Subgroups — #28355925

Order 13851.

944 conjugacy classes of subgroups.

The subgroup lattice has 242 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	#17987415	C₃	1	2	yes	maximal, complemented
3	#17987415	C₃	1	2	yes	maximal, complemented
4	3₋¹⁺²×#4294	C₃	1	2	yes	maximal, complemented
5	C₉×#10587023	C₃	1	3	yes	maximal, complemented
6	C₉×#10587022	C₃	1	3	yes	maximal, complemented
7	C₉×#10587025	C₃	1	2	yes	maximal, complemented
8	#17987410	C₃	1	6	yes	maximal, complemented
9	#17987409	C₃	1	6	yes	maximal, complemented
10	#17987404	C₃	1	6	yes	maximal, complemented
11	#17987403	C₃	1	3	yes	maximal, complemented
12	C₃×#13095303	C₃	1	1	yes	maximal, characteristic
13	C₁₉⋊C₃×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
14	C₁₉⋊C₃×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
15	C₉×C₁₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
16	C₃³×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
17	C₃×#10587025	C₃²	1	2	yes	—
18	C₃×#10587022	C₃²	1	3	yes	—
19	C₃×#10587023	C₃²	1	3	yes	—
20	3₋¹⁺²×C₁₉⋊C₃	C₉	1	27	yes	complemented
21	C₃²×#4294	C₃²	1	2	yes	—
22	C₃×C₉×C₁₉⋊C₃	C₃²	1	3	yes	—
23	C₃×C₉×C₁₉⋊C₃	C₃²	1	3	yes	—
24	C₃×C₉×C₁₉⋊C₃	C₃²	1	3	yes	—
25	#13095307	C₃²	1	6	yes	complemented
26	#13095306	C₃²	1	6	yes	complemented
27	#13095305	C₃²	1	6	yes	complemented
28	#13095305	C₃²	1	6	yes	complemented
29	#13095305	C₃²	1	6	yes	complemented
30	#13095305	C₃²	1	6	yes	complemented
31	#13095305	C₃²	1	6	yes	complemented
32	#13095305	C₃²	1	6	yes	complemented
33	C₃²⋊#4294	C₃²	1	2	yes	—
34	C₃²⋊#4294	C₃²	1	2	yes	—
35	#13095303	C₃²	1	3	yes	—
36	#13095303	C₃²	1	3	yes	—
37	#13095303	C₃²	1	3	yes	—
38	#13095303	C₃²	1	3	yes	—
39	#13095303	C₃²	1	1	yes	characteristic
40	#13095303	C₃²	1	2	yes	—
41	#13095303	C₃²	1	2	yes	—
42	#13095303	C₃²	1	2	yes	—
43	#13095303	C₃²	1	1	yes	characteristic
44	#13095303	C₃²	1	1	yes	characteristic
45	C₉×#4294	C₃²	1	6	yes	complemented
46	C₉×#4294	C₃²	1	6	yes	complemented
47	C₉×#4294	C₃²	1	6	yes	complemented
48	C₉×#4294	C₃²	1	6	yes	complemented
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	6	yes	complemented
52	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
53	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
54	C₉²×C₁₉	C₃²	1	3	yes	complemented
