Subgroups — C₁₉⋊C₃×#6894 #28355965

Order 13851.

932 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×3₋¹⁺²×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₉×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic
3	#17987441	C₃	1	2	yes	maximal, complemented
4	#17987440	C₃	1	2	yes	maximal, complemented
5	#17987439	C₃	1	1	yes	maximal, characteristic, complemented
6	#17987438	C₃	1	1	yes	maximal, characteristic, complemented
7	#17987437	C₃	1	6	yes	maximal, complemented
8	#17987436	C₃	1	6	yes	maximal, complemented
9	#17987435	C₃	1	2	yes	maximal, complemented
10	#17987434	C₃	1	3	yes	maximal, complemented
11	#17987433	C₃	1	3	yes	maximal, complemented
12	C₁₉⋊C₃×C₉⋊C₉	C₃	1	3	yes	maximal, complemented
13	C₁₉⋊C₃×C₉⋊C₉	C₃	1	6	yes	maximal, complemented
14	C₁₉⋊C₃×C₃²⋊C₉	C₃	1	2	yes	maximal
15	C₁₉×#6894	C₃	1	1	yes	maximal, characteristic, complemented
16	C₃³×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
17	C₃×#10587025	C₃²	1	1	yes	characteristic
18	C₃×#10587025	C₃²	1	1	yes	characteristic
19	C₃×#10587022	C₃²	1	3	yes	—
20	C₃×#10587023	C₃²	1	3	yes	—
21	3₋¹⁺²×C₁₉⋊C₃	—	3	9	no	—
22	C₃²×#4294	C₃²	1	2	yes	—
23	C₃×C₉×C₁₉⋊C₃	—	3	9	no	—
24	C₃×C₉×C₁₉⋊C₃	C₃²	1	6	yes	—
25	C₃×C₉×C₁₉⋊C₃	C₃²	1	2	yes	—
26	C₃×C₉×C₁₉⋊C₃	C₃²	1	3	yes	—
27	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
28	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
29	C₃×C₉×C₁₉⋊C₃	C₃²	1	2	yes	—
30	C₃×C₉×C₁₉⋊C₃	C₃²	1	2	yes	—
31	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
32	#13095307	C₃²	1	3	yes	complemented
33	#13095307	C₃²	1	6	yes	complemented
34	#13095306	C₃²	1	3	yes	complemented
35	#13095306	C₃²	1	6	yes	complemented
36	#13095305	C₃²	1	6	yes	complemented
37	#13095305	C₃²	1	6	yes	complemented
38	#13095305	C₃²	1	6	yes	complemented
39	#13095305	C₃²	1	6	yes	complemented
40	#13095305	C₃²	1	6	yes	complemented
41	#13095305	C₃²	1	6	yes	complemented
42	#13095305	C₃²	1	6	yes	complemented
43	#13095305	C₃²	1	6	yes	complemented
44	#13095305	C₃²	1	6	yes	complemented
45	C₃²⋊#4294	C₃²	1	2	yes	—
46	C₃²⋊#4294	C₃²	1	2	yes	—
47	#13095303	C₃²	1	6	yes	—
48	#13095303	C₃²	1	6	yes	—
49	C₃×C₁₉×3₋¹⁺²	C₃²	1	1	yes	characteristic
50	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
51	C₁₉×C₉⋊C₉	C₃²	1	3	yes	complemented
