Subgroups — #28356105

Order 13851.

764 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17987430	C₃	1	2	yes	maximal, complemented
2	#17987430	C₃	1	4	yes	maximal, complemented
3	#17987429	C₃	1	2	yes	maximal, complemented
4	#17987427	C₃	1	4	yes	maximal, complemented
5	#17987426	C₃	1	2	yes	maximal, complemented
6	#17987424	C₃	1	4	yes	maximal, complemented
7	#17987424	C₃	1	4	yes	maximal, complemented
8	#17987424	C₃	1	4	yes	maximal, complemented
9	#17987422	C₃	1	2	yes	maximal, complemented
10	#17987422	C₃	1	4	yes	maximal, complemented
11	#17987419	C₃	1	2	yes	maximal, complemented
12	(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
13	C₉×C₃²⋊(C₁₉⋊C₃)	C₃	1	1	yes	maximal, characteristic, complemented
14	#17987406	C₃	1	2	yes	maximal, complemented
15	C₁₉⋊C₃×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
16	C₁₉×#6893	C₃	1	1	yes	maximal, characteristic, complemented
17	C₃³×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
18	C₃×#10587025	C₃²	1	4	yes	complemented
19	C₃×#10587022	C₃²	1	4	yes	complemented
20	C₃×#10587022	C₃²	1	2	yes	complemented
21	C₃×#10587023	C₃²	1	4	yes	—
22	C₃×#10587023	C₃²	1	2	yes	—
23	C₃×#10587023	C₃²	1	2	yes	complemented
24	C₃×#10587023	C₃²	1	4	yes	complemented
25	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	2	yes	—
26	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	1	yes	characteristic
27	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	2	yes	—
28	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	4	yes	—
29	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	2	yes	—
30	C₃²×#4294	C₃²	1	2	yes	complemented
31	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
32	C₃×C₉×C₁₉⋊C₃	C₃²	1	2	yes	—
33	C₃×C₉×C₁₉⋊C₃	C₃²	1	4	yes	complemented
34	C₃×C₉×C₁₉⋊C₃	C₃²	1	2	yes	complemented
35	C₃²⋊#4294	C₃²	1	2	yes	—
36	C₃²⋊#4294	C₃²	1	4	yes	—
37	C₃²⋊#4294	C₃²	1	2	yes	complemented
38	C₃²⋊#4294	C₃²	1	2	yes	complemented
39	C₃²⋊#4294	C₃²	1	4	yes	complemented
40	C₃²⋊#4294	C₃²	1	4	yes	complemented
41	#13095303	C₃²	1	4	yes	—
42	#13095303	C₃²	1	2	yes	—
43	#13095303	C₃²	1	4	yes	—
44	#13095303	C₃²	1	2	yes	—
45	#13095303	C₃²	1	4	yes	—
46	#13095303	C₃²	1	2	yes	—
47	#13095303	C₃²	1	4	yes	complemented
48	#13095303	C₃²	1	4	yes	complemented
