	1	no	—
15	C₂×D₂₇	A₄	1	1	yes	characteristic
16	C₄×C₂₇	—	3	1	no	—
17	Dic₂₇	—	3	1	no	—
18	C₃×C₂₇	—	4	1	no	Sylow
19	C₃×SL(2,𝔽₃)	D₉	1	1	yes	characteristic
20	Q₈×C₉	C₃×D₃	1	1	yes	characteristic
21	C₄×D₉	—	9	1	no	—
22	Dic₁₈	—	9	1	no	—
23	Q₈⋊C₉	—	2	1	no	—
24	C₂×C₃×C₉	—	4	1	no	—
25	C₃×D₉	—	12	2	no	—
26	C₂×C₂₇	—	8	1	no	—
27	C₂×C₂₇	C₂×A₄	1	1	yes	characteristic
28	D₂₇	SL(2,𝔽₃)	1	2	yes	direct product factor
29	D₃×Q₈	—	9	1	no	—
30	C₂×SL(2,𝔽₃)	—	27	1	no	—
31	C₂×C₃×D₃	—	36	1	no	—
32	C₂×D₉	—	3	1	no	—
33	C₄×C₉	—	3	1	no	—
34	Dic₉	—	9	1	no	—
35	C₃×C₉	—	4	1	no	—
36	C₂₇	—	8	1	no	—
37	C₂₇	C₂×SL(2,𝔽₃)	1	1	yes	characteristic, complemented
38	C₃×Q₈	C₃×D₉	1	1	yes	characteristic
39	C₄×D₃	—	27	1	no	—
40	Dic₆	—	27	1	no	—
41	SL(2,𝔽₃)	—	2	1	no	—
42	SL(2,𝔽₃)	D₂₇	1	1	yes	characteristic, direct product factor
43	C₂×C₃²	—	4	1	no	—
44	C₃×D₃	—	36	2	no	—
45	C₂×C₉	—	8	1	no	—
46	C₂×C₉	D₃×A₄	1	1	yes	characteristic
47	D₉	—	3	2	no	—
48	C₂×Q₈	—	27	1	no	Sylow
49	C₂²×C₃	—	108	1	no	—
50	C₂×D₃	—	9	1	no	—
51	C₃×C₄	—	3	1	no	—
52	Dic₃	—	27	1	no	—
53	C₃²	—	4	1	no	—
54	C₉	—	8	1	no	—
55	C₉	D₃×SL(2,𝔽₃)	1	1	yes	characteristic
56	Q₈	—	81	1	no	—
57	Q₈	C₃×D₂₇	1	1	yes	characteristic, complemented
58	C₂×C₄	—	81	1	no	—
59	C₂×C₃	—	4	1	no	—
60	C₂×C₃	—	8	1	no	—
61	C₂×C₃	—	108	2	no	—
62	C₂×C₃	A₄×D₉	1	1	yes	characteristic
63	D₃	—	9	2	no	—
64	C₂²	—	27	1	no	—
65	C₄	—	3	1	no	—
66	C₄	—	81	1	no	—
67	C₃	—	4	1	no	—
68	C₃	—	8	1	no	—
69	C₃	D₉×SL(2,𝔽₃)	1	1	yes	characteristic
70	C₂	—	27	2	no	—
71	C₂	A₄×D₂₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂	A₄×D₂₇
commutator, Fitting, cosocle, hypocenter	Q₈×C₂₇	C₂×C₃
Frattini	C₂×C₉	D₃×A₄
socle	C₂×C₃	A₄×D₉
LCS	G ▷ Q₈×C₂₇
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. SL(2,𝔽₃)×C₂₇	C₂	semidirect
3. Q₈×D₂₇	C₃	semidirect
6. Q₈×C₂₇	C₂×C₃	semidirect
28. D₂₇	SL(2,𝔽₃)	direct (N × H)
37. C₂₇	C₂×SL(2,𝔽₃)	semidirect
42. SL(2,𝔽₃)	D₂₇	direct (N × H)
57. Q₈	C₃×D₂₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
SL(2,𝔽₃)×C₂₇	2
D₉×SL(2,𝔽₃)	3
Q₈×D₂₇	3
C₂×C₃×D₂₇	4

Subgroups — SL(2,𝔽3)×D27 #13047029

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — SL(2,𝔽₃)×D₂₇ #13047029