Subgroups — #41007907

Order 26973.

644 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582600	C₃	1	3	yes	maximal, complemented
2	#22582598	C₃	1	3	yes	maximal, complemented
3	#22582598	C₃	1	3	yes	maximal, complemented
4	#22582588	C₃	1	3	yes	maximal, complemented
5	#22582588	C₃	1	3	yes	maximal, complemented
6	#22582588	C₃	1	3	yes	maximal, complemented
7	#22582578	C₃	1	1	yes	maximal, characteristic, complemented
8	#22582576	C₃	1	1	yes	maximal, characteristic, complemented
9	#22582576	C₃	1	1	yes	maximal, characteristic, complemented
10	#22582575	C₃	1	1	yes	maximal, characteristic, complemented
11	#22582575	C₃	1	1	yes	maximal, characteristic, complemented
12	#22582570	C₃	1	3	yes	maximal, complemented
13	#22582567	C₃	1	3	yes	maximal, complemented
14	#22582553	C₃	1	1	yes	maximal, characteristic
15	#22582552	C₃	1	1	yes	maximal, characteristic
16	#22582543	C₃	1	3	yes	maximal, complemented
17	#22582538	C₃	1	1	yes	maximal, characteristic, complemented
18	#22582535	C₃	1	1	yes	maximal, characteristic, complemented
19	#22582535	C₃	1	1	yes	maximal, characteristic, complemented
20	C₃×#15688502	C₃	1	1	yes	maximal, characteristic
21	C₃²⋊C₉×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
22	C₃₇×#6895	C₃	1	1	yes	maximal, characteristic, complemented
23	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
24	C₃×#11758605	C₃²	1	3	yes	—
25	C₃×#11758606	C₃²	1	3	yes	—
26	C₃²×#66524	C₃²	1	1	yes	characteristic
27	C₃²×#66524	C₃²	1	1	yes	characteristic
28	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	—
29	#15688506	C₃²	1	3	yes	complemented
30	#15688506	C₃²	1	3	yes	complemented
31	#15688506	C₃²	1	3	yes	complemented
32	#15688505	C₃²	1	3	yes	complemented
33	#15688505	C₃²	1	3	yes	complemented
34	#15688505	C₃²	1	3	yes	complemented
35	#15688504	C₃²	1	3	yes	complemented
36	#15688504	C₃²	1	3	yes	complemented
37	#15688504	C₃²	1	3	yes	complemented
38	#15688504	C₃²	1	3	yes	complemented
39	#15688504	C₃²	1	3	yes	complemented
40	#15688504	C₃²	1	3	yes	complemented
41	#15688504	C₃²	1	3	yes	complemented
42	#15688504	C₃²	1	3	yes	complemented
43	#15688504	C₃²	1	3	yes	complemented
44	#15688504	C₃²	1	3	yes	complemented
45	#15688504	C₃²	1	3	yes	complemented
46	#15688504	C₃²	1	3	yes	complemented
47	#15688504	C₃²	1	3	yes	complemented
48	#15688504	C₃²	1	3	yes	complemented
49	#15688504	C₃²	1	3	yes	complemented
50	C₃²⋊#66524	C₃²	1	1	yes	characteristic
51	C₃²⋊#66524	C₃²	1	1	yes	characteristic
52	C₃²⋊#66524	C₃²	1	1	yes	characteristic
53	C₃²⋊#66524	C₃²	1	1	yes	characteristic
54	C₃²⋊#66524	C₃²	1	1	yes	characteristic
55	C₃²⋊#66524	C₃²	1	1	yes	characteristic
56	#15688502	C₃²	1	3	yes	—
57	#15688502	C₃²	1	3	yes	—
58	#15688502	C₃²	1	1	yes	characteristic
59	#15688502	C₃²	1	1	yes	characteristic
60	#15688502	C₃²	1	1	yes	characteristic
61	#15688502	C₃²	1	1	yes	characteristic
62	#15688502	C₃²	1	1	yes	characteristic
63	#15688502	C₃²	1	1	yes	characteristic
64	#15688502	C₃²	1	1	yes	characteristic
65	#15688502	C₃²	1	1	yes	characteristic
66	#15688502	C₃²	1	1	yes	characteristic
67	#15688502	C₃²	1	3	yes	—
68	#15688502	C₃²	1	3	yes	—
69	#15688502	C₃²	1	3	yes	—
70	#15688502	C₃²	1	3	yes	—
71	C₉×#66524	C₃²	1	3	yes	complemented
72	C₉×#66524	C₃²	1	3	yes	complemented
73	C₉×#66524	C₃²	1	3	yes	complemented
74	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
75	C₃₇×C₉⋊C₉	C₃²	1	3	yes	complemented
76	C₃₇×C₉⋊C₉	C₃²	1	3	yes	complemented
77	C₃₇×C₉⋊C₉	C₃²	1	3	yes	complemented
78	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
