Subgroups — #41007555

Order 26973.

1166 conjugacy classes of subgroups.

The subgroup lattice has 302 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	3	yes	maximal, complemented
2	C₃²×C₉×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582563	C₃	1	6	yes	maximal, complemented
4	#22582561	C₃	1	1	yes	maximal, characteristic, complemented
5	#22582560	C₃	1	1	yes	maximal, characteristic, complemented
6	#22582559	C₃	1	1	yes	maximal, characteristic, complemented
7	#22582558	C₃	1	1	yes	maximal, characteristic, complemented
8	#22582557	C₃	1	6	yes	maximal, complemented
9	#22582556	C₃	1	1	yes	maximal, characteristic, complemented
10	#22582555	C₃	1	6	yes	maximal, complemented
11	#22582553	C₃	1	1	yes	maximal, characteristic, complemented
12	#22582552	C₃	1	1	yes	maximal, characteristic, complemented
13	(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
14	H(3,𝔽₃)×#66524	C₃	1	1	yes	maximal, characteristic, complemented
15	#22582539	C₃	1	1	yes	maximal, characteristic, complemented
16	#22582539	C₃	1	1	yes	maximal, characteristic, complemented
17	#22582539	C₃	1	6	yes	maximal, complemented
18	C₃₇×C₉⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
19	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
20	#15688532	—	3	27	no	—
21	C₃×#11758608	C₃²	1	3	yes	complemented
22	C₃×#11758605	C₃²	1	3	yes	—
23	C₃×#11758605	C₃²	1	3	yes	complemented
24	C₃×#11758605	C₃²	1	3	yes	complemented
25	C₃×#11758606	C₃²	1	3	yes	—
26	C₃×#11758606	C₃²	1	3	yes	complemented
27	C₃×#11758606	C₃²	1	3	yes	complemented
28	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	6	yes	—
29	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	1	yes	characteristic
30	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	1	yes	characteristic
31	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	3	yes	—
32	C₃²×#66524	C₃²	1	1	yes	characteristic
33	C₃²×#66524	C₃²	1	1	yes	characteristic
34	C₃²×#66524	C₃²	1	3	yes	complemented
35	C₃×C₉×C₃₇⋊C₃	—	3	9	no	—
36	C₃×C₉×C₃₇⋊C₃	C₃²	1	2	yes	—
37	C₃×C₉×C₃₇⋊C₃	C₃²	1	2	yes	—
38	C₃×C₉×C₃₇⋊C₃	C₃²	1	2	yes	—
39	C₃×C₉×C₃₇⋊C₃	C₃²	1	2	yes	—
40	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
41	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	complemented
42	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	complemented
43	C₃²⋊#66524	C₃²	1	1	yes	characteristic
44	C₃²⋊#66524	C₃²	1	1	yes	characteristic
45	C₃²⋊#66524	C₃²	1	1	yes	characteristic
46	C₃²⋊#66524	C₃²	1	1	yes	characteristic
47	C₃²⋊#66524	C₃²	1	6	yes	complemented
48	C₃²⋊#66524	C₃²	1	6	yes	complemented
49	#15688502	C₃²	1	6	yes	—
50	#15688502	C₃²	1	6	yes	—
51	#15688502	C₃²	1	6	yes	complemented
52	#15688502	C₃²	1	6	yes	complemented
53	#15688502	C₃²	1	6	yes	complemented
54	#15688502	C₃²	1	6	yes	complemented
55	#15688502	C₃²	1	6	yes	complemented
56	#15688502	C₃²	1	6	yes	complemented
57	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	complemented
58	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
