Subgroups — #18078573

Order 4864.

624 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572199	C₂	1	1	yes	maximal, characteristic, complemented
2	#14562303	C₂	1	2	yes	maximal, complemented
3	#14561949	C₂	1	1	yes	maximal, characteristic, complemented
4	#14561493	C₂	1	2	yes	maximal, complemented
5	C₁₉×#2619	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	—	2	1	no	—
7	#11926597	C₂²	1	2	yes	—
8	#11926571	—	2	4	no	—
9	#11926568	—	2	1	no	—
10	#11926568	C₂²	1	2	yes	—
11	#11926288	—	2	4	no	—
12	#11926270	—	2	1	no	—
13	#11926268	—	2	1	no	—
14	C₂×C₁₉×#127	C₂²	1	1	yes	characteristic
15	C₂×C₁₉×#121	—	2	1	no	—
16	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
17	C₁₉×#394	—	2	1	no	—
18	C₁₉×#394	C₄	1	2	yes	—
19	C₁₉×#392	—	2	1	no	—
20	C₁₉×#392	C₄	1	2	yes	—
21	C₁₉×#390	—	2	1	no	—
22	C₁₉×#382	C₂²	1	1	yes	characteristic
23	C₂²×D₄×C₁₉	—	2	2	no	—
24	C₂²×D₄×C₁₉	—	2	1	no	—
25	C₂²×D₄×C₁₉	—	2	1	no	—
26	C₂²×D₄×C₁₉	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
28	C₁₉×#127	—	2	2	no	—
29	C₁₉×#127	—	2	2	no	—
30	C₁₉×#121	—	4	2	no	—
31	C₁₉×#121	—	4	2	no	—
32	C₁₉×#117	—	2	2	no	—
33	C₁₉×#117	—	2	2	no	—
34	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
35	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
38	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
39	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
40	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
44	C₂×#64520	—	2	2	no	—
45	C₂×#64520	—	2	2	no	—
46	C₂×#64520	—	4	2	no	—
47	C₂×#64520	—	4	2	no	—
48	C₂²×C₁₉⋊C₈	—	2	2	no	—
49	C₁₉×#95	—	2	1	no	—
50	C₁₉×#95	C₂×C₄	1	2	yes	—
51	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
52	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
53	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
54	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
55	C₄⋊(C₁₉⋊C₈)	D₄	1	4	yes	complemented
56	C₄×C₁₉⋊C₈	—	2	2	no	—
57	C₂⁴×C₁₉	—	2	2	no	—
58	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×D₄×C₁₉	—	2	4	no	—
60	C₂×D₄×C₁₉	—	4	4	no	—
61	C₂×D₄×C₁₉	—	4	1	no	—
62	C₂×D₄×C₁₉	—	4	2	no	—
63	C₂×D₄×C₁₉	—	4	1	no	—
64	C₂×D₄×C₁₉	—	4	1	no	—
65	C₂×D₄×C₁₉	—	4	1	no	—
66	C₂×D₄×C₁₉	—	4	1	no	—
67	C₂×D₄×C₁₉	—	4	1	no	—
68	C₂×D₄×C₁₉	—	8	2	no	—
69	C₂×D₄×C₁₉	—	8	2	no	—
70	C₂×D₄×C₁₉	—	8	2	no	—
71	C₂×D₄×C₁₉	—	8	2	no	—
72	C₂²×C₄×C₁₉	—	2	1	no	—
73	C₂²×C₄×C₁₉	—	2	1	no	—
74	C₂²×C₄×C₁₉	—	2	1	no	—
75	C₂²×C₄×C₁₉	—	2	1	no	—
76	C₂²×C₄×C₁₉	—	2	2	no	—
77	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₉	D₈	1	2	yes	—
81	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
82	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
85	C₄⋊C₄×C₁₉	—	2	2	no	—
86	C₄⋊C₄×C₁₉	—	2	2	no	—
87	C₄⋊C₄×C₁₉	—	4	1	no	—
