Subgroups — #41007612

Order 26973.

662 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	#22582594	C₃	1	3	yes	maximal, complemented
2	#22582593	C₃	1	3	yes	maximal, complemented
3	#22582590	C₃	1	3	yes	maximal, complemented
4	#22582585	C₃	1	3	yes	maximal, complemented
5	#22582585	C₃	1	3	yes	maximal, complemented
6	#22582580	C₃	1	3	yes	maximal, complemented
7	#22582576	C₃	1	1	yes	maximal, characteristic, 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	#22582572	C₃	1	1	yes	maximal, characteristic, complemented
12	#22582571	C₃	1	1	yes	maximal, characteristic, complemented
13	#22582562	C₃	1	1	yes	maximal, characteristic
14	#22582556	C₃	1	1	yes	maximal, characteristic
15	#22582548	C₃	1	1	yes	maximal, characteristic, complemented
16	#22582548	C₃	1	1	yes	maximal, characteristic, complemented
17	#22582548	C₃	1	3	yes	maximal, complemented
18	3₋¹⁺²×#66524	C₃	1	3	yes	maximal, complemented
19	H(3,𝔽₃)×#66524	C₃	1	1	yes	maximal, characteristic
20	#22582535	C₃	1	3	yes	maximal, complemented
21	C₃×C₉×#66524	C₃	1	1	yes	maximal, characteristic, complemented
22	C₃₇×C₉⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
23	C₃×#11758608	C₃²	1	3	yes	—
24	C₃×#11758608	C₃²	1	1	yes	characteristic
25	C₃×#11758608	C₃²	1	1	yes	characteristic
26	C₃²×#66524	C₃²	1	3	yes	—
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	#15688506	C₃²	1	3	yes	—
31	#15688506	C₃²	1	3	yes	—
32	#15688506	C₃²	1	3	yes	—
33	#15688505	C₃²	1	3	yes	—
34	#15688505	C₃²	1	3	yes	—
35	#15688505	C₃²	1	3	yes	—
36	#15688504	C₃²	1	3	yes	—
37	#15688504	C₃²	1	3	yes	—
38	#15688504	C₃²	1	3	yes	—
39	#15688504	C₃²	1	3	yes	—
40	#15688504	C₃²	1	3	yes	—
41	#15688504	C₃²	1	3	yes	—
42	#15688504	C₃²	1	3	yes	—
43	#15688504	C₃²	1	3	yes	—
44	#15688504	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	C₃²⋊#66524	C₃²	1	3	yes	—
53	C₃²⋊#66524	C₃²	1	3	yes	—
54	C₃²⋊#66524	C₃²	1	1	yes	characteristic
55	C₃²⋊#66524	C₃²	1	1	yes	characteristic
56	C₃²⋊#66524	C₃²	1	3	yes	—
57	C₃²⋊#66524	C₃²	1	3	yes	—
58	C₃²⋊#66524	C₃²	1	3	yes	—
59	C₃²⋊#66524	C₃²	1	3	yes	—
60	C₃²⋊#66524	C₃²	1	1	yes	characteristic
61	C₃²⋊#66524	C₃²	1	1	yes	characteristic
62	C₃²⋊#66524	C₃²	1	3	yes	—
63	C₉×#66524	C₃²	1	1	yes	characteristic
64	C₉×#66524	C₃²	1	3	yes	—
65	C₉×#66524	C₃²	1	3	yes	—
66	C₉×#66524	C₃²	1	1	yes	characteristic
67	C₉×#66524	C₃²	1	1	yes	characteristic
68	C₉×#66524	C₃²	1	1	yes	characteristic
69	C₉×#66524	C₃²	1	1	yes	characteristic
70	C₉×#66524	C₃²	1	1	yes	characteristic
71	C₉×#66524	C₃²	1	1	yes	characteristic
72	C₉×#66524	C₃²	1	1	yes	characteristic
73	C₉×#66524	C₃²	1	1	yes	characteristic
74	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
75	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
76	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
77	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
78	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
79	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
80	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
81	C₃³×C₃₇	C₃³	1	3	yes	—
