Subgroups — #44535364

Order 31347.

980 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782922	C₃	1	3	yes	maximal, complemented
2	C₃×3₋¹⁺²×C₄₃⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
3	#23782907	C₃	1	2	yes	maximal, complemented
4	#23782905	C₃	1	2	yes	maximal, complemented
5	#23782903	C₃	1	3	yes	maximal, complemented
6	#23782902	C₃	1	1	yes	maximal, characteristic, complemented
7	#23782900	C₃	1	3	yes	maximal, complemented
8	#23782898	C₃	1	3	yes	maximal, complemented
9	#23782897	C₃	1	3	yes	maximal, complemented
10	#23782893	C₃	1	2	yes	maximal, complemented
11	#23782889	C₃	1	1	yes	maximal, characteristic, complemented
12	#23782885	C₃	1	3	yes	maximal, complemented
13	#23782878	C₃	1	2	yes	maximal, complemented
14	#23782876	C₃	1	3	yes	maximal, complemented
15	#23782875	C₃	1	1	yes	maximal, characteristic, complemented
16	#23782874	C₃	1	3	yes	maximal, complemented
17	AΣL(1,𝔽₂₇)×C₄₃⋊C₃	C₃	1	3	yes	maximal, complemented
18	C₄₃×#6913	C₃	1	1	yes	maximal, characteristic, complemented
19	C₃³×C₄₃⋊C₃	C₃²	1	1	yes	characteristic
20	#16812526	—	3	6	no	—
21	#16812526	—	3	6	no	—
22	#16812526	C₃²	1	6	yes	complemented
23	#16812526	C₃²	1	6	yes	complemented
24	#16812525	—	3	6	no	—
25	#16812525	C₃²	1	6	yes	complemented
26	#16812524	C₃²	1	3	yes	complemented
27	#16812524	C₃²	1	3	yes	complemented
28	C₃×#11923958	C₃²	1	1	yes	characteristic, complemented
29	C₃×#11923958	C₃²	1	1	yes	characteristic, complemented
30	C₃×#11923957	—	3	3	no	—
31	C₃×#11923957	C₃²	1	1	yes	characteristic
32	C₃×#11923957	C₃²	1	1	yes	characteristic
33	C₃×#11923957	C₃²	1	1	yes	characteristic
34	C₃×#11923956	C₃²	1	1	yes	characteristic, complemented
35	C₃×#11923956	C₃²	1	1	yes	characteristic, complemented
36	C₃×#11923956	C₃²	1	1	yes	characteristic, complemented
37	3₋¹⁺²×C₄₃⋊C₃	—	3	6	no	—
38	3₋¹⁺²×C₄₃⋊C₃	—	3	3	no	—
39	3₋¹⁺²×C₄₃⋊C₃	—	3	6	no	—
40	3₋¹⁺²×C₄₃⋊C₃	C₃²	1	6	yes	—
41	H(3,𝔽₃)×C₄₃⋊C₃	C₃²	1	3	yes	—
42	C₃×C₉×C₄₃⋊C₃	C₃²	1	1	yes	characteristic, complemented
43	C₃×C₉×C₄₃⋊C₃	C₃²	1	1	yes	characteristic, complemented
44	C₃×C₉×C₄₃⋊C₃	C₃²	1	1	yes	characteristic, complemented
45	#16812515	C₃²	1	2	yes	complemented
46	#16812515	C₃²	1	2	yes	complemented
47	#16812514	C₃²	1	6	yes	complemented
48	#16812514	C₃²	1	6	yes	complemented
49	#16812513	C₃²	1	2	yes	complemented
50	#16812512	C₃²	1	3	yes	complemented
51	#16812512	C₃²	1	3	yes	complemented
52	#16812511	C₃²	1	6	yes	complemented
