Subgroups — #41008225

Order 26973.

948 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582677	C₃	1	3	yes	maximal, complemented
2	C₃×#15688532	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582663	C₃	1	2	yes	maximal, complemented
4	#22582661	C₃	1	2	yes	maximal, complemented
5	#22582660	C₃	1	3	yes	maximal, complemented
6	#22582659	C₃	1	3	yes	maximal, complemented
7	#22582658	C₃	1	1	yes	maximal, characteristic, complemented
8	#22582655	C₃	1	3	yes	maximal, complemented
9	#22582653	C₃	1	3	yes	maximal, complemented
10	#22582649	C₃	1	2	yes	maximal, complemented
11	#22582645	C₃	1	1	yes	maximal, characteristic, complemented
12	#22582636	C₃	1	3	yes	maximal, complemented
13	#22582634	C₃	1	2	yes	maximal, complemented
14	#22582631	C₃	1	1	yes	maximal, characteristic, complemented
15	#22582630	C₃	1	3	yes	maximal, complemented
16	#22582629	C₃	1	3	yes	maximal, complemented
17	#731×C₃₇⋊C₃	C₃	1	3	yes	maximal, complemented
18	C₃₇×#6913	C₃	1	1	yes	maximal, characteristic, complemented
19	#15688532	—	3	6	no	—
20	#15688532	—	3	3	no	—
21	#15688532	—	3	6	no	—
22	#15688532	—	3	6	no	—
23	#15688532	C₃²	1	6	yes	complemented
24	#15688532	C₃²	1	6	yes	complemented
25	#15688531	—	3	6	no	—
26	#15688531	C₃²	1	6	yes	complemented
27	#15688530	C₃²	1	3	yes	complemented
28	#15688530	C₃²	1	3	yes	complemented
29	C₃×#11758608	C₃²	1	1	yes	characteristic, complemented
30	C₃×#11758605	C₃²	1	1	yes	characteristic
31	C₃×#11758605	C₃²	1	1	yes	characteristic, complemented
32	C₃×#11758605	C₃²	1	1	yes	characteristic, complemented
33	C₃×#11758606	—	3	3	no	—
34	C₃×#11758606	C₃²	1	1	yes	characteristic
35	C₃×#11758606	C₃²	1	1	yes	characteristic, complemented
36	C₃×#11758606	C₃²	1	1	yes	characteristic, complemented
37	3₋¹⁺²×C₃₇⋊C₃	—	3	6	no	—
38	3₋¹⁺²×C₃₇⋊C₃	C₃²	1	6	yes	—
39	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	1	yes	characteristic
40	H(3,𝔽₃)×C₃₇⋊C₃	C₃²	1	3	yes	—
41	C₃²×#66524	C₃²	1	1	yes	characteristic, complemented
42	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
43	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic, complemented
44	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic, complemented
45	#15688521	C₃²	1	2	yes	complemented
46	#15688521	C₃²	1	2	yes	complemented
47	#15688520	C₃²	1	6	yes	complemented
48	#15688520	C₃²	1	6	yes	complemented
49	#15688519	C₃²	1	2	yes	—
50	#15688518	C₃²	1	3	yes	—
51	#15688518	C₃²	1	3	yes	—
