Subgroups — #18088150

Order 4864.

918 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926298	C₂	1	2	yes	maximal
3	C₂×#11926298	C₂	1	2	yes	maximal
4	C₄×C₁₉⋊#95	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₉×#336	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
7	C₂³×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₁₉	C₂²	1	2	yes	—
9	C₂×C₄×C₁₉⋊C₈	—	2	4	no	—
10	C₂×C₄×C₁₉⋊C₈	C₄	1	4	yes	complemented
11	C₂×C₁₉⋊#95	C₄	1	2	yes	—
12	C₂×C₁₉⋊#95	C₄	1	2	yes	—
13	#11926298	—	2	4	no	—
14	#11926298	—	2	4	no	—
15	C₂³×C₈×C₁₉	C₂²	1	2	yes	—
16	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#336	C₄	1	4	yes	complemented
18	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
21	C₂³×Dic₁₉	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₁₉	—	2	2	no	—
23	C₂²×C₈×C₁₉	—	2	2	no	—
24	C₂²×C₈×C₁₉	—	2	2	no	—
25	C₂²×C₈×C₁₉	—	2	2	no	—
26	C₂²×C₈×C₁₉	—	2	2	no	—
27	C₂²×C₈×C₁₉	—	2	2	no	—
28	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₁₉	—	2	2	no	—
30	C₂×C₄²×C₁₉	—	2	2	no	—
31	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
33	C₂×C₄×Dic₁₉	—	2	2	no	—
34	C₂×C₄×Dic₁₉	—	2	2	no	—
35	C₂×C₄×Dic₁₉	—	2	2	no	—
36	C₂×C₄×Dic₁₉	—	2	2	no	—
37	C₂×C₄×Dic₁₉	—	2	2	no	—
38	C₂×C₄×Dic₁₉	—	2	2	no	—
39	C₂²×C₁₉⋊C₈	—	2	2	no	—
40	C₂²×C₁₉⋊C₈	—	2	2	no	—
41	C₂²×C₁₉⋊C₈	—	2	2	no	—
42	C₂²×C₁₉⋊C₈	—	2	1	no	—
43	C₂²×C₁₉⋊C₈	—	2	1	no	—
44	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
45	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
46	C₂²×C₁₉⋊C₈	D₄	1	4	yes	—
47	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
48	C₂²×C₁₉⋊C₈	Q₈	1	2	yes	—
49	C₁₉⋊#95	—	2	4	no	—
50	C₁₉⋊#95	C₈	1	8	yes	complemented
51	C₄×C₁₉⋊C₈	—	2	8	no	—
52	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	—	2	2	no	—
54	C₂²×C₄×C₁₉	—	2	2	no	—
55	C₂²×C₄×C₁₉	—	2	2	no	—
56	C₂²×C₄×C₁₉	—	2	2	no	—
57	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
58	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
59	C₂²×C₄×C₁₉	C₂×C₈	1	4	yes	—
60	C₂²×C₄×C₁₉	M₁₆	1	4	yes	—
61	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	#41	1	2	yes	—
67	C₂²×C₄×C₁₉	#41	1	2	yes	—
68	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	#41	1	2	yes	—
70	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
71	C₂²×Dic₁₉	—	2	4	no	—
72	C₂²×Dic₁₉	—	2	4	no	—
73	C₂²×Dic₁₉	—	2	4	no	—
74	C₂²×Dic₁₉	—	2	4	no	—
75	C₂²×Dic₁₉	—	2	4	no	—
76	C₂²×Dic₁₉	—	2	4	no	—
77	C₂²×Dic₁₉	C₂×C₈	1	8	yes	—
78	C₂×C₈×C₁₉	—	2	4	no	—
79	C₂×C₈×C₁₉	—	2	4	no	—
80	C₂×C₈×C₁₉	—	2	4	no	—
81	C₂×C₈×C₁₉	—	4	2	no	—
82	C₂×C₈×C₁₉	—	4	2	no	—
83	C₂×C₈×C₁₉	—	4	2	no	—
84	C₂×C₈×C₁₉	—	4	2	no	—
85	C₄²×C₁₉	—	2	2	no	—
86	C₄²×C₁₉	—	2	2	no	—
87	C₄²×C₁₉	—	2	4	no	—
88	C₄×Dic₁₉	—	4	2	no	—
89	C₄×Dic₁₉	—	4	2	no	—
90	C₄×Dic₁₉	—	4	2	no	—
91	C₄×Dic₁₉	—	4	2	no	—
92	C₂×C₁₉⋊C₈	—	2	2	no	—
93	C₂×C₁₉⋊C₈	—	2	2	no	—
94	C₂×C₁₉⋊C₈	—	2	2	no	—
95	C₂×C₁₉⋊C₈	—	2	2	no	—
96	C₂×C₁₉⋊C₈	—	2	4	no	—
97	C₂×C₁₉⋊C₈	—	2	4	no	—
98	C₂×C₁₉⋊C₈	—	2	4	no	—
