Subgroups — #36379696

Order 22599.

86 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	#21419258	C₃	1	2	yes	maximal, complemented
3	C₃₁×C₈₁⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₉×C₃₁⋊C₃	C₉	1	1	yes	characteristic
5	C₃×C₃₁⋊C₂₇	C₉	1	2	yes	—
6	C₂₇×C₃₁⋊C₃	—	3	1	no	—
7	C₂₇×C₃₁⋊C₃	—	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	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	—	3	1	no	—
16	C₉×C₃₁⋊C₃	—	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	#10624502	—	31	1	no	maximal, Sylow
23	C₃²×C₃₁	C₃×C₂₇	1	1	yes	characteristic
24	C₃×C₃₁⋊C₃	—	3	1	no	—
25	C₃×C₃₁⋊C₃	—	3	3	no	—
26	C₉×C₃₁	C₃²⋊C₉	1	1	yes	characteristic
27	C₉×C₃₁	C₂₇⋊C₃	1	1	yes	characteristic
28	C₉×C₃₁	C₂₇⋊C₃	1	1	yes	characteristic
29	C₃₁⋊C₉	—	3	2	no	—
30	C₃²×C₂₇	—	31	1	no	—
31	C₈₁⋊C₃	—	31	2	no	—
32	C₈₁⋊C₃	C₃₁⋊C₃	1	1	yes	characteristic, complemented
33	C₃×C₃₁	—	3	1	no	—
34	C₃×C₃₁	C₃²⋊C₂₇	1	1	yes	characteristic
35	C₃₁⋊C₃	—	3	3	no	—
36	C₃²×C₉	—	31	1	no	—
37	C₃×C₂₇	—	31	2	no	—
38	C₃×C₂₇	—	93	1	no	—
39	C₃×C₂₇	—	93	1	no	—
40	C₃×C₂₇	—	93	1	no	—
41	C₃×C₂₇	C₃×C₃₁⋊C₃	1	1	yes	characteristic
42	C₈₁	—	3	1	no	—
43	C₈₁	—	93	2	no	—
44	C₃₁	#10624502	1	1	yes	characteristic, complemented, Sylow
45	C₃³	—	31	1	no	—
46	C₃×C₉	—	31	2	no	—
47	C₃×C₉	—	93	1	no	—
48	C₃×C₉	—	93	1	no	—
49	C₃×C₉	—	93	1	no	—
50	C₃×C₉	C₉×C₃₁⋊C₃	1	1	yes	characteristic
51	C₂₇	—	93	2	no	—
52	C₂₇	#11719686	1	1	yes	characteristic
53	C₂₇	#11719687	1	1	yes	characteristic
54	C₂₇	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
55	C₃²	—	93	3	no	—
56	C₃²	—	93	1	no	—
57	C₃²	C₂₇×C₃₁⋊C₃	1	1	yes	characteristic
58	C₉	—	93	2	no	—
59	C₉	(C₃×C₃₁⋊C₃)⋊C₉	1	1	yes	characteristic
60	C₉	#14578890	1	1	yes	characteristic
61	C₉	#14578891	1	1	yes	characteristic
62	C₃	—	3	1	no	—
63	C₃	—	93	3	no	—
64	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	C₃₁×C₈₁⋊C₃	C₃
socle	C₃×C₃₁	C₃²⋊C₂₇
prime radical	C₃²×C₃₁⋊C₃	C₂₇
cosocle	C₃×C₂₇×C₃₁	C₃²
hypercenter	C₈₁⋊C₃	C₃₁⋊C₃
hypocenter	C₃₁	#10624502
LCS	G ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂₇ ◁ C₃×C₂₇ ◁ C₈₁⋊C₃

Complements

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

N	H ≅ G/N	type
2. #21419258	C₃	semidirect
3. C₃₁×C₈₁⋊C₃	C₃	semidirect
32. C₈₁⋊C₃	C₃₁⋊C₃	semidirect
44. C₃₁	#10624502	semidirect

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
#21419258	3
C₃₁×C₈₁⋊C₃	3
#10624502	31