Subgroups — #41007831

Order 26973.

836 conjugacy classes of subgroups.

The subgroup lattice has 298 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	C₃×3₋¹⁺²×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
3	#22582601	C₃	1	2	yes	maximal, complemented
4	#22582601	C₃	1	2	yes	maximal, complemented
5	#22582601	C₃	1	4	yes	maximal, complemented
6	#22582584	C₃	1	2	yes	maximal, complemented
7	#22582583	C₃	1	2	yes	maximal, complemented
8	#22582582	C₃	1	4	yes	maximal, complemented
9	#22582581	C₃	1	1	yes	maximal, characteristic, complemented
10	#22582570	C₃	1	4	yes	maximal, complemented
11	#22582569	C₃	1	4	yes	maximal, complemented
12	#22582567	C₃	1	2	yes	maximal, complemented
13	#22582566	C₃	1	2	yes	maximal, complemented
14	#22582551	C₃	1	2	yes	maximal
15	C₉×#11758606	C₃	1	1	yes	maximal, characteristic, complemented
16	C₉×#11758605	C₃	1	1	yes	maximal, characteristic, complemented
17	#22582543	C₃	1	2	yes	maximal, complemented
18	#22582542	C₃	1	2	yes	maximal, complemented
19	C₃₇×#6900	C₃	1	1	yes	maximal, characteristic, complemented
20	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
21	C₃×#11758608	C₃²	1	2	yes	—
22	C₃×#11758608	C₃²	1	2	yes	—
23	C₃×#11758608	C₃²	1	4	yes	—
24	C₃×#11758605	C₃²	1	2	yes	—
25	C₃×#11758605	C₃²	1	1	yes	characteristic
26	C₃×#11758605	C₃²	1	2	yes	—
27	C₃×#11758605	C₃²	1	1	yes	characteristic
28	C₃×#11758606	C₃²	1	2	yes	—
29	C₃×#11758606	C₃²	1	1	yes	characteristic
30	C₃×#11758606	C₃²	1	2	yes	—
31	C₃×#11758606	C₃²	1	1	yes	characteristic
32	3₋¹⁺²×C₃₇⋊C₃	—	3	9	no	—
33	3₋¹⁺²×C₃₇⋊C₃	—	3	9	no	—
34	C₃×C₉×C₃₇⋊C₃	C₃²	1	2	yes	—
35	C₃×C₉×C₃₇⋊C₃	C₃²	1	2	yes	—
36	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
37	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
38	#15688506	C₃²	1	2	yes	complemented
39	#15688506	C₃²	1	2	yes	complemented
40	#15688506	C₃²	1	4	yes	complemented
41	#15688505	C₃²	1	2	yes	complemented
42	#15688505	C₃²	1	2	yes	complemented
43	#15688505	C₃²	1	4	yes	complemented
44	#15688504	C₃²	1	4	yes	complemented
45	#15688504	C₃²	1	4	yes	complemented
46	#15688504	C₃²	1	4	yes	complemented
47	#15688504	C₃²	1	4	yes	complemented
48	#15688504	C₃²	1	4	yes	complemented
49	#15688504	C₃²	1	4	yes	complemented
50	#15688504	C₃²	1	4	yes	complemented
