Subgroups — #41007579

Order 26973.

824 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₉×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
2	#22582577	C₃	1	3	yes	maximal, complemented
3	#22582577	C₃	1	3	yes	maximal, complemented
4	#22582574	C₃	1	1	yes	maximal, characteristic, complemented
5	#22582573	C₃	1	1	yes	maximal, characteristic, complemented
6	#22582572	C₃	1	1	yes	maximal, characteristic, complemented
7	#22582571	C₃	1	1	yes	maximal, characteristic, complemented
8	#22582570	C₃	1	3	yes	maximal, complemented
9	#22582569	C₃	1	3	yes	maximal, complemented
10	#22582568	C₃	1	1	yes	maximal, characteristic, complemented
11	#22582567	C₃	1	3	yes	maximal, complemented
12	#22582566	C₃	1	3	yes	maximal, complemented
13	#22582553	C₃	1	1	yes	maximal, characteristic
14	#22582552	C₃	1	1	yes	maximal, characteristic
15	#22582551	C₃	1	1	yes	maximal, characteristic
16	#22582548	C₃	1	1	yes	maximal, characteristic, complemented
17	#22582548	C₃	1	1	yes	maximal, characteristic, complemented
18	3₋¹⁺²×#66524	C₃	1	1	yes	maximal, characteristic, complemented
19	#22582543	C₃	1	3	yes	maximal, complemented
20	#22582542	C₃	1	3	yes	maximal, complemented
21	#22582537	C₃	1	3	yes	maximal, complemented
22	C₃₇×#6894	C₃	1	1	yes	maximal, characteristic, complemented
23	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
24	C₃×#11758608	C₃²	1	1	yes	characteristic
25	C₃×#11758608	C₃²	1	1	yes	characteristic
26	C₃×#11758605	C₃²	1	3	yes	—
27	C₃×#11758606	C₃²	1	3	yes	—
28	C₃²×#66524	C₃²	1	1	yes	characteristic
29	C₃²×#66524	C₃²	1	1	yes	characteristic
30	C₃×C₉×C₃₇⋊C₃	—	3	9	no	—
31	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
32	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
33	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
34	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
35	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
36	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
37	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
38	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
39	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
40	#15688506	C₃²	1	3	yes	complemented
41	#15688506	C₃²	1	3	yes	complemented
42	#15688506	C₃²	1	3	yes	complemented
43	#15688505	C₃²	1	3	yes	complemented
44	#15688505	C₃²	1	3	yes	complemented
45	#15688505	C₃²	1	3	yes	complemented
46	#15688504	C₃²	1	3	yes	complemented
47	#15688504	C₃²	1	3	yes	complemented
48	#15688504	C₃²	1	3	yes	complemented
49	#15688504	C₃²	1	3	yes	complemented
50	#15688504	C₃²	1	3	yes	complemented
51	#15688504	C₃²	1	3	yes	complemented
