Subgroups — #44534869

Order 31347.

620 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782844	C₃	1	3	yes	maximal, complemented
2	#23782843	C₃	1	3	yes	maximal, complemented
3	#23782842	C₃	1	3	yes	maximal, complemented
4	#23782842	C₃	1	3	yes	maximal, complemented
5	#23782841	C₃	1	3	yes	maximal, complemented
6	#23782841	C₃	1	3	yes	maximal, complemented
7	#23782836	C₃	1	3	yes	maximal, complemented
8	#23782836	C₃	1	3	yes	maximal, complemented
9	#23782836	C₃	1	3	yes	maximal, complemented
10	#23782818	C₃	1	1	yes	maximal, characteristic, complemented
11	#23782817	C₃	1	1	yes	maximal, characteristic, complemented
12	#23782816	C₃	1	1	yes	maximal, characteristic, complemented
13	#23782815	C₃	1	1	yes	maximal, characteristic, complemented
14	#23782812	C₃	1	1	yes	maximal, characteristic, complemented
15	#23782797	C₃	1	1	yes	maximal, characteristic
16	#23782796	C₃	1	1	yes	maximal, characteristic
17	#23782795	C₃	1	1	yes	maximal, characteristic
18	#23782792	C₃	1	1	yes	maximal, characteristic, complemented
19	#23782792	C₃	1	1	yes	maximal, characteristic, complemented
20	3₋¹⁺²×C₄₃⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
21	C₃×(C₃×C₄₃⋊C₃)⋊C₉	C₃	1	1	yes	maximal, characteristic
22	C₄₃×#6894	C₃	1	1	yes	maximal, characteristic, complemented
23	C₃³×C₄₃⋊C₃	C₃²	1	1	yes	characteristic
24	C₃×#11923958	C₃²	1	1	yes	characteristic
25	C₃×#11923958	C₃²	1	1	yes	characteristic
26	C₃×#11923957	C₃²	1	3	yes	—
27	C₃×#11923956	C₃²	1	3	yes	—
28	C₃²×C₄₃⋊C₉	C₃²	1	1	yes	characteristic
29	C₃²×C₄₃⋊C₉	C₃²	1	1	yes	characteristic
30	#16812500	C₃²	1	3	yes	complemented
31	#16812500	C₃²	1	3	yes	complemented
32	#16812500	C₃²	1	3	yes	complemented
33	#16812499	C₃²	1	3	yes	complemented
34	#16812499	C₃²	1	3	yes	complemented
35	#16812499	C₃²	1	3	yes	complemented
36	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	complemented
37	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	complemented
38	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	complemented
39	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	complemented
40	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	complemented
41	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	complemented
42	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	complemented
43	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	complemented
44	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	complemented
45	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	complemented
46	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	complemented
47	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	complemented
48	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	complemented