82	C₃³×C₃₇	C₃³	1	1	yes	characteristic
83	3₋¹⁺²×C₃₇	—	3	9	no	—
84	H(3,𝔽₃)×C₃₇	—	3	3	no	—
85	C₃×C₉×C₃₇	—	3	3	no	—
86	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
87	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
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	#11758608	—	3	3	no	—
92	#11758608	—	3	3	no	—
93	#11758608	—	3	9	no	—
94	C₃×#66524	—	3	3	no	—
95	C₃×#66524	—	3	3	no	—
96	C₃×#66524	—	3	9	no	—
97	C₃×#66524	—	3	3	no	—
98	C₃×#66524	C₃³	1	1	yes	characteristic
99	C₃×#66524	C₃³	1	1	yes	characteristic
100	C₃×#66524	C₃³	1	1	yes	characteristic
101	C₃×#66524	C₃³	1	1	yes	characteristic
102	C₃×#66524	C₃³	1	1	yes	characteristic
103	C₃×#66524	C₃³	1	1	yes	characteristic
104	C₃×#66524	C₃³	1	1	yes	characteristic
105	C₃×#66524	C₃³	1	3	yes	—
106	C₃×#66524	C₃³	1	3	yes	—
107	C₃×#66524	C₃³	1	3	yes	—
108	C₃×#66524	C₃³	1	3	yes	—
109	C₃×#66524	C₃³	1	3	yes	—
110	C₃×#66524	C₃³	1	3	yes	—
111	C₃×#66524	C₃³	1	1	yes	characteristic
112	C₃×#66524	C₃³	1	1	yes	characteristic
113	#10624877	—	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	—	3	3	no	—
118	C₃²×C₃₇	—	3	3	no	—
119	C₃²×C₃₇	—	3	3	no	—
120	C₃²×C₃₇	—	3	3	no	—
121	C₃²×C₃₇	#737	1	3	yes	—
122	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
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	#737	1	3	yes	—
128	#66524	—	3	1	no	—
129	#66524	—	3	1	no	—
130	#66524	—	3	1	no	—
131	#66524	—	3	1	no	—
132	#66524	—	3	3	no	—
133	#66524	—	3	3	no	—
134	#66524	—	3	1	no	—
135	#66524	—	3	1	no	—
136	#66524	—	3	1	no	—
137	#66524	—	3	3	no	—
138	#66524	—	3	3	no	—
139	#66524	—	3	3	no	—
140	#66524	—	3	3	no	—
141	#66524	#737	1	3	yes	—
142	#66524	#737	1	3	yes	—
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	1	no	—
150	#6895	—	37	1	no	—
151	#6894	—	37	1	no	—
152	#6894	—	37	1	no	—
153	#6894	—	37	1	no	—
154	#6894	—	37	1	no	—
155	#6893	—	37	1	no	—
156	C₉⋊H(3,𝔽₃)	—	37	1	no	—
157	C₉⋊H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic
158	C₉×3₋¹⁺²	—	37	3	no	—
159	C₉×3₋¹⁺²	—	37	3	no	—
160	C₉×3₋¹⁺²	—	37	1	no	—
161	C₉×3₋¹⁺²	—	37	1	no	—
162	C₉×H(3,𝔽₃)	—	37	1	no	—
163	#6887	—	37	3	no	—
164	C₃×C₉²	—	37	1	no	—
165	C₃×C₃₇	—	3	3	no	—
166	C₃×C₃₇	—	9	3	no	—
167	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
168	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
169	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
170	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
171	C₃×3₋¹⁺²	—	37	3	no	—
172	C₃×3₋¹⁺²	—	37	1	no	—
173	C₃×3₋¹⁺²	—	37	1	no	—
174	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	—
175	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
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	3	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	C₃²⋊C₉	—	37	3	no	—
204	C₃²⋊C₉	—	37	3	no	—
205	C₃²⋊C₉	—	37	1	no	—
206	C₃²⋊C₉	—	37	1	no	—
207	C₃²⋊C₉	—	37	3	no	—
208	C₃²⋊C₉	—	37	3	no	—
209	C₃²⋊C₉	—	37	1	no	—
210	C₃²⋊C₉	—	37	1	no	—
211	C₃²⋊C₉	—	37	3	no	—
212	C₃²⋊C₉	—	37	3	no	—
213	C₃²⋊C₉	—	37	3	no	—
214	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
215	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
