Subgroups — C₉×#16812526 #44534662

Order 31347.

986 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#16812526	C₃	1	1	yes	maximal, characteristic
2	#23782792	C₃	1	2	yes	maximal, complemented
3	3₋¹⁺²×C₄₃⋊C₉	C₃	1	2	yes	maximal, complemented
4	C₉×#11923956	C₃	1	3	yes	maximal, complemented
5	C₉×#11923957	C₃	1	3	yes	maximal, complemented
6	C₉×#11923958	C₃	1	1	yes	maximal, characteristic, complemented
7	#11923957⋊C₉	C₃	1	6	yes	maximal, complemented
8	#11923956⋊C₉	C₃	1	6	yes	maximal, complemented
9	C₉×C₃²⋊(C₄₃⋊C₃)	C₃	1	1	yes	maximal, characteristic
10	#23782783	C₃	1	2	yes	maximal
11	#23782781	C₃	1	6	yes	maximal, complemented
12	#23782779	C₃	1	2	yes	maximal, complemented
13	C₃×C₉×C₄₃⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
14	C₉²×C₄₃⋊C₃	C₃	1	3	yes	maximal, complemented
15	C₉×3₋¹⁺²×C₄₃	C₃	1	1	yes	maximal, characteristic, complemented
16	#16812526	C₉	1	27	yes	direct product factor
17	C₃×#11923958	C₃²	1	1	yes	characteristic
18	C₃×#11923957	C₃²	1	3	yes	—
19	C₃×#11923956	C₃²	1	3	yes	—
20	C₃×C₃²⋊(C₄₃⋊C₃)	C₃²	1	1	yes	characteristic
21	C₃²×C₄₃⋊C₉	C₃²	1	2	yes	—
22	C₃²×C₄₃⋊C₉	C₃²	1	1	yes	characteristic
23	C₃×C₉×C₄₃⋊C₃	C₃²	1	6	yes	—
24	C₃×C₉×C₄₃⋊C₃	C₃²	1	3	yes	—
25	C₃×C₉×C₄₃⋊C₃	C₃²	1	3	yes	—
26	#16812500	C₃²	1	6	yes	—
27	#16812499	C₃²	1	6	yes	—
28	(C₄₃⋊C₉)⋊C₉	C₃²	1	6	yes	—
29	(C₄₃⋊C₉)⋊C₉	C₃²	1	6	yes	—
30	(C₄₃⋊C₉)⋊C₉	C₃²	1	6	yes	—
31	(C₄₃⋊C₉)⋊C₉	C₃²	1	6	yes	—
32	(C₄₃⋊C₉)⋊C₉	C₃²	1	6	yes	—
33	C₃²⋊(C₄₃⋊C₉)	C₃²	1	2	yes	—
34	C₃²⋊(C₄₃⋊C₉)	C₃²	1	2	yes	—
35	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	6	yes	—
36	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	6	yes	—
37	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	6	yes	—
38	C₉×C₄₃⋊C₉	C₃²	1	1	yes	characteristic
39	C₉×C₄₃⋊C₉	C₃²	1	6	yes	—
40	C₉×C₄₃⋊C₉	C₃²	1	6	yes	—
41	C₉×C₄₃⋊C₉	C₃²	1	6	yes	—
42	C₉×C₄₃⋊C₉	C₃²	1	3	yes	—
43	C₉×C₄₃⋊C₉	C₃²	1	2	yes	—
44	C₉×C₄₃⋊C₉	C₃²	1	2	yes	—
45	C₉×C₄₃⋊C₉	C₃²	1	2	yes	—
46	C₉×C₄₃⋊C₉	C₃²	1	1	yes	characteristic
47	C₉×C₄₃⋊C₉	C₃²	1	1	yes	characteristic
