Subgroups — #41007845

Order 26973.

740 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×3₋¹⁺²×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
2	#22582598	C₃	1	3	yes	maximal, complemented
3	#22582598	C₃	1	3	yes	maximal, complemented
4	#22582598	C₃	1	3	yes	maximal, complemented
5	#22582583	C₃	1	3	yes	maximal, complemented
6	#22582582	C₃	1	3	yes	maximal, complemented
7	#22582582	C₃	1	3	yes	maximal, complemented
8	#22582574	C₃	1	1	yes	maximal, characteristic, complemented
9	#22582574	C₃	1	1	yes	maximal, characteristic, complemented
10	#22582571	C₃	1	1	yes	maximal, characteristic, complemented
11	#22582569	C₃	1	3	yes	maximal, complemented
12	#22582569	C₃	1	3	yes	maximal, complemented
13	#22582568	C₃	1	1	yes	maximal, characteristic, complemented
14	#22582568	C₃	1	1	yes	maximal, characteristic, complemented
15	#22582552	C₃	1	1	yes	maximal, characteristic
16	#22582552	C₃	1	1	yes	maximal, characteristic
17	#22582548	C₃	1	1	yes	maximal, characteristic, complemented
18	#22582548	C₃	1	1	yes	maximal, characteristic, complemented
19	C₉×#11758608	C₃	1	1	yes	maximal, characteristic, complemented
20	#22582542	C₃	1	3	yes	maximal, 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	C₃×#11758606	C₃²	1	3	yes	—
29	C₃×#11758606	C₃²	1	3	yes	—
30	3₋¹⁺²×C₃₇⋊C₃	—	3	9	no	—
31	C₃²×#66524	C₃²	1	1	yes	characteristic
32	C₃²×#66524	C₃²	1	1	yes	characteristic
33	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	—
34	#15688505	C₃²	1	3	yes	complemented
35	#15688505	C₃²	1	3	yes	complemented
36	#15688505	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	#15688504	C₃²	1	3	yes	complemented
52	#15688504	C₃²	1	3	yes	complemented
53	#15688504	C₃²	1	3	yes	complemented
54	#15688504	C₃²	1	3	yes	complemented
55	C₃²⋊#66524	C₃²	1	1	yes	characteristic
56	C₃²⋊#66524	C₃²	1	1	yes	characteristic
57	C₃²⋊#66524	C₃²	1	1	yes	characteristic
58	C₃²⋊#66524	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	1	yes	characteristic
68	#15688502	C₃²	1	3	yes	—
69	#15688502	C₃²	1	3	yes	—
70	#15688502	C₃²	1	3	yes	—
71	#15688502	C₃²	1	3	yes	—
72	C₉×#66524	C₃²	1	3	yes	complemented
73	C₉×#66524	C₃²	1	3	yes	complemented
74	C₉×#66524	C₃²	1	3	yes	complemented
75	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
76	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
77	C₃₇×C₉⋊C₉	C₃²	1	3	yes	complemented
78	C₃₇×C₉⋊C₉	C₃²	1	3	yes	complemented
79	C₃₇×C₉⋊C₉	C₃²	1	3	yes	complemented
80	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
81	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
82	C₃³×C₃₇	C₃³	1	1	yes	characteristic
83	C₃²×C₃₇⋊C₃	—	3	3	no	—
84	C₃²×C₃₇⋊C₃	—	3	3	no	—
85	C₃²×C₃₇⋊C₃	—	9	1	no	—
86	C₃²×C₃₇⋊C₃	—	9	1	no	—
87	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
88	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
89	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
90	3₋¹⁺²×C₃₇	—	3	3	no	—
91	C₃×C₉×C₃₇	—	9	1	no	—
92	C₃×C₉×C₃₇	C₃³	1	3	yes	—
93	C₃×C₉×C₃₇	C₃³	1	3	yes	—
94	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
95	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
96	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
97	C₃×C₉×C₃₇	C₃³	1	3	yes	—
98	#11758608	—	3	3	no	—
99	#11758608	3₋¹⁺²	1	9	yes	complemented
100	C₃×#66524	—	9	1	no	—
101	C₃×#66524	—	9	1	no	—
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	1	yes	characteristic
109	C₃×#66524	C₃³	1	1	yes	characteristic
110	C₃×#66524	C₃³	1	1	yes	characteristic
111	C₃×#66524	C₃³	1	1	yes	characteristic
