Subgroups — #18082654

Order 4864.

538 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572032	C₂	1	1	yes	maximal, characteristic
2	#14563786	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×Dic₁₉	C₂	1	1	yes	maximal, characteristic
4	#14561519	C₂	1	1	yes	maximal, characteristic
5	#14561519	C₂	1	1	yes	maximal, characteristic
6	#14561518	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1296	C₂	1	1	yes	maximal, characteristic
8	C₂×#10597334	—	2	1	no	—
9	C₂×#10597334	—	2	1	no	—
10	C₂×#10597334	C₄	1	2	yes	—
11	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₉⋊C₈	—	2	1	no	—
14	C₂×Dic₁₉⋊C₈	—	2	1	no	—
15	C₂×C₈×Dic₁₉	—	2	1	no	—
16	#11926414	C₂²	1	1	yes	characteristic
17	#11926413	—	2	1	no	—
18	#11926413	—	2	1	no	—
19	#11926413	C₄	1	2	yes	—
20	C₄²×Dic₁₉	C₂²	1	1	yes	characteristic
21	#11926298	—	2	1	no	—
22	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₁₉×#384	C₂²	1	1	yes	characteristic
25	C₂²×Dic₃₈	—	2	2	no	—
26	C₂²×C₈×C₁₉	—	2	1	no	—
27	C₂²×C₈×C₁₉	—	2	1	no	—
28	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊Dic₁₉	—	2	1	no	—
33	C₂×C₄⋊Dic₁₉	—	2	1	no	—
34	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
35	C₂×Dic₁₉⋊C₄	—	2	2	no	—
36	C₂×Dic₁₉⋊C₄	—	2	2	no	—
37	C₂×Dic₁₉⋊C₄	—	2	2	no	—
38	C₂×C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
41	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
42	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
43	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
44	C₂²×C₁₉⋊C₈	—	2	1	no	—
45	C₂²×C₁₉⋊C₈	—	2	1	no	—
46	#10597334	—	4	2	no	—
47	#10597334	—	4	2	no	—
48	#10597334	—	4	4	no	—
49	#10597334	—	4	2	no	—
50	#10597334	—	4	2	no	—
51	C₁₉×C₄⋊C₈	—	2	1	no	—
52	C₁₉×C₄⋊C₈	—	2	1	no	—
53	C₁₉×C₄⋊C₈	—	2	1	no	—
54	C₁₉×C₄⋊C₈	Q₈	1	2	yes	complemented
55	Dic₁₉⋊C₈	—	4	2	no	—
56	Dic₁₉⋊C₈	—	4	2	no	—
57	C₈×Dic₁₉	—	4	2	no	—
58	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
59	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
60	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
61	C₄⋊(C₁₉⋊C₈)	Q₈	1	2	yes	complemented
62	C₂²×C₄×C₁₉	—	2	2	no	—
63	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×Dic₁₉	—	2	2	no	—
67	C₂²×Dic₁₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×Dic₁₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×Dic₁₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×Dic₁₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×Dic₃₈	—	4	4	no	—
72	C₂×Dic₃₈	—	4	4	no	—
73	C₂×Dic₃₈	—	4	4	no	—
74	C₂×C₈×C₁₉	—	2	2	no	—
75	C₂×C₈×C₁₉	—	4	1	no	—
76	C₂×C₈×C₁₉	—	4	1	no	—
77	C₂×C₈×C₁₉	—	4	1	no	—
78	C₂×C₈×C₁₉	—	4	1	no	—
79	C₂×C₈×C₁₉	—	4	1	no	—
80	C₄⋊C₄×C₁₉	—	4	2	no	—
81	C₄⋊C₄×C₁₉	—	4	1	no	—
82	C₄⋊C₄×C₁₉	—	4	1	no	—
83	C₄⋊C₄×C₁₉	—	4	1	no	—
84	C₄⋊C₄×C₁₉	—	4	1	no	—
85	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
86	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
