Subgroups — #44534979

Order 31347.

836 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	C₃×3₋¹⁺²×C₄₃⋊C₃	C₃	1	1	yes	maximal, characteristic
2	C₃×3₋¹⁺²×C₄₃⋊C₃	C₃	1	1	yes	maximal, characteristic
3	#23782845	C₃	1	2	yes	maximal, complemented
4	#23782845	C₃	1	2	yes	maximal, complemented
5	#23782845	C₃	1	4	yes	maximal, complemented
6	#23782828	C₃	1	2	yes	maximal, complemented
7	#23782827	C₃	1	2	yes	maximal, complemented
8	#23782826	C₃	1	4	yes	maximal, complemented
9	#23782825	C₃	1	1	yes	maximal, characteristic, complemented
10	#23782814	C₃	1	4	yes	maximal, complemented
11	#23782813	C₃	1	4	yes	maximal, complemented
12	#23782811	C₃	1	2	yes	maximal, complemented
13	#23782810	C₃	1	2	yes	maximal, complemented
14	#23782795	C₃	1	2	yes	maximal
15	C₉×#11923956	C₃	1	1	yes	maximal, characteristic, complemented
16	C₉×#11923957	C₃	1	1	yes	maximal, characteristic, complemented
17	#11923957⋊C₉	C₃	1	2	yes	maximal, complemented
18	#11923956⋊C₉	C₃	1	2	yes	maximal, complemented
19	C₄₃×#6900	C₃	1	1	yes	maximal, characteristic, complemented
20	C₃³×C₄₃⋊C₃	C₃²	1	1	yes	characteristic
21	C₃×#11923958	C₃²	1	2	yes	—
22	C₃×#11923958	C₃²	1	2	yes	—
23	C₃×#11923958	C₃²	1	4	yes	—
24	C₃×#11923957	C₃²	1	2	yes	—
25	C₃×#11923957	C₃²	1	1	yes	characteristic
26	C₃×#11923957	C₃²	1	2	yes	—
27	C₃×#11923957	C₃²	1	1	yes	characteristic
28	C₃×#11923956	C₃²	1	2	yes	—
29	C₃×#11923956	C₃²	1	1	yes	characteristic
30	C₃×#11923956	C₃²	1	2	yes	—
31	C₃×#11923956	C₃²	1	1	yes	characteristic
32	3₋¹⁺²×C₄₃⋊C₃	—	3	9	no	—
33	3₋¹⁺²×C₄₃⋊C₃	—	3	9	no	—
34	C₃×C₉×C₄₃⋊C₃	C₃²	1	2	yes	—
35	C₃×C₉×C₄₃⋊C₃	C₃²	1	2	yes	—
36	C₃×C₉×C₄₃⋊C₃	C₃²	1	1	yes	characteristic
37	C₃×C₉×C₄₃⋊C₃	C₃²	1	1	yes	characteristic
38	#16812500	C₃²	1	2	yes	complemented
39	#16812500	C₃²	1	2	yes	complemented
40	#16812500	C₃²	1	4	yes	complemented
41	#16812499	C₃²	1	2	yes	complemented
42	#16812499	C₃²	1	2	yes	complemented
43	#16812499	C₃²	1	4	yes	complemented
44	(C₄₃⋊C₉)⋊C₉	C₃²	1	4	yes	complemented
45	(C₄₃⋊C₉)⋊C₉	C₃²	1	4	yes	complemented
46	(C₄₃⋊C₉)⋊C₉	C₃²	1	4	yes	complemented
47	(C₄₃⋊C₉)⋊C₉	C₃²	1	4	yes	complemented
48	(C₄₃⋊C₉)⋊C₉	C₃²	1	4	yes	complemented
49	(C₄₃⋊C₉)⋊C₉	C₃²	1	4	yes	complemented
50	(C₄₃⋊C₉)⋊C₉	C₃²	1	4	yes	complemented
51	(C₄₃⋊C₉)⋊C₉	C₃²	1	4	yes	complemented
52	(C₄₃⋊C₉)⋊C₉	C₃²	1	4	yes	complemented
