Subgroups — #41007526

Order 26973.

926 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688532	C₃	1	1	yes	maximal, characteristic
2	#22582548	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582548	C₃	1	1	yes	maximal, characteristic, complemented
4	3₋¹⁺²×#66524	C₃	1	1	yes	maximal, characteristic, complemented
5	3₋¹⁺²×#66524	C₃	1	1	yes	maximal, characteristic, complemented
6	C₉×#11758606	C₃	1	3	yes	maximal, complemented
7	C₉×#11758605	C₃	1	3	yes	maximal, complemented
8	C₉×#11758608	C₃	1	1	yes	maximal, characteristic, complemented
9	#22582543	C₃	1	3	yes	maximal, complemented
10	#22582543	C₃	1	3	yes	maximal, complemented
11	#22582542	C₃	1	3	yes	maximal, complemented
12	#22582542	C₃	1	3	yes	maximal, complemented
13	C₉×C₃²⋊(C₃₇⋊C₃)	C₃	1	1	yes	maximal, characteristic
14	#22582539	C₃	1	1	yes	maximal, characteristic
15	#22582539	C₃	1	1	yes	maximal, characteristic
16	#22582537	C₃	1	3	yes	maximal, complemented
17	#22582536	C₃	1	3	yes	maximal, complemented
18	#22582535	C₃	1	1	yes	maximal, characteristic, complemented
19	#22582535	C₃	1	1	yes	maximal, characteristic, complemented
20	C₃×#15688504	C₃	1	1	yes	maximal, characteristic, complemented
21	C₉⋊C₉×C₃₇⋊C₃	C₃	1	3	yes	maximal, complemented
22	C₉×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
23	#15688532	C₉	1	27	yes	complemented
24	C₃×#11758608	C₃²	1	1	yes	characteristic
25	C₃×#11758605	C₃²	1	3	yes	—
26	C₃×#11758606	C₃²	1	3	yes	—
27	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	1	yes	characteristic
28	C₃²×#66524	C₃²	1	1	yes	characteristic
29	C₃²×#66524	C₃²	1	1	yes	characteristic
30	C₃²×#66524	C₃²	1	1	yes	characteristic
31	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	—
32	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	—
33	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	—
34	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	—
35	#15688506	C₃²	1	3	yes	—
36	#15688506	C₃²	1	3	yes	—
37	#15688505	C₃²	1	3	yes	—
38	#15688505	C₃²	1	3	yes	—
39	#15688504	C₃²	1	1	yes	characteristic
40	#15688504	C₃²	1	3	yes	—
41	#15688504	C₃²	1	3	yes	—
42	#15688504	C₃²	1	3	yes	—
43	#15688504	C₃²	1	3	yes	—
44	#15688504	C₃²	1	3	yes	—
45	#15688504	C₃²	1	1	yes	characteristic
46	#15688504	C₃²	1	1	yes	characteristic
47	#15688504	C₃²	1	1	yes	characteristic
48	#15688504	C₃²	1	1	yes	characteristic
49	#15688504	C₃²	1	1	yes	characteristic
50	#15688504	C₃²	1	1	yes	characteristic
51	#15688504	C₃²	1	1	yes	characteristic
52	#15688504	C₃²	1	1	yes	characteristic
53	#15688504	C₃²	1	3	yes	—
54	#15688504	C₃²	1	3	yes	—
55	#15688504	C₃²	1	3	yes	—
56	#15688504	C₃²	1	3	yes	—
57	#15688504	C₃²	1	3	yes	—
58	C₃²⋊#66524	C₃²	1	1	yes	characteristic
