Subgroups — #36382368

Order 22599.

662 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21419356	C₃	1	3	yes	maximal, complemented
2	#21419355	C₃	1	3	yes	maximal, complemented
3	#21419350	C₃	1	3	yes	maximal, complemented
4	#21419347	C₃	1	3	yes	maximal, complemented
5	#21419346	C₃	1	3	yes	maximal, complemented
6	#21419340	C₃	1	3	yes	maximal, complemented
7	#21419334	C₃	1	1	yes	maximal, characteristic, complemented
8	#21419333	C₃	1	1	yes	maximal, characteristic, complemented
9	#21419332	C₃	1	1	yes	maximal, characteristic, complemented
10	#21419331	C₃	1	1	yes	maximal, characteristic, complemented
11	#21419326	C₃	1	1	yes	maximal, characteristic, complemented
12	#21419326	C₃	1	1	yes	maximal, characteristic, complemented
13	#21419323	C₃	1	1	yes	maximal, characteristic
14	#21419319	C₃	1	1	yes	maximal, characteristic
15	#21419318	C₃	1	1	yes	maximal, characteristic
16	C₉×#11719687	C₃	1	3	yes	maximal, complemented
17	C₉×#11719686	C₃	1	3	yes	maximal, complemented
18	C₉×#11719689	C₃	1	1	yes	maximal, characteristic, complemented
19	C₉×C₃²⋊(C₃₁⋊C₃)	C₃	1	1	yes	maximal, characteristic
20	#21419294	C₃	1	3	yes	maximal, complemented
21	C₃×C₉×C₃₁⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
22	C₉×3₋¹⁺²×C₃₁	C₃	1	1	yes	maximal, characteristic, complemented
23	C₃×#11719689	C₃²	1	1	yes	characteristic
24	C₃×#11719686	C₃²	1	3	yes	—
25	C₃×#11719687	C₃²	1	3	yes	—
26	C₃×C₃²⋊(C₃₁⋊C₃)	C₃²	1	1	yes	characteristic
27	C₃²×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
28	C₃²×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
29	C₃²×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
30	C₃×C₉×C₃₁⋊C₃	C₃²	1	3	yes	—
31	#14578887	C₃²	1	3	yes	—
32	#14578887	C₃²	1	3	yes	—
33	#14578886	C₃²	1	3	yes	—
34	#14578886	C₃²	1	3	yes	—
35	(C₃₁⋊C₉)⋊C₉	C₃²	1	3	yes	—
36	(C₃₁⋊C₉)⋊C₉	C₃²	1	3	yes	—
37	(C₃₁⋊C₉)⋊C₉	C₃²	1	3	yes	—
38	(C₃₁⋊C₉)⋊C₉	C₃²	1	3	yes	—
39	(C₃₁⋊C₉)⋊C₉	C₃²	1	3	yes	—
40	(C₃₁⋊C₉)⋊C₉	C₃²	1	3	yes	—
41	(C₃₁⋊C₉)⋊C₉	C₃²	1	3	yes	—
42	(C₃₁⋊C₉)⋊C₉	C₃²	1	3	yes	—
43	(C₃₁⋊C₉)⋊C₉	C₃²	1	3	yes	—
44	(C₃₁⋊C₉)⋊C₉	C₃²	1	3	yes	—
45	(C₃₁⋊C₉)⋊C₉	C₃²	1	3	yes	—
46	(C₃₁⋊C₉)⋊C₉	C₃²	1	3	yes	—
47	(C₃₁⋊C₉)⋊C₉	C₃²	1	3	yes	—