52	#15688517	C₃²	1	6	yes	complemented
53	#15688517	C₃²	1	6	yes	complemented
54	#15688516	C₃²	1	6	yes	complemented
55	#15688515	C₃²	1	3	yes	complemented
56	#15688515	C₃²	1	3	yes	complemented
57	#15688514	C₃²	1	6	yes	complemented
58	#15688514	C₃²	1	6	yes	complemented
59	#15688513	C₃²	1	6	yes	complemented
60	#15688512	C₃²	1	3	yes	complemented
61	#15688512	C₃²	1	3	yes	complemented
62	C₃₇×#737	C₃²	1	3	yes	complemented
63	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic, complemented
64	C₃₇×#733	C₃²	1	2	yes	complemented
65	C₃₇×#732	C₃²	1	1	yes	characteristic
66	C₃₇×#731	C₃²	1	3	yes	complemented
67	C₃₇×AΣL(1,𝔽₂₇)	C₃²	1	3	yes	complemented
68	C₃³×C₃₇	C₃³	1	1	yes	characteristic
69	C₃²×C₃₇⋊C₃	—	3	3	no	—
70	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
71	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
72	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
73	3₋¹⁺²×C₃₇	—	3	3	no	—
74	3₋¹⁺²×C₃₇	—	3	6	no	—
75	3₋¹⁺²×C₃₇	C₃³	1	6	yes	—
76	H(3,𝔽₃)×C₃₇	C₃³	1	3	yes	—
77	C₃×C₉×C₃₇	—	3	3	no	—
78	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
79	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
80	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
81	C₃²⋊(C₃₇⋊C₃)	—	3	6	no	—
82	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
83	C₃²⋊(C₃₇⋊C₃)	C₃³	1	6	yes	—
84	#11758608	—	3	6	no	—
85	#11758608	—	3	6	no	—
86	#11758608	—	3	2	no	—
87	#11758608	—	3	1	no	—
88	#11758608	C₃³	1	6	yes	—
89	#11758608	C₃³	1	6	yes	—
90	C₃×#66524	—	3	3	no	—
91	C₃×#66524	—	3	3	no	—
92	C₃×#66524	—	3	3	no	—
93	C₃×#66524	C₃³	1	1	yes	characteristic
94	C₃×#66524	C₃³	1	1	yes	characteristic
95	C₃×#66524	C₃³	1	1	yes	characteristic
96	C₃×#66524	C₃³	1	1	yes	characteristic
97	C₃×#66524	C₃³	1	1	yes	characteristic
98	C₃×#66524	C₃³	1	1	yes	characteristic
99	#11758606	—	3	6	no	—
100	#11758606	—	3	6	no	—
101	#11758606	—	3	6	no	—
102	#11758606	—	3	6	no	—
103	#11758606	—	3	3	no	—
104	#11758606	—	3	3	no	—
105	#11758606	H(3,𝔽₃)	1	6	yes	—
106	#11758606	H(3,𝔽₃)	1	3	yes	—
107	#11758605	—	3	6	no	—
108	#11758605	—	3	3	no	—
109	#11758605	—	3	6	no	—
110	#11758605	—	3	3	no	—
111	#11758605	—	3	6	no	—
112	#11758605	—	3	6	no	—
113	#11758605	—	3	2	no	—
114	#11758605	—	3	1	no	—
115	C₉×C₃₇⋊C₃	—	3	3	no	—
116	C₉×C₃₇⋊C₃	—	3	6	no	—
117	C₉×C₃₇⋊C₃	—	3	3	no	—
118	C₉×C₃₇⋊C₃	—	3	6	no	—
119	C₉×C₃₇⋊C₃	—	3	2	no	—
120	C₉×C₃₇⋊C₃	—	3	1	no	—
121	C₉×C₃₇⋊C₃	—	3	6	no	—
122	C₉×C₃₇⋊C₃	—	3	6	no	—
123	#10624935	—	37	1	no	maximal, Sylow