51	#15688504	C₃²	1	4	yes	complemented
52	#15688504	C₃²	1	4	yes	complemented
53	#15688504	C₃²	1	4	yes	complemented
54	#15688504	C₃²	1	4	yes	complemented
55	#15688504	C₃²	1	4	yes	complemented
56	#15688502	C₃²	1	4	yes	—
57	#15688502	C₃²	1	4	yes	—
58	#15688502	C₃²	1	2	yes	—
59	#15688502	C₃²	1	2	yes	—
60	#15688502	C₃²	1	2	yes	—
61	#15688502	C₃²	1	4	yes	—
62	C₉×#66524	C₃²	1	4	yes	complemented
63	C₉×#66524	C₃²	1	2	yes	complemented
64	C₉×#66524	C₃²	1	2	yes	complemented
65	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
66	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
67	C₃×3₋¹⁺²×C₃₇	C₃²	1	2	yes	—
68	C₃₇×C₉⋊C₉	C₃²	1	2	yes	complemented
69	C₃₇×C₉⋊C₉	C₃²	1	2	yes	complemented
70	C₃₇×C₉⋊C₉	C₃²	1	4	yes	complemented
71	C₉²×C₃₇	C₃²	1	1	yes	characteristic, complemented
72	C₃³×C₃₇	C₃³	1	1	yes	characteristic
73	C₃²×C₃₇⋊C₃	—	3	3	no	—
74	C₃²×C₃₇⋊C₃	—	3	3	no	—
75	C₃²×C₃₇⋊C₃	—	9	2	no	—
76	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
77	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
78	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
79	3₋¹⁺²×C₃₇	—	3	3	no	—
80	3₋¹⁺²×C₃₇	—	3	3	no	—
81	3₋¹⁺²×C₃₇	—	9	2	no	—
82	C₃×C₉×C₃₇	C₃³	1	2	yes	—
83	C₃×C₉×C₃₇	C₃³	1	4	yes	—
84	C₃×C₉×C₃₇	C₃³	1	2	yes	—
85	C₃×C₉×C₃₇	C₃³	1	2	yes	—
86	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
87	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
88	#11758608	—	3	6	no	—
89	#11758608	—	3	6	no	—
90	#11758608	—	9	4	no	—
91	C₃×#66524	C₃³	1	4	yes	—
92	C₃×#66524	C₃³	1	4	yes	—
93	C₃×#66524	C₃³	1	4	yes	—
94	C₃×#66524	C₃³	1	2	yes	—
95	C₃×#66524	C₃³	1	2	yes	—
96	C₃×#66524	C₃³	1	2	yes	—
97	C₃×#66524	C₃³	1	2	yes	—
98	C₃×#66524	C₃³	1	2	yes	—
99	C₃×#66524	C₃³	1	2	yes	—
100	#11758606	—	3	6	no	—
101	#11758606	—	3	3	no	—
102	#11758606	—	3	6	no	—
103	#11758606	3₋¹⁺²	1	9	yes	complemented
104	#11758605	—	3	6	no	—
105	#11758605	—	3	3	no	—
106	#11758605	—	3	6	no	—
107	#11758605	3₋¹⁺²	1	9	yes	complemented
108	C₉×C₃₇⋊C₃	—	3	3	no	—
109	C₉×C₃₇⋊C₃	—	3	3	no	—
110	C₉×C₃₇⋊C₃	—	3	6	no	—
111	C₉×C₃₇⋊C₃	—	3	6	no	—
112	3₋¹⁺²²	—	37	1	no	maximal, Sylow
113	C₃²×C₃₇	—	3	1	no	—
114	C₃²×C₃₇	—	3	1	no	—
115	C₃²×C₃₇	—	3	2	no	—
116	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
117	C₃×C₃₇⋊C₃	—	3	1	no	—
