Subgroups — #18077739

Order 4864.

476 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14573639	C₂	1	1	yes	maximal, characteristic
2	#14564433	C₂	1	1	yes	maximal, characteristic
3	#14562292	C₂	1	1	yes	maximal, characteristic, complemented
4	#14562059	C₂	1	1	yes	maximal, characteristic
5	#14561530	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561480	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#2961	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₇₆	C₂²	1	1	yes	characteristic
9	#11926872	C₄	1	2	yes	—
10	#11926831	—	2	1	no	—
11	#11926746	—	2	1	no	—
12	#11926694	C₄	1	2	yes	—
13	C₄×#64527	—	2	2	no	—
14	#11926597	C₂²	1	1	yes	characteristic
15	#11926597	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×(C₂×D₁₉)⋊C₈	—	2	1	no	—
18	C₂×(C₂×D₁₉)⋊C₈	—	2	1	no	—
19	#11926293	—	2	1	no	—
20	#11926290	—	2	1	no	—
21	#11926271	—	2	2	no	—
22	C₁₉×#533	C₂²	1	1	yes	characteristic
23	C₁₉×#446	—	2	2	no	—
24	C₁₉×#446	—	2	2	no	—
25	C₁₉×#424	—	2	1	no	—
26	C₁₉×#423	C₂²	1	1	yes	characteristic
27	C₁₉×#405	—	2	1	no	—
28	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
29	C₂²×D₇₆	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×D₁₉	—	2	1	no	—
31	C₂×M₁₆×C₁₉	—	2	1	no	—
32	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
34	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
35	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
36	C₁₉×#125	—	2	1	no	—
37	C₁₉×C₂²⋊Q₈	—	2	1	no	—
38	C₁₉×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
39	C₄×Q₈×C₁₉	—	2	1	no	—
40	C₄×Q₈×C₁₉	D₄	1	2	yes	—
41	C₁₉×#117	—	2	1	no	—
42	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
43	C₂×#64527	—	2	2	no	—
44	C₂×(C₂×D₁₉)⋊C₄	—	2	1	no	—
45	C₂×C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₁₉⋊C₈	—	2	1	no	—
47	C₄×D₇₆	—	2	1	no	—
48	C₄×D₇₆	—	2	1	no	—
49	C₄×D₇₆	—	2	1	no	—
50	C₄×D₇₆	—	2	1	no	—
51	C₁₉×C₄⋊C₈	—	2	4	no	—
52	C₁₉×C₄⋊C₈	—	2	1	no	—
53	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
54	C₁₉×#97	—	2	2	no	—
55	C₁₉×#97	—	2	1	no	—
56	C₁₉×#97	—	2	1	no	—
57	C₄×C₈×C₁₉	—	2	2	no	—
58	#10597292	—	2	2	no	—
59	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
60	C₂²⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
61	(C₂×D₁₉)⋊C₈	—	4	2	no	—
62	(C₂×D₁₉)⋊C₈	—	4	2	no	—
63	#10597266	—	2	2	no	—
64	#10597265	—	2	2	no	—
65	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
66	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
67	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
68	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
69	C₄×C₁₉⋊C₈	—	2	1	no	—
70	C₄×C₁₉⋊C₈	—	2	1	no	—
71	C₂³×D₁₉	—	2	1	no	—
72	C₂×Q₈×C₁₉	—	2	1	no	—
73	C₂×Q₈×C₁₉	C₂²⋊C₄	1	2	yes	—
74	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
77	C₂²×Dic₁₉	—	2	1	no	—
78	C₂×D₇₆	—	2	1	no	—
79	C₂×D₇₆	—	2	1	no	—
80	C₂×D₇₆	—	4	2	no	—
81	C₂×C₄×D₁₉	—	4	1	no	—
82	C₂×C₄×D₁₉	—	4	1	no	—
83	C₂×C₄×D₁₉	—	4	1	no	—
84	C₂×C₄×D₁₉	—	4	1	no	—
