Subgroups — #18088356

Order 4864.

1126 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×Dic₁₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926413	C₂	1	2	yes	maximal
3	C₂×#11926307	C₂	1	1	yes	maximal, characteristic
4	C₂×#11926306	C₂	1	2	yes	maximal
5	C₂×C₁₉×#340	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊Dic₁₉	C₂²	1	2	yes	—
8	C₂²×Dic₁₉⋊C₄	—	2	4	no	—
9	C₂²×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₁₉	—	2	4	no	—
11	C₂×C₈×Dic₁₉	C₄	1	4	yes	complemented
12	#11926413	—	2	4	no	—
13	#11926413	C₄	1	8	yes	complemented
14	#11926307	—	2	2	no	—
15	#11926306	—	2	4	no	—
16	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
18	C₁₉×#340	—	2	2	no	—
19	C₂³×C₄×C₁₉	—	2	2	no	—
20	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
21	C₂³×Dic₁₉	—	2	2	no	—
22	C₂³×Dic₁₉	D₄	1	2	yes	—
23	C₂²×C₈×C₁₉	—	2	2	no	—
24	C₂²×C₈×C₁₉	—	2	2	no	—
25	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₉	D₄	1	2	yes	—
27	C₂²×C₈×C₁₉	Q₈	1	2	yes	—
28	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
31	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
32	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
33	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
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	C₂×C₄⋊Dic₁₉	C₂×C₄	1	4	yes	—
40	C₂×Dic₁₉⋊C₄	—	2	16	no	—
41	C₂×Dic₁₉⋊C₄	—	4	8	no	—
42	C₂×Dic₁₉⋊C₄	—	4	8	no	—
43	C₂×C₄×Dic₁₉	C₂×C₄	1	2	yes	—
44	C₂×C₄×Dic₁₉	C₂×C₄	1	4	yes	—
45	C₂×C₄×Dic₁₉	D₄	1	4	yes	—
46	C₂×C₄×Dic₁₉	Q₈	1	2	yes	—
47	C₂²×C₁₉⋊C₈	—	2	2	no	—
48	C₂²×C₁₉⋊C₈	—	2	2	no	—
49	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
50	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
51	C₂²×C₁₉⋊C₈	Q₈	1	2	yes	—
52	C₈×Dic₁₉	—	2	8	no	—
53	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	—	2	4	no	—
55	C₂²×C₄×C₁₉	—	4	2	no	—
56	C₂²×C₄×C₁₉	—	4	2	no	—
57	C₂²×C₄×C₁₉	—	4	4	no	—
58	C₂²×C₄×C₁₉	—	4	4	no	—
59	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
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	—	4	2	no	—
71	C₂²×Dic₁₉	—	4	2	no	—
72	C₂²×Dic₁₉	—	4	4	no	—
73	C₂²×Dic₁₉	—	4	4	no	—
74	C₂²×Dic₁₉	C₂²⋊C₄	1	8	yes	—
75	C₂²×Dic₁₉	C₄⋊C₄	1	4	yes	—
76	C₂²×Dic₁₉	D₈	1	4	yes	—
77	C₂²×Dic₁₉	SD₁₆	1	8	yes	—
78	C₂²×Dic₁₉	Q₁₆	1	4	yes	—
79	C₂×C₈×C₁₉	—	2	2	no	—
80	C₂×C₈×C₁₉	—	2	2	no	—
81	C₂×C₈×C₁₉	—	2	4	no	—
82	C₂×C₈×C₁₉	—	2	4	no	—
83	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	complemented
84	C₄⋊C₄×C₁₉	—	2	4	no	—
85	C₄⋊C₄×C₁₉	—	2	4	no	—
86	C₄⋊C₄×C₁₉	—	4	2	no	—
87	C₄⋊C₄×C₁₉	—	4	2	no	—
88	C₄⋊Dic₁₉	—	2	4	no	—
89	C₄⋊Dic₁₉	—	2	4	no	—
90	C₄⋊Dic₁₉	—	4	2	no	—
91	C₄⋊Dic₁₉	—	4	2	no	—
92	Dic₁₉⋊C₄	—	4	32	no	—
93	C₄×Dic₁₉	C₄²	1	4	yes	—
94	C₄×Dic₁₉	C₂²⋊C₄	1	4	yes	—
95	C₄×Dic₁₉	C₄⋊C₄	1	8	yes	—
96	C₂×C₁₉⋊C₈	—	2	2	no	—
