Subgroups — #44535195

Order 31347.

740 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23782807	C₃	1	4	yes	maximal, complemented
2	#23782807	C₃	1	2	yes	maximal, complemented
3	#23782806	C₃	1	2	yes	maximal, complemented
4	#23782805	C₃	1	2	yes	maximal, complemented
5	#23782804	C₃	1	4	yes	maximal, complemented
6	#23782803	C₃	1	2	yes	maximal, complemented
7	#23782802	C₃	1	1	yes	maximal, characteristic, complemented
8	#23782801	C₃	1	4	yes	maximal, complemented
9	#23782801	C₃	1	4	yes	maximal, complemented
10	#23782801	C₃	1	4	yes	maximal, complemented
11	#23782799	C₃	1	4	yes	maximal, complemented
12	#23782799	C₃	1	2	yes	maximal, complemented
13	#23782797	C₃	1	1	yes	maximal, characteristic, complemented
14	#23782796	C₃	1	2	yes	maximal, complemented
15	(C₃×C₄₃⋊C₃)⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
16	C₄₃×#6893	C₃	1	1	yes	maximal, characteristic, complemented
17	C₃³×C₄₃⋊C₃	C₃²	1	1	yes	characteristic
18	C₃×#11923958	C₃²	1	4	yes	complemented
19	C₃×#11923957	C₃²	1	2	yes	—
20	C₃×#11923957	C₃²	1	1	yes	characteristic
21	C₃×#11923957	C₃²	1	4	yes	complemented
22	C₃×#11923957	C₃²	1	2	yes	complemented
23	C₃×#11923956	C₃²	1	4	yes	—
24	C₃×#11923956	C₃²	1	2	yes	—
25	C₃×#11923956	C₃²	1	2	yes	complemented
26	C₃×#11923956	C₃²	1	4	yes	complemented
27	C₃×C₃²⋊(C₄₃⋊C₃)	C₃²	1	1	yes	characteristic
28	C₃×C₃²⋊(C₄₃⋊C₃)	C₃²	1	2	yes	—
29	C₃×C₃²⋊(C₄₃⋊C₃)	C₃²	1	4	yes	—
30	C₃×C₃²⋊(C₄₃⋊C₃)	C₃²	1	2	yes	—
31	C₃×C₃²⋊(C₄₃⋊C₃)	C₃²	1	2	yes	—
32	C₃²×C₄₃⋊C₉	C₃²	1	2	yes	complemented
33	C₃×C₉×C₄₃⋊C₃	C₃²	1	4	yes	complemented
34	C₃×C₉×C₄₃⋊C₃	C₃²	1	2	yes	complemented
35	C₃²⋊(C₄₃⋊C₉)	C₃²	1	4	yes	—
36	C₃²⋊(C₄₃⋊C₉)	C₃²	1	2	yes	—
37	C₃²⋊(C₄₃⋊C₉)	C₃²	1	2	yes	complemented
38	C₃²⋊(C₄₃⋊C₉)	C₃²	1	4	yes	complemented
39	C₃²⋊(C₄₃⋊C₉)	C₃²	1	4	yes	complemented
40	C₃²⋊(C₄₃⋊C₉)	C₃²	1	2	yes	complemented
41	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	4	yes	—
42	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	4	yes	—
43	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	2	yes	—
44	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	2	yes	—
45	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	4	yes	—
46	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	2	yes	—
