Subgroups — #36381105

Order 22599.

172 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21419294	C₃	1	1	yes	maximal, characteristic
2	(C₃₁⋊C₉)⋊C₂₇	C₃	1	6	yes	maximal, complemented
3	(C₃×C₃₁⋊C₃)⋊C₂₇	C₃	1	3	yes	maximal
4	#21419222	C₃	1	2	yes	maximal, complemented
5	C₉×C₂₇×C₃₁	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃×C₉×C₃₁⋊C₃	C₃²	1	1	yes	characteristic
7	C₃×C₃₁⋊C₂₇	C₃²	1	6	yes	—
8	(C₃₁⋊C₉)⋊C₉	C₉	1	6	yes	—
9	(C₃×C₃₁⋊C₃)⋊C₉	C₉	1	3	yes	—
10	C₉×C₃₁⋊C₉	C₃²	1	2	yes	—
11	C₃×C₂₇×C₃₁	C₃²	1	3	yes	—
12	C₉²×C₃₁	C₃²	1	1	yes	characteristic
13	C₃²×C₃₁⋊C₃	C₃×C₉	1	1	yes	characteristic
14	C₃×C₉×C₃₁	C₃×C₉	1	3	yes	—
15	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
16	C₃×C₃₁⋊C₉	C₃×C₉	1	6	yes	—
17	C₃×C₃₁⋊C₉	C₃×C₉	1	2	yes	—
18	C₉×C₃₁⋊C₃	—	3	3	no	—
19	C₂₇×C₃₁	—	3	3	no	—
20	C₃₁⋊C₂₇	—	3	6	no	—
21	#10624713	—	31	1	no	maximal, Sylow
22	C₃²×C₃₁	C₃²×C₉	1	1	yes	characteristic
23	C₃×C₃₁⋊C₃	—	3	1	no	—
24	C₃×C₃₁⋊C₃	—	3	3	no	—
25	C₉×C₃₁	—	3	3	no	—
26	C₉×C₃₁	#737	1	3	yes	—
27	C₃₁⋊C₉	—	3	2	no	—
28	C₃₁⋊C₉	—	3	6	no	—
29	#6887	—	31	1	no	—
30	C₉⋊C₂₇	—	31	6	no	—
31	C₃²⋊C₂₇	—	31	3	no	—
32	#6864	—	31	2	no	—
33	C₉×C₂₇	C₃₁⋊C₃	1	1	yes	characteristic, complemented
34	C₃×C₃₁	#6887	1	1	yes	characteristic
35	C₃×C₃₁	#6903	1	3	yes	—
36	C₃₁⋊C₃	—	9	1	no	—
37	C₃²×C₉	—	31	1	no	—
38	C₃×C₂₇	—	31	6	no	—
39	C₃×C₂₇	C₃×C₃₁⋊C₃	1	3	yes	—
40	C₉⋊C₉	—	31	6	no	—
41	C₃²⋊C₉	—	31	3	no	—
42	C₉²	—	31	2	no	—
43	C₉²	C₃×C₃₁⋊C₃	1	1	yes	characteristic
44	C₃₁	#10624713	1	1	yes	characteristic, complemented, Sylow
45	C₃³	—	31	1	no	—
46	C₃×C₉	—	31	6	no	—
47	C₃×C₉	—	31	2	no	—
48	C₃×C₉	—	93	3	no	—
49	C₃×C₉	C₉×C₃₁⋊C₃	1	3	yes	—
50	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
51	C₂₇	—	3	3	no	—
52	C₂₇	—	93	6	no	—
53	C₃²	—	93	1	no	—
54	C₃²	—	93	3	no	—
55	C₃²	C₃×C₉×C₃₁⋊C₃	1	1	yes	characteristic
56	C₉	—	3	3	no	—
57	C₉	—	93	2	no	—
58	C₉	—	93	6	no	—
59	C₉	#14578912	1	3	yes	—
60	C₃	—	279	1	no	—
61	C₃	#21419294	1	1	yes	characteristic
62	C₃	#21419370	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃×C₉	C₃²×C₃₁⋊C₃
commutator, socle	C₃²×C₃₁	C₃²×C₉
Fitting	C₉×C₂₇×C₃₁	C₃
prime radical	C₃²×C₃₁⋊C₃	C₃×C₉
cosocle	C₃×C₉×C₃₁	C₃³
hypercenter	C₉×C₂₇	C₃₁⋊C₃
hypocenter	C₃₁	#10624713
LCS	G ▷ C₃²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃×C₉ ◁ C₉×C₂₇

Complements

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

N	H ≅ G/N	type
2. (C₃₁⋊C₉)⋊C₂₇	C₃	semidirect
4. #21419222	C₃	semidirect
5. C₉×C₂₇×C₃₁	C₃	semidirect
33. C₉×C₂₇	C₃₁⋊C₃	semidirect
44. C₃₁	#10624713	semidirect

Maximal subgroups

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

subgroup	index
#21419294	3
(C₃₁⋊C₉)⋊C₂₇	3
(C₃×C₃₁⋊C₃)⋊C₂₇	3
#21419222	3
C₉×C₂₇×C₃₁	3
#10624713	31