Subgroups — #36382276

Order 22599.

926 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#14578913	C₃	1	1	yes	maximal, characteristic
2	#21419306	C₃	1	1	yes	maximal, characteristic, complemented
3	#21419306	C₃	1	1	yes	maximal, characteristic, complemented
4	3₋¹⁺²×C₃₁⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
5	3₋¹⁺²×C₃₁⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
6	C₉×#11719687	C₃	1	3	yes	maximal, complemented
7	C₉×#11719686	C₃	1	3	yes	maximal, complemented
8	C₉×#11719689	C₃	1	1	yes	maximal, characteristic, complemented
9	#11719686⋊C₉	C₃	1	3	yes	maximal, complemented
10	#11719686⋊C₉	C₃	1	3	yes	maximal, complemented
11	#11719687⋊C₉	C₃	1	3	yes	maximal, complemented
12	#11719687⋊C₉	C₃	1	3	yes	maximal, complemented
13	C₉×C₃²⋊(C₃₁⋊C₃)	C₃	1	1	yes	maximal, characteristic
14	#21419297	C₃	1	1	yes	maximal, characteristic
15	#21419297	C₃	1	1	yes	maximal, characteristic
16	#21419295	C₃	1	3	yes	maximal, complemented
17	#21419294	C₃	1	3	yes	maximal, complemented
18	#21419293	C₃	1	1	yes	maximal, characteristic, complemented
19	#21419293	C₃	1	1	yes	maximal, characteristic, complemented
20	C₃×(C₃₁⋊C₉)⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
21	C₉⋊C₉×C₃₁⋊C₃	C₃	1	3	yes	maximal, complemented
22	C₉×3₋¹⁺²×C₃₁	C₃	1	1	yes	maximal, characteristic, complemented
23	#14578913	C₉	1	27	yes	complemented
24	C₃×#11719689	C₃²	1	1	yes	characteristic
25	C₃×#11719686	C₃²	1	3	yes	—
26	C₃×#11719687	C₃²	1	3	yes	—
27	C₃×C₃²⋊(C₃₁⋊C₃)	C₃²	1	1	yes	characteristic
28	C₃²×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
29	C₃²×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
30	C₃²×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
31	C₃×C₉×C₃₁⋊C₃	C₃²	1	3	yes	—
32	C₃×C₉×C₃₁⋊C₃	C₃²	1	3	yes	—
33	C₃×C₉×C₃₁⋊C₃	C₃²	1	3	yes	—
34	C₃×C₉×C₃₁⋊C₃	C₃²	1	3	yes	—
35	#14578887	C₃²	1	3	yes	—
36	#14578887	C₃²	1	3	yes	—
37	#14578886	C₃²	1	3	yes	—
38	#14578886	C₃²	1	3	yes	—
39	(C₃₁⋊C₉)⋊C₉	C₃²	1	1	yes	characteristic
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	1	yes	characteristic
46	(C₃₁⋊C₉)⋊C₉	C₃²	1	1	yes	characteristic
47	(C₃₁⋊C₉)⋊C₉	C₃²	1	1	yes	characteristic
