Subgroups — #28356038

Order 13851.

758 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#13095333	C₃	1	1	yes	maximal, characteristic
2	#17987470	C₃	1	1	yes	maximal, characteristic, complemented
3	#17987469	C₃	1	1	yes	maximal, characteristic, complemented
4	#17987467	C₃	1	2	yes	maximal, complemented
5	#17987466	C₃	1	2	yes	maximal, complemented
6	#17987461	C₃	1	2	yes	maximal, complemented
7	#17987451	C₃	1	3	yes	maximal, complemented
8	#17987450	C₃	1	3	yes	maximal, complemented
9	#17987447	C₃	1	3	yes	maximal, complemented
10	#17987434	C₃	1	3	yes	maximal, complemented
11	#17987433	C₃	1	3	yes	maximal, complemented
12	#17987422	C₃	1	3	yes	maximal
13	3₋¹⁺²×#4294	C₃	1	3	yes	maximal, complemented
14	#17987410	C₃	1	3	yes	maximal, complemented
15	#17987409	C₃	1	3	yes	maximal, complemented
16	#17987404	C₃	1	2	yes	maximal, complemented
17	C₉²×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
18	C₁₉×#6900	C₃	1	1	yes	maximal, characteristic, complemented
19	#13095333	—	3	9	no	—
20	C₃×#10587025	C₃²	1	1	yes	characteristic
21	C₃×#10587022	C₃²	1	3	yes	—
22	C₃×#10587022	C₃²	1	1	yes	characteristic
23	C₃×#10587022	C₃²	1	1	yes	characteristic
24	C₃×#10587022	C₃²	1	1	yes	characteristic
25	C₃×#10587023	C₃²	1	3	yes	—
26	C₃×#10587023	C₃²	1	1	yes	characteristic
27	C₃×#10587023	C₃²	1	1	yes	characteristic
28	C₃×#10587023	C₃²	1	1	yes	characteristic
29	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	1	yes	characteristic
30	C₃²×#4294	C₃²	1	1	yes	characteristic
31	C₃×C₉×C₁₉⋊C₃	C₃²	1	3	yes	—
32	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
33	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
34	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
35	#13095307	C₃²	1	2	yes	—
36	#13095307	C₃²	1	3	yes	—
37	#13095307	C₃²	1	6	yes	—
38	#13095307	C₃²	1	3	yes	—
39	#13095306	C₃²	1	2	yes	—
40	#13095306	C₃²	1	3	yes	—
41	#13095306	C₃²	1	6	yes	—
42	#13095306	C₃²	1	3	yes	—
43	#13095305	C₃²	1	6	yes	—
44	#13095305	C₃²	1	6	yes	—
45	#13095305	C₃²	1	6	yes	—
46	#13095305	C₃²	1	6	yes	—
47	#13095305	C₃²	1	6	yes	—
48	C₃²⋊#4294	C₃²	1	6	yes	—
49	#13095303	C₃²	1	6	yes	—
50	#13095303	C₃²	1	6	yes	—
51	#13095303	C₃²	1	6	yes	—
52	C₉×#4294	C₃²	1	3	yes	—
53	C₉×#4294	C₃²	1	3	yes	—
54	C₉×#4294	C₃²	1	2	yes	—
55	C₉×#4294	C₃²	1	6	yes	—
56	C₃×C₁₉×3₋¹⁺²	C₃²	1	3	yes	—
57	C₃×C₁₉×3₋¹⁺²	C₃²	1	1	yes	characteristic
58	C₁₉×C₉⋊C₉	C₃²	1	2	yes	—
59	C₁₉×C₉⋊C₉	C₃²	1	3	yes	—
60	C₁₉×C₉⋊C₉	C₃²	1	3	yes	—
61	C₉²×C₁₉	C₃²	1	1	yes	characteristic
62	#10624879	—	19	1	no	maximal, Sylow
63	C₃³×C₁₉	C₃³	1	1	yes	characteristic
64	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
65	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
66	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