124	C₃²×C₃₇	—	3	1	no	—
125	C₃²×C₃₇	—	3	3	no	—
126	C₃²×C₃₇	—	9	1	no	—
127	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
128	C₃×C₃₇⋊C₃	—	3	1	no	—
129	C₃×C₃₇⋊C₃	—	3	1	no	—
130	C₃×C₃₇⋊C₃	—	3	6	no	—
131	C₃×C₃₇⋊C₃	—	9	1	no	—
132	C₃×C₃₇⋊C₃	—	9	1	no	—
133	C₃×C₃₇⋊C₃	—	9	1	no	—
134	C₃×C₃₇⋊C₃	—	27	3	no	—
135	C₃×C₃₇⋊C₃	C₃×H(3,𝔽₃)	1	3	yes	—
136	C₉×C₃₇	—	3	6	no	—
137	C₉×C₃₇	—	3	1	no	—
138	C₉×C₃₇	—	3	1	no	—
139	C₉×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
140	#66524	—	3	1	no	—
141	#66524	—	3	6	no	—
142	#66524	—	3	6	no	—
143	#66524	—	3	1	no	—
144	#66524	—	3	1	no	—
145	#66524	—	3	1	no	—
146	#66524	C₃×H(3,𝔽₃)	1	2	yes	—
147	#66524	C₃×H(3,𝔽₃)	1	3	yes	—
148	#66524	C₃×H(3,𝔽₃)	1	1	yes	characteristic
149	3₋¹⁺⁴	—	37	3	no	—
150	C₃×#737	—	37	1	no	—
151	#6913	—	37	3	no	—
152	#6913	—	37	2	no	—
153	#6913	—	37	2	no	—
154	#6913	—	37	1	no	—
155	#6913	—	37	3	no	—
156	#6913	—	37	3	no	—
157	#6913	—	37	3	no	—
158	#6913	C₃₇⋊C₃	1	1	yes	characteristic, complemented
159	#6912	—	37	2	no	—
160	#6911	—	37	1	no	—
161	#6909	—	37	3	no	—
162	#6908	—	37	3	no	—
163	#6908	—	37	2	no	—
164	#6908	—	37	1	no	—
165	#6908	—	37	3	no	—
166	C₃×#731	—	37	3	no	—
167	C₃×C₃₇	—	3	1	no	—
168	C₃×C₃₇	—	9	3	no	—
169	C₃×C₃₇	—	9	1	no	—
170	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
171	C₃₇⋊C₃	—	3	3	no	—
172	C₃₇⋊C₃	—	9	1	no	—
173	C₃₇⋊C₃	—	9	1	no	—
174	C₃₇⋊C₃	—	9	6	no	—
175	#737	—	37	6	no	—
176	#737	—	37	6	no	—
177	#737	—	37	3	no	—
178	#737	—	37	3	no	—
179	#737	—	37	6	no	—
180	#737	—	111	6	no	—
181	#737	—	111	6	no	—
182	#737	—	111	3	no	—
183	#737	—	111	6	no	—
184	#737	—	111	6	no	—
185	#737	C₃×C₃₇⋊C₃	1	3	yes	complemented
186	C₃×3₋¹⁺²	—	37	1	no	—
187	C₃×3₋¹⁺²	—	37	1	no	—
188	C₃×3₋¹⁺²	—	37	1	no	—
189	C₃×3₋¹⁺²	—	37	1	no	—
190	C₃×3₋¹⁺²	—	37	1	no	—
191	C₃×3₋¹⁺²	—	37	1	no	—
192	C₃×3₋¹⁺²	—	37	1	no	—
193	C₃×3₋¹⁺²	—	37	6	no	—
194	C₃×3₋¹⁺²	—	111	3	no	—
195	C₃×3₋¹⁺²	—	111	6	no	—
196	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic, complemented
197	C₃×H(3,𝔽₃)	—	37	3	no	—
198	C₃×H(3,𝔽₃)	—	37	1	no	—
199	C₃²×C₉	—	37	1	no	—
200	C₃²×C₉	—	37	1	no	—
201	C₃²×C₉	—	37	1	no	—
202	C₃²×C₉	—	37	1	no	—
203	#733	—	37	6	no	—
204	#733	—	37	6	no	—
205	#733	—	37	2	no	—
206	#733	—	37	2	no	—
207	#733	C₃×C₃₇⋊C₃	1	2	yes	complemented
208	#732	—	37	3	no	—
209	#732	—	37	3	no	—
210	#732	—	37	2	no	—
211	#732	C₃×C₃₇⋊C₃	1	1	yes	characteristic
212	#731	—	37	6	no	—
213	#731	—	37	6	no	—
214	#731	—	37	3	no	—