52	C₁₉×C₉⋊C₉	C₃²	1	6	yes	complemented
53	C₁₉×C₃²⋊C₉	C₃²	1	2	yes	—
54	C₃×#6894	—	19	1	no	maximal, Sylow
55	C₃³×C₁₉	C₃³	1	1	yes	characteristic
56	C₃²×C₁₉⋊C₃	—	3	3	no	—
57	C₃²×C₁₉⋊C₃	—	3	6	no	—
58	C₃²×C₁₉⋊C₃	3₋¹⁺²	1	9	yes	—
59	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
60	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
61	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
62	C₁₉×3₋¹⁺²	—	3	3	no	—
63	C₃×C₉×C₁₉	—	3	3	no	—
64	C₃×C₉×C₁₉	C₃³	1	6	yes	—
65	C₃×C₉×C₁₉	C₃³	1	2	yes	—
66	C₃×C₉×C₁₉	C₃³	1	3	yes	—
67	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
68	#10587025	—	3	3	no	—
69	#10587025	—	3	3	no	—
70	C₃×#4294	—	3	6	no	—
71	C₃×#4294	C₃³	1	6	yes	—
72	C₃×#4294	C₃³	1	6	yes	—
73	C₃×#4294	C₃³	1	2	yes	—
74	C₃×#4294	C₃³	1	3	yes	—
75	C₃×#4294	C₃³	1	2	yes	—
76	C₃×#4294	C₃³	1	3	yes	—
77	C₃×#4294	C₃³	1	2	yes	—
78	#10587023	—	3	9	no	—
79	#10587022	—	3	9	no	—
80	C₉×C₁₉⋊C₃	—	3	6	no	—
81	C₉×C₁₉⋊C₃	—	3	9	no	—
82	C₉×C₁₉⋊C₃	—	3	18	no	—
83	C₉×C₁₉⋊C₃	—	3	3	no	—
84	C₉×C₁₉⋊C₃	—	3	3	no	—
85	C₉×C₁₉⋊C₃	—	3	6	no	—
86	C₉×C₁₉⋊C₃	—	3	6	no	—
87	C₉×C₁₉⋊C₃	3₋¹⁺²	1	9	yes	complemented
88	C₃²×3₋¹⁺²	—	19	1	no	—
89	C₃³×C₉	—	19	1	no	—
90	#6894	—	19	2	no	—
91	#6894	—	19	1	no	—
92	#6894	—	19	1	no	—
93	#6894	—	19	3	no	—
94	#6894	—	19	3	no	—
95	#6894	—	19	6	no	—
96	#6894	—	19	6	no	—
97	#6894	—	19	2	no	—
98	#6894	—	19	2	no	—
99	#6894	C₁₉⋊C₃	1	1	yes	characteristic, direct product factor
100	C₃×C₉⋊C₉	—	19	3	no	—
101	C₃×C₉⋊C₉	—	19	6	no	—
102	C₃×C₃²⋊C₉	—	19	2	no	—
103	C₃²×C₁₉	—	3	1	no	—
104	C₃²×C₁₉	—	3	2	no	—
105	C₃²×C₁₉	C₃×3₋¹⁺²	1	3	yes	—
106	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
107	C₃×C₁₉⋊C₃	—	3	1	no	—
108	C₃×C₁₉⋊C₃	—	3	1	no	—
109	C₃×C₁₉⋊C₃	—	3	2	no	—
110	C₃×C₁₉⋊C₃	—	3	2	no	—
111	C₃×C₁₉⋊C₃	—	3	27	no	—
112	C₃×C₁₉⋊C₃	C₃×3₋¹⁺²	1	3	yes	—
113	C₃×C₁₉⋊C₃	C₃×3₋¹⁺²	1	3	yes	—
114	C₃×C₁₉⋊C₃	C₃×3₋¹⁺²	1	3	yes	—
115	C₃×C₁₉⋊C₃	C₃×3₋¹⁺²	1	3	yes	—
116	C₃×C₁₉⋊C₃	#737	1	6	yes	—
117	C₉×C₁₉	—	3	2	no	—
118	C₉×C₁₉	—	3	3	no	—
119	C₉×C₁₉	—	3	6	no	—
120	C₉×C₁₉	C₃×3₋¹⁺²	1	3	yes	complemented
121	#4294	—	3	2	no	—
122	#4294	—	3	2	no	—
123	#4294	—	3	3	no	—
124	#4294	—	3	3	no	—
125	#4294	—	3	6	no	—
126	#4294	—	3	6	no	—
127	#4294	C₃×3₋¹⁺²	1	6	yes	complemented
128	C₃⁴	—	19	1	no	—
129	C₃×3₋¹⁺²	—	19	1	no	—
130	C₃×3₋¹⁺²	—	19	1	no	—
131	C₃×3₋¹⁺²	—	19	3	no	—
