Subgroups — #41007688

Order 26973.

698 conjugacy classes of subgroups.

The subgroup lattice has 310 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	#22582605	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582604	C₃	1	1	yes	maximal, characteristic, complemented
4	#22582600	C₃	1	1	yes	maximal, characteristic, complemented
5	#22582599	C₃	1	1	yes	maximal, characteristic, complemented
6	#22582596	C₃	1	1	yes	maximal, characteristic, complemented
7	#22582595	C₃	1	1	yes	maximal, characteristic, complemented
8	#22582594	C₃	1	1	yes	maximal, characteristic, complemented
9	#22582594	C₃	1	1	yes	maximal, characteristic, complemented
10	#22582582	C₃	1	3	yes	maximal, complemented
11	#22582582	C₃	1	3	yes	maximal, complemented
12	#22582580	C₃	1	3	yes	maximal, complemented
13	#22582570	C₃	1	3	yes	maximal, complemented
14	#22582570	C₃	1	3	yes	maximal, complemented
15	#22582569	C₃	1	3	yes	maximal, complemented
16	#22582569	C₃	1	3	yes	maximal, complemented
17	#22582568	C₃	1	3	yes	maximal, complemented
18	#22582555	C₃	1	3	yes	maximal
19	#22582537	C₃	1	1	yes	maximal, characteristic, complemented
20	#22582536	C₃	1	1	yes	maximal, characteristic, complemented
21	C₉⋊C₉×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
22	C₃₇×#6900	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	1	yes	characteristic
26	C₃×#11758605	C₃²	1	1	yes	characteristic
27	C₃×#11758605	C₃²	1	1	yes	characteristic
28	C₃×#11758605	C₃²	1	3	yes	—
29	C₃×#11758606	C₃²	1	1	yes	characteristic
30	C₃×#11758606	C₃²	1	1	yes	characteristic
31	C₃×#11758606	C₃²	1	1	yes	characteristic
32	C₃×#11758606	C₃²	1	3	yes	—
33	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	1	yes	characteristic
34	C₃²×#66524	C₃²	1	1	yes	characteristic
35	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
36	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
37	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	—
38	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
39	#15688506	C₃²	1	1	yes	characteristic
40	#15688506	C₃²	1	1	yes	characteristic
41	#15688506	C₃²	1	3	yes	—
42	#15688505	C₃²	1	1	yes	characteristic
43	#15688505	C₃²	1	1	yes	characteristic
44	#15688505	C₃²	1	3	yes	—
45	#15688504	C₃²	1	3	yes	—
46	#15688504	C₃²	1	3	yes	—
47	#15688504	C₃²	1	3	yes	—
48	#15688504	C₃²	1	3	yes	—
49	#15688504	C₃²	1	3	yes	—
50	#15688504	C₃²	1	3	yes	—
51	#15688504	C₃²	1	3	yes	—
52	#15688504	C₃²	1	3	yes	—
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	#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	C₃²⋊#66524	C₃²	1	3	yes	—
65	C₃²⋊#66524	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	#15688502	C₃²	1	3	yes	—
72	C₉×#66524	C₃²	1	1	yes	characteristic
73	C₉×#66524	C₃²	1	1	yes	characteristic
74	C₉×#66524	C₃²	1	3	yes	—
75	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
76	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
77	C₃₇×C₉⋊C₉	C₃²	1	1	yes	characteristic
78	C₃₇×C₉⋊C₉	C₃²	1	1	yes	characteristic
79	C₃₇×C₉⋊C₉	C₃²	1	3	yes	—
80	C₃₇×C₉⋊C₉	C₃²	1	3	yes	—
81	C₉²×C₃₇	C₃²	1	1	yes	characteristic
82	C₃³×C₃₇	C₃³	1	1	yes	characteristic
83	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
84	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