48	(C₃₁⋊C₉)⋊C₉	C₃²	1	1	yes	characteristic
49	(C₃₁⋊C₉)⋊C₉	C₃²	1	1	yes	characteristic
50	(C₃₁⋊C₉)⋊C₉	C₃²	1	1	yes	characteristic
51	(C₃₁⋊C₉)⋊C₉	C₃²	1	1	yes	characteristic
52	(C₃₁⋊C₉)⋊C₉	C₃²	1	1	yes	characteristic
53	(C₃₁⋊C₉)⋊C₉	C₃²	1	3	yes	—
54	(C₃₁⋊C₉)⋊C₉	C₃²	1	3	yes	—
55	(C₃₁⋊C₉)⋊C₉	C₃²	1	3	yes	—
56	(C₃₁⋊C₉)⋊C₉	C₃²	1	3	yes	—
57	(C₃₁⋊C₉)⋊C₉	C₃²	1	3	yes	—
58	C₃²⋊(C₃₁⋊C₉)	C₃²	1	1	yes	characteristic
59	C₃²⋊(C₃₁⋊C₉)	C₃²	1	1	yes	characteristic
60	C₃²⋊(C₃₁⋊C₉)	C₃²	1	1	yes	characteristic
61	C₃²⋊(C₃₁⋊C₉)	C₃²	1	1	yes	characteristic
62	(C₃×C₃₁⋊C₃)⋊C₉	C₃²	1	3	yes	—
63	(C₃×C₃₁⋊C₃)⋊C₉	C₃²	1	3	yes	—
64	(C₃×C₃₁⋊C₃)⋊C₉	C₃²	1	3	yes	—
65	(C₃×C₃₁⋊C₃)⋊C₉	C₃²	1	3	yes	—
66	(C₃×C₃₁⋊C₃)⋊C₉	C₃²	1	3	yes	—
67	(C₃×C₃₁⋊C₃)⋊C₉	C₃²	1	3	yes	—
68	C₉×C₃₁⋊C₉	C₃²	1	3	yes	—
69	C₉×C₃₁⋊C₉	C₃²	1	3	yes	—
70	C₉×C₃₁⋊C₉	C₃²	1	3	yes	—
71	C₉×C₃₁⋊C₉	C₃²	1	3	yes	—
72	C₉×C₃₁⋊C₉	C₃²	1	3	yes	—
73	C₉×C₃₁⋊C₉	C₃²	1	3	yes	—
74	C₉×C₃₁⋊C₉	C₃²	1	3	yes	—
75	C₃×3₋¹⁺²×C₃₁	C₃²	1	1	yes	characteristic
76	C₃²×C₉×C₃₁	C₃²	1	1	yes	characteristic
77	C₃₁×C₉⋊C₉	C₃²	1	3	yes	—
78	C₃₁×C₉⋊C₉	C₃²	1	3	yes	—
79	C₃₁×C₃²⋊C₉	C₃²	1	1	yes	characteristic
80	C₃₁×C₃²⋊C₉	C₃²	1	1	yes	characteristic
81	C₉²×C₃₁	C₃²	1	3	yes	—
82	C₃³×C₃₁	C₃³	1	1	yes	characteristic
83	C₃²×C₃₁⋊C₃	C₃³	1	3	yes	—
84	3₋¹⁺²×C₃₁	C₃×C₉	1	9	yes	complemented
85	C₃×C₉×C₃₁	—	9	1	no	—
86	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
87	C₃×C₉×C₃₁	C₃³	1	3	yes	—
88	C₃×C₉×C₃₁	C₃³	1	3	yes	—
89	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
90	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
91	C₃×C₉×C₃₁	C₃³	1	3	yes	—
92	C₃²⋊(C₃₁⋊C₃)	C₃×C₉	1	9	yes	—
93	#11719689	C₃×C₉	1	9	yes	complemented
94	C₃×C₃₁⋊C₉	—	9	1	no	—
95	C₃×C₃₁⋊C₉	—	9	1	no	—
96	C₃×C₃₁⋊C₉	C₃×C₉	1	9	yes	complemented
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	3	yes	—
102	C₃×C₃₁⋊C₉	C₃³	1	3	yes	—
103	C₃×C₃₁⋊C₉	C₃³	1	3	yes	—
104	C₃×C₃₁⋊C₉	C₃³	1	3	yes	—
105	C₃×C₃₁⋊C₉	C₃³	1	1	yes	characteristic
106	C₃×C₃₁⋊C₉	C₃³	1	1	yes	characteristic
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	C₃×C₃₁⋊C₉	C₃³	1	1	yes	characteristic
111	#11719687	C₃×C₉	1	27	yes	complemented
112	#11719686	C₃×C₉	1	27	yes	complemented
113	C₉×C₃₁⋊C₃	—	9	3	no	—
114	C₉×C₃₁⋊C₃	—	9	3	no	—