85	C₂×C₄×D₁₉	—	4	1	no	—
86	C₂×C₄×D₁₉	—	4	1	no	—
87	M₁₆×C₁₉	—	4	1	no	—
88	M₁₆×C₁₉	—	4	1	no	—
89	C₂×C₈×C₁₉	—	2	1	no	—
90	C₂×C₈×C₁₉	—	2	1	no	—
91	C₂×C₈×C₁₉	—	2	1	no	—
92	C₂×C₈×C₁₉	—	2	2	no	—
93	C₂×C₈×C₁₉	—	2	1	no	—
94	C₂×C₈×C₁₉	—	2	1	no	—
95	C₂×C₈×C₁₉	SD₁₆	1	2	yes	complemented
96	C₄⋊C₄×C₁₉	—	2	2	no	—
97	C₄⋊C₄×C₁₉	—	2	1	no	—
98	C₄⋊C₄×C₁₉	—	2	1	no	—
99	C₄⋊C₄×C₁₉	—	2	1	no	—
100	C₄⋊C₄×C₁₉	—	2	2	no	—
101	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
102	C₂²⋊C₄×C₁₉	—	2	2	no	—
103	C₄²×C₁₉	—	2	2	no	—
104	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₁₉	#41	1	1	yes	characteristic
108	#64527	—	4	4	no	—
109	(C₂×D₁₉)⋊C₄	—	4	1	no	—
110	(C₂×D₁₉)⋊C₄	—	4	1	no	—
111	C₄⋊Dic₁₉	—	2	1	no	—
112	C₄⋊Dic₁₉	—	2	1	no	—
113	C₂×C₁₉⋊C₈	—	2	2	no	—
114	C₂×C₁₉⋊C₈	—	4	1	no	—
115	C₂×C₁₉⋊C₈	—	4	1	no	—
116	C₂×C₁₉⋊C₈	—	4	1	no	—
117	C₂×C₁₉⋊C₈	—	4	1	no	—
118	#9050	—	19	1	no	maximal, Sylow
119	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
120	C₂²×D₁₉	—	2	2	no	—
121	C₂²×D₁₉	—	4	1	no	—
122	C₂²×D₁₉	—	4	1	no	—
123	C₂²×D₁₉	—	4	1	no	—
124	C₂²×D₁₉	—	4	1	no	—
125	C₂²×D₁₉	—	4	1	no	—
126	C₂²×D₁₉	—	4	1	no	—
127	Q₈×C₁₉	—	4	2	no	—
128	Q₈×C₁₉	—	8	1	no	—
129	C₂×C₄×C₁₉	—	2	2	no	—
130	C₂×C₄×C₁₉	—	2	2	no	—
131	C₂×C₄×C₁₉	—	4	1	no	—
132	C₂×C₄×C₁₉	—	4	1	no	—
133	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
136	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
137	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
138	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
139	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
140	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
141	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
143	C₂×Dic₁₉	—	4	1	no	—
144	C₂×Dic₁₉	—	4	1	no	—
145	C₂×Dic₁₉	—	4	1	no	—
146	D₇₆	—	2	2	no	—
147	D₇₆	—	2	2	no	—
148	D₇₆	—	4	1	no	—
149	D₇₆	—	4	1	no	—
150	C₄×D₁₉	—	8	1	no	—
151	C₄×D₁₉	—	8	1	no	—
152	C₄×D₁₉	—	8	1	no	—
153	C₄×D₁₉	—	8	1	no	—
154	C₈×C₁₉	—	2	2	no	—
155	C₈×C₁₉	—	4	1	no	—
156	C₈×C₁₉	—	4	2	no	—
157	C₁₉⋊C₈	—	4	2	no	—
158	C₁₉⋊C₈	—	8	1	no	—
159	C₁₉⋊C₈	—	8	1	no	—
160	#3101	—	19	1	no	—
161	#2961	D₁₉	1	1	yes	characteristic, complemented
162	#1955	—	19	1	no	—
163	#1542	—	19	1	no	—
164	#1486	—	19	1	no	—
165	#1301	—	19	1	no	—
166	#1281	—	19	1	no	—
167	C₂²×C₁₉	—	2	1	no	—
168	C₂²×C₁₉	—	2	1	no	—
169	C₂²×C₁₉	#386	1	1	yes	characteristic
170	C₂²×C₁₉	C₄×SD₁₆	1	1	yes	characteristic
171	C₂²×C₁₉	#440	1	1	yes	characteristic
172	C₂×D₁₉	—	4	2	no	—
173	C₂×D₁₉	—	4	2	no	—
174	C₂×D₁₉	—	4	2	no	—
175	C₂×D₁₉	—	8	1	no	—
176	C₂×D₁₉	—	8	1	no	—
177	C₂×D₁₉	—	8	1	no	—
178	C₂×D₁₉	—	8	1	no	—
179	C₄×C₁₉	—	2	1	no	—
180	C₄×C₁₉	—	2	1	no	—
181	C₄×C₁₉	—	2	1	no	—
182	C₄×C₁₉	—	2	1	no	—
183	C₄×C₁₉	—	4	2	no	—
184	C₄×C₁₉	—	4	2	no	—
185	C₄×C₁₉	#448	1	1	yes	characteristic
186	C₄×C₁₉	#449	1	1	yes	characteristic
187	C₄×C₁₉	#460	1	1	yes	characteristic
188	C₄×C₁₉	#464	1	1	yes	characteristic
189	Dic₁₉	—	8	1	no	—