112	C₃×#66524	C₃³	1	3	yes	—
113	C₃×#66524	C₃³	1	3	yes	—
114	#11758606	—	3	9	no	—
115	#11758606	—	3	9	no	—
116	#11758606	—	9	3	no	—
117	#11758605	—	3	9	no	—
118	C₉×C₃₇⋊C₃	—	3	9	no	—
119	#10624899	—	37	1	no	maximal, Sylow
120	C₃²×C₃₇	—	3	1	no	—
121	C₃²×C₃₇	—	3	1	no	—
122	C₃²×C₃₇	—	3	1	no	—
123	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
124	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
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	—	3	1	no	—
134	C₃×C₃₇⋊C₃	—	3	1	no	—
135	C₃×C₃₇⋊C₃	—	3	1	no	—
136	C₃×C₃₇⋊C₃	—	9	9	no	—
137	C₃×C₃₇⋊C₃	#737	1	3	yes	—
138	C₉×C₃₇	—	3	3	no	—
139	C₉×C₃₇	—	3	3	no	—
140	C₉×C₃₇	—	3	3	no	—
141	C₉×C₃₇	—	3	1	no	—
142	C₉×C₃₇	—	3	1	no	—
143	C₉×C₃₇	—	3	1	no	—
144	#66524	—	3	3	no	—
145	#66524	—	3	3	no	—
146	#66524	—	3	3	no	—
147	#66524	—	3	3	no	—
148	#66524	—	3	3	no	—
149	#66524	—	3	1	no	—
150	#66524	—	3	1	no	—
151	#66524	—	3	1	no	—
152	#66524	—	3	1	no	—
153	#66524	—	3	1	no	—
154	#66524	—	3	1	no	—
155	#66524	C₃×3₋¹⁺²	1	9	yes	—
156	C₃²×3₋¹⁺²	—	37	1	no	—
157	#6899	—	37	3	no	—
158	#6899	—	37	3	no	—
159	#6899	—	37	3	no	—
160	#6896	—	37	3	no	—
161	#6896	—	37	3	no	—
162	#6896	—	37	3	no	—
163	#6894	—	37	1	no	—
164	#6894	—	37	1	no	—
165	#6894	—	37	1	no	—
166	#6894	—	37	3	no	—
167	#6894	—	37	3	no	—
168	#6894	—	37	1	no	—
169	#6894	—	37	1	no	—
170	#6894	C₃₇⋊C₃	1	1	yes	characteristic, complemented
171	#6891	—	37	1	no	—
172	#6891	—	37	1	no	—
173	C₉×3₋¹⁺²	—	37	1	no	—
174	C₉×3₋¹⁺²	—	37	3	no	—
175	C₉×3₋¹⁺²	—	37	1	no	—
176	C₉×3₋¹⁺²	—	37	1	no	—
177	C₃×C₃²⋊C₉	—	37	1	no	—
178	C₃×C₃₇	—	3	3	no	—
179	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
180	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
181	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
182	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
183	C₃₇⋊C₃	—	3	3	no	—
184	C₃₇⋊C₃	—	9	1	no	—
185	C₃₇⋊C₃	—	9	1	no	—
186	C₃⁴	—	37	1	no	—
187	C₃×3₋¹⁺²	—	37	1	no	—
188	C₃×3₋¹⁺²	—	37	1	no	—
189	C₃×3₋¹⁺²	—	37	3	no	—
190	C₃×3₋¹⁺²	—	37	3	no	—
191	C₃×3₋¹⁺²	—	37	3	no	—
192	C₃×3₋¹⁺²	—	37	3	no	—
193	C₃×3₋¹⁺²	—	111	9	no	—
194	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
195	C₃²×C₉	—	37	1	no	—
196	C₃²×C₉	—	37	1	no	—
197	C₃²×C₉	—	37	3	no	—
198	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
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	—	37	3	no	—
213	C₉⋊C₉	—	37	3	no	—
214	C₉⋊C₉	—	37	3	no	—
215	C₉⋊C₉	—	37	3	no	—
216	C₉⋊C₉	—	37	3	no	—
217	C₉⋊C₉	—	37	3	no	—
218	C₉⋊C₉	—	37	3	no	—
219	C₉⋊C₉	—	37	3	no	—
220	C₉⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	complemented
221	C₉⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	complemented
222	C₉⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	complemented
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	1	no	—
229	C₃²⋊C₉	—	37	1	no	—
230	C₃²⋊C₉	—	37	1	no	—
231	C₃²⋊C₉	—	37	1	no	—
232	C₃²⋊C₉	—	37	1	no	—
233	C₃²⋊C₉	—	37	1	no	—
234	C₃²⋊C₉	—	37	1	no	—
235	C₃²⋊C₉	—	37	1	no	—
236	C₃²⋊C₉	—	37	3	no	—
237	C₃²⋊C₉	—	37	3	no	—
238	C₃²⋊C₉	—	37	3	no	—
239	C₃²⋊C₉	—	37	3	no	—
240	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
241	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
242	C₉²	—	37	3	no	—
243	C₉²	—	37	3	no	—
244	C₉²	—	37	3	no	—
245	C₃₇	#10624899	1	1	yes	characteristic, complemented, Sylow
246	C₃³	—	37	1	no	—
247	C₃³	—	37	1	no	—
248	C₃³	—	37	1	no	—
249	C₃³	—	333	1	no	—
250	C₃³	—	333	1	no	—
251	C₃³	—	111	3	no	—
252	C₃³	—	111	3	no	—
253	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
