Subgroups — #41007697

Order 26973.

764 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582563	C₃	1	2	yes	maximal, complemented
2	#22582563	C₃	1	4	yes	maximal, complemented
3	#22582562	C₃	1	2	yes	maximal, complemented
4	#22582561	C₃	1	4	yes	maximal, complemented
5	#22582558	C₃	1	2	yes	maximal, complemented
6	#22582557	C₃	1	4	yes	maximal, complemented
7	#22582557	C₃	1	4	yes	maximal, complemented
8	#22582557	C₃	1	4	yes	maximal, complemented
9	#22582555	C₃	1	2	yes	maximal, complemented
10	#22582555	C₃	1	4	yes	maximal, complemented
11	#22582553	C₃	1	2	yes	maximal, complemented
12	(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
13	C₉×C₃²⋊(C₃₇⋊C₃)	C₃	1	1	yes	maximal, characteristic, complemented
14	#22582539	C₃	1	2	yes	maximal, complemented
15	C₃²⋊C₉×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
16	C₃₇×#6893	C₃	1	1	yes	maximal, characteristic, complemented
17	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
18	C₃×#11758608	C₃²	1	4	yes	complemented
19	C₃×#11758605	C₃²	1	4	yes	—
20	C₃×#11758605	C₃²	1	2	yes	—
21	C₃×#11758605	C₃²	1	2	yes	complemented
22	C₃×#11758605	C₃²	1	4	yes	complemented
23	C₃×#11758606	C₃²	1	4	yes	complemented
24	C₃×#11758606	C₃²	1	2	yes	complemented
25	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	2	yes	—
26	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	1	yes	characteristic
27	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	2	yes	—
28	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	4	yes	—
29	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	2	yes	—
30	C₃²×#66524	C₃²	1	2	yes	complemented
31	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
32	C₃×C₉×C₃₇⋊C₃	C₃²	1	2	yes	—
33	C₃×C₉×C₃₇⋊C₃	C₃²	1	4	yes	complemented
34	C₃×C₉×C₃₇⋊C₃	C₃²	1	2	yes	complemented
35	C₃²⋊#66524	C₃²	1	2	yes	—
36	C₃²⋊#66524	C₃²	1	4	yes	—
37	C₃²⋊#66524	C₃²	1	2	yes	complemented
38	C₃²⋊#66524	C₃²	1	2	yes	complemented
39	C₃²⋊#66524	C₃²	1	4	yes	complemented
40	C₃²⋊#66524	C₃²	1	4	yes	complemented
41	#15688502	C₃²	1	4	yes	—
42	#15688502	C₃²	1	2	yes	—
43	#15688502	C₃²	1	4	yes	—
44	#15688502	C₃²	1	2	yes	—
45	#15688502	C₃²	1	4	yes	—
46	#15688502	C₃²	1	2	yes	—
47	#15688502	C₃²	1	4	yes	complemented
48	#15688502	C₃²	1	4	yes	complemented
49	#15688502	C₃²	1	4	yes	complemented
50	#15688502	C₃²	1	4	yes	complemented
51	#15688502	C₃²	1	4	yes	complemented
52	#15688502	C₃²	1	4	yes	complemented
53	#15688502	C₃²	1	4	yes	complemented
54	#15688502	C₃²	1	4	yes	complemented
55	#15688502	C₃²	1	4	yes	complemented
56	C₃×3₋¹⁺²×C₃₇	C₃²	1	2	yes	complemented
57	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
58	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic, complemented
59	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
60	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
61	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	complemented
62	C₃₇×C₃²⋊C₉	C₃²	1	4	yes	complemented
63	C₃³×C₃₇	C₃³	1	1	yes	characteristic
64	C₃³×C₃₇	C₃³	1	2	yes	—
65	C₃³×C₃₇	C₃³	1	1	yes	characteristic
66	C₃²×C₃₇⋊C₃	—	3	3	no	—
67	C₃²×C₃₇⋊C₃	—	9	1	no	—
68	C₃²×C₃₇⋊C₃	—	9	2	no	—
69	C₃²×C₃₇⋊C₃	C₃³	1	2	yes	—
70	C₃²×C₃₇⋊C₃	C₃³	1	4	yes	—
