Subgroups — #44534829

Order 31347.

698 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#16812524	C₃	1	1	yes	maximal, characteristic
2	#23782845	C₃	1	3	yes	maximal, complemented
3	#23782837	C₃	1	3	yes	maximal, complemented
4	#23782837	C₃	1	3	yes	maximal, complemented
5	#23782824	C₃	1	3	yes	maximal, complemented
6	#23782823	C₃	1	3	yes	maximal, complemented
7	#23782823	C₃	1	3	yes	maximal, complemented
8	#23782818	C₃	1	1	yes	maximal, characteristic, complemented
9	#23782818	C₃	1	1	yes	maximal, characteristic, complemented
10	#23782818	C₃	1	3	yes	maximal, complemented
11	#23782817	C₃	1	1	yes	maximal, characteristic, complemented
12	#23782817	C₃	1	1	yes	maximal, characteristic, complemented
13	#23782817	C₃	1	3	yes	maximal, complemented
14	#23782816	C₃	1	1	yes	maximal, characteristic, complemented
15	#23782815	C₃	1	1	yes	maximal, characteristic, complemented
16	#23782812	C₃	1	3	yes	maximal, complemented
17	#23782800	C₃	1	1	yes	maximal, characteristic
18	#23782800	C₃	1	1	yes	maximal, characteristic
19	#23782779	C₃	1	1	yes	maximal, characteristic, complemented
20	#23782779	C₃	1	1	yes	maximal, characteristic, complemented
21	C₃×(C₄₃⋊C₉)⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
22	C₄₃×C₉⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
23	#16812524	—	3	9	no	—
24	C₃×#11923958	C₃²	1	1	yes	characteristic
25	C₃×#11923958	C₃²	1	1	yes	characteristic
26	C₃×#11923958	C₃²	1	3	yes	—
27	C₃×#11923958	C₃²	1	3	yes	—
28	C₃×#11923958	C₃²	1	3	yes	—
29	C₃×#11923958	C₃²	1	3	yes	—
30	C₃²×C₄₃⋊C₉	C₃²	1	3	yes	—
31	C₃²×C₄₃⋊C₉	C₃²	1	1	yes	characteristic
32	C₃²×C₄₃⋊C₉	C₃²	1	1	yes	characteristic
33	C₃²×C₄₃⋊C₉	C₃²	1	1	yes	characteristic
34	#16812500	C₃²	1	3	yes	—
35	#16812500	C₃²	1	3	yes	—
36	#16812500	C₃²	1	3	yes	—
37	#16812499	C₃²	1	3	yes	—
38	#16812499	C₃²	1	3	yes	—
39	#16812499	C₃²	1	3	yes	—
40	(C₄₃⋊C₉)⋊C₉	C₃²	1	1	yes	characteristic
41	(C₄₃⋊C₉)⋊C₉	C₃²	1	1	yes	characteristic
42	(C₄₃⋊C₉)⋊C₉	C₃²	1	1	yes	characteristic
43	(C₄₃⋊C₉)⋊C₉	C₃²	1	1	yes	characteristic
44	(C₄₃⋊C₉)⋊C₉	C₃²	1	1	yes	characteristic
45	(C₄₃⋊C₉)⋊C₉	C₃²	1	1	yes	characteristic
46	(C₄₃⋊C₉)⋊C₉	C₃²	1	1	yes	characteristic
47	(C₄₃⋊C₉)⋊C₉	C₃²	1	1	yes	characteristic
48	(C₄₃⋊C₉)⋊C₉	C₃²	1	1	yes	characteristic
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	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	—
63	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	—