215	#731	—	37	3	no	—
216	#731	—	37	6	no	—
217	#731	C₃×C₃₇⋊C₃	1	3	yes	complemented
218	AΣL(1,𝔽₂₇)	—	37	6	no	—
219	AΣL(1,𝔽₂₇)	—	37	6	no	—
220	AΣL(1,𝔽₂₇)	—	37	6	no	—
221	AΣL(1,𝔽₂₇)	—	37	3	no	—
222	AΣL(1,𝔽₂₇)	—	37	3	no	—
223	AΣL(1,𝔽₂₇)	C₃×C₃₇⋊C₃	1	3	yes	complemented
224	C₃₇	#10624935	1	1	yes	characteristic, complemented, Sylow
225	C₃³	—	37	1	no	—
226	C₃³	—	37	1	no	—
227	C₃³	—	37	1	no	—
228	C₃³	—	111	3	no	—
229	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
230	3₋¹⁺²	—	3	6	no	—
231	3₋¹⁺²	—	3	3	no	—
232	3₋¹⁺²	—	37	6	no	—
233	3₋¹⁺²	—	37	3	no	—
234	3₋¹⁺²	—	37	6	no	—
235	3₋¹⁺²	—	37	6	no	—
236	3₋¹⁺²	—	111	2	no	—
237	3₋¹⁺²	—	111	1	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	3₋¹⁺²	—	111	6	no	—
243	3₋¹⁺²	—	111	6	no	—
244	3₋¹⁺²	—	111	6	no	—
245	3₋¹⁺²	—	111	3	no	—
246	3₋¹⁺²	—	111	6	no	—
247	3₋¹⁺²	—	111	2	no	—
248	3₋¹⁺²	—	111	1	no	—
249	3₋¹⁺²	—	111	6	no	—
250	3₋¹⁺²	—	111	6	no	—
251	3₋¹⁺²	—	111	3	no	—
252	3₋¹⁺²	—	111	6	no	—
253	3₋¹⁺²	—	111	6	no	—
254	3₋¹⁺²	C₃²×C₃₇⋊C₃	1	6	yes	—
255	H(3,𝔽₃)	—	37	6	no	—
256	H(3,𝔽₃)	—	111	6	no	—
257	H(3,𝔽₃)	—	111	3	no	—
258	H(3,𝔽₃)	C₃²×C₃₇⋊C₃	1	3	yes	—
259	C₃×C₉	—	3	3	no	—
260	C₃×C₉	—	37	1	no	—
261	C₃×C₉	—	37	1	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	—	111	2	no	—
267	C₃×C₉	—	111	1	no	—
268	C₃×C₉	—	111	6	no	—
269	C₃×C₉	—	111	6	no	—
270	C₃×C₉	—	111	3	no	—
271	C₃×C₉	—	111	6	no	—
272	C₃×C₉	—	111	3	no	—
273	C₃×C₉	—	111	3	no	—
274	C₃×C₉	—	111	3	no	—
275	C₃×C₉	—	111	3	no	—
276	C₃×C₉	—	111	6	no	—
277	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
278	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
279	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
280	C₃²	—	3	3	no	—
281	C₃²	—	3	1	no	—
282	C₃²	—	9	1	no	—
283	C₃²	—	37	3	no	—
284	C₃²	—	333	1	no	—
285	C₃²	—	333	1	no	—
286	C₃²	—	333	1	no	—
287	C₃²	—	111	6	no	—
288	C₃²	—	111	1	no	—
289	C₃²	—	111	1	no	—
290	C₃²	—	999	3	no	—
291	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
292	C₉	—	3	6	no	—
293	C₉	—	3	1	no	—
294	C₉	—	3	1	no	—
295	C₉	—	37	2	no	—
296	C₉	—	37	1	no	—
297	C₉	—	37	3	no	—
298	C₉	—	111	1	no	—
299	C₉	—	111	1	no	—
300	C₉	—	111	1	no	—
301	C₉	—	111	1	no	—
302	C₉	—	111	6	no	—
303	C₉	—	111	6	no	—
304	C₉	H(3,𝔽₃)×C₃₇⋊C₃	1	3	yes	—
305	C₃	—	3	1	no	—
306	C₃	—	9	1	no	—
307	C₃	—	9	3	no	—
308	C₃	—	333	1	no	—
309	C₃	—	333	1	no	—
310	C₃	—	333	6	no	—
311	C₃	—	111	3	no	—
312	C₃	C₃×H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₃×H(3,𝔽₃)×C₃₇⋊C₃