48	(C₃₁⋊C₉)⋊C₉	C₃²	1	3	yes	—
49	(C₃₁⋊C₉)⋊C₉	C₃²	1	3	yes	—
50	(C₃₁⋊C₉)⋊C₉	C₃²	1	3	yes	—
51	C₃²⋊(C₃₁⋊C₉)	C₃²	1	1	yes	characteristic
52	C₃²⋊(C₃₁⋊C₉)	C₃²	1	1	yes	characteristic
53	C₃²⋊(C₃₁⋊C₉)	C₃²	1	1	yes	characteristic
54	C₃²⋊(C₃₁⋊C₉)	C₃²	1	1	yes	characteristic
55	(C₃×C₃₁⋊C₃)⋊C₉	C₃²	1	3	yes	—
56	(C₃×C₃₁⋊C₃)⋊C₉	C₃²	1	3	yes	—
57	(C₃×C₃₁⋊C₃)⋊C₉	C₃²	1	3	yes	—
58	(C₃×C₃₁⋊C₃)⋊C₉	C₃²	1	3	yes	—
59	(C₃×C₃₁⋊C₃)⋊C₉	C₃²	1	3	yes	—
60	(C₃×C₃₁⋊C₃)⋊C₉	C₃²	1	3	yes	—
61	(C₃×C₃₁⋊C₃)⋊C₉	C₃²	1	3	yes	—
62	(C₃×C₃₁⋊C₃)⋊C₉	C₃²	1	3	yes	—
63	(C₃×C₃₁⋊C₃)⋊C₉	C₃²	1	3	yes	—
64	C₉×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
65	C₉×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
66	C₉×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
67	C₉×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
68	C₉×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
69	C₉×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
70	C₉×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
71	C₉×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
72	C₉×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
73	C₉×C₃₁⋊C₉	C₃²	1	3	yes	—
74	C₃×3₋¹⁺²×C₃₁	C₃²	1	1	yes	characteristic
75	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
76	C₃₁×C₉⋊C₉	C₃²	1	3	yes	—
77	C₃₁×C₉⋊C₉	C₃²	1	3	yes	—
78	C₃₁×C₃²⋊C₉	C₃²	1	1	yes	characteristic
79	C₃₁×C₃²⋊C₉	C₃²	1	1	yes	characteristic
80	C₉²×C₃₁	C₃²	1	3	yes	—
81	C₃³×C₃₁	C₃³	1	1	yes	characteristic
82	C₃²×C₃₁⋊C₃	C₃³	1	3	yes	—
83	3₋¹⁺²×C₃₁	—	3	3	no	—
84	C₃×C₉×C₃₁	—	3	3	no	—
85	C₃×C₉×C₃₁	C₃³	1	3	yes	—
86	C₃×C₉×C₃₁	C₃³	1	3	yes	—
87	C₃×C₉×C₃₁	C₃³	1	3	yes	—
88	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
89	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
90	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
91	C₃²⋊(C₃₁⋊C₃)	—	3	3	no	—
92	#11719689	—	3	3	no	—
93	C₃×C₃₁⋊C₉	—	3	3	no	—
94	C₃×C₃₁⋊C₉	—	3	3	no	—
95	C₃×C₃₁⋊C₉	—	3	3	no	—
96	C₃×C₃₁⋊C₉	C₃³	1	1	yes	characteristic
97	C₃×C₃₁⋊C₉	C₃³	1	1	yes	characteristic
98	C₃×C₃₁⋊C₉	C₃³	1	1	yes	characteristic
99	C₃×C₃₁⋊C₉	C₃³	1	1	yes	characteristic
100	C₃×C₃₁⋊C₉	C₃³	1	1	yes	characteristic
101	C₃×C₃₁⋊C₉	C₃³	1	1	yes	characteristic
