Subgroups — #36382066

Order 22599.

154 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21419370	C₃	1	1	yes	maximal, characteristic
2	#21419258	C₃	1	2	yes	maximal, complemented
3	#21419257	C₃	1	3	yes	maximal, complemented
4	#21419256	C₃	1	3	yes	maximal, complemented
5	C₈₁×C₃₁⋊C₃	C₃	1	3	yes	maximal, complemented
6	C₃×C₃₁×C₈₁	C₃	1	1	yes	maximal, characteristic, complemented
7	#14578912	C₉	1	1	yes	characteristic
8	#14578892	C₉	1	2	yes	—
9	#14578891	C₉	1	3	yes	—
10	#14578890	C₉	1	3	yes	—
11	C₂₇×C₃₁⋊C₃	C₃²	1	3	yes	—
12	C₃₁⋊C₈₁	C₃²	1	6	yes	—
13	C₃×C₂₇×C₃₁	C₃²	1	1	yes	characteristic
14	C₃₁×C₈₁	C₃²	1	1	yes	characteristic
15	C₃₁×C₈₁	C₃²	1	1	yes	characteristic
16	C₃₁×C₈₁	C₃²	1	1	yes	characteristic
17	C₃×C₉×C₃₁	C₃×C₉	1	1	yes	characteristic
18	C₃²⋊(C₃₁⋊C₃)	C₂₇	1	1	yes	characteristic
19	#11719689	C₂₇	1	2	yes	—
20	#11719687	C₂₇	1	3	yes	—
21	#11719686	C₂₇	1	3	yes	—
22	C₉×C₃₁⋊C₃	C₃×C₉	1	3	yes	—
23	C₂₇×C₃₁	C₃×C₉	1	1	yes	characteristic
24	C₂₇×C₃₁	C₃×C₉	1	1	yes	characteristic
25	C₂₇×C₃₁	C₃³	1	1	yes	characteristic
26	C₃₁⋊C₂₇	C₃×C₉	1	6	yes	—
27	#10624833	—	31	1	no	maximal, Sylow
28	C₃²×C₃₁	C₃×C₂₇	1	1	yes	characteristic
29	C₃×C₃₁⋊C₃	C₃×C₂₇	1	3	yes	—
30	C₉×C₃₁	C₃×C₂₇	1	1	yes	characteristic
31	C₉×C₃₁	C₃×C₂₇	1	1	yes	characteristic
32	C₉×C₃₁	C₃²×C₉	1	1	yes	characteristic
33	C₃₁⋊C₉	C₃×C₂₇	1	6	yes	—
34	#6903	—	31	1	no	—
35	C₈₁⋊C₃	—	31	2	no	—
36	C₈₁⋊C₃	—	31	3	no	—
37	C₈₁⋊C₃	—	31	3	no	—
38	C₃×C₈₁	—	31	3	no	—
39	C₃×C₈₁	C₃₁⋊C₃	1	1	yes	characteristic, complemented
40	C₃×C₃₁	—	3	1	no	—
41	C₃×C₃₁	C₃²×C₂₇	1	1	yes	characteristic
42	C₃₁⋊C₃	—	3	3	no	—
43	#737	—	31	1	no	—
44	C₂₇⋊C₃	—	31	2	no	—
45	C₂₇⋊C₃	—	31	3	no	—
46	C₂₇⋊C₃	—	31	3	no	—
47	C₃×C₂₇	—	31	3	no	—
48	C₃×C₂₇	C₃×C₃₁⋊C₃	1	1	yes	characteristic
49	C₈₁	—	31	6	no	—
50	C₈₁	C₃×C₃₁⋊C₃	1	1	yes	characteristic
51	C₈₁	C₃×C₃₁⋊C₃	1	1	yes	characteristic
52	C₈₁	C₃×C₃₁⋊C₃	1	1	yes	characteristic
53	C₃₁	#10624833	1	1	yes	characteristic, complemented, Sylow
54	3₋¹⁺²	—	31	2	no	—
55	3₋¹⁺²	—	31	3	no	—
56	3₋¹⁺²	—	31	3	no	—
57	H(3,𝔽₃)	—	31	1	no	—
58	C₃×C₉	—	31	3	no	—
59	C₃×C₉	C₉×C₃₁⋊C₃	1	1	yes	characteristic
60	C₂₇	—	31	6	no	—
61	C₂₇	C₉×C₃₁⋊C₃	1	1	yes	characteristic
62	C₂₇	C₉×C₃₁⋊C₃	1	1	yes	characteristic
63	C₂₇	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
64	C₃²	—	31	3	no	—
65	C₃²	C₂₇×C₃₁⋊C₃	1	1	yes	characteristic
66	C₉	—	31	6	no	—
67	C₉	C₂₇×C₃₁⋊C₃	1	1	yes	characteristic
68	C₉	C₂₇×C₃₁⋊C₃	1	1	yes	characteristic
69	C₉	C₃×C₉×C₃₁⋊C₃	1	1	yes	characteristic
70	C₃	—	3	1	no	—
71	C₃	—	93	3	no	—
72	C₃	C₃×C₂₇×C₃₁⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₈₁	C₃×C₃₁⋊C₃
commutator, socle	C₃×C₃₁	C₃²×C₂₇
Frattini	C₂₇	C₃²×C₃₁⋊C₃
Fitting	C₃×C₃₁×C₈₁	C₃
prime radical	C₃²⋊(C₃₁⋊C₃)	C₂₇
cosocle	C₂₇×C₃₁	C₃³
hypercenter	C₃×C₈₁	C₃₁⋊C₃
hypocenter	C₃₁	#10624833
LCS	G ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₈₁ ◁ C₃×C₈₁

Complements

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

N	H ≅ G/N	type
2. #21419258	C₃	semidirect
3. #21419257	C₃	semidirect
4. #21419256	C₃	semidirect
5. C₈₁×C₃₁⋊C₃	C₃	semidirect
6. C₃×C₃₁×C₈₁	C₃	semidirect
39. C₃×C₈₁	C₃₁⋊C₃	semidirect
53. C₃₁	#10624833	semidirect

Maximal subgroups

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

subgroup	index
#21419370	3
#21419258	3
#21419257	3
#21419256	3
C₈₁×C₃₁⋊C₃	3
C₃×C₃₁×C₈₁	3
#10624833	31