49	#13095303	C₃²	1	4	yes	complemented
50	#13095303	C₃²	1	4	yes	complemented
51	#13095303	C₃²	1	4	yes	complemented
52	#13095303	C₃²	1	4	yes	complemented
53	#13095303	C₃²	1	4	yes	complemented
54	#13095303	C₃²	1	4	yes	complemented
55	#13095303	C₃²	1	4	yes	complemented
56	C₃×C₁₉×3₋¹⁺²	C₃²	1	2	yes	complemented
57	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
58	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic, complemented
59	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
60	C₁₉×C₃²⋊C₉	C₃²	1	2	yes	—
61	C₁₉×C₃²⋊C₉	C₃²	1	2	yes	complemented
62	C₁₉×C₃²⋊C₉	C₃²	1	4	yes	complemented
63	#10624888	—	19	1	no	maximal, Sylow
64	C₃³×C₁₉	C₃³	1	1	yes	characteristic
65	C₃³×C₁₉	C₃³	1	2	yes	—
66	C₃³×C₁₉	C₃³	1	1	yes	characteristic
67	C₃²×C₁₉⋊C₃	—	3	3	no	—
68	C₃²×C₁₉⋊C₃	—	9	1	no	—
69	C₃²×C₁₉⋊C₃	—	9	2	no	—
70	C₃²×C₁₉⋊C₃	C₃³	1	2	yes	—
71	C₃²×C₁₉⋊C₃	C₃³	1	4	yes	—
72	C₃²×C₁₉⋊C₃	C₃³	1	2	yes	—
73	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
74	C₁₉×3₋¹⁺²	—	9	2	no	—
75	C₁₉×H(3,𝔽₃)	—	9	1	no	—
76	C₃×C₉×C₁₉	—	9	1	no	—
77	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
78	C₃×C₉×C₁₉	C₃³	1	2	yes	—
79	C₃×C₉×C₁₉	C₃³	1	2	yes	—
80	C₃×C₉×C₁₉	C₃³	1	4	yes	—
81	C₃²⋊(C₁₉⋊C₃)	—	3	6	no	—
82	C₃²⋊(C₁₉⋊C₃)	—	3	3	no	—
83	C₃²⋊(C₁₉⋊C₃)	—	9	2	no	—
84	C₃²⋊(C₁₉⋊C₃)	—	9	2	no	—
85	C₃²⋊(C₁₉⋊C₃)	—	9	4	no	—
86	#10587025	—	9	4	no	—
87	C₃×#4294	—	9	2	no	—
88	C₃×#4294	C₃³	1	2	yes	—
89	C₃×#4294	C₃³	1	4	yes	—
90	C₃×#4294	C₃³	1	4	yes	—
91	C₃×#4294	C₃³	1	4	yes	—
92	C₃×#4294	C₃³	1	4	yes	—
93	#10587023	—	9	2	no	—
94	#10587023	—	9	4	no	—
95	#10587023	—	9	4	no	—
96	#10587023	—	9	2	no	—
97	#10587022	—	9	4	no	—
98	#10587022	—	9	2	no	—
99	C₉×C₁₉⋊C₃	—	9	1	no	—
100	C₉×C₁₉⋊C₃	—	9	2	no	—
101	C₉×C₁₉⋊C₃	—	9	4	no	—
102	C₉×C₁₉⋊C₃	—	9	2	no	—
103	#6893	—	19	2	no	—
104	#6893	—	19	4	no	—
105	#6893	—	19	2	no	—
106	#6893	C₁₉⋊C₃	1	1	yes	characteristic, complemented
107	C₉⋊H(3,𝔽₃)	—	19	4	no	—
108	C₉⋊H(3,𝔽₃)	—	19	2	no	—
109	C₉⋊H(3,𝔽₃)	—	19	2	no	—
110	C₉⋊H(3,𝔽₃)	—	19	4	no	—
111	C₉⋊H(3,𝔽₃)	—	19	4	no	—
112	C₉⋊H(3,𝔽₃)	—	19	4	no	—
113	C₉⋊H(3,𝔽₃)	—	19	4	no	—
114	#6891	—	19	2	no	—
115	C₃²⋊H(3,𝔽₃)	—	19	1	no	—
116	C₉×H(3,𝔽₃)	—	19	1	no	—
117	C₉×H(3,𝔽₃)	—	19	2	no	—
118	C₃×C₃²⋊C₉	—	19	1	no	—
119	C₃²×C₁₉	—	3	2	no	—