102	C₃×C₃₁⋊C₉	C₃³	1	1	yes	characteristic
103	C₃×C₃₁⋊C₉	C₃³	1	3	yes	—
104	C₃×C₃₁⋊C₉	C₃³	1	3	yes	—
105	C₃×C₃₁⋊C₉	C₃³	1	3	yes	—
106	C₃×C₃₁⋊C₉	C₃³	1	3	yes	—
107	C₃×C₃₁⋊C₉	C₃³	1	3	yes	—
108	C₃×C₃₁⋊C₉	C₃³	1	1	yes	characteristic
109	C₃×C₃₁⋊C₉	C₃³	1	1	yes	characteristic
110	#11719687	—	3	9	no	—
111	#11719686	—	3	9	no	—
112	C₉×C₃₁⋊C₃	—	3	9	no	—
113	#10624877	—	31	1	no	maximal, Sylow
114	C₃²×C₃₁	—	3	1	no	—
115	C₃²×C₃₁	—	3	1	no	—
116	C₃²×C₃₁	—	3	1	no	—
117	C₃²×C₃₁	#737	1	3	yes	—
118	C₃²×C₃₁	C₃⁴	1	1	yes	characteristic
119	C₃×C₃₁⋊C₃	—	3	3	no	—
120	C₃×C₃₁⋊C₃	—	3	3	no	—
121	C₃×C₃₁⋊C₃	—	3	3	no	—
122	C₃×C₃₁⋊C₃	—	3	3	no	—
123	C₉×C₃₁	—	3	3	no	—
124	C₉×C₃₁	—	3	3	no	—
125	C₉×C₃₁	—	3	1	no	—
126	C₉×C₃₁	—	3	1	no	—
127	C₉×C₃₁	—	3	3	no	—
128	C₉×C₃₁	#737	1	3	yes	—
129	C₃₁⋊C₉	—	3	3	no	—
130	C₃₁⋊C₉	—	3	3	no	—
131	C₃₁⋊C₉	—	3	3	no	—
132	C₃₁⋊C₉	—	3	3	no	—
133	C₃₁⋊C₉	—	3	1	no	—
134	C₃₁⋊C₉	—	3	1	no	—
135	C₃₁⋊C₉	—	3	1	no	—
136	C₃₁⋊C₉	—	3	1	no	—
137	C₃₁⋊C₉	—	3	1	no	—
138	C₃₁⋊C₉	—	3	1	no	—
139	C₃₁⋊C₉	—	3	1	no	—
140	C₃₁⋊C₉	—	3	3	no	—
141	C₃₁⋊C₉	#737	1	3	yes	—
142	C₃₁⋊C₉	#737	1	3	yes	—
143	#6899	—	31	3	no	—
144	#6899	—	31	3	no	—
145	#6898	—	31	3	no	—
146	#6897	—	31	3	no	—
147	#6897	—	31	3	no	—
148	#6896	—	31	3	no	—
149	#6895	—	31	1	no	—
150	#6895	—	31	1	no	—
151	#6894	—	31	1	no	—
152	#6894	—	31	1	no	—
153	#6894	—	31	1	no	—
154	#6894	—	31	1	no	—
155	#6893	—	31	1	no	—
156	C₉⋊H(3,𝔽₃)	—	31	1	no	—
157	C₉⋊H(3,𝔽₃)	—	31	1	no	—
158	C₉×3₋¹⁺²	—	31	1	no	—
159	C₉×3₋¹⁺²	—	31	3	no	—
160	C₉×3₋¹⁺²	—	31	3	no	—
161	C₉×3₋¹⁺²	C₃₁⋊C₃	1	1	yes	characteristic, complemented
162	C₉×H(3,𝔽₃)	—	31	1	no	—
163	#6887	—	31	3	no	—
164	C₃×C₉²	—	31	1	no	—
165	C₃×C₃₁	—	3	3	no	—
166	C₃×C₃₁	C₃×#737	1	1	yes	characteristic
167	C₃×C₃₁	C₃×#737	1	1	yes	characteristic
168	C₃×C₃₁	C₃×#737	1	1	yes	characteristic
169	C₃×C₃₁	C₃×#737	1	1	yes	characteristic
170	C₃₁⋊C₃	—	9	3	no	—
171	C₃×3₋¹⁺²	—	31	1	no	—
172	C₃×3₋¹⁺²	—	31	3	no	—
173	C₃×3₋¹⁺²	—	31	3	no	—
174	C₃×3₋¹⁺²	C₃×C₃₁⋊C₃	1	1	yes	characteristic
175	C₃×H(3,𝔽₃)	—	31	1	no	—
176	C₃²×C₉	—	31	1	no	—
177	C₃²×C₉	—	31	1	no	—
178	C₃²×C₉	—	31	1	no	—
179	C₃²×C₉	—	31	3	no	—
180	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
