Subgroups — #41007862

Order 26973.

730 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	#22582577	C₃	1	3	yes	maximal, complemented
3	#22582577	C₃	1	3	yes	maximal, complemented
4	#22582576	C₃	1	1	yes	maximal, characteristic, complemented
5	#22582576	C₃	1	1	yes	maximal, characteristic, complemented
6	#22582573	C₃	1	1	yes	maximal, characteristic, complemented
7	#22582573	C₃	1	1	yes	maximal, characteristic, complemented
8	#22582572	C₃	1	1	yes	maximal, characteristic, complemented
9	#22582570	C₃	1	3	yes	maximal, complemented
10	#22582570	C₃	1	3	yes	maximal, complemented
11	#22582570	C₃	1	3	yes	maximal, complemented
12	#22582569	C₃	1	3	yes	maximal, complemented
13	#22582569	C₃	1	3	yes	maximal, complemented
14	#22582568	C₃	1	1	yes	maximal, characteristic, complemented
15	#22582568	C₃	1	1	yes	maximal, characteristic, complemented
16	#22582566	C₃	1	3	yes	maximal, complemented
17	C₉×C₃²⋊(C₃₇⋊C₃)	C₃	1	1	yes	maximal, characteristic
18	#22582539	C₃	1	1	yes	maximal, characteristic
19	#22582539	C₃	1	1	yes	maximal, characteristic
20	C₃×#15688504	C₃	1	1	yes	maximal, characteristic, complemented
21	C₉⋊C₉×C₃₇⋊C₃	C₃	1	3	yes	maximal, complemented
22	C₃₇×#6894	C₃	1	1	yes	maximal, characteristic, complemented
23	#15688532	—	3	9	no	—
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	#15688506	C₃²	1	3	yes	—
38	#15688505	C₃²	1	3	yes	—
39	#15688505	C₃²	1	3	yes	—
40	#15688504	C₃²	1	1	yes	characteristic
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	3	yes	—
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	1	yes	characteristic
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	#15688504	C₃²	1	3	yes	—
59	#15688504	C₃²	1	3	yes	—
60	#15688504	C₃²	1	3	yes	—
61	#15688504	C₃²	1	3	yes	—
62	#15688504	C₃²	1	3	yes	—
63	#15688504	C₃²	1	3	yes	—
64	#15688504	C₃²	1	3	yes	—
65	C₃²⋊#66524	C₃²	1	1	yes	characteristic
66	C₃²⋊#66524	C₃²	1	1	yes	characteristic
67	C₃²⋊#66524	C₃²	1	1	yes	characteristic
68	C₃²⋊#66524	C₃²	1	1	yes	characteristic
69	#15688502	C₃²	1	3	yes	—
70	#15688502	C₃²	1	3	yes	—
71	#15688502	C₃²	1	3	yes	—
72	#15688502	C₃²	1	3	yes	—
73	#15688502	C₃²	1	3	yes	—
74	#15688502	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×C₉⋊C₉	C₃²	1	3	yes	—
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	C₃³	1	3	yes	—
84	3₋¹⁺²×C₃₇	—	3	3	no	—
85	C₃×C₉×C₃₇	—	9	1	no	—
86	C₃×C₉×C₃₇	C₃³	1	3	yes	—
87	C₃×C₉×C₃₇	C₃³	1	3	yes	—
88	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
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)	3₋¹⁺²	1	9	yes	—
93	#11758608	—	3	3	no	—
94	C₃×#66524	—	3	3	no	—
95	C₃×#66524	—	3	3	no	—
96	C₃×#66524	—	9	1	no	—
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	1	yes	characteristic
108	C₃×#66524	C₃³	1	1	yes	characteristic
109	C₃×#66524	C₃³	1	3	yes	—
110	C₃×#66524	C₃³	1	1	yes	characteristic
111	#11758606	—	3	9	no	—
112	#11758605	—	3	9	no	—
113	C₉×C₃₇⋊C₃	—	3	9	no	—
114	C₉×C₃₇⋊C₃	—	3	9	no	—
115	C₉×C₃₇⋊C₃	—	9	3	no	—
116	C₉×C₃₇⋊C₃	—	9	3	no	—
117	#10624900	—	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	C₃×3₋¹⁺²	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	C₃×3₋¹⁺²	1	9	yes	—
127	C₉×C₃₇	—	3	3	no	—
128	C₉×C₃₇	—	3	3	no	—
129	C₉×C₃₇	—	3	1	no	—
130	C₉×C₃₇	—	3	3	no	—
131	C₉×C₃₇	—	3	1	no	—
132	C₉×C₃₇	—	3	1	no	—
133	#66524	—	3	1	no	—
134	#66524	—	3	3	no	—
135	#66524	—	3	1	no	—
136	#66524	—	3	1	no	—
137	#66524	—	3	3	no	—
138	#66524	—	3	1	no	—
139	#66524	—	3	1	no	—
140	#66524	—	3	1	no	—
141	#66524	—	3	3	no	—
142	#66524	—	3	3	no	—
143	#66524	—	3	3	no	—
144	#66524	—	3	1	no	—
145	#66524	—	3	1	no	—
146	#66524	#737	1	3	yes	complemented
147	C₃×#737	—	37	1	no	—
148	#6895	—	37	3	no	—
149	#6895	—	37	3	no	—
150	#6895	—	37	1	no	—
151	#6895	—	37	1	no	—
152	#6894	—	37	1	no	—
153	#6894	—	37	1	no	—
154	#6894	—	37	1	no	—
155	#6894	—	37	3	no	—
