Subgroups — C₃×#21419221 #36380944

Order 22599.

332 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₃₁⋊C₂₇	C₃	1	2	yes	maximal
2	C₃×C₉×C₃₁⋊C₉	C₃	1	1	yes	maximal, characteristic
3	#21419221	C₃	1	9	yes	maximal, direct product factor
4	C₃²×C₂₇×C₃₁	C₃	1	1	yes	maximal, characteristic
5	C₃²×C₃₁⋊C₉	C₉	1	3	yes	—
6	C₃×C₃₁⋊C₂₇	C₉	1	18	yes	complemented
7	C₃×C₃₁⋊C₂₇	C₃²	1	6	yes	—
8	C₉×C₃₁⋊C₉	C₉	1	6	yes	—
9	C₉×C₃₁⋊C₉	C₃²	1	3	yes	—
10	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
11	C₃×C₂₇×C₃₁	C₉	1	9	yes	complemented
12	C₃×C₂₇×C₃₁	C₃²	1	3	yes	—
13	C₃³×C₃₁	C₃×C₉	1	1	yes	characteristic
14	C₃×C₉×C₃₁	C₃×C₉	1	2	yes	—
15	C₃×C₉×C₃₁	C₃×C₉	1	6	yes	—
16	C₃×C₉×C₃₁	C₃×C₉	1	3	yes	—
17	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
18	C₃×C₃₁⋊C₉	—	3	9	no	—
19	C₃×C₃₁⋊C₉	C₃×C₉	1	9	yes	—
20	C₂₇×C₃₁	C₃×C₉	1	9	yes	complemented
21	C₃₁⋊C₂₇	C₃×C₉	1	18	yes	complemented
22	C₃×#6864	—	31	1	no	maximal, Sylow
23	C₃²×C₃₁	C₉²	1	3	yes	—
24	C₃²×C₃₁	C₂₇⋊C₃	1	9	yes	—
25	C₃²×C₃₁	C₃²×C₉	1	1	yes	characteristic
26	C₉×C₃₁	C₉²	1	6	yes	—
27	C₉×C₃₁	C₃²×C₉	1	2	yes	—
28	C₉×C₃₁	C₃²×C₉	1	1	yes	characteristic
29	C₃₁⋊C₉	—	3	9	no	—
30	C₃²×C₂₇	—	31	2	no	—
31	C₃²×C₂₇	C₃₁⋊C₃	1	1	yes	characteristic
32	C₃×C₉²	—	31	1	no	—
33	#6864	—	31	9	no	—
34	C₃×C₃₁	#6864	1	9	yes	complemented
35	C₃×C₃₁	C₃×C₉²	1	1	yes	characteristic
36	C₃×C₃₁	C₃×C₂₇⋊C₃	1	3	yes	—
37	C₃²×C₉	—	31	3	no	—
38	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
39	C₃×C₂₇	—	31	18	no	—
40	C₃×C₂₇	—	31	6	no	—
41	C₃×C₂₇	C₃₁⋊C₉	1	9	yes	complemented
42	C₃×C₂₇	C₃×C₃₁⋊C₃	1	3	yes	—
43	C₉²	—	31	6	no	—
44	C₉²	—	31	3	no	—
45	C₃₁	C₃×#6864	1	1	yes	characteristic, complemented, Sylow
46	C₃³	C₉×C₃₁⋊C₃	1	1	yes	characteristic
47	C₃×C₉	—	31	9	no	—
48	C₃×C₉	—	93	9	no	—
49	C₃×C₉	C₉×C₃₁⋊C₃	1	2	yes	—
50	C₃×C₉	C₉×C₃₁⋊C₃	1	6	yes	—
51	C₃×C₉	C₃×C₃₁⋊C₉	1	3	yes	—
52	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
53	C₂₇	—	31	18	no	—
54	C₂₇	C₃×C₃₁⋊C₉	1	9	yes	complemented
55	C₃²	C₉×C₃₁⋊C₉	1	3	yes	—
56	C₃²	#14578891	1	9	yes	—
57	C₃²	C₃×C₉×C₃₁⋊C₃	1	1	yes	characteristic
58	C₉	—	93	9	no	—
59	C₉	C₉×C₃₁⋊C₉	1	6	yes	—
60	C₉	C₃×C₉×C₃₁⋊C₃	1	2	yes	—
61	C₉	C₃²×C₃₁⋊C₉	1	1	yes	characteristic
62	C₃	#21419221	1	9	yes	direct product factor
63	C₃	C₃×C₉×C₃₁⋊C₉	1	1	yes	characteristic
64	C₃	C₃×#14578891	1	3	yes	—

	subgroup	G/H
center	C₃²×C₉	C₃×C₃₁⋊C₃
commutator	C₃×C₃₁	C₃×C₉²
Frattini	C₃×C₉	C₃²×C₃₁⋊C₃
Fitting	C₃²×C₂₇×C₃₁	C₃
socle, prime radical	C₃³×C₃₁	C₃×C₉
cosocle	C₃×C₉×C₃₁	C₃³
hypercenter	C₃²×C₂₇	C₃₁⋊C₃
hypocenter	C₃₁	C₃×#6864
LCS	G ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃²×C₉ ◁ C₃²×C₂₇

Complements

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

N	H ≅ G/N	type
3. #21419221	C₃	direct (N × H)
6. C₃×C₃₁⋊C₂₇	C₉	semidirect
11. C₃×C₂₇×C₃₁	C₉	semidirect
20. C₂₇×C₃₁	C₃×C₉	semidirect
21. C₃₁⋊C₂₇	C₃×C₉	semidirect
34. C₃×C₃₁	#6864	semidirect
41. C₃×C₂₇	C₃₁⋊C₉	semidirect
45. C₃₁	C₃×#6864	semidirect
54. C₂₇	C₃×C₃₁⋊C₉	semidirect
62. C₃	#21419221	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₃₁⋊C₂₇	3
C₃×C₉×C₃₁⋊C₉	3
#21419221	3
C₃²×C₂₇×C₃₁	3
C₃×#6864	31

Subgroups — C3×#21419221 #36380944

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#21419221 #36380944