Subgroups — #18078145

Order 4864.

626 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10597260	C₂	1	1	yes	maximal, characteristic
2	#14563397	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926286	C₂	1	2	yes	maximal, complemented
4	#14561516	C₂	1	2	yes	maximal, complemented
5	C₁₉×#2851	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₁₉⋊C₈	—	2	1	no	—
7	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
8	C₂×#10597260	—	2	4	no	—
9	C₂×#10597260	—	2	1	no	—
10	C₂×#10597260	C₂²	1	2	yes	—
11	C₂×#10597260	C₂²	1	2	yes	—
12	#11926299	—	2	1	no	—
13	#11926286	—	2	4	no	—
14	C₂×C₁₉×#125	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
16	C₁₉×#389	C₄	1	1	yes	characteristic
17	C₁₉×#389	C₄	1	1	yes	characteristic
18	C₁₉×#387	—	2	1	no	—
19	C₁₉×#387	—	2	1	no	—
20	C₁₉×#385	—	2	1	no	—
21	C₁₉×#385	—	2	1	no	—
22	C₁₉×#382	C₂²	1	1	yes	characteristic
23	C₁₉×#380	C₄	1	1	yes	characteristic
24	C₁₉×#380	C₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
28	C₁₉×#125	—	2	2	no	—
29	C₁₉×#125	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
33	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
34	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
39	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
40	C₂×C₄²×C₁₉	—	2	2	no	—
41	C₂×C₄²×C₁₉	—	2	2	no	—
42	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
46	C₂²×C₁₉⋊C₈	—	2	4	no	—
47	C₂²×C₁₉⋊C₈	—	2	4	no	—
48	C₂²×C₁₉⋊C₈	—	2	4	no	—
49	C₂²×C₁₉⋊C₈	—	2	2	no	—
50	C₂²×C₁₉⋊C₈	—	2	2	no	—
51	C₁₉×#95	—	2	1	no	—
52	C₁₉×#95	—	2	1	no	—
53	C₁₉×#95	—	2	1	no	—
54	C₁₉×#95	—	2	1	no	—
55	C₁₉×#95	—	2	1	no	—
56	C₁₉×#95	—	2	1	no	—
57	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
58	#10597260	—	2	4	no	—
59	#10597260	—	2	2	no	—
60	#10597260	—	2	2	no	—
61	#10597260	—	2	2	no	—
62	#10597260	D₄	1	4	yes	—
63	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	—	2	2	no	—
65	C₂²×C₄×C₁₉	—	2	2	no	—
66	C₂²×C₄×C₁₉	—	2	1	no	—
67	C₂²×C₄×C₁₉	—	2	1	no	—
68	C₂²×C₄×C₁₉	—	2	2	no	—
69	C₂²×C₄×C₁₉	—	2	2	no	—
70	C₂²×C₄×C₁₉	—	2	2	no	—
71	C₂²×C₄×C₁₉	—	2	2	no	—
72	C₂²×C₄×C₁₉	—	2	1	no	—
73	C₂²×C₄×C₁₉	—	2	1	no	—
74	C₂²×C₄×C₁₉	—	2	1	no	—
75	C₂²×C₄×C₁₉	—	2	1	no	—
76	C₂²×C₄×C₁₉	—	4	1	no	—
77	C₂²×C₄×C₁₉	—	4	1	no	—
78	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
84	C₄⋊C₄×C₁₉	—	2	2	no	—
85	C₄⋊C₄×C₁₉	—	2	2	no	—
86	C₄⋊C₄×C₁₉	—	4	1	no	—
87	C₄⋊C₄×C₁₉	—	4	1	no	—
88	C₄⋊C₄×C₁₉	—	4	2	no	—
89	C₄⋊C₄×C₁₉	—	4	2	no	—
90	C₄⋊C₄×C₁₉	—	4	1	no	—
91	C₄⋊C₄×C₁₉	—	4	2	no	—
92	C₄⋊C₄×C₁₉	—	4	1	no	—
93	C₂²⋊C₄×C₁₉	—	4	2	no	—
94	C₂²⋊C₄×C₁₉	—	4	2	no	—
95	C₄²×C₁₉	—	2	2	no	—
96	C₄²×C₁₉	—	2	1	no	—
97	C₄²×C₁₉	—	2	2	no	—
98	C₄²×C₁₉	—	2	1	no	—
99	C₄²×C₁₉	C₂×D₄	1	2	yes	—
100	C₄²×C₁₉	#41	1	2	yes	—
101	C₂×C₁₉⋊C₈	—	2	16	no	—
102	C₂×C₁₉⋊C₈	—	2	4	no	—
103	C₂×C₁₉⋊C₈	—	2	4	no	—
104	C₂×C₁₉⋊C₈	—	2	4	no	—
105	C₂×C₁₉⋊C₈	—	2	4	no	—
106	C₂×C₁₉⋊C₈	—	4	2	no	—
107	C₂×C₁₉⋊C₈	—	4	2	no	—
108	#9121	—	19	1	no	maximal, Sylow
109	C₂³×C₁₉	—	2	1	no	—
110	C₂³×C₁₉	—	2	1	no	—
111	C₂³×C₁₉	—	2	1	no	—
112	C₂³×C₁₉	—	2	1	no	—
113	C₂³×C₁₉	#101	1	2	yes	—
114	C₂³×C₁₉	#104	1	2	yes	—
115	C₂³×C₁₉	#104	1	2	yes	—
116	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₉	—	2	4	no	—
118	C₂×C₄×C₁₉	—	2	4	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	2	1	no	—
123	C₂×C₄×C₁₉	—	2	1	no	—
124	C₂×C₄×C₁₉	—	2	1	no	—
125	C₂×C₄×C₁₉	—	2	1	no	—
126	C₂×C₄×C₁₉	—	2	1	no	—
127	C₂×C₄×C₁₉	—	2	1	no	—
128	C₂×C₄×C₁₉	—	2	1	no	—
