Subgroups — #41007969

Order 26973.

622 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	#22582600	C₃	1	3	yes	maximal, complemented
2	#22582599	C₃	1	3	yes	maximal, complemented
3	#22582592	C₃	1	3	yes	maximal, complemented
4	#22582589	C₃	1	3	yes	maximal, complemented
5	#22582588	C₃	1	3	yes	maximal, complemented
6	#22582583	C₃	1	3	yes	maximal, complemented
7	#22582577	C₃	1	3	yes	maximal, complemented
8	#22582575	C₃	1	1	yes	maximal, characteristic, complemented
9	#22582574	C₃	1	1	yes	maximal, characteristic, complemented
10	#22582573	C₃	1	1	yes	maximal, characteristic, complemented
11	#22582568	C₃	1	1	yes	maximal, characteristic, complemented
12	#22582567	C₃	1	3	yes	maximal, complemented
13	#22582565	C₃	1	1	yes	maximal, characteristic
14	#22582561	C₃	1	1	yes	maximal, characteristic
15	#22582559	C₃	1	1	yes	maximal, characteristic
16	3₋¹⁺²×#66524	C₃	1	1	yes	maximal, characteristic, complemented
17	C₉×#11758608	C₃	1	1	yes	maximal, characteristic, complemented
18	#22582542	C₃	1	3	yes	maximal, complemented
19	#22582539	C₃	1	1	yes	maximal, characteristic
20	#22582535	C₃	1	1	yes	maximal, characteristic, complemented
21	C₃×#15688505	C₃	1	1	yes	maximal, characteristic, complemented
22	C₃₇×#6894	C₃	1	1	yes	maximal, characteristic, complemented
23	C₃×#11758608	C₃²	1	1	yes	characteristic
24	C₃×#11758605	C₃²	1	3	yes	—
25	C₃×#11758606	C₃²	1	3	yes	—
26	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	1	yes	characteristic
27	C₃²×#66524	C₃²	1	1	yes	characteristic
28	C₃²×#66524	C₃²	1	1	yes	characteristic
29	C₃²×#66524	C₃²	1	1	yes	characteristic
30	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	—
31	#15688506	C₃²	1	3	yes	—
32	#15688506	C₃²	1	3	yes	—
33	#15688506	C₃²	1	3	yes	—
34	#15688505	C₃²	1	1	yes	characteristic
35	#15688505	C₃²	1	1	yes	characteristic
36	#15688505	C₃²	1	1	yes	characteristic
37	#15688505	C₃²	1	1	yes	characteristic
38	#15688505	C₃²	1	1	yes	characteristic
39	#15688505	C₃²	1	1	yes	characteristic
40	#15688505	C₃²	1	1	yes	characteristic
41	#15688505	C₃²	1	1	yes	characteristic
42	#15688505	C₃²	1	1	yes	characteristic
43	#15688505	C₃²	1	3	yes	—
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	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	#15688502	C₃²	1	3	yes	—
69	#15688502	C₃²	1	3	yes	—
70	#15688502	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	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
75	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
76	C₃₇×C₉⋊C₉	C₃²	1	3	yes	—
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	C₃³×C₃₇	C₃³	1	1	yes	characteristic
82	C₃²×C₃₇⋊C₃	C₃³	1	3	yes	—
83	3₋¹⁺²×C₃₇	—	3	3	no	—
84	C₃×C₉×C₃₇	—	3	3	no	—
85	C₃×C₉×C₃₇	C₃³	1	3	yes	—
86	C₃×C₉×C₃₇	C₃³	1	3	yes	—
87	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
88	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
89	C₃×C₉×C₃₇	C₃³	1	3	yes	—
90	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