59	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
60	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
61	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
62	C₃₇×C₃²⋊C₉	C₃²	1	6	yes	complemented
63	C₃³×C₃₇	C₃³	1	3	yes	—
64	C₃³×C₃₇	C₃³	1	1	yes	characteristic
65	C₃²×C₃₇⋊C₃	—	3	3	no	—
66	C₃²×C₃₇⋊C₃	—	3	3	no	—
67	C₃²×C₃₇⋊C₃	—	3	6	no	—
68	C₃²×C₃₇⋊C₃	C₃³	1	3	yes	—
69	C₃²×C₃₇⋊C₃	C₃³	1	3	yes	—
70	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
71	C₃²×C₃₇⋊C₃	C₃³	1	2	yes	—
72	3₋¹⁺²×C₃₇	—	3	9	no	—
73	H(3,𝔽₃)×C₃₇	—	3	3	no	—
74	C₃×C₉×C₃₇	—	3	3	no	—
75	C₃×C₉×C₃₇	C₃³	1	3	yes	—
76	C₃×C₉×C₃₇	C₃³	1	3	yes	—
77	C₃×C₉×C₃₇	C₃³	1	2	yes	—
78	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
79	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
80	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
81	C₃²⋊(C₃₇⋊C₃)	—	3	9	no	—
82	C₃²⋊(C₃₇⋊C₃)	—	3	18	no	—
83	#11758608	—	3	9	no	—
84	C₃×#66524	—	3	3	no	—
85	C₃×#66524	—	3	3	no	—
86	C₃×#66524	—	3	9	no	—
87	C₃×#66524	C₃³	1	3	yes	—
88	C₃×#66524	C₃³	1	3	yes	—
89	C₃×#66524	C₃³	1	3	yes	—
90	C₃×#66524	C₃³	1	3	yes	—
91	C₃×#66524	C₃³	1	2	yes	—
92	C₃×#66524	C₃³	1	2	yes	—
93	C₃×#66524	C₃³	1	1	yes	characteristic
94	C₃×#66524	C₃³	1	1	yes	characteristic
95	#11758606	—	3	9	no	—
96	#11758606	—	3	9	no	—
97	#11758606	—	3	9	no	—
98	#11758605	—	3	9	no	—
99	#11758605	—	3	9	no	—
100	#11758605	—	3	9	no	—
101	C₉×C₃₇⋊C₃	—	3	6	no	—
102	C₉×C₃₇⋊C₃	—	3	6	no	—
103	C₉×C₃₇⋊C₃	—	3	6	no	—
104	C₉×C₃₇⋊C₃	—	3	6	no	—
105	C₉×C₃₇⋊C₃	—	3	9	no	—
106	C₉×C₃₇⋊C₃	—	3	9	no	—
107	C₉×C₃₇⋊C₃	H(3,𝔽₃)	1	9	yes	complemented
108	#10624869	—	37	1	no	maximal, Sylow
109	C₃²×C₃₇	—	3	1	no	—
110	C₃²×C₃₇	—	3	2	no	—
111	C₃²×C₃₇	—	3	3	no	—
112	C₃²×C₃₇	—	3	6	no	—
113	C₃²×C₃₇	—	3	3	no	—
114	C₃²×C₃₇	#737	1	3	yes	complemented
115	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
116	C₃×C₃₇⋊C₃	—	3	27	no	—
117	C₃×C₃₇⋊C₃	—	3	2	no	—
118	C₃×C₃₇⋊C₃	—	3	2	no	—
119	C₃×C₃₇⋊C₃	—	3	2	no	—
120	C₃×C₃₇⋊C₃	—	3	1	no	—
121	C₃×C₃₇⋊C₃	—	3	2	no	—
122	C₃×C₃₇⋊C₃	—	3	3	no	—
123	C₃×C₃₇⋊C₃	—	3	3	no	—
124	C₃×C₃₇⋊C₃	—	3	6	no	—
125	C₃×C₃₇⋊C₃	—	3	6	no	—
126	C₃×C₃₇⋊C₃	—	3	3	no	—
127	C₃×C₃₇⋊C₃	—	3	3	no	—
128	C₃×C₃₇⋊C₃	C₃×H(3,𝔽₃)	1	3	yes	—
129	C₃×C₃₇⋊C₃	#737	1	6	yes	complemented
130	C₉×C₃₇	—	3	2	no	—
131	C₉×C₃₇	—	3	3	no	—
132	C₉×C₃₇	—	3	3	no	—
133	C₉×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
134	#66524	—	3	2	no	—
135	#66524	—	3	2	no	—
136	#66524	—	3	3	no	—
137	#66524	—	3	3	no	—
138	#66524	—	3	3	no	—
139	#66524	—	3	3	no	—
140	#66524	C₃×H(3,𝔽₃)	1	3	yes	—
141	#66524	C₃×H(3,𝔽₃)	1	3	yes	—
142	C₃×#737	—	37	3	no	—
143	C₃³×C₉	—	37	1	no	—
144	#6893	—	37	6	no	—
145	C₉⋊H(3,𝔽₃)	—	37	6	no	—
146	C₉⋊H(3,𝔽₃)	—	37	1	no	—
147	C₉⋊H(3,𝔽₃)	—	37	1	no	—
148	C₉⋊H(3,𝔽₃)	—	37	1	no	—
149	C₉⋊H(3,𝔽₃)	—	37	1	no	—
150	C₉⋊H(3,𝔽₃)	—	37	1	no	—
151	C₉⋊H(3,𝔽₃)	—	37	6	no	—
152	C₉⋊H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic, complemented
153	#6891	—	37	1	no	—
154	#6891	—	37	1	no	—
155	C₃²⋊H(3,𝔽₃)	—	37	1	no	—
