Subgroups — #18079240

Order 4864.

594 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14566361	C₂	1	1	yes	maximal, characteristic
2	#14564617	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562303	C₂	1	1	yes	maximal, characteristic
4	#14562199	C₂	1	1	yes	maximal, characteristic, complemented
5	#14561494	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561463	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#3208	C₂	1	1	yes	maximal, characteristic
8	C₂×D₄×Dic₁₉	—	2	1	no	—
9	#11926597	C₂²	1	1	yes	characteristic
10	#11926568	C₂²	1	1	yes	characteristic
11	#11926447	—	2	1	no	—
12	#11926424	—	2	2	no	—
13	#11926424	—	2	2	no	—
14	#11926410	C₂²	1	1	yes	characteristic
15	#11926390	C₂²	1	1	yes	characteristic
16	#11926303	—	2	1	no	—
17	#11926303	—	2	1	no	—
18	#11926288	—	2	2	no	—
19	#11926281	—	2	2	no	—
20	C₁₉×#533	C₂²	1	1	yes	characteristic
21	C₂×C₁₉×#127	C₂²	1	1	yes	characteristic
22	C₁₉×#492	C₄	1	2	yes	complemented
23	C₁₉×#486	C₄	1	2	yes	complemented
24	C₁₉×#460	—	2	2	no	—
25	C₁₉×#450	—	2	2	no	—
26	C₁₉×#431	C₂²	1	1	yes	characteristic
27	C₂²×D₄×C₁₉	—	2	1	no	—
28	C₂²×D₄×C₁₉	—	2	1	no	—
29	C₂²×D₄×C₁₉	D₄	1	2	yes	—
30	C₂³×Dic₁₉	—	2	2	no	—
31	C₂×SD₁₆×C₁₉	—	2	4	no	—
32	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
33	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic, complemented
34	C₁₉×#127	—	2	1	no	—
35	C₁₉×#127	—	2	1	no	—
36	C₁₉×#127	C₂×C₄	1	2	yes	—
37	C₁₉×#127	D₄	1	2	yes	—
38	C₁₉×#125	—	2	1	no	—
39	C₁₉×C₂²⋊Q₈	—	2	1	no	—
40	C₁₉×C₂²⋊Q₈	—	2	1	no	—
41	C₄×Q₈×C₁₉	—	2	1	no	—
42	C₄×Q₈×C₁₉	—	2	1	no	—
43	C₁₉×#117	—	2	1	no	—
44	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
45	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
46	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
47	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
48	D₄×Dic₁₉	—	4	2	no	—
49	D₄×Dic₁₉	—	4	2	no	—
50	C₂×C₄⋊Dic₁₉	—	2	1	no	—
51	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
52	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
53	C₂×#64520	—	2	1	no	—
54	C₁₉×C₄⋊C₈	—	2	1	no	—
55	C₁₉×#102	—	2	4	no	—
56	C₁₉×C₂²⋊C₈	—	2	1	no	—
57	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	complemented
58	C₁₉⋊#95	—	2	1	no	—
59	C₁₉⋊#95	—	2	1	no	—
60	C₁₉⋊#95	—	2	1	no	—
61	C₁₉⋊#95	D₄	1	2	yes	—
62	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
63	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