132	C₃×3₋¹⁺²	—	19	3	no	—
133	C₃×3₋¹⁺²	—	57	9	no	—
134	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
135	C₃²×C₉	—	19	1	no	—
136	C₃²×C₉	—	19	2	no	—
137	C₃²×C₉	—	19	2	no	—
138	C₃²×C₉	—	19	2	no	—
139	C₃²×C₉	—	19	2	no	—
140	C₃²×C₉	—	19	1	no	—
141	C₃²×C₉	—	19	1	no	—
142	C₃²×C₉	—	19	3	no	—
143	C₃²×C₉	—	19	6	no	—
144	C₃²×C₉	—	57	9	no	—
145	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
146	C₉⋊C₉	—	19	6	no	—
147	C₉⋊C₉	—	19	6	no	—
148	C₉⋊C₉	—	19	6	no	—
149	C₉⋊C₉	—	19	6	no	—
150	C₉⋊C₉	—	19	6	no	—
151	C₉⋊C₉	—	19	6	no	—
152	C₉⋊C₉	—	19	6	no	—
153	C₉⋊C₉	—	19	6	no	—
154	C₉⋊C₉	—	19	6	no	—
155	C₉⋊C₉	—	19	3	no	—
156	C₉⋊C₉	—	19	6	no	—
157	C₉⋊C₉	—	19	3	no	—
158	C₉⋊C₉	—	19	6	no	—
159	C₉⋊C₉	C₃×C₁₉⋊C₃	1	6	yes	complemented
160	C₉⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	complemented
161	C₃²⋊C₉	—	19	2	no	—
162	C₃²⋊C₉	—	19	2	no	—
163	C₃²⋊C₉	—	19	6	no	—
164	C₃²⋊C₉	—	19	6	no	—
165	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	2	yes	—
166	C₃×C₁₉	—	3	3	no	—
167	C₃×C₁₉	C₃²×3₋¹⁺²	1	1	yes	characteristic
168	C₃×C₁₉	C₃²×3₋¹⁺²	1	1	yes	characteristic
169	C₃×C₁₉	C₃×#737	1	2	yes	—
170	C₁₉⋊C₃	—	3	3	no	—
171	C₁₉⋊C₃	—	3	3	no	—
172	C₁₉⋊C₃	#6894	1	9	yes	direct product factor
173	C₃³	—	19	1	no	—
174	C₃³	—	19	9	no	—
175	C₃³	—	19	1	no	—
176	C₃³	—	19	1	no	—
177	C₃³	—	57	3	no	—
178	C₃³	—	57	6	no	—
179	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
180	3₋¹⁺²	—	3	3	no	—
181	3₋¹⁺²	—	57	9	no	—
182	3₋¹⁺²	—	57	9	no	—
183	3₋¹⁺²	—	57	3	no	—
184	3₋¹⁺²	—	57	3	no	—
185	C₃×C₉	—	3	3	no	—
186	C₃×C₉	—	19	2	no	—
187	C₃×C₉	—	19	9	no	—
188	C₃×C₉	—	19	6	no	—
189	C₃×C₉	—	19	6	no	—
190	C₃×C₉	—	19	2	no	—
191	C₃×C₉	—	19	2	no	—
192	C₃×C₉	—	19	3	no	—
193	C₃×C₉	—	19	3	no	—
194	C₃×C₉	—	57	6	no	—
195	C₃×C₉	—	57	6	no	—
196	C₃×C₉	—	57	18	no	—
197	C₃×C₉	—	57	6	no	—
198	C₃×C₉	—	57	9	no	—
199	C₃×C₉	—	57	6	no	—
200	C₃×C₉	—	57	3	no	—
201	C₃×C₉	—	57	3	no	—
202	C₃×C₉	C₃²×C₁₉⋊C₃	1	6	yes	—
203	C₃×C₉	C₃²×C₁₉⋊C₃	1	2	yes	—
204	C₃×C₉	C₃²×C₁₉⋊C₃	1	3	yes	—
205	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
206	C₁₉	C₃×#6894	1	1	yes	characteristic, complemented, Sylow
207	C₃²	—	3	1	no	—
208	C₃²	—	3	2	no	—
209	C₃²	—	19	3	no	—
210	C₃²	—	19	3	no	—
211	C₃²	—	19	6	no	—
212	C₃²	—	19	3	no	—
213	C₃²	—	19	3	no	—
214	C₃²	—	57	27	no	—
