Subgroups — #28356643

Order 13851.

948 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17987545	C₃	1	3	yes	maximal, complemented
2	C₃×#13095333	C₃	1	1	yes	maximal, characteristic, complemented
3	#17987530	C₃	1	2	yes	maximal, complemented
4	#17987529	C₃	1	2	yes	maximal, complemented
5	#17987528	C₃	1	2	yes	maximal, complemented
6	#17987527	C₃	1	3	yes	maximal, complemented
7	#17987525	C₃	1	1	yes	maximal, characteristic, complemented
8	#17987524	C₃	1	1	yes	maximal, characteristic, complemented
9	#17987523	C₃	1	3	yes	maximal, complemented
10	#17987520	C₃	1	3	yes	maximal, complemented
11	#17987507	C₃	1	3	yes	maximal, complemented
12	#17987503	C₃	1	3	yes	maximal, complemented
13	#17987499	C₃	1	3	yes	maximal, complemented
14	#17987497	C₃	1	3	yes	maximal, complemented
15	#17987496	C₃	1	3	yes	maximal, complemented
16	C₁₉⋊C₃×#733	C₃	1	2	yes	maximal, complemented
17	C₁₉⋊C₃×#732	C₃	1	1	yes	maximal, characteristic, complemented
18	C₁₉×#6913	C₃	1	1	yes	maximal, characteristic, complemented
19	#13095333	—	3	3	no	—
20	#13095333	—	3	6	no	—
21	#13095333	—	3	6	no	—
22	#13095333	—	3	6	no	—
23	#13095333	C₃²	1	6	yes	complemented
24	#13095333	C₃²	1	6	yes	complemented
25	#13095332	—	3	6	no	—
26	#13095332	C₃²	1	6	yes	complemented
27	#13095331	C₃²	1	3	yes	complemented
28	#13095331	C₃²	1	3	yes	complemented
29	C₃×#10587025	C₃²	1	1	yes	characteristic, complemented
30	C₃×#10587022	—	3	3	no	—
31	C₃×#10587022	C₃²	1	1	yes	characteristic
32	C₃×#10587022	C₃²	1	1	yes	characteristic, complemented
33	C₃×#10587022	C₃²	1	1	yes	characteristic, complemented
34	C₃×#10587023	C₃²	1	1	yes	characteristic
35	C₃×#10587023	C₃²	1	1	yes	characteristic, complemented
36	C₃×#10587023	C₃²	1	1	yes	characteristic, complemented
37	3₋¹⁺²×C₁₉⋊C₃	—	3	6	no	—
38	3₋¹⁺²×C₁₉⋊C₃	C₃²	1	6	yes	—
39	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	1	yes	characteristic
40	H(3,𝔽₃)×C₁₉⋊C₃	C₃²	1	3	yes	—
41	C₃²×#4294	C₃²	1	1	yes	characteristic, complemented
42	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
43	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic, complemented
44	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic, complemented
45	#13095322	C₃²	1	2	yes	complemented
46	#13095322	C₃²	1	2	yes	complemented
47	#13095321	C₃²	1	6	yes	complemented
48	#13095321	C₃²	1	6	yes	complemented
49	#13095320	C₃²	1	2	yes	—
50	#13095319	C₃²	1	3	yes	—
51	#13095319	C₃²	1	3	yes	—