48	C₃×3₋¹⁺²×C₄₃	C₃²	1	1	yes	characteristic
49	C₃²×C₉×C₄₃	C₃²	1	1	yes	characteristic
50	C₄₃×C₉⋊C₉	C₃²	1	6	yes	—
51	C₄₃×C₃²⋊C₉	C₃²	1	2	yes	—
52	C₉²×C₄₃	C₃²	1	3	yes	—
53	C₃²×C₄₃⋊C₃	C₃³	1	3	yes	—
54	#11923958	C₃×C₉	1	9	yes	complemented
55	#11923957	C₃×C₉	1	27	yes	complemented
56	#11923956	C₃×C₉	1	27	yes	complemented
57	C₃²⋊(C₄₃⋊C₃)	C₃×C₉	1	9	yes	—
58	C₃×C₄₃⋊C₉	—	3	6	no	—
59	C₃×C₄₃⋊C₉	C₃×C₉	1	9	yes	complemented
60	C₃×C₄₃⋊C₉	C₃³	1	2	yes	—
61	C₃×C₄₃⋊C₉	C₃³	1	2	yes	—
62	C₃×C₄₃⋊C₉	C₃³	1	6	yes	—
63	C₃×C₄₃⋊C₉	C₃³	1	6	yes	—
64	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
65	C₃×C₄₃⋊C₉	C₃³	1	3	yes	—
66	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
67	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
68	C₃×C₄₃⋊C₉	C₃³	1	2	yes	—
69	C₉×C₄₃⋊C₃	—	3	9	no	—
70	C₉×C₄₃⋊C₃	—	3	18	no	—
71	C₉×C₄₃⋊C₃	C₃×C₉	1	27	yes	complemented
72	C₃³×C₄₃	C₃³	1	1	yes	characteristic
73	3₋¹⁺²×C₄₃	C₃×C₉	1	9	yes	complemented
74	C₃×C₉×C₄₃	—	3	3	no	—
75	C₃×C₉×C₄₃	C₃³	1	6	yes	—
76	C₃×C₉×C₄₃	C₃³	1	2	yes	—
77	C₃×C₉×C₄₃	C₃³	1	3	yes	—
78	C₃×C₉×C₄₃	C₃³	1	1	yes	characteristic
79	C₉×#737	—	43	1	no	maximal, Sylow
80	C₃²×C₄₃	—	3	1	no	—
81	C₃²×C₄₃	—	3	2	no	—
82	C₃²×C₄₃	C₃²×C₉	1	3	yes	—
83	C₃²×C₄₃	C₃⁴	1	1	yes	characteristic
84	C₃×C₄₃⋊C₃	—	3	6	no	—
85	C₃×C₄₃⋊C₃	—	3	3	no	—
86	C₃×C₄₃⋊C₃	C₃²×C₉	1	9	yes	—
87	C₉×C₄₃	—	3	2	no	—
88	C₉×C₄₃	—	3	6	no	—
89	C₉×C₄₃	C₃²×C₉	1	9	yes	—
90	C₉×C₄₃	#737	1	3	yes	complemented
91	C₄₃⋊C₉	—	3	6	no	—
92	C₄₃⋊C₉	—	3	6	no	—
93	C₄₃⋊C₉	—	3	2	no	—
94	C₄₃⋊C₉	—	3	2	no	—
95	C₄₃⋊C₉	C₃²×C₉	1	3	yes	—
96	C₄₃⋊C₉	C₃²×C₉	1	9	yes	—
97	C₄₃⋊C₉	C₃²×C₉	1	3	yes	—
98	C₄₃⋊C₉	C₃²×C₉	1	3	yes	—
99	C₄₃⋊C₉	#737	1	6	yes	complemented
100	C₃×#737	—	43	1	no	—
101	C₉×3₋¹⁺²	—	43	2	no	—
102	C₉×3₋¹⁺²	—	43	1	no	—
103	C₉×3₋¹⁺²	—	43	6	no	—
104	C₉×3₋¹⁺²	—	43	6	no	—
105	C₉×3₋¹⁺²	—	43	2	no	—
106	C₉×3₋¹⁺²	—	43	3	no	—
107	C₉×3₋¹⁺²	—	43	3	no	—
108	C₉×3₋¹⁺²	C₄₃⋊C₃	1	1	yes	characteristic, complemented
109	C₉×H(3,𝔽₃)	—	43	1	no	—
110	C₉×H(3,𝔽₃)	—	43	2	no	—
111	#6887	—	43	6	no	—
112	#6887	—	43	2	no	—