129	C₂×C₄×C₁₉	—	2	1	no	—
130	C₂×C₄×C₁₉	—	2	1	no	—
131	C₂×C₄×C₁₉	—	2	1	no	—
132	C₂×C₄×C₁₉	—	2	1	no	—
133	C₂×C₄×C₁₉	—	4	4	no	—
134	C₂×C₄×C₁₉	—	4	4	no	—
135	C₂×C₄×C₁₉	—	4	1	no	—
136	C₂×C₄×C₁₉	—	4	1	no	—
137	C₂×C₄×C₁₉	—	4	1	no	—
138	C₂×C₄×C₁₉	—	4	1	no	—
139	C₂×C₄×C₁₉	—	4	1	no	—
140	C₂×C₄×C₁₉	—	4	1	no	—
141	C₂×C₄×C₁₉	—	4	1	no	—
142	C₂×C₄×C₁₉	—	4	1	no	—
143	C₂×C₄×C₁₉	—	4	1	no	—
144	C₂×C₄×C₁₉	—	4	1	no	—
145	C₂×C₄×C₁₉	—	4	1	no	—
146	C₂×C₄×C₁₉	—	4	1	no	—
147	C₂×C₄×C₁₉	#104	1	2	yes	—
148	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
149	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
150	C₂×C₄×C₁₉	#121	1	2	yes	—
151	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
152	C₁₉⋊C₈	—	2	8	no	—
153	C₁₉⋊C₈	—	4	4	no	—
154	C₂²×#97	—	19	1	no	—
155	#2851	D₁₉	1	1	yes	characteristic
156	#1749	—	19	1	no	—
157	C₂×#330	—	19	2	no	—
158	#1295	—	19	2	no	—
159	C₂²×C₁₉	—	2	2	no	—
160	C₂²×C₁₉	—	2	2	no	—
161	C₂²×C₁₉	—	2	2	no	—
162	C₂²×C₁₉	—	2	2	no	—
163	C₂²×C₁₉	—	2	2	no	—
164	C₂²×C₁₉	—	2	2	no	—
165	C₂²×C₁₉	#330	1	4	yes	complemented
166	C₂²×C₁₉	#386	1	1	yes	characteristic
167	C₂²×C₁₉	C₂×#101	1	1	yes	characteristic
168	C₂²×C₁₉	#413	1	1	yes	characteristic
169	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
170	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
171	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
172	C₂²×C₁₉	#421	1	1	yes	characteristic
173	C₄×C₁₉	—	2	2	no	—
174	C₄×C₁₉	—	2	1	no	—
175	C₄×C₁₉	—	2	2	no	—
176	C₄×C₁₉	—	2	1	no	—
177	C₄×C₁₉	—	2	2	no	—
178	C₄×C₁₉	—	4	1	no	—
179	C₄×C₁₉	—	4	1	no	—
180	C₄×C₁₉	—	8	1	no	—
181	C₄×C₁₉	—	8	1	no	—
182	C₄×C₁₉	—	8	1	no	—
183	C₄×C₁₉	—	8	1	no	—
184	C₂³×C₈	—	38	1	no	—
185	C₂×#125	C₂×D₁₉	1	1	yes	characteristic
186	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
187	C₂×C₂²⋊C₈	—	38	1	no	—
188	C₂×#97	—	19	2	no	—
189	C₂×#97	—	19	2	no	—
190	C₂×#97	—	38	4	no	—
191	C₂×#97	—	38	1	no	—
192	#389	Dic₁₉	1	1	yes	characteristic
193	#389	Dic₁₉	1	1	yes	characteristic
194	#387	—	2	1	no	—
195	#387	—	2	1	no	—
196	#385	—	2	1	no	—
197	#385	—	2	1	no	—
198	#382	C₂×D₁₉	1	1	yes	characteristic
199	#380	Dic₁₉	1	1	yes	characteristic
200	#380	Dic₁₉	1	1	yes	characteristic
201	#336	—	38	1	no	—
202	#330	—	38	4	no	—
203	C₂×C₁₉	—	2	4	no	—
204	C₂×C₁₉	C₂×#330	1	1	yes	characteristic
205	C₂×C₁₉	#1524	1	1	yes	characteristic
206	C₂×C₁₉	#1531	1	1	yes	characteristic
207	C₂×C₁₉	#1532	1	1	yes	characteristic
208	C₂×C₁₉	#1871	1	1	yes	characteristic
209	C₂×C₁₉	#1896	1	1	yes	characteristic
210	C₂×C₁₉	#1901	1	1	yes	characteristic
211	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
212	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
213	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
214	C₂²×C₈	—	38	4	no	—
215	C₂²×C₈	—	38	4	no	—
216	C₂²×C₈	—	38	4	no	—
217	C₂²×C₈	—	38	2	no	—
218	C₂²×C₈	—	38	2	no	—
219	#125	—	2	2	no	—
220	#125	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	—	2	1	no	—
225	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
226	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
227	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
228	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
229	C₂×C₂²⋊C₄	—	2	1	no	—
230	C₂×C₂²⋊C₄	—	2	1	no	—
231	C₂×C₄²	—	2	2	no	—
232	C₂×C₄²	—	2	2	no	—
233	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
234	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
235	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
