Subgroups — #36381107

Order 22599.

172 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21419248	C₃	1	3	yes	maximal, complemented
2	#21419247	C₃	1	3	yes	maximal, complemented
3	C₃²⋊(C₃₁⋊C₂₇)	C₃	1	3	yes	maximal
4	#21419222	C₃	1	2	yes	maximal, complemented
5	C₉×C₃₁⋊C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#6887	C₃	1	1	yes	maximal, characteristic
7	C₃×C₃₁⋊C₂₇	C₃²	1	3	yes	—
8	C₃×C₃₁⋊C₂₇	C₃²	1	6	yes	—
9	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
10	C₃₁×C₉⋊C₉	C₉	1	6	yes	—
11	C₃₁×C₃²⋊C₉	C₉	1	3	yes	—
12	C₉²×C₃₁	C₃²	1	1	yes	characteristic
13	C₉²×C₃₁	C₃²	1	1	yes	characteristic
14	C₉²×C₃₁	C₃²	1	1	yes	characteristic
15	C₃³×C₃₁	C₃×C₉	1	1	yes	characteristic
16	C₃×C₉×C₃₁	—	3	3	no	—
17	C₃×C₉×C₃₁	C₃×C₉	1	3	yes	—
18	C₃×C₉×C₃₁	C₃×C₉	1	3	yes	—
19	C₃×C₉×C₃₁	C₃×C₉	1	3	yes	—
20	C₃×C₉×C₃₁	C₃×C₉	1	2	yes	—
21	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
22	C₃₁⋊C₂₇	—	3	3	no	—
23	C₃₁⋊C₂₇	—	3	6	no	—
24	#10624713	—	31	1	no	maximal, Sylow
25	C₃²×C₃₁	—	3	1	no	—
26	C₃²×C₃₁	—	3	3	no	—
27	C₃²×C₃₁	C₃²×C₉	1	1	yes	characteristic
28	C₉×C₃₁	—	3	2	no	—
29	C₉×C₃₁	—	3	3	no	—
30	C₉×C₃₁	—	3	3	no	—
31	C₉×C₃₁	—	3	3	no	—
32	C₉×C₃₁	#737	1	3	yes	—
33	#6887	C₃₁⋊C₃	1	1	yes	characteristic
34	C₉⋊C₂₇	—	31	3	no	—
35	C₉⋊C₂₇	—	31	3	no	—
36	C₃²⋊C₂₇	—	31	3	no	—
37	#6864	—	31	2	no	—
38	C₉×C₂₇	—	31	1	no	—
39	C₃×C₃₁	—	9	1	no	—
40	C₃×C₃₁	#6887	1	1	yes	characteristic
41	C₃×C₃₁	#6903	1	3	yes	—
42	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
43	C₃×C₂₇	—	31	3	no	—
44	C₃×C₂₇	—	31	6	no	—
45	C₉⋊C₉	C₃₁⋊C₉	1	6	yes	—
46	C₃²⋊C₉	C₃₁⋊C₉	1	3	yes	—
47	C₉²	C₃×C₃₁⋊C₃	1	1	yes	characteristic
48	C₉²	C₃×C₃₁⋊C₃	1	1	yes	characteristic
49	C₉²	C₃×C₃₁⋊C₃	1	1	yes	characteristic
50	C₃₁	#10624713	1	1	yes	characteristic, complemented, Sylow
51	C₃³	C₃×C₃₁⋊C₉	1	1	yes	characteristic
52	C₃×C₉	—	3	3	no	—
53	C₃×C₉	C₃×C₃₁⋊C₉	1	3	yes	—
54	C₃×C₉	C₃×C₃₁⋊C₉	1	3	yes	—
55	C₃×C₉	C₃×C₃₁⋊C₉	1	3	yes	—
56	C₃×C₉	C₃×C₃₁⋊C₉	1	2	yes	—
57	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
58	C₂₇	—	93	3	no	—
59	C₂₇	—	93	6	no	—
60	C₃²	—	3	1	no	—
61	C₃²	—	3	3	no	—
62	C₃²	C₃²×C₃₁⋊C₉	1	1	yes	characteristic
63	C₉	—	3	2	no	—
64	C₉	—	3	3	no	—
65	C₉	—	3	3	no	—
66	C₉	—	3	3	no	—
67	C₉	#14578912	1	3	yes	—
68	C₃	—	9	1	no	—
69	C₃	#21419296	1	1	yes	characteristic
70	C₃	#21419371	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃×C₉	C₃²×C₃₁⋊C₃
commutator, socle	C₃²×C₃₁	C₃²×C₉
Fitting	C₃₁×#6887	C₃
prime radical	C₃³×C₃₁	C₃×C₉
cosocle	C₃×C₉×C₃₁	C₃³
hypercenter	#6887	C₃₁⋊C₃
hypocenter	C₃₁	#10624713
LCS	G ▷ C₃²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃×C₉ ◁ #6887

Complements

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

N	H ≅ G/N	type
1. #21419248	C₃	semidirect
2. #21419247	C₃	semidirect
4. #21419222	C₃	semidirect
5. C₉×C₃₁⋊C₂₇	C₃	semidirect
50. C₃₁	#10624713	semidirect

Maximal subgroups

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

subgroup	index
#21419248	3
#21419247	3
C₃²⋊(C₃₁⋊C₂₇)	3
#21419222	3
C₉×C₃₁⋊C₂₇	3
C₃₁×#6887	3
#10624713	31