67	C₁₉×3₋¹⁺²	—	3	3	no	—
68	C₁₉×3₋¹⁺²	—	3	9	no	—
69	C₃×C₉×C₁₉	C₃³	1	6	yes	—
70	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
71	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
72	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
73	C₃×C₉×C₁₉	C₃³	1	3	yes	—
74	C₃²⋊(C₁₉⋊C₃)	—	3	3	no	—
75	#10587025	—	3	3	no	—
76	C₃×#4294	—	3	3	no	—
77	C₃×#4294	C₃³	1	6	yes	—
78	C₃×#4294	C₃³	1	6	yes	—
79	C₃×#4294	C₃³	1	6	yes	—
80	C₃×#4294	C₃³	1	1	yes	characteristic
81	C₃×#4294	C₃³	1	1	yes	characteristic
82	C₃×#4294	C₃³	1	1	yes	characteristic
83	C₃×#4294	C₃³	1	1	yes	characteristic
84	C₃×#4294	C₃³	1	1	yes	characteristic
85	C₃×#4294	C₃³	1	1	yes	characteristic
86	#10587023	—	3	3	no	—
87	#10587023	—	3	3	no	—
88	#10587023	—	3	9	no	—
89	#10587023	—	3	3	no	—
90	#10587022	—	3	3	no	—
91	#10587022	—	3	3	no	—
92	#10587022	—	3	9	no	—
93	#10587022	—	3	3	no	—
94	C₉×C₁₉⋊C₃	—	3	9	no	—
95	C₉×C₁₉⋊C₃	—	3	3	no	—
96	C₉×C₁₉⋊C₃	—	3	3	no	—
97	C₉×C₁₉⋊C₃	3₋¹⁺²	1	9	yes	complemented
98	C₃×#737	—	19	1	no	—
99	#6900	—	19	1	no	—
100	#6900	—	19	1	no	—
101	#6900	C₁₉⋊C₃	1	1	yes	characteristic, complemented
102	#6899	—	19	2	no	—
103	#6899	—	19	2	no	—
104	#6899	—	19	2	no	—
105	#6896	—	19	3	no	—
106	#6896	—	19	3	no	—
107	#6896	—	19	3	no	—
108	#6894	—	19	3	no	—
109	#6894	—	19	3	no	—
110	C₉⋊H(3,𝔽₃)	—	19	3	no	—
111	C₉×3₋¹⁺²	—	19	3	no	—
112	C₉×3₋¹⁺²	—	19	3	no	—
113	C₉×3₋¹⁺²	—	19	3	no	—
114	#6887	—	19	2	no	—
115	C₃×C₉²	—	19	1	no	—
116	C₃²×C₁₉	—	3	1	no	—
117	C₃²×C₁₉	—	3	3	no	—
118	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
119	C₃×C₁₉⋊C₃	—	3	3	no	—
120	C₃×C₁₉⋊C₃	—	3	1	no	—
121	C₃×C₁₉⋊C₃	—	3	1	no	—
122	C₃×C₁₉⋊C₃	—	3	3	no	—
123	C₃×C₁₉⋊C₃	—	3	3	no	—
124	C₃×C₁₉⋊C₃	C₃×3₋¹⁺²	1	3	yes	—
125	C₉×C₁₉	—	3	1	no	—
126	C₉×C₁₉	—	3	1	no	—
127	C₉×C₁₉	—	3	6	no	—
128	C₉×C₁₉	C₃×3₋¹⁺²	1	3	yes	—
129	C₉×C₁₉	#737	1	9	yes	complemented
130	#4294	—	3	1	no	—
131	#4294	—	3	1	no	—
132	#4294	—	3	1	no	—
133	#4294	—	3	1	no	—
134	#4294	—	3	6	no	—
135	#4294	—	3	6	no	—
136	#4294	—	3	6	no	—
137	#4294	C₃×3₋¹⁺²	1	3	yes	—
138	#4294	C₃×3₋¹⁺²	1	3	yes	—
139	#737	—	57	9	no	—
140	C₃×3₋¹⁺²	—	19	1	no	—
141	C₃×3₋¹⁺²	—	19	1	no	—
142	C₃×3₋¹⁺²	—	19	1	no	—
143	C₃×3₋¹⁺²	—	19	1	no	—
144	C₃×3₋¹⁺²	—	19	1	no	—
145	C₃×3₋¹⁺²	—	19	1	no	—
146	C₃×3₋¹⁺²	—	19	1	no	—
147	C₃×3₋¹⁺²	—	19	3	no	—
148	C₃×3₋¹⁺²	—	19	3	no	—
149	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
150	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	3	yes	—
151	C₃×H(3,𝔽₃)	—	19	1	no	—
152	C₃²×C₉	—	19	1	no	—
153	C₃²×C₉	—	19	1	no	—
154	C₃²×C₉	—	19	1	no	—
155	C₃²×C₉	—	19	1	no	—
156	C₃²×C₉	—	19	3	no	—
157	C₉⋊C₉	—	19	6	no	—
158	C₉⋊C₉	—	19	6	no	—
159	C₉⋊C₉	—	19	6	no	—
160	C₉⋊C₉	—	19	6	no	—
161	C₉⋊C₉	—	19	6	no	—
162	C₉⋊C₉	—	19	2	no	—
