Subgroups — #28356026

Order 13851.

662 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17987467	C₃	1	3	yes	maximal, complemented
2	#17987464	C₃	1	3	yes	maximal, complemented
3	#17987459	C₃	1	3	yes	maximal, complemented
4	#17987456	C₃	1	3	yes	maximal, complemented
5	#17987455	C₃	1	3	yes	maximal, complemented
6	#17987450	C₃	1	3	yes	maximal, complemented
7	#17987443	C₃	1	1	yes	maximal, characteristic, complemented
8	#17987442	C₃	1	1	yes	maximal, characteristic, complemented
9	#17987440	C₃	1	1	yes	maximal, characteristic, complemented
10	#17987438	C₃	1	1	yes	maximal, characteristic, complemented
11	#17987435	C₃	1	1	yes	maximal, characteristic, complemented
12	#17987432	C₃	1	1	yes	maximal, characteristic
13	#17987428	C₃	1	1	yes	maximal, characteristic
14	#17987426	C₃	1	1	yes	maximal, characteristic
15	#17987415	C₃	1	1	yes	maximal, characteristic, complemented
16	3₋¹⁺²×#4294	C₃	1	1	yes	maximal, characteristic, complemented
17	#17987410	C₃	1	3	yes	maximal, complemented
18	#17987409	C₃	1	3	yes	maximal, complemented
19	#17987406	C₃	1	1	yes	maximal, characteristic
20	#17987404	C₃	1	3	yes	maximal, complemented
21	C₃×C₉×#4294	C₃	1	1	yes	maximal, characteristic, complemented
22	C₁₉×#6894	C₃	1	1	yes	maximal, characteristic, complemented
23	C₃×#10587025	C₃²	1	1	yes	characteristic
24	C₃×#10587022	C₃²	1	3	yes	—
25	C₃×#10587023	C₃²	1	3	yes	—
26	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	1	yes	characteristic
27	C₃²×#4294	C₃²	1	1	yes	characteristic
28	C₃²×#4294	C₃²	1	1	yes	characteristic
29	C₃²×#4294	C₃²	1	1	yes	characteristic
30	C₃×C₉×C₁₉⋊C₃	C₃²	1	3	yes	—
31	#13095307	C₃²	1	3	yes	—
32	#13095307	C₃²	1	3	yes	—
33	#13095307	C₃²	1	3	yes	—
34	#13095306	C₃²	1	3	yes	—
35	#13095306	C₃²	1	3	yes	—
36	#13095306	C₃²	1	3	yes	—
37	#13095305	C₃²	1	3	yes	—
38	#13095305	C₃²	1	3	yes	—
39	#13095305	C₃²	1	3	yes	—
40	#13095305	C₃²	1	3	yes	—
41	#13095305	C₃²	1	3	yes	—
42	#13095305	C₃²	1	3	yes	—
43	#13095305	C₃²	1	3	yes	—
44	#13095305	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	C₃²⋊#4294	C₃²	1	1	yes	characteristic
51	C₃²⋊#4294	C₃²	1	1	yes	characteristic
52	C₃²⋊#4294	C₃²	1	1	yes	characteristic
53	C₃²⋊#4294	C₃²	1	1	yes	characteristic
54	#13095303	C₃²	1	3	yes	—
55	#13095303	C₃²	1	3	yes	—
56	#13095303	C₃²	1	3	yes	—
57	#13095303	C₃²	1	3	yes	—
58	#13095303	C₃²	1	3	yes	—
59	#13095303	C₃²	1	3	yes	—
60	#13095303	C₃²	1	3	yes	—
61	#13095303	C₃²	1	3	yes	—
62	#13095303	C₃²	1	3	yes	—
63	C₉×#4294	C₃²	1	1	yes	characteristic
64	C₉×#4294	C₃²	1	1	yes	characteristic
