Subgroups — #28356082

Order 13851.

728 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×3₋¹⁺²×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic
2	#17987444	C₃	1	3	yes	maximal, complemented
3	#17987444	C₃	1	3	yes	maximal, complemented
4	#17987444	C₃	1	3	yes	maximal, complemented
5	#17987441	C₃	1	1	yes	maximal, characteristic, complemented
6	#17987441	C₃	1	1	yes	maximal, characteristic, complemented
7	#17987440	C₃	1	1	yes	maximal, characteristic, complemented
8	#17987440	C₃	1	1	yes	maximal, characteristic, complemented
9	#17987439	C₃	1	1	yes	maximal, characteristic, complemented
10	#17987438	C₃	1	1	yes	maximal, characteristic, complemented
11	#17987437	C₃	1	3	yes	maximal, complemented
12	#17987437	C₃	1	3	yes	maximal, complemented
13	#17987436	C₃	1	3	yes	maximal, complemented
14	#17987436	C₃	1	3	yes	maximal, complemented
15	#17987435	C₃	1	1	yes	maximal, characteristic, complemented
16	#17987435	C₃	1	1	yes	maximal, characteristic, complemented
17	#17987434	C₃	1	3	yes	maximal, complemented
18	#17987433	C₃	1	3	yes	maximal, complemented
19	C₃×#13095303	C₃	1	1	yes	maximal, characteristic
20	C₁₉⋊C₃×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
21	C₁₉⋊C₃×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
22	C₁₉×#6894	C₃	1	1	yes	maximal, characteristic, complemented
23	C₃³×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
24	C₃×#10587025	C₃²	1	1	yes	characteristic
25	C₃×#10587025	C₃²	1	1	yes	characteristic
26	C₃×#10587022	C₃²	1	3	yes	—
27	C₃×#10587023	C₃²	1	3	yes	—
28	3₋¹⁺²×C₁₉⋊C₃	—	3	9	no	—
29	C₃²×#4294	C₃²	1	1	yes	characteristic
30	C₃²×#4294	C₃²	1	1	yes	characteristic
31	C₃×C₉×C₁₉⋊C₃	C₃²	1	3	yes	—
32	C₃×C₉×C₁₉⋊C₃	C₃²	1	3	yes	—
33	C₃×C₉×C₁₉⋊C₃	C₃²	1	3	yes	—
34	#13095307	C₃²	1	3	yes	complemented
35	#13095307	C₃²	1	3	yes	complemented
36	#13095307	C₃²	1	3	yes	complemented
37	#13095306	C₃²	1	3	yes	complemented
38	#13095306	C₃²	1	3	yes	complemented
39	#13095306	C₃²	1	3	yes	complemented
40	#13095305	C₃²	1	3	yes	complemented
41	#13095305	C₃²	1	3	yes	complemented
42	#13095305	C₃²	1	3	yes	complemented
43	#13095305	C₃²	1	3	yes	complemented
44	#13095305	C₃²	1	3	yes	complemented
45	#13095305	C₃²	1	3	yes	complemented
46	#13095305	C₃²	1	3	yes	complemented
47	#13095305	C₃²	1	3	yes	complemented
48	#13095305	C₃²	1	3	yes	complemented
49	#13095305	C₃²	1	3	yes	complemented
50	#13095305	C₃²	1	3	yes	complemented
51	#13095305	C₃²	1	3	yes	complemented
52	#13095305	C₃²	1	3	yes	complemented
