Subgroups — #18091269

Order 4864.

878 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	C₂×#11926299	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926299	C₂	1	4	yes	maximal
3	C₂×#11926299	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₄⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₁₉⋊C₈	C₂²	1	2	yes	—
6	C₂³×C₁₉⋊C₈	C₂²	1	2	yes	—
7	#11926299	—	2	8	no	—
8	#11926299	—	2	2	no	—
9	#11926299	—	2	2	no	—
10	C₂²×C₄⋊C₄×C₁₉	C₄	1	1	yes	characteristic
11	C₂²×C₄⋊C₄×C₁₉	C₄	1	1	yes	characteristic
12	C₂²×C₄²×C₁₉	C₂²	1	2	yes	—
13	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊C₄×C₁₉	—	2	2	no	—
15	C₄×C₄⋊C₄×C₁₉	—	2	2	no	—
16	C₂×C₁₉×#95	C₄	1	2	yes	—
17	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
23	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
24	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
25	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
26	C₂×C₄⋊C₄×C₁₉	C₈	1	4	yes	complemented
27	C₂×C₄²×C₁₉	—	2	4	no	—
28	C₂×C₄²×C₁₉	—	2	4	no	—
29	C₂×C₄²×C₁₉	—	2	2	no	—
30	C₂×C₄²×C₁₉	—	2	2	no	—
31	C₂×C₄²×C₁₉	—	2	4	no	—
32	C₂×C₄²×C₁₉	D₄	1	2	yes	—
33	C₂×C₄²×C₁₉	D₄	1	2	yes	—
34	C₂²×C₁₉⋊C₈	—	2	2	no	—
35	C₂²×C₁₉⋊C₈	—	2	4	no	—
36	C₂²×C₁₉⋊C₈	—	2	2	no	—
37	C₂²×C₁₉⋊C₈	—	2	2	no	—
38	C₂²×C₁₉⋊C₈	—	2	4	no	—
39	C₂²×C₁₉⋊C₈	—	2	2	no	—
40	C₂²×C₁₉⋊C₈	—	2	4	no	—
41	C₂²×C₁₉⋊C₈	—	2	4	no	—
42	C₂²×C₁₉⋊C₈	D₄	1	4	yes	—
43	C₂²×C₁₉⋊C₈	Q₈	1	4	yes	—
44	C₁₉×#95	—	2	4	no	—
45	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₉	—	2	2	no	—
47	C₂²×C₄×C₁₉	—	2	2	no	—
48	C₂²×C₄×C₁₉	—	2	2	no	—
49	C₂²×C₄×C₁₉	—	2	2	no	—
50	C₂²×C₄×C₁₉	—	2	2	no	—
51	C₂²×C₄×C₁₉	—	2	4	no	—
52	C₂²×C₄×C₁₉	—	2	4	no	—
53	C₂²×C₄×C₁₉	—	2	2	no	—
54	C₂²×C₄×C₁₉	—	2	2	no	—
55	C₂²×C₄×C₁₉	—	2	2	no	—
56	C₂²×C₄×C₁₉	—	2	2	no	—
57	C₂²×C₄×C₁₉	—	2	4	no	—
58	C₂²×C₄×C₁₉	—	2	4	no	—
59	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₁₉	C₂×C₈	1	4	yes	—
62	C₂²×C₄×C₁₉	C₂×C₈	1	2	yes	—
63	C₂²×C₄×C₁₉	M₁₆	1	4	yes	—
64	C₂²×C₄×C₁₉	M₁₆	1	2	yes	—
65	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₁₉	#41	1	4	yes	—
70	C₂²×C₄×C₁₉	#41	1	2	yes	—
71	C₂²×C₄×C₁₉	#41	1	2	yes	—
72	C₄⋊C₄×C₁₉	—	2	8	no	—
73	C₄⋊C₄×C₁₉	—	2	8	no	—
74	C₄²×C₁₉	—	2	4	no	—
75	C₄²×C₁₉	—	2	4	no	—
76	C₄²×C₁₉	—	4	4	no	—
77	C₄²×C₁₉	—	4	2	no	—
78	C₄²×C₁₉	—	4	2	no	—
79	C₂×C₁₉⋊C₈	—	2	4	no	—
80	C₂×C₁₉⋊C₈	—	2	4	no	—
81	C₂×C₁₉⋊C₈	—	2	8	no	—
82	C₂×C₁₉⋊C₈	—	2	8	no	—
83	C₂×C₁₉⋊C₈	—	4	4	no	—
84	C₂×C₁₉⋊C₈	—	4	2	no	—
