Subgroups — #41007712

Order 26973.

620 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582589	C₃	1	3	yes	maximal, complemented
2	#22582589	C₃	1	3	yes	maximal, complemented
3	#22582589	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	#22582584	C₃	1	3	yes	maximal, complemented
8	#22582583	C₃	1	3	yes	maximal, complemented
9	#22582582	C₃	1	3	yes	maximal, complemented
10	#22582574	C₃	1	1	yes	maximal, characteristic, complemented
11	#22582573	C₃	1	1	yes	maximal, characteristic, complemented
12	#22582572	C₃	1	1	yes	maximal, characteristic, complemented
13	#22582571	C₃	1	1	yes	maximal, characteristic, complemented
14	#22582568	C₃	1	1	yes	maximal, characteristic, complemented
15	#22582553	C₃	1	1	yes	maximal, characteristic
16	#22582552	C₃	1	1	yes	maximal, characteristic
17	#22582551	C₃	1	1	yes	maximal, characteristic
18	#22582548	C₃	1	1	yes	maximal, characteristic, complemented
19	#22582548	C₃	1	1	yes	maximal, characteristic, complemented
20	3₋¹⁺²×#66524	C₃	1	1	yes	maximal, characteristic, complemented
21	C₃×#15688502	C₃	1	1	yes	maximal, characteristic
22	C₃₇×#6894	C₃	1	1	yes	maximal, characteristic, complemented
23	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
24	C₃×#11758608	C₃²	1	1	yes	characteristic
25	C₃×#11758608	C₃²	1	1	yes	characteristic
26	C₃×#11758605	C₃²	1	3	yes	—
27	C₃×#11758606	C₃²	1	3	yes	—
28	C₃²×#66524	C₃²	1	1	yes	characteristic
29	C₃²×#66524	C₃²	1	1	yes	characteristic
30	#15688506	C₃²	1	3	yes	complemented
31	#15688506	C₃²	1	3	yes	complemented
32	#15688506	C₃²	1	3	yes	complemented
33	#15688505	C₃²	1	3	yes	complemented
34	#15688505	C₃²	1	3	yes	complemented
35	#15688505	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	#15688504	C₃²	1	3	yes	complemented
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	#15688502	C₃²	1	1	yes	characteristic
56	#15688502	C₃²	1	1	yes	characteristic
57	#15688502	C₃²	1	1	yes	characteristic
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	3	yes	—
65	#15688502	C₃²	1	3	yes	—
66	#15688502	C₃²	1	3	yes	—
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	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
75	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
76	C₃₇×C₉⋊C₉	C₃²	1	3	yes	complemented
77	C₃₇×C₉⋊C₉	C₃²	1	3	yes	complemented
78	C₃₇×C₉⋊C₉	C₃²	1	3	yes	complemented
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	—	9	1	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	3₋¹⁺²×C₃₇	—	3	3	no	—
90	C₃×C₉×C₃₇	—	9	1	no	—
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×C₉×C₃₇	C₃³	1	3	yes	—
97	#11758608	—	3	3	no	—
98	#11758608	—	3	3	no	—
99	C₃×#66524	—	9	1	no	—
100	C₃×#66524	—	9	1	no	—
101	C₃×#66524	C₃³	1	1	yes	characteristic
102	C₃×#66524	C₃³	1	1	yes	characteristic
103	C₃×#66524	C₃³	1	1	yes	characteristic
104	C₃×#66524	C₃³	1	3	yes	—
105	C₃×#66524	C₃³	1	3	yes	—
106	C₃×#66524	C₃³	1	3	yes	—
107	C₃×#66524	C₃³	1	3	yes	—
108	C₃×#66524	C₃³	1	3	yes	—
109	C₃×#66524	C₃³	1	3	yes	—
110	C₃×#66524	C₃³	1	1	yes	characteristic
111	C₃×#66524	C₃³	1	1	yes	characteristic
112	C₃×#66524	C₃³	1	1	yes	characteristic
113	#11758606	—	9	3	no	—
114	#11758605	—	9	3	no	—
115	#10624889	—	37	1	no	maximal, Sylow
116	C₃²×C₃₇	—	3	1	no	—
117	C₃²×C₃₇	—	3	1	no	—
118	C₃²×C₃₇	—	3	1	no	—
119	C₃²×C₃₇	#737	1	3	yes	—
120	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
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	—	3	1	no	—
131	C₃×C₃₇⋊C₃	—	3	1	no	—
132	C₃×C₃₇⋊C₃	—	3	1	no	—
133	C₃×C₃₇⋊C₃	—	9	9	no	—
134	C₉×C₃₇	—	3	3	no	—
135	C₉×C₃₇	—	3	3	no	—
136	C₉×C₃₇	—	3	3	no	—
137	C₉×C₃₇	—	3	1	no	—
138	C₉×C₃₇	—	3	1	no	—
139	C₉×C₃₇	—	3	1	no	—
140	#66524	—	3	3	no	—
141	#66524	—	3	3	no	—
142	#66524	—	3	3	no	—
143	#66524	—	3	3	no	—
144	#66524	—	3	3	no	—
145	#66524	—	3	3	no	—
146	#66524	—	3	1	no	—
147	#66524	—	3	1	no	—
148	#66524	—	3	1	no	—
149	#66524	—	3	1	no	—
150	#66524	—	3	1	no	—
151	#66524	—	3	1	no	—
152	#6897	—	37	3	no	—
153	#6897	—	37	3	no	—
154	#6897	—	37	3	no	—
155	#6897	—	37	3	no	—
156	#6897	—	37	3	no	—
157	#6897	—	37	3	no	—
