Subgroups — #18073923

Order 4864.

836 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563615	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊(C₁₉⋊C₈)	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₉×#533	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
6	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
7	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
8	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
9	#11926299	—	2	1	no	—
10	#11926299	—	2	1	no	—
11	Q₈⋊(C₁₉⋊C₈)	—	2	8	no	—
12	C₁₉×#533	—	2	8	no	—
13	C₂×C₁₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
14	C₂×C₁₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
15	C₂×C₁₉×#125	—	2	1	no	—
16	C₂×C₁₉×C₂²⋊Q₈	—	2	1	no	—
17	C₂×C₁₉×C₂²⋊Q₈	C₄	1	2	yes	—
18	C₂×C₄×Q₈×C₁₉	—	2	1	no	—
19	C₂×C₄×Q₈×C₁₉	C₂²	1	2	yes	—
20	C₂×C₁₉×#117	—	2	1	no	—
21	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
22	C₂²×Q₈×C₁₉	—	2	1	no	—
23	C₂²×Q₈×C₁₉	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
27	C₁₉×C₄⋊Q₈	—	2	4	no	—
28	C₁₉×C₄⋊Q₈	—	2	4	no	—
29	C₁₉×#125	—	2	8	no	—
30	C₁₉×C₂²⋊Q₈	—	2	8	no	—
31	C₁₉×C₂²⋊Q₈	—	2	4	no	—
32	C₁₉×C₂²⋊Q₈	—	2	4	no	—
33	C₄×Q₈×C₁₉	—	2	8	no	—
34	C₄×Q₈×C₁₉	—	2	4	no	—
35	C₄×Q₈×C₁₉	D₄	1	8	yes	—
36	C₁₉×#117	—	2	8	no	—
37	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
38	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
39	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
40	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
41	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
43	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
44	C₂×C₄²×C₁₉	—	2	2	no	—
45	C₂×C₄²×C₁₉	—	2	4	no	—
46	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
50	C₂²×C₁₉⋊C₈	—	2	2	no	—
51	C₂²×C₁₉⋊C₈	—	2	2	no	—
52	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
53	C₂²⋊(C₁₉⋊C₈)	—	4	2	no	—
54	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
55	C₄×C₁₉⋊C₈	—	2	4	no	—
56	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×Q₈×C₁₉	—	2	4	no	—
58	C₂×Q₈×C₁₉	—	2	2	no	—
59	C₂×Q₈×C₁₉	—	2	4	no	—
60	C₂×Q₈×C₁₉	—	2	4	no	—
61	C₂×Q₈×C₁₉	—	4	2	no	—
62	C₂×Q₈×C₁₉	—	4	2	no	—
63	C₂×Q₈×C₁₉	C₂²⋊C₄	1	4	yes	—
64	C₂²×C₄×C₁₉	—	2	2	no	—
65	C₂²×C₄×C₁₉	—	2	2	no	—
66	C₂²×C₄×C₁₉	—	2	2	no	—
67	C₂²×C₄×C₁₉	—	2	2	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	2	no	—
73	C₂²×C₄×C₁₉	—	4	1	no	—
74	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
78	C₂²×C₄×C₁₉	Q₁₆	1	2	yes	—
79	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
82	C₄⋊C₄×C₁₉	—	2	4	no	—
83	C₄⋊C₄×C₁₉	—	2	4	no	—
84	C₄⋊C₄×C₁₉	—	2	8	no	—
85	C₄⋊C₄×C₁₉	—	2	4	no	—
86	C₄⋊C₄×C₁₉	—	2	2	no	—
87	C₄⋊C₄×C₁₉	—	2	8	no	—
88	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
89	C₂²⋊C₄×C₁₉	—	2	8	no	—
90	C₄²×C₁₉	—	2	2	no	—
91	C₄²×C₁₉	—	2	2	no	—
92	C₄²×C₁₉	—	2	2	no	—
93	C₄²×C₁₉	—	2	8	no	—
94	C₄²×C₁₉	C₂×D₄	1	4	yes	—
95	C₂×C₁₉⋊C₈	—	2	4	no	—
96	C₂×C₁₉⋊C₈	—	4	2	no	—
97	C₂×C₁₉⋊C₈	—	4	2	no	—
98	C₂×C₁₉⋊C₈	—	4	2	no	—
99	C₂×C₁₉⋊C₈	—	4	2	no	—
100	C₂×C₁₉⋊C₈	—	4	2	no	—
101	#8314	—	19	1	no	maximal, Sylow
102	C₂³×C₁₉	—	2	1	no	—
103	C₂³×C₁₉	—	2	1	no	—
104	C₂³×C₁₉	—	2	1	no	—
105	C₂³×C₁₉	—	2	1	no	—
106	C₂³×C₁₉	—	2	1	no	—
107	C₂³×C₁₉	—	2	1	no	—
108	C₂³×C₁₉	#103	1	2	yes	—
109	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	Q₈×C₁₉	—	2	8	no	—