64	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
65	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
66	C₄×C₁₉⋊C₈	—	2	1	no	—
67	C₂⁴×C₁₉	—	2	2	no	—
68	C₂×Q₈×C₁₉	—	2	1	no	—
69	C₂×Q₈×C₁₉	—	2	1	no	—
70	C₂×D₄×C₁₉	—	2	2	no	—
71	C₂×D₄×C₁₉	—	2	2	no	—
72	C₂×D₄×C₁₉	—	2	2	no	—
73	C₂×D₄×C₁₉	—	2	2	no	—
74	C₂×D₄×C₁₉	—	2	2	no	—
75	C₂×D₄×C₁₉	—	4	1	no	—
76	C₂×D₄×C₁₉	—	4	1	no	—
77	C₂×D₄×C₁₉	—	4	1	no	—
78	C₂×D₄×C₁₉	—	4	1	no	—
79	C₂×D₄×C₁₉	—	8	2	no	—
80	C₂×D₄×C₁₉	—	8	2	no	—
81	C₂×D₄×C₁₉	C₂²⋊C₄	1	4	yes	—
82	C₂²×C₄×C₁₉	C₂×D₄	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	C₂²×Dic₁₉	—	2	1	no	—
86	C₂²×Dic₁₉	—	2	1	no	—
87	C₂²×Dic₁₉	—	2	2	no	—
88	C₂²×Dic₁₉	—	4	4	no	—
89	C₂²×Dic₁₉	—	8	2	no	—
90	C₂²×Dic₁₉	—	8	2	no	—
91	SD₁₆×C₁₉	—	8	4	no	—
92	C₂×C₈×C₁₉	—	2	2	no	—
93	C₄⋊C₄×C₁₉	—	2	1	no	—
94	C₄⋊C₄×C₁₉	—	2	1	no	—
95	C₄⋊C₄×C₁₉	—	2	1	no	—
96	C₄⋊C₄×C₁₉	—	2	1	no	—
97	C₄⋊C₄×C₁₉	—	4	1	no	—
98	C₄⋊C₄×C₁₉	—	4	1	no	—
99	C₂²⋊C₄×C₁₉	—	4	1	no	—
100	C₄²×C₁₉	—	4	1	no	—
101	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₁₉	#41	1	1	yes	characteristic
104	C₄²×C₁₉	#41	1	1	yes	characteristic
105	C₂²⋊Dic₁₉	—	4	4	no	—
106	C₂²⋊Dic₁₉	—	8	2	no	—
107	C₂²⋊Dic₁₉	—	8	2	no	—
108	C₄⋊Dic₁₉	—	4	1	no	—
109	C₄⋊Dic₁₉	—	4	1	no	—
110	C₄×Dic₁₉	—	2	1	no	—
111	C₄×Dic₁₉	—	2	1	no	—
112	C₄×Dic₁₉	—	2	1	no	—
113	C₄×Dic₁₉	SD₁₆	1	2	yes	complemented
114	#64520	—	4	1	no	—
115	#64520	—	4	1	no	—
116	C₂×C₁₉⋊C₈	—	4	1	no	—
117	C₂×C₁₉⋊C₈	—	4	1	no	—
118	#9295	—	19	1	no	maximal, Sylow
119	C₂³×C₁₉	—	2	4	no	—
120	C₂³×C₁₉	—	4	2	no	—
121	C₂³×C₁₉	—	4	2	no	—
122	C₂³×C₁₉	—	4	2	no	—
123	C₂³×C₁₉	—	4	2	no	—
124	C₂³×C₁₉	—	4	2	no	—
125	C₂³×C₁₉	—	4	2	no	—
126	C₂³×C₁₉	#120	1	1	yes	characteristic
127	Q₈×C₁₉	—	4	2	no	—
128	Q₈×C₁₉	—	8	1	no	—
129	D₄×C₁₉	—	4	4	no	—
130	D₄×C₁₉	—	4	2	no	—
131	D₄×C₁₉	—	4	2	no	—
132	D₄×C₁₉	—	4	2	no	—
133	D₄×C₁₉	—	4	2	no	—
134	D₄×C₁₉	—	8	2	no	—
135	D₄×C₁₉	—	8	2	no	—
136	D₄×C₁₉	—	8	2	no	—
137	C₂×C₄×C₁₉	—	4	1	no	—
138	C₂×C₄×C₁₉	—	4	1	no	—
139	C₂×C₄×C₁₉	—	4	1	no	—
140	C₂×C₄×C₁₉	—	4	1	no	—
141	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
145	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
146	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
147	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
148	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
149	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
151	C₂×Dic₁₉	—	2	2	no	—
152	C₂×Dic₁₉	—	4	1	no	—
153	C₂×Dic₁₉	—	4	1	no	—
