Subgroups — #36379574

Order 22599.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21419308	C₃	1	1	yes	maximal, characteristic
2	#21419226	C₃	1	2	yes	maximal, complemented
3	C₃₁×#6891	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₃²⋊(C₃₁⋊C₃)	—	3	1	no	—
5	C₃×C₃²⋊(C₃₁⋊C₃)	—	3	3	no	—
6	C₃²⋊(C₃₁⋊C₉)	—	3	2	no	—
7	C₃⁴×C₃₁	C₃²	1	1	yes	characteristic
8	C₃×3₋¹⁺²×C₃₁	—	3	1	no	—
9	C₃₁×C₃²⋊C₉	—	3	3	no	—
10	C₃³×C₃₁	—	3	1	no	—
11	C₃³×C₃₁	—	3	1	no	—
12	C₃³×C₃₁	—	3	2	no	—
13	C₃³×C₃₁	—	9	3	no	—
14	C₃³×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
15	C₃²×C₃₁⋊C₃	—	3	3	no	—
16	3₋¹⁺²×C₃₁	—	9	1	no	—
17	3₋¹⁺²×C₃₁	—	9	2	no	—
18	C₃×C₉×C₃₁	—	3	3	no	—
19	C₃²⋊(C₃₁⋊C₃)	—	9	9	no	—
20	C₃²⋊(C₃₁⋊C₃)	—	9	1	no	—
21	C₃²⋊(C₃₁⋊C₃)	—	9	2	no	—
22	C₃×C₃₁⋊C₉	—	9	2	no	—
23	#10624475	—	31	1	no	maximal, Sylow
24	C₃²×C₃₁	—	3	3	no	—
25	C₃²×C₃₁	—	3	3	no	—
26	C₃²×C₃₁	—	9	1	no	—
27	C₃²×C₃₁	—	9	6	no	—
28	C₃²×C₃₁	—	9	2	no	—
29	C₃²×C₃₁	—	9	2	no	—
30	C₃²×C₃₁	—	9	1	no	—
31	C₃²×C₃₁	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
32	C₃²×C₃₁	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
33	C₃²×C₃₁	#732	1	2	yes	—
34	C₃×C₃₁⋊C₃	—	9	9	no	—
35	C₃×C₃₁⋊C₃	—	9	3	no	—
36	C₉×C₃₁	—	9	3	no	—
37	C₃₁⋊C₉	—	27	2	no	—
38	#6891	C₃₁⋊C₃	1	1	yes	characteristic, complemented
39	C₃²⋊H(3,𝔽₃)	—	31	1	no	—
40	#6866	—	31	2	no	—
41	C₃×C₃₁	—	3	2	no	—
42	C₃×C₃₁	—	3	1	no	—
43	C₃×C₃₁	—	3	1	no	—
44	C₃×C₃₁	—	9	3	no	—
45	C₃×C₃₁	#6856	1	1	yes	characteristic
46	C₃₁⋊C₃	—	27	3	no	—
47	C₃⁴	C₃×C₃₁⋊C₃	1	1	yes	characteristic
48	C₃×3₋¹⁺²	—	3	1	no	—
49	C₃×H(3,𝔽₃)	—	93	3	no	—
50	C₃×H(3,𝔽₃)	—	93	1	no	—
51	C₃²⋊C₉	—	3	3	no	—
52	C₃²⋊C₉	—	93	2	no	—
53	C₃₁	#10624475	1	1	yes	characteristic, complemented, Sylow
54	C₃³	—	3	1	no	—
55	C₃³	—	3	1	no	—
56	C₃³	—	3	2	no	—
57	C₃³	—	9	3	no	—
58	C₃³	—	93	3	no	—
59	C₃³	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
60	3₋¹⁺²	—	9	1	no	—
61	3₋¹⁺²	—	9	2	no	—
62	H(3,𝔽₃)	—	279	9	no	—
63	H(3,𝔽₃)	—	279	1	no	—
64	H(3,𝔽₃)	—	279	2	no	—
65	C₃×C₉	—	3	3	no	—
66	C₃×C₉	—	279	2	no	—
67	C₃²	—	3	3	no	—
68	C₃²	—	3	3	no	—
69	C₃²	—	9	1	no	—
70	C₃²	—	9	6	no	—
71	C₃²	—	9	2	no	—
72	C₃²	—	9	2	no	—
73	C₃²	—	9	1	no	—
74	C₃²	—	279	9	no	—
75	C₃²	—	279	3	no	—
76	C₃²	#14578893	1	1	yes	characteristic
77	C₃²	#14578894	1	1	yes	characteristic
78	C₃²	#14578899	1	2	yes	—
79	C₉	—	9	3	no	—
80	C₉	—	837	2	no	—
81	C₃	—	3	2	no	—
82	C₃	—	3	1	no	—
83	C₃	—	3	1	no	—
84	C₃	—	9	3	no	—
85	C₃	—	837	3	no	—
86	C₃	#21419200	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#21419200
commutator, cosocle	C₃⁴×C₃₁	C₃²
Frattini	C₃⁴	C₃×C₃₁⋊C₃
Fitting	C₃₁×#6891	C₃
socle	C₃×C₃₁	#6856
prime radical	#21419308	C₃
hypercenter	#6891	C₃₁⋊C₃
hypocenter	C₃₁	#10624475
LCS	G ▷ C₃⁴×C₃₁ ▷ C₃³×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃⁴ ◁ #6891

Complements

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

N	H ≅ G/N	type
2. #21419226	C₃	semidirect
3. C₃₁×#6891	C₃	semidirect
38. #6891	C₃₁⋊C₃	semidirect
53. C₃₁	#10624475	semidirect

Maximal subgroups

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

subgroup	index
#21419308	3
#21419226	3
C₃₁×#6891	3
#10624475	31