156	C₉×H(3,𝔽₃)	—	37	1	no	—
157	C₉×H(3,𝔽₃)	—	37	1	no	—
158	C₉×H(3,𝔽₃)	—	37	1	no	—
159	C₉×H(3,𝔽₃)	—	37	6	no	—
160	C₃×C₃₇	—	3	3	no	—
161	C₃×C₃₇	—	9	3	no	—
162	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
163	C₃×C₃₇	C₃×#737	1	2	yes	—
164	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
165	C₃₇⋊C₃	—	3	3	no	—
166	C₃₇⋊C₃	—	3	6	no	—
167	C₃₇⋊C₃	—	9	3	no	—
168	C₃₇⋊C₃	—	9	3	no	—
169	C₃⁴	—	37	1	no	—
170	#737	—	111	27	no	—
171	C₃×3₋¹⁺²	—	37	3	no	—
172	C₃×3₋¹⁺²	—	37	3	no	—
173	C₃×3₋¹⁺²	—	37	3	no	—
174	C₃×3₋¹⁺²	—	37	3	no	—
175	C₃×3₋¹⁺²	—	37	3	no	—
176	C₃×3₋¹⁺²	—	37	3	no	—
177	C₃×3₋¹⁺²	—	37	3	no	—
178	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	complemented
179	C₃×H(3,𝔽₃)	—	37	3	no	—
180	C₃×H(3,𝔽₃)	—	37	6	no	—
181	C₃×H(3,𝔽₃)	—	37	1	no	—
182	C₃×H(3,𝔽₃)	—	37	1	no	—
183	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
184	C₃²×C₉	—	37	1	no	—
185	C₃²×C₉	—	37	1	no	—
186	C₃²×C₉	—	37	1	no	—
187	C₃²×C₉	—	37	3	no	—
188	C₃²×C₉	—	37	2	no	—
189	C₃²×C₉	—	37	2	no	—
190	C₃²×C₉	—	37	2	no	—
191	C₃²×C₉	—	37	2	no	—
192	C₃²×C₉	—	37	3	no	—
193	C₃²×C₉	—	37	3	no	—
194	C₃²×C₉	—	111	9	no	—
195	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
196	C₃²⋊C₉	—	37	1	no	—
197	C₃²⋊C₉	—	37	1	no	—
198	C₃²⋊C₉	—	37	1	no	—
199	C₃²⋊C₉	—	37	1	no	—
200	C₃²⋊C₉	—	37	6	no	—
201	C₃²⋊C₉	—	37	6	no	—
202	C₃²⋊C₉	—	37	6	no	—
203	C₃²⋊C₉	—	37	6	no	—
204	C₃²⋊C₉	—	37	6	no	—
205	C₃²⋊C₉	—	37	6	no	—
206	C₃²⋊C₉	—	37	6	no	—
207	C₃²⋊C₉	—	37	6	no	—
208	C₃²⋊C₉	—	37	6	no	—
209	C₃²⋊C₉	—	37	6	no	—
210	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
211	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
212	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	6	yes	complemented
213	C₃₇	#10624869	1	1	yes	characteristic, complemented, Sylow
214	C₃³	—	37	2	no	—
215	C₃³	—	37	1	no	—
216	C₃³	—	37	3	no	—
217	C₃³	—	37	3	no	—
218	C₃³	—	111	3	no	—
219	C₃³	—	111	6	no	—
220	C₃³	—	111	3	no	—
221	C₃³	C₃²×C₃₇⋊C₃	1	3	yes	—
222	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
223	3₋¹⁺²	—	3	9	no	—
224	3₋¹⁺²	—	111	9	no	—
225	3₋¹⁺²	—	111	9	no	—
226	3₋¹⁺²	—	111	9	no	—
227	3₋¹⁺²	—	111	9	no	—
228	3₋¹⁺²	—	111	9	no	—
229	3₋¹⁺²	—	111	9	no	—
230	3₋¹⁺²	—	111	9	no	—
231	H(3,𝔽₃)	—	3	3	no	—
232	H(3,𝔽₃)	—	111	18	no	—
233	H(3,𝔽₃)	—	111	9	no	—
234	H(3,𝔽₃)	—	111	3	no	—
235	H(3,𝔽₃)	—	111	3	no	—
236	C₃×C₉	—	3	3	no	—
237	C₃×C₉	—	37	1	no	—
238	C₃×C₉	—	37	1	no	—
239	C₃×C₉	—	37	9	no	—
240	C₃×C₉	—	37	2	no	—
241	C₃×C₉	—	37	2	no	—
242	C₃×C₉	—	37	3	no	—
243	C₃×C₉	—	37	3	no	—
244	C₃×C₉	—	37	3	no	—
245	C₃×C₉	—	37	3	no	—
246	C₃×C₉	—	111	6	no	—
247	C₃×C₉	—	111	6	no	—
248	C₃×C₉	—	111	9	no	—
249	C₃×C₉	—	111	9	no	—
250	C₃×C₉	—	111	6	no	—
251	C₃×C₉	—	111	3	no	—
252	C₃×C₉	—	111	3	no	—
253	C₃×C₉	—	111	9	no	—
254	C₃×C₉	—	111	6	no	—
255	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
256	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
257	C₃×C₉	C₃²×C₃₇⋊C₃	1	2	yes	—
258	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
259	C₃²	—	3	1	no	—
