Subgroups — C₃×SL(2,𝔽₃)×C₂₇ #14517200

Order 1944.

230 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	C₃×Q₈×C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×Q₈⋊C₂₇	C₃	1	2	yes	maximal, complemented
4	SL(2,𝔽₃)×C₂₇	C₃	1	9	yes	maximal, direct product factor
5	C₂×C₃²×C₂₇	—	4	1	no	maximal
6	C₃×C₄×C₂₇	—	3	1	no	—
7	C₃²×C₂₇	—	4	1	no	Sylow
8	C₃²×SL(2,𝔽₃)	C₉	1	1	yes	characteristic
9	C₃×Q₈×C₉	C₃²	1	1	yes	characteristic
10	C₃×Q₈⋊C₉	C₉	1	2	yes	—
11	C₉×SL(2,𝔽₃)	C₉	1	6	yes	—
12	C₉×SL(2,𝔽₃)	C₃²	1	3	yes	—
13	Q₈×C₂₇	C₃²	1	3	yes	complemented
14	Q₈⋊C₂₇	C₃²	1	6	yes	complemented
15	C₂×C₃²×C₉	—	4	1	no	—
16	C₂×C₃×C₂₇	—	4	2	no	—
17	C₂×C₃×C₂₇	—	4	9	no	—
18	C₂×C₃×C₂₇	A₄	1	1	yes	characteristic
19	C₃×C₄×C₉	—	3	1	no	—
20	C₄×C₂₇	—	3	3	no	—
21	C₃²×C₉	—	4	1	no	—
22	C₃×C₂₇	—	4	2	no	—
23	C₃×C₂₇	—	4	9	no	—
24	C₃×C₂₇	SL(2,𝔽₃)	1	1	yes	characteristic, direct product factor
25	C₃²×Q₈	C₃×C₉	1	1	yes	characteristic
26	C₃×SL(2,𝔽₃)	C₂₇	1	9	yes	direct product factor
27	C₃×SL(2,𝔽₃)	C₃×C₉	1	3	yes	—
28	Q₈×C₉	C₃×C₉	1	2	yes	—
29	Q₈×C₉	C₃³	1	1	yes	characteristic
30	Q₈⋊C₉	C₃×C₉	1	6	yes	—
31	C₂×C₃³	—	4	1	no	—
32	C₂×C₃×C₉	—	4	2	no	—
33	C₂×C₃×C₉	—	4	6	no	—
34	C₂×C₃×C₉	—	4	3	no	—
35	C₂×C₃×C₉	C₃×A₄	1	1	yes	characteristic
36	C₂×C₂₇	—	4	6	no	—
37	C₂×C₂₇	C₃×A₄	1	3	yes	—
38	C₃²×C₄	—	3	1	no	—
39	C₄×C₉	—	3	2	no	—
40	C₄×C₉	—	3	1	no	—
41	C₃³	—	4	1	no	—
42	C₃×C₉	—	4	2	no	—
43	C₃×C₉	—	4	6	no	—
44	C₃×C₉	—	4	3	no	—
45	C₃×C₉	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
46	C₂₇	—	4	6	no	—
47	C₂₇	C₃×SL(2,𝔽₃)	1	3	yes	direct product factor
48	C₃×Q₈	C₃×C₂₇	1	3	yes	complemented
49	C₃×Q₈	C₃²×C₉	1	1	yes	characteristic
50	SL(2,𝔽₃)	C₃×C₂₇	1	9	yes	direct product factor
51	C₂×C₃²	—	4	9	no	—
52	C₂×C₃²	—	4	3	no	—
53	C₂×C₃²	C₉×A₄	1	1	yes	characteristic
54	C₂×C₉	—	4	6	no	—
55	C₂×C₉	C₉×A₄	1	2	yes	—
56	C₂×C₉	C₃²×A₄	1	1	yes	characteristic
57	C₃×C₄	—	3	3	no	—
58	C₃×C₄	—	3	1	no	—
59	C₃²	—	4	3	no	—
60	C₃²	—	4	9	no	—
61	C₃²	C₉×SL(2,𝔽₃)	1	1	yes	characteristic
62	C₉	—	4	6	no	—
63	C₉	C₉×SL(2,𝔽₃)	1	2	yes	—
64	C₉	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
65	Q₈	C₃²×C₂₇	1	1	yes	characteristic, complemented, Sylow
66	C₂×C₃	—	4	9	no	—
67	C₂×C₃	A₄×C₂₇	1	3	yes	—
68	C₂×C₃	C₃×C₉×A₄	1	1	yes	characteristic
69	C₄	—	3	1	no	—
70	C₃	—	4	9	no	—
71	C₃	SL(2,𝔽₃)×C₂₇	1	3	yes	direct product factor
72	C₃	C₃×C₉×SL(2,𝔽₃)	1	1	yes	characteristic
73	C₂	C₃×A₄×C₂₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₃×Q₈×C₂₇	C₃	semidirect
3. C₃×Q₈⋊C₂₇	C₃	semidirect
4. SL(2,𝔽₃)×C₂₇	C₃	direct (N × H)
13. Q₈×C₂₇	C₃²	semidirect
14. Q₈⋊C₂₇	C₃²	semidirect
24. C₃×C₂₇	SL(2,𝔽₃)	direct (N × H)
26. C₃×SL(2,𝔽₃)	C₂₇	direct (N × H)
47. C₂₇	C₃×SL(2,𝔽₃)	direct (N × H)
48. C₃×Q₈	C₃×C₂₇	semidirect
50. SL(2,𝔽₃)	C₃×C₂₇	direct (N × H)
65. Q₈	C₃²×C₂₇	semidirect
71. C₃	SL(2,𝔽₃)×C₂₇	direct (N × H)

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
C₃×Q₈×C₂₇	3
C₃×Q₈⋊C₂₇	3
SL(2,𝔽₃)×C₂₇	3
C₂×C₃²×C₂₇	4

Subgroups — C3×SL(2,𝔽3)×C27 #14517200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×SL(2,𝔽₃)×C₂₇ #14517200