113	C₃×C₉²	—	43	1	no	—
114	C₃×C₉²	—	43	3	no	—
115	C₃×C₄₃	—	3	3	no	—
116	C₃×C₄₃	C₃³×C₉	1	1	yes	characteristic
117	C₃×C₄₃	C₃×#737	1	2	yes	—
118	C₃×C₄₃	C₃×#737	1	1	yes	characteristic
119	C₄₃⋊C₃	—	3	9	no	—
120	#737	—	43	27	no	—
121	C₃×3₋¹⁺²	—	43	1	no	—
122	C₃×3₋¹⁺²	—	43	3	no	—
123	C₃×3₋¹⁺²	—	43	3	no	—
124	C₃×3₋¹⁺²	C₃×C₄₃⋊C₃	1	1	yes	characteristic
125	C₃×H(3,𝔽₃)	—	43	1	no	—
126	C₃²×C₉	—	43	1	no	—
127	C₃²×C₉	—	43	3	no	—
128	C₃²×C₉	—	43	2	no	—
129	C₃²×C₉	—	43	6	no	—
130	C₃²×C₉	—	43	3	no	—
131	C₃²×C₉	C₃×C₄₃⋊C₃	1	1	yes	characteristic
132	C₉⋊C₉	—	43	6	no	—
133	C₉⋊C₉	—	43	6	no	—
134	C₉⋊C₉	—	43	6	no	—
135	C₉⋊C₉	—	43	6	no	—
136	C₉⋊C₉	—	43	6	no	—
137	C₉⋊C₉	—	43	6	no	—
138	C₉⋊C₉	—	43	6	no	—
139	C₉⋊C₉	C₃×C₄₃⋊C₃	1	6	yes	—
140	C₃²⋊C₉	—	43	2	no	—
141	C₃²⋊C₉	—	43	2	no	—
142	C₃²⋊C₉	—	43	6	no	—
143	C₃²⋊C₉	—	43	6	no	—
144	C₃²⋊C₉	—	43	6	no	—
145	C₃²⋊C₉	C₃×C₄₃⋊C₃	1	2	yes	—
146	C₉²	—	43	2	no	—
147	C₉²	—	43	6	no	—
148	C₉²	—	43	2	no	—
149	C₉²	—	43	2	no	—
150	C₉²	—	43	6	no	—
151	C₉²	—	43	6	no	—
152	C₉²	—	43	1	no	—
153	C₉²	—	43	3	no	—
154	C₉²	—	43	1	no	—
155	C₉²	—	43	1	no	—
156	C₉²	C₃×C₄₃⋊C₃	1	3	yes	—
157	C₄₃	C₉×#737	1	1	yes	characteristic, complemented, Sylow
158	C₃³	—	43	3	no	—
159	C₃³	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
160	3₋¹⁺²	—	43	9	no	—
161	3₋¹⁺²	—	43	27	no	—
162	3₋¹⁺²	—	43	27	no	—
163	3₋¹⁺²	C₉×C₄₃⋊C₃	1	9	yes	complemented
164	H(3,𝔽₃)	—	43	9	no	—
165	C₃×C₉	—	3	3	no	—
166	C₃×C₉	—	43	9	no	—
167	C₃×C₉	—	43	2	no	—
168	C₃×C₉	—	43	1	no	—
169	C₃×C₉	—	43	1	no	—
170	C₃×C₉	—	43	1	no	—
171	C₃×C₉	—	43	3	no	—
172	C₃×C₉	—	43	27	no	—
173	C₃×C₉	—	43	6	no	—
174	C₃×C₉	—	43	2	no	—
175	C₃×C₉	—	43	2	no	—
176	C₃×C₉	—	43	6	no	—
177	C₃×C₉	—	129	18	no	—
178	C₃×C₉	—	129	6	no	—
179	C₃×C₉	—	129	9	no	—
180	C₃×C₉	C₃²×C₄₃⋊C₃	1	2	yes	—
181	C₃×C₉	C₃²×C₄₃⋊C₃	1	6	yes	—
182	C₃×C₉	C₃²×C₄₃⋊C₃	1	3	yes	—
183	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
184	C₃²	—	3	2	no	—
185	C₃²	—	3	1	no	—
186	C₃²	—	43	9	no	—
187	C₃²	—	129	6	no	—
188	C₃²	—	129	3	no	—
189	C₃²	C₃×C₉×C₄₃⋊C₃	1	3	yes	—
