Subgroups — #28356113

Order 13851.

620 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17987456	C₃	1	4	yes	maximal, complemented
2	#17987456	C₃	1	4	yes	maximal, complemented
3	#17987456	C₃	1	4	yes	maximal, complemented
4	#17987456	C₃	1	4	yes	maximal, complemented
5	#17987453	C₃	1	2	yes	maximal, complemented
6	#17987447	C₃	1	4	yes	maximal, complemented
7	#17987446	C₃	1	2	yes	maximal, complemented
8	#17987446	C₃	1	2	yes	maximal, complemented
9	#17987436	C₃	1	4	yes	maximal, complemented
10	#17987433	C₃	1	2	yes	maximal, complemented
11	#17987419	C₃	1	1	yes	maximal, characteristic
12	#17987418	C₃	1	2	yes	maximal
13	C₉×#10587023	C₃	1	1	yes	maximal, characteristic, complemented
14	#17987409	C₃	1	2	yes	maximal, complemented
15	C₁₉⋊C₃×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
16	C₁₉×#6896	C₃	1	1	yes	maximal, characteristic, complemented
17	C₃³×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
18	C₃×#10587025	C₃²	1	4	yes	—
19	C₃×#10587023	C₃²	1	2	yes	—
20	C₃×#10587023	C₃²	1	1	yes	characteristic
21	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
22	C₃×C₉×C₁₉⋊C₃	C₃²	1	2	yes	—
23	#13095307	C₃²	1	2	yes	complemented
24	#13095307	C₃²	1	2	yes	complemented
25	#13095307	C₃²	1	4	yes	complemented
26	#13095306	C₃²	1	2	yes	complemented
27	#13095306	C₃²	1	2	yes	complemented
28	#13095306	C₃²	1	4	yes	complemented
29	#13095305	C₃²	1	4	yes	complemented
30	#13095305	C₃²	1	4	yes	complemented
31	#13095305	C₃²	1	4	yes	complemented
32	#13095305	C₃²	1	4	yes	complemented
33	#13095305	C₃²	1	4	yes	complemented
34	#13095305	C₃²	1	4	yes	complemented
35	#13095305	C₃²	1	4	yes	complemented
36	#13095305	C₃²	1	4	yes	complemented
37	#13095305	C₃²	1	4	yes	complemented
38	#13095305	C₃²	1	4	yes	complemented
39	#13095305	C₃²	1	4	yes	complemented
40	#13095305	C₃²	1	4	yes	complemented
41	C₃²⋊#4294	C₃²	1	2	yes	—
42	C₃²⋊#4294	C₃²	1	2	yes	—
43	#13095303	C₃²	1	2	yes	—
44	#13095303	C₃²	1	4	yes	—
45	#13095303	C₃²	1	4	yes	—
46	#13095303	C₃²	1	2	yes	—
47	#13095303	C₃²	1	2	yes	—
48	#13095303	C₃²	1	2	yes	—
49	#13095303	C₃²	1	4	yes	—
50	#13095303	C₃²	1	4	yes	—
51	#13095303	C₃²	1	2	yes	—
52	#13095303	C₃²	1	4	yes	—
53	C₉×#4294	C₃²	1	2	yes	complemented
54	C₉×#4294	C₃²	1	4	yes	complemented
55	C₉×#4294	C₃²	1	2	yes	complemented
56	C₃×C₁₉×3₋¹⁺²	C₃²	1	2	yes	—
57	C₁₉×C₉⋊C₉	C₃²	1	2	yes	complemented
58	C₁₉×C₉⋊C₉	C₃²	1	2	yes	complemented
59	C₁₉×C₉⋊C₉	C₃²	1	4	yes	complemented
60	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
61	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
