Subgroups — #36382359

Order 22599.

662 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21419356	C₃	1	2	yes	maximal, complemented
2	#21419355	C₃	1	2	yes	maximal, complemented
3	#21419351	C₃	1	2	yes	maximal, complemented
4	#21419349	C₃	1	2	yes	maximal, complemented
5	#21419347	C₃	1	2	yes	maximal, complemented
6	#21419346	C₃	1	2	yes	maximal, complemented
7	#21419343	C₃	1	2	yes	maximal, complemented
8	#21419340	C₃	1	2	yes	maximal, complemented
9	#21419337	C₃	1	1	yes	maximal, characteristic, complemented
10	#21419335	C₃	1	2	yes	maximal, complemented
11	#21419328	C₃	1	2	yes	maximal, complemented
12	#21419327	C₃	1	2	yes	maximal, complemented
13	#21419321	C₃	1	1	yes	maximal, characteristic
14	#21419315	C₃	1	2	yes	maximal
15	#21419306	C₃	1	2	yes	maximal, complemented
16	#11719686⋊C₉	C₃	1	2	yes	maximal, complemented
17	#11719686⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
18	#11719687⋊C₉	C₃	1	2	yes	maximal, complemented
19	#11719687⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
20	#21419297	C₃	1	1	yes	maximal, characteristic
21	#21419295	C₃	1	2	yes	maximal, complemented
22	#21419293	C₃	1	1	yes	maximal, characteristic, complemented
23	C₉²×C₃₁⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
24	C₃₁×#6898	C₃	1	1	yes	maximal, characteristic, complemented
25	C₃×#11719689	C₃²	1	2	yes	—
26	C₃×#11719686	C₃²	1	2	yes	—
27	C₃×#11719686	C₃²	1	1	yes	characteristic
28	C₃×#11719687	C₃²	1	2	yes	—
29	C₃×#11719687	C₃²	1	1	yes	characteristic
30	C₃×C₃²⋊(C₃₁⋊C₃)	C₃²	1	1	yes	characteristic
31	C₃²×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
32	C₃×C₉×C₃₁⋊C₃	C₃²	1	1	yes	characteristic
33	C₃×C₉×C₃₁⋊C₃	C₃²	1	2	yes	—
34	C₃×C₉×C₃₁⋊C₃	C₃²	1	1	yes	characteristic
35	C₃×C₉×C₃₁⋊C₃	C₃²	1	2	yes	—
36	#14578887	C₃²	1	2	yes	—
37	#14578887	C₃²	1	2	yes	—
38	#14578887	C₃²	1	1	yes	characteristic
39	#14578886	C₃²	1	2	yes	—
40	#14578886	C₃²	1	2	yes	—
41	#14578886	C₃²	1	1	yes	characteristic
42	(C₃₁⋊C₉)⋊C₉	C₃²	1	2	yes	—
43	(C₃₁⋊C₉)⋊C₉	C₃²	1	2	yes	—
44	(C₃₁⋊C₉)⋊C₉	C₃²	1	2	yes	—
45	(C₃₁⋊C₉)⋊C₉	C₃²	1	2	yes	—
46	(C₃₁⋊C₉)⋊C₉	C₃²	1	2	yes	—
47	(C₃₁⋊C₉)⋊C₉	C₃²	1	2	yes	—
48	(C₃₁⋊C₉)⋊C₉	C₃²	1	2	yes	—
49	(C₃₁⋊C₉)⋊C₉	C₃²	1	2	yes	—
50	(C₃₁⋊C₉)⋊C₉	C₃²	1	2	yes	—
51	(C₃₁⋊C₉)⋊C₉	C₃²	1	2	yes	—
52	(C₃₁⋊C₉)⋊C₉	C₃²	1	2	yes	—
53	(C₃₁⋊C₉)⋊C₉	C₃²	1	2	yes	—
54	(C₃₁⋊C₉)⋊C₉	C₃²	1	2	yes	—
