Subgroups — #18081486

Order 4864.

466 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#64527	C₂	1	1	yes	maximal, characteristic
2	#14562381	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562030	C₂	1	1	yes	maximal, characteristic, complemented
4	#14562027	C₂	1	1	yes	maximal, characteristic
5	#14561955	C₂	1	1	yes	maximal, characteristic
6	#14561928	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1461	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#64527	C₄	1	1	yes	characteristic
9	C₂×C₄×D₇₆	C₂²	1	1	yes	characteristic
10	C₄×#64527	—	2	1	no	—
11	C₄×#64527	—	2	1	no	—
12	C₄×#64527	—	2	1	no	—
13	C₄×#64527	—	2	1	no	—
14	#11926597	C₂²	1	1	yes	characteristic
15	#11926597	C₂²	1	1	yes	characteristic
16	C₂×#10597265	C₄	1	1	yes	characteristic
17	C₂×#10597292	C₄	1	1	yes	characteristic
18	C₂×#10597266	C₄	1	1	yes	characteristic
19	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
20	#11926296	—	2	1	no	—
21	#11926294	—	2	1	no	—
22	Q₈⋊(C₁₉⋊C₈)	—	2	2	no	—
23	#11926273	—	2	2	no	—
24	#11926267	—	2	2	no	—
25	#11926266	—	2	1	no	—
26	#11926265	—	2	1	no	—
27	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
28	C₁₉×#431	C₂²	1	1	yes	characteristic
29	C₁₉×#431	C₂²	1	1	yes	characteristic
30	C₁₉×#324	—	2	2	no	—
31	C₂²×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×D₇₆	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×D₁₉	—	2	1	no	—
34	C₄×Q₈×C₁₉	—	2	1	no	—
35	C₄×Q₈×C₁₉	—	2	1	no	—
36	C₄×Q₈×C₁₉	—	2	1	no	—
37	C₄×Q₈×C₁₉	—	2	1	no	—
38	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
39	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₉	—	2	1	no	—
41	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
42	C₂×#64527	—	2	2	no	—
43	C₂×#64527	—	2	2	no	—
44	C₂×#64527	—	2	1	no	—
45	C₂×#64527	—	2	1	no	—
46	C₂×(C₂×D₁₉)⋊C₄	—	2	1	no	—
47	C₂×C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
50	C₄×D₇₆	—	2	1	no	—
51	C₄×D₇₆	—	2	1	no	—
52	C₄×D₇₆	—	2	1	no	—
53	C₄×D₇₆	—	2	1	no	—
54	C₁₉×C₄⋊C₈	—	2	1	no	—
55	C₁₉×C₄⋊C₈	—	2	1	no	—
56	C₁₉×C₂²⋊C₈	—	2	1	no	—
57	C₁₉×C₂²⋊C₈	D₄	1	2	yes	—
58	C₄×C₈×C₁₉	—	2	1	no	—
59	C₄×C₈×C₁₉	—	2	1	no	—
60	#10597292	—	2	1	no	—
61	#10597292	—	2	1	no	—
62	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
63	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
64	#10597266	—	2	1	no	—
65	#10597266	—	2	1	no	—
66	#10597265	—	2	1	no	—
67	#10597265	—	2	1	no	—