59	C₃²⋊#66524	C₃²	1	1	yes	characteristic
60	C₃²⋊#66524	C₃²	1	1	yes	characteristic
61	C₃²⋊#66524	C₃²	1	1	yes	characteristic
62	#15688502	C₃²	1	3	yes	—
63	#15688502	C₃²	1	3	yes	—
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	C₉×#66524	C₃²	1	3	yes	—
69	C₉×#66524	C₃²	1	3	yes	—
70	C₉×#66524	C₃²	1	3	yes	—
71	C₉×#66524	C₃²	1	3	yes	—
72	C₉×#66524	C₃²	1	3	yes	—
73	C₉×#66524	C₃²	1	3	yes	—
74	C₉×#66524	C₃²	1	3	yes	—
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	—
78	C₃₇×C₉⋊C₉	C₃²	1	3	yes	—
79	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
80	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
81	C₉²×C₃₇	C₃²	1	3	yes	—
82	C₃³×C₃₇	C₃³	1	1	yes	characteristic
83	C₃²×C₃₇⋊C₃	C₃³	1	3	yes	—
84	3₋¹⁺²×C₃₇	C₃×C₉	1	9	yes	complemented
85	C₃×C₉×C₃₇	—	9	1	no	—
86	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
87	C₃×C₉×C₃₇	C₃³	1	3	yes	—
88	C₃×C₉×C₃₇	C₃³	1	3	yes	—
89	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
90	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
91	C₃×C₉×C₃₇	C₃³	1	3	yes	—
92	C₃²⋊(C₃₇⋊C₃)	C₃×C₉	1	9	yes	—
93	#11758608	C₃×C₉	1	9	yes	complemented
94	C₃×#66524	—	9	1	no	—
95	C₃×#66524	—	9	1	no	—
96	C₃×#66524	C₃×C₉	1	9	yes	complemented
97	C₃×#66524	C₃³	1	1	yes	characteristic
98	C₃×#66524	C₃³	1	1	yes	characteristic
99	C₃×#66524	C₃³	1	1	yes	characteristic
100	C₃×#66524	C₃³	1	1	yes	characteristic
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	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	#11758606	C₃×C₉	1	27	yes	complemented
112	#11758605	C₃×C₉	1	27	yes	complemented
113	C₉×C₃₇⋊C₃	—	9	3	no	—
114	C₉×C₃₇⋊C₃	—	9	3	no	—
115	C₉×C₃₇⋊C₃	—	9	3	no	—
116	C₉×C₃₇⋊C₃	C₃×C₉	1	27	yes	complemented
117	#10624864	—	37	1	no	maximal, Sylow
118	C₃²×C₃₇	—	3	1	no	—
119	C₃²×C₃₇	—	3	1	no	—
120	C₃²×C₃₇	—	3	1	no	—
121	C₃²×C₃₇	C₃²×C₉	1	3	yes	—
122	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
123	C₃×C₃₇⋊C₃	—	3	3	no	—
124	C₃×C₃₇⋊C₃	—	3	3	no	—
125	C₃×C₃₇⋊C₃	—	3	3	no	—
126	C₃×C₃₇⋊C₃	C₃²×C₉	1	9	yes	—
127	C₉×C₃₇	—	3	1	no	—
128	C₉×C₃₇	—	3	3	no	—
129	C₉×C₃₇	—	3	3	no	—
130	C₉×C₃₇	—	3	1	no	—
131	C₉×C₃₇	—	3	1	no	—
132	C₉×C₃₇	C₃²×C₉	1	9	yes	—
133	#66524	—	3	1	no	—
134	#66524	—	3	1	no	—
135	#66524	—	3	3	no	—
136	#66524	—	3	3	no	—
137	#66524	—	3	3	no	—
138	#66524	—	3	3	no	—
139	#66524	—	3	1	no	—
140	#66524	—	3	1	no	—
141	#66524	—	3	1	no	—
142	#66524	—	3	1	no	—
143	#66524	C₃²×C₉	1	3	yes	—
144	#66524	C₃²×C₉	1	9	yes	—
145	#66524	C₃²×C₉	1	3	yes	—
146	#66524	C₃²×C₉	1	3	yes	—
147	C₃×#737	—	37	1	no	—
148	C₉×3₋¹⁺²	—	37	1	no	—
149	C₉×3₋¹⁺²	—	37	1	no	—
150	C₉×3₋¹⁺²	—	37	1	no	—
151	C₉×3₋¹⁺²	—	37	3	no	—
152	C₉×3₋¹⁺²	—	37	3	no	—
153	C₉×3₋¹⁺²	—	37	3	no	—
154	C₉×3₋¹⁺²	—	37	3	no	—
