Subgroups — C₁₉⋊C₃×#6913 #28356577

Order 13851.

992 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉⋊C₃×#737	C₃	1	3	yes	maximal, complemented
2	C₃×3₋¹⁺²×C₁₉⋊C₃	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	#17987526	C₃	1	3	yes	maximal, complemented
8	#17987525	C₃	1	1	yes	maximal, characteristic, complemented
9	#17987524	C₃	1	1	yes	maximal, characteristic, complemented
10	#17987523	C₃	1	3	yes	maximal, complemented
11	#17987522	C₃	1	3	yes	maximal, complemented
12	#17987521	C₃	1	3	yes	maximal, complemented
13	#17987520	C₃	1	3	yes	maximal, complemented
14	C₁₉⋊C₃×#733	C₃	1	2	yes	maximal, complemented
15	C₁₉⋊C₃×#732	C₃	1	1	yes	maximal, characteristic, complemented
16	C₁₉⋊C₃×#731	C₃	1	3	yes	maximal, complemented
17	C₁₉⋊C₃×AΣL(1,𝔽₂₇)	C₃	1	3	yes	maximal, complemented
18	C₁₉×#6913	C₃	1	1	yes	maximal, characteristic, complemented
19	C₃³×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
20	#13095333	—	3	6	no	—
21	#13095333	—	3	6	no	—
22	#13095333	C₃²	1	6	yes	complemented
23	#13095333	C₃²	1	6	yes	complemented
24	#13095332	—	3	6	no	—
25	#13095332	C₃²	1	6	yes	complemented
26	#13095331	C₃²	1	3	yes	complemented
27	#13095331	C₃²	1	3	yes	complemented
28	C₃×#10587025	C₃²	1	1	yes	characteristic, complemented
29	C₃×#10587025	C₃²	1	1	yes	characteristic, complemented
30	C₃×#10587022	C₃²	1	1	yes	characteristic, complemented
31	C₃×#10587022	C₃²	1	1	yes	characteristic, complemented
32	C₃×#10587022	C₃²	1	1	yes	characteristic, complemented
33	C₃×#10587023	C₃²	1	1	yes	characteristic, complemented
34	C₃×#10587023	C₃²	1	1	yes	characteristic, complemented
35	C₃×#10587023	C₃²	1	1	yes	characteristic, complemented
36	3₋¹⁺²×C₁₉⋊C₃	—	3	3	no	—
37	3₋¹⁺²×C₁₉⋊C₃	—	3	6	no	—
38	3₋¹⁺²×C₁₉⋊C₃	—	3	6	no	—
39	3₋¹⁺²×C₁₉⋊C₃	C₃²	1	6	yes	—
40	H(3,𝔽₃)×C₁₉⋊C₃	C₃²	1	3	yes	—
41	C₃×C₉×C₁₉⋊C₃	—	3	3	no	—
42	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
43	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
44	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
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	complemented
50	#13095319	C₃²	1	3	yes	complemented
51	#13095319	C₃²	1	3	yes	complemented
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, complemented
66	C₁₉×#731	C₃²	1	3	yes	complemented
67	C₁₉×AΣL(1,𝔽₂₇)	C₃²	1	3	yes	complemented
68	C₃×#6913	—	19	1	no	maximal, Sylow
69	C₃³×C₁₉	C₃³	1	1	yes	characteristic
70	C₃²×C₁₉⋊C₃	—	3	3	no	—
71	C₃²×C₁₉⋊C₃	—	3	2	no	—
72	C₃²×C₁₉⋊C₃	—	3	1	no	—
73	C₃²×C₁₉⋊C₃	—	9	3	no	—
74	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
75	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
76	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
77	C₁₉×3₋¹⁺²	—	3	3	no	—
78	C₁₉×3₋¹⁺²	—	3	6	no	—
79	C₁₉×3₋¹⁺²	C₃³	1	6	yes	—
80	C₁₉×H(3,𝔽₃)	C₃³	1	3	yes	—