52	#13095318	C₃²	1	6	yes	complemented
53	#13095318	C₃²	1	6	yes	complemented
54	#13095317	C₃²	1	6	yes	complemented
55	#13095316	C₃²	1	3	yes	complemented
56	#13095316	C₃²	1	3	yes	complemented
57	#13095315	C₃²	1	6	yes	complemented
58	#13095315	C₃²	1	6	yes	complemented
59	#13095314	C₃²	1	6	yes	complemented
60	#13095313	C₃²	1	3	yes	complemented
61	#13095313	C₃²	1	3	yes	complemented
62	C₁₉×#737	C₃²	1	3	yes	complemented
63	C₃×C₁₉×3₋¹⁺²	C₃²	1	1	yes	characteristic, complemented
64	C₁₉×#733	C₃²	1	2	yes	complemented
65	C₁₉×#732	C₃²	1	1	yes	characteristic
66	C₁₉×#731	C₃²	1	3	yes	complemented
67	C₁₉×AΣL(1,𝔽₂₇)	C₃²	1	3	yes	complemented
68	#10624936	—	19	1	no	maximal, Sylow
69	C₃³×C₁₉	C₃³	1	1	yes	characteristic
70	C₃²×C₁₉⋊C₃	—	3	3	no	—
71	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
72	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
73	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
74	C₁₉×3₋¹⁺²	—	3	3	no	—
75	C₁₉×3₋¹⁺²	—	3	6	no	—
76	C₁₉×3₋¹⁺²	C₃³	1	6	yes	—
77	C₁₉×H(3,𝔽₃)	C₃³	1	3	yes	—
78	C₃×C₉×C₁₉	—	3	3	no	—
79	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
80	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
81	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
82	C₃²⋊(C₁₉⋊C₃)	—	3	6	no	—
83	C₃²⋊(C₁₉⋊C₃)	—	3	2	no	—
84	C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
85	C₃²⋊(C₁₉⋊C₃)	C₃³	1	6	yes	—
86	#10587025	—	3	6	no	—
87	#10587025	—	3	6	no	—
88	#10587025	—	3	3	no	—
89	#10587025	C₃³	1	6	yes	—
90	#10587025	C₃³	1	6	yes	—
91	C₃×#4294	—	3	3	no	—
92	C₃×#4294	—	3	3	no	—
93	C₃×#4294	—	3	3	no	—
94	C₃×#4294	C₃³	1	1	yes	characteristic
95	C₃×#4294	C₃³	1	1	yes	characteristic
96	C₃×#4294	C₃³	1	1	yes	characteristic
97	C₃×#4294	C₃³	1	1	yes	characteristic
98	C₃×#4294	C₃³	1	1	yes	characteristic
99	C₃×#4294	C₃³	1	1	yes	characteristic
100	#10587023	—	3	6	no	—
101	#10587023	—	3	2	no	—
102	#10587023	—	3	1	no	—
103	#10587023	—	3	6	no	—
104	#10587023	—	3	3	no	—
105	#10587023	—	3	6	no	—
106	#10587023	—	3	6	no	—
107	#10587023	—	3	3	no	—
108	#10587022	—	3	6	no	—
109	#10587022	—	3	6	no	—
110	#10587022	—	3	6	no	—
111	#10587022	—	3	6	no	—
112	#10587022	—	3	3	no	—
113	#10587022	—	3	3	no	—
114	#10587022	H(3,𝔽₃)	1	3	yes	—
115	#10587022	H(3,𝔽₃)	1	6	yes	—
116	C₉×C₁₉⋊C₃	—	3	2	no	—
117	C₉×C₁₉⋊C₃	—	3	1	no	—
118	C₉×C₁₉⋊C₃	—	3	6	no	—
119	C₉×C₁₉⋊C₃	—	3	3	no	—
120	C₉×C₁₉⋊C₃	—	3	6	no	—
121	C₉×C₁₉⋊C₃	—	3	3	no	—
122	C₉×C₁₉⋊C₃	—	3	6	no	—