155	C₉×3₋¹⁺²	—	37	1	no	—
156	C₉×3₋¹⁺²	—	37	1	no	—
157	C₉×3₋¹⁺²	—	37	3	no	—
158	C₉×3₋¹⁺²	—	37	3	no	—
159	C₉×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic, complemented
160	C₉×H(3,𝔽₃)	—	37	1	no	—
161	C₉×H(3,𝔽₃)	—	37	1	no	—
162	C₉×H(3,𝔽₃)	—	37	1	no	—
163	#6887	—	37	3	no	—
164	#6887	—	37	1	no	—
165	#6887	—	37	1	no	—
166	#6887	—	37	3	no	—
167	C₃×C₉⋊C₉	—	37	1	no	—
168	C₃×C₉⋊C₉	—	37	3	no	—
169	C₃×C₃₇	—	3	3	no	—
170	C₃×C₃₇	C₃³×C₉	1	1	yes	characteristic
171	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
172	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
173	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
174	C₃₇⋊C₃	—	3	9	no	—
175	#737	—	37	27	no	—
176	C₃×3₋¹⁺²	—	37	1	no	—
177	C₃×3₋¹⁺²	—	37	3	no	—
178	C₃×3₋¹⁺²	—	37	3	no	—
179	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
180	C₃×H(3,𝔽₃)	—	37	1	no	—
181	C₃²×C₉	—	37	1	no	—
182	C₃²×C₉	—	37	3	no	—
183	C₃²×C₉	—	37	1	no	—
184	C₃²×C₉	—	37	1	no	—
185	C₃²×C₉	—	37	3	no	—
186	C₃²×C₉	—	37	3	no	—
187	C₃²×C₉	—	37	3	no	—
188	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
189	C₉⋊C₉	—	37	1	no	—
190	C₉⋊C₉	—	37	1	no	—
191	C₉⋊C₉	—	37	3	no	—
192	C₉⋊C₉	—	37	3	no	—
193	C₉⋊C₉	—	37	1	no	—
194	C₉⋊C₉	—	37	3	no	—
195	C₉⋊C₉	—	37	1	no	—
196	C₉⋊C₉	—	37	1	no	—
197	C₉⋊C₉	—	37	1	no	—
198	C₉⋊C₉	—	37	1	no	—
199	C₉⋊C₉	—	37	3	no	—
200	C₉⋊C₉	—	37	3	no	—
201	C₉⋊C₉	—	37	1	no	—
202	C₉⋊C₉	—	37	1	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	—
213	C₉⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
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	3	no	—
219	C₃²⋊C₉	—	37	3	no	—
220	C₃²⋊C₉	—	37	3	no	—
221	C₃²⋊C₉	—	37	3	no	—
222	C₃²⋊C₉	—	37	3	no	—
223	C₃²⋊C₉	—	37	3	no	—
224	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
225	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
226	C₉²	—	37	3	no	—
227	C₉²	—	37	3	no	—
228	C₉²	—	37	3	no	—
229	C₉²	—	37	3	no	—
230	C₉²	—	37	3	no	—
231	C₉²	—	37	3	no	—
232	C₉²	—	37	3	no	—
233	C₉²	C₃×C₃₇⋊C₃	1	3	yes	—
234	C₃₇	#10624864	1	1	yes	characteristic, complemented, Sylow
235	C₃³	—	37	3	no	—
236	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
237	3₋¹⁺²	—	37	9	no	—
238	3₋¹⁺²	—	37	27	no	—
239	3₋¹⁺²	—	37	27	no	—
240	3₋¹⁺²	C₉×C₃₇⋊C₃	1	9	yes	complemented
241	H(3,𝔽₃)	—	37	9	no	—
242	C₃×C₉	—	9	1	no	—
243	C₃×C₉	—	37	9	no	—
244	C₃×C₉	—	37	3	no	—
245	C₃×C₉	—	37	1	no	—
246	C₃×C₉	—	37	1	no	—
247	C₃×C₉	—	37	1	no	—
248	C₃×C₉	—	37	27	no	—
249	C₃×C₉	—	37	1	no	—
250	C₃×C₉	—	37	1	no	—
251	C₃×C₉	—	37	3	no	—
252	C₃×C₉	—	37	1	no	—
253	C₃×C₉	—	37	1	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	—	333	3	no	—
260	C₃×C₉	—	333	3	no	—
261	C₃×C₉	—	333	3	no	—
262	C₃×C₉	—	333	1	no	—
263	C₃×C₉	—	333	1	no	—
