Subgroups — #44534780

Order 31347.

758 conjugacy classes of subgroups.

The subgroup lattice has 310 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	#23782849	C₃	1	1	yes	maximal, characteristic, complemented
3	#23782848	C₃	1	1	yes	maximal, characteristic, complemented
4	#23782845	C₃	1	1	yes	maximal, characteristic, complemented
5	#23782844	C₃	1	1	yes	maximal, characteristic, complemented
6	#23782843	C₃	1	1	yes	maximal, characteristic, complemented
7	#23782840	C₃	1	1	yes	maximal, characteristic, complemented
8	#23782839	C₃	1	1	yes	maximal, characteristic, complemented
9	#23782838	C₃	1	1	yes	maximal, characteristic, complemented
10	#23782826	C₃	1	3	yes	maximal, complemented
11	#23782826	C₃	1	3	yes	maximal, complemented
12	#23782823	C₃	1	3	yes	maximal, complemented
13	#23782814	C₃	1	3	yes	maximal, complemented
14	#23782813	C₃	1	3	yes	maximal, complemented
15	#23782801	C₃	1	3	yes	maximal
16	#23782792	C₃	1	3	yes	maximal, complemented
17	#11923957⋊C₉	C₃	1	3	yes	maximal, complemented
18	#11923956⋊C₉	C₃	1	3	yes	maximal, complemented
19	#23782781	C₃	1	1	yes	maximal, characteristic, complemented
20	#23782781	C₃	1	1	yes	maximal, characteristic, complemented
21	C₉²×C₄₃⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
22	C₄₃×#6899	C₃	1	1	yes	maximal, characteristic, complemented
23	#16812526	—	3	9	no	—
24	C₃×#11923958	C₃²	1	1	yes	characteristic
25	C₃×#11923958	C₃²	1	3	yes	—
26	C₃×#11923957	C₃²	1	1	yes	characteristic
27	C₃×#11923957	C₃²	1	1	yes	characteristic
28	C₃×#11923957	C₃²	1	1	yes	characteristic
29	C₃×#11923957	C₃²	1	3	yes	—
30	C₃×#11923956	C₃²	1	1	yes	characteristic
31	C₃×#11923956	C₃²	1	1	yes	characteristic
32	C₃×#11923956	C₃²	1	1	yes	characteristic
33	C₃×#11923956	C₃²	1	3	yes	—
34	C₃×C₃²⋊(C₄₃⋊C₃)	C₃²	1	1	yes	characteristic
35	C₃²×C₄₃⋊C₉	C₃²	1	1	yes	characteristic
36	C₃×C₉×C₄₃⋊C₃	C₃²	1	1	yes	characteristic
37	C₃×C₉×C₄₃⋊C₃	C₃²	1	1	yes	characteristic
38	C₃×C₉×C₄₃⋊C₃	C₃²	1	3	yes	—
39	C₃×C₉×C₄₃⋊C₃	C₃²	1	1	yes	characteristic
40	#16812500	C₃²	1	1	yes	characteristic
41	#16812500	C₃²	1	1	yes	characteristic
42	#16812500	C₃²	1	3	yes	—
43	#16812499	C₃²	1	1	yes	characteristic
44	#16812499	C₃²	1	1	yes	characteristic
45	#16812499	C₃²	1	3	yes	—
46	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	—
47	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	—
48	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	—
49	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	—
50	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	—
51	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	—
52	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	—
53	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	—
54	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	—
55	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	—
56	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	—
57	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	—
58	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	—
59	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	—
60	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	—
61	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	—