53	#13095305	C₃²	1	3	yes	complemented
54	#13095305	C₃²	1	3	yes	complemented
55	#13095305	C₃²	1	3	yes	complemented
56	#13095305	C₃²	1	3	yes	complemented
57	#13095305	C₃²	1	3	yes	complemented
58	C₃²⋊#4294	C₃²	1	1	yes	characteristic
59	C₃²⋊#4294	C₃²	1	1	yes	characteristic
60	C₃²⋊#4294	C₃²	1	1	yes	characteristic
61	C₃²⋊#4294	C₃²	1	1	yes	characteristic
62	#13095303	C₃²	1	3	yes	—
63	#13095303	C₃²	1	3	yes	—
64	#13095303	C₃²	1	3	yes	—
65	#13095303	C₃²	1	3	yes	—
66	#13095303	C₃²	1	1	yes	characteristic
67	#13095303	C₃²	1	1	yes	characteristic
68	#13095303	C₃²	1	1	yes	characteristic
69	#13095303	C₃²	1	1	yes	characteristic
70	#13095303	C₃²	1	1	yes	characteristic
71	#13095303	C₃²	1	1	yes	characteristic
72	#13095303	C₃²	1	1	yes	characteristic
73	#13095303	C₃²	1	1	yes	characteristic
74	#13095303	C₃²	1	1	yes	characteristic
75	C₃×C₁₉×3₋¹⁺²	C₃²	1	1	yes	characteristic
76	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
77	C₁₉×C₉⋊C₉	C₃²	1	3	yes	complemented
78	C₁₉×C₉⋊C₉	C₃²	1	3	yes	complemented
79	C₁₉×C₉⋊C₉	C₃²	1	3	yes	complemented
80	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
81	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
82	#10624886	—	19	1	no	maximal, Sylow
83	C₃³×C₁₉	C₃³	1	1	yes	characteristic
84	C₃²×C₁₉⋊C₃	—	9	1	no	—
85	C₃²×C₁₉⋊C₃	—	9	1	no	—
86	C₃²×C₁₉⋊C₃	—	9	1	no	—
87	C₃²×C₁₉⋊C₃	3₋¹⁺²	1	9	yes	—
88	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
89	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
90	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
91	C₁₉×3₋¹⁺²	—	3	3	no	—
92	C₃×C₉×C₁₉	—	9	1	no	—
93	C₃×C₉×C₁₉	C₃³	1	3	yes	—
94	C₃×C₉×C₁₉	C₃³	1	3	yes	—
95	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
96	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
97	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
98	C₃×C₉×C₁₉	C₃³	1	3	yes	—
99	#10587025	—	3	3	no	—
100	#10587025	—	3	3	no	—
101	C₃×#4294	—	9	1	no	—
102	C₃×#4294	—	9	1	no	—
103	C₃×#4294	C₃³	1	1	yes	characteristic
104	C₃×#4294	C₃³	1	1	yes	characteristic
105	C₃×#4294	C₃³	1	1	yes	characteristic
106	C₃×#4294	C₃³	1	1	yes	characteristic
107	C₃×#4294	C₃³	1	1	yes	characteristic
108	C₃×#4294	C₃³	1	1	yes	characteristic
109	C₃×#4294	C₃³	1	3	yes	—
110	C₃×#4294	C₃³	1	3	yes	—
111	C₃×#4294	C₃³	1	3	yes	—
112	C₃×#4294	C₃³	1	3	yes	—
113	C₃×#4294	C₃³	1	3	yes	—
114	C₃×#4294	C₃³	1	3	yes	—
115	#10587023	—	3	9	no	—
116	#10587022	—	3	9	no	—
117	C₉×C₁₉⋊C₃	—	3	9	no	—
118	C₉×C₁₉⋊C₃	—	9	3	no	—
119	C₉×C₁₉⋊C₃	—	9	3	no	—
120	C₃²×3₋¹⁺²	—	19	1	no	—
121	#6895	—	19	3	no	—