49	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	complemented
50	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	complemented
51	C₃²⋊(C₄₃⋊C₉)	C₃²	1	1	yes	characteristic
52	C₃²⋊(C₄₃⋊C₉)	C₃²	1	1	yes	characteristic
53	C₃²⋊(C₄₃⋊C₉)	C₃²	1	1	yes	characteristic
54	C₃²⋊(C₄₃⋊C₉)	C₃²	1	1	yes	characteristic
55	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	1	yes	characteristic
56	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	1	yes	characteristic
57	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	1	yes	characteristic
58	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	1	yes	characteristic
59	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	1	yes	characteristic
60	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	1	yes	characteristic
61	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	1	yes	characteristic
62	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	1	yes	characteristic
63	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	1	yes	characteristic
64	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	3	yes	—
65	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	3	yes	—
66	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	3	yes	—
67	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	3	yes	—
68	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	3	yes	—
69	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	3	yes	—
70	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	3	yes	—
71	C₉×C₄₃⋊C₉	C₃²	1	3	yes	complemented
72	C₉×C₄₃⋊C₉	C₃²	1	3	yes	complemented
73	C₉×C₄₃⋊C₉	C₃²	1	3	yes	complemented
74	C₃×3₋¹⁺²×C₄₃	C₃²	1	1	yes	characteristic
75	C₃²×C₉×C₄₃	C₃²	1	1	yes	characteristic
76	C₄₃×C₉⋊C₉	C₃²	1	3	yes	complemented
77	C₄₃×C₉⋊C₉	C₃²	1	3	yes	complemented
78	C₄₃×C₉⋊C₉	C₃²	1	3	yes	complemented
79	C₄₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
80	C₄₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
81	C₃²×C₄₃⋊C₃	—	3	3	no	—
82	C₃²×C₄₃⋊C₃	—	9	1	no	—
83	C₃²×C₄₃⋊C₃	—	9	1	no	—
84	C₃²×C₄₃⋊C₃	—	9	1	no	—
85	C₃²×C₄₃⋊C₃	C₃³	1	1	yes	characteristic
86	C₃²×C₄₃⋊C₃	C₃³	1	1	yes	characteristic
87	C₃²×C₄₃⋊C₃	C₃³	1	1	yes	characteristic
88	#11923958	—	3	3	no	—
89	#11923958	—	3	3	no	—
90	#11923957	—	9	3	no	—
91	#11923956	—	9	3	no	—
92	C₃×C₄₃⋊C₉	—	9	1	no	—
93	C₃×C₄₃⋊C₉	—	9	1	no	—
94	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
95	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
96	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
97	C₃×C₄₃⋊C₉	C₃³	1	3	yes	—
98	C₃×C₄₃⋊C₉	C₃³	1	3	yes	—
99	C₃×C₄₃⋊C₉	C₃³	1	3	yes	—
100	C₃×C₄₃⋊C₉	C₃³	1	3	yes	—
101	C₃×C₄₃⋊C₉	C₃³	1	3	yes	—
102	C₃×C₄₃⋊C₉	C₃³	1	3	yes	—
103	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
