Subgroups — #18081892

Order 4864.

428 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	C₈×C₄⋊Dic₁₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#14562383	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562134	C₂	1	1	yes	maximal, characteristic, complemented
4	#14562130	C₂	1	1	yes	maximal, characteristic
5	#14562038	C₂	1	1	yes	maximal, characteristic, complemented
6	#14562034	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#3131	C₂	1	1	yes	maximal, characteristic
8	#11926597	C₂²	1	1	yes	characteristic
9	#11926597	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₁₉	—	2	1	no	—
12	C₄×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
13	#11926298	—	2	1	no	—
14	#11926297	—	2	1	no	—
15	#11926295	—	2	1	no	—
16	#11926281	—	2	2	no	—
17	#11926278	—	2	2	no	—
18	Q₈⋊(C₁₉⋊C₈)	—	2	2	no	—
19	D₄⋊(C₁₉⋊C₈)	—	2	2	no	—
20	C₁₉×#533	C₂²	1	1	yes	characteristic
21	C₁₉×#532	C₂²	1	1	yes	characteristic
22	C₁₉×#488	—	2	1	no	—
23	C₁₉×#488	—	2	1	no	—
24	C₁₉×#487	C₄	1	1	yes	characteristic, complemented
25	C₁₉×#487	C₄	1	1	yes	characteristic, complemented
26	C₁₉×#486	C₄	1	1	yes	characteristic, complemented
27	C₁₉×#486	C₄	1	1	yes	characteristic, complemented
28	C₁₉×#416	—	2	1	no	—
29	C₁₉×#416	—	2	1	no	—
30	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
31	C₂×#41×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	C₂×C₄	1	1	yes	characteristic
35	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
36	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
37	C₁₉×#127	C₂×C₄	1	1	yes	characteristic
38	C₁₉×#125	—	2	1	no	—
39	C₁₉×#124	—	2	1	no	—
40	C₁₉×C₂²⋊Q₈	—	2	1	no	—
41	C₁₉×C₂²⋊Q₈	—	2	1	no	—
42	C₁₉×#121	—	2	1	no	—
43	C₁₉×#121	—	2	1	no	—
44	C₄×Q₈×C₁₉	—	2	1	no	—
45	C₄×Q₈×C₁₉	—	2	1	no	—
46	C₄×D₄×C₁₉	—	2	1	no	—
47	C₄×D₄×C₁₉	—	2	1	no	—
48	C₁₉×#117	—	2	1	no	—
49	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
50	C₂×C₄⋊Dic₁₉	—	2	1	no	—
51	C₂×C₄×Dic₁₉	—	2	1	no	—
52	C₂²×C₁₉⋊C₈	—	2	1	no	—
53	C₁₉×#103	—	2	1	no	—
54	C₁₉×#103	—	2	1	no	—
55	C₁₉×#103	—	2	1	no	—
56	C₁₉×#103	—	2	1	no	—
57	C₁₉×#102	—	2	1	no	—
58	C₁₉×#102	—	2	1	no	—
59	C₁₉×#102	—	2	1	no	—
60	C₁₉×#102	—	2	1	no	—
61	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
64	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
65	C₁₉⋊#95	D₄	1	2	yes	—
66	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
67	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
68	C₈×Dic₁₉	—	4	2	no	—
69	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
70	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
71	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
72	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
73	C₄×C₁₉⋊C₈	—	2	1	no	—
74	C₄×C₁₉⋊C₈	—	2	1	no	—
75	#41×C₁₉	—	4	1	no	—
76	#41×C₁₉	—	4	1	no	—
77	#41×C₁₉	—	4	1	no	—
78	#41×C₁₉	—	4	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×D₄×C₁₉	—	2	1	no	—
83	C₂×D₄×C₁₉	—	2	1	no	—
84	C₂×D₄×C₁₉	—	2	1	no	—
85	C₂×D₄×C₁₉	—	4	1	no	—