47	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	4	yes	complemented
48	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	4	yes	complemented
49	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	4	yes	complemented
50	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	4	yes	complemented
51	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	4	yes	complemented
52	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	4	yes	complemented
53	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	4	yes	complemented
54	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	4	yes	complemented
55	(C₃×C₄₃⋊C₃)⋊C₉	C₃²	1	4	yes	complemented
56	C₃×3₋¹⁺²×C₄₃	C₃²	1	2	yes	complemented
57	C₃×H(3,𝔽₃)×C₄₃	C₃²	1	1	yes	characteristic
58	C₃²×C₉×C₄₃	C₃²	1	1	yes	characteristic, complemented
59	C₄₃×C₃²⋊C₉	C₃²	1	2	yes	—
60	C₄₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
61	C₄₃×C₃²⋊C₉	C₃²	1	4	yes	complemented
62	C₄₃×C₃²⋊C₉	C₃²	1	2	yes	complemented
63	C₃²×C₄₃⋊C₃	—	9	1	no	—
64	C₃²×C₄₃⋊C₃	—	9	2	no	—
65	C₃²×C₄₃⋊C₃	—	9	1	no	—
66	C₃²×C₄₃⋊C₃	C₃³	1	2	yes	—
67	C₃²×C₄₃⋊C₃	C₃³	1	1	yes	characteristic
68	C₃²×C₄₃⋊C₃	C₃³	1	2	yes	—
69	C₃²×C₄₃⋊C₃	C₃³	1	4	yes	—
70	#11923958	—	9	4	no	—
71	#11923957	—	9	2	no	—
72	#11923957	—	9	1	no	—
73	#11923957	—	9	2	no	—
74	#11923957	—	9	4	no	—
75	#11923956	—	9	2	no	—
76	#11923956	—	9	4	no	—
77	#11923956	—	9	4	no	—
78	#11923956	—	9	2	no	—
79	C₃²⋊(C₄₃⋊C₃)	—	9	2	no	—
80	C₃²⋊(C₄₃⋊C₃)	—	9	1	no	—
81	C₃²⋊(C₄₃⋊C₃)	—	9	2	no	—
82	C₃²⋊(C₄₃⋊C₃)	—	9	4	no	—
83	C₃²⋊(C₄₃⋊C₃)	—	9	2	no	—
84	C₃×C₄₃⋊C₉	—	9	2	no	—
85	C₃×C₄₃⋊C₉	C₃³	1	4	yes	—
86	C₃×C₄₃⋊C₉	C₃³	1	4	yes	—
87	C₃×C₄₃⋊C₉	C₃³	1	4	yes	—
88	C₃×C₄₃⋊C₉	C₃³	1	4	yes	—
89	C₃×C₄₃⋊C₉	C₃³	1	2	yes	—
90	C₉×C₄₃⋊C₃	—	9	2	no	—
91	C₉×C₄₃⋊C₃	—	9	4	no	—
92	C₃³×C₄₃	C₃³	1	1	yes	characteristic
93	C₃³×C₄₃	C₃³	1	2	yes	—
94	C₃³×C₄₃	C₃³	1	1	yes	characteristic
95	3₋¹⁺²×C₄₃	—	9	2	no	—
96	H(3,𝔽₃)×C₄₃	—	9	1	no	—
97	C₃×C₉×C₄₃	—	9	1	no	—
98	C₃×C₉×C₄₃	C₃³	1	2	yes	—
99	C₃×C₉×C₄₃	C₃³	1	1	yes	characteristic
100	C₃×C₉×C₄₃	C₃³	1	4	yes	—
101	C₃×C₉×C₄₃	C₃³	1	2	yes	—
102	#10624910	—	43	1	no	maximal, Sylow
103	C₃²×C₄₃	—	3	1	no	—
104	C₃²×C₄₃	—	3	2	no	—
105	C₃²×C₄₃	—	3	1	no	—
106	C₃²×C₄₃	—	3	4	no	—
107	C₃²×C₄₃	—	3	2	no	—
108	C₃²×C₄₃	—	3	2	no	—
109	C₃²×C₄₃	—	3	1	no	—
110	C₃²×C₄₃	—	3	2	no	—
