Subgroups — #18079226

Order 4864.

558 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572353	C₂	1	1	yes	maximal, characteristic, complemented
2	#14564460	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562303	C₂	1	1	yes	maximal, characteristic, complemented
4	#14562195	C₂	1	1	yes	maximal, characteristic, complemented
5	#14561494	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561465	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#3205	C₂	1	1	yes	maximal, characteristic
8	C₂×#10597394	—	2	1	no	—
9	#11926616	—	2	2	no	—
10	#11926616	—	2	2	no	—
11	#11926597	C₂²	1	1	yes	characteristic
12	#11926568	C₂²	1	1	yes	characteristic
13	C₂×#10597289	—	2	1	no	—
14	C₂×#10597289	—	2	1	no	—
15	#11926511	—	2	1	no	—
16	C₂×#10597260	C₂²	1	1	yes	characteristic
17	#11926410	C₂²	1	1	yes	characteristic
18	#11926288	—	2	2	no	—
19	#11926278	—	2	2	no	—
20	C₁₉×#532	C₂²	1	1	yes	characteristic
21	C₂×C₁₉×#127	C₂²	1	1	yes	characteristic
22	C₁₉×#493	C₄	1	2	yes	complemented
23	C₁₉×#486	C₄	1	2	yes	complemented
24	C₁₉×#459	—	2	2	no	—
25	C₁₉×#447	—	2	2	no	—
26	C₁₉×#431	C₂²	1	1	yes	characteristic
27	C₂×#41×C₁₉	—	2	1	no	—
28	C₂²×D₄×C₁₉	—	2	1	no	—
29	C₂²×D₄×C₁₉	—	2	1	no	—
30	C₂²×D₄×C₁₉	—	2	1	no	—
31	C₂×D₈×C₁₉	—	2	4	no	—
32	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
33	C₁₉×#127	—	2	1	no	—
34	C₁₉×#127	—	2	1	no	—
35	C₁₉×#127	—	2	1	no	—
36	C₁₉×#127	C₂×C₄	1	2	yes	—
37	C₁₉×#127	C₂×C₄	1	1	yes	characteristic
38	C₁₉×#124	—	2	1	no	—
39	C₁₉×#121	—	2	1	no	—
40	C₁₉×#121	D₄	1	2	yes	—
41	C₄×D₄×C₁₉	—	2	1	no	—
42	C₄×D₄×C₁₉	D₄	1	2	yes	—
43	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
44	#10597394	—	4	2	no	—
45	#10597394	—	4	2	no	—
46	C₂×C₄⋊Dic₁₉	—	2	1	no	—
47	C₂×#64520	—	2	2	no	—
48	C₂×#64520	—	2	1	no	—
49	C₂²×C₁₉⋊C₈	—	2	2	no	—
50	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
51	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
52	C₁₉×C₄⋊C₈	—	2	1	no	—
53	C₁₉×#102	—	2	4	no	—
54	C₁₉×C₂²⋊C₈	—	2	1	no	—
55	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
56	#10597289	—	4	2	no	—
57	#10597289	—	4	2	no	—
58	C₁₉⋊#95	—	2	1	no	—
59	C₁₉⋊#95	—	2	1	no	—
60	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
61	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
62	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
63	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