71	C₃²×C₃₇⋊C₃	C₃³	1	2	yes	—
72	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
73	3₋¹⁺²×C₃₇	—	9	2	no	—
74	H(3,𝔽₃)×C₃₇	—	9	1	no	—
75	C₃×C₉×C₃₇	—	9	1	no	—
76	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
77	C₃×C₉×C₃₇	C₃³	1	2	yes	—
78	C₃×C₉×C₃₇	C₃³	1	2	yes	—
79	C₃×C₉×C₃₇	C₃³	1	4	yes	—
80	C₃²⋊(C₃₇⋊C₃)	—	3	6	no	—
81	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
82	C₃²⋊(C₃₇⋊C₃)	—	9	2	no	—
83	C₃²⋊(C₃₇⋊C₃)	—	9	2	no	—
84	C₃²⋊(C₃₇⋊C₃)	—	9	4	no	—
85	#11758608	—	9	4	no	—
86	C₃×#66524	—	9	2	no	—
87	C₃×#66524	C₃³	1	2	yes	—
88	C₃×#66524	C₃³	1	4	yes	—
89	C₃×#66524	C₃³	1	4	yes	—
90	C₃×#66524	C₃³	1	4	yes	—
91	C₃×#66524	C₃³	1	4	yes	—
92	#11758606	—	9	4	no	—
93	#11758606	—	9	2	no	—
94	#11758605	—	9	2	no	—
95	#11758605	—	9	4	no	—
96	#11758605	—	9	4	no	—
97	#11758605	—	9	2	no	—
98	C₉×C₃₇⋊C₃	—	9	1	no	—
99	C₉×C₃₇⋊C₃	—	9	2	no	—
100	C₉×C₃₇⋊C₃	—	9	4	no	—
101	C₉×C₃₇⋊C₃	—	9	2	no	—
102	#10624888	—	37	1	no	maximal, Sylow
103	C₃²×C₃₇	—	3	2	no	—
104	C₃²×C₃₇	—	3	1	no	—
105	C₃²×C₃₇	—	3	1	no	—
106	C₃²×C₃₇	—	3	2	no	—
107	C₃²×C₃₇	—	3	1	no	—
108	C₃²×C₃₇	—	3	1	no	—
109	C₃²×C₃₇	—	3	4	no	—
110	C₃²×C₃₇	—	3	2	no	—
111	C₃²×C₃₇	—	3	2	no	—
112	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
113	C₃×C₃₇⋊C₃	—	3	1	no	—
114	C₃×C₃₇⋊C₃	—	3	2	no	—
115	C₃×C₃₇⋊C₃	—	3	4	no	—
116	C₃×C₃₇⋊C₃	—	3	2	no	—
117	C₃×C₃₇⋊C₃	—	3	2	no	—
118	C₃×C₃₇⋊C₃	—	3	4	no	—
119	C₃×C₃₇⋊C₃	—	3	2	no	—
120	C₃×C₃₇⋊C₃	—	3	2	no	—
121	C₃×C₃₇⋊C₃	—	3	2	no	—
122	C₃×C₃₇⋊C₃	—	3	4	no	—
123	C₃×C₃₇⋊C₃	—	3	4	no	—
124	C₃×C₃₇⋊C₃	—	3	4	no	—
125	C₃×C₃₇⋊C₃	—	3	2	no	—
126	C₃×C₃₇⋊C₃	—	9	9	no	—
127	C₃×C₃₇⋊C₃	#737	1	3	yes	—
128	C₉×C₃₇	—	3	1	no	—
129	C₉×C₃₇	—	3	2	no	—
130	C₉×C₃₇	—	3	4	no	—
131	C₉×C₃₇	—	3	2	no	—
132	#66524	—	3	2	no	—
133	#66524	—	3	4	no	—
134	#66524	—	3	4	no	—
135	#66524	—	3	4	no	—
136	#66524	—	3	4	no	—
137	#6893	—	37	2	no	—
138	#6893	—	37	4	no	—
139	#6893	—	37	2	no	—
140	#6893	C₃₇⋊C₃	1	1	yes	characteristic, complemented
141	C₉⋊H(3,𝔽₃)	—	37	4	no	—
142	C₉⋊H(3,𝔽₃)	—	37	2	no	—
143	C₉⋊H(3,𝔽₃)	—	37	2	no	—
144	C₉⋊H(3,𝔽₃)	—	37	4	no	—
145	C₉⋊H(3,𝔽₃)	—	37	4	no	—
146	C₉⋊H(3,𝔽₃)	—	37	4	no	—
147	C₉⋊H(3,𝔽₃)	—	37	4	no	—
148	#6891	—	37	2	no	—
149	C₃²⋊H(3,𝔽₃)	—	37	1	no	—
150	C₉×H(3,𝔽₃)	—	37	1	no	—
151	C₉×H(3,𝔽₃)	—	37	2	no	—
152	C₃×C₃²⋊C₉	—	37	1	no	—
153	C₃×C₃₇	—	9	1	no	—
154	C₃×C₃₇	—	9	2	no	—
155	C₃×C₃₇	—	9	1	no	—
156	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
157	C₃×C₃₇	H(5,𝔽₃)	1	1	yes	characteristic
158	C₃×C₃₇	3₋¹⁺⁴	1	2	yes	—
159	C₃₇⋊C₃	—	3	3	no	—
160	C₃₇⋊C₃	—	9	4	no	—
161	C₃₇⋊C₃	—	9	2	no	—
162	C₃₇⋊C₃	—	9	2	no	—
163	C₃⁴	—	37	1	no	—
164	C₃×3₋¹⁺²	—	37	4	no	—
165	C₃×3₋¹⁺²	—	37	4	no	—
166	C₃×3₋¹⁺²	—	37	2	no	—
167	C₃×3₋¹⁺²	—	37	4	no	—
168	C₃×3₋¹⁺²	—	37	2	no	—
169	C₃×3₋¹⁺²	—	37	2	no	—
170	C₃×3₋¹⁺²	—	37	4	no	—
171	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	2	yes	complemented
172	C₃×H(3,𝔽₃)	—	37	2	no	—
173	C₃×H(3,𝔽₃)	—	37	4	no	—
174	C₃×H(3,𝔽₃)	—	37	2	no	—
175	C₃×H(3,𝔽₃)	—	37	1	no	—
176	C₃×H(3,𝔽₃)	—	37	2	no	—
177	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
178	C₃²×C₉	—	37	2	no	—
179	C₃²×C₉	—	37	1	no	—
180	C₃²×C₉	—	37	2	no	—
181	C₃²×C₉	—	37	4	no	—
182	C₃²×C₉	—	37	2	no	—