120	C₃²×C₁₉	—	3	1	no	—
121	C₃²×C₁₉	—	3	1	no	—
122	C₃²×C₁₉	—	3	2	no	—
123	C₃²×C₁₉	—	3	1	no	—
124	C₃²×C₁₉	—	3	1	no	—
125	C₃²×C₁₉	—	3	4	no	—
126	C₃²×C₁₉	—	3	2	no	—
127	C₃²×C₁₉	—	3	2	no	—
128	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
129	C₃×C₁₉⋊C₃	—	3	1	no	—
130	C₃×C₁₉⋊C₃	—	3	2	no	—
131	C₃×C₁₉⋊C₃	—	3	4	no	—
132	C₃×C₁₉⋊C₃	—	3	2	no	—
133	C₃×C₁₉⋊C₃	—	3	2	no	—
134	C₃×C₁₉⋊C₃	—	3	4	no	—
135	C₃×C₁₉⋊C₃	—	3	2	no	—
136	C₃×C₁₉⋊C₃	—	3	2	no	—
137	C₃×C₁₉⋊C₃	—	3	2	no	—
138	C₃×C₁₉⋊C₃	—	3	4	no	—
139	C₃×C₁₉⋊C₃	—	3	4	no	—
140	C₃×C₁₉⋊C₃	—	3	4	no	—
141	C₃×C₁₉⋊C₃	—	3	2	no	—
142	C₃×C₁₉⋊C₃	—	9	9	no	—
143	C₃×C₁₉⋊C₃	#737	1	3	yes	—
144	C₉×C₁₉	—	3	1	no	—
145	C₉×C₁₉	—	3	2	no	—
146	C₉×C₁₉	—	3	4	no	—
147	C₉×C₁₉	—	3	2	no	—
148	#4294	—	3	2	no	—
149	#4294	—	3	4	no	—
150	#4294	—	3	4	no	—
151	#4294	—	3	4	no	—
152	#4294	—	3	4	no	—
153	C₃⁴	—	19	1	no	—
154	C₃×3₋¹⁺²	—	19	4	no	—
155	C₃×3₋¹⁺²	—	19	4	no	—
156	C₃×3₋¹⁺²	—	19	2	no	—
157	C₃×3₋¹⁺²	—	19	2	no	—
158	C₃×3₋¹⁺²	—	19	4	no	—
159	C₃×3₋¹⁺²	—	19	4	no	—
160	C₃×3₋¹⁺²	—	19	2	no	—
161	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	2	yes	complemented
162	C₃×H(3,𝔽₃)	—	19	1	no	—
163	C₃×H(3,𝔽₃)	—	19	2	no	—
164	C₃×H(3,𝔽₃)	—	19	2	no	—
165	C₃×H(3,𝔽₃)	—	19	4	no	—
166	C₃×H(3,𝔽₃)	—	19	2	no	—
167	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	1	yes	characteristic
168	C₃²×C₉	—	19	2	no	—
169	C₃²×C₉	—	19	1	no	—
170	C₃²×C₉	—	19	2	no	—
171	C₃²×C₉	—	19	4	no	—
172	C₃²×C₉	—	19	2	no	—
173	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented
174	C₃²⋊C₉	—	19	2	no	—
175	C₃²⋊C₉	—	19	4	no	—
176	C₃²⋊C₉	—	19	2	no	—
177	C₃²⋊C₉	—	19	2	no	—
178	C₃²⋊C₉	—	19	4	no	—
179	C₃²⋊C₉	—	19	4	no	—
180	C₃²⋊C₉	—	19	4	no	—
181	C₃²⋊C₉	—	19	2	no	—
182	C₃²⋊C₉	—	19	4	no	—
183	C₃²⋊C₉	—	19	2	no	—
184	C₃²⋊C₉	—	19	4	no	—
185	C₃²⋊C₉	—	19	2	no	—
186	C₃²⋊C₉	—	19	4	no	—
187	C₃²⋊C₉	—	19	4	no	—
188	C₃²⋊C₉	—	19	4	no	—
189	C₃²⋊C₉	—	19	4	no	—
190	C₃²⋊C₉	—	19	4	no	—
191	C₃²⋊C₉	—	19	4	no	—
192	C₃²⋊C₉	—	19	4	no	—
193	C₃²⋊C₉	—	19	4	no	—
194	C₃²⋊C₉	—	19	4	no	—
195	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
196	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	2	yes	—
197	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	2	yes	complemented
198	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	4	yes	complemented