52	#15688504	C₃²	1	3	yes	complemented
53	#15688504	C₃²	1	3	yes	complemented
54	#15688504	C₃²	1	3	yes	complemented
55	#15688504	C₃²	1	3	yes	complemented
56	#15688504	C₃²	1	3	yes	complemented
57	#15688504	C₃²	1	3	yes	complemented
58	#15688504	C₃²	1	3	yes	complemented
59	#15688504	C₃²	1	3	yes	complemented
60	#15688504	C₃²	1	3	yes	complemented
61	C₃²⋊#66524	C₃²	1	1	yes	characteristic
62	C₃²⋊#66524	C₃²	1	1	yes	characteristic
63	C₃²⋊#66524	C₃²	1	1	yes	characteristic
64	C₃²⋊#66524	C₃²	1	1	yes	characteristic
65	#15688502	C₃²	1	3	yes	—
66	#15688502	C₃²	1	3	yes	—
67	#15688502	C₃²	1	3	yes	—
68	#15688502	C₃²	1	3	yes	—
69	#15688502	C₃²	1	3	yes	—
70	#15688502	C₃²	1	3	yes	—
71	#15688502	C₃²	1	3	yes	—
72	C₉×#66524	C₃²	1	3	yes	complemented
73	C₉×#66524	C₃²	1	3	yes	complemented
74	C₉×#66524	C₃²	1	3	yes	complemented
75	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
76	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
77	C₃₇×C₉⋊C₉	C₃²	1	3	yes	complemented
78	C₃₇×C₉⋊C₉	C₃²	1	3	yes	complemented
79	C₃₇×C₉⋊C₉	C₃²	1	3	yes	complemented
80	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
81	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
82	C₃³×C₃₇	C₃³	1	1	yes	characteristic
83	C₃²×C₃₇⋊C₃	—	3	3	no	—
84	C₃²×C₃₇⋊C₃	—	3	3	no	—
85	C₃²×C₃₇⋊C₃	—	3	3	no	—
86	C₃²×C₃₇⋊C₃	—	3	3	no	—
87	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
88	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
89	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
90	3₋¹⁺²×C₃₇	—	3	3	no	—
91	C₃×C₉×C₃₇	—	3	3	no	—
92	C₃×C₉×C₃₇	C₃³	1	3	yes	—
93	C₃×C₉×C₃₇	C₃³	1	3	yes	—
94	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
95	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
96	C₃×C₉×C₃₇	C₃³	1	3	yes	—
97	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
98	#11758608	—	3	3	no	—
99	#11758608	—	3	3	no	—
100	C₃×#66524	—	3	3	no	—
101	C₃×#66524	—	3	3	no	—
102	C₃×#66524	C₃³	1	3	yes	—
103	C₃×#66524	C₃³	1	3	yes	—
104	C₃×#66524	C₃³	1	3	yes	—
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	1	yes	characteristic
109	C₃×#66524	C₃³	1	1	yes	characteristic
110	C₃×#66524	C₃³	1	1	yes	characteristic
111	C₃×#66524	C₃³	1	1	yes	characteristic
112	C₃×#66524	C₃³	1	1	yes	characteristic
113	C₃×#66524	C₃³	1	1	yes	characteristic
114	#11758606	—	3	9	no	—
115	#11758605	—	3	9	no	—
116	C₉×C₃₇⋊C₃	—	3	3	no	—
117	C₉×C₃₇⋊C₃	—	3	3	no	—
118	C₉×C₃₇⋊C₃	—	3	3	no	—
119	C₉×C₃₇⋊C₃	—	3	3	no	—
120	C₉×C₃₇⋊C₃	—	3	3	no	—
121	C₉×C₃₇⋊C₃	—	3	3	no	—
122	C₉×C₃₇⋊C₃	—	3	3	no	—
123	C₉×C₃₇⋊C₃	—	3	3	no	—
124	C₉×C₃₇⋊C₃	3₋¹⁺²	1	9	yes	complemented
125	#10624872	—	37	1	no	maximal, Sylow
126	C₃²×C₃₇	—	3	1	no	—