183	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic, complemented
184	C₃²⋊C₉	—	37	2	no	—
185	C₃²⋊C₉	—	37	4	no	—
186	C₃²⋊C₉	—	37	2	no	—
187	C₃²⋊C₉	—	37	2	no	—
188	C₃²⋊C₉	—	37	4	no	—
189	C₃²⋊C₉	—	37	4	no	—
190	C₃²⋊C₉	—	37	4	no	—
191	C₃²⋊C₉	—	37	2	no	—
192	C₃²⋊C₉	—	37	4	no	—
193	C₃²⋊C₉	—	37	2	no	—
194	C₃²⋊C₉	—	37	4	no	—
195	C₃²⋊C₉	—	37	2	no	—
196	C₃²⋊C₉	—	37	4	no	—
197	C₃²⋊C₉	—	37	4	no	—
198	C₃²⋊C₉	—	37	4	no	—
199	C₃²⋊C₉	—	37	4	no	—
200	C₃²⋊C₉	—	37	4	no	—
201	C₃²⋊C₉	—	37	4	no	—
202	C₃²⋊C₉	—	37	4	no	—
203	C₃²⋊C₉	—	37	4	no	—
204	C₃²⋊C₉	—	37	4	no	—
205	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
206	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	—
207	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	complemented
208	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	4	yes	complemented
209	C₃₇	#10624888	1	1	yes	characteristic, complemented, Sylow
210	C₃³	—	37	1	no	—
211	C₃³	—	37	4	no	—
212	C₃³	—	37	2	no	—
213	C₃³	—	37	2	no	—
214	C₃³	—	333	1	no	—
215	C₃³	—	333	2	no	—
216	C₃³	—	111	3	no	—
217	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
218	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
219	C₃³	C₃²×C₃₇⋊C₃	1	2	yes	—
220	3₋¹⁺²	—	9	2	no	—
221	3₋¹⁺²	—	333	4	no	—
222	3₋¹⁺²	—	333	2	no	—
223	3₋¹⁺²	—	333	2	no	—
224	3₋¹⁺²	—	333	4	no	—
225	3₋¹⁺²	—	333	4	no	—
226	3₋¹⁺²	—	333	2	no	—
227	3₋¹⁺²	—	333	4	no	—
228	H(3,𝔽₃)	—	9	1	no	—
229	H(3,𝔽₃)	—	333	2	no	—
230	H(3,𝔽₃)	—	333	2	no	—
231	H(3,𝔽₃)	—	333	4	no	—
232	H(3,𝔽₃)	—	111	3	no	—
233	H(3,𝔽₃)	—	111	6	no	—
234	C₃×C₉	—	9	1	no	—
235	C₃×C₉	—	37	2	no	—
236	C₃×C₉	—	37	4	no	—
237	C₃×C₉	—	37	4	no	—
238	C₃×C₉	—	37	4	no	—
239	C₃×C₉	—	37	4	no	—
240	C₃×C₉	—	333	1	no	—
241	C₃×C₉	—	333	2	no	—
242	C₃×C₉	—	333	4	no	—
243	C₃×C₉	—	333	2	no	—
244	C₃×C₉	—	333	2	no	—
245	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
246	C₃×C₉	C₃²×C₃₇⋊C₃	1	2	yes	—
247	C₃×C₉	C₃²×C₃₇⋊C₃	1	4	yes	—
248	C₃×C₉	C₃²×C₃₇⋊C₃	1	2	yes	—
249	C₃²	—	3	1	no	—
250	C₃²	—	3	2	no	—
251	C₃²	—	3	1	no	—
252	C₃²	—	3	1	no	—
253	C₃²	—	3	2	no	—
254	C₃²	—	3	1	no	—
255	C₃²	—	3	2	no	—
256	C₃²	—	3	4	no	—
257	C₃²	—	3	2	no	—
258	C₃²	—	37	3	no	—
259	C₃²	—	333	9	no	—
260	C₃²	—	111	2	no	—
261	C₃²	—	111	4	no	—
262	C₃²	—	111	2	no	—
263	C₃²	—	111	2	no	—
264	C₃²	—	111	4	no	—
265	C₃²	—	111	2	no	—
266	C₃²	—	111	2	no	—
267	C₃²	—	111	4	no	—
268	C₃²	—	111	4	no	—
269	C₃²	—	111	4	no	—
270	C₃²	—	111	2	no	—
271	C₃²	—	111	1	no	—
272	C₃²	—	111	2	no	—
273	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
274	C₉	—	3	1	no	—
275	C₉	—	3	2	no	—
276	C₉	—	3	4	no	—
277	C₉	—	3	2	no	—
278	C₉	—	111	2	no	—
279	C₉	—	111	4	no	—
280	C₉	—	111	4	no	—
281	C₉	—	111	4	no	—
282	C₉	—	111	4	no	—
283	C₃	—	9	1	no	—
284	C₃	—	9	1	no	—
285	C₃	—	9	2	no	—
286	C₃	—	333	2	no	—
287	C₃	—	333	2	no	—
288	C₃	—	333	4	no	—
289	C₃	—	111	3	no	—
290	C₃	#737×C₃₇⋊C₃	1	1	yes	characteristic
291	C₃	C₃₇⋊H(5,𝔽₃)	1	1	yes	characteristic
292	C₃	#22582678	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting	C₃₇×#6893	C₃
prime radical	(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	C₃
hypercenter	#6893	C₃₇⋊C₃
hypocenter	C₃₇	#10624888
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ #6893