62	C₉²×C₁₉	C₃²	1	1	yes	characteristic, complemented
63	#10624889	—	19	1	no	maximal, Sylow
64	C₃³×C₁₉	C₃³	1	1	yes	characteristic
65	C₃²×C₁₉⋊C₃	—	3	3	no	—
66	C₃²×C₁₉⋊C₃	—	9	1	no	—
67	C₃²×C₁₉⋊C₃	—	9	2	no	—
68	C₃²×C₁₉⋊C₃	C₃³	1	2	yes	—
69	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
70	C₁₉×3₋¹⁺²	—	9	2	no	—
71	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
72	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
73	C₃×C₉×C₁₉	C₃³	1	2	yes	—
74	C₃×C₉×C₁₉	C₃³	1	2	yes	—
75	C₃×C₉×C₁₉	C₃³	1	2	yes	—
76	C₃×C₉×C₁₉	C₃³	1	4	yes	—
77	#10587025	—	9	4	no	—
78	C₃×#4294	C₃³	1	2	yes	—
79	C₃×#4294	C₃³	1	4	yes	—
80	C₃×#4294	C₃³	1	4	yes	—
81	C₃×#4294	C₃³	1	4	yes	—
82	C₃×#4294	C₃³	1	4	yes	—
83	C₃×#4294	C₃³	1	2	yes	—
84	C₃×#4294	C₃³	1	4	yes	—
85	#10587023	—	3	6	no	—
86	#10587023	—	3	3	no	—
87	C₉×C₁₉⋊C₃	—	9	1	no	—
88	C₉×C₁₉⋊C₃	—	9	2	no	—
89	#6897	—	19	4	no	—
90	#6897	—	19	4	no	—
91	#6897	—	19	4	no	—
92	#6897	—	19	4	no	—
93	#6897	—	19	2	no	—
94	#6896	—	19	4	no	—
95	#6896	—	19	2	no	—
96	#6896	—	19	2	no	—
97	#6896	C₁₉⋊C₃	1	1	yes	characteristic, complemented
98	#6894	—	19	2	no	—
99	#6894	—	19	4	no	—
100	#6891	—	19	2	no	—
101	#6891	—	19	1	no	—
102	C₉×3₋¹⁺²	—	19	2	no	—
103	C₉×3₋¹⁺²	—	19	1	no	—
104	C₃×C₃²⋊C₉	—	19	1	no	—
105	C₃²×C₁₉	—	3	1	no	—
106	C₃²×C₁₉	—	3	2	no	—
107	C₃²×C₁₉	—	3	1	no	—
108	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
109	C₃×C₁₉⋊C₃	—	3	1	no	—
110	C₃×C₁₉⋊C₃	—	3	2	no	—
111	C₃×C₁₉⋊C₃	—	3	2	no	—
112	C₃×C₁₉⋊C₃	—	3	2	no	—
113	C₃×C₁₉⋊C₃	—	3	4	no	—
114	C₃×C₁₉⋊C₃	—	9	9	no	—
115	C₃×C₁₉⋊C₃	#737	1	3	yes	—
116	C₉×C₁₉	—	3	1	no	—
117	C₉×C₁₉	—	3	2	no	—
118	C₉×C₁₉	—	3	1	no	—
119	C₉×C₁₉	—	3	2	no	—
120	C₉×C₁₉	—	3	2	no	—
121	C₉×C₁₉	—	3	4	no	—
122	#4294	—	3	4	no	—
123	#4294	—	3	2	no	—
124	#4294	—	3	2	no	—
125	#4294	—	3	4	no	—
126	#4294	—	3	4	no	—
127	#4294	—	3	4	no	—
128	#4294	—	3	4	no	—
129	C₃⁴	—	19	1	no	—
130	C₃×3₋¹⁺²	—	19	4	no	—
131	C₃×3₋¹⁺²	—	19	2	no	—
132	C₃×3₋¹⁺²	—	19	1	no	—
133	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	2	yes	—
134	C₃²×C₉	—	19	1	no	—
135	C₃²×C₉	—	19	2	no	—
136	C₉⋊C₉	—	19	4	no	—
137	C₉⋊C₉	—	19	4	no	—
138	C₉⋊C₉	—	19	4	no	—
139	C₉⋊C₉	—	19	4	no	—
140	C₉⋊C₉	—	19	4	no	—
141	C₉⋊C₉	—	19	4	no	—
142	C₉⋊C₉	—	19	4	no	—
143	C₉⋊C₉	—	19	4	no	—
144	C₉⋊C₉	—	19	4	no	—
145	C₉⋊C₉	—	19	4	no	—
146	C₉⋊C₉	—	19	4	no	—
147	C₉⋊C₉	—	19	4	no	—
148	C₉⋊C₉	—	19	2	no	—
149	C₉⋊C₉	—	19	2	no	—
150	C₉⋊C₉	—	19	4	no	—
151	C₉⋊C₉	—	19	2	no	—
152	C₉⋊C₉	—	19	2	no	—
153	C₉⋊C₉	—	19	4	no	—
154	C₉⋊C₉	C₃×C₁₉⋊C₃	1	2	yes	complemented