199	C₃×C₁₉	—	9	1	no	—
200	C₃×C₁₉	—	9	2	no	—
201	C₃×C₁₉	—	9	1	no	—
202	C₃×C₁₉	C₃×#737	1	1	yes	characteristic
203	C₃×C₁₉	H(5,𝔽₃)	1	1	yes	characteristic
204	C₃×C₁₉	3₋¹⁺⁴	1	2	yes	—
205	C₁₉⋊C₃	—	3	3	no	—
206	C₁₉⋊C₃	—	9	4	no	—
207	C₁₉⋊C₃	—	9	2	no	—
208	C₁₉⋊C₃	—	9	2	no	—
209	C₃³	—	19	1	no	—
210	C₃³	—	19	4	no	—
211	C₃³	—	19	2	no	—
212	C₃³	—	19	2	no	—
213	C₃³	—	57	3	no	—
214	C₃³	—	171	1	no	—
215	C₃³	—	171	2	no	—
216	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
217	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
218	C₃³	C₃²×C₁₉⋊C₃	1	2	yes	—
219	3₋¹⁺²	—	9	2	no	—
220	3₋¹⁺²	—	171	4	no	—
221	3₋¹⁺²	—	171	2	no	—
222	3₋¹⁺²	—	171	4	no	—
223	3₋¹⁺²	—	171	2	no	—
224	3₋¹⁺²	—	171	2	no	—
225	3₋¹⁺²	—	171	4	no	—
226	3₋¹⁺²	—	171	4	no	—
227	H(3,𝔽₃)	—	9	1	no	—
228	H(3,𝔽₃)	—	57	3	no	—
229	H(3,𝔽₃)	—	57	6	no	—
230	H(3,𝔽₃)	—	171	2	no	—
231	H(3,𝔽₃)	—	171	2	no	—
232	H(3,𝔽₃)	—	171	4	no	—
233	C₃×C₉	—	9	1	no	—
234	C₃×C₉	—	19	2	no	—
235	C₃×C₉	—	19	4	no	—
236	C₃×C₉	—	19	4	no	—
237	C₃×C₉	—	19	4	no	—
238	C₃×C₉	—	19	4	no	—
239	C₃×C₉	—	171	1	no	—
240	C₃×C₉	—	171	2	no	—
241	C₃×C₉	—	171	4	no	—
242	C₃×C₉	—	171	2	no	—
243	C₃×C₉	—	171	2	no	—
244	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
245	C₃×C₉	C₃²×C₁₉⋊C₃	1	2	yes	—
246	C₃×C₉	C₃²×C₁₉⋊C₃	1	4	yes	—
247	C₃×C₉	C₃²×C₁₉⋊C₃	1	2	yes	—
248	C₁₉	#10624888	1	1	yes	characteristic, complemented, Sylow
249	C₃²	—	3	1	no	—
250	C₃²	—	3	2	no	—
251	C₃²	—	3	1	no	—
252	C₃²	—	3	1	no	—
253	C₃²	—	3	2	no	—
254	C₃²	—	3	1	no	—
255	C₃²	—	3	2	no	—
256	C₃²	—	3	4	no	—
257	C₃²	—	3	2	no	—
258	C₃²	—	19	3	no	—
259	C₃²	—	57	1	no	—
260	C₃²	—	57	2	no	—
261	C₃²	—	57	2	no	—
262	C₃²	—	57	4	no	—
263	C₃²	—	57	2	no	—
264	C₃²	—	57	2	no	—
265	C₃²	—	57	4	no	—
266	C₃²	—	57	2	no	—
267	C₃²	—	57	2	no	—
268	C₃²	—	57	4	no	—
269	C₃²	—	57	4	no	—
270	C₃²	—	57	4	no	—
271	C₃²	—	57	2	no	—
272	C₃²	—	171	9	no	—
273	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
274	C₉	—	3	1	no	—
275	C₉	—	3	2	no	—
276	C₉	—	3	4	no	—
277	C₉	—	3	2	no	—
278	C₉	—	57	2	no	—
279	C₉	—	57	4	no	—
280	C₉	—	57	4	no	—
281	C₉	—	57	4	no	—
282	C₉	—	57	4	no	—
283	C₃	—	9	1	no	—
284	C₃	—	9	1	no	—
285	C₃	—	9	2	no	—
286	C₃	—	57	3	no	—
287	C₃	—	171	2	no	—
288	C₃	—	171	2	no	—
289	C₃	—	171	4	no	—