115	C₉×C₃₁⋊C₃	—	9	3	no	—
116	C₉×C₃₁⋊C₃	C₃×C₉	1	27	yes	complemented
117	#10624864	—	31	1	no	maximal, Sylow
118	C₃²×C₃₁	—	3	1	no	—
119	C₃²×C₃₁	—	3	1	no	—
120	C₃²×C₃₁	—	3	1	no	—
121	C₃²×C₃₁	C₃²×C₉	1	3	yes	—
122	C₃²×C₃₁	C₃⁴	1	1	yes	characteristic
123	C₃×C₃₁⋊C₃	—	3	3	no	—
124	C₃×C₃₁⋊C₃	—	3	3	no	—
125	C₃×C₃₁⋊C₃	—	3	3	no	—
126	C₃×C₃₁⋊C₃	C₃²×C₉	1	9	yes	—
127	C₉×C₃₁	—	3	1	no	—
128	C₉×C₃₁	—	3	3	no	—
129	C₉×C₃₁	—	3	3	no	—
130	C₉×C₃₁	—	3	1	no	—
131	C₉×C₃₁	—	3	1	no	—
132	C₉×C₃₁	C₃²×C₉	1	9	yes	—
133	C₃₁⋊C₉	—	3	1	no	—
134	C₃₁⋊C₉	—	3	1	no	—
135	C₃₁⋊C₉	—	3	3	no	—
136	C₃₁⋊C₉	—	3	3	no	—
137	C₃₁⋊C₉	—	3	3	no	—
138	C₃₁⋊C₉	—	3	3	no	—
139	C₃₁⋊C₉	—	3	1	no	—
140	C₃₁⋊C₉	—	3	1	no	—
141	C₃₁⋊C₉	—	3	1	no	—
142	C₃₁⋊C₉	—	3	1	no	—
143	C₃₁⋊C₉	C₃²×C₉	1	3	yes	—
144	C₃₁⋊C₉	C₃²×C₉	1	9	yes	—
145	C₃₁⋊C₉	C₃²×C₉	1	3	yes	—
146	C₃₁⋊C₉	C₃²×C₉	1	3	yes	—
147	C₃×#737	—	31	1	no	—
148	C₉×3₋¹⁺²	—	31	1	no	—
149	C₉×3₋¹⁺²	—	31	1	no	—
150	C₉×3₋¹⁺²	—	31	1	no	—
151	C₉×3₋¹⁺²	—	31	3	no	—
152	C₉×3₋¹⁺²	—	31	3	no	—
153	C₉×3₋¹⁺²	—	31	3	no	—
154	C₉×3₋¹⁺²	—	31	3	no	—
155	C₉×3₋¹⁺²	—	31	1	no	—
156	C₉×3₋¹⁺²	—	31	1	no	—
157	C₉×3₋¹⁺²	—	31	3	no	—
158	C₉×3₋¹⁺²	—	31	3	no	—
159	C₉×3₋¹⁺²	C₃₁⋊C₃	1	1	yes	characteristic, complemented
160	C₉×H(3,𝔽₃)	—	31	1	no	—
161	C₉×H(3,𝔽₃)	—	31	1	no	—
162	C₉×H(3,𝔽₃)	—	31	1	no	—
163	#6887	—	31	3	no	—
164	#6887	—	31	1	no	—
165	#6887	—	31	1	no	—
166	#6887	—	31	3	no	—
167	C₃×C₉⋊C₉	—	31	1	no	—
168	C₃×C₉⋊C₉	—	31	3	no	—
169	C₃×C₃₁	—	3	3	no	—
170	C₃×C₃₁	C₃³×C₉	1	1	yes	characteristic
171	C₃×C₃₁	3₋¹⁺⁴	1	1	yes	characteristic
172	C₃×C₃₁	3₋¹⁺⁴	1	1	yes	characteristic
173	C₃×C₃₁	3₋¹⁺⁴	1	1	yes	characteristic
174	C₃₁⋊C₃	—	3	9	no	—
175	#737	—	31	27	no	—
176	C₃×3₋¹⁺²	—	31	1	no	—
177	C₃×3₋¹⁺²	—	31	3	no	—
178	C₃×3₋¹⁺²	—	31	3	no	—
179	C₃×3₋¹⁺²	C₃×C₃₁⋊C₃	1	1	yes	characteristic
180	C₃×H(3,𝔽₃)	—	31	1	no	—
181	C₃²×C₉	—	31	1	no	—
182	C₃²×C₉	—	31	1	no	—
183	C₃²×C₉	—	31	3	no	—
184	C₃²×C₉	—	31	3	no	—
185	C₃²×C₉	—	31	3	no	—
186	C₃²×C₉	—	31	1	no	—
187	C₃²×C₉	—	31	3	no	—
188	C₃²×C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
189	C₉⋊C₉	—	31	1	no	—
190	C₉⋊C₉	—	31	1	no	—
191	C₉⋊C₉	—	31	3	no	—
192	C₉⋊C₉	—	31	3	no	—
193	C₉⋊C₉	—	31	1	no	—