99	C₂×C₁₉⋊C₈	—	2	4	no	—
100	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	4	yes	—
101	C₂×C₁₉⋊C₈	C₄⋊C₄	1	4	yes	—
102	#10797	—	19	1	no	maximal, Sylow
103	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
104	C₂³×C₁₉	#117	1	2	yes	—
105	C₂³×C₁₉	C₄×D₄	1	2	yes	—
106	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
107	C₂³×C₁₉	C₄×Q₈	1	1	yes	characteristic
108	C₂³×C₁₉	C₂²×C₈	1	2	yes	—
109	C₂³×C₁₉	C₂×M₁₆	1	2	yes	—
110	C₂³×C₁₉	#131	1	2	yes	—
111	C₂³×C₁₉	#131	1	2	yes	—
112	C₂×C₄×C₁₉	—	2	2	no	—
113	C₂×C₄×C₁₉	—	2	2	no	—
114	C₂×C₄×C₁₉	—	2	2	no	—
115	C₂×C₄×C₁₉	—	2	2	no	—
116	C₂×C₄×C₁₉	—	2	8	no	—
117	C₂×C₄×C₁₉	—	2	8	no	—
118	C₂×C₄×C₁₉	C₄×C₈	1	4	yes	—
119	C₂×C₄×C₁₉	#97	1	4	yes	—
120	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
123	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
124	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
125	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
126	C₂×C₄×C₁₉	#117	1	2	yes	—
127	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#121	1	2	yes	—
131	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
132	C₂×C₄×C₁₉	#123	1	2	yes	—
133	C₂×C₄×C₁₉	#123	1	2	yes	—
134	C₂×C₄×C₁₉	#124	1	2	yes	—
135	C₂×C₄×C₁₉	#125	1	2	yes	—
136	C₂×C₄×C₁₉	#126	1	2	yes	—
137	C₂×C₄×C₁₉	#126	1	2	yes	—
138	C₂×Dic₁₉	—	2	8	no	—
139	C₂×Dic₁₉	—	2	8	no	—
140	C₂×Dic₁₉	—	2	8	no	—
141	C₂×Dic₁₉	—	4	4	no	—
142	C₂×Dic₁₉	—	4	4	no	—
143	C₂×Dic₁₉	—	4	4	no	—
144	C₂×Dic₁₉	—	4	4	no	—
145	C₈×C₁₉	—	4	4	no	—
146	C₁₉⋊C₈	—	2	4	no	—
147	C₁₉⋊C₈	—	2	4	no	—
148	C₂²×C₄×C₈	—	19	1	no	—
149	C₂×#336	—	19	2	no	—
150	C₂×#336	—	19	2	no	—
151	C₂×#336	D₁₉	1	1	yes	characteristic
152	C₄×#95	—	19	1	no	—
153	C₂²×C₁₉	#376	1	1	yes	characteristic
154	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
156	C₂²×C₁₉	#387	1	2	yes	—
157	C₂²×C₁₉	#388	1	2	yes	—
158	C₂²×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
159	C₂²×C₁₉	C₂×#97	1	1	yes	characteristic
160	C₂²×C₁₉	#405	1	1	yes	characteristic
161	C₂²×C₁₉	#405	1	1	yes	characteristic
162	C₂²×C₁₉	#431	1	4	yes	—
163	C₂²×C₁₉	#433	1	4	yes	—
164	C₂²×C₁₉	C₈×D₄	1	4	yes	—
165	C₂²×C₁₉	C₈×D₄	1	2	yes	—
166	C₂²×C₁₉	#435	1	4	yes	—
167	C₂²×C₁₉	#436	1	2	yes	—
168	C₂²×C₁₉	C₈×Q₈	1	2	yes	—
169	C₂²×C₁₉	#446	1	2	yes	—
170	C₄×C₁₉	—	2	4	no	—
171	C₄×C₁₉	—	2	4	no	—
172	C₄×C₁₉	#380	1	1	yes	characteristic
173	C₄×C₁₉	#381	1	1	yes	characteristic
174	C₄×C₁₉	#382	1	1	yes	characteristic
175	C₄×C₁₉	#383	1	1	yes	characteristic
176	C₄×C₁₉	#387	1	2	yes	—
177	C₄×C₁₉	#388	1	2	yes	—
178	Dic₁₉	—	4	8	no	—
179	C₂³×C₈	—	19	1	no	—
180	C₂³×C₈	—	19	1	no	—
181	C₂³×C₈	C₂×D₁₉	1	2	yes	—
182	C₂²×C₄²	—	19	2	no	—
183	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
184	C₂×C₄×C₈	—	19	4	no	—
185	C₂×C₄×C₈	—	38	4	no	—
186	C₂×#95	—	19	2	no	—
187	C₂×#95	—	19	2	no	—
188	#336	—	38	4	no	—
189	#336	—	38	4	no	—
190	#336	Dic₁₉	1	4	yes	complemented
191	C₂×C₁₉	#1434	1	1	yes	characteristic