53	#16812511	C₃²	1	6	yes	complemented
54	#16812510	C₃²	1	6	yes	complemented
55	#16812509	C₃²	1	3	yes	complemented
56	#16812509	C₃²	1	3	yes	complemented
57	#16812508	C₃²	1	6	yes	complemented
58	#16812508	C₃²	1	6	yes	complemented
59	#16812507	C₃²	1	6	yes	complemented
60	#16812506	C₃²	1	3	yes	complemented
61	#16812506	C₃²	1	3	yes	complemented
62	C₄₃×#737	C₃²	1	3	yes	complemented
63	C₃×3₋¹⁺²×C₄₃	C₃²	1	1	yes	characteristic, complemented
64	C₄₃×#733	C₃²	1	2	yes	complemented
65	C₄₃×#732	C₃²	1	1	yes	characteristic, complemented
66	C₄₃×#731	C₃²	1	3	yes	complemented
67	C₄₃×AΣL(1,𝔽₂₇)	C₃²	1	3	yes	complemented
68	C₃²×C₄₃⋊C₃	—	3	3	no	—
69	C₃²×C₄₃⋊C₃	—	3	3	no	—
70	C₃²×C₄₃⋊C₃	—	9	3	no	—
71	C₃²×C₄₃⋊C₃	C₃³	1	1	yes	characteristic
72	C₃²×C₄₃⋊C₃	C₃³	1	1	yes	characteristic
73	C₃²×C₄₃⋊C₃	C₃³	1	1	yes	characteristic
74	#11923958	—	3	2	no	—
75	#11923958	—	3	3	no	—
76	#11923958	—	3	1	no	—
77	#11923958	—	3	6	no	—
78	#11923958	—	3	6	no	—
79	#11923958	C₃³	1	6	yes	—
80	#11923958	C₃³	1	6	yes	—
81	#11923957	—	3	3	no	—
82	#11923957	—	3	3	no	—
83	#11923957	—	3	6	no	—
84	#11923957	—	3	6	no	—
85	#11923957	—	3	6	no	—
86	#11923957	—	3	6	no	—
87	#11923957	H(3,𝔽₃)	1	6	yes	—
88	#11923957	H(3,𝔽₃)	1	3	yes	—
89	#11923956	—	3	6	no	—
90	#11923956	—	3	3	no	—
91	#11923956	—	3	6	no	—
92	#11923956	—	3	2	no	—
93	#11923956	—	3	3	no	—
94	#11923956	—	3	1	no	—
95	#11923956	—	3	6	no	—
96	#11923956	—	3	6	no	—
97	C₃²⋊(C₄₃⋊C₃)	—	3	6	no	—
98	C₃²⋊(C₄₃⋊C₃)	C₃³	1	6	yes	—
99	C₃×C₄₃⋊C₉	—	3	3	no	—
100	C₃×C₄₃⋊C₉	—	3	3	no	—
101	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
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	C₉×C₄₃⋊C₃	—	3	6	no	—
108	C₉×C₄₃⋊C₃	—	3	3	no	—
109	C₉×C₄₃⋊C₃	—	3	6	no	—
110	C₉×C₄₃⋊C₃	—	3	2	no	—
111	C₉×C₄₃⋊C₃	—	3	3	no	—
112	C₉×C₄₃⋊C₃	—	3	1	no	—
113	C₉×C₄₃⋊C₃	—	3	6	no	—
114	C₉×C₄₃⋊C₃	—	3	6	no	—
115	C₃³×C₄₃	C₃³	1	1	yes	characteristic
116	3₋¹⁺²×C₄₃	—	3	3	no	—
117	3₋¹⁺²×C₄₃	—	3	6	no	—
118	3₋¹⁺²×C₄₃	C₃³	1	6	yes	—
119	H(3,𝔽₃)×C₄₃	C₃³	1	3	yes	—
120	C₃×C₉×C₄₃	—	3	3	no	—
121	C₃×C₉×C₄₃	C₃³	1	1	yes	characteristic
122	C₃×C₉×C₄₃	C₃³	1	1	yes	characteristic
123	C₃×C₉×C₄₃	C₃³	1	1	yes	characteristic
124	#10624934	—	43	1	no	maximal, Sylow
125	C₃²×C₄₃	—	3	3	no	—
126	C₃²×C₄₃	—	3	1	no	—
127	C₃²×C₄₃	—	9	1	no	—
128	C₃²×C₄₃	C₃⁴	1	1	yes	characteristic
129	C₃×C₄₃⋊C₃	—	3	1	no	—
130	C₃×C₄₃⋊C₃	—	3	3	no	—
131	C₃×C₄₃⋊C₃	—	3	6	no	—
132	C₃×C₄₃⋊C₃	—	3	1	no	—
133	C₃×C₄₃⋊C₃	—	9	1	no	—