64	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
65	#10597260	—	2	1	no	—
66	#10597260	—	2	1	no	—
67	C₄×C₁₉⋊C₈	—	2	1	no	—
68	C₂⁴×C₁₉	—	4	1	no	—
69	#41×C₁₉	—	4	1	no	—
70	#41×C₁₉	—	4	1	no	—
71	#41×C₁₉	—	4	1	no	—
72	#41×C₁₉	—	4	1	no	—
73	C₂×Q₈×C₁₉	—	2	1	no	—
74	C₂×D₄×C₁₉	—	2	2	no	—
75	C₂×D₄×C₁₉	—	2	2	no	—
76	C₂×D₄×C₁₉	—	2	1	no	—
77	C₂×D₄×C₁₉	—	2	2	no	—
78	C₂×D₄×C₁₉	—	2	1	no	—
79	C₂×D₄×C₁₉	—	2	2	no	—
80	C₂×D₄×C₁₉	—	4	1	no	—
81	C₂×D₄×C₁₉	—	4	1	no	—
82	C₂×D₄×C₁₉	—	4	1	no	—
83	C₂×D₄×C₁₉	—	4	1	no	—
84	C₂×D₄×C₁₉	—	4	1	no	—
85	C₂×D₄×C₁₉	—	4	1	no	—
86	C₂×D₄×C₁₉	—	4	1	no	—
87	C₂×D₄×C₁₉	—	8	2	no	—
88	C₂×D₄×C₁₉	—	8	2	no	—
89	C₂×D₄×C₁₉	C₂²⋊C₄	1	2	yes	—
90	C₂²×C₄×C₁₉	—	2	2	no	—
91	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
94	C₂²×Dic₁₉	—	4	1	no	—
95	D₈×C₁₉	—	8	4	no	—
96	C₂×C₈×C₁₉	—	2	2	no	—
97	C₄⋊C₄×C₁₉	—	2	1	no	—
98	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
99	C₂²⋊C₄×C₁₉	—	2	2	no	—
100	C₂²⋊C₄×C₁₉	—	2	2	no	—
101	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₁₉	#41	1	1	yes	characteristic
105	C₄⋊Dic₁₉	—	4	1	no	—
106	C₄⋊Dic₁₉	—	4	1	no	—
107	#64520	—	4	4	no	—
108	#64520	—	4	1	no	—
109	#64520	—	4	1	no	—
110	C₂×C₁₉⋊C₈	—	2	1	no	—
111	C₂×C₁₉⋊C₈	—	2	1	no	—
112	C₂×C₁₉⋊C₈	—	2	1	no	—
113	C₂×C₁₉⋊C₈	—	2	2	no	—
114	C₂×C₁₉⋊C₈	—	2	1	no	—
115	C₂×C₁₉⋊C₈	—	2	1	no	—
116	C₂×C₁₉⋊C₈	—	4	4	no	—
117	C₂×C₁₉⋊C₈	—	4	1	no	—
118	C₂×C₁₉⋊C₈	D₈	1	2	yes	complemented
119	#9293	—	19	1	no	maximal, Sylow
120	C₂³×C₁₉	—	2	2	no	—
121	C₂³×C₁₉	—	4	2	no	—
122	C₂³×C₁₉	—	8	2	no	—
123	C₂³×C₁₉	—	8	2	no	—
124	C₂³×C₁₉	—	8	1	no	—
125	C₂³×C₁₉	—	8	1	no	—
126	C₂³×C₁₉	#121	1	1	yes	characteristic
127	Q₈×C₁₉	—	4	1	no	—
128	Q₈×C₁₉	—	4	1	no	—
129	D₄×C₁₉	—	4	2	no	—
130	D₄×C₁₉	—	4	4	no	—
131	D₄×C₁₉	—	4	2	no	—
132	D₄×C₁₉	—	4	2	no	—
133	D₄×C₁₉	—	4	1	no	—
134	D₄×C₁₉	—	4	2	no	—
135	D₄×C₁₉	—	4	1	no	—
136	D₄×C₁₉	—	4	2	no	—
137	D₄×C₁₉	—	8	1	no	—
138	D₄×C₁₉	—	8	2	no	—
139	D₄×C₁₉	—	8	2	no	—
140	D₄×C₁₉	—	8	2	no	—
141	D₄×C₁₉	—	8	1	no	—
142	D₄×C₁₉	—	8	1	no	—
143	C₂×C₄×C₁₉	—	2	2	no	—
144	C₂×C₄×C₁₉	—	4	1	no	—
145	C₂×C₄×C₁₉	—	4	1	no	—
146	C₂×C₄×C₁₉	—	4	2	no	—
147	C₂×C₄×C₁₉	—	4	2	no	—
148	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
151	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
152	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
153	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
154	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
