Subgroups — #36381075

Order 22599.

218 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#14578892	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₃₁⋊C₂₇	C₃	1	1	yes	maximal, characteristic
3	#21419296	C₃	1	1	yes	maximal, characteristic
4	(C₃₁⋊C₉)⋊C₂₇	C₃	1	9	yes	maximal, complemented
5	C₃×C₃₁×C₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic
6	#14578892	—	3	3	no	—
7	C₃×C₃₁⋊C₂₇	—	3	3	no	—
8	C₃×C₃₁⋊C₂₇	C₃²	1	3	yes	—
9	C₃×C₃₁⋊C₂₇	C₃²	1	3	yes	—
10	#14578887	C₉	1	3	yes	—
11	#14578886	C₉	1	3	yes	—
12	C₃²⋊(C₃₁⋊C₉)	C₉	1	3	yes	—
13	C₉×C₃₁⋊C₉	C₃²	1	3	yes	—
14	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
15	C₃₁×C₂₇⋊C₃	—	3	3	no	—
16	C₃×C₂₇×C₃₁	C₃²	1	3	yes	—
17	C₃³×C₃₁	C₃×C₉	1	1	yes	characteristic
18	C₃×C₉×C₃₁	C₃×C₉	1	1	yes	characteristic
19	C₃×C₉×C₃₁	C₃×C₉	1	1	yes	characteristic
20	C₃×C₉×C₃₁	3₋¹⁺²	1	3	yes	—
21	C₃×C₉×C₃₁	3₋¹⁺²	1	3	yes	—
22	C₃×C₉×C₃₁	3₋¹⁺²	1	3	yes	—
23	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
24	C₃×C₃₁⋊C₉	C₃×C₉	1	9	yes	—
25	C₂₇×C₃₁	—	3	3	no	—
26	C₃₁⋊C₂₇	—	3	3	no	—
27	C₃₁⋊C₂₇	—	3	3	no	—
28	#10624707	—	31	1	no	maximal, Sylow
29	C₃²×C₃₁	—	3	3	no	—
30	C₃²×C₃₁	C₉⋊C₉	1	3	yes	—
31	C₃²×C₃₁	C₃²×C₉	1	1	yes	characteristic
32	C₉×C₃₁	C₉⋊C₉	1	3	yes	—
33	C₉×C₃₁	C₉⋊C₉	1	3	yes	—
34	C₉×C₃₁	C₃×3₋¹⁺²	1	1	yes	characteristic
35	C₉×C₃₁	C₃×3₋¹⁺²	1	1	yes	characteristic
36	C₉×C₃₁	C₃×3₋¹⁺²	1	1	yes	characteristic
37	C₃₁⋊C₉	—	3	9	no	—
38	C₃×C₂₇⋊C₃	—	31	1	no	—
39	C₃×C₂₇⋊C₃	C₃₁⋊C₃	1	1	yes	characteristic
40	C₃²×C₂₇	—	31	1	no	—
41	#6887	—	31	1	no	—
42	C₉⋊C₂₇	—	31	9	no	—
43	C₃×C₃₁	—	3	3	no	—
44	C₃×C₃₁	C₃×C₉⋊C₉	1	1	yes	characteristic
45	C₃×C₃₁	#6903	1	3	yes	—
46	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
47	C₂₇⋊C₃	—	3	3	no	—
48	C₂₇⋊C₃	—	93	3	no	—
49	C₃×C₂₇	—	31	3	no	—
50	C₃×C₂₇	—	31	3	no	—
51	C₃×C₂₇	—	93	3	no	—
52	C₃×C₂₇	C₃×C₃₁⋊C₃	1	3	yes	—
53	C₉⋊C₉	—	31	3	no	—
54	C₉⋊C₉	—	31	3	no	—
55	C₃²⋊C₉	—	31	3	no	—
56	C₉²	—	31	3	no	—
57	C₃₁	#10624707	1	1	yes	characteristic, complemented, Sylow
58	C₃³	C₉×C₃₁⋊C₃	1	1	yes	characteristic
59	C₃×C₉	—	31	9	no	—
60	C₃×C₉	C₉×C₃₁⋊C₃	1	1	yes	characteristic
61	C₃×C₉	C₉×C₃₁⋊C₃	1	1	yes	characteristic
62	C₃×C₉	#11719686	1	3	yes	—
63	C₃×C₉	#11719687	1	3	yes	—
64	C₃×C₉	#11719689	1	3	yes	—
65	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
66	C₂₇	—	3	3	no	—
67	C₂₇	—	93	3	no	—
68	C₂₇	—	93	3	no	—
69	C₃²	—	3	3	no	—
70	C₃²	(C₃₁⋊C₉)⋊C₉	1	3	yes	—
71	C₃²	C₃×C₉×C₃₁⋊C₃	1	1	yes	characteristic
72	C₉	—	93	9	no	—
73	C₉	(C₃₁⋊C₉)⋊C₉	1	3	yes	—
74	C₉	(C₃₁⋊C₉)⋊C₉	1	3	yes	—
75	C₉	C₃×#11719687	1	1	yes	characteristic
76	C₉	C₃×#11719686	1	1	yes	characteristic
77	C₉	C₃×#11719689	1	1	yes	characteristic
78	C₃	—	3	3	no	—
79	C₃	C₃×(C₃₁⋊C₉)⋊C₉	1	1	yes	characteristic
80	C₃	#21419372	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃×C₉	C₃²×C₃₁⋊C₃
commutator, socle	C₃²×C₃₁	C₃²×C₉
Fitting	C₃×C₃₁×C₂₇⋊C₃	C₃
prime radical	C₃³×C₃₁	C₃×C₉
cosocle	C₃×C₉×C₃₁	C₃³
hypercenter	C₃×C₂₇⋊C₃	C₃₁⋊C₃
hypocenter	C₃₁	#10624707
LCS	G ▷ C₃²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃×C₉ ◁ C₃×C₂₇⋊C₃

Complements

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

N	H ≅ G/N	type
4. (C₃₁⋊C₉)⋊C₂₇	C₃	semidirect
57. C₃₁	#10624707	semidirect

Maximal subgroups

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

subgroup	index
C₃×#14578892	3
C₃²×C₃₁⋊C₂₇	3
#21419296	3
(C₃₁⋊C₉)⋊C₂₇	3
C₃×C₃₁×C₂₇⋊C₃	3
#10624707	31