111	C₃²×C₄₃	—	3	1	no	—
112	C₃²×C₄₃	C₃⁴	1	1	yes	characteristic
113	C₃×C₄₃⋊C₃	—	3	4	no	—
114	C₃×C₄₃⋊C₃	—	3	2	no	—
115	C₃×C₄₃⋊C₃	—	3	2	no	—
116	C₃×C₄₃⋊C₃	—	3	1	no	—
117	C₃×C₄₃⋊C₃	—	3	2	no	—
118	C₃×C₄₃⋊C₃	—	3	1	no	—
119	C₃×C₄₃⋊C₃	—	3	4	no	—
120	C₃×C₄₃⋊C₃	—	3	4	no	—
121	C₃×C₄₃⋊C₃	—	3	4	no	—
122	C₃×C₄₃⋊C₃	—	3	2	no	—
123	C₃×C₄₃⋊C₃	—	3	4	no	—
124	C₃×C₄₃⋊C₃	—	3	2	no	—
125	C₃×C₄₃⋊C₃	—	3	2	no	—
126	C₃×C₄₃⋊C₃	—	3	2	no	—
127	C₃×C₄₃⋊C₃	—	9	9	no	—
128	C₉×C₄₃	—	3	2	no	—
129	C₉×C₄₃	—	3	1	no	—
130	C₉×C₄₃	—	3	4	no	—
131	C₉×C₄₃	—	3	2	no	—
132	C₄₃⋊C₉	—	3	4	no	—
133	C₄₃⋊C₉	—	3	4	no	—
134	C₄₃⋊C₉	—	3	4	no	—
135	C₄₃⋊C₉	—	3	4	no	—
136	C₄₃⋊C₉	—	3	2	no	—
137	#6893	—	43	2	no	—
138	#6893	—	43	2	no	—
139	#6893	—	43	4	no	—
140	#6893	C₄₃⋊C₃	1	1	yes	characteristic, complemented
141	C₉⋊H(3,𝔽₃)	—	43	4	no	—
142	C₉⋊H(3,𝔽₃)	—	43	2	no	—
143	C₉⋊H(3,𝔽₃)	—	43	1	no	—
144	C₉⋊H(3,𝔽₃)	—	43	2	no	—
145	C₉⋊H(3,𝔽₃)	—	43	4	no	—
146	C₉⋊H(3,𝔽₃)	—	43	4	no	—
147	C₉⋊H(3,𝔽₃)	—	43	4	no	—
148	C₉⋊H(3,𝔽₃)	—	43	4	no	—
149	C₉⋊H(3,𝔽₃)	—	43	2	no	—
150	#6891	—	43	2	no	—
151	#6891	—	43	1	no	—
152	C₃²⋊H(3,𝔽₃)	—	43	1	no	—
153	C₃×C₄₃	—	9	1	no	—
154	C₃×C₄₃	—	9	1	no	—
155	C₃×C₄₃	—	9	2	no	—
156	C₃×C₄₃	H(5,𝔽₃)	1	1	yes	characteristic
157	C₃×C₄₃	3₋¹⁺⁴	1	2	yes	—
158	C₃×C₄₃	3₋¹⁺⁴	1	1	yes	characteristic
159	C₄₃⋊C₃	—	9	4	no	—
160	C₄₃⋊C₃	—	9	2	no	—
161	C₄₃⋊C₃	—	9	2	no	—
162	C₄₃⋊C₃	—	9	1	no	—
163	C₃⁴	—	43	1	no	—
164	C₃×3₋¹⁺²	—	43	4	no	—
165	C₃×3₋¹⁺²	—	43	4	no	—
166	C₃×3₋¹⁺²	—	43	2	no	—
167	C₃×3₋¹⁺²	—	43	2	no	—
168	C₃×3₋¹⁺²	—	43	4	no	—
169	C₃×3₋¹⁺²	—	43	2	no	—
170	C₃×3₋¹⁺²	—	43	1	no	—
171	C₃×3₋¹⁺²	—	43	4	no	—
172	C₃×3₋¹⁺²	—	43	2	no	—
173	C₃×3₋¹⁺²	C₃×C₄₃⋊C₃	1	2	yes	complemented
174	C₃×H(3,𝔽₃)	—	43	1	no	—
175	C₃×H(3,𝔽₃)	—	43	2	no	—
176	C₃×H(3,𝔽₃)	—	43	4	no	—
177	C₃×H(3,𝔽₃)	—	43	2	no	—
178	C₃×H(3,𝔽₃)	—	43	2	no	—
179	C₃×H(3,𝔽₃)	C₃×C₄₃⋊C₃	1	1	yes	characteristic
180	C₃²×C₉	—	43	2	no	—
181	C₃²×C₉	—	43	2	no	—
182	C₃²×C₉	—	43	4	no	—
183	C₃²×C₉	C₃×C₄₃⋊C₃	1	1	yes	characteristic, complemented
184	C₃²⋊C₉	—	43	4	no	—
185	C₃²⋊C₉	—	43	2	no	—
186	C₃²⋊C₉	—	43	2	no	—
187	C₃²⋊C₉	—	43	4	no	—
188	C₃²⋊C₉	—	43	4	no	—
189	C₃²⋊C₉	—	43	2	no	—
190	C₃²⋊C₉	—	43	4	no	—
191	C₃²⋊C₉	—	43	4	no	—
192	C₃²⋊C₉	—	43	4	no	—
193	C₃²⋊C₉	—	43	2	no	—
194	C₃²⋊C₉	—	43	2	no	—
195	C₃²⋊C₉	—	43	2	no	—
