Subgroups — #14519158

Order 1944.

202 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11758047	C₂	1	2	yes	maximal
2	H(3,𝔽₃)×#157	C₂	1	1	yes	maximal, characteristic
3	C₃×C₃⋊PSU(3,𝔽₂)	—	3	4	no	maximal
4	C₃⋊D₃×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
5	C₃×C₃⋊#157	—	3	8	no	—
6	C₃²×#157	D₃	1	4	yes	—
7	C₃²×H(3,𝔽₃)	Q₈	1	1	yes	characteristic, complemented, Sylow
8	C₃⋊PSU(3,𝔽₂)	—	3	4	no	—
9	C₃×PSU(3,𝔽₂)	—	9	1	no	—
10	#6229	—	9	1	no	maximal
11	C₃²×C₃⋊D₃	C₂×D₃	1	4	yes	—
12	D₃×H(3,𝔽₃)	—	12	1	no	—
13	C₃⋊#157	—	3	8	no	—
14	C₃×#157	—	3	4	no	—
15	C₃×#157	—	9	2	no	—
16	C₃×#157	C₃⋊D₃	1	1	yes	characteristic
17	C₄×H(3,𝔽₃)	—	9	1	no	—
18	#1084	—	9	2	no	—
19	C₃⁴	Dic₆	1	4	yes	complemented
20	C₃×H(3,𝔽₃)	—	4	1	no	—
21	C₃×H(3,𝔽₃)	—	8	4	no	—
22	PSU(3,𝔽₂)	—	9	1	no	—
23	C₃×Dic₆	—	27	4	no	—
24	C₃×C₃⋊D₃	—	3	4	no	—
25	C₃×C₃⋊D₃	C₂×C₃⋊D₃	1	1	yes	characteristic
26	C₃²×D₃	—	12	4	no	—
27	C₂×H(3,𝔽₃)	—	9	1	no	—
28	#157	—	9	2	no	—
29	#157	#272	1	1	yes	characteristic
30	C₃²×C₄	—	9	4	no	—
31	C₃×Dic₃	—	27	8	no	—
32	C₃³	—	3	4	no	—
33	C₃³	—	4	4	no	—
34	C₃³	—	8	4	no	—
35	C₃³	—	24	4	no	—
36	C₃³	#634	1	1	yes	characteristic
37	H(3,𝔽₃)	—	8	6	no	—
38	H(3,𝔽₃)	—	8	4	no	—
39	H(3,𝔽₃)	PSU(3,𝔽₂)	1	1	yes	characteristic, complemented
40	C₃×Q₈	—	81	1	no	—
41	Dic₆	—	27	4	no	—
42	C₂×C₃²	—	9	4	no	—
43	C₃⋊D₃	C₂×#272	1	1	yes	characteristic
44	C₃×D₃	—	12	1	no	—
45	C₃×D₃	—	36	4	no	—
46	C₃×C₄	—	9	1	no	—
47	C₃×C₄	—	27	4	no	—
48	C₃×C₄	—	81	2	no	—
49	Dic₃	—	27	8	no	—
50	C₃²	—	4	1	no	—
51	C₃²	—	8	1	no	—
52	C₃²	—	8	4	no	—
53	C₃²	—	12	4	no	—
54	C₃²	—	24	4	no	—
55	C₃²	—	24	4	no	—
56	C₃²	—	24	8	no	—
57	C₃²	#6229	1	1	yes	characteristic, complemented
58	C₃²	C₃⋊PSU(3,𝔽₂)	1	4	yes	complemented
59	Q₈	—	81	1	no	Sylow
60	C₂×C₃	—	9	1	no	—
61	C₂×C₃	—	27	4	no	—
62	D₃	—	12	1	no	—
63	C₄	—	9	1	no	—
64	C₄	—	81	2	no	—
65	C₃	—	3	4	no	—
66	C₃	—	4	1	no	—
67	C₃	—	8	1	no	—
68	C₃	—	24	4	no	—
69	C₃	#10620209	1	1	yes	characteristic
70	C₂	—	9	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₃	#10620209
commutator, prime radical, cosocle	C₃⋊D₃×H(3,𝔽₃)	C₂²
Fitting, hypocenter	C₃²×H(3,𝔽₃)	Q₈
socle	C₃³	#634
LCS	G ▷ C₃⋊D₃×H(3,𝔽₃) ▷ C₃²×H(3,𝔽₃)
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
7. C₃²×H(3,𝔽₃)	Q₈	semidirect
19. C₃⁴	Dic₆	semidirect
39. H(3,𝔽₃)	PSU(3,𝔽₂)	semidirect
57. C₃²	#6229	semidirect
58. C₃²	C₃⋊PSU(3,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
#11758047	2
H(3,𝔽₃)×#157	2
C₃×C₃⋊PSU(3,𝔽₂)	3
#6229	9