Subgroups — #36381108

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	1	yes	maximal, characteristic, complemented
5	C₉×C₃₁⋊C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#6864	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃×C₉×C₃₁⋊C₃	C₃²	1	1	yes	characteristic
8	C₃×C₃₁⋊C₂₇	C₃²	1	3	yes	—
9	C₃×C₃₁⋊C₂₇	C₃²	1	3	yes	—
10	(C₃₁⋊C₉)⋊C₉	C₉	1	6	yes	—
11	(C₃×C₃₁⋊C₃)⋊C₉	C₉	1	3	yes	—
12	C₉×C₃₁⋊C₉	C₃²	1	2	yes	—
13	C₃×C₂₇×C₃₁	C₃²	1	3	yes	—
14	C₉²×C₃₁	C₃²	1	1	yes	characteristic
15	C₃²×C₃₁⋊C₃	C₃×C₉	1	1	yes	characteristic
16	C₃×C₉×C₃₁	C₃×C₉	1	3	yes	—
17	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
18	C₃×C₃₁⋊C₉	C₃×C₉	1	6	yes	—
19	C₃×C₃₁⋊C₉	C₃×C₉	1	2	yes	—
20	C₉×C₃₁⋊C₃	—	3	3	no	—
21	C₂₇×C₃₁	—	3	3	no	—
22	C₃₁⋊C₂₇	—	3	3	no	—
23	C₃₁⋊C₂₇	—	3	3	no	—
24	#10624713	—	31	1	no	maximal, Sylow
25	C₃²×C₃₁	C₃²×C₉	1	1	yes	characteristic
26	C₃×C₃₁⋊C₃	—	3	1	no	—
27	C₃×C₃₁⋊C₃	—	3	3	no	—
28	C₉×C₃₁	—	3	3	no	—
29	C₉×C₃₁	#737	1	3	yes	—
30	C₃₁⋊C₉	—	3	2	no	—
31	C₃₁⋊C₉	—	3	6	no	—
32	#6887	—	31	1	no	—
33	C₉⋊C₂₇	—	31	6	no	—
34	C₃²⋊C₂₇	—	31	3	no	—
35	#6864	—	31	1	no	—
36	#6864	C₃₁⋊C₃	1	1	yes	characteristic, complemented
37	C₉×C₂₇	—	31	1	no	—
38	C₃×C₃₁	#6887	1	1	yes	characteristic
39	C₃×C₃₁	#6903	1	3	yes	—
40	C₃₁⋊C₃	—	9	1	no	—
41	C₃²×C₉	—	31	1	no	—
42	C₃×C₂₇	—	31	3	no	—
43	C₃×C₂₇	—	31	3	no	—
44	C₃×C₂₇	C₃×C₃₁⋊C₃	1	3	yes	—
45	C₉⋊C₉	—	31	6	no	—
46	C₃²⋊C₉	—	31	3	no	—
47	C₉²	—	31	2	no	—
48	C₉²	C₃×C₃₁⋊C₃	1	1	yes	characteristic
49	C₃₁	#10624713	1	1	yes	characteristic, complemented, Sylow
50	C₃³	—	31	1	no	—
51	C₃×C₉	—	31	6	no	—
52	C₃×C₉	—	31	2	no	—
53	C₃×C₉	—	93	3	no	—
54	C₃×C₉	C₉×C₃₁⋊C₃	1	3	yes	—
55	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
56	C₂₇	—	3	3	no	—
57	C₂₇	—	93	3	no	—
58	C₂₇	—	93	3	no	—
59	C₃²	—	93	1	no	—
60	C₃²	—	93	3	no	—
61	C₃²	C₃×C₉×C₃₁⋊C₃	1	1	yes	characteristic
62	C₉	—	3	3	no	—
63	C₉	—	93	2	no	—
64	C₉	—	93	6	no	—
65	C₉	#14578912	1	3	yes	—
66	C₃	—	279	1	no	—
67	C₃	#21419294	1	1	yes	characteristic
68	C₃	#21419372	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃×C₉	C₃²×C₃₁⋊C₃
commutator, socle	C₃²×C₃₁	C₃²×C₉
Fitting	C₃₁×#6864	C₃
prime radical	C₃²×C₃₁⋊C₃	C₃×C₉
cosocle	C₃×C₉×C₃₁	C₃³
hypercenter	#6864	C₃₁⋊C₃
hypocenter	C₃₁	#10624713
LCS	G ▷ C₃²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃×C₉ ◁ #6864

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×C₃₁⋊C₂₇	C₃	semidirect
6. C₃₁×#6864	C₃	semidirect
36. #6864	C₃₁⋊C₃	semidirect
49. 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×C₃₁⋊C₂₇	3
C₃₁×#6864	3
#10624713	31