Subgroups — #41007608

Order 26973.

662 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582598	C₃	1	2	yes	maximal, complemented
2	#22582597	C₃	1	2	yes	maximal, complemented
3	#22582593	C₃	1	2	yes	maximal, complemented
4	#22582592	C₃	1	2	yes	maximal, complemented
5	#22582590	C₃	1	1	yes	maximal, characteristic, complemented
6	#22582589	C₃	1	2	yes	maximal, complemented
7	#22582588	C₃	1	2	yes	maximal, complemented
8	#22582585	C₃	1	2	yes	maximal, complemented
9	#22582582	C₃	1	2	yes	maximal, complemented
10	#22582579	C₃	1	1	yes	maximal, characteristic, complemented
11	#22582577	C₃	1	2	yes	maximal, complemented
12	#22582570	C₃	1	2	yes	maximal, complemented
13	#22582569	C₃	1	2	yes	maximal, complemented
14	#22582563	C₃	1	1	yes	maximal, characteristic
15	#22582557	C₃	1	2	yes	maximal
16	C₉×#11758606	C₃	1	1	yes	maximal, characteristic, complemented
17	C₉×#11758606	C₃	1	2	yes	maximal, complemented
18	C₉×#11758605	C₃	1	1	yes	maximal, characteristic, complemented
19	C₉×#11758605	C₃	1	2	yes	maximal, complemented
20	C₉×#11758608	C₃	1	2	yes	maximal, complemented
21	C₉×C₃²⋊(C₃₇⋊C₃)	C₃	1	1	yes	maximal, characteristic
22	#22582536	C₃	1	2	yes	maximal, complemented
23	C₉²×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
24	C₃₇×#6887	C₃	1	1	yes	maximal, characteristic, complemented
25	C₃×#11758608	C₃²	1	2	yes	—
26	C₃×#11758605	C₃²	1	2	yes	—
27	C₃×#11758605	C₃²	1	1	yes	characteristic
28	C₃×#11758606	C₃²	1	2	yes	—
29	C₃×#11758606	C₃²	1	1	yes	characteristic
30	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	1	yes	characteristic
31	C₃×C₉×C₃₇⋊C₃	C₃²	1	2	yes	—
32	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
33	C₃×C₉×C₃₇⋊C₃	C₃²	1	2	yes	—
34	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
35	#15688506	C₃²	1	2	yes	—
36	#15688506	C₃²	1	2	yes	—
37	#15688506	C₃²	1	1	yes	characteristic
38	#15688506	C₃²	1	1	yes	characteristic
39	#15688505	C₃²	1	2	yes	—
40	#15688505	C₃²	1	2	yes	—
41	#15688505	C₃²	1	1	yes	characteristic
42	#15688505	C₃²	1	1	yes	characteristic
43	#15688504	C₃²	1	2	yes	—
44	#15688504	C₃²	1	2	yes	—
45	#15688504	C₃²	1	2	yes	—
46	#15688504	C₃²	1	2	yes	—
47	#15688504	C₃²	1	2	yes	—
48	#15688504	C₃²	1	2	yes	—
49	#15688504	C₃²	1	2	yes	—
50	#15688504	C₃²	1	2	yes	—
51	#15688504	C₃²	1	2	yes	—
52	#15688504	C₃²	1	2	yes	—
53	#15688504	C₃²	1	2	yes	—
54	#15688504	C₃²	1	2	yes	—
55	#15688504	C₃²	1	2	yes	—
56	#15688504	C₃²	1	2	yes	—
