Subgroups — SL(2,𝔽₃)×C₂₇⋊C₃ #14517203

Order 1944.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×SL(2,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	Q₈×C₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
3	#10619570	C₃	1	2	yes	maximal, complemented
4	C₂₇⋊SL(2,𝔽₃)	C₃	1	6	yes	maximal, complemented
5	SL(2,𝔽₃)×C₂₇	C₃	1	3	yes	maximal, complemented
6	C₂×C₃×C₂₇⋊C₃	—	4	1	no	maximal
7	C₄×C₂₇⋊C₃	—	3	1	no	—
8	C₃×C₂₇⋊C₃	—	4	1	no	Sylow
9	C₃²×SL(2,𝔽₃)	C₉	1	1	yes	characteristic
10	C₃×Q₈×C₉	C₃²	1	1	yes	characteristic
11	C₃×Q₈⋊C₉	C₉	1	2	yes	—
12	C₉×SL(2,𝔽₃)	C₉	1	4	yes	—
13	C₉×SL(2,𝔽₃)	C₉	1	2	yes	—
14	C₉×SL(2,𝔽₃)	C₃²	1	1	yes	characteristic
15	C₉×SL(2,𝔽₃)	C₃²	1	2	yes	—
16	Q₈×C₂₇	C₃²	1	3	yes	complemented
17	Q₈⋊C₂₇	C₃²	1	6	yes	complemented
18	C₂×C₃²×C₉	—	4	1	no	—
19	C₂×C₂₇⋊C₃	—	4	2	no	—
20	C₂×C₂₇⋊C₃	—	4	6	no	—
21	C₂×C₂₇⋊C₃	A₄	1	1	yes	characteristic
22	C₂×C₃×C₂₇	—	4	3	no	—
23	C₃×C₄×C₉	—	3	1	no	—
24	C₄×C₂₇	—	3	3	no	—
25	C₃²×C₉	—	4	1	no	—
26	C₂₇⋊C₃	—	4	2	no	—
27	C₂₇⋊C₃	—	4	6	no	—
28	C₂₇⋊C₃	SL(2,𝔽₃)	1	1	yes	characteristic, direct product factor
29	C₃×C₂₇	—	4	3	no	—
30	C₃²×Q₈	C₃×C₉	1	1	yes	characteristic
31	C₃×SL(2,𝔽₃)	—	3	3	no	—
32	C₃×SL(2,𝔽₃)	C₃×C₉	1	2	yes	—
33	C₃×SL(2,𝔽₃)	C₃×C₉	1	1	yes	characteristic
34	Q₈×C₉	C₃×C₉	1	2	yes	—
35	Q₈×C₉	C₃³	1	1	yes	characteristic
36	Q₈⋊C₉	C₃×C₉	1	2	yes	—
37	Q₈⋊C₉	C₃×C₉	1	4	yes	—
38	C₂×C₃³	—	4	1	no	—
39	C₂×C₃×C₉	—	4	2	no	—
40	C₂×C₃×C₉	—	4	2	no	—
41	C₂×C₃×C₉	—	4	4	no	—
42	C₂×C₃×C₉	—	4	1	no	—
43	C₂×C₃×C₉	—	4	2	no	—
44	C₂×C₃×C₉	C₃×A₄	1	1	yes	characteristic
45	C₂×C₂₇	—	4	6	no	—
46	C₂×C₂₇	C₃×A₄	1	3	yes	—
47	C₃²×C₄	—	3	1	no	—
48	C₄×C₉	—	3	2	no	—
49	C₄×C₉	—	3	1	no	—
50	C₃³	—	4	1	no	—
51	C₃×C₉	—	4	2	no	—
52	C₃×C₉	—	4	2	no	—
53	C₃×C₉	—	4	4	no	—
54	C₃×C₉	—	4	1	no	—
55	C₃×C₉	—	4	2	no	—
56	C₃×C₉	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
57	C₂₇	—	4	6	no	—
58	C₂₇	C₃×SL(2,𝔽₃)	1	3	yes	complemented
59	C₃×Q₈	—	3	1	no	—
60	C₃×Q₈	C₃²×C₉	1	1	yes	characteristic
61	SL(2,𝔽₃)	—	3	2	no	—
62	SL(2,𝔽₃)	C₂₇⋊C₃	1	3	yes	direct product factor
63	C₂×C₃²	—	4	1	no	—
64	C₂×C₃²	—	4	2	no	—
65	C₂×C₃²	—	12	3	no	—
66	C₂×C₃²	C₉×A₄	1	1	yes	characteristic
67	C₂×C₉	—	4	2	no	—
68	C₂×C₉	—	4	4	no	—
69	C₂×C₉	C₉×A₄	1	2	yes	—
70	C₂×C₉	C₃²×A₄	1	1	yes	characteristic
71	C₃×C₄	—	3	1	no	—
72	C₃×C₄	—	9	1	no	—
73	C₃²	—	4	1	no	—
74	C₃²	—	4	2	no	—
75	C₃²	—	12	3	no	—
76	C₃²	C₉×SL(2,𝔽₃)	1	1	yes	characteristic
77	C₉	—	4	2	no	—
78	C₉	—	4	4	no	—
79	C₉	C₉×SL(2,𝔽₃)	1	2	yes	—
80	C₉	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
81	Q₈	C₃×C₂₇⋊C₃	1	1	yes	characteristic, complemented, Sylow
82	C₂×C₃	—	3	1	no	—
83	C₂×C₃	—	4	3	no	—
84	C₂×C₃	—	12	2	no	—
85	C₂×C₃	C₃×C₉×A₄	1	1	yes	characteristic
86	C₄	—	3	1	no	—
87	C₃	—	3	1	no	—
88	C₃	—	4	3	no	—
89	C₃	—	12	2	no	—
90	C₃	C₃×C₉×SL(2,𝔽₃)	1	1	yes	characteristic
91	C₂	A₄×C₂₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₉	C₃²×A₄
commutator	C₃×Q₈	C₃²×C₉
Fitting	Q₈×C₂₇⋊C₃	C₃
socle	C₂×C₃	C₃×C₉×A₄
prime radical	C₃²×SL(2,𝔽₃)	C₉
cosocle	Q₈×C₉	C₃³
hypercenter	C₂×C₂₇⋊C₃	A₄
hypocenter	Q₈	C₃×C₂₇⋊C₃
LCS	G ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₉ ◁ C₂×C₂₇⋊C₃

Complements

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

N	H ≅ G/N	type
2. Q₈×C₂₇⋊C₃	C₃	semidirect
3. #10619570	C₃	semidirect
4. C₂₇⋊SL(2,𝔽₃)	C₃	semidirect
5. SL(2,𝔽₃)×C₂₇	C₃	semidirect
16. Q₈×C₂₇	C₃²	semidirect
17. Q₈⋊C₂₇	C₃²	semidirect
28. C₂₇⋊C₃	SL(2,𝔽₃)	direct (N × H)
58. C₂₇	C₃×SL(2,𝔽₃)	semidirect
62. SL(2,𝔽₃)	C₂₇⋊C₃	direct (N × H)
81. Q₈	C₃×C₂₇⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×SL(2,𝔽₃)	3
Q₈×C₂₇⋊C₃	3
#10619570	3
C₂₇⋊SL(2,𝔽₃)	3
SL(2,𝔽₃)×C₂₇	3
C₂×C₃×C₂₇⋊C₃	4

Subgroups — SL(2,𝔽3)×C27⋊C3 #14517203

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — SL(2,𝔽₃)×C₂₇⋊C₃ #14517203