122	#6895	—	19	3	no	—
123	#6895	—	19	3	no	—
124	#6894	—	19	1	no	—
125	#6894	—	19	1	no	—
126	#6894	—	19	1	no	—
127	#6894	—	19	1	no	—
128	#6894	—	19	3	no	—
129	#6894	—	19	3	no	—
130	#6894	—	19	3	no	—
131	#6894	—	19	3	no	—
132	#6894	—	19	3	no	—
133	#6894	—	19	3	no	—
134	#6894	—	19	1	no	—
135	#6894	—	19	1	no	—
136	#6894	—	19	1	no	—
137	#6894	—	19	1	no	—
138	#6894	C₁₉⋊C₃	1	1	yes	characteristic, complemented
139	C₃×C₃²⋊C₉	—	19	1	no	—
140	C₃×C₃²⋊C₉	—	19	1	no	—
141	C₃×C₃²⋊C₉	—	19	1	no	—
142	C₃²×C₁₉	—	3	1	no	—
143	C₃²×C₁₉	—	3	1	no	—
144	C₃²×C₁₉	—	3	1	no	—
145	C₃²×C₁₉	C₃×3₋¹⁺²	1	3	yes	—
146	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
147	C₃×C₁₉⋊C₃	—	3	1	no	—
148	C₃×C₁₉⋊C₃	—	3	1	no	—
149	C₃×C₁₉⋊C₃	—	3	1	no	—
150	C₃×C₁₉⋊C₃	—	3	1	no	—
151	C₃×C₁₉⋊C₃	—	3	1	no	—
152	C₃×C₁₉⋊C₃	—	3	1	no	—
153	C₃×C₁₉⋊C₃	—	3	1	no	—
154	C₃×C₁₉⋊C₃	—	3	1	no	—
155	C₃×C₁₉⋊C₃	—	3	1	no	—
156	C₃×C₁₉⋊C₃	—	9	9	no	—
157	C₃×C₁₉⋊C₃	C₃×3₋¹⁺²	1	3	yes	—
158	C₃×C₁₉⋊C₃	C₃×3₋¹⁺²	1	3	yes	—
159	C₃×C₁₉⋊C₃	C₃×3₋¹⁺²	1	3	yes	—
160	C₉×C₁₉	—	3	3	no	—
161	C₉×C₁₉	—	3	3	no	—
162	C₉×C₁₉	—	3	3	no	—
163	C₉×C₁₉	—	3	1	no	—
164	C₉×C₁₉	—	3	1	no	—
165	C₉×C₁₉	—	3	1	no	—
166	#4294	—	3	3	no	—
167	#4294	—	3	3	no	—
168	#4294	—	3	3	no	—
169	#4294	—	3	3	no	—
170	#4294	—	3	3	no	—
171	#4294	—	3	3	no	—
172	#4294	—	3	1	no	—
173	#4294	—	3	1	no	—
174	#4294	—	3	1	no	—
175	#4294	—	3	1	no	—
176	#4294	—	3	1	no	—
177	#4294	—	3	1	no	—
178	C₃⁴	—	19	1	no	—
179	C₃×3₋¹⁺²	—	19	1	no	—
180	C₃×3₋¹⁺²	—	19	1	no	—
181	C₃×3₋¹⁺²	—	19	3	no	—
182	C₃×3₋¹⁺²	—	19	3	no	—
183	C₃×3₋¹⁺²	—	57	9	no	—
184	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
185	C₃²×C₉	—	19	1	no	—
186	C₃²×C₉	—	19	1	no	—
187	C₃²×C₉	—	19	3	no	—
188	C₃²×C₉	—	19	3	no	—
189	C₃²×C₉	—	19	3	no	—
190	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
191	C₉⋊C₉	—	19	3	no	—
192	C₉⋊C₉	—	19	3	no	—
193	C₉⋊C₉	—	19	3	no	—
194	C₉⋊C₉	—	19	3	no	—
195	C₉⋊C₉	—	19	3	no	—
196	C₉⋊C₉	—	19	3	no	—
197	C₉⋊C₉	—	19	3	no	—
198	C₉⋊C₉	—	19	3	no	—
199	C₉⋊C₉	—	19	3	no	—
200	C₉⋊C₉	—	19	3	no	—
201	C₉⋊C₉	—	19	3	no	—
202	C₉⋊C₉	—	19	3	no	—
203	C₉⋊C₉	—	19	3	no	—
204	C₉⋊C₉	—	19	3	no	—
205	C₉⋊C₉	—	19	3	no	—
206	C₉⋊C₉	—	19	3	no	—
207	C₉⋊C₉	—	19	3	no	—
208	C₉⋊C₉	—	19	3	no	—
209	C₉⋊C₉	—	19	3	no	—
210	C₉⋊C₉	—	19	3	no	—
211	C₉⋊C₉	—	19	3	no	—
212	C₉⋊C₉	—	19	3	no	—
213	C₉⋊C₉	—	19	3	no	—
214	C₉⋊C₉	—	19	3	no	—
215	C₉⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	complemented
216	C₉⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	complemented
