Subgroups — #18088415

Order 4864.

918 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#64520	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926414	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926312	C₂	1	4	yes	maximal
4	C₂×C₁₉×#423	C₂	1	1	yes	maximal, characteristic
5	C₂²×#64520	C₂²	1	2	yes	—
6	C₂²×C₄⋊Dic₁₉	C₄	1	4	yes	—
7	C₂²×C₄⋊Dic₁₉	C₂²	1	2	yes	—
8	C₄×#64520	—	2	2	no	—
9	C₄×#64520	C₄	1	4	yes	complemented
10	C₂×#10597260	—	2	1	no	—
11	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
12	#11926414	—	2	1	no	—
13	#11926414	—	2	1	no	—
14	#11926414	—	2	2	no	—
15	#11926312	—	2	8	no	—
16	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
17	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#423	—	2	2	no	—
19	C₁₉×#423	C₄	1	4	yes	complemented
20	C₂×C₁₉×C₄⋊C₈	—	2	2	no	—
21	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
23	C₂³×Dic₁₉	C₂×C₄	1	4	yes	—
24	C₂×M₁₆×C₁₉	—	2	2	no	—
25	C₂×M₁₆×C₁₉	—	2	4	no	—
26	C₂×M₁₆×C₁₉	—	2	4	no	—
27	C₂×M₁₆×C₁₉	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₁₉	—	2	2	no	—
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	1	yes	characteristic
33	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₉	D₄	1	2	yes	—
35	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
36	C₂×C₄⋊Dic₁₉	—	2	4	no	—
37	C₂×C₄⋊Dic₁₉	—	2	4	no	—
38	C₂×C₄⋊Dic₁₉	—	2	4	no	—
39	C₂×C₄⋊Dic₁₉	—	2	2	no	—
40	C₂×C₄⋊Dic₁₉	—	2	2	no	—
41	C₂×C₄⋊Dic₁₉	—	2	4	no	—
42	C₂×C₄⋊Dic₁₉	—	4	8	no	—
43	C₂×#64520	—	2	2	no	—
44	C₂×#64520	—	2	2	no	—
45	C₂×#64520	—	2	2	no	—
46	C₂×#64520	C₂×C₄	1	4	yes	—
47	C₂×#64520	D₄	1	4	yes	—
48	C₂×#64520	Q₈	1	4	yes	—
49	C₂²×C₁₉⋊C₈	—	2	2	no	—
50	C₁₉×C₄⋊C₈	—	2	8	no	—
51	#10597260	—	2	4	no	—
52	C₄×C₁₉⋊C₈	—	2	4	no	—
53	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	—	2	4	no	—
55	C₂²×C₄×C₁₉	—	2	4	no	—
56	C₂²×C₄×C₁₉	C₄²	1	4	yes	—
57	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
60	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂×D₄	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×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₁₉	#41	1	2	yes	—
69	C₂²×C₄×C₁₉	#41	1	2	yes	—
70	C₂²×Dic₁₉	—	2	4	no	—
71	C₂²×Dic₁₉	—	2	8	no	—
72	C₂²×Dic₁₉	—	4	8	no	—
73	M₁₆×C₁₉	—	2	8	no	—
74	M₁₆×C₁₉	—	4	4	no	—
75	C₂×C₈×C₁₉	—	2	4	no	—
76	C₂×C₈×C₁₉	—	4	2	no	—
77	C₂×C₈×C₁₉	—	4	2	no	—
78	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
79	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
80	C₄²×C₁₉	C₂²⋊C₄	1	4	yes	—
81	C₄²×C₁₉	C₄⋊C₄	1	4	yes	—
82	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
83	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
84	C₄⋊Dic₁₉	—	4	8	no	—
85	C₄⋊Dic₁₉	—	4	8	no	—
86	C₄⋊Dic₁₉	—	4	4	no	—
87	C₄⋊Dic₁₉	—	4	2	no	—
88	C₄⋊Dic₁₉	—	4	2	no	—
89	#64520	—	2	4	no	—
90	#64520	—	2	4	no	—
91	#64520	—	2	4	no	—
92	#64520	C₄⋊C₄	1	8	yes	complemented
93	C₂×C₁₉⋊C₈	—	2	4	no	—
94	C₂×C₁₉⋊C₈	—	2	4	no	—
95	C₂×C₁₉⋊C₈	—	2	4	no	—
96	#10857	—	19	1	no	maximal, Sylow
97	C₂³×C₁₉	—	2	2	no	—
98	C₂³×C₁₉	—	2	2	no	—
99	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
100	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂³×C₁₉	C₄×D₄	1	2	yes	—
103	C₂³×C₁₉	C₄×Q₈	1	2	yes	—
104	C₂×C₄×C₁₉	—	2	2	no	—
105	C₂×C₄×C₁₉	—	2	2	no	—
106	C₂×C₄×C₁₉	—	2	8	no	—
107	C₂×C₄×C₁₉	—	2	8	no	—
108	C₂×C₄×C₁₉	#95	1	4	yes	—
109	C₂×C₄×C₁₉	#95	1	4	yes	—
110	C₂×C₄×C₁₉	#100	1	4	yes	—
111	C₂×C₄×C₁₉	#101	1	4	yes	—