118	C₃×C₃₇⋊C₃	—	3	1	no	—
119	C₃×C₃₇⋊C₃	—	3	2	no	—
120	C₃×C₃₇⋊C₃	—	3	2	no	—
121	C₃×C₃₇⋊C₃	—	3	1	no	—
122	C₃×C₃₇⋊C₃	—	3	1	no	—
123	C₃×C₃₇⋊C₃	—	3	2	no	—
124	C₃×C₃₇⋊C₃	—	9	9	no	—
125	C₃×C₃₇⋊C₃	C₃×3₋¹⁺²	1	3	yes	—
126	C₃×C₃₇⋊C₃	C₃×3₋¹⁺²	1	3	yes	—
127	C₉×C₃₇	—	3	2	no	—
128	C₉×C₃₇	—	3	2	no	—
129	C₉×C₃₇	—	3	2	no	—
130	C₉×C₃₇	—	3	4	no	—
131	C₉×C₃₇	C₃×3₋¹⁺²	1	3	yes	complemented
132	C₉×C₃₇	C₃×3₋¹⁺²	1	3	yes	complemented
133	#66524	—	3	2	no	—
134	#66524	—	3	2	no	—
135	#66524	—	3	2	no	—
136	#66524	—	3	2	no	—
137	#66524	—	3	4	no	—
138	#66524	—	3	4	no	—
139	#66524	—	3	4	no	—
140	#66524	C₃×3₋¹⁺²	1	6	yes	complemented
141	#66524	C₃×3₋¹⁺²	1	6	yes	complemented
142	C₃²×3₋¹⁺²	—	37	1	no	—
143	C₃²×3₋¹⁺²	—	37	1	no	—
144	#6900	—	37	4	no	—
145	#6900	—	37	2	no	—
146	#6900	—	37	2	no	—
147	#6900	C₃₇⋊C₃	1	1	yes	characteristic, complemented
148	#6896	—	37	4	no	—
149	#6896	—	37	2	no	—
150	#6896	—	37	2	no	—
151	#6896	—	37	1	no	—
152	#6894	—	37	2	no	—
153	#6894	—	37	2	no	—
154	#6894	—	37	4	no	—
155	#6894	—	37	4	no	—
156	#6891	—	37	2	no	—
157	C₉×3₋¹⁺²	—	37	2	no	—
158	C₉×3₋¹⁺²	—	37	2	no	—
159	C₉×3₋¹⁺²	—	37	1	no	—
160	C₉×3₋¹⁺²	—	37	1	no	—
161	C₃×C₃₇	—	9	1	no	—
162	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
163	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
164	C₃×C₃₇	3₋¹⁺⁴	1	2	yes	—
165	C₃₇⋊C₃	—	3	3	no	—
166	C₃₇⋊C₃	—	3	3	no	—
167	C₃₇⋊C₃	—	9	1	no	—
168	C₃⁴	—	37	1	no	—
169	C₃×3₋¹⁺²	—	37	4	no	—
170	C₃×3₋¹⁺²	—	37	2	no	—
171	C₃×3₋¹⁺²	—	37	2	no	—
172	C₃×3₋¹⁺²	—	37	1	no	—
173	C₃×3₋¹⁺²	—	37	1	no	—
174	C₃×3₋¹⁺²	—	37	2	no	—
175	C₃×3₋¹⁺²	—	37	2	no	—
176	C₃×3₋¹⁺²	—	37	1	no	—
177	C₃×3₋¹⁺²	—	37	2	no	—
178	C₃×3₋¹⁺²	—	37	2	no	—
179	C₃×3₋¹⁺²	—	37	1	no	—
180	C₃×3₋¹⁺²	—	111	9	no	—
181	C₃×3₋¹⁺²	—	111	9	no	—
182	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
183	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
184	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	2	yes	—
185	C₃²×C₉	—	37	2	no	—
186	C₃²×C₉	—	37	2	no	—
187	C₃²×C₉	—	37	1	no	—
188	C₃²×C₉	—	37	1	no	—
189	C₉⋊C₉	—	37	4	no	—
190	C₉⋊C₉	—	37	4	no	—
191	C₉⋊C₉	—	37	4	no	—
192	C₉⋊C₉	—	37	4	no	—
193	C₉⋊C₉	—	37	4	no	—
194	C₉⋊C₉	—	37	4	no	—
195	C₉⋊C₉	—	37	4	no	—
196	C₉⋊C₉	—	37	4	no	—
197	C₉⋊C₉	—	37	4	no	—
198	C₉⋊C₉	—	37	4	no	—
199	C₉⋊C₉	—	37	4	no	—