192	C₂×C₁₉	#1744	1	1	yes	characteristic
193	C₂×C₁₉	#1746	1	1	yes	characteristic
194	C₂×C₁₉	C₈×C₂²⋊C₄	1	1	yes	characteristic
195	C₂×C₁₉	#1752	1	1	yes	characteristic
196	C₂×C₁₉	C₈×C₄⋊C₄	1	1	yes	characteristic
197	C₂×C₁₉	#1770	1	1	yes	characteristic
198	C₂×C₁₉	#1916	1	4	yes	—
199	C₂×C₁₉	#1923	1	4	yes	—
200	C₂³×C₄	—	19	4	no	—
201	C₂³×C₄	C₄×D₁₉	1	1	yes	characteristic
202	C₂³×C₄	C₄×D₁₉	1	1	yes	characteristic
203	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
204	C₂²×C₈	—	2	2	no	—
205	C₂²×C₈	—	2	2	no	—
206	C₂²×C₈	—	2	2	no	—
207	C₂²×C₈	—	2	2	no	—
208	C₂²×C₈	—	2	2	no	—
209	C₂²×C₈	—	2	2	no	—
210	C₂²×C₈	—	19	2	no	—
211	C₂²×C₈	—	19	2	no	—
212	C₂²×C₈	—	19	2	no	—
213	C₂²×C₈	—	19	4	no	—
214	C₂²×C₈	—	19	2	no	—
215	C₂²×C₈	—	38	2	no	—
216	C₂²×C₈	—	38	2	no	—
217	C₂²×C₈	—	38	2	no	—
218	C₂²×C₈	—	38	1	no	—
219	C₂²×C₈	—	38	1	no	—
220	C₂²×C₈	C₂×Dic₁₉	1	4	yes	—
221	C₂×C₄²	—	2	2	no	—
222	C₂×C₄²	—	2	2	no	—
223	C₂×C₄²	—	38	2	no	—
224	C₂×C₄²	—	38	2	no	—
225	C₂×C₄²	—	38	2	no	—
226	C₂×C₄²	—	38	2	no	—
227	C₂×C₄²	—	38	2	no	—
228	C₂×C₄²	—	38	2	no	—
229	C₂×C₄²	C₄×D₁₉	1	2	yes	—
230	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
231	C₄×C₈	—	38	8	no	—
232	#95	—	19	8	no	—
233	#95	—	38	4	no	—
234	C₁₉	#10797	1	1	yes	characteristic, complemented, Sylow
235	C₂⁴	C₂×C₄×D₁₉	1	1	yes	characteristic
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	19	8	no	—
241	C₂²×C₄	—	38	4	no	—
242	C₂²×C₄	—	38	4	no	—
243	C₂²×C₄	—	38	4	no	—
244	C₂²×C₄	—	38	4	no	—
245	C₂²×C₄	—	38	4	no	—
246	C₂²×C₄	—	38	4	no	—
247	C₂²×C₄	C₈×D₁₉	1	4	yes	—
248	C₂²×C₄	#64515	1	4	yes	—
249	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
250	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
251	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
252	C₂²×C₄	#64541	1	2	yes	—
253	C₂²×C₄	#64541	1	2	yes	—
254	C₂²×C₄	D₄×D₁₉	1	2	yes	—
255	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
256	C₂²×C₄	#64543	1	2	yes	—
257	C₂²×C₄	#64543	1	1	yes	characteristic
258	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
259	C₂²×C₄	#64545	1	1	yes	characteristic
260	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	—	4	2	no	—
265	C₂×C₈	—	4	2	no	—
266	C₂×C₈	—	4	2	no	—
267	C₂×C₈	—	4	2	no	—
268	C₂×C₈	—	19	4	no	—
269	C₂×C₈	—	19	4	no	—
270	C₂×C₈	—	38	2	no	—
271	C₂×C₈	—	38	2	no	—
272	C₂×C₈	—	38	2	no	—
273	C₂×C₈	—	38	4	no	—
274	C₂×C₈	—	38	4	no	—
275	C₂×C₈	—	38	2	no	—
276	C₂×C₈	—	38	4	no	—
277	C₂×C₈	—	38	4	no	—
278	C₄²	—	2	4	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	76	2	no	—
282	C₄²	—	76	2	no	—
283	C₄²	—	76	2	no	—
284	C₄²	—	76	2	no	—
285	C₂³	#10597318	1	2	yes	—
286	C₂³	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
287	C₂³	#10597333	1	1	yes	characteristic
288	C₂³	#10597339	1	1	yes	characteristic
289	C₂³	C₂×C₈×D₁₉	1	2	yes	—
290	C₂³	C₂×#64515	1	2	yes	—
291	C₂³	#10597353	1	2	yes	—
292	C₂³	#10597353	1	2	yes	—
293	C₂³	C₂×C₄×Dic₁₉	1	1	yes	characteristic