64	(C₄₃⋊C₉)⋊C₉	C₃²	1	3	yes	—
65	C₃²⋊(C₄₃⋊C₉)	C₃²	1	1	yes	characteristic
66	C₃²⋊(C₄₃⋊C₉)	C₃²	1	1	yes	characteristic
67	C₃²⋊(C₄₃⋊C₉)	C₃²	1	1	yes	characteristic
68	C₃²⋊(C₄₃⋊C₉)	C₃²	1	1	yes	characteristic
69	C₃²⋊(C₄₃⋊C₉)	C₃²	1	3	yes	—
70	C₃²⋊(C₄₃⋊C₉)	C₃²	1	3	yes	—
71	C₃²⋊(C₄₃⋊C₉)	C₃²	1	3	yes	—
72	C₃²⋊(C₄₃⋊C₉)	C₃²	1	3	yes	—
73	C₉×C₄₃⋊C₉	C₃²	1	3	yes	—
74	C₉×C₄₃⋊C₉	C₃²	1	3	yes	—
75	C₃×3₋¹⁺²×C₄₃	C₃²	1	3	yes	—
76	C₃×H(3,𝔽₃)×C₄₃	C₃²	1	1	yes	characteristic
77	C₃²×C₉×C₄₃	C₃²	1	1	yes	characteristic
78	C₄₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
79	C₄₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
80	C₄₃×C₃²⋊C₉	C₃²	1	3	yes	—
81	C₄₃×C₃²⋊C₉	C₃²	1	3	yes	—
82	#11923958	—	3	3	no	—
83	#11923958	—	3	3	no	—
84	#11923958	—	3	9	no	—
85	#11923958	—	3	9	no	—
86	#11923958	—	9	3	no	—
87	#11923958	—	9	3	no	—
88	C₃×C₄₃⋊C₉	—	3	9	no	—
89	C₃×C₄₃⋊C₉	—	9	1	no	—
90	C₃×C₄₃⋊C₉	—	9	1	no	—
91	C₃×C₄₃⋊C₉	3₋¹⁺²	1	9	yes	complemented
92	C₃×C₄₃⋊C₉	C₃³	1	3	yes	—
93	C₃×C₄₃⋊C₉	C₃³	1	3	yes	—
94	C₃×C₄₃⋊C₉	C₃³	1	3	yes	—
95	C₃×C₄₃⋊C₉	C₃³	1	3	yes	—
96	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
97	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
98	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
99	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
100	C₃×C₄₃⋊C₉	C₃³	1	3	yes	—
101	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
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	C₃×C₄₃⋊C₉	C₃³	1	1	yes	characteristic
107	C₃³×C₄₃	C₃³	1	3	yes	—
108	C₃³×C₄₃	C₃³	1	1	yes	characteristic
109	3₋¹⁺²×C₄₃	—	9	3	no	—
110	H(3,𝔽₃)×C₄₃	—	3	3	no	—
111	C₃×C₉×C₄₃	—	9	1	no	—
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	C₃×C₉×C₄₃	C₃³	1	3	yes	—
116	C₃×C₉×C₄₃	C₃³	1	1	yes	characteristic
117	#10624887	—	43	1	no	maximal, Sylow
118	C₃²×C₄₃	—	3	1	no	—
119	C₃²×C₄₃	—	3	1	no	—
120	C₃²×C₄₃	—	3	1	no	—
121	C₃²×C₄₃	—	3	3	no	—
122	C₃²×C₄₃	—	3	3	no	—
123	C₃²×C₄₃	—	3	3	no	—
124	C₃²×C₄₃	—	3	3	no	—
125	C₃²×C₄₃	C₃×3₋¹⁺²	1	3	yes	—
126	C₃²×C₄₃	C₃⁴	1	1	yes	characteristic
127	C₉×C₄₃	—	3	1	no	—
128	C₉×C₄₃	—	3	1	no	—
129	C₉×C₄₃	—	3	1	no	—
130	C₉×C₄₃	—	3	3	no	—
131	C₉×C₄₃	—	3	3	no	—
132	C₄₃⋊C₉	—	3	3	no	—
133	C₄₃⋊C₉	—	3	3	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	1	no	—
139	C₄₃⋊C₉	—	3	1	no	—
140	C₄₃⋊C₉	—	3	3	no	—
141	C₄₃⋊C₉	—	3	3	no	—
142	C₄₃⋊C₉	—	3	3	no	—
143	C₄₃⋊C₉	—	3	3	no	—
144	C₄₃⋊C₉	C₃×3₋¹⁺²	1	3	yes	—