104	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
105	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
106	C₃³×C₄₃	C₃³	1	1	yes	characteristic
107	3₋¹⁺²×C₄₃	—	3	3	no	—
108	C₃×C₉×C₄₃	—	9	1	no	—
109	C₃×C₉×C₄₃	C₃³	1	3	yes	—
110	C₃×C₉×C₄₃	C₃³	1	3	yes	—
111	C₃×C₉×C₄₃	C₃³	1	1	yes	characteristic
112	C₃×C₉×C₄₃	C₃³	1	1	yes	characteristic
113	C₃×C₉×C₄₃	C₃³	1	1	yes	characteristic
114	C₃×C₉×C₄₃	C₃³	1	3	yes	—
115	#10624890	—	43	1	no	maximal, Sylow
116	C₃²×C₄₃	—	3	1	no	—
117	C₃²×C₄₃	—	3	1	no	—
118	C₃²×C₄₃	—	3	1	no	—
119	C₃²×C₄₃	#737	1	3	yes	—
120	C₃²×C₄₃	C₃⁴	1	1	yes	characteristic
121	C₃×C₄₃⋊C₃	—	3	1	no	—
122	C₃×C₄₃⋊C₃	—	3	1	no	—
123	C₃×C₄₃⋊C₃	—	3	1	no	—
124	C₃×C₄₃⋊C₃	—	3	1	no	—
125	C₃×C₄₃⋊C₃	—	3	1	no	—
126	C₃×C₄₃⋊C₃	—	3	1	no	—
127	C₃×C₄₃⋊C₃	—	3	1	no	—
128	C₃×C₄₃⋊C₃	—	3	1	no	—
129	C₃×C₄₃⋊C₃	—	3	1	no	—
130	C₃×C₄₃⋊C₃	—	3	1	no	—
131	C₃×C₄₃⋊C₃	—	3	1	no	—
132	C₃×C₄₃⋊C₃	—	3	1	no	—
133	C₃×C₄₃⋊C₃	—	9	9	no	—
134	C₉×C₄₃	—	3	3	no	—
135	C₉×C₄₃	—	3	1	no	—
136	C₉×C₄₃	—	3	1	no	—
137	C₉×C₄₃	—	3	1	no	—
138	C₉×C₄₃	—	3	3	no	—
139	C₉×C₄₃	—	3	3	no	—
140	C₄₃⋊C₉	—	3	3	no	—
141	C₄₃⋊C₉	—	3	3	no	—
142	C₄₃⋊C₉	—	3	1	no	—
143	C₄₃⋊C₉	—	3	1	no	—
144	C₄₃⋊C₉	—	3	1	no	—
145	C₄₃⋊C₉	—	3	1	no	—
146	C₄₃⋊C₉	—	3	1	no	—
147	C₄₃⋊C₉	—	3	1	no	—
148	C₄₃⋊C₉	—	3	3	no	—
149	C₄₃⋊C₉	—	3	3	no	—
150	C₄₃⋊C₉	—	3	3	no	—
151	C₄₃⋊C₉	—	3	3	no	—
152	#6899	—	43	3	no	—
153	#6899	—	43	3	no	—
154	#6899	—	43	3	no	—
155	#6899	—	43	3	no	—
156	#6899	—	43	3	no	—
157	#6899	—	43	3	no	—
158	#6898	—	43	3	no	—
159	#6898	—	43	3	no	—
160	#6898	—	43	3	no	—
161	#6894	—	43	1	no	—
162	#6894	—	43	1	no	—
163	#6894	—	43	1	no	—
164	#6894	—	43	1	no	—
165	#6894	—	43	1	no	—
166	#6894	C₄₃⋊C₃	1	1	yes	characteristic, complemented
167	#6891	—	43	1	no	—
168	#6891	—	43	1	no	—
169	#6891	—	43	1	no	—
170	C₉×3₋¹⁺²	—	43	1	no	—
171	C₉×3₋¹⁺²	—	43	1	no	—
172	C₉×3₋¹⁺²	—	43	1	no	—
173	C₃×C₃²⋊C₉	—	43	1	no	—
174	C₃×C₄₃	—	3	3	no	—
175	C₃×C₄₃	C₃×#737	1	1	yes	characteristic
176	C₃×C₄₃	3₋¹⁺⁴	1	1	yes	characteristic
177	C₃×C₄₃	3₋¹⁺⁴	1	1	yes	characteristic
178	C₃×C₄₃	3₋¹⁺⁴	1	1	yes	characteristic
179	C₄₃⋊C₃	—	9	1	no	—
180	C₄₃⋊C₃	—	9	1	no	—
181	C₄₃⋊C₃	—	9	1	no	—
182	C₃⁴	—	43	1	no	—
183	C₃×3₋¹⁺²	—	43	1	no	—
184	C₃×3₋¹⁺²	—	43	1	no	—
185	C₃×3₋¹⁺²	—	43	3	no	—
186	C₃×3₋¹⁺²	—	43	3	no	—
187	C₃×3₋¹⁺²	C₃×C₄₃⋊C₃	1	1	yes	characteristic
188	C₃²×C₉	—	43	1	no	—
189	C₃²×C₉	—	43	1	no	—
190	C₃²×C₉	C₃×C₄₃⋊C₃	1	1	yes	characteristic
191	C₉⋊C₉	—	43	3	no	—
192	C₉⋊C₉	—	43	3	no	—
193	C₉⋊C₉	—	43	3	no	—
194	C₉⋊C₉	—	43	3	no	—
195	C₉⋊C₉	—	43	3	no	—
196	C₉⋊C₉	—	43	3	no	—
197	C₉⋊C₉	—	43	3	no	—
198	C₉⋊C₉	—	43	3	no	—
199	C₉⋊C₉	—	43	3	no	—
200	C₉⋊C₉	—	43	3	no	—
201	C₉⋊C₉	—	43	3	no	—
202	C₉⋊C₉	—	43	3	no	—
203	C₉⋊C₉	—	43	3	no	—
204	C₉⋊C₉	—	43	3	no	—
205	C₉⋊C₉	—	43	3	no	—