65	C₉×#4294	C₃²	1	1	yes	characteristic
66	C₉×#4294	C₃²	1	1	yes	characteristic
67	C₉×#4294	C₃²	1	1	yes	characteristic
68	C₉×#4294	C₃²	1	1	yes	characteristic
69	C₉×#4294	C₃²	1	1	yes	characteristic
70	C₉×#4294	C₃²	1	1	yes	characteristic
71	C₉×#4294	C₃²	1	1	yes	characteristic
72	C₉×#4294	C₃²	1	3	yes	—
73	C₉×#4294	C₃²	1	3	yes	—
74	C₃×C₁₉×3₋¹⁺²	C₃²	1	1	yes	characteristic
75	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
76	C₁₉×C₉⋊C₉	C₃²	1	3	yes	—
77	C₁₉×C₉⋊C₉	C₃²	1	3	yes	—
78	C₁₉×C₉⋊C₉	C₃²	1	3	yes	—
79	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
80	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
81	#10624877	—	19	1	no	maximal, Sylow
82	C₃³×C₁₉	C₃³	1	1	yes	characteristic
83	C₃²×C₁₉⋊C₃	C₃³	1	3	yes	—
84	C₁₉×3₋¹⁺²	—	3	3	no	—
85	C₃×C₉×C₁₉	—	3	3	no	—
86	C₃×C₉×C₁₉	C₃³	1	3	yes	—
87	C₃×C₉×C₁₉	C₃³	1	3	yes	—
88	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
89	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
90	C₃×C₉×C₁₉	C₃³	1	3	yes	—
91	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
92	C₃²⋊(C₁₉⋊C₃)	—	3	3	no	—
93	#10587025	—	3	3	no	—
94	C₃×#4294	—	3	3	no	—
95	C₃×#4294	—	3	3	no	—
96	C₃×#4294	—	3	3	no	—
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	C₃×#4294	C₃³	1	1	yes	characteristic
101	C₃×#4294	C₃³	1	1	yes	characteristic
102	C₃×#4294	C₃³	1	1	yes	characteristic
103	C₃×#4294	C₃³	1	1	yes	characteristic
104	C₃×#4294	C₃³	1	3	yes	—
105	C₃×#4294	C₃³	1	3	yes	—
106	C₃×#4294	C₃³	1	3	yes	—
107	C₃×#4294	C₃³	1	3	yes	—
108	C₃×#4294	C₃³	1	3	yes	—
109	C₃×#4294	C₃³	1	1	yes	characteristic
110	C₃×#4294	C₃³	1	1	yes	characteristic
111	#10587023	—	3	9	no	—
112	#10587022	—	3	9	no	—
113	C₉×C₁₉⋊C₃	—	3	9	no	—
114	#6899	—	19	3	no	—
115	#6899	—	19	3	no	—
116	#6898	—	19	3	no	—
117	#6897	—	19	3	no	—
118	#6897	—	19	3	no	—
119	#6896	—	19	3	no	—
120	#6895	—	19	1	no	—
121	#6895	—	19	1	no	—
122	#6894	—	19	1	no	—
123	#6894	—	19	1	no	—
124	#6894	—	19	1	no	—
125	#6894	C₁₉⋊C₃	1	1	yes	characteristic, complemented
126	#6893	—	19	1	no	—
127	C₉⋊H(3,𝔽₃)	—	19	1	no	—
128	C₉⋊H(3,𝔽₃)	—	19	1	no	—
129	C₉×3₋¹⁺²	—	19	1	no	—
130	C₉×3₋¹⁺²	—	19	3	no	—
131	C₉×3₋¹⁺²	—	19	3	no	—
132	C₉×3₋¹⁺²	—	19	1	no	—
133	C₉×H(3,𝔽₃)	—	19	1	no	—
134	#6887	—	19	3	no	—
135	C₃×C₉²	—	19	1	no	—
136	C₃²×C₁₉	—	3	1	no	—
137	C₃²×C₁₉	—	3	1	no	—
138	C₃²×C₁₉	—	3	1	no	—
139	C₃²×C₁₉	#737	1	3	yes	—
140	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
141	C₃×C₁₉⋊C₃	—	3	3	no	—
142	C₃×C₁₉⋊C₃	—	3	3	no	—
143	C₃×C₁₉⋊C₃	—	3	3	no	—
144	C₃×C₁₉⋊C₃	—	3	3	no	—
145	C₉×C₁₉	—	3	3	no	—
