Subgroups — #44534766

Order 31347.

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	#23782842	C₃	1	3	yes	maximal, complemented
2	#23782841	C₃	1	3	yes	maximal, complemented
3	#23782836	C₃	1	3	yes	maximal, complemented
4	#23782833	C₃	1	3	yes	maximal, complemented
5	#23782832	C₃	1	3	yes	maximal, complemented
6	#23782826	C₃	1	3	yes	maximal, complemented
7	#23782820	C₃	1	1	yes	maximal, characteristic, complemented
8	#23782819	C₃	1	1	yes	maximal, characteristic, complemented
9	#23782818	C₃	1	1	yes	maximal, characteristic, complemented
10	#23782817	C₃	1	1	yes	maximal, characteristic, complemented
11	#23782812	C₃	1	1	yes	maximal, characteristic, complemented
12	#23782812	C₃	1	1	yes	maximal, characteristic, complemented
13	#23782809	C₃	1	1	yes	maximal, characteristic
14	#23782805	C₃	1	1	yes	maximal, characteristic
15	#23782804	C₃	1	1	yes	maximal, characteristic
16	C₉×#11923956	C₃	1	3	yes	maximal, complemented
17	C₉×#11923957	C₃	1	3	yes	maximal, complemented
18	C₉×#11923958	C₃	1	1	yes	maximal, characteristic, complemented
19	C₉×C₃²⋊(C₄₃⋊C₃)	C₃	1	1	yes	maximal, characteristic
20	#23782780	C₃	1	3	yes	maximal, complemented
21	C₃×C₉×C₄₃⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
22	C₉×3₋¹⁺²×C₄₃	C₃	1	1	yes	maximal, characteristic, complemented
23	C₃×#11923958	C₃²	1	1	yes	characteristic
24	C₃×#11923957	C₃²	1	3	yes	—
25	C₃×#11923956	C₃²	1	3	yes	—
26	C₃×C₃²⋊(C₄₃⋊C₃)	C₃²	1	1	yes	characteristic
27	C₃²×C₄₃⋊C₉	C₃²	1	1	yes	characteristic
28	C₃²×C₄₃⋊C₉	C₃²	1	1	yes	characteristic
29	C₃²×C₄₃⋊C₉	C₃²	1	1	yes	characteristic
30	C₃×C₉×C₄₃⋊C₃	C₃²	1	3	yes	—
31	#16812500	C₃²	1	3	yes	—
32	#16812500	C₃²	1	3	yes	—
33	#16812499	C₃²	1	3	yes	—
34	#16812499	C₃²	1	3	yes	—
35	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	—
36	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	—
37	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	—
38	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	—
39	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	—
40	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	—
41	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	—
42	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	—
43	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	—
44	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	—
45	(C₄₃⋊C₉)⋊C₉	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	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	3	yes	—
56	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	3	yes	—
57	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	3	yes	—