145	C₄₃⋊C₉	C₃×3₋¹⁺²	1	3	yes	—
146	C₄₃⋊C₉	C₃×3₋¹⁺²	1	3	yes	—
147	C₃×#737	—	43	1	no	—
148	#6900	—	43	3	no	—
149	#6899	—	43	3	no	—
150	#6899	—	43	3	no	—
151	#6896	—	43	3	no	—
152	#6896	—	43	3	no	—
153	#6896	—	43	3	no	—
154	#6894	—	43	3	no	—
155	#6894	—	43	1	no	—
156	#6894	—	43	1	no	—
157	#6894	—	43	1	no	—
158	#6894	—	43	1	no	—
159	#6894	—	43	3	no	—
160	#6894	—	43	1	no	—
161	#6894	—	43	1	no	—
162	#6894	—	43	3	no	—
163	C₉⋊H(3,𝔽₃)	—	43	1	no	—
164	C₉⋊H(3,𝔽₃)	—	43	1	no	—
165	C₉⋊H(3,𝔽₃)	C₄₃⋊C₃	1	1	yes	characteristic
166	#6887	—	43	1	no	—
167	#6887	—	43	1	no	—
168	C₃×C₉⋊C₉	—	43	1	no	—
169	C₃×C₄₃	—	3	3	no	—
170	C₃×C₄₃	—	9	3	no	—
171	C₃×C₄₃	C₃²×3₋¹⁺²	1	1	yes	characteristic
172	C₃×C₄₃	3₋¹⁺⁴	1	1	yes	characteristic
173	C₃×C₄₃	3₋¹⁺⁴	1	1	yes	characteristic
174	C₃×C₄₃	3₋¹⁺⁴	1	1	yes	characteristic
175	#737	—	129	9	no	—
176	C₃×3₋¹⁺²	—	43	1	no	—
177	C₃×3₋¹⁺²	—	43	1	no	—
178	C₃×3₋¹⁺²	—	43	3	no	—
179	C₃×3₋¹⁺²	—	43	3	no	—
180	C₃×3₋¹⁺²	—	43	3	no	—
181	C₃×3₋¹⁺²	—	43	3	no	—
182	C₃×3₋¹⁺²	C₃×C₄₃⋊C₃	1	3	yes	—
183	C₃×H(3,𝔽₃)	C₃×C₄₃⋊C₃	1	1	yes	characteristic
184	C₃²×C₉	—	43	1	no	—
185	C₃²×C₉	—	43	3	no	—
186	C₃²×C₉	—	43	1	no	—
187	C₃²×C₉	—	43	1	no	—
188	C₃²×C₉	C₃×C₄₃⋊C₃	1	1	yes	characteristic
189	C₉⋊C₉	—	43	1	no	—
190	C₉⋊C₉	—	43	1	no	—
191	C₉⋊C₉	—	43	1	no	—
192	C₉⋊C₉	—	43	1	no	—
193	C₉⋊C₉	—	43	1	no	—
194	C₉⋊C₉	—	43	1	no	—
195	C₉⋊C₉	—	43	1	no	—
196	C₉⋊C₉	—	43	1	no	—
197	C₉⋊C₉	—	43	1	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	—	43	3	no	—
213	C₉⋊C₉	—	43	3	no	—
214	C₉⋊C₉	—	43	3	no	—
215	C₉⋊C₉	—	43	3	no	—
216	C₉⋊C₉	—	43	3	no	—
217	C₉⋊C₉	—	43	3	no	—
218	C₉⋊C₉	—	43	3	no	—
219	C₉⋊C₉	—	43	3	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	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	1	yes	characteristic
229	C₃²⋊C₉	C₃×C₄₃⋊C₃	1	1	yes	characteristic
230	C₃²⋊C₉	C₃×C₄₃⋊C₃	1	3	yes	—
231	C₃²⋊C₉	C₃×C₄₃⋊C₃	1	3	yes	—
232	C₉²	—	43	3	no	—
233	C₉²	—	43	3	no	—
234	C₄₃	#10624887	1	1	yes	characteristic, complemented, Sylow
235	C₃³	C₃²×C₄₃⋊C₃	1	3	yes	—
236	C₃³	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
237	3₋¹⁺²	—	9	3	no	—
238	3₋¹⁺²	—	129	9	no	—
239	3₋¹⁺²	—	129	9	no	—
240	3₋¹⁺²	—	129	3	no	—
241	3₋¹⁺²	—	129	3	no	—
242	3₋¹⁺²	—	387	3	no	—
243	3₋¹⁺²	—	387	3	no	—
244	H(3,𝔽₃)	—	3	3	no	—
245	C₃×C₉	—	9	1	no	—
246	C₃×C₉	—	43	9	no	—
247	C₃×C₉	—	43	1	no	—
248	C₃×C₉	—	43	1	no	—
249	C₃×C₉	—	43	1	no	—
250	C₃×C₉	—	43	3	no	—
251	C₃×C₉	—	43	3	no	—
