Subgroups — #36379698

Order 22599.

76 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	C₃×C₃₁×C₈₁	C₃	1	1	yes	maximal, characteristic, complemented
4	#14578912	C₉	1	1	yes	characteristic
5	#14578892	C₉	1	2	yes	—
6	#14578891	—	3	1	no	—
7	#14578890	—	3	1	no	—
8	C₂₇×C₃₁⋊C₃	—	3	1	no	—
9	C₃₁⋊C₈₁	—	3	2	no	—
10	C₃×C₂₇×C₃₁	C₃²	1	1	yes	characteristic
11	C₃₁×C₈₁	—	3	1	no	—
12	C₃×C₉×C₃₁	C₃×C₉	1	1	yes	characteristic
13	C₃²⋊(C₃₁⋊C₃)	C₂₇	1	1	yes	characteristic
14	#11719689	C₂₇	1	2	yes	—
15	#11719687	—	3	1	no	—
16	#11719686	—	3	1	no	—
17	C₉×C₃₁⋊C₃	—	3	1	no	—
18	C₂₇×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
19	C₂₇×C₃₁	3₋¹⁺²	1	1	yes	characteristic
20	C₂₇×C₃₁	3₋¹⁺²	1	1	yes	characteristic
21	C₃₁⋊C₂₇	—	3	2	no	—
22	#10624503	—	31	1	no	maximal, Sylow
23	C₃²×C₃₁	C₃×C₂₇	1	1	yes	characteristic
24	C₃×C₃₁⋊C₃	—	3	1	no	—
25	C₉×C₃₁	C₃²⋊C₉	1	1	yes	characteristic
26	C₉×C₃₁	C₂₇⋊C₃	1	1	yes	characteristic
27	C₉×C₃₁	C₂₇⋊C₃	1	1	yes	characteristic
28	C₃₁⋊C₉	—	3	2	no	—
29	#6903	—	31	1	no	—
30	C₈₁⋊C₃	—	31	2	no	—
31	C₃×C₈₁	C₃₁⋊C₃	1	1	yes	characteristic, complemented
32	C₃×C₃₁	—	3	1	no	—
33	C₃×C₃₁	C₃²⋊C₂₇	1	1	yes	characteristic
34	C₃₁⋊C₃	—	9	1	no	—
35	#737	—	31	1	no	—
36	C₂₇⋊C₃	—	31	2	no	—
37	C₂₇⋊C₃	—	93	1	no	—
38	C₂₇⋊C₃	—	93	1	no	—
39	C₃×C₂₇	—	93	1	no	—
40	C₃×C₂₇	C₃×C₃₁⋊C₃	1	1	yes	characteristic
41	C₈₁	—	3	1	no	—
42	C₈₁	—	93	2	no	—
43	C₃₁	#10624503	1	1	yes	characteristic, complemented, Sylow
44	3₋¹⁺²	—	31	2	no	—
45	3₋¹⁺²	—	93	1	no	—
46	3₋¹⁺²	—	93	1	no	—
47	H(3,𝔽₃)	—	31	1	no	—
48	C₃×C₉	—	93	1	no	—
49	C₃×C₉	C₉×C₃₁⋊C₃	1	1	yes	characteristic
50	C₂₇	—	93	2	no	—
51	C₂₇	#11719686	1	1	yes	characteristic
52	C₂₇	#11719687	1	1	yes	characteristic
53	C₂₇	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
54	C₃²	—	93	1	no	—
55	C₃²	C₂₇×C₃₁⋊C₃	1	1	yes	characteristic
56	C₉	—	93	2	no	—
57	C₉	(C₃×C₃₁⋊C₃)⋊C₉	1	1	yes	characteristic
58	C₉	#14578890	1	1	yes	characteristic
59	C₉	#14578891	1	1	yes	characteristic
60	C₃	—	3	1	no	—
61	C₃	—	279	1	no	—
62	C₃	(C₃×C₃₁⋊C₃)⋊C₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂₇	C₃²⋊(C₃₁⋊C₃)
commutator	C₃²×C₃₁	C₃×C₂₇
Frattini	C₃×C₂₇	C₃×C₃₁⋊C₃
Fitting	C₃×C₃₁×C₈₁	C₃
socle	C₃×C₃₁	C₃²⋊C₂₇
prime radical	C₃²⋊(C₃₁⋊C₃)	C₂₇
cosocle	C₃×C₂₇×C₃₁	C₃²
hypercenter	C₃×C₈₁	C₃₁⋊C₃
hypocenter	C₃₁	#10624503
LCS	G ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂₇ ◁ C₃×C₂₇ ◁ C₃×C₈₁

Complements

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

N	H ≅ G/N	type
2. #21419258	C₃	semidirect
3. C₃×C₃₁×C₈₁	C₃	semidirect
31. C₃×C₈₁	C₃₁⋊C₃	semidirect
43. C₃₁	#10624503	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
C₃×C₃₁×C₈₁	3
#10624503	31