155	C₉⋊C₉	C₃×C₁₉⋊C₃	1	2	yes	complemented
156	C₉⋊C₉	C₃×C₁₉⋊C₃	1	4	yes	complemented
157	C₃²⋊C₉	—	19	2	no	—
158	C₃²⋊C₉	—	19	2	no	—
159	C₃²⋊C₉	—	19	4	no	—
160	C₃²⋊C₉	—	19	2	no	—
161	C₃²⋊C₉	—	19	4	no	—
162	C₃²⋊C₉	—	19	2	no	—
163	C₃²⋊C₉	—	19	4	no	—
164	C₃²⋊C₉	—	19	4	no	—
165	C₃²⋊C₉	—	19	4	no	—
166	C₃²⋊C₉	—	19	2	no	—
167	C₃²⋊C₉	—	19	2	no	—
168	C₃²⋊C₉	—	19	2	no	—
169	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
170	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
171	C₉²	—	19	4	no	—
172	C₉²	—	19	2	no	—
173	C₉²	—	19	2	no	—
174	C₉²	C₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented
175	C₃×C₁₉	—	9	1	no	—
176	C₃×C₁₉	C₃×#737	1	1	yes	characteristic
177	C₃×C₁₉	3₋¹⁺⁴	1	1	yes	characteristic
178	C₃×C₁₉	3₋¹⁺⁴	1	2	yes	—
179	C₁₉⋊C₃	—	3	3	no	—
180	C₁₉⋊C₃	—	9	2	no	—
181	C₃³	—	19	1	no	—
182	C₃³	—	19	2	no	—
183	C₃³	—	57	3	no	—
184	C₃³	—	171	1	no	—
185	C₃³	—	171	2	no	—
186	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
187	3₋¹⁺²	—	9	2	no	—
188	3₋¹⁺²	—	57	6	no	—
189	3₋¹⁺²	—	57	3	no	—
190	3₋¹⁺²	—	171	4	no	—
191	C₃×C₉	—	19	2	no	—
192	C₃×C₉	—	19	4	no	—
193	C₃×C₉	—	19	2	no	—
194	C₃×C₉	—	19	4	no	—
195	C₃×C₉	—	19	4	no	—
196	C₃×C₉	—	19	4	no	—
197	C₃×C₉	—	19	4	no	—
198	C₃×C₉	—	171	1	no	—
199	C₃×C₉	—	171	2	no	—
200	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
201	C₃×C₉	C₃²×C₁₉⋊C₃	1	4	yes	—
202	C₃×C₉	C₃²×C₁₉⋊C₃	1	2	yes	—
203	C₃×C₉	C₃²×C₁₉⋊C₃	1	2	yes	—
204	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
205	C₃×C₉	C₃²×C₁₉⋊C₃	1	2	yes	—
206	C₁₉	#10624889	1	1	yes	characteristic, complemented, Sylow
207	C₃²	—	3	1	no	—
208	C₃²	—	3	1	no	—
209	C₃²	—	3	2	no	—
210	C₃²	—	19	3	no	—
211	C₃²	—	57	1	no	—
212	C₃²	—	57	2	no	—
213	C₃²	—	57	2	no	—
214	C₃²	—	57	2	no	—
215	C₃²	—	57	4	no	—
216	C₃²	—	171	9	no	—
217	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
218	C₉	—	3	2	no	—
219	C₉	—	3	1	no	—
220	C₉	—	3	1	no	—
221	C₉	—	3	2	no	—
222	C₉	—	3	4	no	—
223	C₉	—	3	2	no	—
224	C₉	—	57	4	no	—
225	C₉	—	57	2	no	—
226	C₉	—	57	2	no	—
227	C₉	—	57	4	no	—
228	C₉	—	57	4	no	—
229	C₉	—	57	4	no	—
230	C₉	—	57	4	no	—
231	C₃	—	9	1	no	—
232	C₃	—	57	3	no	—
233	C₃	—	171	2	no	—
234	C₃	C₁₉⋊C₃×#737	1	1	yes	characteristic
235	C₃	#17987544	1	1	yes	characteristic
236	C₃	#17987546	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₁₉⋊C₃
commutator, socle, cosocle	C₃²×C₁₉	C₃⁴
Fitting	C₁₉×#6896	C₃
prime radical	C₃³×C₁₉⋊C₃	C₃²
hypercenter	#6896	C₁₉⋊C₃
hypocenter	C₁₉	#10624889
LCS	G ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ #6896

