Subgroups — #28356095

Order 13851.

698 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#13095333	C₃	1	1	yes	maximal, characteristic
2	#17987472	C₃	1	1	yes	maximal, characteristic, complemented
3	#17987471	C₃	1	1	yes	maximal, characteristic, complemented
4	#17987467	C₃	1	1	yes	maximal, characteristic, complemented
5	#17987466	C₃	1	1	yes	maximal, characteristic, complemented
6	#17987463	C₃	1	1	yes	maximal, characteristic, complemented
7	#17987462	C₃	1	1	yes	maximal, characteristic, complemented
8	#17987461	C₃	1	1	yes	maximal, characteristic, complemented
9	#17987461	C₃	1	1	yes	maximal, characteristic, complemented
10	#17987449	C₃	1	3	yes	maximal, complemented
11	#17987449	C₃	1	3	yes	maximal, complemented
12	#17987447	C₃	1	3	yes	maximal, complemented
13	#17987437	C₃	1	3	yes	maximal, complemented
14	#17987437	C₃	1	3	yes	maximal, complemented
15	#17987436	C₃	1	3	yes	maximal, complemented
16	#17987436	C₃	1	3	yes	maximal, complemented
17	#17987435	C₃	1	3	yes	maximal, complemented
18	#17987422	C₃	1	3	yes	maximal
19	#17987404	C₃	1	1	yes	maximal, characteristic, complemented
20	#17987403	C₃	1	1	yes	maximal, characteristic, complemented
21	C₁₉⋊C₃×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
22	C₁₉×#6900	C₃	1	1	yes	maximal, characteristic, complemented
23	#13095333	—	3	9	no	—
24	C₃×#10587025	C₃²	1	1	yes	characteristic
25	C₃×#10587022	C₃²	1	1	yes	characteristic
26	C₃×#10587022	C₃²	1	1	yes	characteristic
27	C₃×#10587022	C₃²	1	1	yes	characteristic
28	C₃×#10587022	C₃²	1	3	yes	—
29	C₃×#10587023	C₃²	1	1	yes	characteristic
30	C₃×#10587023	C₃²	1	1	yes	characteristic
31	C₃×#10587023	C₃²	1	1	yes	characteristic
32	C₃×#10587023	C₃²	1	3	yes	—
33	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	1	yes	characteristic
34	C₃²×#4294	C₃²	1	1	yes	characteristic
35	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
36	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
37	C₃×C₉×C₁₉⋊C₃	C₃²	1	3	yes	—
38	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
39	#13095307	C₃²	1	1	yes	characteristic
40	#13095307	C₃²	1	1	yes	characteristic
41	#13095307	C₃²	1	3	yes	—
42	#13095306	C₃²	1	1	yes	characteristic
43	#13095306	C₃²	1	1	yes	characteristic
44	#13095306	C₃²	1	3	yes	—
45	#13095305	C₃²	1	3	yes	—
46	#13095305	C₃²	1	3	yes	—
47	#13095305	C₃²	1	3	yes	—
48	#13095305	C₃²	1	3	yes	—
49	#13095305	C₃²	1	3	yes	—
50	#13095305	C₃²	1	3	yes	—
51	#13095305	C₃²	1	3	yes	—
52	#13095305	C₃²	1	3	yes	—
53	#13095305	C₃²	1	3	yes	—
54	#13095305	C₃²	1	3	yes	—
55	#13095305	C₃²	1	3	yes	—
56	#13095305	C₃²	1	3	yes	—
57	#13095305	C₃²	1	3	yes	—
58	#13095305	C₃²	1	3	yes	—
59	#13095305	C₃²	1	3	yes	—
60	#13095305	C₃²	1	3	yes	—
61	#13095305	C₃²	1	3	yes	—
62	#13095305	C₃²	1	3	yes	—
63	#13095305	C₃²	1	3	yes	—
64	C₃²⋊#4294	C₃²	1	3	yes	—
65	C₃²⋊#4294	C₃²	1	3	yes	—
66	#13095303	C₃²	1	3	yes	—
67	#13095303	C₃²	1	3	yes	—
68	#13095303	C₃²	1	3	yes	—
69	#13095303	C₃²	1	3	yes	—
70	#13095303	C₃²	1	3	yes	—
71	#13095303	C₃²	1	3	yes	—
72	C₉×#4294	C₃²	1	1	yes	characteristic
73	C₉×#4294	C₃²	1	1	yes	characteristic
74	C₉×#4294	C₃²	1	3	yes	—