97	C₂×C₁₉⋊C₈	—	2	2	no	—
98	C₂×C₁₉⋊C₈	—	2	4	no	—
99	C₂×C₁₉⋊C₈	—	2	4	no	—
100	C₂×C₁₉⋊C₈	C₄⋊C₄	1	4	yes	complemented
101	#10841	—	19	1	no	maximal, Sylow
102	C₂³×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
103	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂³×C₁₉	#124	1	2	yes	—
105	C₂³×C₁₉	#127	1	1	yes	characteristic
106	C₂³×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
107	C₂³×C₁₉	C₂×D₈	1	2	yes	—
108	C₂³×C₁₉	C₂×SD₁₆	1	4	yes	—
109	C₂³×C₁₉	C₂×Q₁₆	1	2	yes	—
110	C₂×C₄×C₁₉	—	4	4	no	—
111	C₂×C₄×C₁₉	—	4	8	no	—
112	C₂×C₄×C₁₉	—	8	4	no	—
113	C₂×C₄×C₁₉	—	8	2	no	—
114	C₂×C₄×C₁₉	—	8	2	no	—
115	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
121	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
125	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
126	C₂×C₄×C₁₉	#121	1	2	yes	—
127	C₂×C₄×C₁₉	#121	1	2	yes	—
128	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
129	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
130	C₂×C₄×C₁₉	#125	1	2	yes	—
131	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
133	C₂×Dic₁₉	—	4	4	no	—
134	C₂×Dic₁₉	—	4	8	no	—
135	C₂×Dic₁₉	—	8	4	no	—
136	C₂×Dic₁₉	—	8	2	no	—
137	C₂×Dic₁₉	—	8	2	no	—
138	C₂×Dic₁₉	#95	1	8	yes	—
139	C₂×Dic₁₉	#102	1	16	yes	—
140	C₂×Dic₁₉	#103	1	16	yes	—
141	C₂×Dic₁₉	#106	1	8	yes	—
142	C₂×Dic₁₉	#107	1	8	yes	—
143	C₈×C₁₉	—	2	4	no	—
144	C₁₉⋊C₈	—	2	4	no	—
145	C₂²×C₄×C₈	—	19	1	no	—
146	C₂×#384	—	19	2	no	—
147	C₂×#340	—	19	1	no	—
148	C₂×#340	—	19	2	no	—
149	C₂×#340	D₁₉	1	1	yes	characteristic
150	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
151	C₂²×C₁₉	#382	1	1	yes	characteristic
152	C₂²×C₁₉	#384	1	1	yes	characteristic
153	C₂²×C₁₉	#390	1	2	yes	—
154	C₂²×C₁₉	#391	1	2	yes	—
155	C₂²×C₁₉	C₂×#102	1	4	yes	—
156	C₂²×C₁₉	C₂×#103	1	4	yes	—
157	C₂²×C₁₉	C₂×#106	1	2	yes	—
158	C₂²×C₁₉	C₂×#107	1	2	yes	—
159	C₂²×C₁₉	#486	1	4	yes	—
160	C₂²×C₁₉	#487	1	4	yes	—
161	C₂²×C₁₉	#492	1	2	yes	—
162	C₂²×C₁₉	#493	1	1	yes	characteristic
163	C₂²×C₁₉	#494	1	1	yes	characteristic
164	C₂²×C₁₉	#498	1	2	yes	—
165	C₂²×C₁₉	#500	1	2	yes	—
166	C₄×C₁₉	—	8	4	no	—
167	C₄×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
168	C₄×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
169	C₄×C₁₉	#380	1	1	yes	characteristic
170	C₄×C₁₉	#380	1	1	yes	characteristic
171	C₄×C₁₉	#389	1	2	yes	—
172	C₄×C₁₉	#389	1	2	yes	—
173	Dic₁₉	—	8	4	no	—
174	Dic₁₉	#340	1	16	yes	complemented
175	C₂³×C₈	—	19	1	no	—
176	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
177	C₂²×C₄⋊C₄	—	19	2	no	—
178	C₂²×C₄⋊C₄	—	38	4	no	—
179	C₂²×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
180	C₂²×C₄²	—	19	1	no	—
181	C₂×C₄×C₈	—	19	4	no	—
182	C₂×C₄×C₈	—	38	4	no	—
183	#384	—	19	8	no	—
184	#384	—	38	4	no	—