127	C₃²×C₃₇	—	3	1	no	—
128	C₃²×C₃₇	—	3	1	no	—
129	C₃²×C₃₇	#737	1	3	yes	—
130	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
131	C₃×C₃₇⋊C₃	—	3	27	no	—
132	C₃×C₃₇⋊C₃	—	3	1	no	—
133	C₃×C₃₇⋊C₃	—	3	1	no	—
134	C₃×C₃₇⋊C₃	—	3	1	no	—
135	C₃×C₃₇⋊C₃	—	3	1	no	—
136	C₃×C₃₇⋊C₃	—	3	1	no	—
137	C₃×C₃₇⋊C₃	—	3	1	no	—
138	C₃×C₃₇⋊C₃	—	3	1	no	—
139	C₃×C₃₇⋊C₃	—	3	1	no	—
140	C₃×C₃₇⋊C₃	—	3	1	no	—
141	C₃×C₃₇⋊C₃	C₃×3₋¹⁺²	1	3	yes	—
142	C₃×C₃₇⋊C₃	#737	1	3	yes	—
143	C₃×C₃₇⋊C₃	#737	1	3	yes	—
144	C₉×C₃₇	—	3	1	no	—
145	C₉×C₃₇	—	3	1	no	—
146	C₉×C₃₇	—	3	3	no	—
147	C₉×C₃₇	—	3	3	no	—
148	C₉×C₃₇	—	3	3	no	—
149	C₉×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
150	#66524	—	3	1	no	—
151	#66524	—	3	1	no	—
152	#66524	—	3	1	no	—
153	#66524	—	3	1	no	—
154	#66524	—	3	3	no	—
155	#66524	—	3	3	no	—
156	#66524	—	3	3	no	—
157	#66524	—	3	3	no	—
158	#66524	—	3	3	no	—
159	#66524	—	3	3	no	—
160	#66524	C₃×3₋¹⁺²	1	3	yes	—
161	#66524	C₃×3₋¹⁺²	1	3	yes	—
162	C₃³×C₉	—	37	1	no	—
163	#6895	—	37	3	no	—
164	#6895	—	37	3	no	—
165	#6894	—	37	1	no	—
166	#6894	—	37	1	no	—
167	#6894	—	37	1	no	—
168	#6894	—	37	3	no	—
169	#6894	—	37	3	no	—
170	#6894	—	37	3	no	—
171	#6894	—	37	3	no	—
172	#6894	—	37	1	no	—
173	#6894	—	37	1	no	—
174	#6894	C₃₇⋊C₃	1	1	yes	characteristic, complemented
175	#6891	—	37	1	no	—
176	#6891	—	37	1	no	—
177	#6891	—	37	1	no	—
178	C₉×3₋¹⁺²	—	37	1	no	—
179	C₉×3₋¹⁺²	—	37	3	no	—
180	C₉×3₋¹⁺²	—	37	3	no	—
181	C₉×3₋¹⁺²	—	37	1	no	—
182	C₉×3₋¹⁺²	—	37	1	no	—
183	#6887	—	37	3	no	—
184	C₃×C₃₇	—	3	3	no	—
185	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
186	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
187	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
188	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
189	C₃₇⋊C₃	—	3	3	no	—
190	C₃₇⋊C₃	—	3	3	no	—
191	C₃₇⋊C₃	—	3	3	no	—
192	C₃⁴	—	37	1	no	—
193	C₃×3₋¹⁺²	—	37	1	no	—
194	C₃×3₋¹⁺²	—	37	1	no	—
195	C₃×3₋¹⁺²	—	37	3	no	—
196	C₃×3₋¹⁺²	—	37	3	no	—
197	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
198	C₃²×C₉	—	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	C₃²×C₉	—	37	1	no	—
204	C₃²×C₉	—	37	1	no	—
205	C₃²×C₉	—	37	1	no	—
206	C₃²×C₉	—	37	1	no	—
207	C₃²×C₉	—	37	1	no	—
208	C₃²×C₉	—	37	1	no	—
209	C₃²×C₉	—	111	9	no	—
210	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
211	C₉⋊C₉	—	37	3	no	—
212	C₉⋊C₉	—	37	3	no	—
213	C₉⋊C₉	—	37	3	no	—
214	C₉⋊C₉	—	37	3	no	—
215	C₉⋊C₉	—	37	3	no	—
216	C₉⋊C₉	—	37	3	no	—
217	C₉⋊C₉	—	37	3	no	—
218	C₉⋊C₉	—	37	3	no	—
219	C₉⋊C₉	—	37	3	no	—
220	C₉⋊C₉	—	37	3	no	—
221	C₉⋊C₉	—	37	3	no	—
222	C₉⋊C₉	—	37	3	no	—
223	C₉⋊C₉	—	37	3	no	—
224	C₉⋊C₉	—	37	3	no	—
225	C₉⋊C₉	—	37	3	no	—