163	C₉⋊C₉	—	19	6	no	—
164	C₉⋊C₉	—	19	3	no	—
165	C₉⋊C₉	—	19	3	no	—
166	C₉⋊C₉	—	19	2	no	—
167	C₉⋊C₉	—	19	6	no	—
168	C₉⋊C₉	—	19	3	no	—
169	C₉⋊C₉	—	19	3	no	—
170	C₉⋊C₉	C₃×C₁₉⋊C₃	1	2	yes	—
171	C₉⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	—
172	C₉⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	—
173	C₃²⋊C₉	—	19	6	no	—
174	C₃²⋊C₉	—	19	6	no	—
175	C₃²⋊C₉	—	19	6	no	—
176	C₃²⋊C₉	—	19	6	no	—
177	C₉²	—	19	3	no	—
178	C₉²	—	19	3	no	—
179	C₉²	—	19	2	no	—
180	C₉²	—	19	6	no	—
181	C₉²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
182	C₃×C₁₉	—	9	1	no	—
183	C₃×C₁₉	C₃²×3₋¹⁺²	1	1	yes	characteristic
184	C₃×C₁₉	C₃×#737	1	3	yes	—
185	C₁₉⋊C₃	—	3	3	no	—
186	C₁₉⋊C₃	—	9	1	no	—
187	C₁₉⋊C₃	—	9	1	no	—
188	C₃³	—	19	1	no	—
189	C₃³	—	19	1	no	—
190	C₃³	—	19	1	no	—
191	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
192	3₋¹⁺²	—	3	3	no	—
193	3₋¹⁺²	—	3	9	no	—
194	3₋¹⁺²	—	57	3	no	—
195	3₋¹⁺²	—	57	3	no	—
196	3₋¹⁺²	—	57	3	no	—
197	3₋¹⁺²	—	57	3	no	—
198	3₋¹⁺²	—	57	3	no	—
199	3₋¹⁺²	—	57	3	no	—
200	3₋¹⁺²	—	57	3	no	—
201	3₋¹⁺²	—	57	9	no	—
202	3₋¹⁺²	—	57	9	no	—
203	H(3,𝔽₃)	—	57	3	no	—
204	C₃×C₉	—	19	1	no	—
205	C₃×C₉	—	19	1	no	—
206	C₃×C₉	—	19	9	no	—
207	C₃×C₉	—	19	6	no	—
208	C₃×C₉	—	19	6	no	—
209	C₃×C₉	—	19	6	no	—
210	C₃×C₉	—	19	1	no	—
211	C₃×C₉	—	19	1	no	—
212	C₃×C₉	—	19	1	no	—
213	C₃×C₉	—	19	1	no	—
214	C₃×C₉	—	57	3	no	—
215	C₃×C₉	—	57	3	no	—
216	C₃×C₉	—	57	3	no	—
217	C₃×C₉	—	57	9	no	—
218	C₃×C₉	C₃²×C₁₉⋊C₃	1	6	yes	—
219	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
220	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
221	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
222	C₃×C₉	C₃²×C₁₉⋊C₃	1	3	yes	—
223	C₁₉	#10624879	1	1	yes	characteristic, complemented, Sylow
224	C₃²	—	3	1	no	—
225	C₃²	—	3	3	no	—
226	C₃²	—	19	3	no	—
227	C₃²	—	57	3	no	—
228	C₃²	—	57	1	no	—
229	C₃²	—	57	1	no	—
230	C₃²	—	57	3	no	—
231	C₃²	—	57	3	no	—
232	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
233	C₉	—	3	1	no	—
234	C₉	—	3	1	no	—
235	C₉	—	3	6	no	—
236	C₉	—	19	3	no	—
237	C₉	—	19	3	no	—
238	C₉	—	57	1	no	—
239	C₉	—	57	1	no	—
240	C₉	—	57	1	no	—
241	C₉	—	57	1	no	—
242	C₉	—	57	6	no	—
243	C₉	—	57	6	no	—
244	C₉	—	57	6	no	—
245	C₉	3₋¹⁺²×C₁₉⋊C₃	1	3	yes	—
246	C₉	#13095333	1	9	yes	complemented
247	C₃	—	9	1	no	—
248	C₃	—	57	3	no	—
249	C₃	—	171	1	no	—
250	C₃	—	171	1	no	—
251	C₃	C₃×3₋¹⁺²×C₁₉⋊C₃	1	1	yes	characteristic
252	C₃	C₃×#13095333	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₁₉⋊C₃
commutator, socle, cosocle	C₃²×C₁₉	C₃⁴
Fitting	C₁₉×#6900	C₃
prime radical	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²
hypercenter	#6900	C₁₉⋊C₃
hypocenter	C₁₉	#10624879
LCS	G ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ #6900