260	C₃²	—	3	2	no	—
261	C₃²	—	3	3	no	—
262	C₃²	—	3	6	no	—
263	C₃²	—	3	3	no	—
264	C₃²	—	37	3	no	—
265	C₃²	—	37	6	no	—
266	C₃²	—	111	27	no	—
267	C₃²	—	111	3	no	—
268	C₃²	—	111	3	no	—
269	C₃²	—	111	6	no	—
270	C₃²	—	111	6	no	—
271	C₃²	—	111	3	no	—
272	C₃²	—	111	3	no	—
273	C₃²	—	111	2	no	—
274	C₃²	—	111	2	no	—
275	C₃²	—	111	2	no	—
276	C₃²	—	111	2	no	—
277	C₃²	—	111	1	no	—
278	C₃²	#15688532	1	3	yes	complemented
279	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
280	C₉	—	3	2	no	—
281	C₉	—	3	3	no	—
282	C₉	—	3	3	no	—
283	C₉	—	37	3	no	—
284	C₉	—	37	3	no	—
285	C₉	—	111	2	no	—
286	C₉	—	111	2	no	—
287	C₉	—	111	3	no	—
288	C₉	—	111	3	no	—
289	C₉	—	111	3	no	—
290	C₉	—	111	3	no	—
291	C₉	H(3,𝔽₃)×C₃₇⋊C₃	1	3	yes	—
292	C₃	—	3	3	no	—
293	C₃	—	9	3	no	—
294	C₃	—	333	3	no	—
295	C₃	—	333	3	no	—
296	C₃	—	111	3	no	—
297	C₃	—	111	6	no	—
298	C₃	C₃×H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
299	C₃	#737×C₃₇⋊C₃	1	2	yes	—
300	C₃	C₃×#15688532	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting	C₃₇×C₉⋊H(3,𝔽₃)	C₃
prime radical	(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	C₃
hypercenter	C₉⋊H(3,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	#10624869
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₉⋊H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃×#15688532	C₃	semidirect
2. C₃²×C₉×C₃₇⋊C₃	C₃	semidirect
3. #22582563	C₃	semidirect
4. #22582561	C₃	semidirect
5. #22582560	C₃	semidirect
6. #22582559	C₃	semidirect
7. #22582558	C₃	semidirect
8. #22582557	C₃	semidirect
9. #22582556	C₃	semidirect
10. #22582555	C₃	semidirect
11. #22582553	C₃	semidirect
12. #22582552	C₃	semidirect
14. H(3,𝔽₃)×#66524	C₃	semidirect
15. #22582539	C₃	semidirect
16. #22582539	C₃	semidirect
17. #22582539	C₃	semidirect
18. C₃₇×C₉⋊H(3,𝔽₃)	C₃	semidirect
21. C₃×#11758608	C₃²	semidirect
23. C₃×#11758605	C₃²	semidirect
24. C₃×#11758605	C₃²	semidirect
26. C₃×#11758606	C₃²	semidirect
27. C₃×#11758606	C₃²	semidirect
34. C₃²×#66524	C₃²	semidirect
41. C₃×C₉×C₃₇⋊C₃	C₃²	semidirect
42. C₃×C₉×C₃₇⋊C₃	C₃²	semidirect
47. C₃²⋊#66524	C₃²	semidirect
48. C₃²⋊#66524	C₃²	semidirect
51. #15688502	C₃²	semidirect
52. #15688502	C₃²	semidirect
53. #15688502	C₃²	semidirect
54. #15688502	C₃²	semidirect
55. #15688502	C₃²	semidirect
56. #15688502	C₃²	semidirect
57. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
62. C₃₇×C₃²⋊C₉	C₃²	semidirect
107. C₉×C₃₇⋊C₃	H(3,𝔽₃)	semidirect
114. C₃²×C₃₇	#737	semidirect
129. C₃×C₃₇⋊C₃	#737	semidirect
152. C₉⋊H(3,𝔽₃)	C₃₇⋊C₃	semidirect
178. C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	semidirect
212. C₃²⋊C₉	C₃×C₃₇⋊C₃	semidirect
213. C₃₇	#10624869	semidirect
278. C₃²	#15688532	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
C₃²×C₉×C₃₇⋊C₃	3
#22582563	3
#22582561	3
#22582560	3
#22582559	3
#22582558	3
#22582557	3
#22582556	3
#22582555	3
#22582553	3
#22582552	3
(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	3
H(3,𝔽₃)×#66524	3
#22582539	3
C₃₇×C₉⋊H(3,𝔽₃)	3
#10624869	37