53	(C₄₃⋊C₉)⋊C₉	C₃²	1	4	yes	complemented
54	(C₄₃⋊C₉)⋊C₉	C₃²	1	4	yes	complemented
55	(C₄₃⋊C₉)⋊C₉	C₃²	1	4	yes	complemented
56	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	4	yes	—
57	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	4	yes	—
58	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	2	yes	—
59	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	2	yes	—
60	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	2	yes	—
61	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	4	yes	—
62	C₉×C₄₃⋊C₉	C₃²	1	4	yes	complemented
63	C₉×C₄₃⋊C₉	C₃²	1	2	yes	complemented
64	C₉×C₄₃⋊C₉	C₃²	1	2	yes	complemented
65	C₃×3₋¹⁺²×C₄₃	C₃²	1	1	yes	characteristic
66	C₃×3₋¹⁺²×C₄₃	C₃²	1	1	yes	characteristic
67	C₃×3₋¹⁺²×C₄₃	C₃²	1	2	yes	—
68	C₄₃×C₉⋊C₉	C₃²	1	2	yes	complemented
69	C₄₃×C₉⋊C₉	C₃²	1	2	yes	complemented
70	C₄₃×C₉⋊C₉	C₃²	1	4	yes	complemented
71	C₉²×C₄₃	C₃²	1	1	yes	characteristic, complemented
72	C₃²×C₄₃⋊C₃	—	3	3	no	—
73	C₃²×C₄₃⋊C₃	—	3	3	no	—
74	C₃²×C₄₃⋊C₃	—	9	2	no	—
75	C₃²×C₄₃⋊C₃	C₃³	1	1	yes	characteristic
76	C₃²×C₄₃⋊C₃	C₃³	1	1	yes	characteristic
77	C₃²×C₄₃⋊C₃	C₃³	1	1	yes	characteristic
78	#11923958	—	3	6	no	—
79	#11923958	—	3	6	no	—
80	#11923958	—	9	4	no	—
81	#11923957	—	3	6	no	—
82	#11923957	—	3	3	no	—
83	#11923957	—	3	6	no	—
84	#11923957	3₋¹⁺²	1	9	yes	complemented
85	#11923956	—	3	6	no	—
86	#11923956	—	3	3	no	—
87	#11923956	—	3	6	no	—
88	#11923956	3₋¹⁺²	1	9	yes	complemented
89	C₃×C₄₃⋊C₉	C₃³	1	4	yes	—
90	C₃×C₄₃⋊C₉	C₃³	1	4	yes	—
91	C₃×C₄₃⋊C₉	C₃³	1	4	yes	—
92	C₃×C₄₃⋊C₉	C₃³	1	2	yes	—
93	C₃×C₄₃⋊C₉	C₃³	1	2	yes	—
94	C₃×C₄₃⋊C₉	C₃³	1	2	yes	—
95	C₃×C₄₃⋊C₉	C₃³	1	2	yes	—
96	C₃×C₄₃⋊C₉	C₃³	1	2	yes	—
97	C₃×C₄₃⋊C₉	C₃³	1	2	yes	—
98	C₉×C₄₃⋊C₃	—	3	3	no	—
99	C₉×C₄₃⋊C₃	—	3	3	no	—
100	C₉×C₄₃⋊C₃	—	3	6	no	—
101	C₉×C₄₃⋊C₃	—	3	6	no	—
102	C₃³×C₄₃	C₃³	1	1	yes	characteristic
103	3₋¹⁺²×C₄₃	—	3	3	no	—
104	3₋¹⁺²×C₄₃	—	3	3	no	—
105	3₋¹⁺²×C₄₃	—	9	2	no	—
106	C₃×C₉×C₄₃	C₃³	1	2	yes	—
107	C₃×C₉×C₄₃	C₃³	1	4	yes	—
108	C₃×C₉×C₄₃	C₃³	1	2	yes	—
109	C₃×C₉×C₄₃	C₃³	1	2	yes	—
110	C₃×C₉×C₄₃	C₃³	1	1	yes	characteristic
111	C₃×C₉×C₄₃	C₃³	1	1	yes	characteristic
112	3₋¹⁺²²	—	43	1	no	maximal, Sylow
113	C₃²×C₄₃	—	3	1	no	—
114	C₃²×C₄₃	—	3	1	no	—
115	C₃²×C₄₃	—	3	2	no	—
116	C₃²×C₄₃	C₃⁴	1	1	yes	characteristic
117	C₃×C₄₃⋊C₃	—	3	1	no	—
118	C₃×C₄₃⋊C₃	—	3	1	no	—