85	C₂×C₁₉⋊C₈	—	4	2	no	—
86	C₂×C₁₉⋊C₈	—	4	2	no	—
87	C₂×C₁₉⋊C₈	—	4	4	no	—
88	C₂×C₁₉⋊C₈	—	4	2	no	—
89	#11346	—	19	1	no	maximal, Sylow
90	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂³×C₁₉	#117	1	2	yes	—
92	C₂³×C₁₉	C₄×D₄	1	2	yes	—
93	C₂³×C₁₉	C₄×Q₈	1	2	yes	—
94	C₂³×C₁₉	C₂²×C₈	1	1	yes	characteristic
95	C₂³×C₁₉	C₂×M₁₆	1	1	yes	characteristic
96	C₂³×C₁₉	C₂×M₁₆	1	2	yes	—
97	C₂³×C₁₉	#131	1	2	yes	—
98	C₂³×C₁₉	#131	1	2	yes	—
99	C₂×C₄×C₁₉	—	2	2	no	—
100	C₂×C₄×C₁₉	—	2	2	no	—
101	C₂×C₄×C₁₉	—	2	4	no	—
102	C₂×C₄×C₁₉	—	2	4	no	—
103	C₂×C₄×C₁₉	—	2	2	no	—
104	C₂×C₄×C₁₉	—	2	4	no	—
105	C₂×C₄×C₁₉	—	2	4	no	—
106	C₂×C₄×C₁₉	—	2	2	no	—
107	C₂×C₄×C₁₉	—	2	2	no	—
108	C₂×C₄×C₁₉	—	2	8	no	—
109	C₂×C₄×C₁₉	—	2	8	no	—
110	C₂×C₄×C₁₉	—	4	2	no	—
111	C₂×C₄×C₁₉	—	4	2	no	—
112	C₂×C₄×C₁₉	—	4	2	no	—
113	C₂×C₄×C₁₉	—	4	2	no	—
114	C₂×C₄×C₁₉	—	4	4	no	—
115	C₂×C₄×C₁₉	—	4	4	no	—
116	C₂×C₄×C₁₉	—	4	4	no	—
117	C₂×C₄×C₁₉	C₂²⋊C₈	1	4	yes	—
118	C₂×C₄×C₁₉	#121	1	2	yes	—
119	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₁₉	C₂²⋊Q₈	1	4	yes	—
121	C₂×C₄×C₁₉	#123	1	4	yes	—
122	C₂×C₄×C₁₉	#123	1	2	yes	—
123	C₂×C₄×C₁₉	#123	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	4	yes	—
127	C₂×C₄×C₁₉	#126	1	4	yes	—
128	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
129	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
130	C₁₉⋊C₈	—	4	4	no	—
131	C₁₉⋊C₈	—	4	4	no	—
132	C₂×C₄×C₄⋊C₄	D₁₉	1	1	yes	characteristic
133	C₂×#336	—	19	4	no	—
134	C₂×#336	—	19	1	no	—
135	C₂×#336	—	19	1	no	—
136	C₂²×C₁₉	#381	1	1	yes	characteristic
137	C₂²×C₁₉	#387	1	4	yes	—
138	C₂²×C₁₉	#390	1	1	yes	characteristic
139	C₂²×C₁₉	#391	1	1	yes	characteristic
140	C₂²×C₁₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
141	C₂²×C₁₉	#407	1	1	yes	characteristic
142	C₂²×C₁₉	#408	1	1	yes	characteristic
143	C₂²×C₁₉	#408	1	1	yes	characteristic
144	C₂²×C₁₉	#431	1	2	yes	—
145	C₂²×C₁₉	#432	1	2	yes	—
146	C₂²×C₁₉	#433	1	4	yes	—
147	C₂²×C₁₉	C₈×D₄	1	2	yes	—
148	C₂²×C₁₉	#435	1	4	yes	—
149	C₂²×C₁₉	#436	1	2	yes	—
150	C₂²×C₁₉	C₈×Q₈	1	2	yes	—
151	C₂²×C₁₉	#446	1	4	yes	—
152	C₂²×C₁₉	#446	1	2	yes	—
153	C₄×C₁₉	—	2	4	no	—
154	C₄×C₁₉	—	4	2	no	—
155	C₄×C₁₉	—	4	4	no	—
156	C₄×C₁₉	—	4	4	no	—
157	C₄×C₁₉	#397	1	2	yes	—
158	C₄×C₁₉	#398	1	2	yes	—
159	C₄×C₁₉	#399	1	2	yes	—
160	C₄×C₁₉	#400	1	2	yes	—
161	C₂³×C₈	—	19	2	no	—
162	C₂³×C₈	—	19	2	no	—
163	C₂²×C₄⋊C₄	Dic₁₉	1	1	yes	characteristic
164	C₂²×C₄⋊C₄	Dic₁₉	1	1	yes	characteristic