58	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	3	yes	—
59	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	3	yes	—
60	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	3	yes	—
61	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	3	yes	—
62	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	3	yes	—
63	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	3	yes	—
64	C₉×C₄₃⋊C₉	C₃²	1	1	yes	characteristic
65	C₉×C₄₃⋊C₉	C₃²	1	1	yes	characteristic
66	C₉×C₄₃⋊C₉	C₃²	1	1	yes	characteristic
67	C₉×C₄₃⋊C₉	C₃²	1	1	yes	characteristic
68	C₉×C₄₃⋊C₉	C₃²	1	1	yes	characteristic
69	C₉×C₄₃⋊C₉	C₃²	1	1	yes	characteristic
70	C₉×C₄₃⋊C₉	C₃²	1	1	yes	characteristic
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	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	—
77	C₄₃×C₉⋊C₉	C₃²	1	3	yes	—
78	C₄₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
79	C₄₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
80	C₉²×C₄₃	C₃²	1	3	yes	—
81	C₃²×C₄₃⋊C₃	C₃³	1	3	yes	—
82	#11923958	—	3	3	no	—
83	#11923957	—	3	9	no	—
84	#11923956	—	3	9	no	—
85	C₃²⋊(C₄₃⋊C₃)	—	3	3	no	—
86	C₃×C₄₃⋊C₉	—	3	3	no	—
87	C₃×C₄₃⋊C₉	—	3	3	no	—
88	C₃×C₄₃⋊C₉	—	3	3	no	—
89	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
90	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
91	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
92	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
93	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
94	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
95	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
96	C₃×C₄₃⋊C₉	C₃³	1	3	yes	—
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	1	yes	characteristic
102	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
103	C₉×C₄₃⋊C₃	—	3	9	no	—
104	C₃³×C₄₃	C₃³	1	1	yes	characteristic
105	3₋¹⁺²×C₄₃	—	3	3	no	—
106	C₃×C₉×C₄₃	—	3	3	no	—
107	C₃×C₉×C₄₃	C₃³	1	3	yes	—
108	C₃×C₉×C₄₃	C₃³	1	3	yes	—
109	C₃×C₉×C₄₃	C₃³	1	3	yes	—
110	C₃×C₉×C₄₃	C₃³	1	1	yes	characteristic
111	C₃×C₉×C₄₃	C₃³	1	1	yes	characteristic
112	C₃×C₉×C₄₃	C₃³	1	1	yes	characteristic
113	#10624877	—	43	1	no	maximal, Sylow
114	C₃²×C₄₃	—	3	1	no	—
115	C₃²×C₄₃	—	3	1	no	—
116	C₃²×C₄₃	—	3	1	no	—
117	C₃²×C₄₃	#737	1	3	yes	—
118	C₃²×C₄₃	C₃⁴	1	1	yes	characteristic
119	C₃×C₄₃⋊C₃	—	3	3	no	—
120	C₃×C₄₃⋊C₃	—	3	3	no	—
121	C₃×C₄₃⋊C₃	—	3	3	no	—
122	C₃×C₄₃⋊C₃	—	3	3	no	—
123	C₉×C₄₃	—	3	1	no	—
124	C₉×C₄₃	—	3	1	no	—
125	C₉×C₄₃	—	3	3	no	—
126	C₉×C₄₃	—	3	3	no	—
127	C₉×C₄₃	—	3	3	no	—
128	C₉×C₄₃	#737	1	3	yes	—
129	C₄₃⋊C₉	—	3	3	no	—
130	C₄₃⋊C₉	—	3	3	no	—
131	C₄₃⋊C₉	—	3	1	no	—
132	C₄₃⋊C₉	—	3	1	no	—
133	C₄₃⋊C₉	—	3	1	no	—
134	C₄₃⋊C₉	—	3	1	no	—
