Subgroups — #28355962

Order 13851.

1166 conjugacy classes of subgroups.

The subgroup lattice has 302 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	3	yes	maximal, complemented
2	C₃²×C₉×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
3	#17987430	C₃	1	6	yes	maximal, complemented
4	#17987428	C₃	1	1	yes	maximal, characteristic, complemented
5	#17987427	C₃	1	1	yes	maximal, characteristic, complemented
6	#17987426	C₃	1	1	yes	maximal, characteristic, complemented
7	#17987425	C₃	1	1	yes	maximal, characteristic, complemented
8	#17987424	C₃	1	6	yes	maximal, complemented
9	#17987423	C₃	1	1	yes	maximal, characteristic, complemented
10	#17987422	C₃	1	6	yes	maximal, complemented
11	#17987420	C₃	1	1	yes	maximal, characteristic, complemented
12	#17987419	C₃	1	1	yes	maximal, characteristic, complemented
13	(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
14	H(3,𝔽₃)×#4294	C₃	1	1	yes	maximal, characteristic, complemented
15	#17987406	C₃	1	1	yes	maximal, characteristic, complemented
16	#17987406	C₃	1	1	yes	maximal, characteristic, complemented
17	#17987406	C₃	1	6	yes	maximal, complemented
18	C₁₉×C₉⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
19	C₃³×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
20	#13095333	—	3	27	no	—
21	C₃×#10587025	C₃²	1	3	yes	complemented
22	C₃×#10587022	C₃²	1	3	yes	—
23	C₃×#10587022	C₃²	1	3	yes	complemented
24	C₃×#10587022	C₃²	1	3	yes	complemented
25	C₃×#10587023	C₃²	1	3	yes	—
26	C₃×#10587023	C₃²	1	3	yes	complemented
27	C₃×#10587023	C₃²	1	3	yes	complemented
28	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	6	yes	—
29	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	1	yes	characteristic
30	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	1	yes	characteristic
31	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	3	yes	—
32	C₃²×#4294	C₃²	1	1	yes	characteristic
33	C₃²×#4294	C₃²	1	1	yes	characteristic
34	C₃²×#4294	C₃²	1	3	yes	complemented
35	C₃×C₉×C₁₉⋊C₃	—	3	9	no	—
36	C₃×C₉×C₁₉⋊C₃	C₃²	1	2	yes	—
37	C₃×C₉×C₁₉⋊C₃	C₃²	1	2	yes	—
38	C₃×C₉×C₁₉⋊C₃	C₃²	1	2	yes	—
39	C₃×C₉×C₁₉⋊C₃	C₃²	1	2	yes	—
40	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
41	C₃×C₉×C₁₉⋊C₃	C₃²	1	3	yes	complemented
42	C₃×C₉×C₁₉⋊C₃	C₃²	1	3	yes	complemented
43	C₃²⋊#4294	C₃²	1	1	yes	characteristic