236	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
237	C₂²⋊C₈	—	38	4	no	—
238	#97	—	19	4	no	—
239	#97	—	38	4	no	—
240	#97	—	38	2	no	—
241	#97	—	38	2	no	—
242	#97	—	38	2	no	—
243	#95	—	2	1	no	—
244	#95	—	2	1	no	—
245	#95	—	2	1	no	—
246	#95	—	2	1	no	—
247	#95	—	2	1	no	—
248	#95	—	2	1	no	—
249	C₁₉	#9121	1	1	yes	characteristic, complemented, Sylow
250	C₂⁴	C₂²⋊Dic₁₉	1	1	yes	characteristic
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	—	2	1	no	—
261	C₂²×C₄	—	2	1	no	—
262	C₂²×C₄	—	2	1	no	—
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	—	4	1	no	—
265	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
266	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
267	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
268	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
269	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
270	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
271	C₂×C₈	—	38	16	no	—
272	C₂×C₈	—	38	4	no	—
273	C₂×C₈	—	38	4	no	—
274	C₂×C₈	—	38	4	no	—
275	C₂×C₈	—	38	4	no	—
276	C₂×C₈	—	76	2	no	—
277	C₂×C₈	—	76	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	4	2	no	—
286	C₄⋊C₄	—	4	2	no	—
287	C₂²⋊C₄	—	4	2	no	—
288	C₂²⋊C₄	—	4	2	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	1	no	—
291	C₄²	—	2	1	no	—
292	C₄²	—	2	2	no	—
293	C₄²	D₄×D₁₉	1	2	yes	—
294	C₄²	#64543	1	2	yes	—
295	C₂³	—	2	1	no	—
296	C₂³	—	2	1	no	—
297	C₂³	—	2	1	no	—
298	C₂³	—	2	1	no	—
299	C₂³	#10597293	1	2	yes	—
300	C₂³	#10597294	1	2	yes	—
301	C₂³	#10597294	1	2	yes	—
302	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	4	4	no	—
320	C₂×C₄	—	4	4	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	#10597294	1	2	yes	—
334	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
335	C₂×C₄	#10597381	1	1	yes	characteristic
336	C₂×C₄	#10597383	1	1	yes	characteristic
337	C₂×C₄	#10597385	1	2	yes	—
338	C₈	—	38	8	no	—
339	C₈	—	76	4	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	2	2	no	—
345	C₂²	—	2	2	no	—
346	C₂²	#11926286	1	4	yes	complemented
347	C₂²	#11926424	1	1	yes	characteristic
348	C₂²	C₂×#10597293	1	1	yes	characteristic
349	C₂²	#11926528	1	1	yes	characteristic
350	C₂²	C₂×#10597294	1	1	yes	characteristic
351	C₂²	C₂×#10597294	1	1	yes	characteristic
352	C₂²	C₂×#10597294	1	1	yes	characteristic
353	C₂²	#11926561	1	1	yes	characteristic
354	C₄	—	2	2	no	—
355	C₄	—	2	2	no	—
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	2	no	—
359	C₄	—	4	1	no	—
360	C₄	—	4	1	no	—
361	C₄	—	8	1	no	—
362	C₄	—	8	1	no	—
363	C₄	—	8	1	no	—
364	C₄	—	8	1	no	—
365	C₂	—	2	4	no	—
366	C₂	C₂×#11926286	1	1	yes	characteristic
367	C₂	#14562223	1	1	yes	characteristic
368	C₂	#14562250	1	1	yes	characteristic
369	C₂	#14562255	1	1	yes	characteristic
370	C₂	#14563938	1	1	yes	characteristic
371	C₂	#14564079	1	1	yes	characteristic
372	C₂	#14564104	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2851	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2851	D₁₉
hypocenter	C₁₉	#9121
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2851

Complements

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

N	H ≅ G/N	type
3. C₂×#11926286	C₂	semidirect
4. #14561516	C₂	semidirect
165. C₂²×C₁₉	#330	semidirect
249. C₁₉	#9121	semidirect
346. C₂²	#11926286	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#10597260	2
#14563397	2
C₂×#11926286	2
#14561516	2
C₁₉×#2851	2
#9121	19