75	C₃×C₁₉×3₋¹⁺²	C₃²	1	1	yes	characteristic
76	C₃×C₁₉×3₋¹⁺²	C₃²	1	3	yes	—
77	C₁₉×C₉⋊C₉	C₃²	1	1	yes	characteristic
78	C₁₉×C₉⋊C₉	C₃²	1	1	yes	characteristic
79	C₁₉×C₉⋊C₉	C₃²	1	3	yes	—
80	C₁₉×C₉⋊C₉	C₃²	1	3	yes	—
81	C₉²×C₁₉	C₃²	1	1	yes	characteristic
82	#10624887	—	19	1	no	maximal, Sylow
83	C₃³×C₁₉	C₃³	1	1	yes	characteristic
84	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
85	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
86	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
87	C₁₉×3₋¹⁺²	—	3	3	no	—
88	C₁₉×3₋¹⁺²	—	9	3	no	—
89	C₃×C₉×C₁₉	C₃³	1	3	yes	—
90	C₃×C₉×C₁₉	C₃³	1	3	yes	—
91	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
92	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
93	C₃×C₉×C₁₉	C₃³	1	3	yes	—
94	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
95	C₃²⋊(C₁₉⋊C₃)	—	3	3	no	—
96	#10587025	—	3	3	no	—
97	C₃×#4294	—	3	3	no	—
98	C₃×#4294	C₃³	1	3	yes	—
99	C₃×#4294	C₃³	1	3	yes	—
100	C₃×#4294	C₃³	1	3	yes	—
101	C₃×#4294	C₃³	1	3	yes	—
102	C₃×#4294	C₃³	1	3	yes	—
103	C₃×#4294	C₃³	1	3	yes	—
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	1	yes	characteristic
110	#10587023	—	3	3	no	—
111	#10587023	—	3	3	no	—
112	#10587023	—	3	3	no	—
113	#10587023	—	9	3	no	—
114	#10587022	—	3	3	no	—
115	#10587022	—	3	3	no	—
116	#10587022	—	3	3	no	—
117	#10587022	—	9	3	no	—
118	C₉×C₁₉⋊C₃	—	3	3	no	—
119	C₉×C₁₉⋊C₃	—	3	3	no	—
120	C₉×C₁₉⋊C₃	—	9	3	no	—
121	C₉×C₁₉⋊C₃	3₋¹⁺²	1	9	yes	complemented
122	C₃×#737	—	19	1	no	—
123	#6900	—	19	1	no	—
124	#6900	—	19	1	no	—
125	#6900	C₁₉⋊C₃	1	1	yes	characteristic, complemented
126	#6899	—	19	1	no	—
127	#6899	—	19	1	no	—
128	#6899	—	19	1	no	—
129	#6899	—	19	1	no	—
130	#6899	—	19	1	no	—
131	#6899	—	19	1	no	—
132	#6896	—	19	3	no	—
133	#6896	—	19	3	no	—
134	#6896	—	19	3	no	—
135	#6894	—	19	3	no	—
136	#6894	—	19	3	no	—
137	#6894	—	19	3	no	—
138	#6894	—	19	3	no	—
139	#6894	—	19	3	no	—
140	C₉⋊H(3,𝔽₃)	—	19	3	no	—
141	#6887	—	19	1	no	—
142	#6887	—	19	1	no	—
143	C₃×C₉⋊C₉	—	19	1	no	—
144	C₃²×C₁₉	—	3	1	no	—
145	C₃²×C₁₉	—	3	3	no	—
146	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
147	C₃×C₁₉⋊C₃	—	3	3	no	—
148	C₃×C₁₉⋊C₃	—	3	1	no	—
149	C₃×C₁₉⋊C₃	—	3	1	no	—
150	C₃×C₁₉⋊C₃	—	3	3	no	—
151	C₃×C₁₉⋊C₃	—	3	3	no	—
152	C₃×C₁₉⋊C₃	C₃×3₋¹⁺²	1	3	yes	—
153	C₉×C₁₉	—	3	1	no	—
154	C₉×C₁₉	—	3	1	no	—
155	C₉×C₁₉	—	3	3	no	—
156	C₉×C₁₉	—	3	3	no	—
157	C₉×C₁₉	—	3	3	no	—
158	C₉×C₁₉	C₃×3₋¹⁺²	1	3	yes	—
159	#4294	—	3	1	no	—
160	#4294	—	3	1	no	—
161	#4294	—	3	1	no	—
162	#4294	—	3	1	no	—
163	#4294	—	3	3	no	—
164	#4294	—	3	3	no	—
165	#4294	—	3	3	no	—
166	#4294	—	3	3	no	—
167	#4294	—	3	3	no	—
168	#4294	—	3	3	no	—
169	#4294	C₃×3₋¹⁺²	1	3	yes	—
170	#4294	C₃×3₋¹⁺²	1	3	yes	—
171	#737	—	57	9	no	—
172	C₃×3₋¹⁺²	—	19	1	no	—
173	C₃×3₋¹⁺²	—	19	1	no	—
174	C₃×3₋¹⁺²	—	19	1	no	—
175	C₃×3₋¹⁺²	—	19	1	no	—
176	C₃×3₋¹⁺²	—	19	1	no	—
177	C₃×3₋¹⁺²	—	19	1	no	—
178	C₃×3₋¹⁺²	—	19	1	no	—
179	C₃×3₋¹⁺²	—	19	3	no	—
180	C₃×3₋¹⁺²	—	19	3	no	—
181	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
182	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	3	yes	—
183	C₃×H(3,𝔽₃)	—	19	1	no	—
184	C₃²×C₉	—	19	1	no	—
185	C₃²×C₉	—	19	1	no	—
