Subgroups — #18088328

Order 4864.

950 conjugacy classes of subgroups.

The subgroup lattice has 324 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×#11926390	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926298	C₂	1	4	yes	maximal
4	C₂²×C₁₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₁₉⋊C₈	C₂²	1	2	yes	—
6	C₂²×C₄×Dic₁₉	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
8	C₂×C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
9	C₂×C₄⋊(C₁₉⋊C₈)	C₄	1	4	yes	complemented
10	#11926390	—	2	2	no	—
11	#11926390	—	2	2	no	—
12	C₂×C₁₉⋊#95	C₄	1	2	yes	—
13	C₂×C₁₉⋊#95	C₄	1	2	yes	—
14	#11926298	—	2	8	no	—
15	C₂³×C₈×C₁₉	C₂²	1	2	yes	—
16	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₉×C₄⋊C₈	—	2	2	no	—
18	C₂×C₁₉×C₄⋊C₈	—	2	2	no	—
19	C₂×C₁₉×C₄⋊C₈	C₄	1	4	yes	complemented
20	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
23	C₂³×Dic₁₉	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₁₉	—	2	4	no	—
25	C₂²×C₈×C₁₉	—	2	2	no	—
26	C₂²×C₈×C₁₉	—	2	2	no	—
27	C₂²×C₈×C₁₉	—	2	4	no	—
28	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₉	D₄	1	2	yes	—
32	C₂×C₄²×C₁₉	D₄	1	2	yes	—
33	C₂×C₄²×C₁₉	D₄	1	2	yes	—
34	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
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	C₂×C₄×Dic₁₉	D₄	1	4	yes	—
40	C₂×C₄×Dic₁₉	Q₈	1	4	yes	—
41	C₂²×C₁₉⋊C₈	—	2	4	no	—
42	C₂²×C₁₉⋊C₈	—	2	2	no	—
43	C₂²×C₁₉⋊C₈	—	2	2	no	—
44	C₂²×C₁₉⋊C₈	—	2	4	no	—
45	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
46	C₁₉×C₄⋊C₈	—	2	8	no	—
47	C₁₉⋊#95	—	2	4	no	—
48	C₁₉⋊#95	—	2	4	no	—
49	C₄⋊(C₁₉⋊C₈)	—	2	8	no	—
50	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	—	2	4	no	—
52	C₂²×C₄×C₁₉	—	2	4	no	—
53	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
54	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
55	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₁₉	#41	1	2	yes	—
68	C₂²×C₄×C₁₉	#41	1	2	yes	—
69	C₂²×Dic₁₉	—	2	4	no	—
70	C₂²×Dic₁₉	—	2	4	no	—
71	C₂²×Dic₁₉	—	2	8	no	—
72	C₂²×Dic₁₉	—	2	8	no	—
73	C₂²×Dic₁₉	C₄⋊C₄	1	8	yes	—
74	C₂×C₈×C₁₉	—	2	4	no	—
75	C₂×C₈×C₁₉	—	2	8	no	—
76	C₂×C₈×C₁₉	—	4	4	no	—
77	C₂×C₈×C₁₉	—	4	2	no	—
78	C₂×C₈×C₁₉	—	4	2	no	—
79	C₄²×C₁₉	C₂²⋊C₄	1	4	yes	—
80	C₄²×C₁₉	C₂²⋊C₄	1	4	yes	—
81	C₄²×C₁₉	C₄⋊C₄	1	4	yes	—
82	C₄²×C₁₉	C₄⋊C₄	1	4	yes	—
83	C₄×Dic₁₉	—	2	4	no	—
84	C₄×Dic₁₉	—	2	4	no	—
85	C₄×Dic₁₉	—	2	4	no	—
86	C₄×Dic₁₉	—	2	4	no	—
87	C₂×C₁₉⋊C₈	—	2	4	no	—
88	C₂×C₁₉⋊C₈	—	2	8	no	—
89	C₂×C₁₉⋊C₈	—	4	4	no	—
90	C₂×C₁₉⋊C₈	—	4	2	no	—
91	C₂×C₁₉⋊C₈	—	4	2	no	—
92	#10833	—	19	1	no	maximal, Sylow
93	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
94	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂³×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
96	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂³×C₁₉	#117	1	2	yes	—
98	C₂³×C₁₉	#131	1	4	yes	—
99	C₂³×C₁₉	#131	1	4	yes	—
100	C₂×C₄×C₁₉	—	2	4	no	—
101	C₂×C₄×C₁₉	—	2	4	no	—
102	C₂×C₄×C₁₉	—	2	4	no	—
103	C₂×C₄×C₁₉	—	2	4	no	—
104	C₂×C₄×C₁₉	—	2	4	no	—
105	C₂×C₄×C₁₉	—	2	4	no	—
106	C₂×C₄×C₁₉	#95	1	4	yes	—
107	C₂×C₄×C₁₉	#95	1	4	yes	—
108	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
115	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
116	C₂×C₄×C₁₉	#121	1	2	yes	—
117	C₂×C₄×C₁₉	#121	1	2	yes	—
118	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
121	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