123	C₉×C₁₉⋊C₃	—	3	6	no	—
124	3₋¹⁺⁴	—	19	3	no	—
125	C₃×#737	—	19	1	no	—
126	#6913	—	19	3	no	—
127	#6913	—	19	2	no	—
128	#6913	—	19	2	no	—
129	#6913	—	19	2	no	—
130	#6913	—	19	1	no	—
131	#6913	—	19	1	no	—
132	#6913	—	19	3	no	—
133	#6913	—	19	3	no	—
134	#6913	C₁₉⋊C₃	1	1	yes	characteristic, complemented
135	#6910	—	19	3	no	—
136	#6909	—	19	3	no	—
137	#6908	—	19	3	no	—
138	#6908	—	19	3	no	—
139	#6908	—	19	3	no	—
140	C₃×#733	—	19	2	no	—
141	C₃×#732	—	19	1	no	—
142	C₃²×C₁₉	—	3	1	no	—
143	C₃²×C₁₉	—	3	3	no	—
144	C₃²×C₁₉	—	9	1	no	—
145	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
146	C₃×C₁₉⋊C₃	—	3	1	no	—
147	C₃×C₁₉⋊C₃	—	3	1	no	—
148	C₃×C₁₉⋊C₃	—	3	6	no	—
149	C₃×C₁₉⋊C₃	—	9	1	no	—
150	C₃×C₁₉⋊C₃	—	9	1	no	—
151	C₃×C₁₉⋊C₃	—	9	1	no	—
152	C₃×C₁₉⋊C₃	—	27	3	no	—
153	C₃×C₁₉⋊C₃	C₃×H(3,𝔽₃)	1	2	yes	—
154	C₃×C₁₉⋊C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
155	C₉×C₁₉	—	3	6	no	—
156	C₉×C₁₉	—	3	1	no	—
157	C₉×C₁₉	—	3	1	no	—
158	C₉×C₁₉	C₃×H(3,𝔽₃)	1	3	yes	—
159	#4294	—	3	1	no	—
160	#4294	—	3	6	no	—
161	#4294	—	3	6	no	—
162	#4294	—	3	1	no	—
163	#4294	—	3	1	no	—
164	#4294	—	3	1	no	—
165	#4294	C₃×H(3,𝔽₃)	1	3	yes	—
166	#4294	C₃×H(3,𝔽₃)	1	3	yes	—
167	#737	—	19	6	no	—
168	#737	—	19	6	no	—
169	#737	—	19	3	no	—
170	#737	—	19	3	no	—
171	#737	—	19	6	no	—
172	#737	—	57	6	no	—
173	#737	—	57	3	no	—
174	#737	—	57	6	no	—
175	#737	—	57	6	no	—
176	#737	—	57	6	no	—
177	#737	C₃×C₁₉⋊C₃	1	3	yes	complemented
178	C₃×3₋¹⁺²	—	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	1	no	—
182	C₃×3₋¹⁺²	—	19	1	no	—
183	C₃×3₋¹⁺²	—	19	1	no	—
184	C₃×3₋¹⁺²	—	19	1	no	—
185	C₃×3₋¹⁺²	—	19	6	no	—
186	C₃×3₋¹⁺²	—	57	3	no	—
187	C₃×3₋¹⁺²	—	57	6	no	—
188	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented
189	C₃×H(3,𝔽₃)	—	19	3	no	—
190	C₃×H(3,𝔽₃)	—	19	1	no	—
191	C₃²×C₉	—	19	1	no	—
192	C₃²×C₉	—	19	1	no	—
193	C₃²×C₉	—	19	1	no	—
194	C₃²×C₉	—	19	1	no	—
195	#733	—	19	6	no	—
196	#733	—	19	6	no	—
197	#733	—	19	2	no	—
198	#733	—	19	2	no	—
199	#733	C₃×C₁₉⋊C₃	1	2	yes	complemented
200	#732	—	19	3	no	—
201	#732	—	19	3	no	—
202	#732	—	19	2	no	—
203	#732	C₃×C₁₉⋊C₃	1	1	yes	characteristic
204	#731	—	19	6	no	—
205	#731	—	19	6	no	—
206	#731	—	19	3	no	—
207	#731	—	19	3	no	—
208	#731	—	19	6	no	—
209	#731	C₃×C₁₉⋊C₃	1	3	yes	complemented
210	AΣL(1,𝔽₂₇)	—	19	6	no	—
211	AΣL(1,𝔽₂₇)	—	19	6	no	—
212	AΣL(1,𝔽₂₇)	—	19	6	no	—
213	AΣL(1,𝔽₂₇)	—	19	3	no	—
214	AΣL(1,𝔽₂₇)	—	19	3	no	—