294	C₂×C₄	—	2	8	no	—
295	C₂×C₄	—	2	8	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	38	8	no	—
301	C₂×C₄	—	38	8	no	—
302	C₂×C₄	—	38	8	no	—
303	C₂×C₄	—	76	4	no	—
304	C₂×C₄	—	76	4	no	—
305	C₂×C₄	—	76	4	no	—
306	C₂×C₄	—	76	4	no	—
307	C₂×C₄	C₈×Dic₁₉	1	4	yes	—
308	C₂×C₄	#10597272	1	4	yes	—
309	C₂×C₄	#10597318	1	1	yes	characteristic
310	C₂×C₄	#10597318	1	1	yes	characteristic
311	C₂×C₄	#10597323	1	1	yes	characteristic
312	C₂×C₄	#10597325	1	2	yes	—
313	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
314	C₂×C₄	#10597329	1	2	yes	—
315	C₂×C₄	#10597330	1	2	yes	—
316	C₂×C₄	#10597331	1	2	yes	—
317	C₂×C₄	#10597335	1	2	yes	—
318	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
319	C₂×C₄	#10597338	1	1	yes	characteristic
320	C₂×C₄	#10597340	1	2	yes	—
321	C₂×C₄	#10597342	1	2	yes	—
322	C₂×C₄	#10597344	1	2	yes	—
323	C₂×C₄	#10597372	1	2	yes	—
324	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
325	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
326	C₂×C₄	Q₈×Dic₁₉	1	1	yes	characteristic
327	C₈	—	4	4	no	—
328	C₈	—	38	4	no	—
329	C₈	—	38	4	no	—
330	C₂²	#11926390	1	1	yes	characteristic
331	C₂²	C₂²⋊C₄×Dic₁₉	1	1	yes	characteristic
332	C₂²	C₄⋊C₄×Dic₁₉	1	1	yes	characteristic
333	C₂²	#11926426	1	2	yes	—
334	C₂²	#11926433	1	2	yes	—
335	C₂²	C₂×C₈×Dic₁₉	1	1	yes	characteristic
336	C₂²	C₂×#10597272	1	1	yes	characteristic
337	C₂²	#11926502	1	1	yes	characteristic
338	C₂²	#11926502	1	1	yes	characteristic
339	C₂²	#11926595	1	4	yes	—
340	C₂²	#11926606	1	4	yes	—
341	C₂²	(C₂²⋊D₁₉)⋊C₈	1	4	yes	—
342	C₂²	#11926609	1	2	yes	—
343	C₂²	#11926612	1	4	yes	—
344	C₂²	#11926621	1	2	yes	—
345	C₂²	#11926668	1	2	yes	—
346	C₂²	#11926671	1	2	yes	—
347	C₄	—	2	4	no	—
348	C₄	—	2	4	no	—
349	C₄	—	76	8	no	—
350	C₄	#11926403	1	1	yes	characteristic
351	C₄	#11926406	1	1	yes	characteristic
352	C₄	#11926408	1	1	yes	characteristic
353	C₄	#11926411	1	1	yes	characteristic
354	C₄	#11926430	1	2	yes	—
355	C₄	#11926437	1	2	yes	—
356	C₂	#14561860	1	1	yes	characteristic
357	C₂	#14563380	1	1	yes	characteristic
358	C₂	#14563388	1	1	yes	characteristic
359	C₂	#14563409	1	1	yes	characteristic
360	C₂	#14563414	1	1	yes	characteristic
361	C₂	#14563500	1	1	yes	characteristic
362	C₂	#14563505	1	1	yes	characteristic
363	C₂	#14564190	1	4	yes	—
364	C₂	#14564239	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₁₉
commutator	C₂²×C₁₉	C₂×C₄×C₈
Fitting	C₂×C₁₉×#336	C₂
socle, prime radical	C₂⁴×C₁₉	C₂²×C₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#336	D₁₉
hypocenter	C₁₉	#10797
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×#336

Complements

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

N	H ≅ G/N	type
10. C₂×C₄×C₁₉⋊C₈	C₄	semidirect
17. C₁₉×#336	C₄	semidirect
50. C₁₉⋊#95	C₈	semidirect
190. #336	Dic₁₉	semidirect
234. C₁₉	#10797	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₁₉⋊C₈	2
C₂×#11926298	2
C₄×C₁₉⋊#95	2
C₂×C₁₉×#336	2
#10797	19