122	C₂×C₄×C₁₉	#125	1	2	yes	—
123	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
126	C₂×Dic₁₉	—	2	8	no	—
127	C₂×Dic₁₉	—	2	16	no	—
128	C₂×Dic₁₉	—	4	4	no	—
129	C₂×Dic₁₉	—	4	4	no	—
130	C₂×Dic₁₉	—	4	8	no	—
131	C₈×C₁₉	—	4	4	no	—
132	C₁₉⋊C₈	—	4	4	no	—
133	C₂²×C₄⋊C₈	—	19	1	no	—
134	C₂²×C₄⋊C₈	D₁₉	1	1	yes	characteristic
135	C₂×#376	—	19	1	no	—
136	C₂×#336	—	19	4	no	—
137	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
138	C₂²×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
139	C₂²×C₁₉	#380	1	2	yes	—
140	C₂²×C₁₉	#382	1	1	yes	characteristic
141	C₂²×C₁₉	#384	1	1	yes	characteristic
142	C₂²×C₁₉	#405	1	2	yes	—
143	C₂²×C₁₉	#405	1	2	yes	—
144	C₂²×C₁₉	#408	1	2	yes	—
145	C₂²×C₁₉	#408	1	2	yes	—
146	C₂²×C₁₉	#424	1	8	yes	—
147	C₂²×C₁₉	#424	1	2	yes	—
148	C₂²×C₁₉	#424	1	2	yes	—
149	C₂²×C₁₉	#433	1	8	yes	—
150	C₄×C₁₉	—	2	4	no	—
151	C₄×C₁₉	—	2	4	no	—
152	C₄×C₁₉	#389	1	2	yes	—
153	C₄×C₁₉	#389	1	2	yes	—
154	C₄×C₁₉	#390	1	1	yes	characteristic
155	C₄×C₁₉	#390	1	1	yes	characteristic
156	C₄×C₁₉	#391	1	1	yes	characteristic
157	C₄×C₁₉	#391	1	1	yes	characteristic
158	Dic₁₉	—	4	8	no	—
159	C₂³×C₈	—	19	2	no	—
160	C₂³×C₈	C₂×D₁₉	1	2	yes	—
161	C₂²×C₄²	—	19	2	no	—
162	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
163	C₂×C₄⋊C₈	—	2	2	no	—
164	C₂×C₄⋊C₈	—	2	2	no	—
165	C₂×C₄⋊C₈	—	19	4	no	—
166	C₂×C₄⋊C₈	—	38	2	no	—
167	C₂×C₄⋊C₈	—	38	2	no	—
168	C₂×C₄⋊C₈	Dic₁₉	1	4	yes	complemented
169	#376	—	38	2	no	—
170	#376	—	38	2	no	—
171	C₂×#95	—	19	2	no	—
172	C₂×#95	—	19	2	no	—
173	#336	—	38	8	no	—
174	C₂×C₁₉	C₄⋊#95	1	1	yes	characteristic
175	C₂×C₁₉	#1729	1	1	yes	characteristic
176	C₂×C₁₉	#1729	1	1	yes	characteristic
177	C₂×C₁₉	#1768	1	4	yes	—
178	C₂×C₁₉	#1798	1	4	yes	—
179	C₂×C₁₉	#1832	1	2	yes	—
180	C₂×C₁₉	#1843	1	2	yes	—
181	C₂³×C₄	—	19	4	no	—
182	C₂³×C₄	C₄×D₁₉	1	1	yes	characteristic
183	C₂³×C₄	C₄×D₁₉	1	1	yes	characteristic
184	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
185	C₂²×C₈	—	2	4	no	—
186	C₂²×C₈	—	2	4	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	19	4	no	—
190	C₂²×C₈	—	38	2	no	—
191	C₂²×C₈	—	38	4	no	—
192	C₂²×C₈	—	38	4	no	—
193	C₂²×C₈	—	38	2	no	—
194	C₂²×C₈	C₂×Dic₁₉	1	4	yes	—
195	C₂×C₄²	—	19	4	no	—
196	C₂×C₄²	—	19	4	no	—
197	C₂×C₄²	—	38	2	no	—
198	C₂×C₄²	—	38	2	no	—
199	C₂×C₄²	—	38	2	no	—
200	C₂×C₄²	—	38	2	no	—
201	C₂×C₄²	Dic₃₈	1	2	yes	—
202	C₂×C₄²	C₄×D₁₉	1	2	yes	—
203	C₂×C₄²	D₇₆	1	2	yes	—
204	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
205	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
206	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
207	C₄⋊C₈	—	2	8	no	—
208	C₄⋊C₈	—	38	8	no	—
209	#95	—	38	4	no	—
210	#95	—	38	4	no	—
211	C₁₉	#10833	1	1	yes	characteristic, complemented, Sylow
212	C₂⁴	C₂×C₄×D₁₉	1	1	yes	characteristic
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	19	8	no	—
216	C₂²×C₄	—	38	4	no	—
217	C₂²×C₄	—	38	8	no	—
218	C₂²×C₄	—	38	8	no	—
219	C₂²×C₄	—	38	4	no	—
220	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
221	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
222	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
223	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
224	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
225	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