112	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₉	#117	1	2	yes	—
118	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
120	C₂×C₄×C₁₉	#121	1	2	yes	—
121	C₂×C₄×C₁₉	#121	1	2	yes	—
122	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
124	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
125	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
126	C₂×C₄×C₁₉	#125	1	2	yes	—
127	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
128	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
129	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
130	C₂×Dic₁₉	—	4	4	no	—
131	C₂×Dic₁₉	—	4	8	no	—
132	C₂×Dic₁₉	—	8	8	no	—
133	C₈×C₁₉	—	4	4	no	—
134	C₁₉⋊C₈	—	2	8	no	—
135	C₂×#423	D₁₉	1	1	yes	characteristic
136	C₂×C₄×M₁₆	—	19	1	no	—
137	C₂×#384	—	19	1	no	—
138	C₂×#343	—	19	4	no	—
139	C₂²×C₁₉	—	2	2	no	—
140	C₂²×C₁₉	—	2	2	no	—
141	C₂²×C₁₉	—	2	2	no	—
142	C₂²×C₁₉	—	2	2	no	—
143	C₂²×C₁₉	—	2	4	no	—
144	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
145	C₂²×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
146	C₂²×C₁₉	#389	1	2	yes	—
147	C₂²×C₁₉	#390	1	1	yes	characteristic
148	C₂²×C₁₉	#391	1	1	yes	characteristic
149	C₂²×C₁₉	C₂×#100	1	2	yes	—
150	C₂²×C₁₉	C₂×#101	1	2	yes	—
151	C₄×C₁₉	—	2	4	no	—
152	C₄×C₁₉	#343	1	8	yes	complemented
153	C₄×C₁₉	#380	1	2	yes	—
154	C₄×C₁₉	#382	1	1	yes	characteristic
155	C₄×C₁₉	#384	1	1	yes	characteristic
156	C₄×C₁₉	#389	1	2	yes	—
157	C₄×C₁₉	#390	1	1	yes	characteristic
158	C₄×C₁₉	#391	1	1	yes	characteristic
159	Dic₁₉	—	8	4	no	—
160	C₂²×M₁₆	—	19	2	no	—
161	C₂²×M₁₆	C₂×D₁₉	1	2	yes	—
162	C₂²×C₄⋊C₄	—	19	2	no	—
163	C₂²×C₄⋊C₄	—	19	4	no	—
164	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
165	#423	—	2	2	no	—
166	#423	Dic₁₉	1	4	yes	complemented
167	C₂×C₄⋊C₈	—	2	2	no	—
168	C₄×M₁₆	—	19	4	no	—
169	C₄×M₁₆	—	38	2	no	—
170	C₂×#97	—	38	1	no	—
171	C₂×C₄×C₈	—	38	1	no	—
172	#384	—	38	1	no	—
173	#384	—	38	1	no	—
174	#384	—	38	2	no	—
175	#343	—	38	8	no	—
176	C₂×C₁₉	—	2	2	no	—
177	C₂×C₁₉	—	2	2	no	—
178	C₂×C₁₉	C₄⋊#95	1	1	yes	characteristic
179	C₂×C₁₉	C₂×#343	1	2	yes	—
180	C₂×C₁₉	#1965	1	1	yes	characteristic
181	C₂×C₁₉	#1966	1	1	yes	characteristic
182	C₂×C₁₉	#1997	1	2	yes	—
183	C₂³×C₄	—	19	4	no	—
184	C₂³×C₄	C₄×D₁₉	1	2	yes	—
185	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	4	no	—
188	C₂×M₁₆	—	2	4	no	—
189	C₂×M₁₆	—	19	4	no	—
190	C₂×M₁₆	—	19	4	no	—
191	C₂×M₁₆	—	19	4	no	—
192	C₂×M₁₆	—	38	2	no	—
193	C₂×M₁₆	—	38	2	no	—
194	C₂×M₁₆	—	38	2	no	—
195	C₂×M₁₆	C₂×Dic₁₉	1	4	yes	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	38	2	no	—
198	C₂×C₄⋊C₄	—	38	4	no	—
199	C₂×C₄⋊C₄	—	38	4	no	—
200	C₂×C₄⋊C₄	—	38	2	no	—
201	C₂×C₄⋊C₄	—	38	4	no	—
202	C₂×C₄⋊C₄	—	38	4	no	—
203	C₂×C₄⋊C₄	—	38	2	no	—
204	C₂×C₄⋊C₄	—	76	8	no	—
205	C₂×C₄²	Dic₃₈	1	2	yes	—
206	C₂×C₄²	C₄×D₁₉	1	2	yes	—
207	C₂×C₄²	D₇₆	1	1	yes	characteristic
208	C₂×C₄²	D₇₆	1	1	yes	characteristic
209	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
210	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
211	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
212	C₄⋊C₈	—	2	8	no	—
213	#97	—	38	4	no	—
214	C₄×C₈	—	38	4	no	—
215	C₁₉	#10857	1	1	yes	characteristic, complemented, Sylow
216	C₂⁴	C₂×C₄×D₁₉	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	38	4	no	—
220	C₂²×C₄	—	38	8	no	—
221	C₂²×C₄	—	76	8	no	—
222	C₂²×C₄	C₄×Dic₁₉	1	4	yes	—
223	C₂²×C₄	Dic₁₉⋊C₄	1	4	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	M₁₆	—	2	8	no	—
