Subgroups — #36379720

Order 22599.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#14578912	C₃	1	1	yes	maximal, characteristic
2	#21419232	C₃	1	2	yes	maximal, complemented
3	C₃₁×#6869	C₃	1	1	yes	maximal, characteristic, complemented
4	#14578912	—	3	3	no	—
5	#14578912	—	3	6	no	—
6	C₃×#11719689	C₉	1	2	yes	—
7	C₃×#11719686	—	3	1	no	—
8	C₃×#11719687	—	3	1	no	—
9	C₃×C₃²⋊(C₃₁⋊C₃)	C₉	1	1	yes	characteristic
10	C₃×C₉×C₃₁⋊C₃	—	3	1	no	—
11	#14578892	—	3	2	no	—
12	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
13	C₃₁×C₂₇⋊C₃	—	3	1	no	—
14	C₃³×C₃₁	C₃×C₉	1	1	yes	characteristic
15	C₃²×C₃₁⋊C₃	—	3	1	no	—
16	C₃×C₉×C₃₁	—	3	1	no	—
17	C₃×C₉×C₃₁	—	3	2	no	—
18	C₃×C₉×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
19	C₃×C₉×C₃₁	3₋¹⁺²	1	1	yes	characteristic
20	C₃×C₉×C₃₁	3₋¹⁺²	1	1	yes	characteristic
21	C₃²⋊(C₃₁⋊C₃)	—	3	3	no	—
22	#11719689	—	3	6	no	—
23	C₃×C₃₁⋊C₉	—	3	2	no	—
24	#11719687	—	3	9	no	—
25	#11719686	—	3	3	no	—
26	#11719686	—	3	6	no	—
27	C₉×C₃₁⋊C₃	—	3	9	no	—
28	C₂₇×C₃₁	—	9	1	no	—
29	C₃₁⋊C₂₇	—	9	2	no	—
30	#10624510	—	31	1	no	maximal, Sylow
31	C₃²×C₃₁	—	3	1	no	—
32	C₃²×C₃₁	—	9	1	no	—
33	C₃²×C₃₁	C₃²⋊C₉	1	1	yes	characteristic
34	C₃×C₃₁⋊C₃	—	3	3	no	—
35	C₃×C₃₁⋊C₃	—	9	3	no	—
36	C₉×C₃₁	—	3	1	no	—
37	C₉×C₃₁	—	3	1	no	—
38	C₉×C₃₁	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
39	C₉×C₃₁	AΣL(1,𝔽₂₇)	1	2	yes	—
40	C₃₁⋊C₉	—	3	6	no	—
41	C₃×#737	—	31	1	no	—
42	#6869	—	31	2	no	—
43	#6869	C₃₁⋊C₃	1	1	yes	characteristic, complemented
44	C₃×C₃₁	—	3	1	no	—
45	C₃×C₃₁	—	9	1	no	—
46	C₃×C₃₁	#6866	1	1	yes	characteristic
47	C₃₁⋊C₃	—	9	3	no	—
48	#737	—	93	3	no	—
49	#737	—	93	6	no	—
50	C₃×3₋¹⁺²	—	31	2	no	—
51	C₃×3₋¹⁺²	—	93	1	no	—
52	C₃×3₋¹⁺²	—	93	1	no	—
53	C₃×H(3,𝔽₃)	—	31	1	no	—
54	C₃²×C₉	—	93	1	no	—
55	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
56	C₂₇⋊C₃	—	3	1	no	—
57	C₂₇⋊C₃	—	93	2	no	—
58	C₃₁	#10624510	1	1	yes	characteristic, complemented, Sylow
59	C₃³	—	93	1	no	—
60	C₃³	C₉×C₃₁⋊C₃	1	1	yes	characteristic
61	3₋¹⁺²	—	93	9	no	—
62	3₋¹⁺²	—	93	3	no	—
63	3₋¹⁺²	—	93	6	no	—
64	3₋¹⁺²	—	93	6	no	—
65	H(3,𝔽₃)	—	93	3	no	—
66	C₃×C₉	—	3	1	no	—
67	C₃×C₉	—	3	2	no	—
68	C₃×C₉	—	93	9	no	—
69	C₃×C₉	—	93	2	no	—
70	C₃×C₉	#11719686	1	1	yes	characteristic
71	C₃×C₉	#11719687	1	1	yes	characteristic
72	C₃×C₉	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
73	C₂₇	—	9	1	no	—
74	C₂₇	—	279	2	no	—
75	C₃²	—	3	1	no	—
76	C₃²	—	9	1	no	—
77	C₃²	—	279	3	no	—
78	C₃²	—	93	3	no	—
79	C₃²	(C₃×C₃₁⋊C₃)⋊C₉	1	1	yes	characteristic
80	C₉	—	3	1	no	—
81	C₉	—	3	1	no	—
82	C₉	—	93	6	no	—
83	C₉	#14578893	1	1	yes	characteristic
84	C₉	#14578895	1	2	yes	—
85	C₃	—	3	1	no	—
86	C₃	—	9	1	no	—
87	C₃	—	279	3	no	—
88	C₃	#21419225	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#21419225
commutator	C₃³×C₃₁	C₃×C₉
Frattini	C₃²×C₉	C₃×C₃₁⋊C₃
Fitting	C₃₁×#6869	C₃
socle	C₃×C₃₁	#6866
prime radical	C₃×C₃²⋊(C₃₁⋊C₃)	C₉
cosocle	C₃²×C₉×C₃₁	C₃²
hypercenter	#6869	C₃₁⋊C₃
hypocenter	C₃₁	#10624510
LCS	G ▷ C₃³×C₃₁ ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ #6869

Complements

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

N	H ≅ G/N	type
2. #21419232	C₃	semidirect
3. C₃₁×#6869	C₃	semidirect
43. #6869	C₃₁⋊C₃	semidirect
58. C₃₁	#10624510	semidirect

Maximal subgroups

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

subgroup	index
C₃×#14578912	3
#21419232	3
C₃₁×#6869	3
#10624510	31