87	C₄²×C₁₉	#41	1	1	yes	characteristic
88	C₄²×C₁₉	#41	1	1	yes	characteristic
89	C₄⋊Dic₁₉	—	4	1	no	—
90	C₄⋊Dic₁₉	—	4	1	no	—
91	C₄⋊Dic₁₉	—	4	1	no	—
92	C₄⋊Dic₁₉	—	4	1	no	—
93	C₄⋊Dic₁₉	—	4	2	no	—
94	Dic₁₉⋊C₄	—	4	4	no	—
95	Dic₁₉⋊C₄	—	4	4	no	—
96	Dic₁₉⋊C₄	—	4	4	no	—
97	C₄×Dic₁₉	—	2	1	no	—
98	C₄×Dic₁₉	—	2	1	no	—
99	C₄×Dic₁₉	—	2	2	no	—
100	C₄×Dic₁₉	—	2	2	no	—
101	C₄×Dic₁₉	D₈	1	2	yes	—
102	C₄×Dic₁₉	SD₁₆	1	2	yes	—
103	C₄×Dic₁₉	SD₁₆	1	2	yes	—
104	C₄×Dic₁₉	SD₁₆	1	2	yes	—
105	C₄×Dic₁₉	Q₁₆	1	2	yes	—
106	C₄×Dic₁₉	Q₁₆	1	2	yes	—
107	C₂×C₁₉⋊C₈	—	2	2	no	—
108	C₂×C₁₉⋊C₈	—	4	1	no	—
109	C₂×C₁₉⋊C₈	—	4	1	no	—
110	C₂×C₁₉⋊C₈	—	4	1	no	—
111	C₂×C₁₉⋊C₈	—	4	1	no	—
112	C₂×C₁₉⋊C₈	—	4	1	no	—
113	#9896	—	19	1	no	maximal, Sylow
114	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	4	2	no	—
118	C₂×C₄×C₁₉	—	4	2	no	—
119	C₂×C₄×C₁₉	—	4	2	no	—
120	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
131	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
132	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×Dic₁₉	—	2	1	no	—
135	C₂×Dic₁₉	—	2	1	no	—
136	C₂×Dic₁₉	—	2	1	no	—
137	C₂×Dic₁₉	—	2	1	no	—
138	C₂×Dic₁₉	—	4	2	no	—
139	C₂×Dic₁₉	—	4	2	no	—
140	C₂×Dic₁₉	—	4	2	no	—
141	C₂×Dic₁₉	#100	1	2	yes	—
142	C₂×Dic₁₉	#101	1	2	yes	—
143	C₂×Dic₁₉	#102	1	2	yes	—
144	C₂×Dic₁₉	#102	1	2	yes	—
145	C₂×Dic₁₉	#103	1	2	yes	—
146	C₂×Dic₁₉	#103	1	2	yes	—
147	C₂×Dic₁₉	#103	1	2	yes	—
148	C₂×Dic₁₉	#103	1	2	yes	—
149	Dic₃₈	—	8	8	no	—
150	C₈×C₁₉	—	4	2	no	—
151	C₈×C₁₉	—	8	1	no	—
152	C₁₉⋊C₈	—	4	2	no	—
153	C₁₉⋊C₈	—	8	1	no	—
154	#2868	—	19	1	no	—
155	#1842	—	19	1	no	—
156	C₄×C₄⋊C₈	—	19	1	no	—
157	#1296	—	19	1	no	—
158	#1296	—	19	1	no	—
159	#1296	—	19	1	no	—
160	#1296	D₁₉	1	1	yes	characteristic
161	C₂²×C₁₉	#391	1	1	yes	characteristic
162	C₂²×C₁₉	C₂×#100	1	1	yes	characteristic
163	C₂²×C₁₉	C₂×#101	1	1	yes	characteristic
164	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
165	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
166	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
167	C₂²×C₁₉	#418	1	1	yes	characteristic
168	C₄×C₁₉	—	2	1	no	—
169	C₄×C₁₉	—	2	1	no	—
170	C₄×C₁₉	—	8	2	no	—
171	C₄×C₁₉	#476	1	1	yes	characteristic
172	C₄×C₁₉	#476	1	1	yes	characteristic
173	C₄×C₁₉	#477	1	1	yes	characteristic
174	C₄×C₁₉	#477	1	1	yes	characteristic
175	C₄×C₁₉	#486	1	1	yes	characteristic
176	C₄×C₁₉	#487	1	1	yes	characteristic
177	C₄×C₁₉	#498	1	1	yes	characteristic
178	C₄×C₁₉	#500	1	1	yes	characteristic
179	Dic₁₉	—	2	2	no	—
180	Dic₁₉	—	2	2	no	—
181	Dic₁₉	—	8	2	no	—
182	Dic₁₉	#332	1	4	yes	—
183	Dic₁₉	#333	1	4	yes	—
184	C₂×C₄⋊Q₈	—	19	2	no	—
185	C₂×C₄⋊Q₈	—	38	1	no	—
186	C₂×C₄⋊Q₈	—	38	1	no	—
187	C₂×C₄⋊C₈	—	19	1	no	—
188	C₂×C₄⋊C₈	—	19	1	no	—
189	C₂×C₄⋊C₈	—	38	1	no	—
190	C₂×C₄⋊C₈	—	38	1	no	—
191	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
192	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