119	C₃×C₄₃⋊C₃	—	3	2	no	—
120	C₃×C₄₃⋊C₃	—	3	2	no	—
121	C₃×C₄₃⋊C₃	—	3	1	no	—
122	C₃×C₄₃⋊C₃	—	3	1	no	—
123	C₃×C₄₃⋊C₃	—	3	2	no	—
124	C₃×C₄₃⋊C₃	—	9	9	no	—
125	C₃×C₄₃⋊C₃	C₃×3₋¹⁺²	1	3	yes	—
126	C₃×C₄₃⋊C₃	C₃×3₋¹⁺²	1	3	yes	—
127	C₉×C₄₃	—	3	2	no	—
128	C₉×C₄₃	—	3	2	no	—
129	C₉×C₄₃	—	3	2	no	—
130	C₉×C₄₃	—	3	4	no	—
131	C₉×C₄₃	C₃×3₋¹⁺²	1	3	yes	complemented
132	C₉×C₄₃	C₃×3₋¹⁺²	1	3	yes	complemented
133	C₄₃⋊C₉	—	3	2	no	—
134	C₄₃⋊C₉	—	3	2	no	—
135	C₄₃⋊C₉	—	3	2	no	—
136	C₄₃⋊C₉	—	3	2	no	—
137	C₄₃⋊C₉	—	3	4	no	—
138	C₄₃⋊C₉	—	3	4	no	—
139	C₄₃⋊C₉	—	3	4	no	—
140	C₄₃⋊C₉	C₃×3₋¹⁺²	1	6	yes	complemented
141	C₄₃⋊C₉	C₃×3₋¹⁺²	1	6	yes	complemented
142	C₃²×3₋¹⁺²	—	43	1	no	—
143	C₃²×3₋¹⁺²	—	43	1	no	—
144	#6900	—	43	4	no	—
145	#6900	—	43	2	no	—
146	#6900	—	43	2	no	—
147	#6900	C₄₃⋊C₃	1	1	yes	characteristic, complemented
148	#6896	—	43	4	no	—
149	#6896	—	43	2	no	—
150	#6896	—	43	2	no	—
151	#6896	—	43	1	no	—
152	#6894	—	43	2	no	—
153	#6894	—	43	2	no	—
154	#6894	—	43	4	no	—
155	#6894	—	43	4	no	—
156	#6891	—	43	2	no	—
157	C₉×3₋¹⁺²	—	43	2	no	—
158	C₉×3₋¹⁺²	—	43	2	no	—
159	C₉×3₋¹⁺²	—	43	1	no	—
160	C₉×3₋¹⁺²	—	43	1	no	—
161	C₃×C₄₃	—	9	1	no	—
162	C₃×C₄₃	C₃²×3₋¹⁺²	1	1	yes	characteristic
163	C₃×C₄₃	C₃²×3₋¹⁺²	1	1	yes	characteristic
164	C₃×C₄₃	3₋¹⁺⁴	1	2	yes	—
165	C₄₃⋊C₃	—	3	3	no	—
166	C₄₃⋊C₃	—	3	3	no	—
167	C₄₃⋊C₃	—	9	1	no	—
168	C₃⁴	—	43	1	no	—
169	C₃×3₋¹⁺²	—	43	2	no	—
170	C₃×3₋¹⁺²	—	43	2	no	—
171	C₃×3₋¹⁺²	—	43	4	no	—
172	C₃×3₋¹⁺²	—	43	1	no	—
173	C₃×3₋¹⁺²	—	43	1	no	—
174	C₃×3₋¹⁺²	—	43	2	no	—
175	C₃×3₋¹⁺²	—	43	2	no	—
176	C₃×3₋¹⁺²	—	43	1	no	—
177	C₃×3₋¹⁺²	—	43	2	no	—
178	C₃×3₋¹⁺²	—	43	2	no	—
179	C₃×3₋¹⁺²	—	43	1	no	—
180	C₃×3₋¹⁺²	—	129	9	no	—
181	C₃×3₋¹⁺²	—	129	9	no	—
182	C₃×3₋¹⁺²	C₃×C₄₃⋊C₃	1	1	yes	characteristic
183	C₃×3₋¹⁺²	C₃×C₄₃⋊C₃	1	1	yes	characteristic
184	C₃×3₋¹⁺²	C₃×C₄₃⋊C₃	1	2	yes	—
185	C₃²×C₉	—	43	2	no	—
186	C₃²×C₉	—	43	2	no	—
187	C₃²×C₉	—	43	1	no	—
188	C₃²×C₉	—	43	1	no	—
189	C₉⋊C₉	—	43	4	no	—
190	C₉⋊C₉	—	43	4	no	—
191	C₉⋊C₉	—	43	4	no	—
192	C₉⋊C₉	—	43	4	no	—
193	C₉⋊C₉	—	43	4	no	—
194	C₉⋊C₉	—	43	4	no	—
195	C₉⋊C₉	—	43	4	no	—
196	C₉⋊C₉	—	43	4	no	—
197	C₉⋊C₉	—	43	4	no	—
198	C₉⋊C₉	—	43	4	no	—
199	C₉⋊C₉	—	43	4	no	—
200	C₉⋊C₉	—	43	4	no	—
201	C₉⋊C₉	—	43	2	no	—
202	C₉⋊C₉	—	43	2	no	—
203	C₉⋊C₉	—	43	4	no	—
204	C₉⋊C₉	—	43	2	no	—
205	C₉⋊C₉	—	43	2	no	—