81	C₃×C₉×C₁₉	—	3	3	no	—
82	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
83	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
84	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
85	C₃²⋊(C₁₉⋊C₃)	—	3	6	no	—
86	C₃²⋊(C₁₉⋊C₃)	C₃³	1	6	yes	—
87	#10587025	—	3	3	no	—
88	#10587025	—	3	3	no	—
89	#10587025	—	3	6	no	—
90	#10587025	—	3	6	no	—
91	#10587025	C₃³	1	6	yes	—
92	#10587025	C₃³	1	6	yes	—
93	C₃×#4294	—	3	3	no	—
94	C₃×#4294	—	3	3	no	—
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	C₃×#4294	C₃³	1	1	yes	characteristic
101	#10587023	—	3	6	no	—
102	#10587023	—	3	3	no	—
103	#10587023	—	3	6	no	—
104	#10587023	—	3	3	no	—
105	#10587023	—	3	3	no	—
106	#10587023	—	3	6	no	—
107	#10587023	—	3	6	no	—
108	#10587022	—	3	6	no	—
109	#10587022	—	3	3	no	—
110	#10587022	—	3	6	no	—
111	#10587022	—	3	3	no	—
112	#10587022	—	3	3	no	—
113	#10587022	—	3	6	no	—
114	#10587022	—	3	6	no	—
115	C₉×C₁₉⋊C₃	—	3	6	no	—
116	C₉×C₁₉⋊C₃	—	3	2	no	—
117	C₉×C₁₉⋊C₃	—	3	2	no	—
118	C₉×C₁₉⋊C₃	—	3	1	no	—
119	C₉×C₁₉⋊C₃	—	3	1	no	—
120	C₉×C₁₉⋊C₃	—	3	6	no	—
121	C₉×C₁₉⋊C₃	—	3	6	no	—
122	C₉×C₁₉⋊C₃	—	3	6	no	—
123	C₉×C₁₉⋊C₃	H(3,𝔽₃)	1	3	yes	—
124	C₉×C₁₉⋊C₃	H(3,𝔽₃)	1	6	yes	—
125	C₃×#737	—	19	3	no	—
126	C₃²×3₋¹⁺²	—	19	1	no	—
127	#6913	—	19	3	no	—
128	#6913	—	19	3	no	—
129	#6913	—	19	2	no	—
130	#6913	—	19	2	no	—
131	#6913	—	19	2	no	—
132	#6913	—	19	1	no	—
133	#6913	—	19	1	no	—
134	#6913	—	19	3	no	—
135	#6913	—	19	3	no	—
136	#6913	—	19	3	no	—
137	#6913	—	19	3	no	—
138	#6913	C₁₉⋊C₃	1	1	yes	characteristic, direct product factor
139	C₃×#733	—	19	2	no	—
140	C₃×#732	—	19	1	no	—
141	C₃×#731	—	19	3	no	—
142	C₃×AΣL(1,𝔽₂₇)	—	19	3	no	—
143	C₃²×C₁₉	—	3	3	no	—
144	C₃²×C₁₉	—	3	1	no	—
145	C₃²×C₁₉	—	9	1	no	—
146	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
147	C₃×C₁₉⋊C₃	—	3	3	no	—
148	C₃×C₁₉⋊C₃	—	3	6	no	—
149	C₃×C₁₉⋊C₃	—	3	1	no	—
150	C₃×C₁₉⋊C₃	—	3	1	no	—
151	C₃×C₁₉⋊C₃	—	9	6	no	—
152	C₃×C₁₉⋊C₃	—	9	9	no	—
153	C₃×C₁₉⋊C₃	—	9	3	no	—
154	C₃×C₁₉⋊C₃	—	9	1	no	—
155	C₃×C₁₉⋊C₃	—	9	1	no	—
156	C₃×C₁₉⋊C₃	C₃×H(3,𝔽₃)	1	2	yes	—
157	C₃×C₁₉⋊C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
158	C₉×C₁₉	—	3	6	no	—
159	C₉×C₁₉	—	3	1	no	—
160	C₉×C₁₉	—	3	1	no	—
161	C₉×C₁₉	C₃×H(3,𝔽₃)	1	3	yes	—
162	#4294	—	3	6	no	—
163	#4294	—	3	6	no	—
164	#4294	—	3	1	no	—
165	#4294	—	3	1	no	—
166	#4294	—	3	1	no	—
167	#4294	—	3	1	no	—
168	#4294	C₃×H(3,𝔽₃)	1	3	yes	—
169	#4294	C₃×H(3,𝔽₃)	1	3	yes	—
170	C₃⁴	—	19	1	no	—
171	#737	—	19	6	no	—
172	#737	—	19	6	no	—
173	#737	—	19	3	no	—
174	#737	—	19	3	no	—
175	#737	—	19	6	no	—
176	#737	—	57	6	no	—
177	#737	—	57	6	no	—
178	#737	—	57	6	no	—
179	#737	C₃×C₁₉⋊C₃	1	3	yes	complemented
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	1	no	—
186	C₃×3₋¹⁺²	—	19	1	no	—
187	C₃×3₋¹⁺²	—	19	1	no	—
188	C₃×3₋¹⁺²	—	19	6	no	—