215	AΣL(1,𝔽₂₇)	C₃×C₁₉⋊C₃	1	3	yes	complemented
216	C₃×C₁₉	—	3	1	no	—
217	C₃×C₁₉	—	9	3	no	—
218	C₃×C₁₉	—	9	1	no	—
219	C₃×C₁₉	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
220	C₁₉⋊C₃	—	3	1	no	—
221	C₁₉⋊C₃	—	3	2	no	—
222	C₁₉⋊C₃	—	9	1	no	—
223	C₁₉⋊C₃	—	9	1	no	—
224	C₁₉⋊C₃	—	9	6	no	—
225	C₃³	—	19	1	no	—
226	C₃³	—	19	1	no	—
227	C₃³	—	19	1	no	—
228	C₃³	—	57	3	no	—
229	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
230	3₋¹⁺²	—	3	6	no	—
231	3₋¹⁺²	—	3	3	no	—
232	3₋¹⁺²	—	19	3	no	—
233	3₋¹⁺²	—	19	6	no	—
234	3₋¹⁺²	—	19	6	no	—
235	3₋¹⁺²	—	19	6	no	—
236	3₋¹⁺²	—	57	3	no	—
237	3₋¹⁺²	—	57	3	no	—
238	3₋¹⁺²	—	57	6	no	—
239	3₋¹⁺²	—	57	6	no	—
240	3₋¹⁺²	—	57	3	no	—
241	3₋¹⁺²	—	57	6	no	—
242	3₋¹⁺²	—	57	6	no	—
243	3₋¹⁺²	—	57	6	no	—
244	3₋¹⁺²	—	57	6	no	—
245	3₋¹⁺²	—	57	2	no	—
246	3₋¹⁺²	—	57	1	no	—
247	3₋¹⁺²	—	57	3	no	—
248	3₋¹⁺²	—	57	6	no	—
249	3₋¹⁺²	—	57	6	no	—
250	3₋¹⁺²	—	57	6	no	—
251	3₋¹⁺²	—	57	3	no	—
252	3₋¹⁺²	—	57	6	no	—
253	3₋¹⁺²	C₃²×C₁₉⋊C₃	1	6	yes	—
254	H(3,𝔽₃)	—	19	6	no	—
255	H(3,𝔽₃)	—	57	6	no	—
256	H(3,𝔽₃)	—	57	2	no	—
257	H(3,𝔽₃)	—	57	1	no	—
258	H(3,𝔽₃)	C₃²×C₁₉⋊C₃	1	3	yes	—
259	C₃×C₉	—	3	3	no	—
260	C₃×C₉	—	19	1	no	—
261	C₃×C₉	—	19	1	no	—
262	C₃×C₉	—	19	1	no	—
263	C₃×C₉	—	19	1	no	—
264	C₃×C₉	—	19	1	no	—
265	C₃×C₉	—	19	1	no	—
266	C₃×C₉	—	57	3	no	—
267	C₃×C₉	—	57	6	no	—
268	C₃×C₉	—	57	6	no	—
269	C₃×C₉	—	57	2	no	—
270	C₃×C₉	—	57	1	no	—
271	C₃×C₉	—	57	6	no	—
272	C₃×C₉	—	57	3	no	—
273	C₃×C₉	—	57	3	no	—
274	C₃×C₉	—	57	3	no	—
275	C₃×C₉	—	57	3	no	—
276	C₃×C₉	—	57	6	no	—
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	1	yes	characteristic
280	C₁₉	#10624936	1	1	yes	characteristic, complemented, Sylow
281	C₃²	—	513	3	no	—
282	C₃²	—	3	3	no	—
283	C₃²	—	3	1	no	—
284	C₃²	—	9	1	no	—
285	C₃²	—	19	2	no	—
286	C₃²	—	19	1	no	—
287	C₃²	—	57	6	no	—
288	C₃²	—	57	1	no	—
289	C₃²	—	57	1	no	—
290	C₃²	—	171	1	no	—
291	C₃²	—	171	1	no	—
292	C₃²	—	171	1	no	—
293	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
294	C₉	—	3	6	no	—
295	C₉	—	3	1	no	—
296	C₉	—	3	1	no	—
297	C₉	—	19	3	no	—
298	C₉	—	19	3	no	—
299	C₉	—	57	1	no	—
300	C₉	—	57	1	no	—
301	C₉	—	57	1	no	—
302	C₉	—	57	1	no	—
303	C₉	—	57	6	no	—
304	C₉	—	57	6	no	—
305	C₉	H(3,𝔽₃)×C₁₉⋊C₃	1	3	yes	—
306	C₃	—	3	1	no	—
307	C₃	—	9	3	no	—
308	C₃	—	9	1	no	—
309	C₃	—	57	1	no	—
310	C₃	—	57	2	no	—
311	C₃	—	171	6	no	—
312	C₃	—	171	1	no	—
313	C₃	—	171	1	no	—
314	C₃	C₃×H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₃×H(3,𝔽₃)×C₁₉⋊C₃