254	3₋¹⁺²	—	3	3	no	—
255	3₋¹⁺²	—	37	9	no	—
256	3₋¹⁺²	—	333	3	no	—
257	3₋¹⁺²	—	111	9	no	—
258	3₋¹⁺²	—	111	9	no	—
259	3₋¹⁺²	—	111	9	no	—
260	3₋¹⁺²	—	111	3	no	—
261	C₃×C₉	—	9	1	no	—
262	C₃×C₉	—	37	3	no	—
263	C₃×C₉	—	37	1	no	—
264	C₃×C₉	—	37	1	no	—
265	C₃×C₉	—	37	3	no	—
266	C₃×C₉	—	37	3	no	—
267	C₃×C₉	—	37	3	no	—
268	C₃×C₉	—	37	3	no	—
269	C₃×C₉	—	37	1	no	—
270	C₃×C₉	—	37	1	no	—
271	C₃×C₉	—	37	1	no	—
272	C₃×C₉	—	37	1	no	—
273	C₃×C₉	—	37	3	no	—
274	C₃×C₉	—	333	1	no	—
275	C₃×C₉	—	333	1	no	—
276	C₃×C₉	—	111	9	no	—
277	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
278	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
279	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
280	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
281	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
282	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
283	C₃²	—	3	1	no	—
284	C₃²	—	3	1	no	—
285	C₃²	—	3	1	no	—
286	C₃²	—	37	3	no	—
287	C₃²	—	333	9	no	—
288	C₃²	—	111	1	no	—
289	C₃²	—	111	1	no	—
290	C₃²	—	111	1	no	—
291	C₃²	—	111	1	no	—
292	C₃²	—	111	1	no	—
293	C₃²	—	111	1	no	—
294	C₃²	—	111	1	no	—
295	C₃²	—	111	1	no	—
296	C₃²	—	111	1	no	—
297	C₃²	—	111	1	no	—
298	C₃²	—	111	1	no	—
299	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	3	yes	—
300	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
301	C₉	—	3	3	no	—
302	C₉	—	3	3	no	—
303	C₉	—	3	3	no	—
304	C₉	—	3	1	no	—
305	C₉	—	3	1	no	—
306	C₉	—	3	1	no	—
307	C₉	—	37	9	no	—
308	C₉	—	111	3	no	—
309	C₉	—	111	3	no	—
310	C₉	—	111	3	no	—
311	C₉	—	111	3	no	—
312	C₉	—	111	3	no	—
313	C₉	—	111	1	no	—
314	C₉	—	111	1	no	—
315	C₉	—	111	1	no	—
316	C₉	—	111	1	no	—
317	C₉	—	111	1	no	—
318	C₉	—	111	1	no	—
319	C₃	—	3	3	no	—
320	C₃	—	333	1	no	—
321	C₃	—	333	1	no	—
322	C₃	—	111	3	no	—
323	C₃	C₃×3₋¹⁺²×C₃₇⋊C₃	1	1	yes	characteristic
324	C₃	#737×C₃₇⋊C₃	1	1	yes	characteristic
325	C₃	#22582678	1	1	yes	characteristic
326	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	#10624899
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
2. #22582598	C₃	semidirect
3. #22582598	C₃	semidirect
4. #22582598	C₃	semidirect
5. #22582583	C₃	semidirect
6. #22582582	C₃	semidirect
7. #22582582	C₃	semidirect
8. #22582574	C₃	semidirect
9. #22582574	C₃	semidirect
10. #22582571	C₃	semidirect
11. #22582569	C₃	semidirect
12. #22582569	C₃	semidirect
13. #22582568	C₃	semidirect
14. #22582568	C₃	semidirect
17. #22582548	C₃	semidirect
18. #22582548	C₃	semidirect
19. C₉×#11758608	C₃	semidirect
20. #22582542	C₃	semidirect
22. C₃₇×#6894	C₃	semidirect
34. #15688505	C₃²	semidirect
35. #15688505	C₃²	semidirect
36. #15688505	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
51. #15688504	C₃²	semidirect
52. #15688504	C₃²	semidirect
53. #15688504	C₃²	semidirect
54. #15688504	C₃²	semidirect
72. C₉×#66524	C₃²	semidirect
73. C₉×#66524	C₃²	semidirect
74. C₉×#66524	C₃²	semidirect
77. C₃₇×C₉⋊C₉	C₃²	semidirect
78. C₃₇×C₉⋊C₉	C₃²	semidirect
79. C₃₇×C₉⋊C₉	C₃²	semidirect
99. #11758608	3₋¹⁺²	semidirect
170. #6894	C₃₇⋊C₃	semidirect
220. C₉⋊C₉	C₃×C₃₇⋊C₃	semidirect
221. C₉⋊C₉	C₃×C₃₇⋊C₃	semidirect
222. C₉⋊C₉	C₃×C₃₇⋊C₃	semidirect
245. C₃₇	#10624899	semidirect

Maximal subgroups

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

subgroup	index
C₃×3₋¹⁺²×C₃₇⋊C₃	3
#22582598	3
#22582583	3
#22582582	3
#22582574	3
#22582571	3
#22582569	3
#22582568	3
#22582552	3
#22582548	3
C₉×#11758608	3
#22582542	3
C₃×#15688502	3
C₃₇×#6894	3
#10624899	37