156	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
157	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
158	C₂×Dic₁₉	—	8	1	no	—
159	C₂×Dic₁₉	—	8	1	no	—
160	C₂×Dic₁₉	—	8	1	no	—
161	C₈×C₁₉	—	4	2	no	—
162	C₁₉⋊C₈	—	2	2	no	—
163	C₁₉⋊C₈	—	4	2	no	—
164	C₁₉⋊C₈	—	4	1	no	—
165	C₁₉⋊C₈	—	8	1	no	—
166	#3205	D₁₉	1	1	yes	characteristic
167	#2932	—	19	1	no	—
168	#1961	—	19	1	no	—
169	#1545	—	19	1	no	—
170	#1517	—	19	1	no	—
171	#1286	—	19	1	no	—
172	#1275	—	19	1	no	—
173	C₂²×C₁₉	—	2	1	no	—
174	C₂²×C₁₉	—	2	1	no	—
175	C₂²×C₁₉	—	4	2	no	—
176	C₂²×C₁₉	—	4	2	no	—
177	C₂²×C₁₉	—	4	2	no	—
178	C₂²×C₁₉	—	8	2	no	—
179	C₂²×C₁₉	—	8	2	no	—
180	C₂²×C₁₉	—	8	2	no	—
181	C₂²×C₁₉	—	16	2	no	—
182	C₂²×C₁₉	—	16	2	no	—
183	C₂²×C₁₉	#386	1	1	yes	characteristic
184	C₂²×C₁₉	#459	1	1	yes	characteristic
185	C₂²×C₁₉	#463	1	1	yes	characteristic
186	C₄×C₁₉	—	2	1	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	C₄×D₈	1	1	yes	characteristic
192	C₄×C₁₉	#442	1	1	yes	characteristic
193	C₄×C₁₉	#447	1	1	yes	characteristic
194	C₄×C₁₉	#449	1	1	yes	characteristic
195	Dic₁₉	—	16	1	no	—
196	C₂×#131	—	38	1	no	—
197	#532	C₂×D₁₉	1	1	yes	characteristic
198	C₂×#127	C₂×D₁₉	1	1	yes	characteristic
199	#493	Dic₁₉	1	2	yes	complemented
200	#486	Dic₁₉	1	2	yes	complemented
201	#459	—	2	2	no	—
202	#447	—	2	2	no	—
203	#435	—	38	2	no	—
204	#435	—	38	2	no	—
205	#431	—	19	1	no	—
206	#431	C₂×D₁₉	1	1	yes	characteristic
207	#423	—	19	1	no	—
208	C₂×#102	—	38	1	no	—
209	C₂×#102	—	38	1	no	—
210	#408	—	38	1	no	—
211	C₂×#97	—	19	1	no	—
212	#382	—	19	1	no	—
213	#331	—	38	2	no	—
214	#327	—	38	2	no	—
215	C₂×C₁₉	—	2	1	no	—
216	C₂×C₁₉	—	2	1	no	—
217	C₂×C₁₉	—	8	2	no	—
218	C₂×C₁₉	—	16	2	no	—
219	C₂×C₁₉	#1879	1	1	yes	characteristic
220	C₂×C₁₉	#1903	1	1	yes	characteristic
221	C₂×C₁₉	#1910	1	1	yes	characteristic
222	C₂×#41	—	2	1	no	—
223	C₂²×D₄	—	2	1	no	—
224	C₂²×D₄	—	2	1	no	—
225	C₂²×D₄	—	2	1	no	—
226	C₂×D₈	—	2	4	no	—
227	#131	—	76	2	no	—
228	#131	—	76	2	no	—
229	C₂×M₁₆	—	38	2	no	—
230	C₂×M₁₆	—	38	1	no	—
231	C₂²×C₈	—	19	1	no	—
232	C₂²×C₈	—	19	1	no	—
233	C₂²×C₈	—	38	2	no	—
234	C₄⋊Q₈	C₂×Dic₁₉	1	1	yes	characteristic
235	#127	—	2	1	no	—
236	#127	—	2	1	no	—
237	#127	—	2	1	no	—
238	#127	C₂×Dic₁₉	1	2	yes	—
239	#127	C₂×Dic₁₉	1	1	yes	characteristic
240	#124	—	2	1	no	—
241	#121	—	2	1	no	—
242	#121	C₂²⋊D₁₉	1	2	yes	—
243	C₄×D₄	—	2	1	no	—