190	#533	C₂×D₁₉	1	1	yes	characteristic
191	C₂×C₄×D₄	—	19	1	no	—
192	#483	—	19	2	no	—
193	#476	—	38	1	no	—
194	#461	—	38	1	no	—
195	#450	—	19	2	no	—
196	#446	—	2	2	no	—
197	#446	—	2	2	no	—
198	C₄×SD₁₆	—	38	2	no	—
199	#431	—	19	1	no	—
200	#431	—	19	1	no	—
201	#424	—	2	1	no	—
202	#423	C₂×D₁₉	1	1	yes	characteristic
203	C₂×C₄⋊C₈	—	19	1	no	—
204	C₂×C₂²⋊C₈	—	38	1	no	—
205	C₂×C₂²⋊C₈	—	38	1	no	—
206	#405	—	2	1	no	—
207	C₂×#97	C₂×D₁₉	1	1	yes	characteristic
208	#333	—	38	1	no	—
209	#332	—	38	1	no	—
210	D₄⋊C₈	—	38	2	no	—
211	C₂×C₁₉	—	2	1	no	—
212	C₂×C₁₉	—	2	1	no	—
213	C₂×C₁₉	#1880	1	1	yes	characteristic
214	C₂×C₁₉	#1881	1	1	yes	characteristic
215	C₂×C₁₉	#1891	1	1	yes	characteristic
216	D₁₉	—	8	2	no	—
217	C₂²×D₄	—	19	1	no	—
218	C₂³×C₄	—	38	1	no	—
219	C₂×SD₁₆	—	38	2	no	—
220	C₂×M₁₆	—	2	1	no	—
221	C₂²×C₈	—	38	1	no	—
222	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
223	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
224	C₄⋊Q₈	D₇₆	1	1	yes	characteristic
225	C₄⋊Q₈	D₇₆	1	1	yes	characteristic
226	#125	—	2	1	no	—
227	C₂²⋊Q₈	—	2	1	no	—
228	C₂²⋊Q₈	C₄×D₁₉	1	2	yes	—
229	C₄×Q₈	—	2	1	no	—
230	C₄×Q₈	C₂²⋊D₁₉	1	2	yes	—
231	C₄×D₄	—	38	1	no	—
232	C₄×D₄	—	38	1	no	—
233	C₄×D₄	—	38	1	no	—
234	C₄×D₄	—	38	1	no	—
235	#117	—	2	1	no	—
236	C₂×C₄⋊C₄	—	19	1	no	—
237	C₂×C₂²⋊C₄	—	38	1	no	—
238	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
239	#106	—	38	2	no	—
240	C₄⋊C₈	—	2	4	no	—
241	C₄⋊C₈	—	2	1	no	—
242	C₄⋊C₈	—	38	1	no	—
243	C₄⋊C₈	—	38	1	no	—
244	C₄⋊C₈	—	38	1	no	—
245	C₄⋊C₈	—	38	1	no	—
246	C₄⋊C₈	C₂²⋊D₁₉	1	2	yes	—
247	#103	—	38	2	no	—
248	#102	—	38	2	no	—
249	C₂²⋊C₈	—	19	2	no	—
250	C₂²⋊C₈	—	38	1	no	—
251	C₂²⋊C₈	—	76	2	no	—
252	C₂²⋊C₈	—	76	2	no	—
253	#97	—	2	2	no	—
254	#97	—	2	1	no	—
255	#97	—	2	1	no	—
256	C₄×C₈	—	2	2	no	—
257	C₄×C₈	—	38	1	no	—
258	C₄×C₈	—	38	1	no	—
259	C₁₉	#9050	1	1	yes	characteristic, complemented, Sylow
260	C₂⁴	—	38	1	no	—
261	C₂×Q₈	—	2	1	no	—
262	C₂×Q₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
263	C₂×D₄	—	38	1	no	—
264	C₂×D₄	—	38	1	no	—
265	C₂×D₄	—	76	2	no	—
266	C₂²×C₄	—	38	1	no	—
267	C₂²×C₄	—	76	1	no	—
268	C₂²×C₄	—	76	1	no	—
269	C₂²×C₄	—	76	1	no	—
270	C₂²×C₄	—	76	1	no	—
271	C₂²×C₄	—	76	1	no	—
272	C₂²×C₄	—	76	1	no	—
273	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
274	C₂²×C₄	#64541	1	1	yes	characteristic
275	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
276	SD₁₆	—	76	4	no	—
277	M₁₆	—	4	1	no	—
278	M₁₆	—	4	1	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	38	2	no	—
286	C₂×C₈	—	76	1	no	—
287	C₂×C₈	—	76	1	no	—
288	C₂×C₈	—	76	1	no	—
289	C₂×C₈	—	76	1	no	—
290	C₂×C₈	#64527	1	2	yes	complemented
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	2	1	no	—
293	C₄⋊C₄	—	2	1	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	38	1	no	—
297	C₄⋊C₄	—	38	1	no	—
298	C₄⋊C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
299	C₂²⋊C₄	—	2	2	no	—
300	C₂²⋊C₄	—	76	1	no	—