62	C₃²⋊(C₄₃⋊C₉)	C₃²	1	3	yes	—
63	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	3	yes	—
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	C₉×C₄₃⋊C₉	C₃²	1	3	yes	—
70	C₉×C₄₃⋊C₉	C₃²	1	3	yes	—
71	C₉×C₄₃⋊C₉	C₃²	1	1	yes	characteristic
72	C₉×C₄₃⋊C₉	C₃²	1	1	yes	characteristic
73	C₉×C₄₃⋊C₉	C₃²	1	3	yes	—
74	C₉×C₄₃⋊C₉	C₃²	1	3	yes	—
75	C₃×3₋¹⁺²×C₄₃	C₃²	1	1	yes	characteristic
76	C₄₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
77	C₄₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
78	C₄₃×C₉⋊C₉	C₃²	1	3	yes	—
79	C₄₃×C₉⋊C₉	C₃²	1	3	yes	—
80	C₄₃×C₃²⋊C₉	C₃²	1	3	yes	—
81	C₉²×C₄₃	C₃²	1	1	yes	characteristic
82	C₃²×C₄₃⋊C₃	C₃³	1	1	yes	characteristic
83	C₃²×C₄₃⋊C₃	C₃³	1	1	yes	characteristic
84	C₃²×C₄₃⋊C₃	C₃³	1	1	yes	characteristic
85	#11923958	—	3	3	no	—
86	#11923958	—	3	9	no	—
87	#11923957	—	3	3	no	—
88	#11923957	—	3	3	no	—
89	#11923957	—	3	3	no	—
90	#11923957	—	3	9	no	—
91	#11923956	—	3	3	no	—
92	#11923956	—	3	3	no	—
93	#11923956	—	3	3	no	—
94	#11923956	—	3	9	no	—
95	C₃²⋊(C₄₃⋊C₃)	—	3	3	no	—
96	C₃×C₄₃⋊C₉	—	3	3	no	—
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	1	yes	characteristic
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	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
107	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
108	C₃×C₄₃⋊C₉	C₃³	1	3	yes	—
109	C₉×C₄₃⋊C₃	—	3	3	no	—
110	C₉×C₄₃⋊C₃	—	3	3	no	—
111	C₉×C₄₃⋊C₃	—	3	9	no	—
112	C₉×C₄₃⋊C₃	3₋¹⁺²	1	9	yes	complemented
113	C₃³×C₄₃	C₃³	1	1	yes	characteristic
114	3₋¹⁺²×C₄₃	—	3	3	no	—
115	C₃×C₉×C₄₃	C₃³	1	3	yes	—
116	C₃×C₉×C₄₃	C₃³	1	3	yes	—
117	C₃×C₉×C₄₃	C₃³	1	3	yes	—
118	C₃×C₉×C₄₃	C₃³	1	1	yes	characteristic
119	C₃×C₉×C₄₃	C₃³	1	1	yes	characteristic
120	C₃×C₉×C₄₃	C₃³	1	1	yes	characteristic
121	#10624879	—	43	1	no	maximal, Sylow
122	C₃²×C₄₃	—	3	1	no	—
123	C₃²×C₄₃	—	3	3	no	—
124	C₃²×C₄₃	C₃⁴	1	1	yes	characteristic
125	C₃×C₄₃⋊C₃	—	3	3	no	—
126	C₃×C₄₃⋊C₃	—	3	1	no	—
127	C₃×C₄₃⋊C₃	—	3	1	no	—
128	C₃×C₄₃⋊C₃	—	3	3	no	—
129	C₃×C₄₃⋊C₃	—	3	3	no	—
130	C₃×C₄₃⋊C₃	C₃×3₋¹⁺²	1	3	yes	—
131	C₉×C₄₃	—	3	1	no	—
132	C₉×C₄₃	—	3	1	no	—
133	C₉×C₄₃	—	3	3	no	—
134	C₉×C₄₃	—	3	3	no	—
135	C₉×C₄₃	—	3	3	no	—
136	C₉×C₄₃	C₃×3₋¹⁺²	1	3	yes	—
137	C₄₃⋊C₉	—	3	1	no	—
138	C₄₃⋊C₉	—	3	1	no	—
139	C₄₃⋊C₉	—	3	1	no	—
140	C₄₃⋊C₉	—	3	1	no	—
141	C₄₃⋊C₉	—	3	3	no	—
142	C₄₃⋊C₉	—	3	3	no	—
143	C₄₃⋊C₉	—	3	3	no	—
144	C₄₃⋊C₉	—	3	3	no	—
145	C₄₃⋊C₉	—	3	3	no	—
146	C₄₃⋊C₉	C₃×3₋¹⁺²	1	3	yes	—
147	C₄₃⋊C₉	C₃×3₋¹⁺²	1	3	yes	—
148	C₄₃⋊C₉	#737	1	9	yes	complemented
149	C₃×#737	—	43	1	no	—
150	#6900	—	43	1	no	—
151	#6900	—	43	1	no	—
152	#6900	—	43	1	no	—
153	#6899	—	43	1	no	—
154	#6899	—	43	1	no	—
155	#6899	—	43	1	no	—
156	#6899	—	43	1	no	—
157	#6899	—	43	1	no	—
158	#6899	C₄₃⋊C₃	1	1	yes	characteristic, complemented
159	#6896	—	43	3	no	—
160	#6896	—	43	3	no	—
161	#6896	—	43	3	no	—
162	#6894	—	43	3	no	—
163	#6894	—	43	3	no	—
164	C₉⋊H(3,𝔽₃)	—	43	3	no	—
165	C₉×3₋¹⁺²	—	43	3	no	—
166	C₉×3₋¹⁺²	—	43	3	no	—
167	C₉×3₋¹⁺²	—	43	3	no	—
168	#6887	—	43	1	no	—
169	#6887	—	43	1	no	—
170	C₃×C₉²	—	43	1	no	—
171	C₃×C₄₃	—	9	1	no	—
172	C₃×C₄₃	C₃²×3₋¹⁺²	1	1	yes	characteristic
173	C₃×C₄₃	C₃×#737	1	3	yes	—
174	C₄₃⋊C₃	—	3	3	no	—
175	C₄₃⋊C₃	—	9	1	no	—
176	C₄₃⋊C₃	—	9	1	no	—