commutator, cosocle	C₃²×C₁₉	C₃⁴
Frattini	C₃²	C₃³×C₁₉⋊C₃
Fitting	C₁₉×#6913	C₃
socle	C₃×C₁₉	C₃²×H(3,𝔽₃)
hypercenter	#6913	C₁₉⋊C₃
hypocenter	C₁₉	#10624936
LCS	G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ #6913

Complements

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

N	H ≅ G/N	type
1. #17987545	C₃	semidirect
2. C₃×#13095333	C₃	semidirect
3. #17987530	C₃	semidirect
4. #17987529	C₃	semidirect
5. #17987528	C₃	semidirect
6. #17987527	C₃	semidirect
7. #17987525	C₃	semidirect
8. #17987524	C₃	semidirect
9. #17987523	C₃	semidirect
10. #17987520	C₃	semidirect
11. #17987507	C₃	semidirect
12. #17987503	C₃	semidirect
13. #17987499	C₃	semidirect
14. #17987497	C₃	semidirect
15. #17987496	C₃	semidirect
16. C₁₉⋊C₃×#733	C₃	semidirect
17. C₁₉⋊C₃×#732	C₃	semidirect
18. C₁₉×#6913	C₃	semidirect
23. #13095333	C₃²	semidirect
24. #13095333	C₃²	semidirect
26. #13095332	C₃²	semidirect
27. #13095331	C₃²	semidirect
28. #13095331	C₃²	semidirect
29. C₃×#10587025	C₃²	semidirect
32. C₃×#10587022	C₃²	semidirect
33. C₃×#10587022	C₃²	semidirect
35. C₃×#10587023	C₃²	semidirect
36. C₃×#10587023	C₃²	semidirect
41. C₃²×#4294	C₃²	semidirect
43. C₃×C₉×C₁₉⋊C₃	C₃²	semidirect
44. C₃×C₉×C₁₉⋊C₃	C₃²	semidirect
45. #13095322	C₃²	semidirect
46. #13095322	C₃²	semidirect
47. #13095321	C₃²	semidirect
48. #13095321	C₃²	semidirect
52. #13095318	C₃²	semidirect
53. #13095318	C₃²	semidirect
54. #13095317	C₃²	semidirect
55. #13095316	C₃²	semidirect
56. #13095316	C₃²	semidirect
57. #13095315	C₃²	semidirect
58. #13095315	C₃²	semidirect
59. #13095314	C₃²	semidirect
60. #13095313	C₃²	semidirect
61. #13095313	C₃²	semidirect
62. C₁₉×#737	C₃²	semidirect
63. C₃×C₁₉×3₋¹⁺²	C₃²	semidirect
64. C₁₉×#733	C₃²	semidirect
66. C₁₉×#731	C₃²	semidirect
67. C₁₉×AΣL(1,𝔽₂₇)	C₃²	semidirect
134. #6913	C₁₉⋊C₃	semidirect
177. #737	C₃×C₁₉⋊C₃	semidirect
188. C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	semidirect
199. #733	C₃×C₁₉⋊C₃	semidirect
209. #731	C₃×C₁₉⋊C₃	semidirect
215. AΣL(1,𝔽₂₇)	C₃×C₁₉⋊C₃	semidirect
280. C₁₉	#10624936	semidirect

Maximal subgroups

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

subgroup	index
#17987545	3
C₃×#13095333	3
#17987530	3
#17987529	3
#17987528	3
#17987527	3
#17987525	3
#17987524	3
#17987523	3
#17987520	3
#17987507	3
#17987503	3
#17987499	3
#17987497	3
#17987496	3
C₁₉⋊C₃×#733	3
C₁₉⋊C₃×#732	3
C₁₉×#6913	3
#10624936	19