135	C₄₃⋊C₉	—	3	1	no	—
136	C₄₃⋊C₉	—	3	1	no	—
137	C₄₃⋊C₉	—	3	1	no	—
138	C₄₃⋊C₉	—	3	3	no	—
139	C₄₃⋊C₉	—	3	3	no	—
140	C₄₃⋊C₉	—	3	3	no	—
141	C₄₃⋊C₉	#737	1	3	yes	—
142	C₄₃⋊C₉	#737	1	3	yes	—
143	#6899	—	43	3	no	—
144	#6899	—	43	3	no	—
145	#6898	—	43	3	no	—
146	#6897	—	43	3	no	—
147	#6897	—	43	3	no	—
148	#6896	—	43	3	no	—
149	#6895	—	43	1	no	—
150	#6895	—	43	1	no	—
151	#6894	—	43	1	no	—
152	#6894	—	43	1	no	—
153	#6894	—	43	1	no	—
154	#6894	—	43	1	no	—
155	#6893	—	43	1	no	—
156	C₉⋊H(3,𝔽₃)	—	43	1	no	—
157	C₉⋊H(3,𝔽₃)	—	43	1	no	—
158	C₉×3₋¹⁺²	—	43	1	no	—
159	C₉×3₋¹⁺²	—	43	3	no	—
160	C₉×3₋¹⁺²	—	43	3	no	—
161	C₉×3₋¹⁺²	C₄₃⋊C₃	1	1	yes	characteristic, complemented
162	C₉×H(3,𝔽₃)	—	43	1	no	—
163	#6887	—	43	3	no	—
164	C₃×C₉²	—	43	1	no	—
165	C₃×C₄₃	—	3	3	no	—
166	C₃×C₄₃	C₃×#737	1	1	yes	characteristic
167	C₃×C₄₃	C₃×#737	1	1	yes	characteristic
168	C₃×C₄₃	C₃×#737	1	1	yes	characteristic
169	C₃×C₄₃	C₃×#737	1	1	yes	characteristic
170	C₄₃⋊C₃	—	9	3	no	—
171	C₃×3₋¹⁺²	—	43	1	no	—
172	C₃×3₋¹⁺²	—	43	3	no	—
173	C₃×3₋¹⁺²	—	43	3	no	—
174	C₃×3₋¹⁺²	C₃×C₄₃⋊C₃	1	1	yes	characteristic
175	C₃×H(3,𝔽₃)	—	43	1	no	—
176	C₃²×C₉	—	43	1	no	—
177	C₃²×C₉	—	43	1	no	—
178	C₃²×C₉	—	43	1	no	—
179	C₃²×C₉	—	43	3	no	—
180	C₃²×C₉	C₃×C₄₃⋊C₃	1	1	yes	characteristic
181	C₉⋊C₉	—	43	3	no	—
182	C₉⋊C₉	—	43	3	no	—
183	C₉⋊C₉	—	43	3	no	—
184	C₉⋊C₉	—	43	3	no	—
185	C₉⋊C₉	—	43	3	no	—
186	C₉⋊C₉	—	43	3	no	—
187	C₉⋊C₉	—	43	3	no	—
188	C₉⋊C₉	—	43	3	no	—
189	C₉⋊C₉	—	43	3	no	—
190	C₉⋊C₉	—	43	3	no	—
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	C₃×C₄₃⋊C₃	1	3	yes	—
202	C₉⋊C₉	C₃×C₄₃⋊C₃	1	3	yes	—
203	C₃²⋊C₉	—	43	1	no	—
204	C₃²⋊C₉	—	43	1	no	—
205	C₃²⋊C₉	—	43	1	no	—
206	C₃²⋊C₉	—	43	1	no	—
207	C₃²⋊C₉	—	43	3	no	—
208	C₃²⋊C₉	—	43	3	no	—
209	C₃²⋊C₉	—	43	3	no	—
210	C₃²⋊C₉	—	43	3	no	—
211	C₃²⋊C₉	—	43	3	no	—
212	C₃²⋊C₉	—	43	3	no	—
213	C₃²⋊C₉	—	43	3	no	—
214	C₃²⋊C₉	—	43	3	no	—
215	C₃²⋊C₉	—	43	3	no	—
216	C₃²⋊C₉	C₃×C₄₃⋊C₃	1	1	yes	characteristic
217	C₃²⋊C₉	C₃×C₄₃⋊C₃	1	1	yes	characteristic
218	C₉²	—	43	1	no	—
219	C₉²	—	43	1	no	—
220	C₉²	—	43	1	no	—
221	C₉²	—	43	1	no	—
222	C₉²	—	43	1	no	—
223	C₉²	—	43	1	no	—
224	C₉²	—	43	1	no	—
225	C₉²	—	43	1	no	—
226	C₉²	—	43	1	no	—
227	C₉²	—	43	3	no	—
228	C₉²	C₃×C₄₃⋊C₃	1	3	yes	—
229	C₄₃	#10624877	1	1	yes	characteristic, complemented, Sylow
230	C₃³	—	43	3	no	—
231	C₃³	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
232	3₋¹⁺²	—	3	3	no	—
233	3₋¹⁺²	—	129	9	no	—
234	3₋¹⁺²	—	129	9	no	—
235	3₋¹⁺²	—	129	3	no	—
236	H(3,𝔽₃)	—	129	3	no	—
237	C₃×C₉	—	3	3	no	—
238	C₃×C₉	—	43	1	no	—
239	C₃×C₉	—	43	1	no	—
240	C₃×C₉	—	43	1	no	—
241	C₃×C₉	—	43	1	no	—
242	C₃×C₉	—	43	1	no	—