186	C₃²×C₉	—	19	1	no	—
187	C₃²×C₉	—	19	1	no	—
188	C₃²×C₉	—	19	3	no	—
189	C₉⋊C₉	—	19	3	no	—
190	C₉⋊C₉	—	19	3	no	—
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	1	no	—
209	C₉⋊C₉	—	19	1	no	—
210	C₉⋊C₉	—	19	3	no	—
211	C₉⋊C₉	—	19	1	no	—
212	C₉⋊C₉	—	19	1	no	—
213	C₉⋊C₉	—	19	3	no	—
214	C₉⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
215	C₉⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
216	C₉⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	—
217	C₉⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	—
218	C₃²⋊C₉	—	19	3	no	—
219	C₃²⋊C₉	—	19	3	no	—
220	C₃²⋊C₉	—	19	3	no	—
221	C₃²⋊C₉	—	19	3	no	—
222	C₃²⋊C₉	—	19	3	no	—
223	C₃²⋊C₉	—	19	3	no	—
224	C₃²⋊C₉	—	19	3	no	—
225	C₃²⋊C₉	—	19	3	no	—
226	C₉²	—	19	3	no	—
227	C₉²	—	19	1	no	—
228	C₉²	—	19	1	no	—
229	C₉²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
230	C₃×C₁₉	—	9	1	no	—
231	C₃×C₁₉	C₃²×3₋¹⁺²	1	1	yes	characteristic
232	C₃×C₁₉	3₋¹⁺⁴	1	3	yes	—
233	C₁₉⋊C₃	—	3	3	no	—
234	C₁₉⋊C₃	—	9	1	no	—
235	C₁₉⋊C₃	—	9	1	no	—
236	C₃³	—	19	1	no	—
237	C₃³	—	19	1	no	—
238	C₃³	—	19	1	no	—
239	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
240	3₋¹⁺²	—	3	3	no	—
241	3₋¹⁺²	—	9	3	no	—
242	3₋¹⁺²	—	57	3	no	—
243	3₋¹⁺²	—	57	3	no	—
244	3₋¹⁺²	—	57	3	no	—
245	3₋¹⁺²	—	57	3	no	—
246	3₋¹⁺²	—	57	3	no	—
247	3₋¹⁺²	—	57	3	no	—
248	3₋¹⁺²	—	57	3	no	—
249	3₋¹⁺²	—	171	3	no	—
250	3₋¹⁺²	—	171	3	no	—
251	H(3,𝔽₃)	—	57	3	no	—
252	C₃×C₉	—	19	1	no	—
253	C₃×C₉	—	19	1	no	—
254	C₃×C₉	—	19	9	no	—
255	C₃×C₉	—	19	3	no	—
256	C₃×C₉	—	19	3	no	—
257	C₃×C₉	—	19	3	no	—
258	C₃×C₉	—	19	3	no	—
259	C₃×C₉	—	19	3	no	—
260	C₃×C₉	—	19	3	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	—	57	3	no	—
266	C₃×C₉	—	57	3	no	—
267	C₃×C₉	—	57	3	no	—
268	C₃×C₉	—	171	3	no	—
269	C₃×C₉	C₃²×C₁₉⋊C₃	1	3	yes	—
270	C₃×C₉	C₃²×C₁₉⋊C₃	1	3	yes	—
271	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
272	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
273	C₃×C₉	C₃²×C₁₉⋊C₃	1	3	yes	—
274	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
275	C₁₉	#10624887	1	1	yes	characteristic, complemented, Sylow
276	C₃²	—	3	1	no	—
277	C₃²	—	3	3	no	—
278	C₃²	—	19	3	no	—
279	C₃²	—	57	3	no	—
280	C₃²	—	57	1	no	—
281	C₃²	—	57	1	no	—
282	C₃²	—	57	3	no	—
283	C₃²	—	57	3	no	—
284	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
285	C₉	—	3	1	no	—
286	C₉	—	3	1	no	—
287	C₉	—	3	3	no	—
288	C₉	—	3	3	no	—
289	C₉	—	3	3	no	—
290	C₉	—	19	3	no	—
291	C₉	—	19	3	no	—
292	C₉	—	57	1	no	—
293	C₉	—	57	1	no	—
294	C₉	—	57	1	no	—
295	C₉	—	57	1	no	—
296	C₉	—	57	3	no	—
297	C₉	—	57	3	no	—
298	C₉	—	57	3	no	—
299	C₉	—	57	3	no	—
300	C₉	—	57	3	no	—
301	C₉	—	57	3	no	—
302	C₉	3₋¹⁺²×C₁₉⋊C₃	1	3	yes	—
303	C₃	—	9	1	no	—
304	C₃	—	57	3	no	—
305	C₃	—	171	1	no	—
306	C₃	—	171	1	no	—
307	C₃	C₃×3₋¹⁺²×C₁₉⋊C₃	1	1	yes	characteristic
308	C₃	#17987546	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₁₉⋊C₃
commutator, socle, cosocle	C₃²×C₁₉	C₃⁴
Fitting	C₁₉×#6900	C₃
prime radical	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²
hypercenter	#6900	C₁₉⋊C₃
hypocenter	C₁₉	#10624887
LCS	G ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ #6900