44	C₃²⋊#4294	C₃²	1	1	yes	characteristic
45	C₃²⋊#4294	C₃²	1	1	yes	characteristic
46	C₃²⋊#4294	C₃²	1	1	yes	characteristic
47	C₃²⋊#4294	C₃²	1	6	yes	complemented
48	C₃²⋊#4294	C₃²	1	6	yes	complemented
49	#13095303	C₃²	1	6	yes	—
50	#13095303	C₃²	1	6	yes	—
51	#13095303	C₃²	1	6	yes	complemented
52	#13095303	C₃²	1	6	yes	complemented
53	#13095303	C₃²	1	6	yes	complemented
54	#13095303	C₃²	1	6	yes	complemented
55	#13095303	C₃²	1	6	yes	complemented
56	#13095303	C₃²	1	6	yes	complemented
57	C₃×C₁₉×3₋¹⁺²	C₃²	1	3	yes	complemented
58	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
59	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
60	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
61	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
62	C₁₉×C₃²⋊C₉	C₃²	1	6	yes	complemented
63	#10624869	—	19	1	no	maximal, Sylow
64	C₃³×C₁₉	C₃³	1	3	yes	—
65	C₃³×C₁₉	C₃³	1	1	yes	characteristic
66	C₃²×C₁₉⋊C₃	—	3	3	no	—
67	C₃²×C₁₉⋊C₃	—	3	3	no	—
68	C₃²×C₁₉⋊C₃	—	3	6	no	—
69	C₃²×C₁₉⋊C₃	C₃³	1	3	yes	—
70	C₃²×C₁₉⋊C₃	C₃³	1	3	yes	—
71	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
72	C₃²×C₁₉⋊C₃	C₃³	1	2	yes	—
73	C₁₉×3₋¹⁺²	—	3	9	no	—
74	C₁₉×H(3,𝔽₃)	—	3	3	no	—
75	C₃×C₉×C₁₉	—	3	3	no	—
76	C₃×C₉×C₁₉	C₃³	1	3	yes	—
77	C₃×C₉×C₁₉	C₃³	1	3	yes	—
78	C₃×C₉×C₁₉	C₃³	1	2	yes	—
79	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
80	C₃²⋊(C₁₉⋊C₃)	—	3	3	no	—
81	C₃²⋊(C₁₉⋊C₃)	—	3	3	no	—
82	C₃²⋊(C₁₉⋊C₃)	—	3	9	no	—
83	C₃²⋊(C₁₉⋊C₃)	—	3	18	no	—
84	#10587025	—	3	9	no	—
85	C₃×#4294	—	3	3	no	—
86	C₃×#4294	—	3	3	no	—
87	C₃×#4294	—	3	9	no	—
88	C₃×#4294	C₃³	1	3	yes	—
89	C₃×#4294	C₃³	1	3	yes	—
90	C₃×#4294	C₃³	1	3	yes	—
91	C₃×#4294	C₃³	1	3	yes	—
92	C₃×#4294	C₃³	1	2	yes	—
93	C₃×#4294	C₃³	1	2	yes	—
94	C₃×#4294	C₃³	1	1	yes	characteristic
95	C₃×#4294	C₃³	1	1	yes	characteristic
96	#10587023	—	3	9	no	—
97	#10587023	—	3	9	no	—
98	#10587023	—	3	9	no	—
99	#10587022	—	3	9	no	—
100	#10587022	—	3	9	no	—
101	#10587022	—	3	9	no	—
102	C₉×C₁₉⋊C₃	—	3	6	no	—
103	C₉×C₁₉⋊C₃	—	3	6	no	—
104	C₉×C₁₉⋊C₃	—	3	6	no	—
105	C₉×C₁₉⋊C₃	—	3	6	no	—
106	C₉×C₁₉⋊C₃	—	3	9	no	—
107	C₉×C₁₉⋊C₃	—	3	9	no	—
108	C₉×C₁₉⋊C₃	H(3,𝔽₃)	1	9	yes	complemented
109	C₃×#737	—	19	3	no	—
110	C₃³×C₉	—	19	1	no	—
111	#6893	—	19	6	no	—
112	C₉⋊H(3,𝔽₃)	—	19	6	no	—
113	C₉⋊H(3,𝔽₃)	—	19	1	no	—
114	C₉⋊H(3,𝔽₃)	—	19	1	no	—
115	C₉⋊H(3,𝔽₃)	—	19	1	no	—