88	C₄⋊C₄×C₁₉	—	4	1	no	—
89	C₂²⋊C₄×C₁₉	—	2	2	no	—
90	C₂²⋊C₄×C₁₉	—	2	2	no	—
91	C₂²⋊C₄×C₁₉	—	4	4	no	—
92	C₂²⋊C₄×C₁₉	—	8	2	no	—
93	C₂²⋊C₄×C₁₉	—	8	2	no	—
94	C₄²×C₁₉	—	2	1	no	—
95	C₄²×C₁₉	—	2	1	no	—
96	C₄²×C₁₉	C₂×D₄	1	2	yes	—
97	C₄²×C₁₉	#41	1	2	yes	—
98	#64520	—	4	4	no	—
99	#64520	—	4	2	no	—
100	#64520	—	4	2	no	—
101	C₂×C₁₉⋊C₈	—	2	4	no	—
102	C₂×C₁₉⋊C₈	—	4	4	no	—
103	C₂×C₁₉⋊C₈	—	4	2	no	—
104	#9197	—	19	1	no	maximal, Sylow
105	C₂³×C₁₉	—	2	1	no	—
106	C₂³×C₁₉	—	2	1	no	—
107	C₂³×C₁₉	—	4	1	no	—
108	C₂³×C₁₉	—	4	1	no	—
109	C₂³×C₁₉	—	4	2	no	—
110	C₂³×C₁₉	—	4	2	no	—
111	C₂³×C₁₉	—	4	2	no	—
112	C₂³×C₁₉	—	4	2	no	—
113	C₂³×C₁₉	—	4	2	no	—
114	C₂³×C₁₉	—	4	2	no	—
115	C₂³×C₁₉	—	4	2	no	—
116	C₂³×C₁₉	#102	1	2	yes	—
117	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	D₄×C₁₉	—	8	4	no	—
119	D₄×C₁₉	—	8	4	no	—
120	D₄×C₁₉	—	8	8	no	—
121	D₄×C₁₉	—	8	1	no	—
122	D₄×C₁₉	—	8	1	no	—
123	D₄×C₁₉	—	8	1	no	—
124	D₄×C₁₉	—	8	1	no	—
125	C₂×C₄×C₁₉	—	2	1	no	—
126	C₂×C₄×C₁₉	—	2	1	no	—
127	C₂×C₄×C₁₉	—	2	2	no	—
128	C₂×C₄×C₁₉	—	2	2	no	—
129	C₂×C₄×C₁₉	—	2	2	no	—
130	C₂×C₄×C₁₉	—	2	2	no	—
131	C₂×C₄×C₁₉	—	2	1	no	—
132	C₂×C₄×C₁₉	—	2	1	no	—
133	C₂×C₄×C₁₉	—	4	1	no	—
134	C₂×C₄×C₁₉	—	4	1	no	—
135	C₂×C₄×C₁₉	—	4	1	no	—
136	C₂×C₄×C₁₉	—	4	1	no	—
137	C₂×C₄×C₁₉	—	4	2	no	—
138	C₂×C₄×C₁₉	—	4	2	no	—
139	C₂×C₄×C₁₉	—	4	2	no	—
140	C₂×C₄×C₁₉	#102	1	2	yes	—
141	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
142	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
143	C₂×C₄×C₁₉	#121	1	2	yes	—
144	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
145	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
147	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
149	C₁₉⋊C₈	—	4	4	no	—
150	C₁₉⋊C₈	—	8	2	no	—
151	#2905	—	19	1	no	—
152	#2619	D₁₉	1	1	yes	characteristic
153	#1545	—	19	2	no	—
154	#1458	—	19	1	no	—
155	#1286	—	19	2	no	—
156	C₂²×C₁₉	—	2	1	no	—
157	C₂²×C₁₉	—	2	1	no	—
158	C₂²×C₁₉	—	2	2	no	—
159	C₂²×C₁₉	—	4	2	no	—
160	C₂²×C₁₉	—	4	1	no	—
161	C₂²×C₁₉	—	4	1	no	—
162	C₂²×C₁₉	—	8	1	no	—
163	C₂²×C₁₉	—	8	2	no	—
164	C₂²×C₁₉	—	8	2	no	—
165	C₂²×C₁₉	—	8	2	no	—
166	C₂²×C₁₉	—	8	2	no	—
167	C₂²×C₁₉	—	8	2	no	—
168	C₂²×C₁₉	—	8	2	no	—
169	C₂²×C₁₉	—	8	2	no	—
170	C₂²×C₁₉	#386	1	1	yes	characteristic
171	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
172	C₂²×C₁₉	#417	1	1	yes	characteristic
173	C₄×C₁₉	—	2	2	no	—
174	C₄×C₁₉	—	2	2	no	—
175	C₄×C₁₉	—	2	2	no	—
176	C₄×C₁₉	—	4	1	no	—
177	C₄×C₁₉	—	4	1	no	—
178	C₄×C₁₉	—	8	2	no	—
179	C₄×C₁₉	#447	1	1	yes	characteristic
180	C₄×C₁₉	#448	1	1	yes	characteristic
181	C₄×C₁₉	#480	1	1	yes	characteristic
182	C₄×C₁₉	#481	1	1	yes	characteristic
183	C₂²×M₁₆	—	38	1	no	—
184	C₂×#127	C₂×D₁₉	1	1	yes	characteristic
185	C₂×#121	—	2	1	no	—