189	C₃×3₋¹⁺²	—	57	3	no	—
190	C₃×3₋¹⁺²	—	57	6	no	—
191	C₃×3₋¹⁺²	—	57	6	no	—
192	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented
193	C₃×H(3,𝔽₃)	—	19	3	no	—
194	C₃²×C₉	—	19	1	no	—
195	C₃²×C₉	—	19	1	no	—
196	C₃²×C₉	—	19	1	no	—
197	C₃²×C₉	—	57	3	no	—
198	#733	—	19	6	no	—
199	#733	—	19	6	no	—
200	#733	—	19	2	no	—
201	#733	—	19	2	no	—
202	#733	C₃×C₁₉⋊C₃	1	2	yes	complemented
203	#732	—	19	3	no	—
204	#732	—	19	3	no	—
205	#732	—	19	2	no	—
206	#732	C₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented
207	#731	—	19	6	no	—
208	#731	—	19	6	no	—
209	#731	—	19	3	no	—
210	#731	—	19	3	no	—
211	#731	—	19	6	no	—
212	#731	C₃×C₁₉⋊C₃	1	3	yes	complemented
213	AΣL(1,𝔽₂₇)	—	19	6	no	—
214	AΣL(1,𝔽₂₇)	—	19	6	no	—
215	AΣL(1,𝔽₂₇)	—	19	6	no	—
216	AΣL(1,𝔽₂₇)	—	19	3	no	—
217	AΣL(1,𝔽₂₇)	—	19	3	no	—
218	AΣL(1,𝔽₂₇)	C₃×C₁₉⋊C₃	1	3	yes	complemented
219	C₃×C₁₉	—	3	1	no	—
220	C₃×C₁₉	—	9	1	no	—
221	C₃×C₁₉	—	9	3	no	—
222	C₃×C₁₉	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
223	C₁₉⋊C₃	—	3	2	no	—
224	C₁₉⋊C₃	—	9	6	no	—
225	C₁₉⋊C₃	—	9	1	no	—
226	C₁₉⋊C₃	—	9	1	no	—
227	C₁₉⋊C₃	#6913	1	3	yes	direct product factor
228	C₃³	—	19	1	no	—
229	C₃³	—	19	1	no	—
230	C₃³	—	19	1	no	—
231	C₃³	—	57	3	no	—
232	C₃³	—	57	1	no	—
233	C₃³	—	57	2	no	—
234	C₃³	—	171	3	no	—
235	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
236	3₋¹⁺²	—	3	3	no	—
237	3₋¹⁺²	—	3	6	no	—
238	3₋¹⁺²	—	19	6	no	—
239	3₋¹⁺²	—	19	6	no	—
240	3₋¹⁺²	—	57	3	no	—
241	3₋¹⁺²	—	57	3	no	—
242	3₋¹⁺²	—	57	6	no	—
243	3₋¹⁺²	—	57	6	no	—
244	3₋¹⁺²	—	57	3	no	—
245	3₋¹⁺²	—	57	3	no	—
246	3₋¹⁺²	—	57	6	no	—
247	3₋¹⁺²	—	57	6	no	—
248	3₋¹⁺²	—	57	6	no	—
249	3₋¹⁺²	—	57	6	no	—
250	3₋¹⁺²	—	57	3	no	—
251	3₋¹⁺²	—	57	3	no	—
252	3₋¹⁺²	—	57	6	no	—
253	3₋¹⁺²	—	57	6	no	—
254	3₋¹⁺²	—	57	3	no	—
255	3₋¹⁺²	—	57	6	no	—
256	3₋¹⁺²	—	57	3	no	—
257	3₋¹⁺²	—	57	6	no	—
258	3₋¹⁺²	C₃²×C₁₉⋊C₃	1	6	yes	—
259	H(3,𝔽₃)	—	19	6	no	—
260	H(3,𝔽₃)	—	57	6	no	—
261	H(3,𝔽₃)	C₃²×C₁₉⋊C₃	1	3	yes	—
262	C₃×C₉	—	3	3	no	—
263	C₃×C₉	—	19	1	no	—
264	C₃×C₉	—	19	1	no	—
265	C₃×C₉	—	19	3	no	—
266	C₃×C₉	—	19	6	no	—
267	C₃×C₉	—	19	1	no	—
268	C₃×C₉	—	19	1	no	—
269	C₃×C₉	—	19	1	no	—
270	C₃×C₉	—	19	1	no	—
271	C₃×C₉	—	57	6	no	—
272	C₃×C₉	—	57	6	no	—
273	C₃×C₉	—	57	1	no	—
274	C₃×C₉	—	57	1	no	—
275	C₃×C₉	—	57	2	no	—
276	C₃×C₉	—	57	2	no	—
277	C₃×C₉	—	57	6	no	—
278	C₃×C₉	—	57	3	no	—
279	C₃×C₉	—	57	3	no	—
280	C₃×C₉	—	57	6	no	—
281	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
282	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
283	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
284	C₁₉	C₃×#6913	1	1	yes	characteristic, complemented, Sylow
285	C₃²	—	3	3	no	—
286	C₃²	—	3	1	no	—
287	C₃²	—	9	1	no	—
288	C₃²	—	19	1	no	—
289	C₃²	—	19	2	no	—
290	C₃²	—	57	3	no	—
291	C₃²	—	57	6	no	—
292	C₃²	—	57	1	no	—
