Subgroups — #18072094

Order 4864.

798 conjugacy classes of subgroups.

The subgroup lattice has 332 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	#14566042	C₂	1	1	yes	maximal, characteristic
3	#14561474	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₉×#423	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₁₉⋊C₈	—	2	1	no	—
6	C₂×#10597372	—	2	1	no	—
7	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
8	C₂×C₄×C₁₉⋊C₈	—	2	2	no	—
9	C₂×C₄×C₁₉⋊C₈	C₄	1	4	yes	complemented
10	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
11	#11926414	C₄	1	2	yes	—
12	#11926410	—	2	1	no	—
13	#11926404	—	2	1	no	—
14	#11926404	C₄	1	2	yes	—
15	C₄×C₄⋊Dic₁₉	—	2	1	no	—
16	C₄×C₄⋊Dic₁₉	C₂²	1	2	yes	—
17	C₂²×M₁₆×C₁₉	—	2	1	no	—
18	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
19	C₁₉×#423	—	2	2	no	—
20	C₁₉×#423	C₄	1	4	yes	complemented
21	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
22	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
26	C₂×M₁₆×C₁₉	—	2	4	no	—
27	C₂×M₁₆×C₁₉	—	4	2	no	—
28	C₂×M₁₆×C₁₉	—	4	2	no	—
29	C₂²×C₈×C₁₉	—	2	2	no	—
30	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₉	D₄	1	2	yes	—
36	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
37	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
38	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
39	#10597372	—	4	2	no	—
40	#10597372	—	4	2	no	—
41	C₂×C₄⋊Dic₁₉	—	2	2	no	—
42	C₂×C₄⋊Dic₁₉	—	2	2	no	—
43	C₂×C₄⋊Dic₁₉	C₂×C₄	1	4	yes	—
44	C₂×C₄×Dic₁₉	—	2	2	no	—
45	C₂²×C₁₉⋊C₈	—	2	8	no	—
46	C₂²×C₁₉⋊C₈	—	2	4	no	—
47	C₂²×C₁₉⋊C₈	—	2	2	no	—
48	C₁₉×C₄⋊C₈	—	2	4	no	—
49	C₁₉×C₄⋊C₈	—	2	2	no	—
50	C₁₉×C₄⋊C₈	C₂×C₄	1	4	yes	—
51	C₁₉⋊#95	—	2	2	no	—
52	C₁₉⋊#95	—	2	2	no	—
53	C₄×C₁₉⋊C₈	—	2	4	no	—
54	C₄×C₁₉⋊C₈	—	2	2	no	—
55	C₄×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
56	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	—	2	4	no	—
58	C₂²×C₄×C₁₉	—	2	4	no	—
59	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
60	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₁₉	D₈	1	2	yes	—
69	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
70	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
71	C₂²×C₄×C₁₉	Q₁₆	1	2	yes	—
72	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×Dic₁₉	—	2	4	no	—
76	M₁₆×C₁₉	—	4	8	no	—
77	C₂×C₈×C₁₉	—	2	4	no	—
78	C₂×C₈×C₁₉	—	4	2	no	—
79	C₂×C₈×C₁₉	—	4	2	no	—
80	C₄²×C₁₉	C₄²	1	2	yes	—
81	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
84	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
85	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
86	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
90	C₂²⋊Dic₁₉	—	4	4	no	—
91	C₄⋊Dic₁₉	—	2	8	no	—
92	C₄⋊Dic₁₉	—	4	4	no	—
93	C₄⋊Dic₁₉	—	4	4	no	—
94	C₄×Dic₁₉	—	4	2	no	—
95	C₄×Dic₁₉	—	4	2	no	—
96	C₂×C₁₉⋊C₈	—	2	8	no	—
97	C₂×C₁₉⋊C₈	—	2	16	no	—
98	C₂×C₁₉⋊C₈	—	2	4	no	—
99	#7897	—	19	1	no	maximal, Sylow
100	C₂³×C₁₉	—	2	2	no	—
101	C₂³×C₁₉	—	2	2	no	—
102	C₂³×C₁₉	#95	1	2	yes	—
103	C₂³×C₁₉	#102	1	2	yes	—
104	C₂³×C₁₉	#103	1	2	yes	—
105	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₉	—	2	2	no	—
107	C₂×C₄×C₁₉	—	2	2	no	—
108	C₂×C₄×C₁₉	—	2	4	no	—
109	C₂×C₄×C₁₉	—	2	4	no	—
110	C₂×C₄×C₁₉	—	2	4	no	—
111	C₂×C₄×C₁₉	—	2	4	no	—
112	C₂×C₄×C₁₉	#95	1	2	yes	—
113	C₂×C₄×C₁₉	#95	1	2	yes	—
114	C₂×C₄×C₁₉	#95	1	2	yes	—
115	C₂×C₄×C₁₉	#102	1	2	yes	—
116	C₂×C₄×C₁₉	#102	1	4	yes	—
117	C₂×C₄×C₁₉	#103	1	2	yes	—
118	C₂×C₄×C₁₉	#103	1	4	yes	—
119	C₂×C₄×C₁₉	#104	1	4	yes	—
120	C₂×C₄×C₁₉	#106	1	4	yes	—