177	#737	—	129	9	no	—
178	C₃×3₋¹⁺²	—	43	1	no	—
179	C₃×3₋¹⁺²	—	43	3	no	—
180	C₃×3₋¹⁺²	—	43	1	no	—
181	C₃×3₋¹⁺²	—	43	1	no	—
182	C₃×3₋¹⁺²	—	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	1	no	—
186	C₃×3₋¹⁺²	—	43	3	no	—
187	C₃×3₋¹⁺²	—	43	3	no	—
188	C₃×3₋¹⁺²	C₃×C₄₃⋊C₃	1	1	yes	characteristic
189	C₃×H(3,𝔽₃)	—	43	1	no	—
190	C₃²×C₉	—	43	1	no	—
191	C₃²×C₉	—	43	1	no	—
192	C₃²×C₉	—	43	1	no	—
193	C₃²×C₉	—	43	1	no	—
194	C₃²×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	1	no	—
212	C₉⋊C₉	—	43	1	no	—
213	C₉⋊C₉	—	43	3	no	—
214	C₉⋊C₉	—	43	1	no	—
215	C₉⋊C₉	—	43	1	no	—
216	C₉⋊C₉	—	43	3	no	—
217	C₉⋊C₉	C₃×C₄₃⋊C₃	1	1	yes	characteristic
218	C₉⋊C₉	C₃×C₄₃⋊C₃	1	1	yes	characteristic
219	C₉⋊C₉	C₃×C₄₃⋊C₃	1	3	yes	—
220	C₉⋊C₉	C₃×C₄₃⋊C₃	1	3	yes	—
221	C₃²⋊C₉	—	43	3	no	—
222	C₃²⋊C₉	—	43	3	no	—
223	C₃²⋊C₉	—	43	3	no	—
224	C₃²⋊C₉	—	43	3	no	—
225	C₃²⋊C₉	—	43	3	no	—
226	C₃²⋊C₉	—	43	3	no	—
227	C₃²⋊C₉	—	43	3	no	—
228	C₃²⋊C₉	C₃×C₄₃⋊C₃	1	3	yes	—
229	C₉²	—	43	3	no	—
230	C₉²	—	43	3	no	—
231	C₉²	—	43	1	no	—
232	C₉²	—	43	1	no	—
233	C₉²	—	43	3	no	—
234	C₉²	—	43	3	no	—
235	C₉²	C₃×C₄₃⋊C₃	1	1	yes	characteristic
236	C₄₃	#10624879	1	1	yes	characteristic, complemented, Sylow
237	C₃³	—	43	1	no	—
238	C₃³	—	43	1	no	—
239	C₃³	—	43	1	no	—
240	C₃³	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
241	3₋¹⁺²	—	3	3	no	—
242	3₋¹⁺²	—	129	3	no	—
243	3₋¹⁺²	—	129	3	no	—
244	3₋¹⁺²	—	129	9	no	—
245	3₋¹⁺²	—	129	9	no	—
246	3₋¹⁺²	—	129	3	no	—
247	3₋¹⁺²	—	129	3	no	—
248	3₋¹⁺²	—	129	3	no	—
249	3₋¹⁺²	—	129	9	no	—
250	3₋¹⁺²	—	129	3	no	—
251	3₋¹⁺²	—	129	3	no	—
252	H(3,𝔽₃)	—	129	3	no	—
253	C₃×C₉	—	43	1	no	—
254	C₃×C₉	—	43	1	no	—
255	C₃×C₉	—	43	3	no	—
256	C₃×C₉	—	43	9	no	—
257	C₃×C₉	—	43	3	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	1	no	—
263	C₃×C₉	—	43	1	no	—
264	C₃×C₉	—	43	1	no	—
265	C₃×C₉	—	43	1	no	—
266	C₃×C₉	—	129	3	no	—
267	C₃×C₉	—	129	3	no	—
268	C₃×C₉	—	129	9	no	—
269	C₃×C₉	—	129	3	no	—
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	3	yes	—
273	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
274	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
275	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
276	C₃²	—	3	1	no	—
277	C₃²	—	3	3	no	—
278	C₃²	—	43	3	no	—
279	C₃²	—	129	3	no	—
280	C₃²	—	129	1	no	—
281	C₃²	—	129	1	no	—
282	C₃²	—	129	3	no	—
283	C₃²	—	129	3	no	—
284	C₃²	C₃³×C₄₃⋊C₃	1	1	yes	characteristic
285	C₉	—	3	1	no	—
286	C₉	—	3	1	no	—
287	C₉	—	3	3	no	—
288	C₉	—	3	3	no	—
289	C₉	—	3	3	no	—
290	C₉	—	43	3	no	—
291	C₉	—	43	3	no	—
292	C₉	—	43	9	no	—
293	C₉	—	129	1	no	—
294	C₉	—	129	1	no	—
295	C₉	—	129	1	no	—
296	C₉	—	129	1	no	—
297	C₉	—	129	3	no	—
298	C₉	—	129	3	no	—
299	C₉	—	129	3	no	—
300	C₉	—	129	3	no	—
301	C₉	—	129	3	no	—
302	C₉	3₋¹⁺²×C₄₃⋊C₃	1	3	yes	—
303	C₃	—	9	1	no	—
304	C₃	—	129	3	no	—
305	C₃	—	387	1	no	—
306	C₃	—	387	1	no	—
307	C₃	C₃×3₋¹⁺²×C₄₃⋊C₃	1	1	yes	characteristic
308	C₃	#737×C₄₃⋊C₃	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₄₃⋊C₃
commutator, socle, cosocle	C₃²×C₄₃	C₃⁴
Fitting	C₄₃×#6899	C₃
prime radical	C₃×C₃²⋊(C₄₃⋊C₃)	C₃²
hypercenter	#6899	C₄₃⋊C₃
hypocenter	C₄₃	#10624879
LCS	G ▷ C₃²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₃² ◁ #6899