55	(C₃₁⋊C₉)⋊C₉	C₃²	1	2	yes	—
56	(C₃₁⋊C₉)⋊C₉	C₃²	1	2	yes	—
57	(C₃₁⋊C₉)⋊C₉	C₃²	1	2	yes	—
58	(C₃₁⋊C₉)⋊C₉	C₃²	1	2	yes	—
59	(C₃₁⋊C₉)⋊C₉	C₃²	1	2	yes	—
60	(C₃₁⋊C₉)⋊C₉	C₃²	1	2	yes	—
61	(C₃₁⋊C₉)⋊C₉	C₃²	1	2	yes	—
62	(C₃₁⋊C₉)⋊C₉	C₃²	1	2	yes	—
63	(C₃₁⋊C₉)⋊C₉	C₃²	1	2	yes	—
64	(C₃₁⋊C₉)⋊C₉	C₃²	1	2	yes	—
65	(C₃₁⋊C₉)⋊C₉	C₃²	1	2	yes	—
66	C₃²⋊(C₃₁⋊C₉)	C₃²	1	2	yes	—
67	C₃²⋊(C₃₁⋊C₉)	C₃²	1	1	yes	characteristic
68	C₃²⋊(C₃₁⋊C₉)	C₃²	1	2	yes	—
69	(C₃×C₃₁⋊C₃)⋊C₉	C₃²	1	2	yes	—
70	(C₃×C₃₁⋊C₃)⋊C₉	C₃²	1	2	yes	—
71	(C₃×C₃₁⋊C₃)⋊C₉	C₃²	1	2	yes	—
72	(C₃×C₃₁⋊C₃)⋊C₉	C₃²	1	2	yes	—
73	(C₃×C₃₁⋊C₃)⋊C₉	C₃²	1	2	yes	—
74	(C₃×C₃₁⋊C₃)⋊C₉	C₃²	1	2	yes	—
75	(C₃×C₃₁⋊C₃)⋊C₉	C₃²	1	2	yes	—
76	(C₃×C₃₁⋊C₃)⋊C₉	C₃²	1	2	yes	—
77	(C₃×C₃₁⋊C₃)⋊C₉	C₃²	1	2	yes	—
78	(C₃×C₃₁⋊C₃)⋊C₉	C₃²	1	2	yes	—
79	(C₃×C₃₁⋊C₃)⋊C₉	C₃²	1	2	yes	—
80	(C₃×C₃₁⋊C₃)⋊C₉	C₃²	1	2	yes	—
81	C₉×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
82	C₉×C₃₁⋊C₉	C₃²	1	2	yes	—
83	C₉×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
84	C₉×C₃₁⋊C₉	C₃²	1	2	yes	—
85	C₉×C₃₁⋊C₉	C₃²	1	2	yes	—
86	C₉×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
87	C₉×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
88	C₉×C₃₁⋊C₉	C₃²	1	2	yes	—
89	C₉×C₃₁⋊C₉	C₃²	1	2	yes	—
90	C₃₁×C₉⋊C₉	C₃²	1	2	yes	—
91	C₃₁×C₉⋊C₉	C₃²	1	2	yes	—
92	C₃₁×C₉⋊C₉	C₃²	1	1	yes	characteristic
93	C₃₁×C₉⋊C₉	C₃²	1	2	yes	—
94	C₃₁×C₉⋊C₉	C₃²	1	1	yes	characteristic
95	C₃₁×C₃²⋊C₉	C₃²	1	1	yes	characteristic
96	C₃₁×C₃²⋊C₉	C₃²	1	1	yes	characteristic
97	C₃₁×C₃²⋊C₉	C₃²	1	2	yes	—
98	C₉²×C₃₁	C₃²	1	1	yes	characteristic
99	C₃³×C₃₁	C₃³	1	1	yes	characteristic
100	C₃²×C₃₁⋊C₃	C₃³	1	2	yes	—
101	C₃²×C₃₁⋊C₃	C₃³	1	1	yes	characteristic
102	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
103	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
104	C₃×C₉×C₃₁	C₃³	1	2	yes	—
105	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
106	C₃×C₉×C₃₁	C₃³	1	2	yes	—
107	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
108	C₃×C₉×C₃₁	C₃³	1	2	yes	—
109	C₃×C₉×C₃₁	C₃³	1	2	yes	—
110	C₃²⋊(C₃₁⋊C₃)	—	3	3	no	—
111	#11719689	—	3	6	no	—
112	C₃×C₃₁⋊C₉	—	3	3	no	—
113	C₃×C₃₁⋊C₉	C₃³	1	2	yes	—
114	C₃×C₃₁⋊C₉	C₃³	1	2	yes	—
115	C₃×C₃₁⋊C₉	C₃³	1	1	yes	characteristic
116	C₃×C₃₁⋊C₉	C₃³	1	2	yes	—
117	C₃×C₃₁⋊C₉	C₃³	1	2	yes	—
118	C₃×C₃₁⋊C₉	C₃³	1	2	yes	—
119	C₃×C₃₁⋊C₉	C₃³	1	2	yes	—
120	C₃×C₃₁⋊C₉	C₃³	1	2	yes	—