244	C₄×D₄	C₂²⋊D₁₉	1	2	yes	—
245	C₂×C₄⋊C₄	—	38	1	no	—
246	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
247	C₄⋊C₈	—	2	1	no	—
248	C₄⋊C₈	—	19	2	no	—
249	C₄⋊C₈	—	38	1	no	—
250	C₄⋊C₈	—	38	1	no	—
251	#102	—	2	4	no	—
252	#102	—	76	2	no	—
253	#102	—	76	2	no	—
254	C₂²⋊C₈	—	2	1	no	—
255	C₂²⋊C₈	—	38	4	no	—
256	C₂²⋊C₈	—	38	1	no	—
257	#97	—	38	1	no	—
258	#97	—	38	1	no	—
259	C₄×C₈	—	38	1	no	—
260	C₄×C₈	C₂×Dic₁₉	1	2	yes	—
261	#95	—	38	1	no	—
262	#95	—	38	1	no	—
263	C₁₉	#9293	1	1	yes	characteristic, complemented, Sylow
264	C₂⁴	—	4	1	no	—
265	#41	—	4	1	no	—
266	#41	—	4	1	no	—
267	#41	—	4	1	no	—
268	#41	—	4	1	no	—
269	C₂×Q₈	—	2	1	no	—
270	C₂×D₄	—	2	2	no	—
271	C₂×D₄	—	2	1	no	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	—	2	1	no	—
274	C₂×D₄	—	2	2	no	—
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	—	4	1	no	—
277	C₂×D₄	—	4	1	no	—
278	C₂×D₄	—	4	1	no	—
279	C₂×D₄	—	4	1	no	—
280	C₂×D₄	—	4	1	no	—
281	C₂×D₄	—	4	1	no	—
282	C₂×D₄	—	4	1	no	—
283	C₂×D₄	—	8	2	no	—
284	C₂×D₄	—	8	2	no	—
285	C₂×D₄	C₂²⋊Dic₁₉	1	2	yes	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	76	1	no	—
288	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
289	C₂²×C₄	#64543	1	1	yes	characteristic
290	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
291	D₈	—	8	4	no	—
292	M₁₆	—	76	4	no	—
293	M₁₆	—	76	1	no	—
294	M₁₆	—	76	1	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	19	2	no	—
297	C₂×C₈	—	38	1	no	—
298	C₂×C₈	—	38	1	no	—
299	C₂×C₈	—	38	1	no	—
300	C₂×C₈	—	38	2	no	—
301	C₂×C₈	—	38	1	no	—
302	C₂×C₈	—	38	1	no	—
303	C₂×C₈	—	76	4	no	—
304	C₂×C₈	—	76	1	no	—
305	C₄⋊C₄	—	2	1	no	—
306	C₄⋊C₄	—	76	1	no	—
307	C₄⋊C₄	—	76	1	no	—
308	C₄⋊C₄	C₂²⋊Dic₁₉	1	2	yes	—
309	C₂²⋊C₄	—	2	2	no	—
310	C₂²⋊C₄	—	2	2	no	—
311	C₄²	D₄×D₁₉	1	1	yes	characteristic
312	C₄²	#64543	1	1	yes	characteristic
313	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
314	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
315	C₂³	—	2	2	no	—
316	C₂³	—	4	2	no	—
317	C₂³	—	8	2	no	—
318	C₂³	—	8	2	no	—
319	C₂³	—	8	1	no	—
320	C₂³	—	8	1	no	—
321	C₂³	#10597385	1	1	yes	characteristic
322	Q₈	—	4	1	no	—
323	Q₈	—	4	1	no	—
324	D₄	—	4	2	no	—
325	D₄	—	4	4	no	—
326	D₄	—	4	2	no	—
327	D₄	—	4	2	no	—
328	D₄	—	4	1	no	—
329	D₄	—	4	2	no	—
330	D₄	—	4	1	no	—
331	D₄	—	4	2	no	—
332	D₄	—	8	1	no	—
333	D₄	—	8	2	no	—
334	D₄	—	8	2	no	—
335	D₄	—	8	2	no	—
336	D₄	—	8	1	no	—