252	C₃×C₉	—	43	3	no	—
253	C₃×C₉	—	43	3	no	—
254	C₃×C₉	—	43	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	1	no	—
261	C₃×C₉	—	43	1	no	—
262	C₃×C₉	—	129	9	no	—
263	C₃×C₉	—	387	1	no	—
264	C₃×C₉	—	387	1	no	—
265	C₃×C₉	C₃²×C₄₃⋊C₃	1	3	yes	—
266	C₃×C₉	C₃²×C₄₃⋊C₃	1	3	yes	—
267	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
268	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
269	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
270	C₃²	—	3	1	no	—
271	C₃²	—	3	1	no	—
272	C₃²	—	3	1	no	—
273	C₃²	—	3	3	no	—
274	C₃²	—	3	3	no	—
275	C₃²	—	3	3	no	—
276	C₃²	—	3	3	no	—
277	C₃²	3₋¹⁺²×C₄₃⋊C₃	1	3	yes	—
278	C₃²	C₃³×C₄₃⋊C₃	1	1	yes	characteristic
279	C₉	—	3	1	no	—
280	C₉	—	3	1	no	—
281	C₉	—	3	1	no	—
282	C₉	—	3	3	no	—
283	C₉	—	3	3	no	—
284	C₉	—	43	3	no	—
285	C₉	—	43	3	no	—
286	C₉	—	43	3	no	—
287	C₉	—	129	3	no	—
288	C₉	—	129	3	no	—
289	C₉	—	129	1	no	—
290	C₉	—	129	1	no	—
291	C₉	—	129	1	no	—
292	C₉	—	129	3	no	—
293	C₉	—	129	3	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	C₃	—	3	3	no	—
300	C₃	—	9	3	no	—
301	C₃	C₃×3₋¹⁺²×C₄₃⋊C₃	1	1	yes	characteristic
302	C₃	#23782921	1	1	yes	characteristic
303	C₃	#23782922	1	1	yes	characteristic
304	C₃	#23782924	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₄₃⋊C₃
commutator, socle, cosocle	C₃²×C₄₃	C₃⁴
Fitting	C₄₃×C₉⋊H(3,𝔽₃)	C₃
prime radical	C₃×H(3,𝔽₃)×C₄₃	C₃²
hypercenter	C₉⋊H(3,𝔽₃)	C₄₃⋊C₃
hypocenter	C₄₃	#10624887
LCS	G ▷ C₃²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₃² ◁ C₉⋊H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
2. #23782845	C₃	semidirect
3. #23782837	C₃	semidirect
4. #23782837	C₃	semidirect
5. #23782824	C₃	semidirect
6. #23782823	C₃	semidirect
7. #23782823	C₃	semidirect
8. #23782818	C₃	semidirect
9. #23782818	C₃	semidirect
10. #23782818	C₃	semidirect
11. #23782817	C₃	semidirect
12. #23782817	C₃	semidirect
13. #23782817	C₃	semidirect
14. #23782816	C₃	semidirect
15. #23782815	C₃	semidirect
16. #23782812	C₃	semidirect
19. #23782779	C₃	semidirect
20. #23782779	C₃	semidirect
21. C₃×(C₄₃⋊C₉)⋊C₉	C₃	semidirect
91. C₃×C₄₃⋊C₉	3₋¹⁺²	semidirect
234. C₄₃	#10624887	semidirect

Maximal subgroups

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

subgroup	index
C₃×#16812524	3
#23782845	3
#23782837	3
#23782824	3
#23782823	3
#23782818	3
#23782817	3
#23782816	3
#23782815	3
#23782812	3
#23782800	3
#23782779	3
C₃×(C₄₃⋊C₉)⋊C₉	3
C₄₃×C₉⋊H(3,𝔽₃)	3
#10624887	43