68	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
69	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
70	C₄×C₁₉⋊C₈	—	2	1	no	—
71	C₄×C₁₉⋊C₈	—	2	1	no	—
72	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
73	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
74	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
75	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
76	C₂³×D₁₉	—	2	1	no	—
77	C₂×Q₈×C₁₉	—	2	1	no	—
78	C₂×Q₈×C₁₉	—	2	1	no	—
79	C₂×Q₈×C₁₉	—	2	1	no	—
80	C₂×Q₈×C₁₉	—	2	1	no	—
81	C₂×Q₈×C₁₉	—	2	1	no	—
82	C₂×Q₈×C₁₉	M₁₆	1	2	yes	—
83	C₂²×C₄×C₁₉	—	2	1	no	—
84	C₂²×C₄×C₁₉	—	2	1	no	—
85	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
88	C₂²×Dic₁₉	—	2	1	no	—
89	C₂×D₇₆	—	2	1	no	—
90	C₂×D₇₆	—	2	1	no	—
91	C₂×D₇₆	—	2	1	no	—
92	C₂×D₇₆	—	2	1	no	—
93	C₂×D₇₆	—	2	1	no	—
94	C₂×D₇₆	M₁₆	1	2	yes	—
95	C₂×C₄×D₁₉	—	4	1	no	—
96	C₂×C₄×D₁₉	—	4	1	no	—
97	C₂×C₄×D₁₉	—	8	2	no	—
98	C₂×C₈×C₁₉	—	2	2	no	—
99	C₂×C₈×C₁₉	—	4	1	no	—
100	C₄⋊C₄×C₁₉	—	2	1	no	—
101	C₄⋊C₄×C₁₉	—	2	1	no	—
102	C₄⋊C₄×C₁₉	—	4	1	no	—
103	C₄⋊C₄×C₁₉	—	4	1	no	—
104	C₄²×C₁₉	—	4	1	no	—
105	C₄²×C₁₉	—	4	1	no	—
106	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₁₉	#41	1	1	yes	characteristic
110	#64527	—	2	4	no	—
111	#64527	—	2	4	no	—
112	(C₂×D₁₉)⋊C₄	—	4	1	no	—
113	(C₂×D₁₉)⋊C₄	—	4	1	no	—
114	C₄⋊Dic₁₉	—	2	1	no	—
115	C₄⋊Dic₁₉	—	2	1	no	—
116	C₂×C₁₉⋊C₈	—	2	1	no	—
117	C₂×C₁₉⋊C₈	—	2	1	no	—
118	C₂×C₁₉⋊C₈	—	2	1	no	—
119	C₂×C₁₉⋊C₈	—	2	1	no	—
120	C₂×C₁₉⋊C₈	—	2	2	no	—
121	C₂×C₁₉⋊C₈	—	4	1	no	—
122	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
126	#9672	—	19	1	no	maximal, Sylow
127	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
128	C₂²×D₁₉	—	2	2	no	—
129	C₂²×D₁₉	—	4	1	no	—
130	C₂²×D₁₉	—	4	1	no	—
131	C₂²×D₁₉	—	8	1	no	—
132	C₂²×D₁₉	—	8	1	no	—
133	Q₈×C₁₉	—	2	2	no	—
134	Q₈×C₁₉	—	2	2	no	—
135	Q₈×C₁₉	—	4	1	no	—
136	Q₈×C₁₉	—	4	1	no	—
137	C₂×C₄×C₁₉	—	2	2	no	—
138	C₂×C₄×C₁₉	—	2	2	no	—
139	C₂×C₄×C₁₉	—	4	1	no	—
140	C₂×C₄×C₁₉	—	4	1	no	—
141	C₂×C₄×C₁₉	—	4	1	no	—
142	C₂×C₄×C₁₉	—	4	1	no	—
143	C₂×C₄×C₁₉	—	8	1	no	—
144	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
147	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
148	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
149	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
150	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
151	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