337	D₄	—	8	1	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	4	2	no	—
342	C₂×C₄	—	4	2	no	—
343	C₂×C₄	—	152	1	no	—
344	C₂×C₄	—	152	1	no	—
345	C₂×C₄	—	152	1	no	—
346	C₂×C₄	D₈×D₁₉	1	1	yes	characteristic
347	C₂×C₄	#10597357	1	1	yes	characteristic
348	C₂×C₄	#10597357	1	1	yes	characteristic
349	C₂×C₄	#10597358	1	1	yes	characteristic
350	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
351	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
352	C₂×C₄	#10597381	1	1	yes	characteristic
353	C₂×C₄	#10597383	1	1	yes	characteristic
354	C₂×C₄	#10597385	1	1	yes	characteristic
355	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
356	C₈	—	4	2	no	—
357	C₈	—	38	2	no	—
358	C₈	—	76	2	no	—
359	C₈	—	76	1	no	—
360	C₈	—	152	1	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	4	2	no	—
364	C₂²	—	4	2	no	—
365	C₂²	—	4	2	no	—
366	C₂²	—	8	2	no	—
367	C₂²	—	8	2	no	—
368	C₂²	—	8	2	no	—
369	C₂²	—	16	2	no	—
370	C₂²	—	16	2	no	—
371	C₂²	#11926424	1	1	yes	characteristic
372	C₂²	#11926737	1	1	yes	characteristic
373	C₂²	#11926757	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	4	2	no	—
379	C₄	—	304	1	no	—
380	C₄	D₈×Dic₁₉	1	1	yes	characteristic
381	C₄	#11926656	1	1	yes	characteristic
382	C₄	#11926676	1	1	yes	characteristic
383	C₄	#11926686	1	1	yes	characteristic
384	C₂	—	2	1	no	—
385	C₂	—	2	1	no	—
386	C₂	—	8	2	no	—
387	C₂	—	16	2	no	—
388	C₂	#14563985	1	1	yes	characteristic
389	C₂	#14564117	1	1	yes	characteristic
390	C₂	#14564163	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926424
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting, prime radical	C₁₉×#3205	C₂
socle	C₂²×C₁₉	#386
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#3205	D₁₉
hypocenter	C₁₉	#9293
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3205

Complements

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

N	H ≅ G/N	type
1. #14572353	C₂	semidirect
2. #14564460	C₂	semidirect
3. #14562303	C₂	semidirect
4. #14562195	C₂	semidirect
5. #14561494	C₂	semidirect
6. #14561465	C₂	semidirect
22. C₁₉×#493	C₄	semidirect
23. C₁₉×#486	C₄	semidirect
118. C₂×C₁₉⋊C₈	D₈	semidirect
199. #493	Dic₁₉	semidirect
200. #486	Dic₁₉	semidirect
263. C₁₉	#9293	semidirect

Maximal subgroups

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

subgroup	index
#14572353	2
#14564460	2
#14562303	2
#14562195	2
#14561494	2
#14561465	2
C₁₉×#3205	2
#9293	19