293	C₃²	—	57	1	no	—
294	C₃²	—	171	1	no	—
295	C₃²	—	171	1	no	—
296	C₃²	—	171	3	no	—
297	C₃²	—	171	9	no	—
298	C₃²	—	171	6	no	—
299	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
300	C₉	—	3	6	no	—
301	C₉	—	3	1	no	—
302	C₉	—	3	1	no	—
303	C₉	—	19	3	no	—
304	C₉	—	19	3	no	—
305	C₉	—	57	1	no	—
306	C₉	—	57	1	no	—
307	C₉	—	57	1	no	—
308	C₉	—	57	1	no	—
309	C₉	—	57	6	no	—
310	C₉	—	57	6	no	—
311	C₉	H(3,𝔽₃)×C₁₉⋊C₃	1	3	yes	—
312	C₃	—	3	1	no	—
313	C₃	—	9	1	no	—
314	C₃	—	9	3	no	—
315	C₃	—	19	3	no	—
316	C₃	—	57	2	no	—
317	C₃	—	171	1	no	—
318	C₃	—	171	1	no	—
319	C₃	—	171	6	no	—
320	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	C₃×#6913
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. C₁₉⋊C₃×#737	C₃	semidirect
2. C₃×3₋¹⁺²×C₁₉⋊C₃	C₃	semidirect
3. #17987530	C₃	semidirect
4. #17987529	C₃	semidirect
5. #17987528	C₃	semidirect
6. #17987527	C₃	semidirect
7. #17987526	C₃	semidirect
8. #17987525	C₃	semidirect
9. #17987524	C₃	semidirect
10. #17987523	C₃	semidirect
11. #17987522	C₃	semidirect
12. #17987521	C₃	semidirect
13. #17987520	C₃	semidirect
14. C₁₉⋊C₃×#733	C₃	semidirect
15. C₁₉⋊C₃×#732	C₃	semidirect
16. C₁₉⋊C₃×#731	C₃	semidirect
17. C₁₉⋊C₃×AΣL(1,𝔽₂₇)	C₃	semidirect
18. C₁₉×#6913	C₃	semidirect
22. #13095333	C₃²	semidirect
23. #13095333	C₃²	semidirect
25. #13095332	C₃²	semidirect
26. #13095331	C₃²	semidirect
27. #13095331	C₃²	semidirect
28. C₃×#10587025	C₃²	semidirect
29. C₃×#10587025	C₃²	semidirect
30. C₃×#10587022	C₃²	semidirect
31. C₃×#10587022	C₃²	semidirect
32. C₃×#10587022	C₃²	semidirect
33. C₃×#10587023	C₃²	semidirect
34. C₃×#10587023	C₃²	semidirect
35. C₃×#10587023	C₃²	semidirect
45. #13095322	C₃²	semidirect
46. #13095322	C₃²	semidirect
47. #13095321	C₃²	semidirect
48. #13095321	C₃²	semidirect
49. #13095320	C₃²	semidirect
50. #13095319	C₃²	semidirect
51. #13095319	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
65. C₁₉×#732	C₃²	semidirect
66. C₁₉×#731	C₃²	semidirect
67. C₁₉×AΣL(1,𝔽₂₇)	C₃²	semidirect
138. #6913	C₁₉⋊C₃	direct (N × H)
179. #737	C₃×C₁₉⋊C₃	semidirect
192. C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	semidirect
202. #733	C₃×C₁₉⋊C₃	semidirect
206. #732	C₃×C₁₉⋊C₃	semidirect
212. #731	C₃×C₁₉⋊C₃	semidirect
218. AΣL(1,𝔽₂₇)	C₃×C₁₉⋊C₃	semidirect
227. C₁₉⋊C₃	#6913	direct (N × H)
284. C₁₉	C₃×#6913	semidirect

Maximal subgroups

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

subgroup	index
C₁₉⋊C₃×#737	3
C₃×3₋¹⁺²×C₁₉⋊C₃	3
#17987530	3
#17987529	3
#17987528	3
#17987527	3
#17987526	3
#17987525	3
#17987524	3
#17987523	3
#17987522	3
#17987521	3
#17987520	3
C₁₉⋊C₃×#733	3
C₁₉⋊C₃×#732	3
C₁₉⋊C₃×#731	3
C₁₉⋊C₃×AΣL(1,𝔽₂₇)	3
C₁₉×#6913	3
C₃×#6913	19

Subgroups — C19⋊C3×#6913 #28356577

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉⋊C₃×#6913 #28356577