Subgroups — #36379857

Order 22599.

374 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×(C₃₁⋊C₉)⋊C₉	C₃	1	6	yes	maximal
2	C₃×C₉×C₃₁⋊C₉	C₃	1	1	yes	maximal, characteristic
3	C₃₁⋊#6855	C₃	1	3	yes	maximal
4	C₃₁⋊#6855	C₃	1	2	yes	maximal
5	C₃₁×#6855	C₃	1	1	yes	maximal, characteristic
6	C₃²×C₃₁⋊C₉	C₃²	1	6	yes	—
7	C₃²×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
8	C₃²×C₃₁⋊C₉	C₃²	1	2	yes	—
9	(C₃₁⋊C₉)⋊C₉	—	3	18	no	—
10	C₉×C₃₁⋊C₉	C₉	1	9	yes	complemented
11	C₃²×C₉×C₃₁	C₃²	1	3	yes	—
12	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
13	C₃³×C₃₁	C₃³	1	1	yes	characteristic
14	C₃×C₉×C₃₁	—	3	1	no	—
15	C₃×C₉×C₃₁	—	3	2	no	—
16	C₃×C₉×C₃₁	—	3	6	no	—
17	C₃×C₉×C₃₁	—	3	3	no	—
18	C₃×C₉×C₃₁	—	3	3	no	—
19	C₃×C₉×C₃₁	C₃×C₉	1	3	yes	—
20	C₃×C₃₁⋊C₉	—	3	3	no	—
21	C₃×C₃₁⋊C₉	—	3	12	no	—
22	C₃×C₃₁⋊C₉	—	3	6	no	—
23	C₃×C₃₁⋊C₉	—	3	6	no	—
24	C₃×C₃₁⋊C₉	C₃×C₉	1	3	yes	—
25	C₃×C₃₁⋊C₉	C₃×C₉	1	6	yes	—
26	C₃×C₃₁⋊C₉	3₋¹⁺²	1	18	yes	—
27	#10624550	—	31	1	no	maximal, Sylow
28	C₃²×C₃₁	C₃²×C₉	1	1	yes	characteristic
29	C₃²×C₃₁	C₃×3₋¹⁺²	1	6	yes	—
30	C₃²×C₃₁	#737	1	1	yes	characteristic
31	C₃²×C₃₁	#737	1	3	yes	—
32	C₃²×C₃₁	#737	1	2	yes	—
33	C₉×C₃₁	—	3	3	no	—
34	C₉×C₃₁	—	9	3	no	—
35	C₃₁⋊C₉	—	3	3	no	—
36	C₃₁⋊C₉	—	9	6	no	—
37	C₃₁⋊C₉	C₉⋊C₉	1	18	yes	complemented
38	C₃×C₉⋊C₉	—	31	6	no	—
39	C₃×C₉²	—	31	1	no	—
40	#6855	—	31	3	no	—
41	#6855	—	31	2	no	—
42	#6855	C₃₁⋊C₃	1	1	yes	characteristic
43	C₃×C₃₁	C₃×C₉⋊C₉	1	2	yes	—
44	C₃×C₃₁	#6887	1	1	yes	characteristic
45	C₃×C₃₁	#6887	1	1	yes	characteristic
46	C₃×C₃₁	#6896	1	3	yes	—
47	C₃×C₃₁	#6896	1	6	yes	—
48	C₃²×C₉	—	31	2	no	—
49	C₃²×C₉	—	31	1	no	—
50	C₃²×C₉	—	31	6	no	—
51	C₃²×C₉	C₃×C₃₁⋊C₃	1	3	yes	—
52	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
53	C₉⋊C₉	—	93	18	no	—
54	C₉²	—	31	9	no	—
55	C₃₁	#10624550	1	1	yes	characteristic, complemented, Sylow
56	C₃³	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
57	C₃×C₉	—	3	1	no	—
58	C₃×C₉	—	3	2	no	—
59	C₃×C₉	—	3	6	no	—
60	C₃×C₉	—	3	3	no	—
61	C₃×C₉	—	3	3	no	—
62	C₃×C₉	—	31	6	no	—
63	C₃×C₉	—	31	18	no	—
64	C₃×C₉	—	31	3	no	—
65	C₃×C₉	—	93	12	no	—
66	C₃×C₉	—	93	6	no	—
67	C₃×C₉	—	93	6	no	—
68	C₃×C₉	—	93	3	no	—
69	C₃×C₉	C₉×C₃₁⋊C₃	1	3	yes	—
70	C₃²	C₃×C₉×C₃₁⋊C₃	1	1	yes	characteristic
71	C₃²	3₋¹⁺²×C₃₁⋊C₃	1	6	yes	—
72	C₃²	#14578911	1	1	yes	characteristic
73	C₃²	#14578912	1	3	yes	—
74	C₃²	#14578913	1	2	yes	—
75	C₉	—	3	3	no	—
76	C₉	—	9	3	no	—
77	C₉	—	279	6	no	—
78	C₉	—	31	18	no	—
79	C₉	—	93	3	no	—
80	C₃	C₉⋊C₉×C₃₁⋊C₃	1	2	yes	—
81	C₃	#21419293	1	1	yes	characteristic
82	C₃	#21419294	1	1	yes	characteristic
83	C₃	#21419337	1	3	yes	—
84	C₃	#21419340	1	6	yes	—

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₃₁⋊C₃
commutator	C₃²×C₃₁	C₃²×C₉
Fitting	C₃₁×#6855	C₃
socle, prime radical, cosocle	C₃³×C₃₁	C₃³
hypercenter	#6855	C₃₁⋊C₃
hypocenter	C₃₁	#10624550
LCS	G ▷ C₃²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃³ ◁ #6855

Complements

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

N	H ≅ G/N	type
10. C₉×C₃₁⋊C₉	C₉	semidirect
37. C₃₁⋊C₉	C₉⋊C₉	semidirect
55. C₃₁	#10624550	semidirect

Maximal subgroups

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

subgroup	index
C₃×(C₃₁⋊C₉)⋊C₉	3
C₃×C₉×C₃₁⋊C₉	3
C₃₁⋊#6855	3
C₃₁×#6855	3
#10624550	31