206	C₉⋊C₉	—	43	3	no	—
207	C₉⋊C₉	—	43	3	no	—
208	C₉⋊C₉	—	43	3	no	—
209	C₉⋊C₉	—	43	3	no	—
210	C₉⋊C₉	—	43	3	no	—
211	C₉⋊C₉	—	43	3	no	—
212	C₉⋊C₉	C₃×C₄₃⋊C₃	1	3	yes	complemented
213	C₉⋊C₉	C₃×C₄₃⋊C₃	1	3	yes	complemented
214	C₉⋊C₉	C₃×C₄₃⋊C₃	1	3	yes	complemented
215	C₃²⋊C₉	—	43	1	no	—
216	C₃²⋊C₉	—	43	1	no	—
217	C₃²⋊C₉	—	43	1	no	—
218	C₃²⋊C₉	—	43	1	no	—
219	C₃²⋊C₉	—	43	1	no	—
220	C₃²⋊C₉	—	43	1	no	—
221	C₃²⋊C₉	—	43	1	no	—
222	C₃²⋊C₉	—	43	1	no	—
223	C₃²⋊C₉	—	43	1	no	—
224	C₃²⋊C₉	—	43	1	no	—
225	C₃²⋊C₉	—	43	1	no	—
226	C₃²⋊C₉	—	43	1	no	—
227	C₃²⋊C₉	—	43	1	no	—
228	C₃²⋊C₉	—	43	3	no	—
229	C₃²⋊C₉	—	43	3	no	—
230	C₃²⋊C₉	—	43	3	no	—
231	C₃²⋊C₉	—	43	3	no	—
232	C₃²⋊C₉	—	43	3	no	—
233	C₃²⋊C₉	—	43	3	no	—
234	C₃²⋊C₉	—	43	3	no	—
235	C₃²⋊C₉	C₃×C₄₃⋊C₃	1	1	yes	characteristic
236	C₃²⋊C₉	C₃×C₄₃⋊C₃	1	1	yes	characteristic
237	C₉²	—	43	3	no	—
238	C₉²	—	43	3	no	—
239	C₉²	—	43	3	no	—
240	C₄₃	#10624890	1	1	yes	characteristic, complemented, Sylow
241	C₃³	—	43	1	no	—
242	C₃³	—	43	1	no	—
243	C₃³	—	43	1	no	—
244	C₃³	—	129	3	no	—
245	C₃³	—	387	1	no	—
246	C₃³	—	387	1	no	—
247	C₃³	—	387	1	no	—
248	C₃³	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
249	3₋¹⁺²	—	3	3	no	—
250	3₋¹⁺²	—	129	3	no	—
251	3₋¹⁺²	—	129	3	no	—
252	3₋¹⁺²	—	387	3	no	—
253	3₋¹⁺²	—	387	3	no	—
254	C₃×C₉	—	9	1	no	—
255	C₃×C₉	—	43	1	no	—
256	C₃×C₉	—	43	1	no	—
257	C₃×C₉	—	43	1	no	—
258	C₃×C₉	—	43	3	no	—
259	C₃×C₉	—	43	3	no	—
260	C₃×C₉	—	43	3	no	—
261	C₃×C₉	—	43	3	no	—
262	C₃×C₉	—	43	3	no	—
263	C₃×C₉	—	43	3	no	—
264	C₃×C₉	—	43	1	no	—
265	C₃×C₉	—	43	1	no	—
266	C₃×C₉	—	43	1	no	—
267	C₃×C₉	—	387	1	no	—
268	C₃×C₉	—	387	1	no	—
269	C₃×C₉	C₃²×C₄₃⋊C₃	1	3	yes	—
270	C₃×C₉	C₃²×C₄₃⋊C₃	1	3	yes	—
271	C₃×C₉	C₃²×C₄₃⋊C₃	1	3	yes	—
272	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
273	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
274	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
275	C₃²	—	3	1	no	—
276	C₃²	—	3	1	no	—
277	C₃²	—	3	1	no	—
278	C₃²	—	129	1	no	—
279	C₃²	—	129	1	no	—
280	C₃²	—	129	1	no	—
281	C₃²	—	129	1	no	—
282	C₃²	—	129	1	no	—
283	C₃²	—	129	1	no	—
284	C₃²	—	129	1	no	—
285	C₃²	—	129	1	no	—
286	C₃²	—	129	1	no	—
287	C₃²	—	129	1	no	—
288	C₃²	—	129	1	no	—
289	C₃²	—	129	1	no	—
290	C₃²	—	387	9	no	—
291	C₃²	#16812526	1	3	yes	—
292	C₃²	C₃³×C₄₃⋊C₃	1	1	yes	characteristic
293	C₉	—	3	3	no	—
294	C₉	—	3	3	no	—
295	C₉	—	3	3	no	—
296	C₉	—	3	1	no	—
297	C₉	—	3	1	no	—
298	C₉	—	3	1	no	—
299	C₉	—	129	3	no	—
300	C₉	—	129	3	no	—
301	C₉	—	129	3	no	—
302	C₉	—	129	3	no	—
303	C₉	—	129	3	no	—
304	C₉	—	129	3	no	—
305	C₉	—	129	1	no	—
306	C₉	—	129	1	no	—
307	C₉	—	129	1	no	—
308	C₉	—	129	1	no	—
309	C₉	—	129	1	no	—
310	C₉	—	129	1	no	—
311	C₃	—	3	3	no	—
312	C₃	—	387	1	no	—
313	C₃	—	387	1	no	—
314	C₃	—	387	1	no	—