91	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
92	#11758608	—	3	3	no	—
93	C₃×#66524	—	3	3	no	—
94	C₃×#66524	—	9	1	no	—
95	C₃×#66524	—	9	1	no	—
96	C₃×#66524	C₃³	1	1	yes	characteristic
97	C₃×#66524	C₃³	1	1	yes	characteristic
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	C₃×#66524	C₃³	1	1	yes	characteristic
110	#11758606	—	3	9	no	—
111	#11758605	—	9	3	no	—
112	C₉×C₃₇⋊C₃	—	9	3	no	—
113	#10624907	—	37	1	no	maximal, Sylow
114	C₃²×C₃₇	—	3	1	no	—
115	C₃²×C₃₇	—	3	1	no	—
116	C₃²×C₃₇	—	3	1	no	—
117	C₃²×C₃₇	#737	1	3	yes	—
118	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
119	C₃×C₃₇⋊C₃	—	3	3	no	—
120	C₃×C₃₇⋊C₃	—	3	3	no	—
121	C₃×C₃₇⋊C₃	—	3	3	no	—
122	C₃×C₃₇⋊C₃	—	3	3	no	—
123	C₉×C₃₇	—	3	1	no	—
124	C₉×C₃₇	—	3	1	no	—
125	C₉×C₃₇	—	3	3	no	—
126	C₉×C₃₇	—	3	3	no	—
127	C₉×C₃₇	—	3	3	no	—
128	C₉×C₃₇	#737	1	3	yes	—
129	#66524	—	3	3	no	—
130	#66524	—	3	3	no	—
131	#66524	—	3	1	no	—
132	#66524	—	3	1	no	—
133	#66524	—	3	1	no	—
134	#66524	—	3	3	no	—
135	#66524	—	3	1	no	—
136	#66524	—	3	3	no	—
137	#66524	—	3	1	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	#6899	—	37	3	no	—
144	#6899	—	37	3	no	—
145	#6898	—	37	3	no	—
146	#6897	—	37	3	no	—
147	#6897	—	37	3	no	—
148	#6896	—	37	3	no	—
149	#6895	—	37	3	no	—
150	#6895	—	37	1	no	—
151	#6894	—	37	1	no	—
152	#6894	—	37	3	no	—
153	#6894	—	37	1	no	—
154	#6894	—	37	1	no	—
155	#6894	C₃₇⋊C₃	1	1	yes	characteristic, complemented
156	#6893	—	37	1	no	—
157	C₉⋊H(3,𝔽₃)	—	37	1	no	—
158	C₉⋊H(3,𝔽₃)	—	37	1	no	—
159	C₉×3₋¹⁺²	—	37	1	no	—
160	C₉×3₋¹⁺²	—	37	1	no	—
161	C₉×3₋¹⁺²	—	37	3	no	—
162	C₉×H(3,𝔽₃)	—	37	1	no	—
163	#6887	—	37	1	no	—
164	C₃×C₉⋊C₉	—	37	1	no	—
165	C₃×C₃₇	—	3	3	no	—
166	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
167	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
168	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
169	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
170	C₃₇⋊C₃	—	9	3	no	—
171	C₃×3₋¹⁺²	—	37	1	no	—
172	C₃×3₋¹⁺²	—	37	3	no	—
173	C₃×3₋¹⁺²	—	37	3	no	—
174	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
175	C₃×H(3,𝔽₃)	—	37	1	no	—
176	C₃²×C₉	—	37	1	no	—
177	C₃²×C₉	—	37	1	no	—
178	C₃²×C₉	—	37	1	no	—
179	C₃²×C₉	—	37	3	no	—
180	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
181	C₉⋊C₉	—	37	3	no	—
182	C₉⋊C₉	—	37	3	no	—
183	C₉⋊C₉	—	37	3	no	—
184	C₉⋊C₉	—	37	3	no	—
185	C₉⋊C₉	—	37	3	no	—
186	C₉⋊C₉	—	37	3	no	—
187	C₉⋊C₉	—	37	3	no	—
188	C₉⋊C₉	—	37	3	no	—
189	C₉⋊C₉	—	37	3	no	—
190	C₉⋊C₉	—	37	3	no	—
191	C₉⋊C₉	—	37	3	no	—
192	C₉⋊C₉	—	37	3	no	—
193	C₉⋊C₉	—	37	3	no	—
194	C₉⋊C₉	—	37	1	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	1	no	—
200	C₉⋊C₉	—	37	1	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	C₃×C₃₇⋊C₃	1	3	yes	—
209	C₉⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
210	C₉⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
211	C₃²⋊C₉	—	37	1	no	—
212	C₃²⋊C₉	—	37	1	no	—
213	C₃²⋊C₉	—	37	1	no	—
214	C₃²⋊C₉	—	37	1	no	—
215	C₃²⋊C₉	—	37	3	no	—
216	C₃²⋊C₉	—	37	3	no	—