290	C₃	C₁₉⋊C₃×#737	1	1	yes	characteristic
291	C₃	C₁₉⋊H(5,𝔽₃)	1	1	yes	characteristic
292	C₃	#17987544	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₁₉⋊C₃
commutator, socle, cosocle	C₃²×C₁₉	C₃⁴
Fitting	C₁₉×#6893	C₃
prime radical	(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)	C₃
hypercenter	#6893	C₁₉⋊C₃
hypocenter	C₁₉	#10624888
LCS	G ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ #6893

Complements

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

N	H ≅ G/N	type
1. #17987430	C₃	semidirect
2. #17987430	C₃	semidirect
3. #17987429	C₃	semidirect
4. #17987427	C₃	semidirect
5. #17987426	C₃	semidirect
6. #17987424	C₃	semidirect
7. #17987424	C₃	semidirect
8. #17987424	C₃	semidirect
9. #17987422	C₃	semidirect
10. #17987422	C₃	semidirect
11. #17987419	C₃	semidirect
13. C₉×C₃²⋊(C₁₉⋊C₃)	C₃	semidirect
14. #17987406	C₃	semidirect
15. C₁₉⋊C₃×C₃²⋊C₉	C₃	semidirect
16. C₁₉×#6893	C₃	semidirect
18. C₃×#10587025	C₃²	semidirect
19. C₃×#10587022	C₃²	semidirect
20. C₃×#10587022	C₃²	semidirect
23. C₃×#10587023	C₃²	semidirect
24. C₃×#10587023	C₃²	semidirect
30. C₃²×#4294	C₃²	semidirect
33. C₃×C₉×C₁₉⋊C₃	C₃²	semidirect
34. C₃×C₉×C₁₉⋊C₃	C₃²	semidirect
37. C₃²⋊#4294	C₃²	semidirect
38. C₃²⋊#4294	C₃²	semidirect
39. C₃²⋊#4294	C₃²	semidirect
40. C₃²⋊#4294	C₃²	semidirect
47. #13095303	C₃²	semidirect
48. #13095303	C₃²	semidirect
49. #13095303	C₃²	semidirect
50. #13095303	C₃²	semidirect
51. #13095303	C₃²	semidirect
52. #13095303	C₃²	semidirect
53. #13095303	C₃²	semidirect
54. #13095303	C₃²	semidirect
55. #13095303	C₃²	semidirect
56. C₃×C₁₉×3₋¹⁺²	C₃²	semidirect
58. C₃²×C₉×C₁₉	C₃²	semidirect
61. C₁₉×C₃²⋊C₉	C₃²	semidirect
62. C₁₉×C₃²⋊C₉	C₃²	semidirect
106. #6893	C₁₉⋊C₃	semidirect
161. C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	semidirect
173. C₃²×C₉	C₃×C₁₉⋊C₃	semidirect
197. C₃²⋊C₉	C₃×C₁₉⋊C₃	semidirect
198. C₃²⋊C₉	C₃×C₁₉⋊C₃	semidirect
248. C₁₉	#10624888	semidirect

Maximal subgroups

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

subgroup	index
#17987430	3
#17987429	3
#17987427	3
#17987426	3
#17987424	3
#17987422	3
#17987419	3
(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)	3
C₉×C₃²⋊(C₁₉⋊C₃)	3
#17987406	3
C₁₉⋊C₃×C₃²⋊C₉	3
C₁₉×#6893	3
#10624888	19