154	C₂×Dic₁₉	—	4	1	no	—
155	C₂×Dic₁₉	—	4	1	no	—
156	C₂×Dic₁₉	—	4	2	no	—
157	C₂×Dic₁₉	—	4	2	no	—
158	C₂×Dic₁₉	—	4	2	no	—
159	C₂×Dic₁₉	—	4	1	no	—
160	C₂×Dic₁₉	—	8	8	no	—
161	C₈×C₁₉	—	4	2	no	—
162	C₁₉⋊C₈	—	8	1	no	—
163	C₁₉⋊C₈	—	8	1	no	—
164	#3208	D₁₉	1	1	yes	characteristic
165	#2328	—	19	1	no	—
166	#1995	—	19	1	no	—
167	#1545	—	19	1	no	—
168	#1518	—	19	1	no	—
169	#1286	—	19	1	no	—
170	#1274	—	19	1	no	—
171	C₂²×C₁₉	—	2	1	no	—
172	C₂²×C₁₉	—	2	1	no	—
173	C₂²×C₁₉	—	4	4	no	—
174	C₂²×C₁₉	—	4	4	no	—
175	C₂²×C₁₉	—	4	4	no	—
176	C₂²×C₁₉	—	8	2	no	—
177	C₂²×C₁₉	—	8	2	no	—
178	C₂²×C₁₉	—	8	2	no	—
179	C₂²×C₁₉	—	8	2	no	—
180	C₂²×C₁₉	#386	1	1	yes	characteristic
181	C₂²×C₁₉	#450	1	1	yes	characteristic
182	C₂²×C₁₉	#452	1	1	yes	characteristic
183	C₄×C₁₉	—	2	1	no	—
184	C₄×C₁₉	—	2	1	no	—
185	C₄×C₁₉	—	2	1	no	—
186	C₄×C₁₉	—	2	1	no	—
187	C₄×C₁₉	—	8	1	no	—
188	C₄×C₁₉	—	8	1	no	—
189	C₄×C₁₉	C₄×SD₁₆	1	1	yes	characteristic
190	C₄×C₁₉	#440	1	1	yes	characteristic
191	C₄×C₁₉	#461	1	1	yes	characteristic
192	C₄×C₁₉	#463	1	1	yes	characteristic
193	Dic₁₉	—	4	2	no	—
194	Dic₁₉	—	8	1	no	—
195	Dic₁₉	—	8	2	no	—
196	#533	C₂×D₁₉	1	1	yes	characteristic
197	C₂×#127	C₂×D₁₉	1	1	yes	characteristic
198	C₂×C₄×D₄	—	38	1	no	—
199	#492	Dic₁₉	1	2	yes	complemented
200	#486	Dic₁₉	1	2	yes	complemented
201	#460	—	2	2	no	—
202	#450	—	2	2	no	—
203	#431	—	19	1	no	—
204	#431	C₂×D₁₉	1	1	yes	characteristic
205	#423	—	19	1	no	—
206	#390	—	38	1	no	—
207	#386	—	38	2	no	—
208	#386	—	38	2	no	—
209	#382	—	19	1	no	—
210	#376	—	19	1	no	—
211	#339	—	38	1	no	—
212	#339	—	38	1	no	—
213	#331	—	38	2	no	—
214	#328	—	38	2	no	—
215	C₂×C₁₉	—	2	1	no	—
216	C₂×C₁₉	—	2	1	no	—
217	C₂×C₁₉	—	8	4	no	—
218	C₂×C₁₉	#1877	1	1	yes	characteristic
219	C₂×C₁₉	#1904	1	1	yes	characteristic
220	C₂×C₁₉	#1911	1	1	yes	characteristic
221	C₂²×D₄	—	2	1	no	—
222	C₂²×D₄	—	2	1	no	—
223	C₂²×D₄	C₂²⋊D₁₉	1	2	yes	—
224	C₂³×C₄	—	38	2	no	—
225	C₂×SD₁₆	—	2	4	no	—
226	C₂×M₁₆	—	38	1	no	—
227	C₄⋊Q₈	C₂×Dic₁₉	1	1	yes	characteristic, complemented
228	C₄⋊Q₈	C₂×Dic₁₉	1	1	yes	characteristic, complemented
229	#127	—	2	1	no	—
230	#127	—	2	1	no	—
231	#127	C₂×Dic₁₉	1	2	yes	—
232	#127	C₂²⋊D₁₉	1	2	yes	—
233	#125	—	2	1	no	—
234	C₂²⋊Q₈	—	2	1	no	—
235	C₂²⋊Q₈	—	2	1	no	—
236	C₄×Q₈	—	2	1	no	—
237	C₄×Q₈	—	2	1	no	—
238	C₄×D₄	—	76	2	no	—
239	C₄×D₄	—	76	2	no	—
240	#117	—	2	1	no	—
241	C₂×C₄⋊C₄	—	38	1	no	—
242	C₂×C₂²⋊C₄	—	38	2	no	—
243	C₂×C₂²⋊C₄	—	38	2	no	—
244	C₂×C₂²⋊C₄	—	38	2	no	—