206	C₉⋊C₉	—	43	4	no	—
207	C₉⋊C₉	C₃×C₄₃⋊C₃	1	2	yes	complemented
208	C₉⋊C₉	C₃×C₄₃⋊C₃	1	2	yes	complemented
209	C₉⋊C₉	C₃×C₄₃⋊C₃	1	4	yes	complemented
210	C₃²⋊C₉	—	43	4	no	—
211	C₃²⋊C₉	—	43	4	no	—
212	C₃²⋊C₉	—	43	2	no	—
213	C₃²⋊C₉	—	43	2	no	—
214	C₃²⋊C₉	—	43	4	no	—
215	C₃²⋊C₉	—	43	2	no	—
216	C₉²	—	43	4	no	—
217	C₉²	—	43	2	no	—
218	C₉²	—	43	2	no	—
219	C₉²	C₃×C₄₃⋊C₃	1	1	yes	characteristic, complemented
220	C₄₃	3₋¹⁺²²	1	1	yes	characteristic, complemented, Sylow
221	C₃³	—	43	1	no	—
222	C₃³	—	43	1	no	—
223	C₃³	—	43	1	no	—
224	C₃³	—	129	3	no	—
225	C₃³	—	129	3	no	—
226	C₃³	—	387	2	no	—
227	C₃³	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
228	3₋¹⁺²	—	3	3	no	—
229	3₋¹⁺²	—	3	3	no	—
230	3₋¹⁺²	—	9	2	no	—
231	3₋¹⁺²	—	43	9	no	—
232	3₋¹⁺²	—	43	9	no	—
233	3₋¹⁺²	—	129	6	no	—
234	3₋¹⁺²	—	129	6	no	—
235	3₋¹⁺²	—	129	6	no	—
236	3₋¹⁺²	—	129	6	no	—
237	3₋¹⁺²	—	129	6	no	—
238	3₋¹⁺²	—	129	6	no	—
239	3₋¹⁺²	—	129	3	no	—
240	3₋¹⁺²	—	129	3	no	—
241	3₋¹⁺²	—	387	4	no	—
242	C₃×C₉	—	43	2	no	—
243	C₃×C₉	—	43	2	no	—
244	C₃×C₉	—	43	4	no	—
245	C₃×C₉	—	43	4	no	—
246	C₃×C₉	—	43	4	no	—
247	C₃×C₉	—	43	2	no	—
248	C₃×C₉	—	43	2	no	—
249	C₃×C₉	—	43	2	no	—
250	C₃×C₉	—	43	2	no	—
251	C₃×C₉	—	129	6	no	—
252	C₃×C₉	—	129	6	no	—
253	C₃×C₉	—	129	3	no	—
254	C₃×C₉	—	129	3	no	—
255	C₃×C₉	C₃²×C₄₃⋊C₃	1	4	yes	—
256	C₃×C₉	C₃²×C₄₃⋊C₃	1	2	yes	—
257	C₃×C₉	C₃²×C₄₃⋊C₃	1	2	yes	—
258	C₃×C₉	C₃²×C₄₃⋊C₃	1	2	yes	—
259	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
260	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
261	C₃²	—	3	1	no	—
262	C₃²	—	3	1	no	—
263	C₃²	—	3	2	no	—
264	C₃²	—	43	3	no	—
265	C₃²	—	43	3	no	—
266	C₃²	—	129	1	no	—
267	C₃²	—	129	1	no	—
268	C₃²	—	129	2	no	—
269	C₃²	—	129	2	no	—
270	C₃²	—	129	1	no	—
271	C₃²	—	129	1	no	—
272	C₃²	—	129	2	no	—
273	C₃²	—	387	9	no	—
274	C₃²	C₃³×C₄₃⋊C₃	1	1	yes	characteristic
275	C₉	—	3	2	no	—
276	C₉	—	3	2	no	—
277	C₉	—	3	4	no	—
278	C₉	—	3	2	no	—
279	C₉	—	43	6	no	—
280	C₉	—	43	6	no	—
281	C₉	—	129	2	no	—
282	C₉	—	129	2	no	—
283	C₉	—	129	2	no	—
284	C₉	—	129	2	no	—
285	C₉	—	129	4	no	—
286	C₉	—	129	4	no	—
287	C₉	—	129	4	no	—
288	C₉	3₋¹⁺²×C₄₃⋊C₃	1	3	yes	complemented
289	C₉	3₋¹⁺²×C₄₃⋊C₃	1	3	yes	complemented
290	C₃	—	9	1	no	—
291	C₃	—	129	3	no	—
292	C₃	—	129	3	no	—
293	C₃	—	387	1	no	—
294	C₃	C₃×3₋¹⁺²×C₄₃⋊C₃	1	1	yes	characteristic