152	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
153	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
154	C₂×Dic₁₉	—	4	1	no	—
155	C₂×Dic₁₉	—	8	1	no	—
156	D₇₆	—	2	2	no	—
157	D₇₆	—	2	2	no	—
158	D₇₆	—	4	1	no	—
159	D₇₆	—	4	1	no	—
160	C₄×D₁₉	—	8	2	no	—
161	C₄×D₁₉	—	8	2	no	—
162	C₈×C₁₉	—	4	2	no	—
163	C₈×C₁₉	—	8	1	no	—
164	C₁₉⋊C₈	—	2	1	no	—
165	C₁₉⋊C₈	—	2	1	no	—
166	C₁₉⋊C₈	—	2	1	no	—
167	C₁₉⋊C₈	—	2	1	no	—
168	C₁₉⋊C₈	—	4	2	no	—
169	C₁₉⋊C₈	—	8	1	no	—
170	C₂×C₄×SD₁₆	—	19	1	no	—
171	#1564	—	19	1	no	—
172	#1479	—	19	1	no	—
173	#1478	—	19	1	no	—
174	#1461	D₁₉	1	1	yes	characteristic, complemented
175	#1460	—	19	1	no	—
176	#1452	—	19	1	no	—
177	C₂²×C₁₉	—	2	1	no	—
178	C₂²×C₁₉	—	2	1	no	—
179	C₂²×C₁₉	#390	1	1	yes	characteristic
180	C₂²×C₁₉	#435	1	1	yes	characteristic
181	C₂²×C₁₉	#435	1	1	yes	characteristic
182	C₂×D₁₉	—	4	2	no	—
183	C₂×D₁₉	—	8	2	no	—
184	C₂×D₁₉	—	8	1	no	—
185	C₂×D₁₉	—	8	1	no	—
186	C₂×D₁₉	—	8	2	no	—
187	C₄×C₁₉	—	2	1	no	—
188	C₄×C₁₉	—	2	1	no	—
189	C₄×C₁₉	—	2	1	no	—
190	C₄×C₁₉	—	4	2	no	—
191	C₄×C₁₉	—	8	1	no	—
192	C₄×C₁₉	#407	1	1	yes	characteristic
193	C₄×C₁₉	#408	1	1	yes	characteristic
194	C₄×C₁₉	#435	1	1	yes	characteristic
195	C₄×C₁₉	#435	1	1	yes	characteristic
196	C₄×C₁₉	#471	1	2	yes	—
197	Dic₁₉	—	8	1	no	—
198	C₂²×SD₁₆	—	19	1	no	—
199	C₂×C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
200	C₂×C₄×D₄	—	19	1	no	—
201	C₄×SD₁₆	—	38	1	no	—
202	C₄×SD₁₆	—	38	1	no	—
203	C₄×SD₁₆	—	38	1	no	—
204	C₄×SD₁₆	—	38	1	no	—
205	#431	—	19	1	no	—
206	#431	—	19	1	no	—
207	#431	C₂×D₁₉	1	1	yes	characteristic
208	#431	C₂×D₁₉	1	1	yes	characteristic
209	C₂×#106	—	19	1	no	—
210	C₂×#103	—	19	1	no	—
211	C₂×#102	—	19	1	no	—
212	C₂×C₄×C₈	—	19	1	no	—
213	#335	—	38	1	no	—
214	#334	—	38	1	no	—
215	Q₈⋊C₈	—	38	2	no	—
216	D₄⋊C₈	—	38	2	no	—
217	#324	—	2	2	no	—
218	#323	—	38	2	no	—
219	#322	—	38	1	no	—
220	#322	—	38	1	no	—
221	C₂×C₁₉	—	2	1	no	—
222	C₂×C₁₉	—	2	1	no	—
223	C₂×C₁₉	#1955	1	1	yes	characteristic
224	C₂×C₁₉	#1976	1	1	yes	characteristic
225	C₂×C₁₉	#1977	1	1	yes	characteristic
226	D₁₉	—	8	2	no	—
227	C₂²×Q₈	C₄×D₁₉	1	1	yes	characteristic
228	C₂²×D₄	—	19	1	no	—
229	C₂³×C₄	—	38	1	no	—
230	C₂×SD₁₆	—	38	1	no	—
231	C₂×SD₁₆	—	38	1	no	—
232	C₂×SD₁₆	—	38	2	no	—
233	C₂×SD₁₆	—	38	2	no	—
234	C₂²×C₈	—	19	1	no	—
235	C₂²×C₈	—	19	1	no	—
236	C₄×Q₈	—	2	1	no	—
237	C₄×Q₈	—	2	1	no	—