55	#10624863	—	19	1	no	maximal, Sylow
56	C₃³×C₁₉	C₃³	1	1	yes	characteristic
57	C₃²×C₁₉⋊C₃	—	9	1	no	—
58	C₃²×C₁₉⋊C₃	—	9	1	no	—
59	C₃²×C₁₉⋊C₃	—	9	1	no	—
60	C₃²×C₁₉⋊C₃	C₃×C₉	1	9	yes	—
61	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
62	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
63	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
64	C₁₉×3₋¹⁺²	C₃×C₉	1	9	yes	complemented
65	C₃×C₉×C₁₉	—	9	1	no	—
66	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
67	C₃×C₉×C₁₉	C₃³	1	3	yes	—
68	C₃×C₉×C₁₉	C₃³	1	3	yes	—
69	C₃×C₉×C₁₉	C₃³	1	2	yes	—
70	C₃×C₉×C₁₉	C₃³	1	3	yes	—
71	#10587025	C₃×C₉	1	18	yes	complemented
72	C₃×#4294	—	9	2	no	—
73	C₃×#4294	C₃³	1	2	yes	—
74	C₃×#4294	C₃³	1	2	yes	—
75	C₃×#4294	C₃³	1	2	yes	—
76	C₃×#4294	C₃³	1	6	yes	—
77	C₃×#4294	C₃³	1	6	yes	—
78	C₃×#4294	C₃³	1	6	yes	—
79	#10587023	C₃×C₉	1	27	yes	complemented
80	#10587022	C₃×C₉	1	27	yes	complemented
81	C₉×C₁₉⋊C₃	—	9	3	no	—
82	C₉×C₁₉⋊C₃	—	9	3	no	—
83	C₉×C₁₉⋊C₃	C₃×C₉	1	27	yes	complemented
84	C₃²×3₋¹⁺²	—	19	1	no	—
85	C₉×3₋¹⁺²	—	19	2	no	—
86	C₉×3₋¹⁺²	—	19	2	no	—
87	C₉×3₋¹⁺²	—	19	6	no	—
88	C₉×3₋¹⁺²	—	19	6	no	—
89	C₉×3₋¹⁺²	—	19	2	no	—
90	C₉×3₋¹⁺²	—	19	2	no	—
91	C₉×3₋¹⁺²	—	19	3	no	—
92	C₉×3₋¹⁺²	—	19	3	no	—
93	C₉×3₋¹⁺²	C₁₉⋊C₃	1	1	yes	characteristic, complemented
94	#6887	—	19	6	no	—
95	#6887	—	19	3	no	—
96	C₃×C₃²⋊C₉	—	19	1	no	—
97	C₃×C₃²⋊C₉	—	19	1	no	—
98	C₃×C₃²⋊C₉	—	19	1	no	—
99	C₃²×C₁₉	—	3	1	no	—
100	C₃²×C₁₉	—	3	1	no	—
101	C₃²×C₁₉	—	3	1	no	—
102	C₃²×C₁₉	C₃²×C₉	1	3	yes	—
103	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
104	C₃×C₁₉⋊C₃	—	3	1	no	—
105	C₃×C₁₉⋊C₃	—	3	1	no	—
106	C₃×C₁₉⋊C₃	—	3	1	no	—
107	C₃×C₁₉⋊C₃	—	3	1	no	—
108	C₃×C₁₉⋊C₃	—	3	1	no	—
109	C₃×C₁₉⋊C₃	—	3	1	no	—
110	C₃×C₁₉⋊C₃	—	3	1	no	—
111	C₃×C₁₉⋊C₃	—	3	1	no	—
112	C₃×C₁₉⋊C₃	—	3	1	no	—
113	C₃×C₁₉⋊C₃	—	9	9	no	—
114	C₃×C₁₉⋊C₃	C₃²×C₉	1	3	yes	—
115	C₃×C₁₉⋊C₃	C₃²×C₉	1	3	yes	—
116	C₃×C₁₉⋊C₃	C₃²×C₉	1	3	yes	—
117	C₉×C₁₉	—	3	3	no	—
118	C₉×C₁₉	—	3	3	no	—
119	C₉×C₁₉	—	3	1	no	—
120	C₉×C₁₉	—	3	2	no	—
121	C₉×C₁₉	C₃²×C₉	1	9	yes	—
122	#4294	—	3	6	no	—
123	#4294	—	3	6	no	—
124	#4294	—	3	2	no	—
125	#4294	—	3	2	no	—
126	#4294	—	3	2	no	—
127	#4294	C₃²×C₉	1	18	yes	—
128	C₃⁴	—	19	1	no	—
129	C₃×3₋¹⁺²	—	19	2	no	—
130	C₃×3₋¹⁺²	—	19	27	no	—
131	C₃×3₋¹⁺²	—	19	3	no	—
132	C₃×3₋¹⁺²	—	19	3	no	—