158	#6896	—	37	3	no	—
159	#6896	—	37	3	no	—
160	#6896	—	37	3	no	—
161	#6894	—	37	1	no	—
162	#6894	—	37	1	no	—
163	#6894	—	37	1	no	—
164	#6894	—	37	1	no	—
165	#6894	—	37	1	no	—
166	#6894	C₃₇⋊C₃	1	1	yes	characteristic, complemented
167	#6891	—	37	1	no	—
168	#6891	—	37	1	no	—
169	#6891	—	37	1	no	—
170	C₉×3₋¹⁺²	—	37	1	no	—
171	C₉×3₋¹⁺²	—	37	1	no	—
172	C₉×3₋¹⁺²	—	37	1	no	—
173	C₃×C₃²⋊C₉	—	37	1	no	—
174	C₃×C₃₇	—	3	3	no	—
175	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
176	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
177	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
178	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
179	C₃₇⋊C₃	—	9	1	no	—
180	C₃₇⋊C₃	—	9	1	no	—
181	C₃₇⋊C₃	—	9	1	no	—
182	C₃⁴	—	37	1	no	—
183	C₃×3₋¹⁺²	—	37	1	no	—
184	C₃×3₋¹⁺²	—	37	1	no	—
185	C₃×3₋¹⁺²	—	37	3	no	—
186	C₃×3₋¹⁺²	—	37	3	no	—
187	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
188	C₃²×C₉	—	37	1	no	—
189	C₃²×C₉	—	37	1	no	—
190	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
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	—	37	3	no	—
209	C₉⋊C₉	—	37	3	no	—
210	C₉⋊C₉	—	37	3	no	—
211	C₉⋊C₉	—	37	3	no	—
212	C₉⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	complemented
213	C₉⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	complemented
214	C₉⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	complemented
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	1	no	—
223	C₃²⋊C₉	—	37	1	no	—
224	C₃²⋊C₉	—	37	1	no	—
225	C₃²⋊C₉	—	37	1	no	—
226	C₃²⋊C₉	—	37	1	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	—	37	3	no	—
233	C₃²⋊C₉	—	37	3	no	—
234	C₃²⋊C₉	—	37	3	no	—
235	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
236	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
237	C₉²	—	37	3	no	—
238	C₉²	—	37	3	no	—
239	C₉²	—	37	3	no	—
240	C₃₇	#10624889	1	1	yes	characteristic, complemented, Sylow
241	C₃³	—	37	1	no	—
242	C₃³	—	37	1	no	—
243	C₃³	—	37	1	no	—
244	C₃³	—	333	1	no	—
245	C₃³	—	333	1	no	—
246	C₃³	—	333	1	no	—
247	C₃³	—	111	3	no	—
248	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
249	3₋¹⁺²	—	3	3	no	—
250	3₋¹⁺²	—	333	3	no	—
251	3₋¹⁺²	—	333	3	no	—
252	3₋¹⁺²	—	111	3	no	—
253	3₋¹⁺²	—	111	3	no	—
254	C₃×C₉	—	9	1	no	—
255	C₃×C₉	—	37	1	no	—
256	C₃×C₉	—	37	1	no	—
257	C₃×C₉	—	37	1	no	—
258	C₃×C₉	—	37	3	no	—
259	C₃×C₉	—	37	3	no	—
260	C₃×C₉	—	37	3	no	—
261	C₃×C₉	—	37	3	no	—
262	C₃×C₉	—	37	3	no	—
263	C₃×C₉	—	37	3	no	—
264	C₃×C₉	—	37	1	no	—
265	C₃×C₉	—	37	1	no	—
266	C₃×C₉	—	37	1	no	—
267	C₃×C₉	—	333	1	no	—
268	C₃×C₉	—	333	1	no	—
269	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
270	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
271	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
272	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
273	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
274	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
275	C₃²	—	3	1	no	—
276	C₃²	—	3	1	no	—
277	C₃²	—	3	1	no	—
278	C₃²	—	333	9	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	—	111	1	no	—
288	C₃²	—	111	1	no	—
289	C₃²	—	111	1	no	—
290	C₃²	—	111	1	no	—
291	C₃²	#15688532	1	3	yes	—
292	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
293	C₉	—	3	3	no	—
294	C₉	—	3	3	no	—
295	C₉	—	3	3	no	—
296	C₉	—	3	1	no	—
297	C₉	—	3	1	no	—
298	C₉	—	3	1	no	—
299	C₉	—	111	3	no	—
300	C₉	—	111	3	no	—
301	C₉	—	111	3	no	—
302	C₉	—	111	3	no	—
303	C₉	—	111	3	no	—
304	C₉	—	111	3	no	—
305	C₉	—	111	1	no	—
306	C₉	—	111	1	no	—
307	C₉	—	111	1	no	—
308	C₉	—	111	1	no	—
309	C₉	—	111	1	no	—
310	C₉	—	111	1	no	—
311	C₃	—	3	3	no	—
312	C₃	—	333	1	no	—
313	C₃	—	333	1	no	—
314	C₃	—	333	1	no	—
315	C₃	C₃×#15688532	1	1	yes	characteristic
316	C₃	#22582677	1	1	yes	characteristic
317	C₃	#22582678	1	1	yes	characteristic
318	C₃	#22582679	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting	C₃₇×#6894	C₃
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	#6894	C₃₇⋊C₃
hypocenter	C₃₇	#10624889
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ #6894