commutator, cosocle	C₃²×C₃₇	C₃⁴
Frattini	C₃²	C₃³×C₃₇⋊C₃
Fitting	C₃₇×#6913	C₃
socle	C₃×C₃₇	C₃²×H(3,𝔽₃)
hypercenter	#6913	C₃₇⋊C₃
hypocenter	C₃₇	#10624935
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ #6913

Complements

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

N	H ≅ G/N	type
1. #22582677	C₃	semidirect
2. C₃×#15688532	C₃	semidirect
3. #22582663	C₃	semidirect
4. #22582661	C₃	semidirect
5. #22582660	C₃	semidirect
6. #22582659	C₃	semidirect
7. #22582658	C₃	semidirect
8. #22582655	C₃	semidirect
9. #22582653	C₃	semidirect
10. #22582649	C₃	semidirect
11. #22582645	C₃	semidirect
12. #22582636	C₃	semidirect
13. #22582634	C₃	semidirect
14. #22582631	C₃	semidirect
15. #22582630	C₃	semidirect
16. #22582629	C₃	semidirect
17. #731×C₃₇⋊C₃	C₃	semidirect
18. C₃₇×#6913	C₃	semidirect
23. #15688532	C₃²	semidirect
24. #15688532	C₃²	semidirect
26. #15688531	C₃²	semidirect
27. #15688530	C₃²	semidirect
28. #15688530	C₃²	semidirect
29. C₃×#11758608	C₃²	semidirect
31. C₃×#11758605	C₃²	semidirect
32. C₃×#11758605	C₃²	semidirect
35. C₃×#11758606	C₃²	semidirect
36. C₃×#11758606	C₃²	semidirect
41. C₃²×#66524	C₃²	semidirect
43. C₃×C₉×C₃₇⋊C₃	C₃²	semidirect
44. C₃×C₉×C₃₇⋊C₃	C₃²	semidirect
45. #15688521	C₃²	semidirect
46. #15688521	C₃²	semidirect
47. #15688520	C₃²	semidirect
48. #15688520	C₃²	semidirect
52. #15688517	C₃²	semidirect
53. #15688517	C₃²	semidirect
54. #15688516	C₃²	semidirect
55. #15688515	C₃²	semidirect
56. #15688515	C₃²	semidirect
57. #15688514	C₃²	semidirect
58. #15688514	C₃²	semidirect
59. #15688513	C₃²	semidirect
60. #15688512	C₃²	semidirect
61. #15688512	C₃²	semidirect
62. C₃₇×#737	C₃²	semidirect
63. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
64. C₃₇×#733	C₃²	semidirect
66. C₃₇×#731	C₃²	semidirect
67. C₃₇×AΣL(1,𝔽₂₇)	C₃²	semidirect
158. #6913	C₃₇⋊C₃	semidirect
185. #737	C₃×C₃₇⋊C₃	semidirect
196. C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	semidirect
207. #733	C₃×C₃₇⋊C₃	semidirect
217. #731	C₃×C₃₇⋊C₃	semidirect
223. AΣL(1,𝔽₂₇)	C₃×C₃₇⋊C₃	semidirect
224. C₃₇	#10624935	semidirect

Maximal subgroups

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

subgroup	index
#22582677	3
C₃×#15688532	3
#22582663	3
#22582661	3
#22582660	3
#22582659	3
#22582658	3
#22582655	3
#22582653	3
#22582649	3
#22582645	3
#22582636	3
#22582634	3
#22582631	3
#22582630	3
#22582629	3
#731×C₃₇⋊C₃	3
C₃₇×#6913	3
#10624935	37