Subgroups — #36382277

Order 22599.

926 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	3₋¹⁺²×C₃₁⋊C₉	C₃	1	24	yes	maximal, complemented
2	H(3,𝔽₃)×C₃₁⋊C₉	C₃	1	3	yes	maximal
3	#21419296	C₃	1	8	yes	maximal, complemented
4	C₃×#14578886	C₃	1	4	yes	maximal, complemented
5	C₃×C₃₁×#737	C₃	1	1	yes	maximal, characteristic
6	C₃²×C₃₁⋊C₉	C₃²	1	12	yes	—
7	#14578887	C₃²	1	24	yes	—
8	#14578886	C₃²	1	1	yes	characteristic
9	#14578886	C₃²	1	8	yes	—
10	#14578886	C₃²	1	24	yes	—
11	C₃²⋊(C₃₁⋊C₉)	C₃²	1	24	yes	—
12	C₉×C₃₁⋊C₉	C₃²	1	24	yes	—
13	C₃₁×#737	C₉	1	27	yes	complemented
14	C₃×3₋¹⁺²×C₃₁	C₃²	1	8	yes	—
15	C₃×H(3,𝔽₃)×C₃₁	C₃²	1	1	yes	characteristic
16	C₃²×C₉×C₃₁	C₃²	1	4	yes	—
17	C₃³×C₃₁	C₃³	1	4	yes	—
18	3₋¹⁺²×C₃₁	C₃×C₉	1	72	yes	complemented
19	H(3,𝔽₃)×C₃₁	C₃×C₉	1	9	yes	—
20	C₃×C₉×C₃₁	C₃×C₉	1	36	yes	complemented
21	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
22	C₃×C₉×C₃₁	C₃³	1	8	yes	—
23	C₃×C₃₁⋊C₉	—	9	12	no	—
24	C₃×C₃₁⋊C₉	C₃³	1	24	yes	—
25	C₃×C₃₁⋊C₉	C₃³	1	3	yes	—
26	#10624864	—	31	1	no	maximal, Sylow
27	C₃²×C₃₁	—	3	12	no	—
28	C₃²×C₃₁	C₃²×C₉	1	12	yes	—
29	C₃²×C₃₁	C₃⁴	1	1	yes	characteristic
30	C₉×C₃₁	C₃²×C₉	1	3	yes	—
31	C₉×C₃₁	C₃²×C₉	1	24	yes	—
32	C₃₁⋊C₉	—	3	3	no	—
33	C₃₁⋊C₉	—	3	24	no	—
34	C₃×#737	C₃₁⋊C₃	1	1	yes	characteristic
35	C₉×3₋¹⁺²	—	31	24	no	—
36	C₉×H(3,𝔽₃)	—	31	3	no	—
37	#6887	—	31	8	no	—
38	C₃×C₉⋊C₉	—	31	4	no	—
39	C₃×C₃₁	—	3	12	no	—
40	C₃×C₃₁	C₃³×C₉	1	1	yes	characteristic
41	C₃×C₃₁	3₋¹⁺⁴	1	3	yes	—
42	#737	C₃₁⋊C₉	1	27	yes	complemented
43	C₃×3₋¹⁺²	C₃×C₃₁⋊C₃	1	8	yes	—
44	C₃×H(3,𝔽₃)	C₃×C₃₁⋊C₃	1	1	yes	characteristic
45	C₃²×C₉	—	31	12	no	—
46	C₃²×C₉	C₃×C₃₁⋊C₃	1	4	yes	—
47	C₉⋊C₉	—	31	1	no	—
48	C₉⋊C₉	—	31	8	no	—
49	C₉⋊C₉	—	31	24	no	—
50	C₉⋊C₉	—	31	24	no	—
51	C₃²⋊C₉	—	31	24	no	—
52	C₉²	—	31	24	no	—
53	C₃₁	#10624864	1	1	yes	characteristic, complemented, Sylow
54	C₃³	C₃²×C₃₁⋊C₃	1	4	yes	—
55	3₋¹⁺²	C₃×C₃₁⋊C₉	1	72	yes	complemented
56	H(3,𝔽₃)	C₃×C₃₁⋊C₉	1	9	yes	—
57	C₃×C₉	—	279	12	no	—
58	C₃×C₉	—	31	24	no	—
59	C₃×C₉	—	31	3	no	—
60	C₃×C₉	C₃×C₃₁⋊C₉	1	36	yes	complemented
61	C₃×C₉	C₃²×C₃₁⋊C₃	1	8	yes	—
62	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
63	C₃²	—	3	12	no	—
64	C₃²	C₃²×C₃₁⋊C₉	1	12	yes	—
65	C₃²	C₃³×C₃₁⋊C₃	1	1	yes	characteristic
66	C₉	—	93	3	no	—
67	C₉	—	93	24	no	—
68	C₉	C₃²×C₃₁⋊C₉	1	3	yes	—
69	C₉	C₃²×C₃₁⋊C₉	1	24	yes	—
70	C₃	—	3	12	no	—
71	C₃	C₃³×C₃₁⋊C₉	1	1	yes	characteristic
72	C₃	#21419436	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₁⋊C₃
commutator	C₃×C₃₁	C₃³×C₉
Fitting	C₃×C₃₁×#737	C₃
socle, cosocle	C₃²×C₃₁	C₃⁴
prime radical	C₃×H(3,𝔽₃)×C₃₁	C₃²
hypercenter	C₃×#737	C₃₁⋊C₃
hypocenter	C₃₁	#10624864
LCS	G ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃² ◁ C₃×#737

Complements

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

N	H ≅ G/N	type
1. 3₋¹⁺²×C₃₁⋊C₉	C₃	semidirect
3. #21419296	C₃	semidirect
4. C₃×#14578886	C₃	semidirect
13. C₃₁×#737	C₉	semidirect
18. 3₋¹⁺²×C₃₁	C₃×C₉	semidirect
20. C₃×C₉×C₃₁	C₃×C₉	semidirect
42. #737	C₃₁⋊C₉	semidirect
53. C₃₁	#10624864	semidirect
55. 3₋¹⁺²	C₃×C₃₁⋊C₉	semidirect
60. C₃×C₉	C₃×C₃₁⋊C₉	semidirect

Maximal subgroups

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

subgroup	index
3₋¹⁺²×C₃₁⋊C₉	3
H(3,𝔽₃)×C₃₁⋊C₉	3
#21419296	3
C₃×#14578886	3
C₃×C₃₁×#737	3
#10624864	31