186	C₂×#117	C₂×D₁₉	1	1	yes	characteristic
187	#431	—	19	2	no	—
188	#424	—	38	4	no	—
189	#423	—	19	2	no	—
190	#423	—	38	1	no	—
191	#394	—	2	1	no	—
192	#394	Dic₁₉	1	2	yes	—
193	#392	—	2	1	no	—
194	#392	Dic₁₉	1	2	yes	—
195	#390	—	2	1	no	—
196	#382	C₂×D₁₉	1	1	yes	characteristic
197	#331	—	38	4	no	—
198	#324	—	38	1	no	—
199	#323	—	38	1	no	—
200	C₂×C₁₉	—	2	1	no	—
201	C₂×C₁₉	—	2	1	no	—
202	C₂×C₁₉	—	4	2	no	—
203	C₂×C₁₉	—	16	2	no	—
204	C₂×C₁₉	#1874	1	1	yes	characteristic
205	C₂×C₁₉	#1910	1	1	yes	characteristic
206	C₂×C₁₉	#1914	1	1	yes	characteristic
207	C₂²×D₄	—	2	2	no	—
208	C₂²×D₄	—	2	1	no	—
209	C₂²×D₄	—	2	1	no	—
210	C₂²×D₄	C₂×Dic₁₉	1	2	yes	—
211	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
212	C₂×M₁₆	—	38	2	no	—
213	C₂×M₁₆	—	38	2	no	—
214	C₂×M₁₆	—	76	2	no	—
215	C₂×M₁₆	—	76	2	no	—
216	C₂²×C₈	—	38	2	no	—
217	#127	—	2	2	no	—
218	#127	—	2	2	no	—
219	#121	—	4	2	no	—
220	#121	—	4	2	no	—
221	#117	—	2	2	no	—
222	#117	—	2	2	no	—
223	C₂×C₄⋊C₄	—	2	1	no	—
224	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
225	C₂×C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
226	C₂×C₂²⋊C₄	—	2	2	no	—
227	C₂×C₂²⋊C₄	—	2	2	no	—
228	C₂×C₂²⋊C₄	—	2	2	no	—
229	C₂×C₂²⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
230	C₂×C₂²⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
231	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
232	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
233	C₄⋊C₈	—	19	4	no	—
234	C₄⋊C₈	—	38	4	no	—
235	C₄⋊C₈	—	38	2	no	—
236	C₄⋊C₈	—	38	2	no	—
237	C₂²⋊C₈	—	38	2	no	—
238	C₄×C₈	—	38	2	no	—
239	#95	—	2	1	no	—
240	#95	C₂×Dic₁₉	1	2	yes	—
241	C₁₉	#9197	1	1	yes	characteristic, complemented, Sylow
242	C₂⁴	—	2	2	no	—
243	C₂⁴	C₂²⋊Dic₁₉	1	1	yes	characteristic
244	C₂×D₄	—	2	4	no	—
245	C₂×D₄	—	4	4	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂×D₄	—	4	1	no	—
250	C₂×D₄	—	4	2	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	8	2	no	—
254	C₂×D₄	—	8	2	no	—
255	C₂×D₄	—	8	2	no	—
256	C₂×D₄	—	8	2	no	—
257	C₂²×C₄	—	2	1	no	—
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	—	2	1	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	#64525	1	2	yes	—
263	C₂²×C₄	#64526	1	2	yes	—
264	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
265	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
266	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
267	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
268	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
269	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
270	M₁₆	—	76	4	no	—
271	M₁₆	—	76	2	no	—
272	M₁₆	—	76	2	no	—
273	C₂×C₈	—	38	4	no	—
274	C₂×C₈	—	76	4	no	—
275	C₂×C₈	—	76	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₂²⋊C₄	—	2	2	no	—