216	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
217	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
218	C₉²	—	37	1	no	—
219	C₉²	—	37	1	no	—
220	C₉²	—	37	1	no	—
221	C₉²	—	37	1	no	—
222	C₉²	—	37	1	no	—
223	C₉²	—	37	1	no	—
224	C₉²	—	37	1	no	—
225	C₉²	—	37	1	no	—
226	C₉²	—	37	1	no	—
227	C₉²	—	37	3	no	—
228	C₉²	—	37	3	no	—
229	C₃₇	#10624877	1	1	yes	characteristic, complemented, Sylow
230	C₃³	C₃²×C₃₇⋊C₃	1	3	yes	—
231	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
232	3₋¹⁺²	—	3	9	no	—
233	3₋¹⁺²	—	111	9	no	—
234	3₋¹⁺²	—	111	3	no	—
235	3₋¹⁺²	—	111	3	no	—
236	H(3,𝔽₃)	—	3	3	no	—
237	C₃×C₉	—	3	3	no	—
238	C₃×C₉	—	37	1	no	—
239	C₃×C₉	—	37	1	no	—
240	C₃×C₉	—	37	1	no	—
241	C₃×C₉	—	37	1	no	—
242	C₃×C₉	—	37	1	no	—
243	C₃×C₉	—	37	1	no	—
244	C₃×C₉	—	37	1	no	—
245	C₃×C₉	—	37	1	no	—
246	C₃×C₉	—	37	1	no	—
247	C₃×C₉	—	37	3	no	—
248	C₃×C₉	—	37	3	no	—
249	C₃×C₉	—	37	3	no	—
250	C₃×C₉	—	37	3	no	—
251	C₃×C₉	—	37	3	no	—
252	C₃×C₉	—	37	3	no	—
253	C₃×C₉	—	111	3	no	—
254	C₃×C₉	—	111	9	no	—
255	C₃×C₉	—	111	3	no	—
256	C₃×C₉	—	111	3	no	—
257	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
258	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
259	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
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	—	3	3	no	—
266	C₃²	—	3	3	no	—
267	C₃²	—	3	3	no	—
268	C₃²	—	3	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	—	37	3	no	—
276	C₉	—	37	3	no	—
277	C₉	—	111	1	no	—
278	C₉	—	111	1	no	—
279	C₉	—	111	1	no	—
280	C₉	—	111	1	no	—
281	C₉	—	111	3	no	—
282	C₉	—	111	3	no	—
283	C₉	—	111	3	no	—
284	C₉	—	111	3	no	—
285	C₉	—	111	1	no	—
286	C₉	—	111	1	no	—
287	C₉	—	111	1	no	—
288	C₉	—	111	3	no	—
289	C₉	—	111	3	no	—
290	C₉	#15688530	1	3	yes	—
291	C₃	—	3	3	no	—
292	C₃	—	9	3	no	—
293	C₃	#737×C₃₇⋊C₃	1	1	yes	characteristic
294	C₃	#737×C₃₇⋊C₃	1	1	yes	characteristic
295	C₃	C₃×#15688530	1	1	yes	characteristic
296	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×H(3,𝔽₃)×C₃₇	C₃²
hypercenter	C₉⋊H(3,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	#10624877
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. #22582594	C₃	semidirect
2. #22582593	C₃	semidirect
3. #22582590	C₃	semidirect
4. #22582585	C₃	semidirect
5. #22582585	C₃	semidirect
6. #22582580	C₃	semidirect
7. #22582576	C₃	semidirect
8. #22582575	C₃	semidirect
9. #22582574	C₃	semidirect
10. #22582573	C₃	semidirect
11. #22582572	C₃	semidirect
12. #22582571	C₃	semidirect
15. #22582548	C₃	semidirect
16. #22582548	C₃	semidirect
17. #22582548	C₃	semidirect
18. 3₋¹⁺²×#66524	C₃	semidirect
20. #22582535	C₃	semidirect
21. C₃×C₉×#66524	C₃	semidirect
229. C₃₇	#10624877	semidirect

Maximal subgroups

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

subgroup	index
#22582594	3
#22582593	3
#22582590	3
#22582585	3
#22582580	3
#22582576	3
#22582575	3
#22582574	3
#22582573	3
#22582572	3
#22582571	3
#22582562	3
#22582556	3
#22582548	3
3₋¹⁺²×#66524	3
H(3,𝔽₃)×#66524	3
#22582535	3
C₃×C₉×#66524	3
C₃₇×C₉⋊H(3,𝔽₃)	3
#10624877	37