146	C₉×C₁₉	—	3	3	no	—
147	C₉×C₁₉	—	3	1	no	—
148	C₉×C₁₉	—	3	1	no	—
149	C₉×C₁₉	—	3	3	no	—
150	C₉×C₁₉	#737	1	3	yes	—
151	#4294	—	3	3	no	—
152	#4294	—	3	3	no	—
153	#4294	—	3	3	no	—
154	#4294	—	3	3	no	—
155	#4294	—	3	1	no	—
156	#4294	—	3	1	no	—
157	#4294	—	3	1	no	—
158	#4294	—	3	1	no	—
159	#4294	—	3	1	no	—
160	#4294	—	3	1	no	—
161	#4294	—	3	1	no	—
162	#4294	—	3	3	no	—
163	#4294	#737	1	3	yes	—
164	#4294	#737	1	3	yes	—
165	C₃×3₋¹⁺²	—	19	1	no	—
166	C₃×3₋¹⁺²	—	19	3	no	—
167	C₃×3₋¹⁺²	—	19	3	no	—
168	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
169	C₃×H(3,𝔽₃)	—	19	1	no	—
170	C₃²×C₉	—	19	1	no	—
171	C₃²×C₉	—	19	1	no	—
172	C₃²×C₉	—	19	1	no	—
173	C₃²×C₉	—	19	3	no	—
174	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
175	C₉⋊C₉	—	19	3	no	—
176	C₉⋊C₉	—	19	3	no	—
177	C₉⋊C₉	—	19	3	no	—
178	C₉⋊C₉	—	19	3	no	—
179	C₉⋊C₉	—	19	3	no	—
180	C₉⋊C₉	—	19	3	no	—
181	C₉⋊C₉	—	19	3	no	—
182	C₉⋊C₉	—	19	3	no	—
183	C₉⋊C₉	—	19	3	no	—
184	C₉⋊C₉	—	19	3	no	—
185	C₉⋊C₉	—	19	3	no	—
186	C₉⋊C₉	—	19	3	no	—
187	C₉⋊C₉	—	19	3	no	—
188	C₉⋊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	C₃×C₁₉⋊C₃	1	3	yes	—
195	C₉⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	—
196	C₉⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	—
197	C₃²⋊C₉	—	19	1	no	—
198	C₃²⋊C₉	—	19	1	no	—
199	C₃²⋊C₉	—	19	1	no	—
200	C₃²⋊C₉	—	19	1	no	—
201	C₃²⋊C₉	—	19	3	no	—
202	C₃²⋊C₉	—	19	3	no	—
203	C₃²⋊C₉	—	19	3	no	—
204	C₃²⋊C₉	—	19	3	no	—
205	C₃²⋊C₉	—	19	3	no	—
206	C₃²⋊C₉	—	19	3	no	—
207	C₃²⋊C₉	—	19	3	no	—
208	C₃²⋊C₉	—	19	3	no	—
209	C₃²⋊C₉	—	19	3	no	—
210	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
211	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
212	C₉²	—	19	1	no	—
213	C₉²	—	19	1	no	—
214	C₉²	—	19	1	no	—
215	C₉²	—	19	1	no	—
216	C₉²	—	19	1	no	—
217	C₉²	—	19	1	no	—
218	C₉²	—	19	1	no	—
219	C₉²	—	19	1	no	—
220	C₉²	—	19	1	no	—
221	C₉²	—	19	3	no	—
222	C₉²	—	19	3	no	—
223	C₃×C₁₉	—	3	3	no	—
224	C₃×C₁₉	C₃×#737	1	1	yes	characteristic
225	C₃×C₁₉	C₃×#737	1	1	yes	characteristic
226	C₃×C₁₉	C₃×#737	1	1	yes	characteristic
227	C₃×C₁₉	C₃×#737	1	1	yes	characteristic
228	C₁₉⋊C₃	—	9	3	no	—
229	C₃³	—	19	3	no	—
230	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
231	3₋¹⁺²	—	3	3	no	—
232	3₋¹⁺²	—	57	9	no	—
233	3₋¹⁺²	—	57	3	no	—
234	3₋¹⁺²	—	57	9	no	—
235	H(3,𝔽₃)	—	57	3	no	—
236	C₃×C₉	—	3	3	no	—
237	C₃×C₉	—	19	1	no	—
238	C₃×C₉	—	19	1	no	—
239	C₃×C₉	—	19	1	no	—
240	C₃×C₉	—	19	1	no	—
241	C₃×C₉	—	19	1	no	—
242	C₃×C₉	—	19	1	no	—
243	C₃×C₉	—	19	1	no	—
244	C₃×C₉	—	19	1	no	—
245	C₃×C₉	—	19	1	no	—