196	C₃²⋊C₉	—	43	4	no	—
197	C₃²⋊C₉	—	43	4	no	—
198	C₃²⋊C₉	—	43	4	no	—
199	C₃²⋊C₉	—	43	4	no	—
200	C₃²⋊C₉	—	43	4	no	—
201	C₃²⋊C₉	—	43	4	no	—
202	C₃²⋊C₉	—	43	4	no	—
203	C₃²⋊C₉	—	43	4	no	—
204	C₃²⋊C₉	—	43	4	no	—
205	C₃²⋊C₉	C₃×C₄₃⋊C₃	1	2	yes	—
206	C₃²⋊C₉	C₃×C₄₃⋊C₃	1	1	yes	characteristic
207	C₃²⋊C₉	C₃×C₄₃⋊C₃	1	4	yes	complemented
208	C₃²⋊C₉	C₃×C₄₃⋊C₃	1	2	yes	complemented
209	C₄₃	#10624910	1	1	yes	characteristic, complemented, Sylow
210	C₃³	—	43	2	no	—
211	C₃³	—	43	4	no	—
212	C₃³	—	43	2	no	—
213	C₃³	—	43	1	no	—
214	C₃³	—	387	1	no	—
215	C₃³	—	387	2	no	—
216	C₃³	—	387	1	no	—
217	C₃³	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
218	C₃³	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
219	C₃³	C₃²×C₄₃⋊C₃	1	2	yes	—
220	3₋¹⁺²	—	9	2	no	—
221	3₋¹⁺²	—	387	4	no	—
222	3₋¹⁺²	—	387	2	no	—
223	3₋¹⁺²	—	387	2	no	—
224	3₋¹⁺²	—	387	4	no	—
225	3₋¹⁺²	—	387	2	no	—
226	3₋¹⁺²	—	387	1	no	—
227	3₋¹⁺²	—	387	4	no	—
228	3₋¹⁺²	—	387	2	no	—
229	3₋¹⁺²	—	387	4	no	—
230	H(3,𝔽₃)	—	9	1	no	—
231	H(3,𝔽₃)	—	387	2	no	—
232	H(3,𝔽₃)	—	387	2	no	—
233	H(3,𝔽₃)	—	387	4	no	—
234	H(3,𝔽₃)	—	387	1	no	—
235	H(3,𝔽₃)	—	387	2	no	—
236	C₃×C₉	—	9	1	no	—
237	C₃×C₉	—	43	4	no	—
238	C₃×C₉	—	43	2	no	—
239	C₃×C₉	—	43	4	no	—
240	C₃×C₉	—	43	4	no	—
241	C₃×C₉	—	43	4	no	—
242	C₃×C₉	—	387	4	no	—
243	C₃×C₉	—	387	2	no	—
244	C₃×C₉	—	387	2	no	—
245	C₃×C₉	C₃²×C₄₃⋊C₃	1	4	yes	—
246	C₃×C₉	C₃²×C₄₃⋊C₃	1	2	yes	—
247	C₃×C₉	C₃²×C₄₃⋊C₃	1	2	yes	—
248	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
249	C₃²	—	3	1	no	—
250	C₃²	—	3	2	no	—
251	C₃²	—	3	1	no	—
252	C₃²	—	3	2	no	—
253	C₃²	—	3	4	no	—
254	C₃²	—	3	2	no	—
255	C₃²	—	3	1	no	—
256	C₃²	—	3	2	no	—
257	C₃²	—	3	1	no	—
258	C₃²	—	129	4	no	—
259	C₃²	—	129	2	no	—
260	C₃²	—	129	2	no	—
261	C₃²	—	129	1	no	—
262	C₃²	—	129	2	no	—
263	C₃²	—	129	4	no	—
264	C₃²	—	129	4	no	—
265	C₃²	—	129	4	no	—
266	C₃²	—	129	2	no	—
267	C₃²	—	129	2	no	—
268	C₃²	—	129	1	no	—
269	C₃²	—	129	2	no	—
270	C₃²	—	129	4	no	—
271	C₃²	—	129	2	no	—
272	C₃²	—	387	9	no	—
273	C₃²	C₃³×C₄₃⋊C₃	1	1	yes	characteristic
274	C₉	—	3	2	no	—
275	C₉	—	3	1	no	—
276	C₉	—	3	4	no	—
277	C₉	—	3	2	no	—
278	C₉	—	129	4	no	—
279	C₉	—	129	4	no	—
280	C₉	—	129	4	no	—
281	C₉	—	129	4	no	—
282	C₉	—	129	2	no	—
283	C₃	—	9	1	no	—
284	C₃	—	9	2	no	—
285	C₃	—	9	1	no	—