Complements

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

N	H ≅ G/N	type
2. #17987472	C₃	semidirect
3. #17987471	C₃	semidirect
4. #17987467	C₃	semidirect
5. #17987466	C₃	semidirect
6. #17987463	C₃	semidirect
7. #17987462	C₃	semidirect
8. #17987461	C₃	semidirect
9. #17987461	C₃	semidirect
10. #17987449	C₃	semidirect
11. #17987449	C₃	semidirect
12. #17987447	C₃	semidirect
13. #17987437	C₃	semidirect
14. #17987437	C₃	semidirect
15. #17987436	C₃	semidirect
16. #17987436	C₃	semidirect
17. #17987435	C₃	semidirect
19. #17987404	C₃	semidirect
20. #17987403	C₃	semidirect
21. C₁₉⋊C₃×C₉⋊C₉	C₃	semidirect
22. C₁₉×#6900	C₃	semidirect
121. C₉×C₁₉⋊C₃	3₋¹⁺²	semidirect
125. #6900	C₁₉⋊C₃	semidirect
275. C₁₉	#10624887	semidirect

Maximal subgroups

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

subgroup	index
C₃×#13095333	3
#17987472	3
#17987471	3
#17987467	3
#17987466	3
#17987463	3
#17987462	3
#17987461	3
#17987449	3
#17987447	3
#17987437	3
#17987436	3
#17987435	3
#17987422	3
#17987404	3
#17987403	3
C₁₉⋊C₃×C₉⋊C₉	3
C₁₉×#6900	3
#10624887	19