121	C₃×C₃₁⋊C₉	C₃³	1	2	yes	—
122	C₃×C₃₁⋊C₉	C₃³	1	2	yes	—
123	C₃×C₃₁⋊C₉	C₃³	1	2	yes	—
124	C₃×C₃₁⋊C₉	C₃³	1	1	yes	characteristic
125	C₃×C₃₁⋊C₉	C₃³	1	2	yes	—
126	#11719687	—	3	3	no	—
127	#11719687	—	3	6	no	—
128	#11719686	—	3	3	no	—
129	#11719686	—	3	6	no	—
130	C₉×C₃₁⋊C₃	—	3	3	no	—
131	C₉×C₃₁⋊C₃	—	3	3	no	—
132	C₉×C₃₁⋊C₃	—	3	6	no	—
133	C₉×C₃₁⋊C₃	—	3	6	no	—
134	#10624877	—	31	1	no	maximal, Sylow
135	C₃²×C₃₁	—	3	1	no	—
136	C₃²×C₃₁	—	3	2	no	—
137	C₃²×C₃₁	—	3	1	no	—
138	C₃²×C₃₁	C₃⁴	1	1	yes	characteristic
139	C₃×C₃₁⋊C₃	—	3	1	no	—
140	C₃×C₃₁⋊C₃	—	3	2	no	—
141	C₃×C₃₁⋊C₃	—	3	2	no	—
142	C₃×C₃₁⋊C₃	—	3	2	no	—
143	C₃×C₃₁⋊C₃	—	3	2	no	—
144	C₃×C₃₁⋊C₃	—	3	2	no	—
145	C₃×C₃₁⋊C₃	#737	1	3	yes	—
146	C₉×C₃₁	—	3	1	no	—
147	C₉×C₃₁	—	3	1	no	—
148	C₉×C₃₁	—	3	2	no	—
149	C₉×C₃₁	—	3	2	no	—
150	C₉×C₃₁	—	3	1	no	—
151	C₉×C₃₁	—	3	2	no	—
152	C₉×C₃₁	—	3	1	no	—
153	C₉×C₃₁	—	3	2	no	—
154	C₃₁⋊C₉	—	3	2	no	—
155	C₃₁⋊C₉	—	3	1	no	—
156	C₃₁⋊C₉	—	3	2	no	—
157	C₃₁⋊C₉	—	3	2	no	—
158	C₃₁⋊C₉	—	3	2	no	—
159	C₃₁⋊C₉	—	3	2	no	—
160	C₃₁⋊C₉	—	3	2	no	—
161	C₃₁⋊C₉	—	3	2	no	—
162	C₃₁⋊C₉	—	3	2	no	—
163	C₃₁⋊C₉	—	3	2	no	—
164	C₃₁⋊C₉	—	3	2	no	—
165	C₃₁⋊C₉	#737	1	3	yes	—
166	C₃₁⋊C₉	#737	1	6	yes	—
167	#6899	—	31	2	no	—
168	#6899	—	31	2	no	—
169	#6899	—	31	2	no	—
170	#6898	—	31	2	no	—
171	#6898	C₃₁⋊C₃	1	1	yes	characteristic, complemented
172	#6897	—	31	2	no	—
173	#6897	—	31	2	no	—
174	#6897	—	31	2	no	—
175	#6896	—	31	2	no	—
176	#6896	—	31	1	no	—
177	#6895	—	31	2	no	—
178	#6894	—	31	2	no	—
179	#6894	—	31	2	no	—
180	#6893	—	31	1	no	—
181	C₉⋊H(3,𝔽₃)	—	31	2	no	—
182	C₉×3₋¹⁺²	—	31	2	no	—
183	C₉×3₋¹⁺²	—	31	1	no	—
184	C₉×3₋¹⁺²	—	31	2	no	—
185	C₉×3₋¹⁺²	—	31	1	no	—
186	C₉×3₋¹⁺²	—	31	2	no	—
187	C₉×H(3,𝔽₃)	—	31	1	no	—
188	#6887	—	31	2	no	—
189	#6887	—	31	1	no	—
190	C₃×C₉²	—	31	1	no	—
191	C₃×C₃₁	—	9	1	no	—
192	C₃×C₃₁	C₃×#737	1	1	yes	characteristic
193	C₃×C₃₁	C₃×#737	1	1	yes	characteristic
194	C₃×C₃₁	C₃×#737	1	2	yes	—
195	C₃₁⋊C₃	—	3	3	no	—
196	C₃₁⋊C₃	—	9	2	no	—
197	C₃×3₋¹⁺²	—	31	2	no	—
198	C₃×3₋¹⁺²	—	31	2	no	—
199	C₃×3₋¹⁺²	—	31	1	no	—
200	C₃×3₋¹⁺²	—	31	2	no	—
201	C₃×3₋¹⁺²	—	31	1	no	—
202	C₃×H(3,𝔽₃)	—	31	1	no	—
203	C₃²×C₉	—	31	1	no	—
204	C₃²×C₉	—	31	1	no	—
205	C₃²×C₉	—	31	2	no	—
206	C₃²×C₉	—	31	1	no	—
207	C₃²×C₉	—	31	2	no	—
208	C₉⋊C₉	—	31	2	no	—
209	C₉⋊C₉	—	31	2	no	—
210	C₉⋊C₉	—	31	2	no	—
211	C₉⋊C₉	—	31	2	no	—