57	#15688504	C₃²	1	2	yes	—
58	#15688504	C₃²	1	2	yes	—
59	#15688504	C₃²	1	2	yes	—
60	#15688504	C₃²	1	2	yes	—
61	#15688504	C₃²	1	2	yes	—
62	#15688504	C₃²	1	2	yes	—
63	#15688504	C₃²	1	2	yes	—
64	#15688504	C₃²	1	2	yes	—
65	#15688504	C₃²	1	2	yes	—
66	#15688504	C₃²	1	2	yes	—
67	C₃²⋊#66524	C₃²	1	1	yes	characteristic
68	C₃²⋊#66524	C₃²	1	1	yes	characteristic
69	C₃²⋊#66524	C₃²	1	2	yes	—
70	C₃²⋊#66524	C₃²	1	2	yes	—
71	#15688502	C₃²	1	2	yes	—
72	#15688502	C₃²	1	2	yes	—
73	#15688502	C₃²	1	2	yes	—
74	#15688502	C₃²	1	2	yes	—
75	#15688502	C₃²	1	2	yes	—
76	#15688502	C₃²	1	2	yes	—
77	#15688502	C₃²	1	2	yes	—
78	#15688502	C₃²	1	2	yes	—
79	#15688502	C₃²	1	2	yes	—
80	#15688502	C₃²	1	2	yes	—
81	#15688502	C₃²	1	2	yes	—
82	#15688502	C₃²	1	2	yes	—
83	C₉×#66524	C₃²	1	2	yes	—
84	C₉×#66524	C₃²	1	2	yes	—
85	C₉×#66524	C₃²	1	2	yes	—
86	C₉×#66524	C₃²	1	1	yes	characteristic
87	C₉×#66524	C₃²	1	2	yes	—
88	C₉×#66524	C₃²	1	1	yes	characteristic
89	C₉×#66524	C₃²	1	2	yes	—
90	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
91	C₃₇×C₉⋊C₉	C₃²	1	2	yes	—
92	C₃₇×C₉⋊C₉	C₃²	1	2	yes	—
93	C₃₇×C₉⋊C₉	C₃²	1	2	yes	—
94	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
95	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
96	C₉²×C₃₇	C₃²	1	1	yes	characteristic
97	C₉²×C₃₇	C₃²	1	1	yes	characteristic
98	C₉²×C₃₇	C₃²	1	1	yes	characteristic
99	C₃³×C₃₇	C₃³	1	1	yes	characteristic
100	C₃²×C₃₇⋊C₃	C₃³	1	2	yes	—
101	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
102	C₃×C₉×C₃₇	—	3	3	no	—
103	C₃×C₉×C₃₇	C₃³	1	2	yes	—
104	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
105	C₃×C₉×C₃₇	C₃³	1	2	yes	—
106	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
107	C₃×C₉×C₃₇	C₃³	1	2	yes	—
108	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
109	C₃×C₉×C₃₇	C₃³	1	2	yes	—
110	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
111	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
112	#11758608	—	3	6	no	—
113	C₃×#66524	C₃³	1	2	yes	—
114	C₃×#66524	C₃³	1	2	yes	—
115	C₃×#66524	C₃³	1	1	yes	characteristic
116	C₃×#66524	C₃³	1	1	yes	characteristic
117	C₃×#66524	C₃³	1	2	yes	—
118	C₃×#66524	C₃³	1	2	yes	—
119	C₃×#66524	C₃³	1	2	yes	—
120	C₃×#66524	C₃³	1	2	yes	—
121	C₃×#66524	C₃³	1	2	yes	—
122	C₃×#66524	C₃³	1	2	yes	—
123	C₃×#66524	C₃³	1	2	yes	—
124	C₃×#66524	C₃³	1	2	yes	—
125	C₃×#66524	C₃³	1	2	yes	—
126	#11758606	—	3	3	no	—
127	#11758606	—	3	6	no	—
128	#11758605	—	3	3	no	—