134	C₃×C₄₃⋊C₃	—	9	9	no	—
135	C₃×C₄₃⋊C₃	—	9	1	no	—
136	C₃×C₄₃⋊C₃	—	27	3	no	—
137	C₃×C₄₃⋊C₃	C₃×H(3,𝔽₃)	1	3	yes	—
138	C₉×C₄₃	—	3	6	no	—
139	C₉×C₄₃	—	3	1	no	—
140	C₉×C₄₃	—	3	1	no	—
141	C₉×C₄₃	C₃×H(3,𝔽₃)	1	3	yes	—
142	C₄₃⋊C₉	—	3	6	no	—
143	C₄₃⋊C₉	—	3	6	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	C₃×H(3,𝔽₃)	1	2	yes	—
149	C₄₃⋊C₉	C₃×H(3,𝔽₃)	1	3	yes	—
150	C₄₃⋊C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
151	3₋¹⁺⁴	—	43	3	no	—
152	C₃²×3₋¹⁺²	—	43	1	no	—
153	#6913	—	43	3	no	—
154	#6913	—	43	2	no	—
155	#6913	—	43	2	no	—
156	#6913	—	43	1	no	—
157	#6913	—	43	3	no	—
158	#6913	—	43	3	no	—
159	#6913	—	43	3	no	—
160	#6913	C₄₃⋊C₃	1	1	yes	characteristic, complemented
161	#6912	—	43	2	no	—
162	#6911	—	43	1	no	—
163	#6910	—	43	3	no	—
164	#6908	—	43	3	no	—
165	#6908	—	43	2	no	—
166	#6908	—	43	1	no	—
167	#6908	—	43	3	no	—
168	C₃×AΣL(1,𝔽₂₇)	—	43	3	no	—
169	C₃×C₄₃	—	3	1	no	—
170	C₃×C₄₃	—	9	1	no	—
171	C₃×C₄₃	—	9	3	no	—
172	C₃×C₄₃	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
173	C₄₃⋊C₃	—	3	3	no	—
174	C₄₃⋊C₃	—	3	3	no	—
175	C₄₃⋊C₃	—	9	1	no	—
176	C₄₃⋊C₃	—	9	6	no	—
177	C₃⁴	—	43	1	no	—
178	#737	—	43	6	no	—
179	#737	—	43	6	no	—
180	#737	—	43	3	no	—
181	#737	—	43	3	no	—
182	#737	—	43	6	no	—
183	#737	—	129	6	no	—
184	#737	—	129	6	no	—
185	#737	—	129	6	no	—
186	#737	C₃×C₄₃⋊C₃	1	3	yes	complemented
187	C₃×3₋¹⁺²	—	43	1	no	—
188	C₃×3₋¹⁺²	—	43	1	no	—
189	C₃×3₋¹⁺²	—	43	1	no	—
190	C₃×3₋¹⁺²	—	43	1	no	—
191	C₃×3₋¹⁺²	—	43	1	no	—
192	C₃×3₋¹⁺²	—	43	1	no	—
193	C₃×3₋¹⁺²	—	43	1	no	—
194	C₃×3₋¹⁺²	—	43	1	no	—
195	C₃×3₋¹⁺²	—	43	6	no	—
196	C₃×3₋¹⁺²	—	129	3	no	—
197	C₃×3₋¹⁺²	—	129	3	no	—
198	C₃×3₋¹⁺²	—	129	6	no	—
199	C₃×3₋¹⁺²	—	129	6	no	—
200	C₃×3₋¹⁺²	C₃×C₄₃⋊C₃	1	1	yes	characteristic, complemented
201	C₃×H(3,𝔽₃)	—	43	3	no	—
202	C₃²×C₉	—	43	1	no	—
203	C₃²×C₉	—	43	1	no	—
204	C₃²×C₉	—	43	1	no	—
205	#733	—	43	6	no	—
206	#733	—	43	6	no	—
207	#733	—	43	2	no	—
208	#733	—	43	2	no	—
209	#733	C₃×C₄₃⋊C₃	1	2	yes	complemented
210	#732	—	43	3	no	—
211	#732	—	43	3	no	—
212	#732	—	43	2	no	—
213	#732	C₃×C₄₃⋊C₃	1	1	yes	characteristic, complemented
214	#731	—	43	6	no	—
215	#731	—	43	6	no	—
216	#731	—	43	3	no	—
217	#731	—	43	3	no	—
218	#731	—	43	6	no	—
219	#731	C₃×C₄₃⋊C₃	1	3	yes	complemented
220	AΣL(1,𝔽₂₇)	—	43	6	no	—
221	AΣL(1,𝔽₂₇)	—	43	6	no	—
222	AΣL(1,𝔽₂₇)	—	43	6	no	—
223	AΣL(1,𝔽₂₇)	—	43	3	no	—
224	AΣL(1,𝔽₂₇)	—	43	3	no	—
225	AΣL(1,𝔽₂₇)	C₃×C₄₃⋊C₃	1	3	yes	complemented