116	C₉⋊H(3,𝔽₃)	—	19	1	no	—
117	C₉⋊H(3,𝔽₃)	—	19	1	no	—
118	C₉⋊H(3,𝔽₃)	—	19	6	no	—
119	C₉⋊H(3,𝔽₃)	C₁₉⋊C₃	1	1	yes	characteristic, complemented
120	#6891	—	19	1	no	—
121	#6891	—	19	1	no	—
122	C₃²⋊H(3,𝔽₃)	—	19	1	no	—
123	C₉×H(3,𝔽₃)	—	19	1	no	—
124	C₉×H(3,𝔽₃)	—	19	1	no	—
125	C₉×H(3,𝔽₃)	—	19	1	no	—
126	C₉×H(3,𝔽₃)	—	19	6	no	—
127	C₃²×C₁₉	—	3	1	no	—
128	C₃²×C₁₉	—	3	2	no	—
129	C₃²×C₁₉	—	3	3	no	—
130	C₃²×C₁₉	—	3	6	no	—
131	C₃²×C₁₉	—	3	3	no	—
132	C₃²×C₁₉	#737	1	3	yes	complemented
133	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
134	C₃×C₁₉⋊C₃	—	3	27	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	1	no	—
139	C₃×C₁₉⋊C₃	—	3	2	no	—
140	C₃×C₁₉⋊C₃	—	3	3	no	—
141	C₃×C₁₉⋊C₃	—	3	3	no	—
142	C₃×C₁₉⋊C₃	—	3	6	no	—
143	C₃×C₁₉⋊C₃	—	3	6	no	—
144	C₃×C₁₉⋊C₃	—	3	3	no	—
145	C₃×C₁₉⋊C₃	—	3	3	no	—
146	C₃×C₁₉⋊C₃	C₃×H(3,𝔽₃)	1	3	yes	—
147	C₃×C₁₉⋊C₃	#737	1	6	yes	complemented
148	C₉×C₁₉	—	3	2	no	—
149	C₉×C₁₉	—	3	3	no	—
150	C₉×C₁₉	—	3	3	no	—
151	C₉×C₁₉	C₃×H(3,𝔽₃)	1	3	yes	—
152	#4294	—	3	2	no	—
153	#4294	—	3	2	no	—
154	#4294	—	3	3	no	—
155	#4294	—	3	3	no	—
156	#4294	—	3	3	no	—
157	#4294	—	3	3	no	—
158	#4294	C₃×H(3,𝔽₃)	1	3	yes	—
159	#4294	C₃×H(3,𝔽₃)	1	3	yes	—
160	C₃⁴	—	19	1	no	—
161	#737	—	57	27	no	—
162	C₃×3₋¹⁺²	—	19	3	no	—
163	C₃×3₋¹⁺²	—	19	3	no	—
164	C₃×3₋¹⁺²	—	19	3	no	—
165	C₃×3₋¹⁺²	—	19	3	no	—
166	C₃×3₋¹⁺²	—	19	3	no	—
167	C₃×3₋¹⁺²	—	19	3	no	—
168	C₃×3₋¹⁺²	—	19	3	no	—
169	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	3	yes	complemented
170	C₃×H(3,𝔽₃)	—	19	3	no	—
171	C₃×H(3,𝔽₃)	—	19	6	no	—
172	C₃×H(3,𝔽₃)	—	19	1	no	—
173	C₃×H(3,𝔽₃)	—	19	1	no	—
174	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	1	yes	characteristic
175	C₃²×C₉	—	19	1	no	—
176	C₃²×C₉	—	19	1	no	—
177	C₃²×C₉	—	19	1	no	—
178	C₃²×C₉	—	19	3	no	—
179	C₃²×C₉	—	19	2	no	—
180	C₃²×C₉	—	19	2	no	—
181	C₃²×C₉	—	19	2	no	—
182	C₃²×C₉	—	19	2	no	—
183	C₃²×C₉	—	19	3	no	—
184	C₃²×C₉	—	19	3	no	—
185	C₃²×C₉	—	57	9	no	—
186	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
187	C₃²⋊C₉	—	19	1	no	—
188	C₃²⋊C₉	—	19	1	no	—
189	C₃²⋊C₉	—	19	1	no	—
190	C₃²⋊C₉	—	19	1	no	—
191	C₃²⋊C₉	—	19	6	no	—
192	C₃²⋊C₉	—	19	6	no	—
193	C₃²⋊C₉	—	19	6	no	—
194	C₃²⋊C₉	—	19	6	no	—
195	C₃²⋊C₉	—	19	6	no	—
196	C₃²⋊C₉	—	19	6	no	—
197	C₃²⋊C₉	—	19	6	no	—
198	C₃²⋊C₉	—	19	6	no	—
199	C₃²⋊C₉	—	19	6	no	—
200	C₃²⋊C₉	—	19	6	no	—