194	C₉⋊C₉	—	31	3	no	—
195	C₉⋊C₉	—	31	1	no	—
196	C₉⋊C₉	—	31	1	no	—
197	C₉⋊C₉	—	31	1	no	—
198	C₉⋊C₉	—	31	1	no	—
199	C₉⋊C₉	—	31	3	no	—
200	C₉⋊C₉	—	31	3	no	—
201	C₉⋊C₉	—	31	1	no	—
202	C₉⋊C₉	—	31	1	no	—
203	C₉⋊C₉	—	31	3	no	—
204	C₉⋊C₉	—	31	3	no	—
205	C₉⋊C₉	—	31	3	no	—
206	C₉⋊C₉	—	31	3	no	—
207	C₉⋊C₉	—	31	3	no	—
208	C₉⋊C₉	—	31	3	no	—
209	C₉⋊C₉	—	31	3	no	—
210	C₉⋊C₉	—	31	3	no	—
211	C₉⋊C₉	—	31	3	no	—
212	C₉⋊C₉	C₃×C₃₁⋊C₃	1	3	yes	—
213	C₉⋊C₉	C₃×C₃₁⋊C₃	1	3	yes	—
214	C₃²⋊C₉	—	31	1	no	—
215	C₃²⋊C₉	—	31	1	no	—
216	C₃²⋊C₉	—	31	1	no	—
217	C₃²⋊C₉	—	31	1	no	—
218	C₃²⋊C₉	—	31	3	no	—
219	C₃²⋊C₉	—	31	3	no	—
220	C₃²⋊C₉	—	31	3	no	—
221	C₃²⋊C₉	—	31	3	no	—
222	C₃²⋊C₉	—	31	3	no	—
223	C₃²⋊C₉	—	31	3	no	—
224	C₃²⋊C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
225	C₃²⋊C₉	C₃×C₃₁⋊C₃	1	1	yes	characteristic
226	C₉²	—	31	3	no	—
227	C₉²	—	31	3	no	—
228	C₉²	—	31	3	no	—
229	C₉²	—	31	3	no	—
230	C₉²	—	31	3	no	—
231	C₉²	—	31	3	no	—
232	C₉²	—	31	3	no	—
233	C₉²	C₃×C₃₁⋊C₃	1	3	yes	—
234	C₃₁	#10624864	1	1	yes	characteristic, complemented, Sylow
235	C₃³	—	31	3	no	—
236	C₃³	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
237	3₋¹⁺²	—	31	9	no	—
238	3₋¹⁺²	—	31	27	no	—
239	3₋¹⁺²	—	31	27	no	—
240	3₋¹⁺²	C₉×C₃₁⋊C₃	1	9	yes	complemented
241	H(3,𝔽₃)	—	31	9	no	—
242	C₃×C₉	—	9	1	no	—
243	C₃×C₉	—	279	3	no	—
244	C₃×C₉	—	279	3	no	—
245	C₃×C₉	—	279	3	no	—
246	C₃×C₉	—	279	1	no	—
247	C₃×C₉	—	279	1	no	—
248	C₃×C₉	—	31	9	no	—
249	C₃×C₉	—	31	3	no	—
250	C₃×C₉	—	31	1	no	—
251	C₃×C₉	—	31	1	no	—
252	C₃×C₉	—	31	1	no	—
253	C₃×C₉	—	31	27	no	—
254	C₃×C₉	—	31	1	no	—
255	C₃×C₉	—	31	1	no	—
256	C₃×C₉	—	31	3	no	—
257	C₃×C₉	—	31	1	no	—
258	C₃×C₉	—	31	1	no	—
259	C₃×C₉	—	31	3	no	—
260	C₃×C₉	—	31	3	no	—
261	C₃×C₉	—	31	3	no	—
262	C₃×C₉	—	31	1	no	—
263	C₃×C₉	—	31	1	no	—
264	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
265	C₃×C₉	C₃²×C₃₁⋊C₃	1	3	yes	—
266	C₃×C₉	C₃²×C₃₁⋊C₃	1	3	yes	—
267	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
268	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
269	C₃×C₉	C₃²×C₃₁⋊C₃	1	3	yes	—
270	C₃²	—	3	1	no	—
271	C₃²	—	3	1	no	—
272	C₃²	—	3	1	no	—
273	C₃²	—	31	9	no	—
274	C₃²	—	93	3	no	—
275	C₃²	—	93	3	no	—