86	C₂²×C₄×C₁₉	—	4	1	no	—
87	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
90	C₂²×Dic₁₉	—	2	2	no	—
91	C₂×C₈×C₁₉	—	2	1	no	—
92	C₂×C₈×C₁₉	—	2	1	no	—
93	C₂×C₈×C₁₉	—	2	1	no	—
94	C₂×C₈×C₁₉	—	2	1	no	—
95	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
99	C₄⋊C₄×C₁₉	—	2	1	no	—
100	C₄⋊C₄×C₁₉	—	2	1	no	—
101	C₄⋊C₄×C₁₉	—	2	1	no	—
102	C₄⋊C₄×C₁₉	—	2	1	no	—
103	C₄⋊C₄×C₁₉	—	2	1	no	—
104	C₄⋊C₄×C₁₉	—	2	1	no	—
105	C₄⋊C₄×C₁₉	—	4	1	no	—
106	C₄⋊C₄×C₁₉	—	4	1	no	—
107	C₂²⋊C₄×C₁₉	—	4	1	no	—
108	C₂²⋊C₄×C₁₉	—	4	1	no	—
109	C₂²⋊C₄×C₁₉	—	4	1	no	—
110	C₄²×C₁₉	—	4	1	no	—
111	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₁₉	#41	1	1	yes	characteristic
115	C₄⋊Dic₁₉	—	2	4	no	—
116	C₄×Dic₁₉	—	4	1	no	—
117	C₄×Dic₁₉	—	4	1	no	—
118	C₂×C₁₉⋊C₈	—	2	2	no	—
119	C₂×C₁₉⋊C₈	—	4	1	no	—
120	C₂×C₁₉⋊C₈	—	4	1	no	—
121	C₂×C₁₉⋊C₈	—	4	1	no	—
122	C₂×C₁₉⋊C₈	—	4	1	no	—
123	#9751	—	19	1	no	maximal, Sylow
124	C₂³×C₁₉	—	4	1	no	—
125	C₂³×C₁₉	#121	1	1	yes	characteristic
126	Q₈×C₁₉	—	4	1	no	—
127	Q₈×C₁₉	—	4	1	no	—
128	Q₈×C₁₉	—	4	2	no	—
129	Q₈×C₁₉	—	8	1	no	—
130	D₄×C₁₉	—	4	1	no	—
131	D₄×C₁₉	—	4	1	no	—
132	D₄×C₁₉	—	4	2	no	—
133	D₄×C₁₉	—	8	1	no	—
134	D₄×C₁₉	—	8	1	no	—
135	D₄×C₁₉	—	8	1	no	—
136	C₂×C₄×C₁₉	—	2	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	—	8	1	no	—
142	C₂×C₄×C₁₉	—	8	1	no	—
143	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₁₉	C₄×D₄	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	#124	1	1	yes	characteristic
148	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
149	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
150	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
151	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
153	C₂×Dic₁₉	—	2	4	no	—
154	C₂×Dic₁₉	—	4	2	no	—
155	C₂×Dic₁₉	—	4	2	no	—
156	C₈×C₁₉	—	2	1	no	—
157	C₈×C₁₉	—	2	1	no	—
158	C₈×C₁₉	—	2	1	no	—
159	C₈×C₁₉	—	2	1	no	—
160	C₁₉⋊C₈	—	4	2	no	—
161	C₁₉⋊C₈	—	8	1	no	—
162	C₁₉⋊C₈	—	8	1	no	—
163	#3131	D₁₉	1	1	yes	characteristic
164	C₈×C₄⋊C₄	—	19	1	no	—
165	#1564	—	19	1	no	—
166	#1503	—	19	1	no	—
167	#1502	—	19	1	no	—
168	#1481	—	19	1	no	—
169	#1480	—	19	1	no	—
170	C₂²×C₁₉	—	2	1	no	—
171	C₂²×C₁₉	—	2	1	no	—
172	C₂²×C₁₉	—	8	1	no	—
173	C₂²×C₁₉	—	8	1	no	—
174	C₂²×C₁₉	—	8	1	no	—
175	C₂²×C₁₉	#390	1	1	yes	characteristic
176	C₂²×C₁₉	#463	1	1	yes	characteristic
177	C₂²×C₁₉	#464	1	1	yes	characteristic
178	C₄×C₁₉	—	2	1	no	—
179	C₄×C₁₉	—	2	1	no	—
180	C₄×C₁₉	—	2	1	no	—
181	C₄×C₁₉	—	8	1	no	—
182	C₄×C₁₉	—	8	1	no	—
183	C₄×C₁₉	—	8	1	no	—
184	C₄×C₁₉	#416	1	1	yes	characteristic
185	C₄×C₁₉	#417	1	1	yes	characteristic
186	C₄×C₁₉	#443	1	2	yes	—
187	C₄×C₁₉	#463	1	1	yes	characteristic
188	C₄×C₁₉	#464	1	1	yes	characteristic
189	Dic₁₉	—	4	4	no	—
190	#533	C₂×D₁₉	1	1	yes	characteristic
191	#532	C₂×D₁₉	1	1	yes	characteristic
192	#488	—	2	1	no	—
193	#488	—	2	1	no	—
194	#487	Dic₁₉	1	1	yes	characteristic, complemented
195	#487	Dic₁₉	1	1	yes	characteristic, complemented
196	#486	Dic₁₉	1	1	yes	characteristic, complemented