111	Q₈×C₁₉	—	4	8	no	—
112	Q₈×C₁₉	—	4	4	no	—
113	C₂×C₄×C₁₉	—	2	1	no	—
114	C₂×C₄×C₁₉	—	2	1	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	2	1	no	—
118	C₂×C₄×C₁₉	—	2	1	no	—
119	C₂×C₄×C₁₉	—	2	4	no	—
120	C₂×C₄×C₁₉	—	2	4	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	2	4	no	—
123	C₂×C₄×C₁₉	—	2	4	no	—
124	C₂×C₄×C₁₉	—	2	1	no	—
125	C₂×C₄×C₁₉	—	2	1	no	—
126	C₂×C₄×C₁₉	—	4	4	no	—
127	C₂×C₄×C₁₉	—	4	2	no	—
128	C₂×C₄×C₁₉	—	4	2	no	—
129	C₂×C₄×C₁₉	—	4	2	no	—
130	C₂×C₄×C₁₉	—	4	2	no	—
131	C₂×C₄×C₁₉	—	4	4	no	—
132	C₂×C₄×C₁₉	#103	1	2	yes	—
133	C₂×C₄×C₁₉	#104	1	2	yes	—
134	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
135	C₂×C₄×C₁₉	#120	1	2	yes	—
136	C₂×C₄×C₁₉	#120	1	2	yes	—
137	C₂×C₄×C₁₉	#131	1	2	yes	—
138	C₂×C₄×C₁₉	#131	1	2	yes	—
139	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
140	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
141	C₂×C₄×C₁₉	#137	1	2	yes	—
142	C₁₉⋊C₈	—	4	4	no	—
143	C₁₉⋊C₈	—	8	2	no	—
144	C₂×#533	D₁₉	1	1	yes	characteristic
145	#1797	—	19	1	no	—
146	C₄×C₂²⋊C₈	—	19	1	no	—
147	C₂×Q₈⋊C₈	—	19	4	no	—
148	C₂²×C₁₉	—	2	2	no	—
149	C₂²×C₁₉	—	2	2	no	—
150	C₂²×C₁₉	—	2	2	no	—
151	C₂²×C₁₉	—	4	1	no	—
152	C₂²×C₁₉	—	4	1	no	—
153	C₂²×C₁₉	—	4	1	no	—
154	C₂²×C₁₉	—	4	1	no	—
155	C₂²×C₁₉	#379	1	1	yes	characteristic
156	C₂²×C₁₉	#408	1	1	yes	characteristic
157	C₂²×C₁₉	#408	1	1	yes	characteristic
158	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
159	C₂²×C₁₉	#419	1	1	yes	characteristic
160	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
161	C₂²×C₁₉	#421	1	1	yes	characteristic
162	C₄×C₁₉	—	2	2	no	—
163	C₄×C₁₉	—	2	2	no	—
164	C₄×C₁₉	—	2	2	no	—
165	C₄×C₁₉	—	4	1	no	—
166	C₄×C₁₉	—	4	4	no	—
167	C₄×C₁₉	—	4	4	no	—
168	C₄×C₁₉	#408	1	4	yes	—
169	C₄×C₁₉	#448	1	2	yes	—
170	C₄×C₁₉	#451	1	2	yes	—
171	#533	—	2	8	no	—
172	C₂×C₄⋊Q₈	Dic₁₉	1	1	yes	characteristic
173	C₂×C₄⋊Q₈	Dic₁₉	1	1	yes	characteristic
174	C₂×#125	—	2	1	no	—
175	C₂×C₂²⋊Q₈	—	2	1	no	—
176	C₂×C₂²⋊Q₈	Dic₁₉	1	2	yes	—
177	C₂×C₄×Q₈	—	2	1	no	—
178	C₂×C₄×Q₈	C₂×D₁₉	1	2	yes	—
179	C₂×#117	—	2	1	no	—
180	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
181	C₂×C₄⋊C₈	—	19	2	no	—
182	C₂×C₂²⋊C₈	—	38	1	no	—
183	C₂×C₂²⋊C₈	—	38	1	no	—
184	C₂×C₄×C₈	—	19	2	no	—
185	#336	—	38	1	no	—
186	#336	—	38	1	no	—
187	Q₈⋊C₈	—	38	8	no	—
188	C₂×C₁₉	—	4	2	no	—
189	C₂×C₁₉	#1481	1	1	yes	characteristic
190	C₂×C₁₉	#1485	1	1	yes	characteristic
191	C₂×C₁₉	#1493	1	1	yes	characteristic
192	C₂×C₁₉	#1497	1	1	yes	characteristic
193	C₂×C₁₉	#1780	1	1	yes	characteristic
194	C₂×C₁₉	#1787	1	1	yes	characteristic
195	C₂×C₁₉	#1789	1	1	yes	characteristic
196	C₂²×Q₈	—	2	1	no	—
197	C₂²×Q₈	C₂×Dic₁₉	1	2	yes	—
198	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
199	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
200	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
201	C₂²×C₈	—	38	2	no	—
202	C₂²×C₈	—	38	2	no	—
203	C₄⋊Q₈	—	2	4	no	—
204	C₄⋊Q₈	—	2	4	no	—
205	#125	—	2	8	no	—
206	C₂²⋊Q₈	—	2	8	no	—
207	C₂²⋊Q₈	—	2	4	no	—
208	C₂²⋊Q₈	—	2	4	no	—
209	C₄×Q₈	—	2	8	no	—
210	C₄×Q₈	—	2	4	no	—
211	C₄×Q₈	C₂²⋊D₁₉	1	8	yes	—
212	#117	—	2	8	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	—	2	1	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
219	C₂×C₂²⋊C₄	—	2	2	no	—
220	C₂×C₄²	—	2	2	no	—
221	C₂×C₄²	—	2	4	no	—
