Subgroups — #36379721

Order 22599.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21419232	C₃	1	3	yes	maximal, complemented
2	C₃×C₃₁×#737	C₃	1	1	yes	maximal, characteristic
3	#14578892	—	3	3	no	—
4	C₃₁×#737	—	3	9	no	—
5	C₃×3₋¹⁺²×C₃₁	—	3	1	no	—
6	C₃×3₋¹⁺²×C₃₁	—	3	1	no	—
7	C₃×3₋¹⁺²×C₃₁	C₉	1	2	yes	—
8	C₃×H(3,𝔽₃)×C₃₁	C₉	1	1	yes	characteristic
9	C₃²×C₉×C₃₁	—	3	1	no	—
10	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
11	C₃³×C₃₁	—	3	1	no	—
12	C₃³×C₃₁	C₃×C₉	1	1	yes	characteristic
13	3₋¹⁺²×C₃₁	—	3	9	no	—
14	3₋¹⁺²×C₃₁	—	3	9	no	—
15	3₋¹⁺²×C₃₁	—	3	6	no	—
16	H(3,𝔽₃)×C₃₁	—	3	3	no	—
17	C₃×C₉×C₃₁	—	3	9	no	—
18	C₃×C₉×C₃₁	—	3	2	no	—
19	C₃×C₉×C₃₁	—	3	3	no	—
20	C₃×C₉×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃×C₉×C₃₁	3₋¹⁺²	1	1	yes	characteristic
22	C₃×C₉×C₃₁	3₋¹⁺²	1	1	yes	characteristic
23	C₃₁⋊C₂₇	—	9	3	no	—
24	#10624510	—	31	1	no	maximal, Sylow
25	C₃²×C₃₁	—	3	1	no	—
26	C₃²×C₃₁	—	3	3	no	—
27	C₃²×C₃₁	—	9	1	no	—
28	C₃²×C₃₁	—	9	3	no	—
29	C₃²×C₃₁	C₃²⋊C₉	1	1	yes	characteristic
30	C₉×C₃₁	—	3	1	no	—
31	C₉×C₃₁	—	3	6	no	—
32	C₉×C₃₁	—	3	1	no	—
33	C₉×C₃₁	AΣL(1,𝔽₂₇)	1	3	yes	—
34	C₃×#737	C₃₁⋊C₃	1	1	yes	characteristic
35	#6869	—	31	3	no	—
36	C₃×C₃₁	—	3	1	no	—
37	C₃×C₃₁	—	9	1	no	—
38	C₃×C₃₁	—	9	3	no	—
39	C₃×C₃₁	#6866	1	1	yes	characteristic
40	#737	—	3	9	no	—
41	C₃×3₋¹⁺²	—	3	1	no	—
42	C₃×3₋¹⁺²	—	3	1	no	—
43	C₃×3₋¹⁺²	C₃₁⋊C₉	1	2	yes	—
44	C₃×H(3,𝔽₃)	C₃₁⋊C₉	1	1	yes	characteristic
45	C₃²×C₉	—	3	1	no	—
46	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
47	C₂₇⋊C₃	—	93	3	no	—
48	C₃₁	#10624510	1	1	yes	characteristic, complemented, Sylow
49	C₃³	—	3	1	no	—
50	C₃³	C₃×C₃₁⋊C₉	1	1	yes	characteristic
51	3₋¹⁺²	—	3	9	no	—
52	3₋¹⁺²	—	3	9	no	—
53	3₋¹⁺²	—	3	6	no	—
54	H(3,𝔽₃)	—	3	3	no	—
55	C₃×C₉	—	3	9	no	—
56	C₃×C₉	—	3	2	no	—
57	C₃×C₉	—	3	3	no	—
58	C₃×C₉	#11719689	1	1	yes	characteristic
59	C₃×C₉	#11719689	1	1	yes	characteristic
60	C₃×C₉	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
61	C₂₇	—	279	3	no	—
62	C₃²	—	3	1	no	—
63	C₃²	—	3	3	no	—
64	C₃²	—	9	1	no	—
65	C₃²	—	9	3	no	—
66	C₃²	C₃²⋊(C₃₁⋊C₉)	1	1	yes	characteristic
67	C₉	—	3	1	no	—
68	C₉	—	3	6	no	—
69	C₉	—	3	1	no	—
70	C₉	#14578894	1	3	yes	—
71	C₃	—	3	1	no	—
72	C₃	—	9	3	no	—
73	C₃	—	9	1	no	—
74	C₃	#21419226	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#21419226
commutator	C₃³×C₃₁	C₃×C₉
Frattini	C₃²×C₉	C₃×C₃₁⋊C₃
Fitting	C₃×C₃₁×#737	C₃
socle	C₃×C₃₁	#6866
prime radical	C₃×H(3,𝔽₃)×C₃₁	C₉
cosocle	C₃²×C₉×C₃₁	C₃²
hypercenter	C₃×#737	C₃₁⋊C₃
hypocenter	C₃₁	#10624510
LCS	G ▷ C₃³×C₃₁ ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ C₃×#737

Complements

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

N	H ≅ G/N	type
1. #21419232	C₃	semidirect
48. C₃₁	#10624510	semidirect

Maximal subgroups

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

subgroup	index
#21419232	3
C₃×C₃₁×#737	3
#10624510	31