215	C₃²	—	57	1	no	—
216	C₃²	—	57	1	no	—
217	C₃²	—	57	2	no	—
218	C₃²	—	57	2	no	—
219	C₃²	3₋¹⁺²×C₁₉⋊C₃	1	3	yes	—
220	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
221	C₉	—	3	2	no	—
222	C₉	—	3	3	no	—
223	C₉	—	3	6	no	—
224	C₉	—	19	6	no	—
225	C₉	—	57	2	no	—
226	C₉	—	57	2	no	—
227	C₉	—	57	3	no	—
228	C₉	—	57	3	no	—
229	C₉	—	57	6	no	—
230	C₉	—	57	6	no	—
231	C₉	3₋¹⁺²×C₁₉⋊C₃	1	3	yes	complemented
232	C₃	—	3	3	no	—
233	C₃	—	19	9	no	—
234	C₃	—	57	3	no	—
235	C₃	—	57	3	no	—
236	C₃	C₃×3₋¹⁺²×C₁₉⋊C₃	1	1	yes	characteristic
237	C₃	C₃×3₋¹⁺²×C₁₉⋊C₃	1	1	yes	characteristic
238	C₃	C₁₉⋊C₃×#737	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₁₉⋊C₃
commutator, socle, cosocle	C₃²×C₁₉	C₃⁴
Fitting	C₁₉×#6894	C₃
prime radical	C₃³×C₁₉⋊C₃	C₃²
hypercenter	#6894	C₁₉⋊C₃
hypocenter	C₁₉	C₃×#6894
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
3. #17987441	C₃	semidirect
4. #17987440	C₃	semidirect
5. #17987439	C₃	semidirect
6. #17987438	C₃	semidirect
7. #17987437	C₃	semidirect
8. #17987436	C₃	semidirect
9. #17987435	C₃	semidirect
10. #17987434	C₃	semidirect
11. #17987433	C₃	semidirect
12. C₁₉⋊C₃×C₉⋊C₉	C₃	semidirect
13. C₁₉⋊C₃×C₉⋊C₉	C₃	semidirect
15. C₁₉×#6894	C₃	semidirect
32. #13095307	C₃²	semidirect
33. #13095307	C₃²	semidirect
34. #13095306	C₃²	semidirect
35. #13095306	C₃²	semidirect
36. #13095305	C₃²	semidirect
37. #13095305	C₃²	semidirect
38. #13095305	C₃²	semidirect
39. #13095305	C₃²	semidirect
40. #13095305	C₃²	semidirect
41. #13095305	C₃²	semidirect
42. #13095305	C₃²	semidirect
43. #13095305	C₃²	semidirect
44. #13095305	C₃²	semidirect
51. C₁₉×C₉⋊C₉	C₃²	semidirect
52. C₁₉×C₉⋊C₉	C₃²	semidirect
87. C₉×C₁₉⋊C₃	3₋¹⁺²	semidirect
99. #6894	C₁₉⋊C₃	direct (N × H)
120. C₉×C₁₉	C₃×3₋¹⁺²	semidirect
127. #4294	C₃×3₋¹⁺²	semidirect
159. C₉⋊C₉	C₃×C₁₉⋊C₃	semidirect
160. C₉⋊C₉	C₃×C₁₉⋊C₃	semidirect
172. C₁₉⋊C₃	#6894	direct (N × H)
206. C₁₉	C₃×#6894	semidirect
231. C₉	3₋¹⁺²×C₁₉⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×3₋¹⁺²×C₁₉⋊C₃	3
C₃²×C₉×C₁₉⋊C₃	3
#17987441	3
#17987440	3
#17987439	3
#17987438	3
#17987437	3
#17987436	3
#17987435	3
#17987434	3
#17987433	3
C₁₉⋊C₃×C₉⋊C₉	3
C₁₉⋊C₃×C₃²⋊C₉	3
C₁₉×#6894	3
C₃×#6894	19

Subgroups — C19⋊C3×#6894 #28355965

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉⋊C₃×#6894 #28355965