282	C₂²⋊C₄	—	4	4	no	—
283	C₂²⋊C₄	—	8	2	no	—
284	C₂²⋊C₄	—	8	2	no	—
285	C₄²	—	2	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	D₄×D₁₉	1	2	yes	—
288	C₄²	#64543	1	2	yes	—
289	C₂³	—	2	1	no	—
290	C₂³	—	2	1	no	—
291	C₂³	—	4	1	no	—
292	C₂³	—	4	1	no	—
293	C₂³	—	4	2	no	—
294	C₂³	—	4	2	no	—
295	C₂³	—	4	2	no	—
296	C₂³	—	4	2	no	—
297	C₂³	—	4	2	no	—
298	C₂³	—	4	2	no	—
299	C₂³	—	4	2	no	—
300	C₂³	#10597289	1	2	yes	—
301	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
302	D₄	—	8	4	no	—
303	D₄	—	8	4	no	—
304	D₄	—	8	8	no	—
305	D₄	—	8	1	no	—
306	D₄	—	8	1	no	—
307	D₄	—	8	1	no	—
308	D₄	—	8	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	#10597289	1	2	yes	—
325	C₂×C₄	C₂×#64525	1	1	yes	characteristic
326	C₂×C₄	C₂×#64526	1	1	yes	characteristic
327	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
328	C₂×C₄	#10597381	1	1	yes	characteristic
329	C₂×C₄	#10597383	1	1	yes	characteristic
330	C₂×C₄	#10597385	1	2	yes	—
331	C₂×C₄	#10597395	1	1	yes	characteristic
332	C₂×C₄	#10597397	1	1	yes	characteristic
333	C₈	—	76	4	no	—
334	C₈	—	152	2	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	4	2	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	8	1	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	8	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	—	8	2	no	—
347	C₂²	—	8	2	no	—
348	C₂²	—	8	2	no	—
349	C₂²	#11926424	1	1	yes	characteristic
350	C₂²	C₂×#10597289	1	1	yes	characteristic
351	C₂²	#11926545	1	1	yes	characteristic
352	C₄	—	2	2	no	—
353	C₄	—	2	2	no	—
354	C₄	—	2	2	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	8	2	no	—
358	C₄	#11926677	1	1	yes	characteristic
359	C₄	#11926682	1	1	yes	characteristic
360	C₄	#11926855	1	1	yes	characteristic
361	C₄	#11926860	1	1	yes	characteristic
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	4	2	no	—
365	C₂	—	16	2	no	—
366	C₂	#14563953	1	1	yes	characteristic
367	C₂	#14564163	1	1	yes	characteristic
368	C₂	#14564183	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926424
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting, prime radical	C₁₉×#2619	C₂
socle	C₂²×C₁₉	#386
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2619	D₁₉
hypocenter	C₁₉	#9197
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2619

Complements

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

N	H ≅ G/N	type
1. #14572199	C₂	semidirect
2. #14562303	C₂	semidirect
3. #14561949	C₂	semidirect
4. #14561493	C₂	semidirect
55. C₄⋊(C₁₉⋊C₈)	D₄	semidirect
241. C₁₉	#9197	semidirect

Maximal subgroups

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

subgroup	index
#14572199	2
#14562303	2
#14561949	2
#14561493	2
C₁₉×#2619	2
#9197	19