Complements

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

N	H ≅ G/N	type
2. #17987470	C₃	semidirect
3. #17987469	C₃	semidirect
4. #17987467	C₃	semidirect
5. #17987466	C₃	semidirect
6. #17987461	C₃	semidirect
7. #17987451	C₃	semidirect
8. #17987450	C₃	semidirect
9. #17987447	C₃	semidirect
10. #17987434	C₃	semidirect
11. #17987433	C₃	semidirect
13. 3₋¹⁺²×#4294	C₃	semidirect
14. #17987410	C₃	semidirect
15. #17987409	C₃	semidirect
16. #17987404	C₃	semidirect
17. C₉²×C₁₉⋊C₃	C₃	semidirect
18. C₁₉×#6900	C₃	semidirect
97. C₉×C₁₉⋊C₃	3₋¹⁺²	semidirect
101. #6900	C₁₉⋊C₃	semidirect
129. C₉×C₁₉	#737	semidirect
223. C₁₉	#10624879	semidirect
246. C₉	#13095333	semidirect

Maximal subgroups

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

subgroup	index
C₃×#13095333	3
#17987470	3
#17987469	3
#17987467	3
#17987466	3
#17987461	3
#17987451	3
#17987450	3
#17987447	3
#17987434	3
#17987433	3
#17987422	3
3₋¹⁺²×#4294	3
#17987410	3
#17987409	3
#17987404	3
C₉²×C₁₉⋊C₃	3
C₁₉×#6900	3
#10624879	19