Subgroups — #36379583

Order 22599.

154 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21419312	C₃	1	2	yes	maximal
2	#21419226	C₃	1	1	yes	maximal, characteristic
3	C₃₁×#6866	C₃	1	1	yes	maximal, characteristic
4	C₃×#11719689	—	3	2	no	—
5	C₃²⋊(C₃₁⋊C₉)	—	3	6	no	—
6	C₃²⋊(C₃₁⋊C₉)	—	3	1	no	—
7	C₃⁴×C₃₁	C₃²	1	1	yes	characteristic
8	C₃₁×C₃²⋊C₉	—	3	1	no	—
9	C₃³×C₃₁	—	3	2	no	—
10	C₃³×C₃₁	—	3	1	no	—
11	C₃³×C₃₁	—	3	1	no	—
12	C₃³×C₃₁	—	9	3	no	—
13	C₃³×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
14	C₃×C₉×C₃₁	—	9	1	no	—
15	#11719689	—	9	2	no	—
16	#11719689	—	9	2	no	—
17	#11719689	—	9	2	no	—
18	C₃×C₃₁⋊C₉	—	3	6	no	—
19	C₃×C₃₁⋊C₉	—	9	1	no	—
20	#10624476	—	31	1	no	maximal, Sylow
21	C₃²×C₃₁	—	3	6	no	—
22	C₃²×C₃₁	—	9	1	no	—
23	C₃²×C₃₁	—	9	3	no	—
24	C₃²×C₃₁	—	9	3	no	—
25	C₃²×C₃₁	—	9	4	no	—
26	C₃²×C₃₁	—	9	1	no	—
27	C₃²×C₃₁	AΣL(1,𝔽₂₇)	1	2	yes	—
28	C₃²×C₃₁	#732	1	1	yes	characteristic
29	C₃²×C₃₁	#732	1	1	yes	characteristic
30	C₉×C₃₁	—	27	1	no	—
31	C₃₁⋊C₉	—	9	6	no	—
32	C₃₁⋊C₉	—	27	1	no	—
33	#6891	—	31	2	no	—
34	#6866	—	31	1	no	—
35	#6866	C₃₁⋊C₃	1	1	yes	characteristic
36	C₃×C₃₁	—	3	1	no	—
37	C₃×C₃₁	—	3	1	no	—
38	C₃×C₃₁	—	3	2	no	—
39	C₃×C₃₁	—	9	3	no	—
40	C₃×C₃₁	#6856	1	1	yes	characteristic
41	C₃⁴	C₃×C₃₁⋊C₃	1	1	yes	characteristic
42	C₃×3₋¹⁺²	—	93	2	no	—
43	C₃²⋊C₉	—	3	1	no	—
44	C₃²⋊C₉	—	93	1	no	—
45	C₃²⋊C₉	—	93	6	no	—
46	C₃₁	#10624476	1	1	yes	characteristic, complemented, Sylow
47	C₃³	—	3	2	no	—
48	C₃³	—	3	1	no	—
49	C₃³	—	3	1	no	—
50	C₃³	—	9	3	no	—
51	C₃³	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
52	3₋¹⁺²	—	279	2	no	—
53	3₋¹⁺²	—	279	2	no	—
54	3₋¹⁺²	—	279	2	no	—
55	C₃×C₉	—	9	1	no	—
56	C₃×C₉	—	279	1	no	—
57	C₃×C₉	—	93	6	no	—
58	C₃²	—	3	6	no	—
59	C₃²	—	9	1	no	—
60	C₃²	—	9	3	no	—
61	C₃²	—	9	3	no	—
62	C₃²	—	9	4	no	—
63	C₃²	—	9	1	no	—
64	C₃²	#14578895	1	2	yes	—
65	C₃²	#14578899	1	1	yes	characteristic
66	C₃²	#14578900	1	1	yes	characteristic
67	C₉	—	279	6	no	—
68	C₉	—	27	1	no	—
69	C₉	—	837	1	no	—
70	C₃	—	3	1	no	—
71	C₃	—	3	1	no	—
72	C₃	—	3	2	no	—
73	C₃	—	9	3	no	—
74	C₃	#21419201	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#21419201
commutator, prime radical, cosocle	C₃⁴×C₃₁	C₃²
Frattini	C₃⁴	C₃×C₃₁⋊C₃
Fitting	C₃₁×#6866	C₃
socle	C₃×C₃₁	#6856
hypercenter	#6866	C₃₁⋊C₃
hypocenter	C₃₁	#10624476
LCS	G ▷ C₃⁴×C₃₁ ▷ C₃³×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃⁴ ◁ #6866

Complements

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

N	H ≅ G/N	type
46. C₃₁	#10624476	semidirect

Maximal subgroups

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

subgroup	index
#21419312	3
#21419226	3
C₃₁×#6866	3
#10624476	31