226	C₄₃	#10624934	1	1	yes	characteristic, complemented, Sylow
227	C₃³	—	43	1	no	—
228	C₃³	—	43	1	no	—
229	C₃³	—	43	1	no	—
230	C₃³	—	129	3	no	—
231	C₃³	—	129	3	no	—
232	C₃³	—	387	3	no	—
233	C₃³	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
234	3₋¹⁺²	—	3	3	no	—
235	3₋¹⁺²	—	3	6	no	—
236	3₋¹⁺²	—	43	3	no	—
237	3₋¹⁺²	—	43	6	no	—
238	3₋¹⁺²	—	43	6	no	—
239	3₋¹⁺²	—	43	6	no	—
240	3₋¹⁺²	—	129	2	no	—
241	3₋¹⁺²	—	129	1	no	—
242	3₋¹⁺²	—	129	3	no	—
243	3₋¹⁺²	—	129	6	no	—
244	3₋¹⁺²	—	129	6	no	—
245	3₋¹⁺²	—	129	6	no	—
246	3₋¹⁺²	—	129	6	no	—
247	3₋¹⁺²	—	129	6	no	—
248	3₋¹⁺²	—	129	3	no	—
249	3₋¹⁺²	—	129	3	no	—
250	3₋¹⁺²	—	129	6	no	—
251	3₋¹⁺²	—	129	2	no	—
252	3₋¹⁺²	—	129	3	no	—
253	3₋¹⁺²	—	129	1	no	—
254	3₋¹⁺²	—	129	6	no	—
255	3₋¹⁺²	—	129	6	no	—
256	3₋¹⁺²	—	129	3	no	—
257	3₋¹⁺²	—	129	6	no	—
258	3₋¹⁺²	—	129	6	no	—
259	3₋¹⁺²	C₃²×C₄₃⋊C₃	1	6	yes	—
260	H(3,𝔽₃)	—	43	6	no	—
261	H(3,𝔽₃)	—	129	6	no	—
262	H(3,𝔽₃)	C₃²×C₄₃⋊C₃	1	3	yes	—
263	C₃×C₉	—	3	3	no	—
264	C₃×C₉	—	43	1	no	—
265	C₃×C₉	—	43	1	no	—
266	C₃×C₉	—	43	1	no	—
267	C₃×C₉	—	43	1	no	—
268	C₃×C₉	—	43	1	no	—
269	C₃×C₉	—	43	1	no	—
270	C₃×C₉	—	129	3	no	—
271	C₃×C₉	—	129	2	no	—
272	C₃×C₉	—	129	1	no	—
273	C₃×C₉	—	129	6	no	—
274	C₃×C₉	—	129	6	no	—
275	C₃×C₉	—	129	6	no	—
276	C₃×C₉	—	129	3	no	—
277	C₃×C₉	—	129	3	no	—
278	C₃×C₉	—	129	3	no	—
279	C₃×C₉	—	129	6	no	—
280	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
281	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
282	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
283	C₃²	—	3	3	no	—
284	C₃²	—	3	1	no	—
285	C₃²	—	9	1	no	—
286	C₃²	—	43	3	no	—
287	C₃²	—	129	3	no	—
288	C₃²	—	129	1	no	—
289	C₃²	—	129	6	no	—
290	C₃²	—	129	1	no	—
291	C₃²	—	387	1	no	—
292	C₃²	—	387	1	no	—
293	C₃²	—	387	9	no	—
294	C₃²	—	1161	3	no	—
295	C₃²	C₃³×C₄₃⋊C₃	1	1	yes	characteristic
296	C₉	—	3	6	no	—
297	C₉	—	3	1	no	—
298	C₉	—	3	1	no	—
299	C₉	—	43	2	no	—
300	C₉	—	43	3	no	—
301	C₉	—	43	1	no	—
302	C₉	—	129	1	no	—
303	C₉	—	129	1	no	—
304	C₉	—	129	1	no	—
305	C₉	—	129	1	no	—
306	C₉	—	129	6	no	—
307	C₉	—	129	6	no	—
308	C₉	H(3,𝔽₃)×C₄₃⋊C₃	1	3	yes	—
309	C₃	—	3	1	no	—
310	C₃	—	9	1	no	—
311	C₃	—	9	3	no	—
312	C₃	—	129	3	no	—
313	C₃	—	129	3	no	—
314	C₃	—	387	1	no	—
315	C₃	—	387	6	no	—
316	C₃	C₃×H(3,𝔽₃)×C₄₃⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₃×H(3,𝔽₃)×C₄₃⋊C₃
commutator, cosocle	C₃²×C₄₃	C₃⁴
Frattini	C₃²	C₃³×C₄₃⋊C₃
Fitting	C₄₃×#6913	C₃
socle	C₃×C₄₃	C₃²×H(3,𝔽₃)
hypercenter	#6913	C₄₃⋊C₃