129	#11758605	—	3	6	no	—
130	C₉×C₃₇⋊C₃	—	3	3	no	—
131	C₉×C₃₇⋊C₃	—	3	3	no	—
132	C₉×C₃₇⋊C₃	—	3	6	no	—
133	C₉×C₃₇⋊C₃	—	3	6	no	—
134	#10624877	—	37	1	no	maximal, Sylow
135	C₃²×C₃₇	—	3	2	no	—
136	C₃²×C₃₇	—	3	1	no	—
137	C₃²×C₃₇	—	3	1	no	—
138	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
139	C₃×C₃₇⋊C₃	—	3	2	no	—
140	C₃×C₃₇⋊C₃	—	3	1	no	—
141	C₃×C₃₇⋊C₃	—	3	2	no	—
142	C₃×C₃₇⋊C₃	—	3	2	no	—
143	C₃×C₃₇⋊C₃	—	3	2	no	—
144	C₃×C₃₇⋊C₃	—	3	2	no	—
145	C₃×C₃₇⋊C₃	#737	1	3	yes	—
146	C₉×C₃₇	—	3	1	no	—
147	C₉×C₃₇	—	3	2	no	—
148	C₉×C₃₇	—	3	1	no	—
149	C₉×C₃₇	—	3	2	no	—
150	C₉×C₃₇	—	3	1	no	—
151	C₉×C₃₇	—	3	2	no	—
152	C₉×C₃₇	—	3	2	no	—
153	C₉×C₃₇	#737	1	3	yes	—
154	#66524	—	3	2	no	—
155	#66524	—	3	2	no	—
156	#66524	—	3	1	no	—
157	#66524	—	3	1	no	—
158	#66524	—	3	2	no	—
159	#66524	—	3	2	no	—
160	#66524	—	3	2	no	—
161	#66524	—	3	2	no	—
162	#66524	—	3	2	no	—
163	#66524	—	3	2	no	—
164	#66524	—	3	2	no	—
165	#66524	—	3	2	no	—
166	#66524	#737	1	6	yes	—
167	#6899	—	37	2	no	—
168	#6899	—	37	2	no	—
169	#6899	—	37	2	no	—
170	#6898	—	37	2	no	—
171	#6898	—	37	1	no	—
172	#6897	—	37	2	no	—
173	#6897	—	37	2	no	—
174	#6897	—	37	2	no	—
175	#6896	—	37	2	no	—
176	#6896	—	37	1	no	—
177	#6895	—	37	2	no	—
178	#6894	—	37	2	no	—
179	#6894	—	37	2	no	—
180	#6893	—	37	1	no	—
181	C₉⋊H(3,𝔽₃)	—	37	2	no	—
182	C₉×3₋¹⁺²	—	37	2	no	—
183	C₉×3₋¹⁺²	—	37	2	no	—
184	C₉×3₋¹⁺²	—	37	1	no	—
185	C₉×3₋¹⁺²	—	37	2	no	—
186	C₉×3₋¹⁺²	—	37	1	no	—
187	C₉×H(3,𝔽₃)	—	37	1	no	—
188	#6887	—	37	2	no	—
189	#6887	C₃₇⋊C₃	1	1	yes	characteristic, complemented
190	C₃×C₉²	—	37	1	no	—
191	C₃×C₃₇	—	9	1	no	—
192	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
193	C₃×C₃₇	C₃×#737	1	2	yes	—
194	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
195	C₃₇⋊C₃	—	3	3	no	—
196	C₃₇⋊C₃	—	9	2	no	—
197	C₃×3₋¹⁺²	—	37	2	no	—
198	C₃×3₋¹⁺²	—	37	2	no	—
199	C₃×3₋¹⁺²	—	37	1	no	—
200	C₃×3₋¹⁺²	—	37	2	no	—
201	C₃×3₋¹⁺²	—	37	1	no	—
202	C₃×H(3,𝔽₃)	—	37	1	no	—
203	C₃²×C₉	—	37	2	no	—
204	C₃²×C₉	—	37	1	no	—
205	C₃²×C₉	—	37	1	no	—
206	C₃²×C₉	—	37	2	no	—
207	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
208	C₉⋊C₉	—	37	2	no	—
209	C₉⋊C₉	—	37	2	no	—
210	C₉⋊C₉	—	37	2	no	—
211	C₉⋊C₉	—	37	2	no	—
212	C₉⋊C₉	—	37	2	no	—
213	C₉⋊C₉	—	37	2	no	—
214	C₉⋊C₉	—	37	2	no	—
215	C₉⋊C₉	—	37	2	no	—
216	C₉⋊C₉	—	37	2	no	—
217	C₉⋊C₉	—	37	2	no	—
218	C₉⋊C₉	—	37	2	no	—
219	C₉⋊C₉	—	37	2	no	—
220	C₉⋊C₉	—	37	2	no	—
221	C₉⋊C₉	—	37	2	no	—