301	C₂²⋊C₄	—	76	1	no	—
302	C₄²	—	2	2	no	—
303	C₄²	C₂×D₇₆	1	1	yes	characteristic
304	C₄²	#64541	1	1	yes	characteristic
305	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
306	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
307	C₂³	—	38	2	no	—
308	C₂³	—	76	1	no	—
309	C₂³	—	76	1	no	—
310	C₂³	—	76	1	no	—
311	C₂³	—	76	1	no	—
312	C₂³	—	76	1	no	—
313	C₂³	—	76	1	no	—
314	C₂³	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
315	Q₈	—	4	2	no	—
316	Q₈	—	8	1	no	—
317	D₄	—	38	2	no	—
318	D₄	—	38	2	no	—
319	D₄	—	76	1	no	—
320	D₄	—	76	1	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	76	1	no	—
326	C₂×C₄	—	76	1	no	—
327	C₂×C₄	—	76	1	no	—
328	C₂×C₄	—	152	1	no	—
329	C₂×C₄	—	152	1	no	—
330	C₂×C₄	—	152	1	no	—
331	C₂×C₄	—	152	1	no	—
332	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
333	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
334	C₂×C₄	#10597375	1	1	yes	characteristic
335	C₂×C₄	#10597376	1	1	yes	characteristic
336	C₂×C₄	#10597376	1	1	yes	characteristic
337	C₂×C₄	C₂×#64527	1	1	yes	characteristic
338	C₂×C₄	#10597388	1	1	yes	characteristic
339	C₂×C₄	#10597395	1	1	yes	characteristic
340	C₂×C₄	#10597396	1	1	yes	characteristic
341	C₂×C₄	#10597399	1	1	yes	characteristic
342	C₈	—	2	2	no	—
343	C₈	—	4	1	no	—
344	C₈	—	4	2	no	—
345	C₈	—	76	2	no	—
346	C₈	—	152	1	no	—
347	C₈	—	152	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	76	2	no	—
351	C₂²	—	76	2	no	—
352	C₂²	—	76	2	no	—
353	C₂²	—	152	1	no	—
354	C₂²	—	152	1	no	—
355	C₂²	—	152	1	no	—
356	C₂²	—	152	1	no	—
357	C₂²	#11926423	1	1	yes	characteristic
358	C₂²	C₄×#64527	1	1	yes	characteristic
359	C₂²	#11926642	1	1	yes	characteristic
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	4	2	no	—
365	C₄	—	4	2	no	—
366	C₄	—	152	1	no	—
367	C₄	#11926683	1	1	yes	characteristic
368	C₄	#11926689	1	1	yes	characteristic
369	C₄	#11926744	1	1	yes	characteristic
370	C₄	#11926766	1	1	yes	characteristic
371	C₂	—	2	1	no	—
372	C₂	—	2	1	no	—
373	C₂	—	152	2	no	—
374	C₂	#14563990	1	1	yes	characteristic
375	C₂	#14563997	1	1	yes	characteristic
376	C₂	#14564053	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597399
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2961	C₂
socle	C₂²×C₁₉	#386
prime radical	C₂²×D₇₆	C₂×C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2961	D₁₉
hypocenter	C₁₉	#9050
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2961

Complements

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

N	H ≅ G/N	type
3. #14562292	C₂	semidirect
5. #14561530	C₂	semidirect
6. #14561480	C₂	semidirect
7. C₁₉×#2961	C₂	semidirect
95. C₂×C₈×C₁₉	SD₁₆	semidirect
161. #2961	D₁₉	semidirect
259. C₁₉	#9050	semidirect
290. C₂×C₈	#64527	semidirect

Maximal subgroups

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

subgroup	index
#14573639	2
#14564433	2
#14562292	2
#14562059	2
#14561530	2
#14561480	2
C₁₉×#2961	2
#9050	19