85	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
86	3₋¹⁺²×C₃₇	—	3	3	no	—
87	3₋¹⁺²×C₃₇	—	9	3	no	—
88	C₃×C₉×C₃₇	C₃³	1	3	yes	—
89	C₃×C₉×C₃₇	C₃³	1	3	yes	—
90	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
91	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
92	C₃×C₉×C₃₇	C₃³	1	3	yes	—
93	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
94	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
95	#11758608	—	3	3	no	—
96	C₃×#66524	—	3	3	no	—
97	C₃×#66524	C₃³	1	3	yes	—
98	C₃×#66524	C₃³	1	3	yes	—
99	C₃×#66524	C₃³	1	3	yes	—
100	C₃×#66524	C₃³	1	3	yes	—
101	C₃×#66524	C₃³	1	3	yes	—
102	C₃×#66524	C₃³	1	3	yes	—
103	C₃×#66524	C₃³	1	1	yes	characteristic
104	C₃×#66524	C₃³	1	1	yes	characteristic
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	#11758606	—	3	3	no	—
110	#11758606	—	3	3	no	—
111	#11758606	—	3	3	no	—
112	#11758606	—	9	3	no	—
113	#11758605	—	3	3	no	—
114	#11758605	—	3	3	no	—
115	#11758605	—	3	3	no	—
116	#11758605	—	9	3	no	—
117	C₉×C₃₇⋊C₃	—	3	3	no	—
118	C₉×C₃₇⋊C₃	—	3	3	no	—
119	C₉×C₃₇⋊C₃	—	9	3	no	—
120	C₉×C₃₇⋊C₃	3₋¹⁺²	1	9	yes	complemented
121	#10624887	—	37	1	no	maximal, Sylow
122	C₃²×C₃₇	—	3	1	no	—
123	C₃²×C₃₇	—	3	3	no	—
124	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
125	C₃×C₃₇⋊C₃	—	3	3	no	—
126	C₃×C₃₇⋊C₃	—	3	1	no	—
127	C₃×C₃₇⋊C₃	—	3	1	no	—
128	C₃×C₃₇⋊C₃	—	3	3	no	—
129	C₃×C₃₇⋊C₃	—	3	3	no	—
130	C₃×C₃₇⋊C₃	C₃×3₋¹⁺²	1	3	yes	—
131	C₉×C₃₇	—	3	1	no	—
132	C₉×C₃₇	—	3	1	no	—
133	C₉×C₃₇	—	3	3	no	—
134	C₉×C₃₇	—	3	3	no	—
135	C₉×C₃₇	—	3	3	no	—
136	C₉×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
137	#66524	—	3	1	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	3	no	—
145	#66524	—	3	3	no	—
146	#66524	—	3	3	no	—
147	#66524	C₃×3₋¹⁺²	1	3	yes	—
148	#66524	C₃×3₋¹⁺²	1	3	yes	—
149	C₃×#737	—	37	1	no	—
150	#6900	—	37	1	no	—
151	#6900	—	37	1	no	—
152	#6900	C₃₇⋊C₃	1	1	yes	characteristic, complemented
153	#6899	—	37	1	no	—
154	#6899	—	37	1	no	—
155	#6899	—	37	1	no	—
156	#6899	—	37	1	no	—
157	#6899	—	37	1	no	—
158	#6899	—	37	1	no	—
159	#6896	—	37	3	no	—
160	#6896	—	37	3	no	—
161	#6896	—	37	3	no	—
162	#6894	—	37	3	no	—
163	#6894	—	37	3	no	—
164	#6894	—	37	3	no	—
165	#6894	—	37	3	no	—
166	#6894	—	37	3	no	—
167	C₉⋊H(3,𝔽₃)	—	37	3	no	—
168	#6887	—	37	1	no	—
169	#6887	—	37	1	no	—
170	C₃×C₉⋊C₉	—	37	1	no	—
171	C₃×C₃₇	—	9	1	no	—
172	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
173	C₃×C₃₇	3₋¹⁺⁴	1	3	yes	—
174	C₃₇⋊C₃	—	3	3	no	—
175	C₃₇⋊C₃	—	9	1	no	—
176	C₃₇⋊C₃	—	9	1	no	—
177	#737	—	111	9	no	—
178	C₃×3₋¹⁺²	—	37	1	no	—
179	C₃×3₋¹⁺²	—	37	3	no	—
180	C₃×3₋¹⁺²	—	37	3	no	—
181	C₃×3₋¹⁺²	—	37	1	no	—
182	C₃×3₋¹⁺²	—	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	1	no	—
186	C₃×3₋¹⁺²	—	37	1	no	—
187	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
188	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	—
189	C₃×H(3,𝔽₃)	—	37	1	no	—
190	C₃²×C₉	—	37	1	no	—
191	C₃²×C₉	—	37	1	no	—
192	C₃²×C₉	—	37	3	no	—
193	C₃²×C₉	—	37	1	no	—
194	C₃²×C₉	—	37	1	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	—	37	3	no	—
213	C₉⋊C₉	—	37	3	no	—
214	C₉⋊C₉	—	37	1	no	—
215	C₉⋊C₉	—	37	1	no	—
216	C₉⋊C₉	—	37	3	no	—
217	C₉⋊C₉	—	37	1	no	—
218	C₉⋊C₉	—	37	1	no	—
219	C₉⋊C₉	—	37	3	no	—
220	C₉⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