222	C₉⋊C₉	—	37	2	no	—
223	C₉⋊C₉	—	37	2	no	—
224	C₉⋊C₉	—	37	2	no	—
225	C₉⋊C₉	—	37	2	no	—
226	C₉⋊C₉	—	37	2	no	—
227	C₉⋊C₉	—	37	2	no	—
228	C₉⋊C₉	—	37	2	no	—
229	C₉⋊C₉	—	37	2	no	—
230	C₉⋊C₉	—	37	2	no	—
231	C₉⋊C₉	—	37	2	no	—
232	C₉⋊C₉	—	37	2	no	—
233	C₉⋊C₉	—	37	2	no	—
234	C₉⋊C₉	—	37	1	no	—
235	C₉⋊C₉	—	37	1	no	—
236	C₉⋊C₉	—	37	2	no	—
237	C₉⋊C₉	—	37	2	no	—
238	C₉⋊C₉	—	37	1	no	—
239	C₉⋊C₉	—	37	1	no	—
240	C₉⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	—
241	C₉⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	—
242	C₉⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	—
243	C₃²⋊C₉	—	37	1	no	—
244	C₃²⋊C₉	—	37	1	no	—
245	C₃²⋊C₉	—	37	2	no	—
246	C₃²⋊C₉	—	37	2	no	—
247	C₃²⋊C₉	—	37	2	no	—
248	C₃²⋊C₉	—	37	2	no	—
249	C₃²⋊C₉	—	37	2	no	—
250	C₃²⋊C₉	—	37	2	no	—
251	C₃²⋊C₉	—	37	2	no	—
252	C₃²⋊C₉	—	37	2	no	—
253	C₃²⋊C₉	—	37	2	no	—
254	C₃²⋊C₉	—	37	2	no	—
255	C₃²⋊C₉	—	37	2	no	—
256	C₃²⋊C₉	—	37	2	no	—
257	C₃²⋊C₉	—	37	2	no	—
258	C₃²⋊C₉	—	37	2	no	—
259	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	—
260	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
261	C₉²	—	37	1	no	—
262	C₉²	—	37	2	no	—
263	C₉²	—	37	1	no	—
264	C₉²	—	37	2	no	—
265	C₉²	—	37	2	no	—
266	C₉²	—	37	2	no	—
267	C₉²	—	37	2	no	—
268	C₉²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
269	C₉²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
270	C₉²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
271	C₃₇	#10624877	1	1	yes	characteristic, complemented, Sylow
272	C₃³	—	37	1	no	—
273	C₃³	—	37	2	no	—
274	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
275	3₋¹⁺²	—	111	6	no	—
276	3₋¹⁺²	—	111	6	no	—
277	3₋¹⁺²	—	111	3	no	—
278	3₋¹⁺²	—	111	3	no	—
279	3₋¹⁺²	—	111	6	no	—
280	H(3,𝔽₃)	—	111	3	no	—
281	C₃×C₉	—	3	3	no	—
282	C₃×C₉	—	37	2	no	—
283	C₃×C₉	—	37	2	no	—
284	C₃×C₉	—	37	2	no	—
285	C₃×C₉	—	37	1	no	—
286	C₃×C₉	—	37	1	no	—
287	C₃×C₉	—	37	2	no	—
288	C₃×C₉	—	37	2	no	—
289	C₃×C₉	—	37	2	no	—
290	C₃×C₉	—	37	2	no	—
291	C₃×C₉	—	37	2	no	—
292	C₃×C₉	—	37	2	no	—
293	C₃×C₉	—	37	2	no	—
294	C₃×C₉	—	37	2	no	—
295	C₃×C₉	—	111	6	no	—
296	C₃×C₉	—	111	3	no	—
297	C₃×C₉	—	111	6	no	—
298	C₃×C₉	—	111	3	no	—
299	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
300	C₃×C₉	C₃²×C₃₇⋊C₃	1	2	yes	—
301	C₃×C₉	C₃²×C₃₇⋊C₃	1	2	yes	—
302	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
303	C₃×C₉	C₃²×C₃₇⋊C₃	1	2	yes	—
304	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