156	#6894	—	37	3	no	—
157	#6894	—	37	3	no	—
158	#6894	—	37	3	no	—
159	#6894	—	37	3	no	—
160	#6894	—	37	3	no	—
161	#6894	—	37	1	no	—
162	#6894	—	37	1	no	—
163	#6894	C₃₇⋊C₃	1	1	yes	characteristic, complemented
164	C₉×H(3,𝔽₃)	—	37	1	no	—
165	C₉×H(3,𝔽₃)	—	37	1	no	—
166	C₉×H(3,𝔽₃)	—	37	1	no	—
167	C₃×C₉⋊C₉	—	37	1	no	—
168	C₃×C₉⋊C₉	—	37	3	no	—
169	C₃×C₃₇	—	3	3	no	—
170	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
171	C₃×C₃₇	C₃×#737	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	—	111	9	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	1	no	—
183	C₃²×C₉	—	37	1	no	—
184	C₃²×C₉	—	37	3	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	3	no	—
191	C₉⋊C₉	—	37	1	no	—
192	C₉⋊C₉	—	37	1	no	—
193	C₉⋊C₉	—	37	3	no	—
194	C₉⋊C₉	—	37	3	no	—
195	C₉⋊C₉	—	37	3	no	—
196	C₉⋊C₉	—	37	1	no	—
197	C₉⋊C₉	—	37	1	no	—
198	C₉⋊C₉	—	37	1	no	—
199	C₉⋊C₉	—	37	1	no	—
200	C₉⋊C₉	—	37	1	no	—
201	C₉⋊C₉	—	37	1	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	C₃×C₃₇⋊C₃	1	3	yes	—
220	C₉⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
221	C₉⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
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	3	no	—
227	C₃²⋊C₉	—	37	3	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	C₃₇	#10624900	1	1	yes	characteristic, complemented, Sylow
235	C₃³	—	37	3	no	—
236	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
237	3₋¹⁺²	—	3	3	no	—
238	3₋¹⁺²	—	111	9	no	—
239	3₋¹⁺²	—	111	9	no	—
240	3₋¹⁺²	—	111	3	no	—
241	H(3,𝔽₃)	—	37	9	no	—
242	C₃×C₉	—	9	1	no	—
243	C₃×C₉	—	37	1	no	—
244	C₃×C₉	—	37	1	no	—
245	C₃×C₉	—	37	1	no	—
246	C₃×C₉	—	37	3	no	—
247	C₃×C₉	—	37	1	no	—
248	C₃×C₉	—	37	1	no	—
249	C₃×C₉	—	37	3	no	—
250	C₃×C₉	—	37	3	no	—
251	C₃×C₉	—	37	3	no	—
252	C₃×C₉	—	37	1	no	—
253	C₃×C₉	—	37	1	no	—
254	C₃×C₉	—	37	1	no	—
255	C₃×C₉	—	37	1	no	—
256	C₃×C₉	—	37	3	no	—
257	C₃×C₉	—	333	3	no	—
258	C₃×C₉	—	333	3	no	—
259	C₃×C₉	—	333	1	no	—
260	C₃×C₉	—	111	9	no	—
261	C₃×C₉	—	111	9	no	—
262	C₃×C₉	—	111	3	no	—
263	C₃×C₉	—	111	3	no	—
264	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
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	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	3₋¹⁺²×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	3	no	—
282	C₉	—	3	1	no	—
283	C₉	—	3	1	no	—
284	C₉	—	3	1	no	—
285	C₉	—	37	3	no	—
286	C₉	—	111	1	no	—
287	C₉	—	111	3	no	—
288	C₉	—	111	1	no	—
289	C₉	—	111	1	no	—
290	C₉	—	111	3	no	—
291	C₉	—	111	1	no	—
292	C₉	—	111	1	no	—
293	C₉	—	111	1	no	—
294	C₉	—	111	3	no	—
295	C₉	—	111	1	no	—
296	C₉	—	111	1	no	—
297	C₉	—	111	3	no	—
298	C₉	—	111	3	no	—
299	C₃	—	3	3	no	—
300	C₃	—	111	9	no	—
301	C₃	C₃×3₋¹⁺²×C₃₇⋊C₃	1	1	yes	characteristic
302	C₃	#737×C₃₇⋊C₃	1	1	yes	characteristic
303	C₃	#22582677	1	1	yes	characteristic
304	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	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²
hypercenter	#6894	C₃₇⋊C₃
hypocenter	C₃₇	#10624900
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. #22582577	C₃	semidirect
3. #22582577	C₃	semidirect
4. #22582576	C₃	semidirect
5. #22582576	C₃	semidirect
6. #22582573	C₃	semidirect
7. #22582573	C₃	semidirect
8. #22582572	C₃	semidirect
9. #22582570	C₃	semidirect
10. #22582570	C₃	semidirect
11. #22582570	C₃	semidirect
12. #22582569	C₃	semidirect
13. #22582569	C₃	semidirect
14. #22582568	C₃	semidirect
15. #22582568	C₃	semidirect
16. #22582566	C₃	semidirect
20. C₃×#15688504	C₃	semidirect
21. C₉⋊C₉×C₃₇⋊C₃	C₃	semidirect
22. C₃₇×#6894	C₃	semidirect
146. #66524	#737	semidirect
163. #6894	C₃₇⋊C₃	semidirect
234. C₃₇	#10624900	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
#22582577	3
#22582576	3
#22582573	3
#22582572	3
#22582570	3
#22582569	3
#22582568	3
#22582566	3
C₉×C₃²⋊(C₃₇⋊C₃)	3
#22582539	3
C₃×#15688504	3
C₉⋊C₉×C₃₇⋊C₃	3
C₃₇×#6894	3
#10624900	37