197	#486	Dic₁₉	1	1	yes	characteristic, complemented
198	#431	—	19	1	no	—
199	#431	—	19	1	no	—
200	C₂×C₄⋊C₈	—	19	1	no	—
201	#416	—	2	1	no	—
202	#416	—	2	1	no	—
203	C₂×C₄×C₈	—	38	1	no	—
204	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
205	C₄×C₄⋊C₄	—	19	1	no	—
206	#336	—	38	1	no	—
207	#335	—	38	1	no	—
208	#334	—	38	1	no	—
209	#328	—	38	2	no	—
210	#327	—	38	2	no	—
211	Q₈⋊C₈	—	38	2	no	—
212	D₄⋊C₈	—	38	2	no	—
213	C₂×C₁₉	—	2	1	no	—
214	C₂×C₁₉	—	2	1	no	—
215	C₂×C₁₉	—	16	1	no	—
216	C₂×C₁₉	#1974	1	1	yes	characteristic
217	C₂×C₁₉	#1976	1	1	yes	characteristic
218	C₂×C₁₉	#1983	1	1	yes	characteristic
219	C₂×#41	—	2	1	no	—
220	C₂²×C₈	—	38	1	no	—
221	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
222	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
223	C₄⋊Q₈	C₂×Dic₁₉	1	1	yes	characteristic
224	C₄⋊Q₈	C₂×Dic₁₉	1	1	yes	characteristic
225	C₄⋊Q₈	C₂×Dic₁₉	1	1	yes	characteristic
226	#127	C₂×Dic₁₉	1	1	yes	characteristic
227	#125	—	2	1	no	—
228	#124	—	2	1	no	—
229	C₂²⋊Q₈	—	2	1	no	—
230	C₂²⋊Q₈	—	2	1	no	—
231	#121	—	2	1	no	—
232	#121	—	2	1	no	—
233	C₄×Q₈	—	2	1	no	—
234	C₄×Q₈	—	2	1	no	—
235	C₄×D₄	—	2	1	no	—
236	C₄×D₄	—	2	1	no	—
237	#117	—	2	1	no	—
238	C₂×C₄⋊C₄	—	38	1	no	—
239	C₂×C₄²	—	38	1	no	—
240	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
241	C₄⋊C₈	—	19	2	no	—
242	C₄⋊C₈	—	38	1	no	—
243	C₄⋊C₈	—	38	1	no	—
244	C₄⋊C₈	—	38	1	no	—
245	#103	—	2	1	no	—
246	#103	—	2	1	no	—
247	#103	—	2	1	no	—
248	#103	—	2	1	no	—
249	#102	—	2	1	no	—
250	#102	—	2	1	no	—
251	#102	—	2	1	no	—
252	#102	—	2	1	no	—
253	C₂²⋊C₈	—	38	1	no	—
254	C₂²⋊C₈	—	38	1	no	—
255	C₄×C₈	—	38	1	no	—
256	C₄×C₈	—	38	1	no	—
257	C₄×C₈	—	76	2	no	—
258	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
259	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
260	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
261	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
262	#95	—	19	2	no	—
263	C₁₉	#9751	1	1	yes	characteristic, complemented, Sylow
264	#41	—	4	1	no	—
265	#41	—	4	1	no	—
266	#41	—	4	1	no	—
267	#41	—	4	1	no	—
268	C₂×Q₈	—	2	1	no	—
269	C₂×Q₈	—	2	1	no	—
270	C₂×Q₈	—	2	1	no	—
271	C₂×D₄	—	2	1	no	—
272	C₂×D₄	—	2	1	no	—
273	C₂×D₄	—	2	1	no	—
274	C₂×D₄	—	4	1	no	—
275	C₂²×C₄	—	4	1	no	—
276	C₂²×C₄	—	38	2	no	—
277	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
278	C₂²×C₄	#64543	1	1	yes	characteristic
279	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	38	2	no	—
285	C₂×C₈	—	76	1	no	—
286	C₂×C₈	—	76	1	no	—
287	C₂×C₈	—	76	1	no	—
288	C₂×C₈	—	76	1	no	—
289	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
290	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
291	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
292	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
293	C₄⋊C₄	—	2	1	no	—
294	C₄⋊C₄	—	2	1	no	—
295	C₄⋊C₄	—	2	1	no	—
296	C₄⋊C₄	—	2	1	no	—
297	C₄⋊C₄	—	2	1	no	—
298	C₄⋊C₄	—	2	1	no	—
299	C₄⋊C₄	—	4	1	no	—
300	C₄⋊C₄	—	4	1	no	—
301	C₄⋊C₄	—	38	4	no	—
302	C₂²⋊C₄	—	4	1	no	—