237	M₁₆	—	4	4	no	—
238	M₁₆	—	19	8	no	—
239	M₁₆	—	38	4	no	—
240	M₁₆	—	38	4	no	—
241	M₁₆	—	38	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	4	2	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	38	4	no	—
246	C₂×C₈	—	38	4	no	—
247	C₂×C₈	—	38	4	no	—
248	C₄⋊C₄	—	76	8	no	—
249	C₄⋊C₄	—	76	2	no	—
250	C₄⋊C₄	—	76	2	no	—
251	C₄⋊C₄	—	76	8	no	—
252	C₄⋊C₄	—	76	4	no	—
253	C₄²	Dic₁₉⋊C₄	1	4	yes	—
254	C₄²	C₄⋊Dic₁₉	1	2	yes	—
255	C₄²	C₄⋊Dic₁₉	1	2	yes	—
256	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
257	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
258	C₄²	C₂²⋊Dic₁₉	1	4	yes	—
259	C₂³	—	2	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	#10597333	1	2	yes	—
262	C₂³	#10597339	1	2	yes	—
263	C₂³	C₂×C₄×Dic₁₉	1	1	yes	characteristic
264	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
265	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	76	4	no	—
271	C₂×C₄	—	76	8	no	—
272	C₂×C₄	—	152	8	no	—
273	C₂×C₄	#10597280	1	4	yes	—
274	C₂×C₄	#10597281	1	4	yes	—
275	C₂×C₄	C₁₉⋊#95	1	4	yes	—
276	C₂×C₄	C₁₉⋊#95	1	4	yes	—
277	C₂×C₄	#10597334	1	2	yes	—
278	C₂×C₄	#10597336	1	2	yes	—
279	C₂×C₄	C₄⋊C₄×D₁₉	1	2	yes	—
280	C₂×C₄	#10597338	1	2	yes	—
281	C₂×C₄	#10597341	1	2	yes	—
282	C₂×C₄	#10597343	1	2	yes	—
283	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
284	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
285	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
286	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
287	C₂×C₄	#10597382	1	1	yes	characteristic
288	C₂×C₄	#10597384	1	2	yes	—
289	C₂×C₄	#10597386	1	1	yes	characteristic
290	C₂×C₄	#10597390	1	1	yes	characteristic
291	C₂×C₄	Q₈×Dic₁₉	1	2	yes	—
292	C₂×C₄	#10597392	1	2	yes	—
293	C₂×C₄	#10597393	1	1	yes	characteristic
294	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
295	C₈	—	4	4	no	—
296	C₈	—	38	8	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	2	no	—
302	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
303	C₂²	C₄⋊C₄×Dic₁₉	1	2	yes	—
304	C₂²	#11926440	1	2	yes	—
305	C₂²	#11926445	1	1	yes	characteristic
306	C₂²	#11926449	1	1	yes	characteristic
307	C₂²	C₂×#10597280	1	2	yes	—
308	C₂²	C₂×#10597281	1	2	yes	—
309	C₄	—	2	4	no	—
310	C₄	—	152	4	no	—
311	C₄	#11926312	1	8	yes	complemented
312	C₄	#11926401	1	2	yes	—
313	C₄	#11926409	1	1	yes	characteristic
314	C₄	#11926413	1	1	yes	characteristic
315	C₄	#11926443	1	2	yes	—
316	C₄	#11926447	1	1	yes	characteristic
317	C₄	#11926451	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	2	2	no	—
320	C₂	#14561864	1	1	yes	characteristic
321	C₂	C₂×#11926312	1	2	yes	—
322	C₂	#14564474	1	1	yes	characteristic
323	C₂	#14564478	1	1	yes	characteristic
324	C₂	#14564623	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×(C₂×D₁₉)⋊C₄
commutator	C₂³×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×#423	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂²×C₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#423	D₁₉
hypocenter	C₁₉	#10857
LCS	G ▷ C₂³×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
9. C₄×#64520	C₄	semidirect
19. C₁₉×#423	C₄	semidirect
92. #64520	C₄⋊C₄	semidirect
152. C₄×C₁₉	#343	semidirect
166. #423	Dic₁₉	semidirect
215. C₁₉	#10857	semidirect
311. C₄	#11926312	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#64520	2
C₂×#11926414	2
C₂×#11926312	2
C₂×C₁₉×#423	2
#10857	19