226	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
229	C₂²×C₄	D₄×D₁₉	1	2	yes	—
230	C₂²×C₄	#64543	1	2	yes	—
231	C₂²×C₄	Q₈×D₁₉	1	2	yes	—
232	C₂²×C₄	#64545	1	2	yes	—
233	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
236	C₂×C₈	—	2	8	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	4	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	38	8	no	—
242	C₂×C₈	—	38	4	no	—
243	C₂×C₈	—	76	2	no	—
244	C₂×C₈	—	76	2	no	—
245	C₂×C₈	—	76	4	no	—
246	C₄²	—	38	4	no	—
247	C₄²	—	38	4	no	—
248	C₄²	—	38	4	no	—
249	C₄²	—	38	4	no	—
250	C₄²	Dic₁₉⋊C₄	1	4	yes	—
251	C₄²	C₄⋊Dic₁₉	1	4	yes	—
252	C₄²	(C₂×D₁₉)⋊C₄	1	4	yes	—
253	C₄²	C₂²⋊Dic₁₉	1	4	yes	—
254	C₂³	C₄⋊C₄×D₁₉	1	2	yes	—
255	C₂³	#10597338	1	2	yes	—
256	C₂³	#10597347	1	4	yes	—
257	C₂³	#10597353	1	4	yes	—
258	C₂³	C₂×C₄×Dic₁₉	1	1	yes	characteristic
259	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
260	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	38	16	no	—
268	C₂×C₄	—	38	8	no	—
269	C₂×C₄	—	76	8	no	—
270	C₂×C₄	—	76	4	no	—
271	C₂×C₄	—	76	4	no	—
272	C₂×C₄	C₁₉⋊#95	1	4	yes	—
273	C₂×C₄	C₁₉⋊#95	1	4	yes	—
274	C₂×C₄	#10597333	1	2	yes	—
275	C₂×C₄	#10597334	1	2	yes	—
276	C₂×C₄	#10597336	1	2	yes	—
277	C₂×C₄	#10597339	1	2	yes	—
278	C₂×C₄	#10597341	1	2	yes	—
279	C₂×C₄	#10597343	1	2	yes	—
280	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
281	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
282	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
283	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
284	C₂×C₄	#10597382	1	1	yes	characteristic
285	C₂×C₄	#10597384	1	2	yes	—
286	C₂×C₄	#10597386	1	1	yes	characteristic
287	C₂×C₄	#10597390	1	1	yes	characteristic
288	C₂×C₄	Q₈×Dic₁₉	1	2	yes	—
289	C₂×C₄	#10597392	1	2	yes	—
290	C₂×C₄	#10597393	1	1	yes	characteristic
291	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
292	C₈	—	4	4	no	—
293	C₈	—	76	4	no	—
294	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
295	C₂²	C₄⋊C₄×Dic₁₉	1	2	yes	—
296	C₂²	#11926401	1	2	yes	—
297	C₂²	#11926409	1	1	yes	characteristic
298	C₂²	#11926413	1	1	yes	characteristic
299	C₂²	#11926499	1	2	yes	—
300	C₂²	#11926502	1	2	yes	—
301	C₂²	#11926510	1	2	yes	—
302	C₂²	#11926512	1	2	yes	—
303	C₂²	#11926569	1	8	yes	—
304	C₂²	#11926570	1	2	yes	—
305	C₂²	#11926572	1	2	yes	—
306	C₂²	#11926604	1	8	yes	—
307	C₄	—	2	4	no	—
308	C₄	—	2	4	no	—
309	C₄	—	76	8	no	—
310	C₄	#11926440	1	2	yes	—
311	C₄	#11926443	1	2	yes	—
312	C₄	#11926445	1	1	yes	characteristic
313	C₄	#11926447	1	1	yes	characteristic
314	C₄	#11926449	1	1	yes	characteristic
315	C₄	#11926451	1	1	yes	characteristic
316	C₂	#14561864	1	1	yes	characteristic
317	C₂	#14563331	1	1	yes	characteristic
318	C₂	#14563333	1	1	yes	characteristic
319	C₂	#14563494	1	4	yes	—
320	C₂	#14563616	1	4	yes	—
321	C₂	#14563754	1	2	yes	—
322	C₂	#14563788	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₁₉
commutator	C₂³×C₁₉	C₂×C₄²
Fitting	C₂²×C₁₉×C₄⋊C₈	C₂
socle, prime radical	C₂⁴×C₁₉	C₂²×C₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂²×C₄⋊C₈	D₁₉
hypocenter	C₁₉	#10833
LCS	G ▷ C₂³×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
9. C₂×C₄⋊(C₁₉⋊C₈)	C₄	semidirect
19. C₂×C₁₉×C₄⋊C₈	C₄	semidirect
168. C₂×C₄⋊C₈	Dic₁₉	semidirect
211. C₁₉	#10833	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×#11926390	2
C₂×#11926298	2
C₂²×C₁₉×C₄⋊C₈	2
#10833	19