276	C₃²	—	93	3	no	—
277	C₃²	C₃×C₉×C₃₁⋊C₃	1	3	yes	—
278	C₃²	C₃³×C₃₁⋊C₃	1	1	yes	characteristic
279	C₉	—	3	3	no	—
280	C₉	—	3	3	no	—
281	C₉	—	3	1	no	—
282	C₉	—	3	1	no	—
283	C₉	—	3	1	no	—
284	C₉	—	31	3	no	—
285	C₉	—	31	9	no	—
286	C₉	—	31	3	no	—
287	C₉	—	31	3	no	—
288	C₉	—	93	1	no	—
289	C₉	—	93	3	no	—
290	C₉	—	93	1	no	—
291	C₉	—	93	1	no	—
292	C₉	—	93	1	no	—
293	C₉	—	93	3	no	—
294	C₉	—	93	1	no	—
295	C₉	—	93	1	no	—
296	C₉	—	93	3	no	—
297	C₉	—	93	3	no	—
298	C₉	C₃×C₉×C₃₁⋊C₃	1	9	yes	—
299	C₃	—	3	3	no	—
300	C₃	—	93	9	no	—
301	C₃	C₃²×C₉×C₃₁⋊C₃	1	1	yes	characteristic
302	C₃	#21419435	1	1	yes	characteristic
303	C₃	#21419436	1	1	yes	characteristic
304	C₃	#21419437	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₁⋊C₃
commutator	C₃×C₃₁	C₃³×C₉
Fitting	C₉×3₋¹⁺²×C₃₁	C₃
socle, cosocle	C₃²×C₃₁	C₃⁴
prime radical	C₃×C₃²⋊(C₃₁⋊C₃)	C₃²
hypercenter	C₉×3₋¹⁺²	C₃₁⋊C₃
hypocenter	C₃₁	#10624864
LCS	G ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃² ◁ C₉×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
2. #21419306	C₃	semidirect
3. #21419306	C₃	semidirect
4. 3₋¹⁺²×C₃₁⋊C₉	C₃	semidirect
5. 3₋¹⁺²×C₃₁⋊C₉	C₃	semidirect
6. C₉×#11719687	C₃	semidirect
7. C₉×#11719686	C₃	semidirect
8. C₉×#11719689	C₃	semidirect
9. #11719686⋊C₉	C₃	semidirect
10. #11719686⋊C₉	C₃	semidirect
11. #11719687⋊C₉	C₃	semidirect
12. #11719687⋊C₉	C₃	semidirect
16. #21419295	C₃	semidirect
17. #21419294	C₃	semidirect
18. #21419293	C₃	semidirect
19. #21419293	C₃	semidirect
20. C₃×(C₃₁⋊C₉)⋊C₉	C₃	semidirect
21. C₉⋊C₉×C₃₁⋊C₃	C₃	semidirect
22. C₉×3₋¹⁺²×C₃₁	C₃	semidirect
23. #14578913	C₉	semidirect
84. 3₋¹⁺²×C₃₁	C₃×C₉	semidirect
93. #11719689	C₃×C₉	semidirect
96. C₃×C₃₁⋊C₉	C₃×C₉	semidirect
111. #11719687	C₃×C₉	semidirect
112. #11719686	C₃×C₉	semidirect
116. C₉×C₃₁⋊C₃	C₃×C₉	semidirect
159. C₉×3₋¹⁺²	C₃₁⋊C₃	semidirect
234. C₃₁	#10624864	semidirect
240. 3₋¹⁺²	C₉×C₃₁⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×#14578913	3
#21419306	3
3₋¹⁺²×C₃₁⋊C₉	3
C₉×#11719687	3
C₉×#11719686	3
C₉×#11719689	3
#11719686⋊C₉	3
#11719687⋊C₉	3
C₉×C₃²⋊(C₃₁⋊C₃)	3
#21419297	3
#21419295	3
#21419294	3
#21419293	3
C₃×(C₃₁⋊C₉)⋊C₉	3
C₉⋊C₉×C₃₁⋊C₃	3
C₉×3₋¹⁺²×C₃₁	3
#10624864	31