238	C₄×Q₈	—	2	1	no	—
239	C₄×Q₈	—	2	1	no	—
240	C₄×D₄	—	38	1	no	—
241	C₄×D₄	—	38	1	no	—
242	C₄×D₄	—	38	1	no	—
243	C₄×D₄	—	38	1	no	—
244	C₂×C₄⋊C₄	—	2	1	no	—
245	C₂×C₄⋊C₄	—	19	1	no	—
246	C₂×C₄⋊C₄	C₄×D₁₉	1	1	yes	characteristic
247	C₂×C₂²⋊C₄	—	38	1	no	—
248	C₂×C₄²	—	2	1	no	—
249	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
250	#106	—	38	1	no	—
251	#106	—	38	1	no	—
252	C₄⋊C₈	—	2	1	no	—
253	C₄⋊C₈	—	2	1	no	—
254	C₄⋊C₈	—	19	2	no	—
255	C₄⋊C₈	—	38	1	no	—
256	#103	—	38	1	no	—
257	#103	—	38	1	no	—
258	#102	—	38	1	no	—
259	#102	—	38	1	no	—
260	C₂²⋊C₈	—	2	1	no	—
261	C₂²⋊C₈	—	38	1	no	—
262	C₂²⋊C₈	—	38	1	no	—
263	C₂²⋊C₈	C₂²⋊D₁₉	1	2	yes	—
264	C₄×C₈	—	2	1	no	—
265	C₄×C₈	—	2	1	no	—
266	C₄×C₈	—	19	1	no	—
267	C₄×C₈	—	19	1	no	—
268	C₄×C₈	—	19	1	no	—
269	C₄×C₈	—	19	1	no	—
270	C₄×C₈	—	38	1	no	—
271	C₄×C₈	—	38	1	no	—
272	C₁₉	#9672	1	1	yes	characteristic, complemented, Sylow
273	C₂⁴	—	38	1	no	—
274	C₂×Q₈	—	2	1	no	—
275	C₂×Q₈	—	2	1	no	—
276	C₂×Q₈	—	2	1	no	—
277	C₂×Q₈	—	2	1	no	—
278	C₂×Q₈	—	2	1	no	—
279	C₂×Q₈	#64515	1	2	yes	—
280	C₂×D₄	—	19	2	no	—
281	C₂×D₄	—	38	1	no	—
282	C₂×D₄	—	38	1	no	—
283	C₂×D₄	—	38	1	no	—
284	C₂×D₄	—	38	1	no	—
285	C₂×D₄	—	38	1	no	—
286	C₂²×C₄	—	2	1	no	—
287	C₂²×C₄	—	2	1	no	—
288	C₂²×C₄	—	38	1	no	—
289	C₂²×C₄	—	76	1	no	—
290	C₂²×C₄	—	76	1	no	—
291	C₂²×C₄	—	152	2	no	—
292	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
293	C₂²×C₄	#64541	1	1	yes	characteristic
294	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
295	SD₁₆	—	38	4	no	—
296	SD₁₆	—	38	4	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	4	1	no	—
299	C₂×C₈	—	19	1	no	—
300	C₂×C₈	—	19	1	no	—
301	C₂×C₈	—	19	1	no	—
302	C₂×C₈	—	19	1	no	—
303	C₂×C₈	—	38	1	no	—
304	C₂×C₈	—	38	1	no	—
305	C₂×C₈	—	38	1	no	—
306	C₂×C₈	—	38	1	no	—
307	C₂×C₈	—	38	2	no	—
308	C₂×C₈	—	76	1	no	—
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	4	1	no	—
312	C₄⋊C₄	—	4	1	no	—
313	C₄⋊C₄	—	38	1	no	—
314	C₄⋊C₄	—	38	1	no	—
315	C₂²⋊C₄	—	76	1	no	—
316	C₂²⋊C₄	—	76	1	no	—
317	C₄²	—	4	1	no	—
318	C₄²	—	4	1	no	—
319	C₄²	D₄×D₁₉	1	1	yes	characteristic
320	C₄²	D₄×D₁₉	1	1	yes	characteristic
321	C₄²	D₄×D₁₉	1	1	yes	characteristic
322	C₄²	#64543	1	1	yes	characteristic
323	C₂³	—	38	2	no	—
324	C₂³	—	76	1	no	—
325	C₂³	—	76	1	no	—
326	C₂³	—	152	1	no	—
327	C₂³	—	152	1	no	—
328	C₂³	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
329	Q₈	—	2	2	no	—
330	Q₈	—	2	2	no	—