264	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
265	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
266	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
267	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
268	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
269	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
270	C₃²	—	3	1	no	—
271	C₃²	—	3	1	no	—
272	C₃²	—	3	1	no	—
273	C₃²	—	37	9	no	—
274	C₃²	—	111	3	no	—
275	C₃²	—	111	3	no	—
276	C₃²	—	111	3	no	—
277	C₃²	C₃×C₉×C₃₇⋊C₃	1	3	yes	—
278	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
279	C₉	—	3	3	no	—
280	C₉	—	3	3	no	—
281	C₉	—	3	1	no	—
282	C₉	—	3	1	no	—
283	C₉	—	3	1	no	—
284	C₉	—	37	3	no	—
285	C₉	—	37	3	no	—
286	C₉	—	37	3	no	—
287	C₉	—	37	9	no	—
288	C₉	—	111	1	no	—
289	C₉	—	111	3	no	—
290	C₉	—	111	1	no	—
291	C₉	—	111	1	no	—
292	C₉	—	111	1	no	—
293	C₉	—	111	3	no	—
294	C₉	—	111	1	no	—
295	C₉	—	111	1	no	—
296	C₉	—	111	3	no	—
297	C₉	—	111	3	no	—
298	C₉	C₃×C₉×C₃₇⋊C₃	1	9	yes	—
299	C₃	—	3	3	no	—
300	C₃	—	111	9	no	—
301	C₃	C₃²×C₉×C₃₇⋊C₃	1	1	yes	characteristic
302	C₃	#22582677	1	1	yes	characteristic
303	C₃	#22582678	1	1	yes	characteristic
304	C₃	#22582679	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator	C₃×C₃₇	C₃³×C₉
Fitting	C₉×3₋¹⁺²×C₃₇	C₃
socle, cosocle	C₃²×C₃₇	C₃⁴
prime radical	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²
hypercenter	C₉×3₋¹⁺²	C₃₇⋊C₃
hypocenter	C₃₇	#10624864
LCS	G ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₉×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
2. #22582548	C₃	semidirect
3. #22582548	C₃	semidirect
4. 3₋¹⁺²×#66524	C₃	semidirect
5. 3₋¹⁺²×#66524	C₃	semidirect
6. C₉×#11758606	C₃	semidirect
7. C₉×#11758605	C₃	semidirect
8. C₉×#11758608	C₃	semidirect
9. #22582543	C₃	semidirect
10. #22582543	C₃	semidirect
11. #22582542	C₃	semidirect
12. #22582542	C₃	semidirect
16. #22582537	C₃	semidirect
17. #22582536	C₃	semidirect
18. #22582535	C₃	semidirect
19. #22582535	C₃	semidirect
20. C₃×#15688504	C₃	semidirect
21. C₉⋊C₉×C₃₇⋊C₃	C₃	semidirect
22. C₉×3₋¹⁺²×C₃₇	C₃	semidirect
23. #15688532	C₉	semidirect
84. 3₋¹⁺²×C₃₇	C₃×C₉	semidirect
93. #11758608	C₃×C₉	semidirect
96. C₃×#66524	C₃×C₉	semidirect
111. #11758606	C₃×C₉	semidirect
112. #11758605	C₃×C₉	semidirect
116. C₉×C₃₇⋊C₃	C₃×C₉	semidirect
159. C₉×3₋¹⁺²	C₃₇⋊C₃	semidirect
234. C₃₇	#10624864	semidirect
240. 3₋¹⁺²	C₉×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×#15688532	3
#22582548	3
3₋¹⁺²×#66524	3
C₉×#11758606	3
C₉×#11758605	3
C₉×#11758608	3
#22582543	3
#22582542	3
C₉×C₃²⋊(C₃₇⋊C₃)	3
#22582539	3
#22582537	3
#22582536	3
#22582535	3
C₃×#15688504	3
C₉⋊C₉×C₃₇⋊C₃	3
C₉×3₋¹⁺²×C₃₇	3
#10624864	37