222	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
223	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
224	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
225	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
226	C₄⋊C₈	—	38	4	no	—
227	C₂²⋊C₈	—	38	4	no	—
228	C₂²⋊C₈	—	76	2	no	—
229	C₄×C₈	—	38	4	no	—
230	C₁₉	#8314	1	1	yes	characteristic, complemented, Sylow
231	C₂⁴	C₂²⋊Dic₁₉	1	1	yes	characteristic
232	C₂×Q₈	—	2	4	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	2	4	no	—
235	C₂×Q₈	—	2	4	no	—
236	C₂×Q₈	—	4	2	no	—
237	C₂×Q₈	—	4	2	no	—
238	C₂×Q₈	C₂²⋊Dic₁₉	1	4	yes	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	#64527	1	2	yes	—
250	C₂²×C₄	#64528	1	2	yes	—
251	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
252	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
253	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
254	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
257	C₂×C₈	—	38	4	no	—
258	C₂×C₈	—	76	2	no	—
259	C₂×C₈	—	76	2	no	—
260	C₂×C₈	—	76	2	no	—
261	C₂×C₈	—	76	2	no	—
262	C₂×C₈	—	76	2	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	8	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	8	no	—
269	C₄⋊C₄	C₂²⋊Dic₁₉	1	4	yes	—
270	C₂²⋊C₄	—	2	8	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	2	8	no	—
275	C₄²	C₂×C₂²⋊D₁₉	1	4	yes	—
276	C₂³	—	2	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	—	2	1	no	—
279	C₂³	—	2	1	no	—
280	C₂³	—	2	1	no	—
281	C₂³	—	2	1	no	—
282	C₂³	#10597292	1	2	yes	—
283	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
284	Q₈	—	2	8	no	—
285	Q₈	—	4	8	no	—
286	Q₈	—	4	4	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	#10597292	1	2	yes	—
307	C₂×C₄	#10597294	1	2	yes	—
308	C₂×C₄	C₂×#64527	1	1	yes	characteristic
309	C₂×C₄	#10597388	1	2	yes	—
310	C₂×C₄	C₂×#64528	1	1	yes	characteristic
311	C₂×C₄	#10597394	1	2	yes	—
312	C₂×C₄	#10597394	1	2	yes	—
313	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	2	yes	—
314	C₂×C₄	#10597399	1	2	yes	—
315	C₂×C₄	#10597399	1	2	yes	—
316	C₈	—	76	4	no	—
317	C₈	—	152	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	#11926400	1	1	yes	characteristic
326	C₂²	#11926511	1	1	yes	characteristic
327	C₂²	#11926511	1	1	yes	characteristic
328	C₂²	C₂×#10597292	1	1	yes	characteristic
329	C₂²	#11926553	1	1	yes	characteristic
330	C₂²	C₂×#10597294	1	1	yes	characteristic
331	C₂²	#11926561	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	2	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	4	no	—
337	C₄	—	4	4	no	—
338	C₄	#11926511	1	4	yes	—
339	C₄	#11926683	1	2	yes	—
340	C₄	#11926700	1	2	yes	—
341	C₂	—	4	2	no	—
342	C₂	#14562038	1	1	yes	characteristic
343	C₂	#14562056	1	1	yes	characteristic
344	C₂	#14562090	1	1	yes	characteristic
345	C₂	#14562108	1	1	yes	characteristic
346	C₂	#14563547	1	1	yes	characteristic
347	C₂	#14563570	1	1	yes	characteristic
348	C₂	#14563579	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#533	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#533	D₁₉
hypocenter	C₁₉	#8314
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
3. C₂×Q₈⋊(C₁₉⋊C₈)	C₂	semidirect
230. C₁₉	#8314	semidirect

Maximal subgroups

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

subgroup	index
#14563615	2
C₄×C₂²⋊(C₁₉⋊C₈)	2
C₂×Q₈⋊(C₁₉⋊C₈)	2
C₂×C₁₉×#533	2
#8314	19