226	C₉⋊C₉	—	37	3	no	—
227	C₉⋊C₉	—	37	3	no	—
228	C₉⋊C₉	—	37	3	no	—
229	C₉⋊C₉	—	37	3	no	—
230	C₉⋊C₉	—	37	3	no	—
231	C₉⋊C₉	—	37	3	no	—
232	C₉⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	complemented
233	C₉⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	complemented
234	C₉⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	complemented
235	C₃²⋊C₉	—	37	1	no	—
236	C₃²⋊C₉	—	37	1	no	—
237	C₃²⋊C₉	—	37	1	no	—
238	C₃²⋊C₉	—	37	1	no	—
239	C₃²⋊C₉	—	37	3	no	—
240	C₃²⋊C₉	—	37	3	no	—
241	C₃²⋊C₉	—	37	3	no	—
242	C₃²⋊C₉	—	37	3	no	—
243	C₃²⋊C₉	—	37	3	no	—
244	C₃²⋊C₉	—	37	3	no	—
245	C₃²⋊C₉	—	37	3	no	—
246	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
247	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
248	C₉²	—	37	3	no	—
249	C₉²	—	37	3	no	—
250	C₉²	—	37	3	no	—
251	C₃₇	#10624872	1	1	yes	characteristic, complemented, Sylow
252	C₃³	—	37	1	no	—
253	C₃³	—	37	1	no	—
254	C₃³	—	37	1	no	—
255	C₃³	—	111	3	no	—
256	C₃³	—	111	3	no	—
257	C₃³	—	111	3	no	—
258	C₃³	—	111	3	no	—
259	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
260	3₋¹⁺²	—	3	3	no	—
261	3₋¹⁺²	—	111	9	no	—
262	3₋¹⁺²	—	111	9	no	—
263	3₋¹⁺²	—	111	3	no	—
264	3₋¹⁺²	—	111	3	no	—
265	C₃×C₉	—	3	3	no	—
266	C₃×C₉	—	37	1	no	—
267	C₃×C₉	—	37	1	no	—
268	C₃×C₉	—	37	9	no	—
269	C₃×C₉	—	37	3	no	—
270	C₃×C₉	—	37	3	no	—
271	C₃×C₉	—	37	3	no	—
272	C₃×C₉	—	37	3	no	—
273	C₃×C₉	—	37	3	no	—
274	C₃×C₉	—	37	3	no	—
275	C₃×C₉	—	37	1	no	—
276	C₃×C₉	—	37	1	no	—
277	C₃×C₉	—	37	1	no	—
278	C₃×C₉	—	37	1	no	—
279	C₃×C₉	—	111	3	no	—
280	C₃×C₉	—	111	3	no	—
281	C₃×C₉	—	111	3	no	—
282	C₃×C₉	—	111	3	no	—
283	C₃×C₉	—	111	3	no	—
284	C₃×C₉	—	111	3	no	—
285	C₃×C₉	—	111	3	no	—
286	C₃×C₉	—	111	3	no	—
287	C₃×C₉	—	111	3	no	—
288	C₃×C₉	—	111	3	no	—
289	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
290	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
291	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
292	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
293	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
294	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
295	C₃²	—	3	1	no	—
296	C₃²	—	3	1	no	—
297	C₃²	—	3	1	no	—
298	C₃²	—	37	3	no	—
299	C₃²	—	37	3	no	—
300	C₃²	—	37	3	no	—
301	C₃²	—	111	27	no	—
302	C₃²	—	111	1	no	—
303	C₃²	—	111	1	no	—
304	C₃²	—	111	1	no	—
305	C₃²	—	111	1	no	—
306	C₃²	—	111	1	no	—
307	C₃²	—	111	1	no	—
308	C₃²	—	111	1	no	—
309	C₃²	—	111	1	no	—
310	C₃²	—	111	1	no	—
311	C₃²	#15688532	1	3	yes	—
312	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
313	C₉	—	3	1	no	—
314	C₉	—	3	1	no	—
315	C₉	—	3	3	no	—
316	C₉	—	3	3	no	—
317	C₉	—	3	3	no	—