200	C₉⋊C₉	—	37	4	no	—
201	C₉⋊C₉	—	37	2	no	—
202	C₉⋊C₉	—	37	2	no	—
203	C₉⋊C₉	—	37	4	no	—
204	C₉⋊C₉	—	37	2	no	—
205	C₉⋊C₉	—	37	2	no	—
206	C₉⋊C₉	—	37	4	no	—
207	C₉⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	complemented
208	C₉⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	complemented
209	C₉⋊C₉	C₃×C₃₇⋊C₃	1	4	yes	complemented
210	C₃²⋊C₉	—	37	4	no	—
211	C₃²⋊C₉	—	37	4	no	—
212	C₃²⋊C₉	—	37	2	no	—
213	C₃²⋊C₉	—	37	2	no	—
214	C₃²⋊C₉	—	37	4	no	—
215	C₃²⋊C₉	—	37	2	no	—
216	C₉²	—	37	4	no	—
217	C₉²	—	37	2	no	—
218	C₉²	—	37	2	no	—
219	C₉²	C₃×C₃₇⋊C₃	1	1	yes	characteristic, complemented
220	C₃₇	3₋¹⁺²²	1	1	yes	characteristic, complemented, Sylow
221	C₃³	—	37	1	no	—
222	C₃³	—	37	1	no	—
223	C₃³	—	37	1	no	—
224	C₃³	—	333	2	no	—
225	C₃³	—	111	3	no	—
226	C₃³	—	111	3	no	—
227	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
228	3₋¹⁺²	—	3	3	no	—
229	3₋¹⁺²	—	3	3	no	—
230	3₋¹⁺²	—	9	2	no	—
231	3₋¹⁺²	—	37	9	no	—
232	3₋¹⁺²	—	37	9	no	—
233	3₋¹⁺²	—	333	4	no	—
234	3₋¹⁺²	—	111	6	no	—
235	3₋¹⁺²	—	111	6	no	—
236	3₋¹⁺²	—	111	6	no	—
237	3₋¹⁺²	—	111	6	no	—
238	3₋¹⁺²	—	111	6	no	—
239	3₋¹⁺²	—	111	6	no	—
240	3₋¹⁺²	—	111	3	no	—
241	3₋¹⁺²	—	111	3	no	—
242	C₃×C₉	—	37	2	no	—
243	C₃×C₉	—	37	2	no	—
244	C₃×C₉	—	37	4	no	—
245	C₃×C₉	—	37	4	no	—
246	C₃×C₉	—	37	4	no	—
247	C₃×C₉	—	37	2	no	—
248	C₃×C₉	—	37	2	no	—
249	C₃×C₉	—	37	2	no	—
250	C₃×C₉	—	37	2	no	—
251	C₃×C₉	—	111	6	no	—
252	C₃×C₉	—	111	6	no	—
253	C₃×C₉	—	111	3	no	—
254	C₃×C₉	—	111	3	no	—
255	C₃×C₉	C₃²×C₃₇⋊C₃	1	4	yes	—
256	C₃×C₉	C₃²×C₃₇⋊C₃	1	2	yes	—
257	C₃×C₉	C₃²×C₃₇⋊C₃	1	2	yes	—
258	C₃×C₉	C₃²×C₃₇⋊C₃	1	2	yes	—
259	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
260	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
261	C₃²	—	3	1	no	—
262	C₃²	—	3	1	no	—
263	C₃²	—	3	2	no	—
264	C₃²	—	37	3	no	—
265	C₃²	—	37	3	no	—
266	C₃²	—	333	9	no	—
267	C₃²	—	111	1	no	—
268	C₃²	—	111	1	no	—
269	C₃²	—	111	2	no	—
270	C₃²	—	111	1	no	—
271	C₃²	—	111	1	no	—
272	C₃²	—	111	2	no	—
273	C₃²	—	111	2	no	—
274	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
275	C₉	—	3	2	no	—
276	C₉	—	3	2	no	—
277	C₉	—	3	4	no	—
278	C₉	—	3	2	no	—
279	C₉	—	37	6	no	—
280	C₉	—	37	6	no	—
281	C₉	—	111	2	no	—
282	C₉	—	111	2	no	—
283	C₉	—	111	2	no	—
284	C₉	—	111	2	no	—