315	C₃	C₃×#16812526	1	1	yes	characteristic
316	C₃	#23782921	1	1	yes	characteristic
317	C₃	#23782922	1	1	yes	characteristic
318	C₃	#23782923	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₄₃⋊C₃
commutator, socle, cosocle	C₃²×C₄₃	C₃⁴
Fitting	C₄₃×#6894	C₃
prime radical	C₃³×C₄₃⋊C₃	C₃²
hypercenter	#6894	C₄₃⋊C₃
hypocenter	C₄₃	#10624890
LCS	G ▷ C₃²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₃² ◁ #6894

Complements

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

N	H ≅ G/N	type
1. #23782844	C₃	semidirect
2. #23782843	C₃	semidirect
3. #23782842	C₃	semidirect
4. #23782842	C₃	semidirect
5. #23782841	C₃	semidirect
6. #23782841	C₃	semidirect
7. #23782836	C₃	semidirect
8. #23782836	C₃	semidirect
9. #23782836	C₃	semidirect
10. #23782818	C₃	semidirect
11. #23782817	C₃	semidirect
12. #23782816	C₃	semidirect
13. #23782815	C₃	semidirect
14. #23782812	C₃	semidirect
18. #23782792	C₃	semidirect
19. #23782792	C₃	semidirect
20. 3₋¹⁺²×C₄₃⋊C₉	C₃	semidirect
22. C₄₃×#6894	C₃	semidirect
30. #16812500	C₃²	semidirect
31. #16812500	C₃²	semidirect
32. #16812500	C₃²	semidirect
33. #16812499	C₃²	semidirect
34. #16812499	C₃²	semidirect
35. #16812499	C₃²	semidirect
36. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
37. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
38. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
39. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
40. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
41. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
42. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
43. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
44. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
45. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
46. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
47. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
48. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
49. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
50. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
71. C₉×C₄₃⋊C₉	C₃²	semidirect
72. C₉×C₄₃⋊C₉	C₃²	semidirect
73. C₉×C₄₃⋊C₉	C₃²	semidirect
76. C₄₃×C₉⋊C₉	C₃²	semidirect
77. C₄₃×C₉⋊C₉	C₃²	semidirect
78. C₄₃×C₉⋊C₉	C₃²	semidirect
166. #6894	C₄₃⋊C₃	semidirect
212. C₉⋊C₉	C₃×C₄₃⋊C₃	semidirect
213. C₉⋊C₉	C₃×C₄₃⋊C₃	semidirect
214. C₉⋊C₉	C₃×C₄₃⋊C₃	semidirect
240. C₄₃	#10624890	semidirect

Maximal subgroups

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

subgroup	index
#23782844	3
#23782843	3
#23782842	3
#23782841	3
#23782836	3
#23782818	3
#23782817	3
#23782816	3
#23782815	3
#23782812	3
#23782797	3
#23782796	3
#23782795	3
#23782792	3
3₋¹⁺²×C₄₃⋊C₉	3
C₃×(C₃×C₄₃⋊C₃)⋊C₉	3
C₄₃×#6894	3
#10624890	43