133	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
134	C₃²×C₉	—	19	3	no	—
135	C₃²×C₉	—	19	2	no	—
136	C₃²×C₉	—	19	3	no	—
137	C₃²×C₉	—	19	3	no	—
138	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
139	C₉⋊C₉	—	19	6	no	—
140	C₉⋊C₉	—	19	6	no	—
141	C₉⋊C₉	—	19	6	no	—
142	C₉⋊C₉	—	19	6	no	—
143	C₉⋊C₉	—	19	6	no	—
144	C₉⋊C₉	—	19	6	no	—
145	C₉⋊C₉	—	19	6	no	—
146	C₉⋊C₉	—	19	6	no	—
147	C₉⋊C₉	C₃×C₁₉⋊C₃	1	6	yes	complemented
148	C₃²⋊C₉	—	19	2	no	—
149	C₃²⋊C₉	—	19	2	no	—
150	C₃²⋊C₉	—	19	2	no	—
151	C₃²⋊C₉	—	19	3	no	—
152	C₃²⋊C₉	—	19	3	no	—
153	C₃²⋊C₉	—	19	1	no	—
154	C₃²⋊C₉	—	19	2	no	—
155	C₃²⋊C₉	—	19	2	no	—
156	C₃²⋊C₉	—	19	3	no	—
157	C₃²⋊C₉	—	19	3	no	—
158	C₃²⋊C₉	—	19	1	no	—
159	C₃²⋊C₉	—	19	1	no	—
160	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
161	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
162	C₉²	—	19	6	no	—
163	C₉²	—	19	6	no	—
164	C₉²	—	19	6	no	—
165	C₉²	—	19	6	no	—
166	C₉²	C₃×C₁₉⋊C₃	1	3	yes	complemented
167	C₃×C₁₉	—	3	3	no	—
168	C₃×C₁₉	C₃³×C₉	1	1	yes	characteristic
169	C₃×C₁₉	3₋¹⁺⁴	1	1	yes	characteristic
170	C₃×C₁₉	3₋¹⁺⁴	1	1	yes	characteristic
171	C₃×C₁₉	3₋¹⁺⁴	1	1	yes	characteristic
172	C₁₉⋊C₃	—	3	3	no	—
173	C₁₉⋊C₃	—	3	3	no	—
174	C₁₉⋊C₃	—	3	3	no	—
175	C₃³	—	19	1	no	—
176	C₃³	—	19	1	no	—
177	C₃³	—	19	1	no	—
178	C₃³	—	19	9	no	—
179	C₃³	—	171	1	no	—
180	C₃³	—	171	1	no	—
181	C₃³	—	171	1	no	—
182	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
183	3₋¹⁺²	—	19	18	no	—
184	3₋¹⁺²	—	19	27	no	—
185	3₋¹⁺²	—	19	27	no	—
186	3₋¹⁺²	C₉×C₁₉⋊C₃	1	9	yes	complemented
187	C₃×C₉	—	9	1	no	—
188	C₃×C₉	—	19	6	no	—
189	C₃×C₉	—	19	27	no	—
190	C₃×C₉	—	19	6	no	—
191	C₃×C₉	—	19	6	no	—
192	C₃×C₉	—	19	2	no	—
193	C₃×C₉	—	19	2	no	—
194	C₃×C₉	—	19	2	no	—
195	C₃×C₉	—	171	3	no	—
196	C₃×C₉	—	171	3	no	—
197	C₃×C₉	—	171	2	no	—
198	C₃×C₉	C₃²×C₁₉⋊C₃	1	2	yes	—
199	C₃×C₉	C₃²×C₁₉⋊C₃	1	3	yes	—
200	C₃×C₉	C₃²×C₁₉⋊C₃	1	3	yes	—
201	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
202	C₃×C₉	C₃²×C₁₉⋊C₃	1	3	yes	—
203	C₁₉	#10624863	1	1	yes	characteristic, complemented, Sylow
204	C₃²	—	3	1	no	—
205	C₃²	—	3	1	no	—
206	C₃²	—	3	1	no	—
207	C₃²	—	19	3	no	—
208	C₃²	—	19	3	no	—
209	C₃²	—	19	3	no	—
210	C₃²	—	57	1	no	—
211	C₃²	—	57	1	no	—
212	C₃²	—	57	1	no	—
213	C₃²	—	57	1	no	—
214	C₃²	—	57	1	no	—
215	C₃²	—	57	1	no	—
