Subgroups — #36379568

Order 22599.

156 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#14578892	C₃	1	2	yes	maximal
2	#21419296	C₃	1	1	yes	maximal, characteristic
3	C₃×C₃₁×C₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic
4	#14578892	—	3	2	no	—
5	#14578892	—	3	2	no	—
6	#14578892	C₉	1	6	yes	complemented
7	C₃×C₃₁⋊C₂₇	—	3	2	no	—
8	#14578887	C₉	1	3	yes	—
9	#14578886	—	3	1	no	—
10	C₃²⋊(C₃₁⋊C₉)	—	3	1	no	—
11	C₉×C₃₁⋊C₉	—	3	1	no	—
12	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
13	C₃₁×C₂₇⋊C₃	—	3	1	no	—
14	C₃₁×C₂₇⋊C₃	—	3	2	no	—
15	C₃×C₂₇×C₃₁	C₉	1	3	yes	complemented
16	C₃³×C₃₁	3₋¹⁺²	1	1	yes	characteristic
17	C₃×C₉×C₃₁	C₃×C₉	1	2	yes	—
18	C₃×C₉×C₃₁	C₃×C₉	1	1	yes	characteristic
19	C₃×C₉×C₃₁	C₃×C₉	1	1	yes	characteristic
20	C₃×C₉×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃×C₉×C₃₁	3₋¹⁺²	1	2	yes	—
22	C₃×C₉×C₃₁	3₋¹⁺²	1	2	yes	—
23	C₃×C₉×C₃₁	3₋¹⁺²	1	2	yes	—
24	C₃×C₉×C₃₁	3₋¹⁺²	1	1	yes	characteristic
25	C₃×C₃₁⋊C₉	—	3	3	no	—
26	C₂₇×C₃₁	—	3	3	no	—
27	C₃₁⋊C₂₇	—	3	6	no	—
28	#10624473	—	31	1	no	maximal, Sylow
29	C₃²×C₃₁	—	9	1	no	—
30	C₃²×C₃₁	C₃²⋊C₉	1	1	yes	characteristic
31	C₃²×C₃₁	C₉⋊C₉	1	2	yes	—
32	C₃²×C₃₁	C₉⋊C₉	1	1	yes	characteristic
33	C₉×C₃₁	C₉²	1	1	yes	characteristic
34	C₉×C₃₁	C₃²⋊C₉	1	2	yes	—
35	C₉×C₃₁	C₃²⋊C₉	1	1	yes	characteristic
36	C₉×C₃₁	C₉⋊C₉	1	2	yes	—
37	C₉×C₃₁	C₉⋊C₉	1	2	yes	—
38	C₉×C₃₁	C₉⋊C₉	1	1	yes	characteristic
39	C₃₁⋊C₉	—	9	3	no	—
40	C₃×C₂₇⋊C₃	—	31	2	no	—
41	C₃×C₂₇⋊C₃	C₃₁⋊C₃	1	1	yes	characteristic
42	#6887	—	31	1	no	—
43	C₃×C₃₁	—	3	1	no	—
44	C₃×C₃₁	—	3	2	no	—
45	C₃×C₃₁	—	3	1	no	—
46	C₃×C₃₁	#6855	1	1	yes	characteristic
47	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
48	C₂₇⋊C₃	—	3	2	no	—
49	C₂₇⋊C₃	—	3	1	no	—
50	C₂₇⋊C₃	—	31	6	no	—
51	C₂₇⋊C₃	—	93	2	no	—
52	C₂₇⋊C₃	—	93	2	no	—
53	C₃×C₂₇	—	93	2	no	—
54	C₃×C₂₇	C₃₁⋊C₉	1	3	yes	complemented
55	C₉⋊C₉	—	31	3	no	—
56	C₉⋊C₉	—	93	1	no	—
57	C₃²⋊C₉	—	93	1	no	—
58	C₉²	—	93	1	no	—
59	C₃₁	#10624473	1	1	yes	characteristic, complemented, Sylow
60	C₃³	#11719686	1	1	yes	characteristic
61	C₃×C₉	—	93	3	no	—
62	C₃×C₉	C₉×C₃₁⋊C₃	1	2	yes	—
63	C₃×C₉	C₉×C₃₁⋊C₃	1	1	yes	characteristic
64	C₃×C₉	#11719686	1	2	yes	—
65	C₃×C₉	#11719687	1	2	yes	—
66	C₃×C₉	#11719687	1	1	yes	characteristic
67	C₃×C₉	C₃×C₃₁⋊C₉	1	1	yes	characteristic
68	C₃×C₉	#11719689	1	2	yes	—
69	C₃×C₉	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
70	C₂₇	—	3	3	no	—
71	C₂₇	—	93	6	no	—
72	C₃²	—	9	1	no	—
73	C₃²	(C₃×C₃₁⋊C₃)⋊C₉	1	1	yes	characteristic
74	C₃²	(C₃₁⋊C₉)⋊C₉	1	2	yes	—
75	C₃²	#14578886	1	1	yes	characteristic
76	C₉	—	279	3	no	—
77	C₉	C₉×C₃₁⋊C₉	1	1	yes	characteristic
78	C₉	(C₃×C₃₁⋊C₃)⋊C₉	1	2	yes	—
79	C₉	C₃²⋊(C₃₁⋊C₉)	1	1	yes	characteristic
80	C₉	(C₃₁⋊C₉)⋊C₉	1	2	yes	—
81	C₉	(C₃₁⋊C₉)⋊C₉	1	2	yes	—
82	C₉	#14578887	1	1	yes	characteristic
83	C₃	—	3	1	no	—
84	C₃	—	3	2	no	—
85	C₃	—	3	1	no	—
86	C₃	C₃₁⋊#6855	1	1	yes	characteristic

	subgroup	G/H
center	C₉	#14578887
commutator	C₉×C₃₁	C₉²
Frattini	C₃²×C₉	C₃×C₃₁⋊C₃
Fitting	C₃×C₃₁×C₂₇⋊C₃	C₃
socle	C₃×C₃₁	#6855
prime radical	C₃³×C₃₁	3₋¹⁺²
cosocle	C₃²×C₉×C₃₁	C₃²
hypercenter	C₃×C₂₇⋊C₃	C₃₁⋊C₃
hypocenter	C₃₁	#10624473
LCS	G ▷ C₉×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₉ ◁ C₃²×C₉ ◁ C₃×C₂₇⋊C₃

Complements

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

N	H ≅ G/N	type
6. #14578892	C₉	semidirect
15. C₃×C₂₇×C₃₁	C₉	semidirect
54. C₃×C₂₇	C₃₁⋊C₉	semidirect
59. C₃₁	#10624473	semidirect

Maximal subgroups

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

subgroup	index
C₃×#14578892	3
#21419296	3
C₃×C₃₁×C₂₇⋊C₃	3
#10624473	31