217	C₉⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	complemented
218	C₃²⋊C₉	—	19	1	no	—
219	C₃²⋊C₉	—	19	1	no	—
220	C₃²⋊C₉	—	19	1	no	—
221	C₃²⋊C₉	—	19	1	no	—
222	C₃²⋊C₉	—	19	1	no	—
223	C₃²⋊C₉	—	19	1	no	—
224	C₃²⋊C₉	—	19	1	no	—
225	C₃²⋊C₉	—	19	1	no	—
226	C₃²⋊C₉	—	19	1	no	—
227	C₃²⋊C₉	—	19	1	no	—
228	C₃²⋊C₉	—	19	3	no	—
229	C₃²⋊C₉	—	19	3	no	—
230	C₃²⋊C₉	—	19	1	no	—
231	C₃²⋊C₉	—	19	1	no	—
232	C₃²⋊C₉	—	19	1	no	—
233	C₃²⋊C₉	—	19	3	no	—
234	C₃²⋊C₉	—	19	3	no	—
235	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
236	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
237	C₃×C₁₉	—	3	3	no	—
238	C₃×C₁₉	C₃²×3₋¹⁺²	1	1	yes	characteristic
239	C₃×C₁₉	3₋¹⁺⁴	1	1	yes	characteristic
240	C₃×C₁₉	3₋¹⁺⁴	1	1	yes	characteristic
241	C₃×C₁₉	3₋¹⁺⁴	1	1	yes	characteristic
242	C₁₉⋊C₃	—	3	3	no	—
243	C₁₉⋊C₃	—	3	3	no	—
244	C₁₉⋊C₃	—	3	3	no	—
245	C₃³	—	19	1	no	—
246	C₃³	—	19	1	no	—
247	C₃³	—	19	1	no	—
248	C₃³	—	19	9	no	—
249	C₃³	—	171	1	no	—
250	C₃³	—	171	1	no	—
251	C₃³	—	171	1	no	—
252	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
253	3₋¹⁺²	—	3	3	no	—
254	3₋¹⁺²	—	57	9	no	—
255	3₋¹⁺²	—	57	9	no	—
256	3₋¹⁺²	—	57	3	no	—
257	3₋¹⁺²	—	57	3	no	—
258	C₃×C₉	—	9	1	no	—
259	C₃×C₉	—	19	3	no	—
260	C₃×C₉	—	19	3	no	—
261	C₃×C₉	—	19	3	no	—
262	C₃×C₉	—	19	3	no	—
263	C₃×C₉	—	19	1	no	—
264	C₃×C₉	—	19	1	no	—
265	C₃×C₉	—	19	1	no	—
266	C₃×C₉	—	19	1	no	—
267	C₃×C₉	—	19	1	no	—
268	C₃×C₉	—	19	1	no	—
269	C₃×C₉	—	19	3	no	—
270	C₃×C₉	—	19	3	no	—
271	C₃×C₉	—	57	9	no	—
272	C₃×C₉	—	171	3	no	—
273	C₃×C₉	—	171	3	no	—
274	C₃×C₉	—	171	1	no	—
275	C₃×C₉	—	171	1	no	—
276	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
277	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
278	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
279	C₃×C₉	C₃²×C₁₉⋊C₃	1	3	yes	—
280	C₃×C₉	C₃²×C₁₉⋊C₃	1	3	yes	—
281	C₃×C₉	C₃²×C₁₉⋊C₃	1	3	yes	—
282	C₁₉	#10624886	1	1	yes	characteristic, complemented, Sylow
283	C₃²	—	3	1	no	—
284	C₃²	—	3	1	no	—
285	C₃²	—	3	1	no	—
286	C₃²	—	19	3	no	—
287	C₃²	—	19	3	no	—
288	C₃²	—	19	3	no	—
289	C₃²	—	57	1	no	—
290	C₃²	—	57	1	no	—
291	C₃²	—	57	1	no	—
292	C₃²	—	57	1	no	—
293	C₃²	—	57	1	no	—
294	C₃²	—	57	1	no	—
295	C₃²	—	57	1	no	—
296	C₃²	—	57	1	no	—
297	C₃²	—	57	1	no	—
298	C₃²	—	171	9	no	—
299	C₃²	3₋¹⁺²×C₁₉⋊C₃	1	3	yes	—
300	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
301	C₉	—	3	3	no	—
302	C₉	—	3	3	no	—
303	C₉	—	3	3	no	—
304	C₉	—	3	1	no	—
305	C₉	—	3	1	no	—
306	C₉	—	3	1	no	—
307	C₉	—	57	3	no	—
308	C₉	—	57	3	no	—
309	C₉	—	57	3	no	—
310	C₉	—	57	3	no	—