Complements

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

N	H ≅ G/N	type
1. #17987456	C₃	semidirect
2. #17987456	C₃	semidirect
3. #17987456	C₃	semidirect
4. #17987456	C₃	semidirect
5. #17987453	C₃	semidirect
6. #17987447	C₃	semidirect
7. #17987446	C₃	semidirect
8. #17987446	C₃	semidirect
9. #17987436	C₃	semidirect
10. #17987433	C₃	semidirect
13. C₉×#10587023	C₃	semidirect
14. #17987409	C₃	semidirect
16. C₁₉×#6896	C₃	semidirect
23. #13095307	C₃²	semidirect
24. #13095307	C₃²	semidirect
25. #13095307	C₃²	semidirect
26. #13095306	C₃²	semidirect
27. #13095306	C₃²	semidirect
28. #13095306	C₃²	semidirect
29. #13095305	C₃²	semidirect
30. #13095305	C₃²	semidirect
31. #13095305	C₃²	semidirect
32. #13095305	C₃²	semidirect
33. #13095305	C₃²	semidirect
34. #13095305	C₃²	semidirect
35. #13095305	C₃²	semidirect
36. #13095305	C₃²	semidirect
37. #13095305	C₃²	semidirect
38. #13095305	C₃²	semidirect
39. #13095305	C₃²	semidirect
40. #13095305	C₃²	semidirect
53. C₉×#4294	C₃²	semidirect
54. C₉×#4294	C₃²	semidirect
55. C₉×#4294	C₃²	semidirect
57. C₁₉×C₉⋊C₉	C₃²	semidirect
58. C₁₉×C₉⋊C₉	C₃²	semidirect
59. C₁₉×C₉⋊C₉	C₃²	semidirect
62. C₉²×C₁₉	C₃²	semidirect
97. #6896	C₁₉⋊C₃	semidirect
154. C₉⋊C₉	C₃×C₁₉⋊C₃	semidirect
155. C₉⋊C₉	C₃×C₁₉⋊C₃	semidirect
156. C₉⋊C₉	C₃×C₁₉⋊C₃	semidirect
174. C₉²	C₃×C₁₉⋊C₃	semidirect
206. C₁₉	#10624889	semidirect

Maximal subgroups

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

subgroup	index
#17987456	3
#17987453	3
#17987447	3
#17987446	3
#17987436	3
#17987433	3
#17987419	3
#17987418	3
C₉×#10587023	3
#17987409	3
C₁₉⋊C₃×C₃²⋊C₉	3
C₁₉×#6896	3
#10624889	19