79	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
80	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
81	C₃³×C₃₇	C₃³	1	1	yes	characteristic
82	C₃²×C₃₇⋊C₃	—	3	3	no	—
83	C₃²×C₃₇⋊C₃	—	3	3	no	—
84	C₃²×C₃₇⋊C₃	—	9	1	no	—
85	C₃²×C₃₇⋊C₃	—	9	1	no	—
86	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
87	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
88	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
89	C₃×C₉×C₃₇	—	3	3	no	—
90	C₃×C₉×C₃₇	C₃³	1	3	yes	—
91	C₃×C₉×C₃₇	C₃³	1	3	yes	—
92	C₃×C₉×C₃₇	C₃³	1	3	yes	—
93	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
94	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
95	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
96	C₃×#66524	—	3	3	no	—
97	C₃×#66524	—	3	3	no	—
98	C₃×#66524	C₃³	1	1	yes	characteristic
99	C₃×#66524	C₃³	1	1	yes	characteristic
100	C₃×#66524	C₃³	1	3	yes	—
101	C₃×#66524	C₃³	1	3	yes	—
102	C₃×#66524	C₃³	1	3	yes	—
103	C₃×#66524	C₃³	1	3	yes	—
104	C₃×#66524	C₃³	1	3	yes	—
105	C₃×#66524	C₃³	1	1	yes	characteristic
106	C₃×#66524	C₃³	1	1	yes	characteristic
107	C₃×#66524	C₃³	1	1	yes	characteristic
108	C₃×#66524	C₃³	1	1	yes	characteristic
109	C₃×#66524	C₃³	1	3	yes	—
110	#11758606	—	9	3	no	—
111	#11758605	—	3	9	no	—
112	C₉×C₃₇⋊C₃	—	9	3	no	—
113	#10624903	—	37	1	no	maximal, Sylow
114	C₃²×C₃₇	—	3	1	no	—
115	C₃²×C₃₇	—	3	1	no	—
116	C₃²×C₃₇	—	3	1	no	—
117	C₃²×C₃₇	#737	1	3	yes	—
118	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
119	C₃×C₃₇⋊C₃	—	3	1	no	—
120	C₃×C₃₇⋊C₃	—	3	1	no	—
121	C₃×C₃₇⋊C₃	—	3	1	no	—
122	C₃×C₃₇⋊C₃	—	3	1	no	—
123	C₃×C₃₇⋊C₃	—	3	1	no	—
124	C₃×C₃₇⋊C₃	—	3	1	no	—
125	C₃×C₃₇⋊C₃	—	3	1	no	—
126	C₃×C₃₇⋊C₃	—	3	1	no	—
127	C₃×C₃₇⋊C₃	—	3	1	no	—
128	C₃×C₃₇⋊C₃	—	3	1	no	—
129	C₃×C₃₇⋊C₃	—	3	1	no	—
130	C₃×C₃₇⋊C₃	—	9	9	no	—
131	C₃×C₃₇⋊C₃	#737	1	3	yes	—
132	C₉×C₃₇	—	3	3	no	—
133	C₉×C₃₇	—	3	1	no	—
134	C₉×C₃₇	—	3	1	no	—
135	C₉×C₃₇	—	3	1	no	—
136	C₉×C₃₇	—	3	3	no	—
137	C₉×C₃₇	—	3	3	no	—
138	#66524	—	3	3	no	—
139	#66524	—	3	3	no	—
140	#66524	—	3	1	no	—
141	#66524	—	3	1	no	—
142	#66524	—	3	1	no	—
143	#66524	—	3	1	no	—
144	#66524	—	3	1	no	—
145	#66524	—	3	1	no	—
146	#66524	—	3	3	no	—
147	#66524	—	3	3	no	—
148	#66524	—	3	3	no	—
149	#66524	—	3	3	no	—
150	#6899	—	37	3	no	—
151	#6899	—	37	3	no	—
152	#6899	—	37	3	no	—
153	#6897	—	37	3	no	—
154	#6897	—	37	3	no	—
155	#6897	—	37	3	no	—
156	#6895	—	37	1	no	—
157	#6895	—	37	1	no	—
158	#6895	—	37	1	no	—
159	#6895	—	37	1	no	—
160	#6895	—	37	1	no	—
161	#6895	C₃₇⋊C₃	1	1	yes	characteristic, complemented
162	#6894	—	37	3	no	—
163	#6894	—	37	3	no	—
164	#6891	—	37	1	no	—
165	#6891	—	37	1	no	—
166	C₉×3₋¹⁺²	—	37	3	no	—
167	#6887	—	37	1	no	—
168	#6887	—	37	1	no	—
169	#6887	—	37	1	no	—
170	C₃×C₃²⋊C₉	—	37	1	no	—
171	C₃×C₃²⋊C₉	—	37	1	no	—
172	C₃×C₃₇	—	3	3	no	—
173	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
174	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
175	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
176	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
177	C₃₇⋊C₃	—	3	3	no	—
178	C₃₇⋊C₃	—	9	1	no	—
179	C₃₇⋊C₃	—	9	1	no	—
180	C₃⁴	—	37	1	no	—
181	C₃×3₋¹⁺²	—	37	3	no	—
182	C₃×3₋¹⁺²	—	37	3	no	—
183	C₃²×C₉	—	37	1	no	—
184	C₃²×C₉	—	37	1	no	—
185	C₃²×C₉	—	37	3	no	—
186	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
187	C₉⋊C₉	—	37	3	no	—
188	C₉⋊C₉	—	37	3	no	—
189	C₉⋊C₉	—	37	3	no	—
190	C₉⋊C₉	—	37	3	no	—
191	C₉⋊C₉	—	37	3	no	—
192	C₉⋊C₉	—	37	3	no	—
193	C₉⋊C₉	—	37	3	no	—
194	C₉⋊C₉	—	37	3	no	—
195	C₉⋊C₉	—	37	3	no	—