246	C₃×C₉	—	19	3	no	—
247	C₃×C₉	—	19	3	no	—
248	C₃×C₉	—	19	3	no	—
249	C₃×C₉	—	19	3	no	—
250	C₃×C₉	—	19	3	no	—
251	C₃×C₉	—	57	9	no	—
252	C₃×C₉	—	57	3	no	—
253	C₃×C₉	—	57	3	no	—
254	C₃×C₉	—	57	3	no	—
255	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
256	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
257	C₃×C₉	C₃²×C₁₉⋊C₃	1	3	yes	—
258	C₃×C₉	C₃²×C₁₉⋊C₃	1	3	yes	—
259	C₃×C₉	C₃²×C₁₉⋊C₃	1	3	yes	—
260	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
261	C₁₉	#10624877	1	1	yes	characteristic, complemented, Sylow
262	C₃²	—	3	1	no	—
263	C₃²	—	3	1	no	—
264	C₃²	—	3	1	no	—
265	C₃²	—	57	3	no	—
266	C₃²	—	57	3	no	—
267	C₃²	—	57	3	no	—
268	C₃²	—	57	3	no	—
269	C₃²	#13095333	1	3	yes	—
270	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
271	C₉	—	3	3	no	—
272	C₉	—	3	3	no	—
273	C₉	—	3	1	no	—
274	C₉	—	3	1	no	—
275	C₉	—	3	3	no	—
276	C₉	—	19	3	no	—
277	C₉	—	19	3	no	—
278	C₉	—	57	1	no	—
279	C₉	—	57	1	no	—
280	C₉	—	57	1	no	—
281	C₉	—	57	1	no	—
282	C₉	—	57	3	no	—
283	C₉	—	57	3	no	—
284	C₉	—	57	1	no	—
285	C₉	—	57	1	no	—
286	C₉	—	57	1	no	—
287	C₉	—	57	3	no	—
288	C₉	—	57	3	no	—
289	C₉	—	57	3	no	—
290	C₉	#13095333	1	3	yes	—
291	C₃	—	3	3	no	—
292	C₃	—	171	3	no	—
293	C₃	C₁₉⋊C₃×#737	1	1	yes	characteristic
294	C₃	C₁₉⋊C₃×#737	1	1	yes	characteristic
295	C₃	C₃×#13095333	1	1	yes	characteristic
296	C₃	C₃×#13095333	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	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²
hypercenter	#6894	C₁₉⋊C₃
hypocenter	C₁₉	#10624877
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
1. #17987467	C₃	semidirect
2. #17987464	C₃	semidirect
3. #17987459	C₃	semidirect
4. #17987456	C₃	semidirect
5. #17987455	C₃	semidirect
6. #17987450	C₃	semidirect
7. #17987443	C₃	semidirect
8. #17987442	C₃	semidirect
9. #17987440	C₃	semidirect
10. #17987438	C₃	semidirect
11. #17987435	C₃	semidirect
15. #17987415	C₃	semidirect
16. 3₋¹⁺²×#4294	C₃	semidirect
17. #17987410	C₃	semidirect
18. #17987409	C₃	semidirect
20. #17987404	C₃	semidirect
21. C₃×C₉×#4294	C₃	semidirect
22. C₁₉×#6894	C₃	semidirect
125. #6894	C₁₉⋊C₃	semidirect
261. C₁₉	#10624877	semidirect

Maximal subgroups

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

subgroup	index
#17987467	3
#17987464	3
#17987459	3
#17987456	3
#17987455	3
#17987450	3
#17987443	3
#17987442	3
#17987440	3
#17987438	3
#17987435	3
#17987432	3
#17987428	3
#17987426	3
#17987415	3
3₋¹⁺²×#4294	3
#17987410	3
#17987409	3
#17987406	3
#17987404	3
C₃×C₉×#4294	3
C₁₉×#6894	3
#10624877	19