243	C₃×C₉	—	43	1	no	—
244	C₃×C₉	—	43	1	no	—
245	C₃×C₉	—	43	1	no	—
246	C₃×C₉	—	43	1	no	—
247	C₃×C₉	—	43	3	no	—
248	C₃×C₉	—	43	3	no	—
249	C₃×C₉	—	43	3	no	—
250	C₃×C₉	—	43	3	no	—
251	C₃×C₉	—	43	3	no	—
252	C₃×C₉	—	129	3	no	—
253	C₃×C₉	—	129	3	no	—
254	C₃×C₉	—	129	3	no	—
255	C₃×C₉	—	129	9	no	—
256	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
257	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
258	C₃×C₉	C₃²×C₄₃⋊C₃	1	3	yes	—
259	C₃×C₉	C₃²×C₄₃⋊C₃	1	3	yes	—
260	C₃×C₉	C₃²×C₄₃⋊C₃	1	3	yes	—
261	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
262	C₃²	—	3	1	no	—
263	C₃²	—	3	1	no	—
264	C₃²	—	3	1	no	—
265	C₃²	—	129	3	no	—
266	C₃²	—	129	3	no	—
267	C₃²	—	129	3	no	—
268	C₃²	—	129	3	no	—
269	C₃²	#16812525	1	3	yes	—
270	C₃²	C₃³×C₄₃⋊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	—	43	3	no	—
277	C₉	—	43	3	no	—
278	C₉	—	129	1	no	—
279	C₉	—	129	1	no	—
280	C₉	—	129	1	no	—
281	C₉	—	129	1	no	—
282	C₉	—	129	3	no	—
283	C₉	—	129	3	no	—
284	C₉	—	129	1	no	—
285	C₉	—	129	1	no	—
286	C₉	—	129	1	no	—
287	C₉	—	129	3	no	—
288	C₉	—	129	3	no	—
289	C₉	—	129	3	no	—
290	C₉	#16812526	1	3	yes	—
291	C₃	—	3	3	no	—
292	C₃	—	387	3	no	—
293	C₃	#737×C₄₃⋊C₃	1	1	yes	characteristic
294	C₃	#737×C₄₃⋊C₃	1	1	yes	characteristic
295	C₃	C₃×#16812525	1	1	yes	characteristic
296	C₃	C₃×#16812526	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₉	C₃²×C₄₃⋊C₃
commutator, socle, cosocle	C₃²×C₄₃	C₃⁴
Frattini	C₃²	C₃³×C₄₃⋊C₃
Fitting	C₉×3₋¹⁺²×C₄₃	C₃
prime radical	C₃×C₃²⋊(C₄₃⋊C₃)	C₃²
hypercenter	C₉×3₋¹⁺²	C₄₃⋊C₃
hypocenter	C₄₃	#10624877
LCS	G ▷ C₃²×C₄₃ ▷ C₄₃
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. #23782842	C₃	semidirect
2. #23782841	C₃	semidirect
3. #23782836	C₃	semidirect
4. #23782833	C₃	semidirect
5. #23782832	C₃	semidirect
6. #23782826	C₃	semidirect
7. #23782820	C₃	semidirect
8. #23782819	C₃	semidirect
9. #23782818	C₃	semidirect
10. #23782817	C₃	semidirect
11. #23782812	C₃	semidirect
12. #23782812	C₃	semidirect
16. C₉×#11923956	C₃	semidirect
17. C₉×#11923957	C₃	semidirect
18. C₉×#11923958	C₃	semidirect
20. #23782780	C₃	semidirect
21. C₃×C₉×C₄₃⋊C₉	C₃	semidirect
22. C₉×3₋¹⁺²×C₄₃	C₃	semidirect
161. C₉×3₋¹⁺²	C₄₃⋊C₃	semidirect
229. C₄₃	#10624877	semidirect

Maximal subgroups

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

subgroup	index
#23782842	3
#23782841	3
#23782836	3
#23782833	3
#23782832	3
#23782826	3
#23782820	3
#23782819	3
#23782818	3
#23782817	3
#23782812	3
#23782809	3
#23782805	3
#23782804	3
C₉×#11923956	3
C₉×#11923957	3
C₉×#11923958	3
C₉×C₃²⋊(C₄₃⋊C₃)	3
#23782780	3
C₃×C₉×C₄₃⋊C₉	3
C₉×3₋¹⁺²×C₄₃	3
#10624877	43