201	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
202	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
203	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	6	yes	complemented
204	C₃×C₁₉	—	3	3	no	—
205	C₃×C₁₉	—	9	3	no	—
206	C₃×C₁₉	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
207	C₃×C₁₉	C₃×#737	1	2	yes	—
208	C₃×C₁₉	C₃×#737	1	1	yes	characteristic
209	C₁₉⋊C₃	—	3	3	no	—
210	C₁₉⋊C₃	—	3	6	no	—
211	C₁₉⋊C₃	—	9	3	no	—
212	C₁₉⋊C₃	—	9	3	no	—
213	C₃³	—	19	2	no	—
214	C₃³	—	19	1	no	—
215	C₃³	—	19	3	no	—
216	C₃³	—	19	3	no	—
217	C₃³	—	57	3	no	—
218	C₃³	—	57	6	no	—
219	C₃³	—	57	3	no	—
220	C₃³	C₃²×C₁₉⋊C₃	1	3	yes	—
221	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
222	3₋¹⁺²	—	3	9	no	—
223	3₋¹⁺²	—	57	9	no	—
224	3₋¹⁺²	—	57	9	no	—
225	3₋¹⁺²	—	57	9	no	—
226	3₋¹⁺²	—	57	9	no	—
227	3₋¹⁺²	—	57	9	no	—
228	3₋¹⁺²	—	57	9	no	—
229	3₋¹⁺²	—	57	9	no	—
230	H(3,𝔽₃)	—	3	3	no	—
231	H(3,𝔽₃)	—	57	18	no	—
232	H(3,𝔽₃)	—	57	9	no	—
233	H(3,𝔽₃)	—	57	3	no	—
234	H(3,𝔽₃)	—	57	3	no	—
235	C₃×C₉	—	3	3	no	—
236	C₃×C₉	—	19	1	no	—
237	C₃×C₉	—	19	1	no	—
238	C₃×C₉	—	19	9	no	—
239	C₃×C₉	—	19	2	no	—
240	C₃×C₉	—	19	2	no	—
241	C₃×C₉	—	19	3	no	—
242	C₃×C₉	—	19	3	no	—
243	C₃×C₉	—	19	3	no	—
244	C₃×C₉	—	19	3	no	—
245	C₃×C₉	—	57	6	no	—
246	C₃×C₉	—	57	6	no	—
247	C₃×C₉	—	57	9	no	—
248	C₃×C₉	—	57	9	no	—
249	C₃×C₉	—	57	6	no	—
250	C₃×C₉	—	57	3	no	—
251	C₃×C₉	—	57	3	no	—
252	C₃×C₉	—	57	9	no	—
253	C₃×C₉	—	57	6	no	—
254	C₃×C₉	C₃²×C₁₉⋊C₃	1	3	yes	—
255	C₃×C₉	C₃²×C₁₉⋊C₃	1	3	yes	—
256	C₃×C₉	C₃²×C₁₉⋊C₃	1	2	yes	—
257	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
258	C₁₉	#10624869	1	1	yes	characteristic, complemented, Sylow
259	C₃²	—	3	1	no	—
260	C₃²	—	3	2	no	—
261	C₃²	—	3	3	no	—
262	C₃²	—	3	6	no	—
263	C₃²	—	3	3	no	—
264	C₃²	—	19	3	no	—
265	C₃²	—	19	6	no	—
266	C₃²	—	57	27	no	—
267	C₃²	—	57	2	no	—
268	C₃²	—	57	2	no	—
269	C₃²	—	57	2	no	—
270	C₃²	—	57	2	no	—
271	C₃²	—	57	1	no	—
272	C₃²	—	57	3	no	—
273	C₃²	—	57	3	no	—
274	C₃²	—	57	6	no	—
275	C₃²	—	57	6	no	—
276	C₃²	—	57	3	no	—
277	C₃²	—	57	3	no	—
278	C₃²	#13095333	1	3	yes	complemented
279	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
280	C₉	—	3	2	no	—
281	C₉	—	3	3	no	—
282	C₉	—	3	3	no	—
283	C₉	—	19	3	no	—
284	C₉	—	19	3	no	—
285	C₉	—	57	2	no	—
286	C₉	—	57	2	no	—
287	C₉	—	57	3	no	—
288	C₉	—	57	3	no	—
289	C₉	—	57	3	no	—