245	C₂×C₄²	—	19	1	no	—
246	C₂×C₄²	—	19	1	no	—
247	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
248	C₄⋊C₈	—	2	1	no	—
249	C₄⋊C₈	—	38	1	no	—
250	C₄⋊C₈	—	38	1	no	—
251	C₄⋊C₈	—	38	1	no	—
252	#102	—	2	4	no	—
253	C₂²⋊C₈	—	2	1	no	—
254	C₂²⋊C₈	—	38	1	no	—
255	C₄×C₈	—	38	1	no	—
256	C₄×C₈	C₂×Dic₁₉	1	2	yes	complemented
257	#95	—	19	2	no	—
258	#95	—	38	1	no	—
259	#95	—	38	1	no	—
260	#95	—	38	1	no	—
261	C₁₉	#9295	1	1	yes	characteristic, complemented, Sylow
262	C₂⁴	—	2	2	no	—
263	C₂×Q₈	—	2	1	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	—	2	2	no	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	—	4	1	no	—
271	C₂×D₄	—	4	1	no	—
272	C₂×D₄	—	4	1	no	—
273	C₂×D₄	—	4	1	no	—
274	C₂×D₄	—	8	2	no	—
275	C₂×D₄	—	8	2	no	—
276	C₂×D₄	C₂²⋊Dic₁₉	1	4	yes	—
277	C₂²×C₄	—	38	1	no	—
278	C₂²×C₄	—	38	2	no	—
279	C₂²×C₄	—	38	1	no	—
280	C₂²×C₄	—	76	4	no	—
281	C₂²×C₄	—	152	2	no	—
282	C₂²×C₄	—	152	2	no	—
283	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
284	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
285	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
286	SD₁₆	—	8	4	no	—
287	M₁₆	—	76	1	no	—
288	M₁₆	—	76	1	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	76	1	no	—
291	C₂×C₈	—	76	1	no	—
292	C₄⋊C₄	—	2	1	no	—
293	C₄⋊C₄	—	2	1	no	—
294	C₄⋊C₄	—	2	1	no	—
295	C₄⋊C₄	—	2	1	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₄⋊C₄	—	4	1	no	—
298	C₄⋊C₄	—	76	1	no	—
299	C₄⋊C₄	—	76	1	no	—
300	C₂²⋊C₄	—	4	1	no	—
301	C₂²⋊C₄	—	76	4	no	—
302	C₂²⋊C₄	—	152	2	no	—
303	C₂²⋊C₄	—	152	2	no	—
304	C₄²	—	4	1	no	—
305	C₄²	—	19	2	no	—
306	C₄²	—	38	1	no	—
307	C₄²	—	38	1	no	—
308	C₄²	—	38	1	no	—
309	C₄²	#64543	1	1	yes	characteristic
310	C₄²	#64543	1	1	yes	characteristic
311	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
312	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
313	C₂³	—	2	4	no	—
314	C₂³	—	4	2	no	—
315	C₂³	—	4	2	no	—
316	C₂³	—	4	2	no	—
317	C₂³	—	4	2	no	—
318	C₂³	—	4	2	no	—
319	C₂³	—	4	2	no	—
320	C₂³	#10597383	1	1	yes	characteristic
321	Q₈	—	4	2	no	—
322	Q₈	—	8	1	no	—
323	D₄	—	4	4	no	—
324	D₄	—	4	2	no	—
325	D₄	—	4	2	no	—
326	D₄	—	4	2	no	—
327	D₄	—	4	2	no	—
328	D₄	—	8	2	no	—
329	D₄	—	8	2	no	—
330	D₄	—	8	2	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	38	2	no	—
336	C₂×C₄	—	76	1	no	—
337	C₂×C₄	—	76	1	no	—
338	C₂×C₄	—	76	1	no	—
339	C₂×C₄	—	76	1	no	—
340	C₂×C₄	—	76	2	no	—
341	C₂×C₄	—	76	2	no	—
342	C₂×C₄	—	76	2	no	—