185	#340	—	2	2	no	—
186	#340	—	38	4	no	—
187	#340	—	38	2	no	—
188	C₂×C₁₉	C₄⋊#95	1	1	yes	characteristic
189	C₂×C₁₉	C₂×#340	1	2	yes	—
190	C₂×C₁₉	#1834	1	2	yes	—
191	C₂×C₁₉	#1844	1	1	yes	characteristic
192	C₂×C₁₉	#1845	1	1	yes	characteristic
193	C₂×C₁₉	#1956	1	2	yes	—
194	C₂×C₁₉	#1957	1	2	yes	—
195	C₂×C₁₉	#1990	1	4	yes	—
196	C₂³×C₄	—	2	2	no	—
197	C₂³×C₄	—	19	2	no	—
198	C₂³×C₄	—	38	2	no	—
199	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
200	C₂²×C₈	—	2	2	no	—
201	C₂²×C₈	—	2	2	no	—
202	C₂²×C₈	—	19	2	no	—
203	C₂²×C₈	—	19	2	no	—
204	C₂²×C₈	—	19	2	no	—
205	C₂²×C₈	—	38	2	no	—
206	C₂²×C₈	—	38	2	no	—
207	C₂²×C₈	Dic₃₈	1	2	yes	—
208	C₂²×C₈	C₄×D₁₉	1	2	yes	—
209	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	19	4	no	—
216	C₂×C₄⋊C₄	—	38	2	no	—
217	C₂×C₄⋊C₄	—	38	2	no	—
218	C₂×C₄⋊C₄	—	38	2	no	—
219	C₂×C₄⋊C₄	—	38	2	no	—
220	C₂×C₄⋊C₄	—	38	16	no	—
221	C₂×C₄⋊C₄	—	38	2	no	—
222	C₂×C₄⋊C₄	—	76	8	no	—
223	C₂×C₄⋊C₄	—	76	8	no	—
224	C₂×C₄⋊C₄	C₄×D₁₉	1	4	yes	—
225	C₂×C₄²	—	19	4	no	—
226	C₂×C₄²	—	19	2	no	—
227	C₂×C₄²	—	19	4	no	—
228	C₂×C₄²	—	19	2	no	—
229	C₄×C₈	—	38	8	no	—
230	C₁₉	#10841	1	1	yes	characteristic, complemented, Sylow
231	C₂⁴	D₄×D₁₉	1	1	yes	characteristic
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	—	4	4	no	—
236	C₂²×C₄	—	4	4	no	—
237	C₂²×C₄	—	19	8	no	—
238	C₂²×C₄	—	19	4	no	—
239	C₂²×C₄	—	19	4	no	—
240	C₂²×C₄	—	19	8	no	—
241	C₂²×C₄	—	19	4	no	—
242	C₂²×C₄	—	38	4	no	—
243	C₂²×C₄	—	76	2	no	—
244	C₂²×C₄	—	76	2	no	—
245	C₂²×C₄	—	76	4	no	—
246	C₂²×C₄	—	76	4	no	—
247	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
249	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
250	C₂²×C₄	#64541	1	2	yes	—
251	C₂²×C₄	D₄×D₁₉	1	2	yes	—
252	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
253	C₂²×C₄	#64543	1	2	yes	—
254	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
255	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	19	4	no	—
262	C₂×C₈	—	38	2	no	—
263	C₂×C₈	—	38	2	no	—
264	C₂×C₈	—	38	4	no	—
265	C₂×C₈	—	38	4	no	—
266	C₂×C₈	Dic₁₉⋊C₄	1	4	yes	complemented
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	38	4	no	—
272	C₄⋊C₄	—	38	4	no	—
273	C₄⋊C₄	—	76	32	no	—
274	C₄⋊C₄	—	76	2	no	—
275	C₄⋊C₄	—	76	2	no	—
276	C₄²	—	19	4	no	—
277	C₄²	—	19	4	no	—
278	C₄²	—	19	8	no	—
279	C₂³	C₂²⋊C₄×D₁₉	1	2	yes	—
280	C₂³	#10597331	1	2	yes	—
281	C₂³	#10597334	1	1	yes	characteristic
282	C₂³	C₄⋊C₄×D₁₉	1	1	yes	characteristic
283	C₂³	D₈×D₁₉	1	2	yes	—
284	C₂³	SD₁₆×D₁₉	1	4	yes	—
285	C₂³	Q₁₆×D₁₉	1	2	yes	—
286	C₂³	#10597386	1	1	yes	characteristic
287	C₂×C₄	—	4	8	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	8	4	no	—
290	C₂×C₄	—	8	2	no	—
291	C₂×C₄	—	8	2	no	—
292	C₂×C₄	—	19	16	no	—
293	C₂×C₄	—	19	16	no	—
294	C₂×C₄	—	19	8	no	—