212	C₉⋊C₉	—	31	2	no	—
213	C₉⋊C₉	—	31	2	no	—
214	C₉⋊C₉	—	31	2	no	—
215	C₉⋊C₉	—	31	2	no	—
216	C₉⋊C₉	—	31	2	no	—
217	C₉⋊C₉	—	31	2	no	—
218	C₉⋊C₉	—	31	2	no	—
219	C₉⋊C₉	—	31	2	no	—
220	C₉⋊C₉	—	31	2	no	—
221	C₉⋊C₉	—	31	2	no	—
222	C₉⋊C₉	—	31	2	no	—
223	C₉⋊C₉	—	31	2	no	—
224	C₉⋊C₉	—	31	2	no	—
225	C₉⋊C₉	—	31	2	no	—
226	C₉⋊C₉	—	31	2	no	—
227	C₉⋊C₉	—	31	2	no	—
228	C₉⋊C₉	—	31	2	no	—
229	C₉⋊C₉	—	31	2	no	—
230	C₉⋊C₉	—	31	2	no	—
231	C₉⋊C₉	—	31	2	no	—
232	C₉⋊C₉	—	31	2	no	—
233	C₉⋊C₉	—	31	2	no	—
234	C₉⋊C₉	—	31	1	no	—
235	C₉⋊C₉	—	31	2	no	—
236	C₉⋊C₉	—	31	2	no	—
237	C₉⋊C₉	—	31	1	no	—
238	C₉⋊C₉	C₃×C₃₁⋊C₃	1	2	yes	—
239	C₉⋊C₉	C₃×C₃₁⋊C₃	1	2	yes	—
240	C₉⋊C₉	C₃×C₃₁⋊C₃	1	2	yes	—
241	C₉⋊C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
242	C₉⋊C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
243	C₃²⋊C₉	—	31	2	no	—
244	C₃²⋊C₉	—	31	1	no	—
245	C₃²⋊C₉	—	31	2	no	—
246	C₃²⋊C₉	—	31	2	no	—
247	C₃²⋊C₉	—	31	2	no	—
248	C₃²⋊C₉	—	31	2	no	—
249	C₃²⋊C₉	—	31	2	no	—
250	C₃²⋊C₉	—	31	2	no	—
251	C₃²⋊C₉	—	31	2	no	—
252	C₃²⋊C₉	—	31	2	no	—
253	C₃²⋊C₉	—	31	2	no	—
254	C₃²⋊C₉	—	31	2	no	—
255	C₃²⋊C₉	—	31	2	no	—
256	C₃²⋊C₉	—	31	2	no	—
257	C₃²⋊C₉	—	31	2	no	—
258	C₃²⋊C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
259	C₃²⋊C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
260	C₃²⋊C₉	C₃×C₃₁⋊C₃	1	2	yes	—
261	C₉²	—	31	1	no	—
262	C₉²	—	31	2	no	—
263	C₉²	—	31	1	no	—
264	C₉²	—	31	2	no	—
265	C₉²	—	31	2	no	—
266	C₉²	—	31	1	no	—
267	C₉²	—	31	1	no	—
268	C₉²	—	31	2	no	—
269	C₉²	—	31	2	no	—
270	C₉²	C₃×C₃₁⋊C₃	1	1	yes	characteristic
271	C₃₁	#10624877	1	1	yes	characteristic, complemented, Sylow
272	C₃³	—	31	1	no	—
273	C₃³	—	31	2	no	—
274	C₃³	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
275	3₋¹⁺²	—	93	6	no	—
276	3₋¹⁺²	—	93	6	no	—
277	3₋¹⁺²	—	93	3	no	—
278	3₋¹⁺²	—	93	3	no	—
279	3₋¹⁺²	—	93	6	no	—
280	H(3,𝔽₃)	—	93	3	no	—
281	C₃×C₉	—	31	1	no	—
282	C₃×C₉	—	31	2	no	—
283	C₃×C₉	—	31	2	no	—
284	C₃×C₉	—	31	2	no	—
285	C₃×C₉	—	31	1	no	—
286	C₃×C₉	—	31	2	no	—
287	C₃×C₉	—	31	2	no	—
288	C₃×C₉	—	31	2	no	—
289	C₃×C₉	—	31	2	no	—
290	C₃×C₉	—	31	2	no	—
291	C₃×C₉	—	31	2	no	—
292	C₃×C₉	—	31	2	no	—
293	C₃×C₉	—	31	2	no	—
294	C₃×C₉	—	93	6	no	—
295	C₃×C₉	—	93	3	no	—
296	C₃×C₉	—	93	6	no	—
297	C₃×C₉	—	93	3	no	—
298	C₃×C₉	—	93	3	no	—
299	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