121	C₂×C₄×C₁₉	#107	1	4	yes	—
122	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₉	#131	1	4	yes	—
129	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
131	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
133	C₂×Dic₁₉	—	4	4	no	—
134	C₂×Dic₁₉	—	4	4	no	—
135	C₂×Dic₁₉	—	4	4	no	—
136	C₈×C₁₉	—	4	4	no	—
137	C₁₉⋊C₈	—	2	8	no	—
138	C₂×#423	D₁₉	1	1	yes	characteristic
139	C₂²×C₄×C₈	—	19	1	no	—
140	#2285	—	19	1	no	—
141	#1279	—	19	4	no	—
142	C₂²×C₁₉	—	2	2	no	—
143	C₂²×C₁₉	—	2	2	no	—
144	C₂²×C₁₉	—	2	4	no	—
145	C₂²×C₁₉	—	2	4	no	—
146	C₂²×C₁₉	#340	1	4	yes	—
147	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
148	C₂²×C₁₉	#408	1	2	yes	—
149	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
150	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
151	C₂²×C₁₉	C₂×#104	1	2	yes	—
152	C₄×C₁₉	—	2	2	no	—
153	C₄×C₁₉	—	2	1	no	—
154	C₄×C₁₉	—	2	1	no	—
155	C₄×C₁₉	#339	1	4	yes	—
156	C₄×C₁₉	#340	1	4	yes	—
157	C₄×C₁₉	#405	1	2	yes	—
158	C₄×C₁₉	C₂×#102	1	1	yes	characteristic
159	C₄×C₁₉	C₂×#103	1	1	yes	characteristic
160	C₄×C₁₉	#424	1	2	yes	—
161	C₄×C₁₉	C₂×#106	1	1	yes	characteristic
162	C₄×C₁₉	C₂×#107	1	1	yes	characteristic
163	Dic₁₉	—	8	4	no	—
164	C₂²×M₁₆	—	2	1	no	—
165	C₂³×C₈	—	38	1	no	—
166	C₂×#117	—	38	1	no	—
167	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
168	#423	—	2	2	no	—
169	#423	Dic₁₉	1	4	yes	complemented
170	C₂×C₄⋊C₈	—	2	1	no	—
171	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
172	C₂×C₄×C₈	—	19	4	no	—
173	C₂×C₄×C₈	—	19	2	no	—
174	C₂×C₄×C₈	—	38	1	no	—
175	C₂×C₄×C₈	—	38	2	no	—
176	#384	—	19	2	no	—
177	#382	—	38	1	no	—
178	#380	—	19	2	no	—
179	#380	—	38	1	no	—
180	C₄×C₄⋊C₄	—	19	2	no	—
181	C₄×C₄⋊C₄	—	38	1	no	—
182	C₂×C₁₉	—	2	4	no	—
183	C₂×C₁₉	#1492	1	2	yes	—
184	C₂×C₁₉	#1493	1	2	yes	—
185	C₂×C₁₉	#1729	1	1	yes	characteristic
186	C₂×C₁₉	C₂×#339	1	1	yes	characteristic
187	C₂×C₁₉	C₂×#340	1	1	yes	characteristic
188	C₂³×C₄	C₄×D₁₉	1	1	yes	characteristic
189	C₂³×C₄	D₇₆	1	1	yes	characteristic
190	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
191	C₂×M₁₆	—	2	4	no	—
192	C₂×M₁₆	—	4	2	no	—
193	C₂×M₁₆	—	4	2	no	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	—	38	8	no	—
196	C₂²×C₈	—	38	4	no	—
197	C₂²×C₈	—	38	2	no	—
198	#117	—	76	2	no	—
199	#117	—	76	2	no	—
200	C₂×C₄⋊C₄	—	19	4	no	—
201	C₂×C₄⋊C₄	—	38	2	no	—
202	C₂×C₄⋊C₄	—	38	2	no	—
203	C₂×C₂²⋊C₄	—	38	2	no	—
204	C₂×C₄²	—	38	2	no	—
205	C₂×C₄²	Dic₃₈	1	2	yes	—
206	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
207	C₂×C₄²	C₄×D₁₉	1	2	yes	—
208	C₂×C₄²	D₇₆	1	1	yes	characteristic
209	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
210	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
211	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
212	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
213	C₄⋊C₈	—	2	4	no	—
214	C₄⋊C₈	—	2	2	no	—
215	C₄⋊C₈	C₂×Dic₁₉	1	4	yes	—
216	C₄×C₈	—	19	4	no	—
217	C₄×C₈	—	38	4	no	—
218	C₄×C₈	—	38	2	no	—
219	#95	—	38	2	no	—
220	#95	—	38	2	no	—
221	C₁₉	#7897	1	1	yes	characteristic, complemented, Sylow
222	C₂⁴	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	38	4	no	—
226	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
227	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
228	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
229	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
230	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
231	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