217	C₃²⋊C₉	—	37	3	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	C₃₇	#10624907	1	1	yes	characteristic, complemented, Sylow
230	C₃³	—	37	3	no	—
231	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
232	3₋¹⁺²	—	3	3	no	—
233	3₋¹⁺²	—	333	3	no	—
234	3₋¹⁺²	—	111	3	no	—
235	3₋¹⁺²	—	111	9	no	—
236	H(3,𝔽₃)	—	111	3	no	—
237	C₃×C₉	—	3	3	no	—
238	C₃×C₉	—	37	1	no	—
239	C₃×C₉	—	37	1	no	—
240	C₃×C₉	—	37	3	no	—
241	C₃×C₉	—	37	3	no	—
242	C₃×C₉	—	37	3	no	—
243	C₃×C₉	—	37	3	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	1	no	—
249	C₃×C₉	—	37	1	no	—
250	C₃×C₉	—	37	1	no	—
251	C₃×C₉	—	37	1	no	—
252	C₃×C₉	—	333	3	no	—
253	C₃×C₉	—	333	1	no	—
254	C₃×C₉	—	333	1	no	—
255	C₃×C₉	—	111	3	no	—
256	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
257	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
258	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
259	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
260	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
261	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
262	C₃²	—	3	1	no	—
263	C₃²	—	3	1	no	—
264	C₃²	—	3	1	no	—
265	C₃²	—	111	3	no	—
266	C₃²	—	111	3	no	—
267	C₃²	—	111	3	no	—
268	C₃²	—	111	3	no	—
269	C₃²	#15688532	1	3	yes	—
270	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
271	C₉	—	3	1	no	—
272	C₉	—	3	1	no	—
273	C₉	—	3	3	no	—
274	C₉	—	3	3	no	—
275	C₉	—	3	3	no	—
276	C₉	—	111	1	no	—
277	C₉	—	111	1	no	—
278	C₉	—	111	1	no	—
279	C₉	—	111	3	no	—
280	C₉	—	111	3	no	—
281	C₉	—	111	3	no	—
282	C₉	—	111	1	no	—
283	C₉	—	111	1	no	—
284	C₉	—	111	3	no	—
285	C₉	—	111	1	no	—
286	C₉	—	111	1	no	—
287	C₉	—	111	1	no	—
288	C₉	—	111	1	no	—
289	C₉	—	111	3	no	—
290	C₉	#15688532	1	3	yes	—
291	C₃	—	3	3	no	—
292	C₃	—	333	3	no	—
293	C₃	C₃×#15688532	1	1	yes	characteristic
294	C₃	C₃×#15688532	1	1	yes	characteristic
295	C₃	#22582678	1	1	yes	characteristic
296	C₃	#22582678	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	#10624907
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
1. #22582600	C₃	semidirect
2. #22582599	C₃	semidirect
3. #22582592	C₃	semidirect
4. #22582589	C₃	semidirect
5. #22582588	C₃	semidirect
6. #22582583	C₃	semidirect
7. #22582577	C₃	semidirect
8. #22582575	C₃	semidirect
9. #22582574	C₃	semidirect
10. #22582573	C₃	semidirect
11. #22582568	C₃	semidirect
12. #22582567	C₃	semidirect
16. 3₋¹⁺²×#66524	C₃	semidirect
17. C₉×#11758608	C₃	semidirect
18. #22582542	C₃	semidirect
20. #22582535	C₃	semidirect
21. C₃×#15688505	C₃	semidirect
22. C₃₇×#6894	C₃	semidirect
155. #6894	C₃₇⋊C₃	semidirect
229. C₃₇	#10624907	semidirect

Maximal subgroups

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

subgroup	index
#22582600	3
#22582599	3
#22582592	3
#22582589	3
#22582588	3
#22582583	3
#22582577	3
#22582575	3
#22582574	3
#22582573	3
#22582568	3
#22582567	3
#22582565	3
#22582561	3
#22582559	3
3₋¹⁺²×#66524	3
C₉×#11758608	3
#22582542	3
#22582539	3
#22582535	3
C₃×#15688505	3
C₃₇×#6894	3
#10624907	37