285	C₉	—	111	4	no	—
286	C₉	—	111	4	no	—
287	C₉	—	111	4	no	—
288	C₉	3₋¹⁺²×C₃₇⋊C₃	1	3	yes	complemented
289	C₉	3₋¹⁺²×C₃₇⋊C₃	1	3	yes	complemented
290	C₃	—	9	1	no	—
291	C₃	—	333	1	no	—
292	C₃	—	111	3	no	—
293	C₃	—	111	3	no	—
294	C₃	C₃×3₋¹⁺²×C₃₇⋊C₃	1	1	yes	characteristic
295	C₃	C₃×3₋¹⁺²×C₃₇⋊C₃	1	1	yes	characteristic
296	C₃	#22582679	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting	C₃₇×#6900	C₃
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	#6900	C₃₇⋊C₃
hypocenter	C₃₇	3₋¹⁺²²
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ #6900

Complements

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

N	H ≅ G/N	type
3. #22582601	C₃	semidirect
4. #22582601	C₃	semidirect
5. #22582601	C₃	semidirect
6. #22582584	C₃	semidirect
7. #22582583	C₃	semidirect
8. #22582582	C₃	semidirect
9. #22582581	C₃	semidirect
10. #22582570	C₃	semidirect
11. #22582569	C₃	semidirect
12. #22582567	C₃	semidirect
13. #22582566	C₃	semidirect
15. C₉×#11758606	C₃	semidirect
16. C₉×#11758605	C₃	semidirect
17. #22582543	C₃	semidirect
18. #22582542	C₃	semidirect
19. C₃₇×#6900	C₃	semidirect
38. #15688506	C₃²	semidirect
39. #15688506	C₃²	semidirect
40. #15688506	C₃²	semidirect
41. #15688505	C₃²	semidirect
42. #15688505	C₃²	semidirect
43. #15688505	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
55. #15688504	C₃²	semidirect
62. C₉×#66524	C₃²	semidirect
63. C₉×#66524	C₃²	semidirect
64. C₉×#66524	C₃²	semidirect
68. C₃₇×C₉⋊C₉	C₃²	semidirect
69. C₃₇×C₉⋊C₉	C₃²	semidirect
70. C₃₇×C₉⋊C₉	C₃²	semidirect
71. C₉²×C₃₇	C₃²	semidirect
103. #11758606	3₋¹⁺²	semidirect
107. #11758605	3₋¹⁺²	semidirect
131. C₉×C₃₇	C₃×3₋¹⁺²	semidirect
132. C₉×C₃₇	C₃×3₋¹⁺²	semidirect
140. #66524	C₃×3₋¹⁺²	semidirect
141. #66524	C₃×3₋¹⁺²	semidirect
147. #6900	C₃₇⋊C₃	semidirect
207. C₉⋊C₉	C₃×C₃₇⋊C₃	semidirect
208. C₉⋊C₉	C₃×C₃₇⋊C₃	semidirect
209. C₉⋊C₉	C₃×C₃₇⋊C₃	semidirect
219. C₉²	C₃×C₃₇⋊C₃	semidirect
220. C₃₇	3₋¹⁺²²	semidirect
288. C₉	3₋¹⁺²×C₃₇⋊C₃	semidirect
289. C₉	3₋¹⁺²×C₃₇⋊C₃	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
#22582601	3
#22582584	3
#22582583	3
#22582582	3
#22582581	3
#22582570	3
#22582569	3
#22582567	3
#22582566	3
#22582551	3
C₉×#11758606	3
C₉×#11758605	3
#22582543	3
#22582542	3
C₃₇×#6900	3
3₋¹⁺²²	37