300	C₃×C₉	C₃²×C₃₁⋊C₃	1	2	yes	—
301	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
302	C₃×C₉	C₃²×C₃₁⋊C₃	1	2	yes	—
303	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
304	C₃×C₉	C₃²×C₃₁⋊C₃	1	2	yes	—
305	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
306	C₃×C₉	C₃²×C₃₁⋊C₃	1	2	yes	—
307	C₃²	—	3	1	no	—
308	C₃²	—	3	1	no	—
309	C₃²	—	3	2	no	—
310	C₃²	—	31	3	no	—
311	C₃²	—	93	2	no	—
312	C₃²	—	93	2	no	—
313	C₃²	—	93	2	no	—
314	C₃²	—	93	2	no	—
315	C₃²	—	93	1	no	—
316	C₃²	—	93	2	no	—
317	C₃²	C₃³×C₃₁⋊C₃	1	1	yes	characteristic
318	C₉	—	3	1	no	—
319	C₉	—	3	2	no	—
320	C₉	—	3	2	no	—
321	C₉	—	3	1	no	—
322	C₉	—	3	2	no	—
323	C₉	—	3	2	no	—
324	C₉	—	3	1	no	—
325	C₉	—	3	1	no	—
326	C₉	—	31	3	no	—
327	C₉	—	31	6	no	—
328	C₉	—	93	1	no	—
329	C₉	—	93	2	no	—
330	C₉	—	93	2	no	—
331	C₉	—	93	2	no	—
332	C₉	—	93	2	no	—
333	C₉	—	93	2	no	—
334	C₉	—	93	2	no	—
335	C₉	—	93	2	no	—
336	C₉	—	93	2	no	—
337	C₉	—	93	2	no	—
338	C₉	—	93	2	no	—
339	C₃	—	9	1	no	—
340	C₃	—	279	2	no	—
341	C₃	—	93	3	no	—
342	C₃	#737×C₃₁⋊C₃	1	1	yes	characteristic
343	C₃	#737×C₃₁⋊C₃	1	2	yes	—
344	C₃	#737×C₃₁⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₁⋊C₃
commutator, socle, cosocle	C₃²×C₃₁	C₃⁴
Fitting	C₃₁×#6898	C₃
prime radical	C₃×C₃²⋊(C₃₁⋊C₃)	C₃²
hypercenter	#6898	C₃₁⋊C₃
hypocenter	C₃₁	#10624877
LCS	G ▷ C₃²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃² ◁ #6898

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. #21419351	C₃	semidirect
4. #21419349	C₃	semidirect
5. #21419347	C₃	semidirect
6. #21419346	C₃	semidirect
7. #21419343	C₃	semidirect
8. #21419340	C₃	semidirect
9. #21419337	C₃	semidirect
10. #21419335	C₃	semidirect
11. #21419328	C₃	semidirect
12. #21419327	C₃	semidirect
15. #21419306	C₃	semidirect
16. #11719686⋊C₉	C₃	semidirect
17. #11719686⋊C₉	C₃	semidirect
18. #11719687⋊C₉	C₃	semidirect
19. #11719687⋊C₉	C₃	semidirect
21. #21419295	C₃	semidirect
22. #21419293	C₃	semidirect
23. C₉²×C₃₁⋊C₃	C₃	semidirect
24. C₃₁×#6898	C₃	semidirect
171. #6898	C₃₁⋊C₃	semidirect
271. 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
#21419351	3
#21419349	3
#21419347	3
#21419346	3
#21419343	3
#21419340	3
#21419337	3
#21419335	3
#21419328	3
#21419327	3
#21419321	3
#21419315	3
#21419306	3
#11719686⋊C₉	3
#11719687⋊C₉	3
#21419297	3
#21419295	3
#21419293	3
C₉²×C₃₁⋊C₃	3
C₃₁×#6898	3
#10624877	31