Complements

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

N	H ≅ G/N	type
1. #22582589	C₃	semidirect
2. #22582589	C₃	semidirect
3. #22582589	C₃	semidirect
4. #22582588	C₃	semidirect
5. #22582588	C₃	semidirect
6. #22582588	C₃	semidirect
7. #22582584	C₃	semidirect
8. #22582583	C₃	semidirect
9. #22582582	C₃	semidirect
10. #22582574	C₃	semidirect
11. #22582573	C₃	semidirect
12. #22582572	C₃	semidirect
13. #22582571	C₃	semidirect
14. #22582568	C₃	semidirect
18. #22582548	C₃	semidirect
19. #22582548	C₃	semidirect
20. 3₋¹⁺²×#66524	C₃	semidirect
22. C₃₇×#6894	C₃	semidirect
30. #15688506	C₃²	semidirect
31. #15688506	C₃²	semidirect
32. #15688506	C₃²	semidirect
33. #15688505	C₃²	semidirect
34. #15688505	C₃²	semidirect
35. #15688505	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
50. #15688504	C₃²	semidirect
71. C₉×#66524	C₃²	semidirect
72. C₉×#66524	C₃²	semidirect
73. C₉×#66524	C₃²	semidirect
76. C₃₇×C₉⋊C₉	C₃²	semidirect
77. C₃₇×C₉⋊C₉	C₃²	semidirect
78. C₃₇×C₉⋊C₉	C₃²	semidirect
166. #6894	C₃₇⋊C₃	semidirect
212. C₉⋊C₉	C₃×C₃₇⋊C₃	semidirect
213. C₉⋊C₉	C₃×C₃₇⋊C₃	semidirect
214. C₉⋊C₉	C₃×C₃₇⋊C₃	semidirect
240. C₃₇	#10624889	semidirect

Maximal subgroups

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

subgroup	index
#22582589	3
#22582588	3
#22582584	3
#22582583	3
#22582582	3
#22582574	3
#22582573	3
#22582572	3
#22582571	3
#22582568	3
#22582553	3
#22582552	3
#22582551	3
#22582548	3
3₋¹⁺²×#66524	3
C₃×#15688502	3
C₃₇×#6894	3
#10624889	37