Subgroups — #36379525

Order 22599.

208 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²⋊(C₃₁⋊C₂₇)	C₃	1	3	yes	maximal, complemented
2	C₃×C₃₁×#737	C₃	1	1	yes	maximal, characteristic
3	C₃×C₃₁⋊C₂₇	—	3	3	no	—
4	C₃₁×#737	—	3	9	no	—
5	C₃×3₋¹⁺²×C₃₁	—	3	2	no	—
6	C₃×3₋¹⁺²×C₃₁	C₉	1	2	yes	—
7	C₃×H(3,𝔽₃)×C₃₁	C₉	1	1	yes	characteristic
8	C₃²×C₉×C₃₁	—	3	1	no	—
9	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
10	C₃³×C₃₁	—	3	1	no	—
11	C₃³×C₃₁	C₃×C₉	1	1	yes	characteristic
12	3₋¹⁺²×C₃₁	—	3	18	no	—
13	3₋¹⁺²×C₃₁	—	3	6	no	—
14	H(3,𝔽₃)×C₃₁	—	3	3	no	—
15	C₃×C₉×C₃₁	—	3	2	no	—
16	C₃×C₉×C₃₁	—	3	9	no	—
17	C₃×C₉×C₃₁	—	3	3	no	—
18	C₃×C₉×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
19	C₃×C₉×C₃₁	3₋¹⁺²	1	2	yes	—
20	C₃₁⋊C₂₇	—	9	3	no	—
21	#10624456	—	31	1	no	maximal, Sylow
22	C₃²×C₃₁	—	3	1	no	—
23	C₃²×C₃₁	—	3	3	no	—
24	C₃²×C₃₁	—	3	3	no	—
25	C₃²×C₃₁	—	9	3	no	—
26	C₃²×C₃₁	C₃²⋊C₉	1	1	yes	characteristic
27	C₉×C₃₁	—	3	6	no	—
28	C₉×C₃₁	—	3	2	no	—
29	C₉×C₃₁	AΣL(1,𝔽₂₇)	1	3	yes	—
30	C₃×#737	C₃₁⋊C₃	1	1	yes	characteristic
31	C₃²⋊C₂₇	—	31	3	no	—
32	C₃×C₃₁	—	9	1	no	—
33	C₃×C₃₁	—	9	3	no	—
34	C₃×C₃₁	#6866	1	1	yes	characteristic
35	C₃×C₃₁	#6872	1	3	yes	—
36	#737	—	3	9	no	—
37	C₃×3₋¹⁺²	—	3	2	no	—
38	C₃×3₋¹⁺²	C₃₁⋊C₉	1	2	yes	—
39	C₃×H(3,𝔽₃)	C₃₁⋊C₉	1	1	yes	characteristic
40	C₃²×C₉	—	3	1	no	—
41	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
42	C₃×C₂₇	—	93	3	no	—
43	C₃₁	#10624456	1	1	yes	characteristic, complemented, Sylow
44	C₃³	—	3	1	no	—
45	C₃³	C₃×C₃₁⋊C₉	1	1	yes	characteristic
46	3₋¹⁺²	—	3	18	no	—
47	3₋¹⁺²	—	3	6	no	—
48	H(3,𝔽₃)	—	3	3	no	—
49	C₃×C₉	—	3	2	no	—
50	C₃×C₉	—	3	9	no	—
51	C₃×C₉	—	3	3	no	—
52	C₃×C₉	#11719689	1	2	yes	—
53	C₃×C₉	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
54	C₂₇	—	279	3	no	—
55	C₃²	—	3	1	no	—
56	C₃²	—	3	3	no	—
57	C₃²	—	3	3	no	—
58	C₃²	—	9	3	no	—
59	C₃²	C₃²⋊(C₃₁⋊C₉)	1	1	yes	characteristic
60	C₉	—	3	6	no	—
61	C₉	—	3	2	no	—
62	C₉	#14578894	1	3	yes	—
63	C₃	—	9	1	no	—
64	C₃	—	9	3	no	—
65	C₃	#21419226	1	1	yes	characteristic
66	C₃	#21419241	1	3	yes	—

	subgroup	G/H
center	C₃×C₉	C₃²⋊(C₃₁⋊C₃)
commutator	C₃³×C₃₁	C₃×C₉
Frattini	C₃²×C₉	C₃×C₃₁⋊C₃
Fitting	C₃×C₃₁×#737	C₃
socle	C₃²×C₃₁	C₃²⋊C₉
prime radical	C₃×H(3,𝔽₃)×C₃₁	C₉
cosocle	C₃²×C₉×C₃₁	C₃²
hypercenter	C₃×#737	C₃₁⋊C₃
hypocenter	C₃₁	#10624456
LCS	G ▷ C₃³×C₃₁ ▷ C₃²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ C₃×#737

Complements

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

N	H ≅ G/N	type
1. C₃²⋊(C₃₁⋊C₂₇)	C₃	semidirect
43. C₃₁	#10624456	semidirect

Maximal subgroups

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

subgroup	index
C₃²⋊(C₃₁⋊C₂₇)	3
C₃×C₃₁×#737	3
#10624456	31