232	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
233	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
234	C₂²×C₄	#64525	1	2	yes	—
235	C₂²×C₄	#64526	1	2	yes	—
236	C₂²×C₄	#64527	1	2	yes	—
237	C₂²×C₄	#64528	1	2	yes	—
238	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
239	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
242	M₁₆	—	4	8	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	4	2	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	38	8	no	—
247	C₂×C₈	—	38	16	no	—
248	C₂×C₈	—	38	4	no	—
249	C₄⋊C₄	—	38	8	no	—
250	C₄⋊C₄	—	76	4	no	—
251	C₄⋊C₄	—	76	4	no	—
252	C₂²⋊C₄	—	76	4	no	—
253	C₄²	—	76	2	no	—
254	C₄²	—	76	2	no	—
255	C₄²	C₄×Dic₁₉	1	2	yes	—
256	C₄²	Dic₁₉⋊C₄	1	2	yes	—
257	C₄²	Dic₁₉⋊C₄	1	2	yes	—
258	C₄²	C₄⋊Dic₁₉	1	2	yes	—
259	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
260	C₄²	C₂²⋊Dic₁₉	1	2	yes	—
261	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
262	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
263	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
264	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	#10597266	1	2	yes	—
268	C₂³	#10597267	1	2	yes	—
269	C₂³	C₁₉⋊#95	1	2	yes	—
270	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	76	4	no	—
278	C₂×C₄	—	76	4	no	—
279	C₂×C₄	—	76	4	no	—
280	C₂×C₄	#10597264	1	4	yes	—
281	C₂×C₄	#10597265	1	4	yes	—
282	C₂×C₄	#10597266	1	2	yes	—
283	C₂×C₄	#10597267	1	2	yes	—
284	C₂×C₄	#10597282	1	4	yes	—
285	C₂×C₄	C₁₉⋊#95	1	2	yes	—
286	C₂×C₄	C₁₉⋊#95	1	2	yes	—
287	C₂×C₄	C₁₉⋊#95	1	2	yes	—
288	C₂×C₄	#10597289	1	4	yes	—
289	C₂×C₄	#10597292	1	4	yes	—
290	C₂×C₄	#10597353	1	4	yes	—
291	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
292	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
293	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
294	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
295	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
296	C₂×C₄	C₂×#64525	1	1	yes	characteristic
297	C₂×C₄	C₂×#64526	1	1	yes	characteristic
298	C₂×C₄	C₂×#64527	1	1	yes	characteristic
299	C₂×C₄	C₂×#64528	1	1	yes	characteristic
300	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
301	C₈	—	4	4	no	—
302	C₈	—	38	8	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	2	4	no	—
307	C₂²	#11926306	1	4	yes	—
308	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
309	C₂²	#11926512	1	2	yes	—
310	C₂²	C₂×#10597266	1	1	yes	characteristic
311	C₂²	C₂×#10597267	1	1	yes	characteristic
312	C₂²	C₂×#10597282	1	2	yes	—
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	152	4	no	—
317	C₄	#11926305	1	4	yes	—
318	C₄	#11926306	1	4	yes	—
319	C₄	#11926502	1	2	yes	—
320	C₄	C₂×#10597289	1	1	yes	characteristic
321	C₄	C₂×#10597292	1	1	yes	characteristic
322	C₄	#11926572	1	2	yes	—
323	C₄	C₂×#10597265	1	1	yes	characteristic
324	C₄	C₂×#10597264	1	1	yes	characteristic
325	C₂	—	2	4	no	—
326	C₂	#14562085	1	2	yes	—
327	C₂	#14562089	1	2	yes	—
328	C₂	#14563333	1	1	yes	characteristic
329	C₂	C₂×#11926305	1	1	yes	characteristic
330	C₂	C₂×#11926306	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#423	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#423	D₁₉
hypocenter	C₁₉	#7897
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
3. #14561474	C₂	semidirect
9. C₂×C₄×C₁₉⋊C₈	C₄	semidirect
20. C₁₉×#423	C₄	semidirect
169. #423	Dic₁₉	semidirect
221. C₁₉	#7897	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
#14566042	2
#14561474	2
C₂×C₁₉×#423	2
#7897	19