196	C₉⋊C₉	—	37	3	no	—
197	C₉⋊C₉	—	37	3	no	—
198	C₉⋊C₉	—	37	3	no	—
199	C₉⋊C₉	—	37	3	no	—
200	C₉⋊C₉	—	37	3	no	—
201	C₉⋊C₉	—	37	3	no	—
202	C₉⋊C₉	—	37	3	no	—
203	C₉⋊C₉	—	37	3	no	—
204	C₉⋊C₉	—	37	3	no	—
205	C₉⋊C₉	—	37	3	no	—
206	C₉⋊C₉	—	37	3	no	—
207	C₉⋊C₉	—	37	3	no	—
208	C₉⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	complemented
209	C₉⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	complemented
210	C₉⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	complemented
211	C₃²⋊C₉	—	37	1	no	—
212	C₃²⋊C₉	—	37	1	no	—
213	C₃²⋊C₉	—	37	1	no	—
214	C₃²⋊C₉	—	37	1	no	—
215	C₃²⋊C₉	—	37	1	no	—
216	C₃²⋊C₉	—	37	1	no	—
217	C₃²⋊C₉	—	37	1	no	—
218	C₃²⋊C₉	—	37	1	no	—
219	C₃²⋊C₉	—	37	1	no	—
220	C₃²⋊C₉	—	37	1	no	—
221	C₃²⋊C₉	—	37	1	no	—
222	C₃²⋊C₉	—	37	3	no	—
223	C₃²⋊C₉	—	37	1	no	—
224	C₃²⋊C₉	—	37	1	no	—
225	C₃²⋊C₉	—	37	1	no	—
226	C₃²⋊C₉	—	37	3	no	—
227	C₃²⋊C₉	—	37	1	no	—
228	C₃²⋊C₉	—	37	3	no	—
229	C₃²⋊C₉	—	37	3	no	—
230	C₃²⋊C₉	—	37	3	no	—
231	C₃²⋊C₉	—	37	3	no	—
232	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
233	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
234	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
235	C₉²	—	37	3	no	—
236	C₉²	—	37	3	no	—
237	C₉²	—	37	3	no	—
238	C₃₇	#10624903	1	1	yes	characteristic, complemented, Sylow
239	C₃³	—	37	1	no	—
240	C₃³	—	37	1	no	—
241	C₃³	—	37	1	no	—
242	C₃³	—	333	1	no	—
243	C₃³	—	333	1	no	—
244	C₃³	—	111	3	no	—
245	C₃³	—	111	3	no	—
246	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
247	3₋¹⁺²	—	333	3	no	—
248	3₋¹⁺²	—	111	9	no	—
249	C₃×C₉	—	3	3	no	—
250	C₃×C₉	—	37	1	no	—
251	C₃×C₉	—	37	1	no	—
252	C₃×C₉	—	37	3	no	—
253	C₃×C₉	—	37	3	no	—
254	C₃×C₉	—	37	3	no	—
255	C₃×C₉	—	37	3	no	—
256	C₃×C₉	—	37	3	no	—
257	C₃×C₉	—	37	1	no	—
258	C₃×C₉	—	37	1	no	—
259	C₃×C₉	—	37	1	no	—
260	C₃×C₉	—	37	1	no	—
261	C₃×C₉	—	37	3	no	—
262	C₃×C₉	—	333	3	no	—
263	C₃×C₉	—	111	3	no	—
264	C₃×C₉	—	111	3	no	—
265	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
266	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
267	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
268	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
269	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
270	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
271	C₃²	—	3	1	no	—
272	C₃²	—	3	1	no	—
273	C₃²	—	3	1	no	—
274	C₃²	—	37	3	no	—
275	C₃²	—	333	9	no	—
276	C₃²	—	111	1	no	—
277	C₃²	—	111	1	no	—
278	C₃²	—	111	1	no	—
279	C₃²	—	111	1	no	—
280	C₃²	—	111	1	no	—
281	C₃²	—	111	1	no	—
282	C₃²	—	111	1	no	—
283	C₃²	—	111	1	no	—
284	C₃²	—	111	1	no	—
285	C₃²	—	111	1	no	—
286	C₃²	—	111	1	no	—
287	C₃²	#15688532	1	3	yes	—
288	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
289	C₉	—	3	3	no	—
290	C₉	—	3	1	no	—
291	C₉	—	3	1	no	—
292	C₉	—	3	1	no	—
293	C₉	—	3	3	no	—
294	C₉	—	3	3	no	—
295	C₉	—	111	3	no	—
296	C₉	—	111	3	no	—
297	C₉	—	111	1	no	—
298	C₉	—	111	1	no	—
299	C₉	—	111	1	no	—
300	C₉	—	111	1	no	—
301	C₉	—	111	1	no	—
302	C₉	—	111	1	no	—
303	C₉	—	111	3	no	—
304	C₉	—	111	3	no	—
305	C₉	—	111	3	no	—
306	C₉	—	111	3	no	—
307	C₃	—	3	3	no	—
308	C₃	—	333	1	no	—
309	C₃	—	333	1	no	—
310	C₃	—	111	3	no	—
311	C₃	#737×C₃₇⋊C₃	1	1	yes	characteristic
312	C₃	C₃×#15688532	1	1	yes	characteristic
313	C₃	#22582677	1	1	yes	characteristic
314	C₃	#22582678	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting	C₃₇×#6895	C₃
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	#6895	C₃₇⋊C₃
hypocenter	C₃₇	#10624903
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ #6895