181	C₉⋊C₉	—	31	3	no	—
182	C₉⋊C₉	—	31	3	no	—
183	C₉⋊C₉	—	31	3	no	—
184	C₉⋊C₉	—	31	3	no	—
185	C₉⋊C₉	—	31	3	no	—
186	C₉⋊C₉	—	31	3	no	—
187	C₉⋊C₉	—	31	3	no	—
188	C₉⋊C₉	—	31	3	no	—
189	C₉⋊C₉	—	31	3	no	—
190	C₉⋊C₉	—	31	3	no	—
191	C₉⋊C₉	—	31	3	no	—
192	C₉⋊C₉	—	31	3	no	—
193	C₉⋊C₉	—	31	3	no	—
194	C₉⋊C₉	—	31	3	no	—
195	C₉⋊C₉	—	31	3	no	—
196	C₉⋊C₉	—	31	3	no	—
197	C₉⋊C₉	—	31	3	no	—
198	C₉⋊C₉	—	31	3	no	—
199	C₉⋊C₉	—	31	3	no	—
200	C₉⋊C₉	—	31	3	no	—
201	C₉⋊C₉	C₃×C₃₁⋊C₃	1	3	yes	—
202	C₉⋊C₉	C₃×C₃₁⋊C₃	1	3	yes	—
203	C₃²⋊C₉	—	31	1	no	—
204	C₃²⋊C₉	—	31	1	no	—
205	C₃²⋊C₉	—	31	1	no	—
206	C₃²⋊C₉	—	31	1	no	—
207	C₃²⋊C₉	—	31	3	no	—
208	C₃²⋊C₉	—	31	3	no	—
209	C₃²⋊C₉	—	31	3	no	—
210	C₃²⋊C₉	—	31	3	no	—
211	C₃²⋊C₉	—	31	3	no	—
212	C₃²⋊C₉	—	31	3	no	—
213	C₃²⋊C₉	—	31	3	no	—
214	C₃²⋊C₉	—	31	3	no	—
215	C₃²⋊C₉	—	31	3	no	—
216	C₃²⋊C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
217	C₃²⋊C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
218	C₉²	—	31	1	no	—
219	C₉²	—	31	1	no	—
220	C₉²	—	31	1	no	—
221	C₉²	—	31	1	no	—
222	C₉²	—	31	1	no	—
223	C₉²	—	31	1	no	—
224	C₉²	—	31	1	no	—
225	C₉²	—	31	1	no	—
226	C₉²	—	31	1	no	—
227	C₉²	—	31	3	no	—
228	C₉²	C₃×C₃₁⋊C₃	1	3	yes	—
229	C₃₁	#10624877	1	1	yes	characteristic, complemented, Sylow
230	C₃³	—	31	3	no	—
231	C₃³	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
232	3₋¹⁺²	—	3	3	no	—
233	3₋¹⁺²	—	93	9	no	—
234	3₋¹⁺²	—	93	9	no	—
235	3₋¹⁺²	—	93	3	no	—
236	H(3,𝔽₃)	—	93	3	no	—
237	C₃×C₉	—	3	3	no	—
238	C₃×C₉	—	31	1	no	—
239	C₃×C₉	—	31	1	no	—
240	C₃×C₉	—	31	1	no	—
241	C₃×C₉	—	31	1	no	—
242	C₃×C₉	—	31	1	no	—
243	C₃×C₉	—	31	1	no	—
244	C₃×C₉	—	31	1	no	—
245	C₃×C₉	—	31	1	no	—
246	C₃×C₉	—	31	1	no	—
247	C₃×C₉	—	31	3	no	—
248	C₃×C₉	—	31	3	no	—
249	C₃×C₉	—	31	3	no	—
250	C₃×C₉	—	31	3	no	—
251	C₃×C₉	—	31	3	no	—
252	C₃×C₉	—	93	3	no	—
253	C₃×C₉	—	93	3	no	—
254	C₃×C₉	—	93	3	no	—
255	C₃×C₉	—	93	9	no	—
256	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
257	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
258	C₃×C₉	C₃²×C₃₁⋊C₃	1	3	yes	—
259	C₃×C₉	C₃²×C₃₁⋊C₃	1	3	yes	—
260	C₃×C₉	C₃²×C₃₁⋊C₃	1	3	yes	—
261	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