343	C₂×C₄	—	76	1	no	—
344	C₂×C₄	—	152	8	no	—
345	C₂×C₄	SD₁₆×D₁₉	1	1	yes	characteristic
346	C₂×C₄	#10597360	1	1	yes	characteristic
347	C₂×C₄	#10597361	1	1	yes	characteristic
348	C₂×C₄	#10597362	1	1	yes	characteristic
349	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
350	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
351	C₂×C₄	#10597381	1	1	yes	characteristic
352	C₂×C₄	#10597385	1	1	yes	characteristic
353	C₂×C₄	#10597385	1	1	yes	characteristic
354	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
355	C₈	—	4	2	no	—
356	C₈	—	152	1	no	—
357	C₈	—	152	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	4	4	no	—
361	C₂²	—	4	4	no	—
362	C₂²	—	4	4	no	—
363	C₂²	—	8	2	no	—
364	C₂²	—	8	2	no	—
365	C₂²	—	8	2	no	—
366	C₂²	—	8	2	no	—
367	C₂²	#11926424	1	1	yes	characteristic
368	C₂²	#11926693	1	1	yes	characteristic
369	C₂²	#11926703	1	1	yes	characteristic
370	C₄	—	2	1	no	—
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	8	1	no	—
375	C₄	—	8	1	no	—
376	C₄	—	76	2	no	—
377	C₄	—	152	2	no	—
378	C₄	—	152	1	no	—
379	C₄	SD₁₆×Dic₁₉	1	1	yes	characteristic
380	C₄	#11926645	1	1	yes	characteristic
381	C₄	#11926747	1	1	yes	characteristic
382	C₄	#11926761	1	1	yes	characteristic
383	C₂	—	2	1	no	—
384	C₂	—	2	1	no	—
385	C₂	—	8	4	no	—
386	C₂	#14563971	1	1	yes	characteristic
387	C₂	#14564124	1	1	yes	characteristic
388	C₂	#14564168	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926424
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#3208	C₂
socle	C₂²×C₁₉	#386
prime radical	C₂×C₁₉×#127	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#3208	D₁₉
hypocenter	C₁₉	#9295
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3208

Complements

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

N	H ≅ G/N	type
2. #14564617	C₂	semidirect
4. #14562199	C₂	semidirect
5. #14561494	C₂	semidirect
6. #14561463	C₂	semidirect
22. C₁₉×#492	C₄	semidirect
23. C₁₉×#486	C₄	semidirect
32. C₁₉×C₄⋊Q₈	C₂×C₄	semidirect
33. C₁₉×C₄⋊Q₈	C₂×C₄	semidirect
57. C₄×C₈×C₁₉	C₂×C₄	semidirect
113. C₄×Dic₁₉	SD₁₆	semidirect
199. #492	Dic₁₉	semidirect
200. #486	Dic₁₉	semidirect
227. C₄⋊Q₈	C₂×Dic₁₉	semidirect
228. C₄⋊Q₈	C₂×Dic₁₉	semidirect
256. C₄×C₈	C₂×Dic₁₉	semidirect
261. C₁₉	#9295	semidirect

Maximal subgroups

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

subgroup	index
#14566361	2
#14564617	2
#14562303	2
#14562199	2
#14561494	2
#14561463	2
C₁₉×#3208	2
#9295	19