331	Q₈	—	4	1	no	—
332	Q₈	—	4	1	no	—
333	D₄	—	38	2	no	—
334	D₄	—	38	2	no	—
335	D₄	—	76	1	no	—
336	D₄	—	76	1	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	4	1	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	8	1	no	—
344	C₂×C₄	—	76	1	no	—
345	C₂×C₄	—	152	2	no	—
346	C₂×C₄	—	152	2	no	—
347	C₂×C₄	—	152	1	no	—
348	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
349	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
350	C₂×C₄	#10597330	1	1	yes	characteristic
351	C₂×C₄	#10597331	1	1	yes	characteristic
352	C₂×C₄	C₂×#64515	1	1	yes	characteristic
353	C₂×C₄	#10597347	1	1	yes	characteristic
354	C₂×C₄	M₁₆×D₁₉	1	1	yes	characteristic
355	C₂×C₄	#10597353	1	1	yes	characteristic
356	C₂×C₄	#10597385	1	1	yes	characteristic
357	C₂×C₄	#10597386	1	1	yes	characteristic
358	C₈	—	4	2	no	—
359	C₈	—	8	1	no	—
360	C₈	—	38	1	no	—
361	C₈	—	38	1	no	—
362	C₈	—	38	1	no	—
363	C₈	—	38	1	no	—
364	C₈	—	76	2	no	—
365	C₈	—	152	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	76	2	no	—
369	C₂²	—	152	2	no	—
370	C₂²	—	152	2	no	—
371	C₂²	—	152	1	no	—
372	C₂²	—	152	1	no	—
373	C₂²	#11926444	1	1	yes	characteristic
374	C₂²	#11926613	1	1	yes	characteristic
375	C₂²	#11926615	1	1	yes	characteristic
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	4	2	no	—
380	C₄	—	8	1	no	—
381	C₄	—	152	1	no	—
382	C₄	#11926506	1	1	yes	characteristic
383	C₄	#11926509	1	1	yes	characteristic
384	C₄	#11926612	1	1	yes	characteristic
385	C₄	#11926618	1	1	yes	characteristic
386	C₄	#11926810	1	2	yes	—
387	C₂	—	2	1	no	—
388	C₂	—	2	1	no	—
389	C₂	—	152	2	no	—
390	C₂	#14564431	1	1	yes	characteristic
391	C₂	#14564530	1	1	yes	characteristic
392	C₂	#14564539	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597330
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1461	C₂
socle	C₂²×C₁₉	#390
prime radical	C₂²×D₇₆	C₂×C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1461	D₁₉
hypocenter	C₁₉	#9672
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1461

Complements

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

N	H ≅ G/N	type
2. #14562381	C₂	semidirect
3. #14562030	C₂	semidirect
6. #14561928	C₂	semidirect
7. C₁₉×#1461	C₂	semidirect
174. #1461	D₁₉	semidirect
272. C₁₉	#9672	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#64527	2
#14562381	2
#14562030	2
#14562027	2
#14561955	2
#14561928	2
C₁₉×#1461	2
#9672	19