190	C₃²	C₃³×C₄₃⋊C₃	1	1	yes	characteristic
191	C₉	—	3	6	no	—
192	C₉	—	3	2	no	—
193	C₉	—	43	6	no	—
194	C₉	—	43	3	no	—
195	C₉	—	43	9	no	—
196	C₉	—	43	3	no	—
197	C₉	—	43	3	no	—
198	C₉	—	129	6	no	—
199	C₉	—	129	2	no	—
200	C₉	—	129	2	no	—
201	C₉	—	129	6	no	—
202	C₉	C₃×C₉×C₄₃⋊C₃	1	9	yes	—
203	C₉	#16812526	1	3	yes	direct product factor
204	C₃	—	3	3	no	—
205	C₃	—	129	9	no	—
206	C₃	C₃²×C₉×C₄₃⋊C₃	1	1	yes	characteristic
207	C₃	#737×C₄₃⋊C₃	1	2	yes	—
208	C₃	C₃×#16812526	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₉	C₃²×C₄₃⋊C₃
commutator	C₃×C₄₃	C₃³×C₉
Frattini	C₃²	C₃³×C₄₃⋊C₃
Fitting	C₉×3₋¹⁺²×C₄₃	C₃
socle, cosocle	C₃²×C₄₃	C₃⁴
prime radical	C₃×C₃²⋊(C₄₃⋊C₃)	C₃²
hypercenter	C₉×3₋¹⁺²	C₄₃⋊C₃
hypocenter	C₄₃	C₉×#737
LCS	G ▷ C₃×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₃×C₉ ◁ C₉×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
2. #23782792	C₃	semidirect
3. 3₋¹⁺²×C₄₃⋊C₉	C₃	semidirect
4. C₉×#11923956	C₃	semidirect
5. C₉×#11923957	C₃	semidirect
6. C₉×#11923958	C₃	semidirect
7. #11923957⋊C₉	C₃	semidirect
8. #11923956⋊C₉	C₃	semidirect
11. #23782781	C₃	semidirect
12. #23782779	C₃	semidirect
13. C₃×C₉×C₄₃⋊C₉	C₃	semidirect
14. C₉²×C₄₃⋊C₃	C₃	semidirect
15. C₉×3₋¹⁺²×C₄₃	C₃	semidirect
16. #16812526	C₉	direct (N × H)
54. #11923958	C₃×C₉	semidirect
55. #11923957	C₃×C₉	semidirect
56. #11923956	C₃×C₉	semidirect
59. C₃×C₄₃⋊C₉	C₃×C₉	semidirect
71. C₉×C₄₃⋊C₃	C₃×C₉	semidirect
73. 3₋¹⁺²×C₄₃	C₃×C₉	semidirect
90. C₉×C₄₃	#737	semidirect
99. C₄₃⋊C₉	#737	semidirect
108. C₉×3₋¹⁺²	C₄₃⋊C₃	semidirect
157. C₄₃	C₉×#737	semidirect
163. 3₋¹⁺²	C₉×C₄₃⋊C₃	semidirect
203. C₉	#16812526	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#16812526	3
#23782792	3
3₋¹⁺²×C₄₃⋊C₉	3
C₉×#11923956	3
C₉×#11923957	3
C₉×#11923958	3
#11923957⋊C₉	3
#11923956⋊C₉	3
C₉×C₃²⋊(C₄₃⋊C₃)	3
#23782783	3
#23782781	3
#23782779	3
C₃×C₉×C₄₃⋊C₉	3
C₉²×C₄₃⋊C₃	3
C₉×3₋¹⁺²×C₄₃	3
C₉×#737	43

Subgroups — C9×#16812526 #44534662

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#16812526 #44534662