165	C₂²×C₄²	C₂×D₁₉	1	2	yes	—
166	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
167	C₄×C₄⋊C₄	—	2	2	no	—
168	C₄×C₄⋊C₄	—	2	2	no	—
169	C₂×#95	Dic₁₉	1	2	yes	—
170	#336	—	38	8	no	—
171	#336	—	38	2	no	—
172	#336	—	38	2	no	—
173	C₂×C₁₉	#1445	1	1	yes	characteristic
174	C₂×C₁₉	#1817	1	1	yes	characteristic
175	C₂×C₁₉	#1818	1	1	yes	characteristic
176	C₂×C₁₉	#1916	1	4	yes	—
177	C₂×C₁₉	#1917	1	4	yes	—
178	C₂×C₁₉	#1954	1	1	yes	characteristic
179	C₂×C₁₉	#1955	1	1	yes	characteristic
180	C₂×C₁₉	#1984	1	1	yes	characteristic
181	C₂×C₁₉	#1985	1	1	yes	characteristic
182	C₂³×C₄	C₂×Dic₁₉	1	2	yes	—
183	C₂³×C₄	C₂×Dic₁₉	1	2	yes	—
184	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
185	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
186	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
187	C₂²×C₈	—	19	4	no	—
188	C₂²×C₈	—	19	4	no	—
189	C₂²×C₈	—	38	4	no	—
190	C₂²×C₈	—	38	4	no	—
191	C₂²×C₈	—	38	4	no	—
192	C₂²×C₈	—	38	2	no	—
193	C₂²×C₈	—	38	2	no	—
194	C₂²×C₈	—	38	4	no	—
195	C₂²×C₈	—	38	2	no	—
196	C₂²×C₈	—	38	2	no	—
197	C₂×C₄⋊C₄	—	2	4	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	C₁₉⋊C₈	1	4	yes	complemented
202	C₂×C₄²	—	2	4	no	—
203	C₂×C₄²	—	2	4	no	—
204	C₂×C₄²	—	2	4	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
208	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
209	#95	—	2	4	no	—
210	C₁₉	#11346	1	1	yes	characteristic, complemented, Sylow
211	C₂⁴	C₂²×Dic₁₉	1	1	yes	characteristic
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	C₂×C₁₉⋊C₈	1	4	yes	—
226	C₂²×C₄	C₂×C₁₉⋊C₈	1	2	yes	—
227	C₂²×C₄	#64520	1	4	yes	—
228	C₂²×C₄	#64520	1	2	yes	—
229	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
230	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
231	C₂²×C₄	#64541	1	4	yes	—
232	C₂²×C₄	D₄×D₁₉	1	2	yes	—
233	C₂²×C₄	#64543	1	2	yes	—
234	C₂²×C₄	Q₈×D₁₉	1	2	yes	—
235	C₂²×C₄	#64545	1	2	yes	—
236	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
238	C₂×C₈	—	38	4	no	—
239	C₂×C₈	—	38	4	no	—
240	C₂×C₈	—	38	8	no	—
241	C₂×C₈	—	38	8	no	—
242	C₂×C₈	—	76	2	no	—
243	C₂×C₈	—	76	2	no	—
244	C₂×C₈	—	76	4	no	—
245	C₂×C₈	—	76	4	no	—
246	C₂×C₈	—	76	2	no	—
247	C₂×C₈	—	76	2	no	—
248	C₄⋊C₄	—	2	8	no	—
249	C₄⋊C₄	—	2	8	no	—
250	C₄²	—	2	4	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	4	2	no	—
253	C₄²	—	4	4	no	—
254	C₄²	—	4	2	no	—
255	C₂³	C₂²×C₁₉⋊C₈	1	1	yes	characteristic
256	C₂³	C₂×#64520	1	2	yes	—