262	C₃²	—	3	1	no	—
263	C₃²	—	3	1	no	—
264	C₃²	—	3	1	no	—
265	C₃²	—	93	3	no	—
266	C₃²	—	93	3	no	—
267	C₃²	—	93	3	no	—
268	C₃²	—	93	3	no	—
269	C₃²	#14578912	1	3	yes	—
270	C₃²	C₃³×C₃₁⋊C₃	1	1	yes	characteristic
271	C₉	—	3	3	no	—
272	C₉	—	3	3	no	—
273	C₉	—	3	3	no	—
274	C₉	—	3	1	no	—
275	C₉	—	3	1	no	—
276	C₉	—	31	3	no	—
277	C₉	—	31	3	no	—
278	C₉	—	93	1	no	—
279	C₉	—	93	1	no	—
280	C₉	—	93	1	no	—
281	C₉	—	93	1	no	—
282	C₉	—	93	3	no	—
283	C₉	—	93	3	no	—
284	C₉	—	93	1	no	—
285	C₉	—	93	1	no	—
286	C₉	—	93	1	no	—
287	C₉	—	93	3	no	—
288	C₉	—	93	3	no	—
289	C₉	—	93	3	no	—
290	C₉	#14578913	1	3	yes	—
291	C₃	—	3	3	no	—
292	C₃	—	279	3	no	—
293	C₃	#737×C₃₁⋊C₃	1	1	yes	characteristic
294	C₃	#737×C₃₁⋊C₃	1	1	yes	characteristic
295	C₃	C₃×#14578912	1	1	yes	characteristic
296	C₃	C₃×#14578913	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₉	C₃²×C₃₁⋊C₃
commutator, socle, cosocle	C₃²×C₃₁	C₃⁴
Frattini	C₃²	C₃³×C₃₁⋊C₃
Fitting	C₉×3₋¹⁺²×C₃₁	C₃
prime radical	C₃×C₃²⋊(C₃₁⋊C₃)	C₃²
hypercenter	C₉×3₋¹⁺²	C₃₁⋊C₃
hypocenter	C₃₁	#10624877
LCS	G ▷ C₃²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃×C₉ ◁ C₉×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. #21419356	C₃	semidirect
2. #21419355	C₃	semidirect
3. #21419350	C₃	semidirect
4. #21419347	C₃	semidirect
5. #21419346	C₃	semidirect
6. #21419340	C₃	semidirect
7. #21419334	C₃	semidirect
8. #21419333	C₃	semidirect
9. #21419332	C₃	semidirect
10. #21419331	C₃	semidirect
11. #21419326	C₃	semidirect
12. #21419326	C₃	semidirect
16. C₉×#11719687	C₃	semidirect
17. C₉×#11719686	C₃	semidirect
18. C₉×#11719689	C₃	semidirect
20. #21419294	C₃	semidirect
21. C₃×C₉×C₃₁⋊C₉	C₃	semidirect
22. C₉×3₋¹⁺²×C₃₁	C₃	semidirect
161. C₉×3₋¹⁺²	C₃₁⋊C₃	semidirect
229. C₃₁	#10624877	semidirect

Maximal subgroups

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

subgroup	index
#21419356	3
#21419355	3
#21419350	3
#21419347	3
#21419346	3
#21419340	3
#21419334	3
#21419333	3
#21419332	3
#21419331	3
#21419326	3
#21419323	3
#21419319	3
#21419318	3
C₉×#11719687	3
C₉×#11719686	3
C₉×#11719689	3
C₉×C₃²⋊(C₃₁⋊C₃)	3
#21419294	3
C₃×C₉×C₃₁⋊C₉	3
C₉×3₋¹⁺²×C₃₁	3
#10624877	31