295	C₂×C₄	—	19	8	no	—
296	C₂×C₄	—	19	8	no	—
297	C₂×C₄	—	76	8	no	—
298	C₂×C₄	—	76	4	no	—
299	C₂×C₄	—	152	4	no	—
300	C₂×C₄	—	152	2	no	—
301	C₂×C₄	—	152	2	no	—
302	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
303	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
304	C₂×C₄	#10597318	1	1	yes	characteristic
305	C₂×C₄	#10597323	1	1	yes	characteristic
306	C₂×C₄	#10597324	1	2	yes	—
307	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
308	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
309	C₂×C₄	#10597330	1	2	yes	—
310	C₂×C₄	#10597333	1	2	yes	—
311	C₂×C₄	#10597334	1	1	yes	characteristic
312	C₂×C₄	#10597335	1	2	yes	—
313	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
314	C₂×C₄	C₄⋊C₄×D₁₉	1	2	yes	—
315	C₂×C₄	#10597342	1	2	yes	—
316	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
317	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
318	C₂×C₄	#10597385	1	2	yes	—
319	C₂×C₄	#10597390	1	1	yes	characteristic
320	C₈	—	2	4	no	—
321	C₈	—	38	4	no	—
322	C₂²	#95×D₁₉	1	1	yes	characteristic
323	C₂²	#11926408	1	1	yes	characteristic
324	C₂²	#11926413	1	1	yes	characteristic
325	C₂²	#11926444	1	2	yes	—
326	C₂²	#11926448	1	2	yes	—
327	C₂²	#102×D₁₉	1	4	yes	—
328	C₂²	#103×D₁₉	1	4	yes	—
329	C₂²	#106×D₁₉	1	2	yes	—
330	C₂²	#107×D₁₉	1	2	yes	—
331	C₂²	#11926889	1	4	yes	—
332	C₂²	#11926893	1	4	yes	—
333	C₂²	#11926915	1	2	yes	—
334	C₂²	#11926919	1	1	yes	characteristic
335	C₂²	#11926922	1	1	yes	characteristic
336	C₂²	#11926941	1	2	yes	—
337	C₂²	#11926948	1	2	yes	—
338	C₄	—	8	4	no	—
339	C₄	—	19	16	no	—
340	C₄	—	152	4	no	—
341	C₄	#11926395	1	1	yes	characteristic
342	C₄	C₄×Dic₁₉⋊C₄	1	1	yes	characteristic
343	C₄	#11926402	1	1	yes	characteristic
344	C₄	#11926403	1	1	yes	characteristic
345	C₄	#11926439	1	2	yes	—
346	C₄	#11926441	1	2	yes	—
347	C₂	Dic₁₉⋊#95	1	1	yes	characteristic
348	C₂	D₁₉×#340	1	2	yes	—
349	C₂	#14563761	1	2	yes	—
350	C₂	#14563790	1	1	yes	characteristic
351	C₂	#14563793	1	1	yes	characteristic
352	C₂	#14564435	1	2	yes	—
353	C₂	#14564439	1	2	yes	—
354	C₂	#14564594	1	4	yes	—

	subgroup	G/H
center	C₂⁴	D₄×D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×#340	C₂
socle, prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#340	D₁₉
hypocenter	C₁₉	#10841
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂×#340

Complements

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

N	H ≅ G/N	type
11. C₂×C₈×Dic₁₉	C₄	semidirect
13. #11926413	C₄	semidirect
83. C₂×C₈×C₁₉	C₄⋊C₄	semidirect
100. C₂×C₁₉⋊C₈	C₄⋊C₄	semidirect
174. Dic₁₉	#340	semidirect
230. C₁₉	#10841	semidirect
266. C₂×C₈	Dic₁₉⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈×Dic₁₉	2
C₂×#11926413	2
C₂×#11926307	2
C₂×#11926306	2
C₂×C₁₉×#340	2
#10841	19