257	C₂³	C₂×#64520	1	1	yes	characteristic
258	C₂³	#10597372	1	2	yes	—
259	C₂³	D₄×Dic₁₉	1	2	yes	—
260	C₂³	Q₈×Dic₁₉	1	2	yes	—
261	C₂³	#10597394	1	2	yes	—
262	C₂³	#10597394	1	2	yes	—
263	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
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	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	4	yes	—
283	C₂×C₄	#10597335	1	4	yes	—
284	C₂×C₄	#10597340	1	4	yes	—
285	C₂×C₄	#10597342	1	4	yes	—
286	C₂×C₄	#10597344	1	4	yes	—
287	C₂×C₄	#10597375	1	2	yes	—
288	C₂×C₄	#10597375	1	2	yes	—
289	C₂×C₄	#10597382	1	1	yes	characteristic
290	C₂×C₄	#10597384	1	2	yes	—
291	C₂×C₄	#10597386	1	1	yes	characteristic
292	C₂×C₄	#10597390	1	1	yes	characteristic
293	C₂×C₄	#10597392	1	2	yes	—
294	C₂×C₄	#10597393	1	1	yes	characteristic
295	C₈	—	76	4	no	—
296	C₈	—	76	4	no	—
297	C₂²	#11926407	1	1	yes	characteristic
298	C₂²	#11926430	1	4	yes	—
299	C₂²	#11926447	1	1	yes	characteristic
300	C₂²	#11926451	1	1	yes	characteristic
301	C₂²	C₂×C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
302	C₂²	#11926508	1	1	yes	characteristic
303	C₂²	#11926511	1	1	yes	characteristic
304	C₂²	#11926511	1	1	yes	characteristic
305	C₂²	#11926597	1	2	yes	—
306	C₂²	#11926601	1	2	yes	—
307	C₂²	#11926605	1	4	yes	—
308	C₂²	D₄×C₁₉⋊C₈	1	2	yes	—
309	C₂²	#11926616	1	4	yes	—
310	C₂²	#11926622	1	2	yes	—
311	C₂²	Q₈×C₁₉⋊C₈	1	2	yes	—
312	C₂²	#11926675	1	4	yes	—
313	C₂²	#11926675	1	2	yes	—
314	C₄	—	2	4	no	—
315	C₄	—	4	4	no	—
316	C₄	—	4	4	no	—
317	C₄	—	4	2	no	—
318	C₄	#11926470	1	2	yes	—
319	C₄	#11926475	1	2	yes	—
320	C₄	#11926479	1	2	yes	—
321	C₄	#11926482	1	2	yes	—
322	C₂	#14561903	1	1	yes	characteristic
323	C₂	#14563703	1	1	yes	characteristic
324	C₂	#14563706	1	1	yes	characteristic
325	C₂	#14564194	1	4	yes	—
326	C₂	#14564201	1	4	yes	—
327	C₂	#14564430	1	1	yes	characteristic
328	C₂	#14564434	1	1	yes	characteristic
329	C₂	#14564572	1	1	yes	characteristic
330	C₂	#14564576	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₁₉
commutator	C₂³×C₁₉	C₂²×C₈
Fitting	C₂×C₄×C₄⋊C₄×C₁₉	C₂
socle, prime radical	C₂⁴×C₁₉	C₂²×C₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×C₄×C₄⋊C₄	D₁₉
hypocenter	C₁₉	#11346
LCS	G ▷ C₂³×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
26. C₂×C₄⋊C₄×C₁₉	C₈	semidirect
201. C₂×C₄⋊C₄	C₁₉⋊C₈	semidirect
210. C₁₉	#11346	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926299	2
C₂×C₄×C₄⋊C₄×C₁₉	2
#11346	19