305	C₃×C₉	C₃²×C₃₇⋊C₃	1	2	yes	—
306	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
307	C₃²	—	3	1	no	—
308	C₃²	—	3	2	no	—
309	C₃²	—	3	1	no	—
310	C₃²	—	37	3	no	—
311	C₃²	—	111	2	no	—
312	C₃²	—	111	2	no	—
313	C₃²	—	111	2	no	—
314	C₃²	—	111	2	no	—
315	C₃²	—	111	2	no	—
316	C₃²	—	111	1	no	—
317	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
318	C₉	—	3	2	no	—
319	C₉	—	3	1	no	—
320	C₉	—	3	2	no	—
321	C₉	—	3	2	no	—
322	C₉	—	3	1	no	—
323	C₉	—	3	1	no	—
324	C₉	—	3	2	no	—
325	C₉	—	37	6	no	—
326	C₉	—	111	2	no	—
327	C₉	—	111	2	no	—
328	C₉	—	111	2	no	—
329	C₉	—	111	2	no	—
330	C₉	—	111	2	no	—
331	C₉	—	111	2	no	—
332	C₉	—	111	2	no	—
333	C₉	—	111	2	no	—
334	C₉	—	111	1	no	—
335	C₉	—	111	1	no	—
336	C₉	—	111	2	no	—
337	C₉	—	111	2	no	—
338	C₉	#15688531	1	3	yes	—
339	C₃	—	9	1	no	—
340	C₃	—	333	2	no	—
341	C₃	—	111	3	no	—
342	C₃	#737×C₃₇⋊C₃	1	2	yes	—
343	C₃	#737×C₃₇⋊C₃	1	1	yes	characteristic
344	C₃	C₃×#15688531	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₉	C₃²×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Frattini	C₃²	C₃³×C₃₇⋊C₃
Fitting	C₃₇×#6887	C₃
prime radical	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²
hypercenter	#6887	C₃₇⋊C₃
hypocenter	C₃₇	#10624877
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ #6887

Complements

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

N	H ≅ G/N	type
1. #22582598	C₃	semidirect
2. #22582597	C₃	semidirect
3. #22582593	C₃	semidirect
4. #22582592	C₃	semidirect
5. #22582590	C₃	semidirect
6. #22582589	C₃	semidirect
7. #22582588	C₃	semidirect
8. #22582585	C₃	semidirect
9. #22582582	C₃	semidirect
10. #22582579	C₃	semidirect
11. #22582577	C₃	semidirect
12. #22582570	C₃	semidirect
13. #22582569	C₃	semidirect
16. C₉×#11758606	C₃	semidirect
17. C₉×#11758606	C₃	semidirect
18. C₉×#11758605	C₃	semidirect
19. C₉×#11758605	C₃	semidirect
20. C₉×#11758608	C₃	semidirect
22. #22582536	C₃	semidirect
23. C₉²×C₃₇⋊C₃	C₃	semidirect
24. C₃₇×#6887	C₃	semidirect
189. #6887	C₃₇⋊C₃	semidirect
271. C₃₇	#10624877	semidirect

Maximal subgroups

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

subgroup	index
#22582598	3
#22582597	3
#22582593	3
#22582592	3
#22582590	3
#22582589	3
#22582588	3
#22582585	3
#22582582	3
#22582579	3
#22582577	3
#22582570	3
#22582569	3
#22582563	3
#22582557	3
C₉×#11758606	3
C₉×#11758605	3
C₉×#11758608	3
C₉×C₃²⋊(C₃₇⋊C₃)	3
#22582536	3
C₉²×C₃₇⋊C₃	3
C₃₇×#6887	3
#10624877	37