318	C₉	—	37	3	no	—
319	C₉	—	37	3	no	—
320	C₉	—	111	1	no	—
321	C₉	—	111	1	no	—
322	C₉	—	111	1	no	—
323	C₉	—	111	1	no	—
324	C₉	—	111	3	no	—
325	C₉	—	111	3	no	—
326	C₉	—	111	3	no	—
327	C₉	—	111	3	no	—
328	C₉	—	111	3	no	—
329	C₉	—	111	3	no	—
330	C₉	3₋¹⁺²×C₃₇⋊C₃	1	3	yes	—
331	C₃	—	3	3	no	—
332	C₃	—	111	3	no	—
333	C₃	—	111	3	no	—
334	C₃	—	111	3	no	—
335	C₃	C₃×3₋¹⁺²×C₃₇⋊C₃	1	1	yes	characteristic
336	C₃	#737×C₃₇⋊C₃	1	1	yes	characteristic
337	C₃	#737×C₃₇⋊C₃	1	1	yes	characteristic
338	C₃	C₃×#15688532	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting	C₃₇×#6894	C₃
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	#6894	C₃₇⋊C₃
hypocenter	C₃₇	#10624872
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ #6894

Complements

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

N	H ≅ G/N	type
2. #22582577	C₃	semidirect
3. #22582577	C₃	semidirect
4. #22582574	C₃	semidirect
5. #22582573	C₃	semidirect
6. #22582572	C₃	semidirect
7. #22582571	C₃	semidirect
8. #22582570	C₃	semidirect
9. #22582569	C₃	semidirect
10. #22582568	C₃	semidirect
11. #22582567	C₃	semidirect
12. #22582566	C₃	semidirect
16. #22582548	C₃	semidirect
17. #22582548	C₃	semidirect
18. 3₋¹⁺²×#66524	C₃	semidirect
19. #22582543	C₃	semidirect
20. #22582542	C₃	semidirect
21. #22582537	C₃	semidirect
22. C₃₇×#6894	C₃	semidirect
40. #15688506	C₃²	semidirect
41. #15688506	C₃²	semidirect
42. #15688506	C₃²	semidirect
43. #15688505	C₃²	semidirect
44. #15688505	C₃²	semidirect
45. #15688505	C₃²	semidirect
46. #15688504	C₃²	semidirect
47. #15688504	C₃²	semidirect
48. #15688504	C₃²	semidirect
49. #15688504	C₃²	semidirect
50. #15688504	C₃²	semidirect
51. #15688504	C₃²	semidirect
52. #15688504	C₃²	semidirect
53. #15688504	C₃²	semidirect
54. #15688504	C₃²	semidirect
55. #15688504	C₃²	semidirect
56. #15688504	C₃²	semidirect
57. #15688504	C₃²	semidirect
58. #15688504	C₃²	semidirect
59. #15688504	C₃²	semidirect
60. #15688504	C₃²	semidirect
72. C₉×#66524	C₃²	semidirect
73. C₉×#66524	C₃²	semidirect
74. C₉×#66524	C₃²	semidirect
77. C₃₇×C₉⋊C₉	C₃²	semidirect
78. C₃₇×C₉⋊C₉	C₃²	semidirect
79. C₃₇×C₉⋊C₉	C₃²	semidirect
124. C₉×C₃₇⋊C₃	3₋¹⁺²	semidirect
174. #6894	C₃₇⋊C₃	semidirect
232. C₉⋊C₉	C₃×C₃₇⋊C₃	semidirect
233. C₉⋊C₉	C₃×C₃₇⋊C₃	semidirect
234. C₉⋊C₉	C₃×C₃₇⋊C₃	semidirect
251. C₃₇	#10624872	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₉×C₃₇⋊C₃	3
#22582577	3
#22582574	3
#22582573	3
#22582572	3
#22582571	3
#22582570	3
#22582569	3
#22582568	3
#22582567	3
#22582566	3
#22582553	3
#22582552	3
#22582551	3
#22582548	3
3₋¹⁺²×#66524	3
#22582543	3
#22582542	3
#22582537	3
C₃₇×#6894	3
#10624872	37