221	C₉⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
222	C₉⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
223	C₉⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
224	C₃²⋊C₉	—	37	3	no	—
225	C₃²⋊C₉	—	37	3	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	C₉²	—	37	3	no	—
233	C₉²	—	37	1	no	—
234	C₉²	—	37	1	no	—
235	C₉²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
236	C₃₇	#10624887	1	1	yes	characteristic, complemented, Sylow
237	C₃³	—	37	1	no	—
238	C₃³	—	37	1	no	—
239	C₃³	—	37	1	no	—
240	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
241	3₋¹⁺²	—	3	3	no	—
242	3₋¹⁺²	—	9	3	no	—
243	3₋¹⁺²	—	333	3	no	—
244	3₋¹⁺²	—	333	3	no	—
245	3₋¹⁺²	—	111	3	no	—
246	3₋¹⁺²	—	111	3	no	—
247	3₋¹⁺²	—	111	3	no	—
248	3₋¹⁺²	—	111	3	no	—
249	3₋¹⁺²	—	111	3	no	—
250	3₋¹⁺²	—	111	3	no	—
251	3₋¹⁺²	—	111	3	no	—
252	H(3,𝔽₃)	—	111	3	no	—
253	C₃×C₉	—	37	1	no	—
254	C₃×C₉	—	37	1	no	—
255	C₃×C₉	—	37	9	no	—
256	C₃×C₉	—	37	3	no	—
257	C₃×C₉	—	37	3	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	1	no	—
263	C₃×C₉	—	37	1	no	—
264	C₃×C₉	—	37	1	no	—
265	C₃×C₉	—	37	1	no	—
266	C₃×C₉	—	333	3	no	—
267	C₃×C₉	—	111	3	no	—
268	C₃×C₉	—	111	3	no	—
269	C₃×C₉	—	111	3	no	—
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	3	yes	—
275	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
276	C₃²	—	3	1	no	—
277	C₃²	—	3	3	no	—
278	C₃²	—	37	3	no	—
279	C₃²	—	111	1	no	—
280	C₃²	—	111	1	no	—
281	C₃²	—	111	3	no	—
282	C₃²	—	111	3	no	—
283	C₃²	—	111	3	no	—
284	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
285	C₉	—	3	1	no	—
286	C₉	—	3	1	no	—
287	C₉	—	3	3	no	—
288	C₉	—	3	3	no	—
289	C₉	—	3	3	no	—
290	C₉	—	37	3	no	—
291	C₉	—	37	3	no	—
292	C₉	—	111	1	no	—
293	C₉	—	111	1	no	—
294	C₉	—	111	1	no	—
295	C₉	—	111	1	no	—
296	C₉	—	111	3	no	—
297	C₉	—	111	3	no	—
298	C₉	—	111	3	no	—
299	C₉	—	111	3	no	—
300	C₉	—	111	3	no	—
301	C₉	—	111	3	no	—
302	C₉	3₋¹⁺²×C₃₇⋊C₃	1	3	yes	—
303	C₃	—	9	1	no	—
304	C₃	—	333	1	no	—
305	C₃	—	333	1	no	—
306	C₃	—	111	3	no	—
307	C₃	C₃×3₋¹⁺²×C₃₇⋊C₃	1	1	yes	characteristic
308	C₃	#22582679	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting	C₃₇×#6900	C₃
prime radical	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²
hypercenter	#6900	C₃₇⋊C₃
hypocenter	C₃₇	#10624887
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
2. #22582605	C₃	semidirect
3. #22582604	C₃	semidirect
4. #22582600	C₃	semidirect
5. #22582599	C₃	semidirect
6. #22582596	C₃	semidirect
7. #22582595	C₃	semidirect
8. #22582594	C₃	semidirect
9. #22582594	C₃	semidirect
10. #22582582	C₃	semidirect
11. #22582582	C₃	semidirect
12. #22582580	C₃	semidirect
13. #22582570	C₃	semidirect
14. #22582570	C₃	semidirect
15. #22582569	C₃	semidirect
16. #22582569	C₃	semidirect
17. #22582568	C₃	semidirect
19. #22582537	C₃	semidirect
20. #22582536	C₃	semidirect
21. C₉⋊C₉×C₃₇⋊C₃	C₃	semidirect
22. C₃₇×#6900	C₃	semidirect
120. C₉×C₃₇⋊C₃	3₋¹⁺²	semidirect
152. #6900	C₃₇⋊C₃	semidirect
236. C₃₇	#10624887	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
#22582605	3
#22582604	3
#22582600	3
#22582599	3
#22582596	3
#22582595	3
#22582594	3
#22582582	3
#22582580	3
#22582570	3
#22582569	3
#22582568	3
#22582555	3
#22582537	3
#22582536	3
C₉⋊C₉×C₃₇⋊C₃	3
C₃₇×#6900	3
#10624887	37