Complements

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

N	H ≅ G/N	type
2. #23782849	C₃	semidirect
3. #23782848	C₃	semidirect
4. #23782845	C₃	semidirect
5. #23782844	C₃	semidirect
6. #23782843	C₃	semidirect
7. #23782840	C₃	semidirect
8. #23782839	C₃	semidirect
9. #23782838	C₃	semidirect
10. #23782826	C₃	semidirect
11. #23782826	C₃	semidirect
12. #23782823	C₃	semidirect
13. #23782814	C₃	semidirect
14. #23782813	C₃	semidirect
16. #23782792	C₃	semidirect
17. #11923957⋊C₉	C₃	semidirect
18. #11923956⋊C₉	C₃	semidirect
19. #23782781	C₃	semidirect
20. #23782781	C₃	semidirect
21. C₉²×C₄₃⋊C₃	C₃	semidirect
22. C₄₃×#6899	C₃	semidirect
112. C₉×C₄₃⋊C₃	3₋¹⁺²	semidirect
148. C₄₃⋊C₉	#737	semidirect
158. #6899	C₄₃⋊C₃	semidirect
236. C₄₃	#10624879	semidirect

Maximal subgroups

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

subgroup	index
C₃×#16812526	3
#23782849	3
#23782848	3
#23782845	3
#23782844	3
#23782843	3
#23782840	3
#23782839	3
#23782838	3
#23782826	3
#23782823	3
#23782814	3
#23782813	3
#23782801	3
#23782792	3
#11923957⋊C₉	3
#11923956⋊C₉	3
#23782781	3
C₉²×C₄₃⋊C₃	3
C₄₃×#6899	3
#10624879	43