Complements

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

N	H ≅ G/N	type
1. #22582600	C₃	semidirect
2. #22582598	C₃	semidirect
3. #22582598	C₃	semidirect
4. #22582588	C₃	semidirect
5. #22582588	C₃	semidirect
6. #22582588	C₃	semidirect
7. #22582578	C₃	semidirect
8. #22582576	C₃	semidirect
9. #22582576	C₃	semidirect
10. #22582575	C₃	semidirect
11. #22582575	C₃	semidirect
12. #22582570	C₃	semidirect
13. #22582567	C₃	semidirect
16. #22582543	C₃	semidirect
17. #22582538	C₃	semidirect
18. #22582535	C₃	semidirect
19. #22582535	C₃	semidirect
22. C₃₇×#6895	C₃	semidirect
29. #15688506	C₃²	semidirect
30. #15688506	C₃²	semidirect
31. #15688506	C₃²	semidirect
32. #15688505	C₃²	semidirect
33. #15688505	C₃²	semidirect
34. #15688505	C₃²	semidirect
35. #15688504	C₃²	semidirect
36. #15688504	C₃²	semidirect
37. #15688504	C₃²	semidirect
38. #15688504	C₃²	semidirect
39. #15688504	C₃²	semidirect
40. #15688504	C₃²	semidirect
41. #15688504	C₃²	semidirect
42. #15688504	C₃²	semidirect
43. #15688504	C₃²	semidirect
44. #15688504	C₃²	semidirect
45. #15688504	C₃²	semidirect
46. #15688504	C₃²	semidirect
47. #15688504	C₃²	semidirect
48. #15688504	C₃²	semidirect
49. #15688504	C₃²	semidirect
71. C₉×#66524	C₃²	semidirect
72. C₉×#66524	C₃²	semidirect
73. C₉×#66524	C₃²	semidirect
75. C₃₇×C₉⋊C₉	C₃²	semidirect
76. C₃₇×C₉⋊C₉	C₃²	semidirect
77. C₃₇×C₉⋊C₉	C₃²	semidirect
161. #6895	C₃₇⋊C₃	semidirect
208. C₉⋊C₉	C₃×C₃₇⋊C₃	semidirect
209. C₉⋊C₉	C₃×C₃₇⋊C₃	semidirect
210. C₉⋊C₉	C₃×C₃₇⋊C₃	semidirect
238. C₃₇	#10624903	semidirect

Maximal subgroups

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

subgroup	index
#22582600	3
#22582598	3
#22582588	3
#22582578	3
#22582576	3
#22582575	3
#22582570	3
#22582567	3
#22582553	3
#22582552	3
#22582543	3
#22582538	3
#22582535	3
C₃×#15688502	3
C₃²⋊C₉×C₃₇⋊C₃	3
C₃₇×#6895	3
#10624903	37