216	C₃²	—	57	1	no	—
217	C₃²	—	57	1	no	—
218	C₃²	—	57	1	no	—
219	C₃²	—	171	9	no	—
220	C₃²	C₃×C₉×C₁₉⋊C₃	1	3	yes	—
221	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
222	C₉	—	3	3	no	—
223	C₉	—	3	3	no	—
224	C₉	—	3	2	no	—
225	C₉	—	3	1	no	—
226	C₉	—	19	18	no	—
227	C₉	—	57	6	no	—
228	C₉	—	57	6	no	—
229	C₉	—	57	2	no	—
230	C₉	—	57	2	no	—
231	C₉	—	57	2	no	—
232	C₉	C₃×C₉×C₁₉⋊C₃	1	9	yes	—
233	C₃	—	3	3	no	—
234	C₃	—	57	3	no	—
235	C₃	—	57	3	no	—
236	C₃	—	57	3	no	—
237	C₃	C₃²×C₉×C₁₉⋊C₃	1	1	yes	characteristic
238	C₃	#17987544	1	1	yes	characteristic
239	C₃	#17987545	1	1	yes	characteristic
240	C₃	#17987546	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₁₉⋊C₃
commutator	C₃×C₁₉	C₃³×C₉
Fitting	C₉×C₁₉×3₋¹⁺²	C₃
socle, cosocle	C₃²×C₁₉	C₃⁴
prime radical	C₃³×C₁₉⋊C₃	C₃²
hypercenter	C₉×3₋¹⁺²	C₁₉⋊C₃
hypocenter	C₁₉	#10624863
LCS	G ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₉×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
2. #17987415	C₃	semidirect
3. #17987415	C₃	semidirect
4. 3₋¹⁺²×#4294	C₃	semidirect
5. C₉×#10587023	C₃	semidirect
6. C₉×#10587022	C₃	semidirect
7. C₉×#10587025	C₃	semidirect
8. #17987410	C₃	semidirect
9. #17987409	C₃	semidirect
10. #17987404	C₃	semidirect
11. #17987403	C₃	semidirect
15. C₉×C₁₉×3₋¹⁺²	C₃	semidirect
20. 3₋¹⁺²×C₁₉⋊C₃	C₉	semidirect
25. #13095307	C₃²	semidirect
26. #13095306	C₃²	semidirect
27. #13095305	C₃²	semidirect
28. #13095305	C₃²	semidirect
29. #13095305	C₃²	semidirect
30. #13095305	C₃²	semidirect
31. #13095305	C₃²	semidirect
32. #13095305	C₃²	semidirect
45. C₉×#4294	C₃²	semidirect
46. C₉×#4294	C₃²	semidirect
47. C₉×#4294	C₃²	semidirect
48. C₉×#4294	C₃²	semidirect
51. C₁₉×C₉⋊C₉	C₃²	semidirect
54. C₉²×C₁₉	C₃²	semidirect
64. C₁₉×3₋¹⁺²	C₃×C₉	semidirect
71. #10587025	C₃×C₉	semidirect
79. #10587023	C₃×C₉	semidirect
80. #10587022	C₃×C₉	semidirect
83. C₉×C₁₉⋊C₃	C₃×C₉	semidirect
93. C₉×3₋¹⁺²	C₁₉⋊C₃	semidirect
147. C₉⋊C₉	C₃×C₁₉⋊C₃	semidirect
166. C₉²	C₃×C₁₉⋊C₃	semidirect
186. 3₋¹⁺²	C₉×C₁₉⋊C₃	semidirect
203. C₁₉	#10624863	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
#17987415	3
3₋¹⁺²×#4294	3
C₉×#10587023	3
C₉×#10587022	3
C₉×#10587025	3
#17987410	3
#17987409	3
#17987404	3
#17987403	3
C₃×#13095303	3
C₁₉⋊C₃×C₃²⋊C₉	3
C₉×C₁₉×3₋¹⁺²	3
#10624863	19