311	C₉	—	57	3	no	—
312	C₉	—	57	3	no	—
313	C₉	—	57	1	no	—
314	C₉	—	57	1	no	—
315	C₉	—	57	1	no	—
316	C₉	—	57	1	no	—
317	C₉	—	57	1	no	—
318	C₉	—	57	1	no	—
319	C₃	—	3	3	no	—
320	C₃	—	57	3	no	—
321	C₃	—	57	3	no	—
322	C₃	—	57	3	no	—
323	C₃	C₃×3₋¹⁺²×C₁₉⋊C₃	1	1	yes	characteristic
324	C₃	#17987544	1	1	yes	characteristic
325	C₃	#17987545	1	1	yes	characteristic
326	C₃	#17987546	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₁₉⋊C₃
commutator, socle, cosocle	C₃²×C₁₉	C₃⁴
Fitting	C₁₉×#6894	C₃
prime radical	C₃³×C₁₉⋊C₃	C₃²
hypercenter	#6894	C₁₉⋊C₃
hypocenter	C₁₉	#10624886
LCS	G ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ #6894

Complements

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

N	H ≅ G/N	type
2. #17987444	C₃	semidirect
3. #17987444	C₃	semidirect
4. #17987444	C₃	semidirect
5. #17987441	C₃	semidirect
6. #17987441	C₃	semidirect
7. #17987440	C₃	semidirect
8. #17987440	C₃	semidirect
9. #17987439	C₃	semidirect
10. #17987438	C₃	semidirect
11. #17987437	C₃	semidirect
12. #17987437	C₃	semidirect
13. #17987436	C₃	semidirect
14. #17987436	C₃	semidirect
15. #17987435	C₃	semidirect
16. #17987435	C₃	semidirect
17. #17987434	C₃	semidirect
18. #17987433	C₃	semidirect
22. C₁₉×#6894	C₃	semidirect
34. #13095307	C₃²	semidirect
35. #13095307	C₃²	semidirect
36. #13095307	C₃²	semidirect
37. #13095306	C₃²	semidirect
38. #13095306	C₃²	semidirect
39. #13095306	C₃²	semidirect
40. #13095305	C₃²	semidirect
41. #13095305	C₃²	semidirect
42. #13095305	C₃²	semidirect
43. #13095305	C₃²	semidirect
44. #13095305	C₃²	semidirect
45. #13095305	C₃²	semidirect
46. #13095305	C₃²	semidirect
47. #13095305	C₃²	semidirect
48. #13095305	C₃²	semidirect
49. #13095305	C₃²	semidirect
50. #13095305	C₃²	semidirect
51. #13095305	C₃²	semidirect
52. #13095305	C₃²	semidirect
53. #13095305	C₃²	semidirect
54. #13095305	C₃²	semidirect
55. #13095305	C₃²	semidirect
56. #13095305	C₃²	semidirect
57. #13095305	C₃²	semidirect
77. C₁₉×C₉⋊C₉	C₃²	semidirect
78. C₁₉×C₉⋊C₉	C₃²	semidirect
79. C₁₉×C₉⋊C₉	C₃²	semidirect
138. #6894	C₁₉⋊C₃	semidirect
215. C₉⋊C₉	C₃×C₁₉⋊C₃	semidirect
216. C₉⋊C₉	C₃×C₁₉⋊C₃	semidirect
217. C₉⋊C₉	C₃×C₁₉⋊C₃	semidirect
282. C₁₉	#10624886	semidirect

Maximal subgroups

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

subgroup	index
C₃×3₋¹⁺²×C₁₉⋊C₃	3
#17987444	3
#17987441	3
#17987440	3
#17987439	3
#17987438	3
#17987437	3
#17987436	3
#17987435	3
#17987434	3
#17987433	3
C₃×#13095303	3
C₁₉⋊C₃×C₃²⋊C₉	3
C₁₉×#6894	3
#10624886	19