290	C₉	—	57	3	no	—
291	C₉	H(3,𝔽₃)×C₁₉⋊C₃	1	3	yes	—
292	C₃	—	3	3	no	—
293	C₃	—	9	3	no	—
294	C₃	—	57	3	no	—
295	C₃	—	57	6	no	—
296	C₃	—	171	3	no	—
297	C₃	—	171	3	no	—
298	C₃	C₃×H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
299	C₃	C₁₉⋊C₃×#737	1	2	yes	—
300	C₃	C₃×#13095333	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₁₉⋊C₃
commutator, socle, cosocle	C₃²×C₁₉	C₃⁴
Fitting	C₁₉×C₉⋊H(3,𝔽₃)	C₃
prime radical	(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)	C₃
hypercenter	C₉⋊H(3,𝔽₃)	C₁₉⋊C₃
hypocenter	C₁₉	#10624869
LCS	G ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₉⋊H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃×#13095333	C₃	semidirect
2. C₃²×C₉×C₁₉⋊C₃	C₃	semidirect
3. #17987430	C₃	semidirect
4. #17987428	C₃	semidirect
5. #17987427	C₃	semidirect
6. #17987426	C₃	semidirect
7. #17987425	C₃	semidirect
8. #17987424	C₃	semidirect
9. #17987423	C₃	semidirect
10. #17987422	C₃	semidirect
11. #17987420	C₃	semidirect
12. #17987419	C₃	semidirect
14. H(3,𝔽₃)×#4294	C₃	semidirect
15. #17987406	C₃	semidirect
16. #17987406	C₃	semidirect
17. #17987406	C₃	semidirect
18. C₁₉×C₉⋊H(3,𝔽₃)	C₃	semidirect
21. C₃×#10587025	C₃²	semidirect
23. C₃×#10587022	C₃²	semidirect
24. C₃×#10587022	C₃²	semidirect
26. C₃×#10587023	C₃²	semidirect
27. C₃×#10587023	C₃²	semidirect
34. C₃²×#4294	C₃²	semidirect
41. C₃×C₉×C₁₉⋊C₃	C₃²	semidirect
42. C₃×C₉×C₁₉⋊C₃	C₃²	semidirect
47. C₃²⋊#4294	C₃²	semidirect
48. C₃²⋊#4294	C₃²	semidirect
51. #13095303	C₃²	semidirect
52. #13095303	C₃²	semidirect
53. #13095303	C₃²	semidirect
54. #13095303	C₃²	semidirect
55. #13095303	C₃²	semidirect
56. #13095303	C₃²	semidirect
57. C₃×C₁₉×3₋¹⁺²	C₃²	semidirect
62. C₁₉×C₃²⋊C₉	C₃²	semidirect
108. C₉×C₁₉⋊C₃	H(3,𝔽₃)	semidirect
119. C₉⋊H(3,𝔽₃)	C₁₉⋊C₃	semidirect
132. C₃²×C₁₉	#737	semidirect
147. C₃×C₁₉⋊C₃	#737	semidirect
169. C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	semidirect
203. C₃²⋊C₉	C₃×C₁₉⋊C₃	semidirect
258. C₁₉	#10624869	semidirect
278. C₃²	#13095333	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
C₃²×C₉×C₁₉⋊C₃	3
#17987430	3
#17987428	3
#17987427	3
#17987426	3
#17987425	3
#17987424	3
#17987423	3
#17987422	3
#17987420	3
#17987419	3
(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)	3
H(3,𝔽₃)×#4294	3
#17987406	3
C₁₉×C₉⋊H(3,𝔽₃)	3
#10624869	19