Complements

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

N	H ≅ G/N	type
1. #22582563	C₃	semidirect
2. #22582563	C₃	semidirect
3. #22582562	C₃	semidirect
4. #22582561	C₃	semidirect
5. #22582558	C₃	semidirect
6. #22582557	C₃	semidirect
7. #22582557	C₃	semidirect
8. #22582557	C₃	semidirect
9. #22582555	C₃	semidirect
10. #22582555	C₃	semidirect
11. #22582553	C₃	semidirect
13. C₉×C₃²⋊(C₃₇⋊C₃)	C₃	semidirect
14. #22582539	C₃	semidirect
15. C₃²⋊C₉×C₃₇⋊C₃	C₃	semidirect
16. C₃₇×#6893	C₃	semidirect
18. C₃×#11758608	C₃²	semidirect
21. C₃×#11758605	C₃²	semidirect
22. C₃×#11758605	C₃²	semidirect
23. C₃×#11758606	C₃²	semidirect
24. C₃×#11758606	C₃²	semidirect
30. C₃²×#66524	C₃²	semidirect
33. C₃×C₉×C₃₇⋊C₃	C₃²	semidirect
34. C₃×C₉×C₃₇⋊C₃	C₃²	semidirect
37. C₃²⋊#66524	C₃²	semidirect
38. C₃²⋊#66524	C₃²	semidirect
39. C₃²⋊#66524	C₃²	semidirect
40. C₃²⋊#66524	C₃²	semidirect
47. #15688502	C₃²	semidirect
48. #15688502	C₃²	semidirect
49. #15688502	C₃²	semidirect
50. #15688502	C₃²	semidirect
51. #15688502	C₃²	semidirect
52. #15688502	C₃²	semidirect
53. #15688502	C₃²	semidirect
54. #15688502	C₃²	semidirect
55. #15688502	C₃²	semidirect
56. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
58. C₃²×C₉×C₃₇	C₃²	semidirect
61. C₃₇×C₃²⋊C₉	C₃²	semidirect
62. C₃₇×C₃²⋊C₉	C₃²	semidirect
140. #6893	C₃₇⋊C₃	semidirect
171. C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	semidirect
183. C₃²×C₉	C₃×C₃₇⋊C₃	semidirect
207. C₃²⋊C₉	C₃×C₃₇⋊C₃	semidirect
208. C₃²⋊C₉	C₃×C₃₇⋊C₃	semidirect
209. C₃₇	#10624888	semidirect

Maximal subgroups

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

subgroup	index
#22582563	3
#22582562	3
#22582561	3
#22582558	3
#22582557	3
#22582555	3
#22582553	3
(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	3
C₉×C₃²⋊(C₃₇⋊C₃)	3
#22582539	3
C₃²⋊C₉×C₃₇⋊C₃	3
C₃₇×#6893	3
#10624888	37