hypocenter	C₄₃	#10624934
LCS	G ▷ C₃²×C₄₃ ▷ C₃×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ #6913

Complements

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

N	H ≅ G/N	type
1. #23782922	C₃	semidirect
2. C₃×3₋¹⁺²×C₄₃⋊C₃	C₃	semidirect
3. #23782907	C₃	semidirect
4. #23782905	C₃	semidirect
5. #23782903	C₃	semidirect
6. #23782902	C₃	semidirect
7. #23782900	C₃	semidirect
8. #23782898	C₃	semidirect
9. #23782897	C₃	semidirect
10. #23782893	C₃	semidirect
11. #23782889	C₃	semidirect
12. #23782885	C₃	semidirect
13. #23782878	C₃	semidirect
14. #23782876	C₃	semidirect
15. #23782875	C₃	semidirect
16. #23782874	C₃	semidirect
17. AΣL(1,𝔽₂₇)×C₄₃⋊C₃	C₃	semidirect
18. C₄₃×#6913	C₃	semidirect
22. #16812526	C₃²	semidirect
23. #16812526	C₃²	semidirect
25. #16812525	C₃²	semidirect
26. #16812524	C₃²	semidirect
27. #16812524	C₃²	semidirect
28. C₃×#11923958	C₃²	semidirect
29. C₃×#11923958	C₃²	semidirect
34. C₃×#11923956	C₃²	semidirect
35. C₃×#11923956	C₃²	semidirect
36. C₃×#11923956	C₃²	semidirect
42. C₃×C₉×C₄₃⋊C₃	C₃²	semidirect
43. C₃×C₉×C₄₃⋊C₃	C₃²	semidirect
44. C₃×C₉×C₄₃⋊C₃	C₃²	semidirect
45. #16812515	C₃²	semidirect
46. #16812515	C₃²	semidirect
47. #16812514	C₃²	semidirect
48. #16812514	C₃²	semidirect
49. #16812513	C₃²	semidirect
50. #16812512	C₃²	semidirect
51. #16812512	C₃²	semidirect
52. #16812511	C₃²	semidirect
53. #16812511	C₃²	semidirect
54. #16812510	C₃²	semidirect
55. #16812509	C₃²	semidirect
56. #16812509	C₃²	semidirect
57. #16812508	C₃²	semidirect
58. #16812508	C₃²	semidirect
59. #16812507	C₃²	semidirect
60. #16812506	C₃²	semidirect
61. #16812506	C₃²	semidirect
62. C₄₃×#737	C₃²	semidirect
63. C₃×3₋¹⁺²×C₄₃	C₃²	semidirect
64. C₄₃×#733	C₃²	semidirect
65. C₄₃×#732	C₃²	semidirect
66. C₄₃×#731	C₃²	semidirect
67. C₄₃×AΣL(1,𝔽₂₇)	C₃²	semidirect
160. #6913	C₄₃⋊C₃	semidirect
186. #737	C₃×C₄₃⋊C₃	semidirect
200. C₃×3₋¹⁺²	C₃×C₄₃⋊C₃	semidirect
209. #733	C₃×C₄₃⋊C₃	semidirect
213. #732	C₃×C₄₃⋊C₃	semidirect
219. #731	C₃×C₄₃⋊C₃	semidirect
225. AΣL(1,𝔽₂₇)	C₃×C₄₃⋊C₃	semidirect
226. C₄₃	#10624934	semidirect

Maximal subgroups

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

subgroup	index
#23782922	3
C₃×3₋¹⁺²×C₄₃⋊C₃	3
#23782907	3
#23782905	3
#23782903	3
#23782902	3
#23782900	3
#23782898	3
#23782897	3
#23782893	3
#23782889	3
#23782885	3
#23782878	3
#23782876	3
#23782875	3
#23782874	3
AΣL(1,𝔽₂₇)×C₄₃⋊C₃	3
C₄₃×#6913	3
#10624934	43