Subgroups — C₃×#21419256 #36382063

Order 22599.

164 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₂₇×C₃₁⋊C₃	C₃	1	1	yes	maximal, characteristic
2	#21419256	C₃	1	9	yes	maximal, direct product factor
3	C₃×C₃₁⋊C₈₁	C₃	1	2	yes	maximal, complemented
4	C₃×C₃₁×C₈₁	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×C₉×C₃₁⋊C₃	C₉	1	1	yes	characteristic
6	C₃×C₃₁⋊C₂₇	C₉	1	2	yes	—
7	C₂₇×C₃₁⋊C₃	C₉	1	6	yes	—
8	C₂₇×C₃₁⋊C₃	C₃²	1	3	yes	—
9	C₃₁⋊C₈₁	C₃²	1	6	yes	complemented
10	C₃×C₂₇×C₃₁	C₃²	1	1	yes	characteristic
11	C₃₁×C₈₁	C₃²	1	3	yes	complemented
12	C₃²×C₃₁⋊C₃	C₂₇	1	1	yes	characteristic
13	C₃×C₉×C₃₁	C₃×C₉	1	1	yes	characteristic
14	C₃×C₃₁⋊C₉	C₂₇	1	2	yes	—
15	C₉×C₃₁⋊C₃	C₂₇	1	6	yes	—
16	C₉×C₃₁⋊C₃	C₃×C₉	1	3	yes	—
17	C₂₇×C₃₁	C₃×C₉	1	2	yes	—
18	C₂₇×C₃₁	C₃³	1	1	yes	characteristic
19	C₃₁⋊C₂₇	C₃×C₉	1	6	yes	—
20	C₃×C₈₁⋊C₃	—	31	1	no	maximal, Sylow
21	C₃²×C₃₁	C₃×C₂₇	1	1	yes	characteristic
22	C₃×C₃₁⋊C₃	—	3	3	no	—
23	C₃×C₃₁⋊C₃	C₃×C₂₇	1	3	yes	—
24	C₉×C₃₁	C₃×C₂₇	1	2	yes	—
25	C₉×C₃₁	C₃²×C₉	1	1	yes	characteristic
26	C₃₁⋊C₉	C₃×C₂₇	1	6	yes	—
27	C₃²×C₂₇	—	31	1	no	—
28	C₈₁⋊C₃	—	31	9	no	—
29	C₃×C₈₁	—	31	2	no	—
30	C₃×C₈₁	C₃₁⋊C₃	1	1	yes	characteristic, complemented
31	C₃×C₃₁	C₈₁⋊C₃	1	3	yes	complemented
32	C₃×C₃₁	C₃²×C₂₇	1	1	yes	characteristic
33	C₃₁⋊C₃	—	3	3	no	—
34	C₃²×C₉	—	31	1	no	—
35	C₃×C₂₇	—	31	2	no	—
36	C₃×C₂₇	—	31	6	no	—
37	C₃×C₂₇	—	31	3	no	—
38	C₃×C₂₇	C₃×C₃₁⋊C₃	1	1	yes	characteristic
39	C₈₁	—	31	6	no	—
40	C₈₁	C₃×C₃₁⋊C₃	1	3	yes	complemented
41	C₃₁	C₃×C₈₁⋊C₃	1	1	yes	characteristic, complemented, Sylow
42	C₃³	—	31	1	no	—
43	C₃×C₉	—	31	2	no	—
44	C₃×C₉	—	31	6	no	—
45	C₃×C₉	—	31	3	no	—
46	C₃×C₉	C₉×C₃₁⋊C₃	1	1	yes	characteristic
47	C₂₇	—	31	6	no	—
48	C₂₇	C₉×C₃₁⋊C₃	1	2	yes	—
49	C₂₇	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
50	C₃²	—	31	3	no	—
51	C₃²	—	93	3	no	—
52	C₃²	C₂₇×C₃₁⋊C₃	1	1	yes	characteristic
53	C₉	—	31	6	no	—
54	C₉	C₂₇×C₃₁⋊C₃	1	2	yes	—
55	C₉	C₃×C₉×C₃₁⋊C₃	1	1	yes	characteristic
56	C₃	—	93	3	no	—
57	C₃	#21419256	1	3	yes	direct product factor
58	C₃	C₃×C₂₇×C₃₁⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₂₇	C₃×C₃₁⋊C₃
commutator	C₃×C₃₁	C₃²×C₂₇
Frattini	C₂₇	C₃²×C₃₁⋊C₃
Fitting	C₃×C₃₁×C₈₁	C₃
socle	C₃²×C₃₁	C₃×C₂₇
prime radical	C₃²×C₃₁⋊C₃	C₂₇
cosocle	C₂₇×C₃₁	C₃³
hypercenter	C₃×C₈₁	C₃₁⋊C₃
hypocenter	C₃₁	C₃×C₈₁⋊C₃
LCS	G ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃×C₂₇ ◁ C₃×C₈₁

Complements

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

N	H ≅ G/N	type
2. #21419256	C₃	direct (N × H)
3. C₃×C₃₁⋊C₈₁	C₃	semidirect
4. C₃×C₃₁×C₈₁	C₃	semidirect
9. C₃₁⋊C₈₁	C₃²	semidirect
11. C₃₁×C₈₁	C₃²	semidirect
30. C₃×C₈₁	C₃₁⋊C₃	semidirect
31. C₃×C₃₁	C₈₁⋊C₃	semidirect
40. C₈₁	C₃×C₃₁⋊C₃	semidirect
41. C₃₁	C₃×C₈₁⋊C₃	semidirect
57. C₃	#21419256	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×C₂₇×C₃₁⋊C₃	3
#21419256	3
C₃×C₃₁⋊C₈₁	3
C₃×C₃₁×C₈₁	3
C₃×C₈₁⋊C₃	31

Subgroups — C3×#21419256 #36382063

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#21419256 #36382063