303	C₂²⋊C₄	—	4	1	no	—
304	C₂²⋊C₄	—	4	1	no	—
305	C₄²	—	4	1	no	—
306	C₄²	—	76	1	no	—
307	C₄²	—	76	1	no	—
308	C₄²	D₄×D₁₉	1	1	yes	characteristic
309	C₄²	#64543	1	1	yes	characteristic
310	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
311	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
312	C₂³	—	4	1	no	—
313	C₂³	#10597384	1	1	yes	characteristic
314	Q₈	—	4	1	no	—
315	Q₈	—	4	1	no	—
316	Q₈	—	4	2	no	—
317	Q₈	—	8	1	no	—
318	D₄	—	4	1	no	—
319	D₄	—	4	1	no	—
320	D₄	—	4	2	no	—
321	D₄	—	8	1	no	—
322	D₄	—	8	1	no	—
323	D₄	—	8	1	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	8	1	no	—
330	C₂×C₄	—	8	1	no	—
331	C₂×C₄	—	38	4	no	—
332	C₂×C₄	—	76	2	no	—
333	C₂×C₄	—	76	2	no	—
334	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
335	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
336	C₂×C₄	#10597382	1	1	yes	characteristic
337	C₂×C₄	#10597384	1	1	yes	characteristic
338	C₂×C₄	#10597386	1	1	yes	characteristic
339	C₂×C₄	#10597395	1	1	yes	characteristic
340	C₂×C₄	#10597396	1	1	yes	characteristic
341	C₂×C₄	#10597396	1	1	yes	characteristic
342	C₂×C₄	#10597397	1	1	yes	characteristic
343	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
344	C₈	—	2	1	no	—
345	C₈	—	2	1	no	—
346	C₈	—	2	1	no	—
347	C₈	—	2	1	no	—
348	C₈	—	76	2	no	—
349	C₈	—	152	1	no	—
350	C₈	—	152	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	8	1	no	—
354	C₂²	—	8	1	no	—
355	C₂²	—	8	1	no	—
356	C₂²	#11926447	1	1	yes	characteristic
357	C₂²	#11926756	1	1	yes	characteristic
358	C₂²	#11926763	1	1	yes	characteristic
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	8	1	no	—
363	C₄	—	8	1	no	—
364	C₄	—	8	1	no	—
365	C₄	—	76	4	no	—
366	C₄	#11926540	1	1	yes	characteristic
367	C₄	#11926545	1	1	yes	characteristic
368	C₄	#11926661	1	2	yes	—
369	C₄	#11926756	1	1	yes	characteristic
370	C₄	#11926763	1	1	yes	characteristic
371	C₂	—	2	1	no	—
372	C₂	—	2	1	no	—
373	C₂	—	16	1	no	—
374	C₂	#14564524	1	1	yes	characteristic
375	C₂	#14564534	1	1	yes	characteristic
376	C₂	#14564568	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597384
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#3131	C₂
socle	C₂²×C₁₉	#390
prime radical	C₁₉×#532	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#3131	D₁₉
hypocenter	C₁₉	#9751
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3131

Complements

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

N	H ≅ G/N	type
1. C₈×C₄⋊Dic₁₉	C₂	semidirect
2. #14562383	C₂	semidirect
3. #14562134	C₂	semidirect
5. #14562038	C₂	semidirect
24. C₁₉×#487	C₄	semidirect
25. C₁₉×#487	C₄	semidirect
26. C₁₉×#486	C₄	semidirect
27. C₁₉×#486	C₄	semidirect
194. #487	Dic₁₉	semidirect
195. #487	Dic₁₉	semidirect
196. #486	Dic₁₉	semidirect
197. #486	Dic₁₉	semidirect
263. C₁₉	#9751	semidirect

Maximal subgroups

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

subgroup	index
C₈×C₄⋊Dic₁₉	2
#14562383	2
#14562134	2
#14562130	2
#14562038	2
#14562034	2
C₁₉×#3131	2
#9751	19