295	C₃	C₃×3₋¹⁺²×C₄₃⋊C₃	1	1	yes	characteristic
296	C₃	#23782923	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₄₃⋊C₃
commutator, socle, cosocle	C₃²×C₄₃	C₃⁴
Fitting	C₄₃×#6900	C₃
prime radical	C₃³×C₄₃⋊C₃	C₃²
hypercenter	#6900	C₄₃⋊C₃
hypocenter	C₄₃	3₋¹⁺²²
LCS	G ▷ C₃²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₃² ◁ #6900

Complements

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

N	H ≅ G/N	type
3. #23782845	C₃	semidirect
4. #23782845	C₃	semidirect
5. #23782845	C₃	semidirect
6. #23782828	C₃	semidirect
7. #23782827	C₃	semidirect
8. #23782826	C₃	semidirect
9. #23782825	C₃	semidirect
10. #23782814	C₃	semidirect
11. #23782813	C₃	semidirect
12. #23782811	C₃	semidirect
13. #23782810	C₃	semidirect
15. C₉×#11923956	C₃	semidirect
16. C₉×#11923957	C₃	semidirect
17. #11923957⋊C₉	C₃	semidirect
18. #11923956⋊C₉	C₃	semidirect
19. C₄₃×#6900	C₃	semidirect
38. #16812500	C₃²	semidirect
39. #16812500	C₃²	semidirect
40. #16812500	C₃²	semidirect
41. #16812499	C₃²	semidirect
42. #16812499	C₃²	semidirect
43. #16812499	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
51. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
52. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
53. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
54. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
55. (C₄₃⋊C₉)⋊C₉	C₃²	semidirect
62. C₉×C₄₃⋊C₉	C₃²	semidirect
63. C₉×C₄₃⋊C₉	C₃²	semidirect
64. C₉×C₄₃⋊C₉	C₃²	semidirect
68. C₄₃×C₉⋊C₉	C₃²	semidirect
69. C₄₃×C₉⋊C₉	C₃²	semidirect
70. C₄₃×C₉⋊C₉	C₃²	semidirect
71. C₉²×C₄₃	C₃²	semidirect
84. #11923957	3₋¹⁺²	semidirect
88. #11923956	3₋¹⁺²	semidirect
131. C₉×C₄₃	C₃×3₋¹⁺²	semidirect
132. C₉×C₄₃	C₃×3₋¹⁺²	semidirect
140. C₄₃⋊C₉	C₃×3₋¹⁺²	semidirect
141. C₄₃⋊C₉	C₃×3₋¹⁺²	semidirect
147. #6900	C₄₃⋊C₃	semidirect
207. C₉⋊C₉	C₃×C₄₃⋊C₃	semidirect
208. C₉⋊C₉	C₃×C₄₃⋊C₃	semidirect
209. C₉⋊C₉	C₃×C₄₃⋊C₃	semidirect
219. C₉²	C₃×C₄₃⋊C₃	semidirect
220. C₄₃	3₋¹⁺²²	semidirect
288. C₉	3₋¹⁺²×C₄₃⋊C₃	semidirect
289. C₉	3₋¹⁺²×C₄₃⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×3₋¹⁺²×C₄₃⋊C₃	3
#23782845	3
#23782828	3
#23782827	3
#23782826	3
#23782825	3
#23782814	3
#23782813	3
#23782811	3
#23782810	3
#23782795	3
C₉×#11923956	3
C₉×#11923957	3
#11923957⋊C₉	3
#11923956⋊C₉	3
C₄₃×#6900	3
3₋¹⁺²²	43