286	C₃	—	387	4	no	—
287	C₃	—	387	2	no	—
288	C₃	—	387	2	no	—
289	C₃	—	387	1	no	—
290	C₃	C₄₃⋊H(5,𝔽₃)	1	1	yes	characteristic
291	C₃	#23782921	1	2	yes	—
292	C₃	#23782922	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₄₃⋊C₃
commutator, socle, cosocle	C₃²×C₄₃	C₃⁴
Fitting	C₄₃×#6893	C₃
prime radical	(C₃×C₄₃⋊C₃)⋊H(3,𝔽₃)	C₃
hypercenter	#6893	C₄₃⋊C₃
hypocenter	C₄₃	#10624910
LCS	G ▷ C₃²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₃² ◁ #6893

Complements

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

N	H ≅ G/N	type
1. #23782807	C₃	semidirect
2. #23782807	C₃	semidirect
3. #23782806	C₃	semidirect
4. #23782805	C₃	semidirect
5. #23782804	C₃	semidirect
6. #23782803	C₃	semidirect
7. #23782802	C₃	semidirect
8. #23782801	C₃	semidirect
9. #23782801	C₃	semidirect
10. #23782801	C₃	semidirect
11. #23782799	C₃	semidirect
12. #23782799	C₃	semidirect
13. #23782797	C₃	semidirect
14. #23782796	C₃	semidirect
16. C₄₃×#6893	C₃	semidirect
18. C₃×#11923958	C₃²	semidirect
21. C₃×#11923957	C₃²	semidirect
22. C₃×#11923957	C₃²	semidirect
25. C₃×#11923956	C₃²	semidirect
26. C₃×#11923956	C₃²	semidirect
32. C₃²×C₄₃⋊C₉	C₃²	semidirect
33. C₃×C₉×C₄₃⋊C₃	C₃²	semidirect
34. C₃×C₉×C₄₃⋊C₃	C₃²	semidirect
37. C₃²⋊(C₄₃⋊C₉)	C₃²	semidirect
38. C₃²⋊(C₄₃⋊C₉)	C₃²	semidirect
39. C₃²⋊(C₄₃⋊C₉)	C₃²	semidirect
40. C₃²⋊(C₄₃⋊C₉)	C₃²	semidirect
47. (C₃×C₄₃⋊C₃)⋊C₉	C₃²	semidirect
48. (C₃×C₄₃⋊C₃)⋊C₉	C₃²	semidirect
49. (C₃×C₄₃⋊C₃)⋊C₉	C₃²	semidirect
50. (C₃×C₄₃⋊C₃)⋊C₉	C₃²	semidirect
51. (C₃×C₄₃⋊C₃)⋊C₉	C₃²	semidirect
52. (C₃×C₄₃⋊C₃)⋊C₉	C₃²	semidirect
53. (C₃×C₄₃⋊C₃)⋊C₉	C₃²	semidirect
54. (C₃×C₄₃⋊C₃)⋊C₉	C₃²	semidirect
55. (C₃×C₄₃⋊C₃)⋊C₉	C₃²	semidirect
56. C₃×3₋¹⁺²×C₄₃	C₃²	semidirect
58. C₃²×C₉×C₄₃	C₃²	semidirect
61. C₄₃×C₃²⋊C₉	C₃²	semidirect
62. C₄₃×C₃²⋊C₉	C₃²	semidirect
140. #6893	C₄₃⋊C₃	semidirect
173. C₃×3₋¹⁺²	C₃×C₄₃⋊C₃	semidirect
183. C₃²×C₉	C₃×C₄₃⋊C₃	semidirect
207. C₃²⋊C₉	C₃×C₄₃⋊C₃	semidirect
208. C₃²⋊C₉	C₃×C₄₃⋊C₃	semidirect
209. C₄₃	#10624910	semidirect

Maximal subgroups

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

subgroup	index
#23782807	3
#23782806	3
#23782805	3
#23782804	3
#23782803	3
#23782802	3
#23782801	3
#23782799	3
#23782797	3
#23782796	3
(C₃×C₄₃⋊C₃)⋊H(3,𝔽₃)	3
C₄₃×#6893	3
#10624910	43