Subgroups — C₃×#21419371 #36382871

Order 22599.

848 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21419371	C₃	1	27	yes	maximal, direct product factor
2	C₃×#14578892	C₃	1	8	yes	maximal, complemented
3	C₃²×C₃₁⋊C₂₇	C₃	1	4	yes	maximal, complemented
4	C₃×C₃₁×#737	C₃	1	1	yes	maximal, characteristic
5	#14578892	C₃²	1	72	yes	complemented
6	C₃×C₃₁⋊C₂₇	C₃²	1	1	yes	characteristic
7	C₃×C₃₁⋊C₂₇	C₃²	1	8	yes	—
8	C₃×C₃₁⋊C₂₇	C₃²	1	36	yes	complemented
9	C₃₁×#737	C₉	1	18	yes	—
10	C₃₁×#737	C₃²	1	9	yes	—
11	C₃×3₋¹⁺²×C₃₁	C₉	1	8	yes	—
12	C₃×H(3,𝔽₃)×C₃₁	C₉	1	1	yes	characteristic
13	C₃²×C₉×C₃₁	C₃²	1	4	yes	—
14	C₃³×C₃₁	C₃×C₉	1	4	yes	—
15	3₋¹⁺²×C₃₁	C₃×C₉	1	72	yes	—
16	H(3,𝔽₃)×C₃₁	C₃×C₉	1	9	yes	—
17	C₃×C₉×C₃₁	C₃×C₉	1	8	yes	—
18	C₃×C₉×C₃₁	C₃×C₉	1	24	yes	—
19	C₃×C₉×C₃₁	C₃³	1	12	yes	—
20	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
21	C₃₁⋊C₂₇	C₃³	1	3	yes	—
22	C₃₁⋊C₂₇	C₃³	1	24	yes	—
23	C₃×#6903	—	31	1	no	maximal, Sylow
24	C₃²×C₃₁	—	3	12	no	—
25	C₃²×C₃₁	C₃²×C₉	1	12	yes	—
26	C₃²×C₃₁	C₃²×C₉	1	1	yes	characteristic
27	C₉×C₃₁	C₃²×C₉	1	2	yes	—
28	C₉×C₃₁	C₃²×C₉	1	24	yes	—
29	C₉×C₃₁	C₃⁴	1	1	yes	characteristic
30	C₃×#737	C₃₁⋊C₃	1	1	yes	characteristic
31	#6903	—	31	27	no	—
32	C₃×C₂₇⋊C₃	—	31	8	no	—
33	C₃²×C₂₇	—	31	4	no	—
34	C₃×C₃₁	—	3	12	no	—
35	C₃×C₃₁	#6903	1	3	yes	complemented
36	C₃×C₃₁	C₃³×C₉	1	1	yes	characteristic
37	#737	C₃₁⋊C₉	1	18	yes	—
38	#737	C₃×C₃₁⋊C₃	1	9	yes	—
39	C₃×3₋¹⁺²	C₃₁⋊C₉	1	8	yes	—
40	C₃×H(3,𝔽₃)	C₃₁⋊C₉	1	1	yes	characteristic
41	C₃²×C₉	C₃×C₃₁⋊C₃	1	4	yes	—
42	C₂₇⋊C₃	—	31	72	no	—
43	C₃×C₂₇	—	31	36	no	—
44	C₃×C₂₇	—	31	1	no	—
45	C₃×C₂₇	—	31	8	no	—
46	C₃₁	C₃×#6903	1	1	yes	characteristic, complemented, Sylow
47	C₃³	C₃×C₃₁⋊C₉	1	4	yes	—
48	3₋¹⁺²	C₃×C₃₁⋊C₉	1	72	yes	—
49	H(3,𝔽₃)	C₃×C₃₁⋊C₉	1	9	yes	—
50	C₃×C₉	C₃×C₃₁⋊C₉	1	8	yes	—
51	C₃×C₉	C₃×C₃₁⋊C₉	1	24	yes	—
52	C₃×C₉	C₃²×C₃₁⋊C₃	1	12	yes	—
53	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
54	C₂₇	—	31	3	no	—
55	C₂₇	—	31	24	no	—
56	C₃²	—	3	12	no	—
57	C₃²	C₃²×C₃₁⋊C₉	1	12	yes	—
58	C₃²	C₃²×C₃₁⋊C₉	1	1	yes	characteristic
59	C₉	C₃²×C₃₁⋊C₉	1	2	yes	—
60	C₉	C₃²×C₃₁⋊C₉	1	24	yes	—
61	C₉	C₃³×C₃₁⋊C₃	1	1	yes	characteristic
62	C₃	—	3	12	no	—
63	C₃	#21419371	1	3	yes	direct product factor
64	C₃	C₃³×C₃₁⋊C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₉	C₃²×C₃₁⋊C₃
commutator	C₃×C₃₁	C₃³×C₉
Frattini	C₉	C₃³×C₃₁⋊C₃
Fitting	C₃×C₃₁×#737	C₃
socle	C₃²×C₃₁	C₃²×C₉
prime radical	C₃×H(3,𝔽₃)×C₃₁	C₉
cosocle	C₉×C₃₁	C₃⁴
hypercenter	C₃×#737	C₃₁⋊C₃
hypocenter	C₃₁	C₃×#6903
LCS	G ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃×C₉ ◁ C₃×#737

Complements

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

N	H ≅ G/N	type
1. #21419371	C₃	direct (N × H)
2. C₃×#14578892	C₃	semidirect
3. C₃²×C₃₁⋊C₂₇	C₃	semidirect
5. #14578892	C₃²	semidirect
8. C₃×C₃₁⋊C₂₇	C₃²	semidirect
35. C₃×C₃₁	#6903	semidirect
46. C₃₁	C₃×#6903	semidirect
63. C₃	#21419371	direct (N × H)

Maximal subgroups

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

subgroup	index
#21419371	3
C₃×#14578892	3
C₃²×C₃₁⋊C₂₇	3
C₃×C₃₁×#737	3
C₃×#6903	31

Subgroups — C3×#21419371 #36382871

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#21419371 #36382871