193	C₂×C₄×C₈	—	38	1	no	—
194	#384	—	19	2	no	—
195	#384	—	19	1	no	—
196	#384	—	38	1	no	—
197	#384	—	38	1	no	—
198	#384	C₂×D₁₉	1	1	yes	characteristic
199	C₄³	—	19	1	no	—
200	#336	—	38	1	no	—
201	C₂×C₁₉	C₂×#332	1	1	yes	characteristic
202	C₂×C₁₉	C₂×#333	1	1	yes	characteristic
203	C₂×C₁₉	#1545	1	1	yes	characteristic
204	C₂×C₁₉	#1549	1	1	yes	characteristic
205	C₂×C₁₉	#1987	1	1	yes	characteristic
206	C₂×C₁₉	#1990	1	1	yes	characteristic
207	C₂×C₁₉	#1997	1	1	yes	characteristic
208	C₂²×Q₈	—	38	2	no	—
209	C₂²×C₈	—	2	1	no	—
210	C₂²×C₈	—	2	1	no	—
211	C₂²×C₈	—	38	1	no	—
212	C₂²×C₈	—	38	1	no	—
213	C₄⋊Q₈	—	76	2	no	—
214	C₄⋊Q₈	—	76	2	no	—
215	C₄⋊Q₈	—	76	2	no	—
216	C₄⋊Q₈	—	76	2	no	—
217	C₄⋊Q₈	—	76	4	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	—	19	2	no	—
221	C₂×C₄⋊C₄	—	38	2	no	—
222	C₂×C₄⋊C₄	—	38	2	no	—
223	C₂×C₄⋊C₄	—	38	2	no	—
224	C₂×C₄⋊C₄	—	38	1	no	—
225	C₂×C₄⋊C₄	—	38	1	no	—
226	C₂×C₄⋊C₄	C₄×D₁₉	1	2	yes	—
227	C₂×C₄²	—	19	1	no	—
228	C₂×C₄²	—	19	1	no	—
229	C₂×C₄²	—	19	1	no	—
230	C₂×C₄²	—	19	1	no	—
231	C₂×C₄²	—	19	1	no	—
232	C₂×C₄²	—	19	1	no	—
233	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
234	C₄⋊C₈	—	2	1	no	—
235	C₄⋊C₈	—	2	1	no	—
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	—	19	2	no	—
238	C₄⋊C₈	—	38	1	no	—
239	C₄⋊C₈	—	38	1	no	—
240	C₄⋊C₈	—	38	1	no	—
241	C₄⋊C₈	—	76	2	no	—
242	C₄⋊C₈	—	76	2	no	—
243	C₄⋊C₈	Dic₃₈	1	2	yes	complemented
244	C₄×C₈	—	76	2	no	—
245	C₁₉	#9896	1	1	yes	characteristic, complemented, Sylow
246	C₂×Q₈	—	76	4	no	—
247	C₂×Q₈	—	76	4	no	—
248	C₂×Q₈	—	76	4	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	19	1	no	—
251	C₂²×C₄	—	19	1	no	—
252	C₂²×C₄	—	19	1	no	—
253	C₂²×C₄	—	19	1	no	—
254	C₂²×C₄	—	38	2	no	—
255	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
256	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
257	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	38	2	no	—
265	C₂×C₈	—	76	1	no	—
266	C₂×C₈	—	76	1	no	—
267	C₂×C₈	—	76	1	no	—
268	C₂×C₈	—	76	1	no	—
269	C₂×C₈	—	76	1	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	76	2	no	—
276	C₄⋊C₄	—	76	4	no	—
277	C₄⋊C₄	—	76	4	no	—
278	C₄⋊C₄	—	76	4	no	—
279	C₄⋊C₄	—	76	1	no	—
280	C₄⋊C₄	—	76	1	no	—
281	C₄⋊C₄	—	76	1	no	—
282	C₄⋊C₄	—	76	1	no	—
283	C₄²	—	19	2	no	—
284	C₄²	—	19	2	no	—
285	C₄²	—	19	2	no	—
286	C₄²	—	19	2	no	—
287	C₄²	—	19	2	no	—
288	C₄²	—	19	2	no	—
289	C₄²	—	38	2	no	—
290	C₄²	—	38	2	no	—
291	C₄²	—	38	1	no	—
292	C₄²	—	38	1	no	—
293	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
294	C₄²	#64541	1	1	yes	characteristic
295	C₄²	#64543	1	1	yes	characteristic
296	C₄²	Q₈×D₁₉	1	1	yes	characteristic
297	C₂³	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
298	Q₈	—	152	8	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	19	2	no	—
305	C₂×C₄	—	19	2	no	—
306	C₂×C₄	—	19	2	no	—
307	C₂×C₄	—	19	2	no	—
308	C₂×C₄	—	19	2	no	—
309	C₂×C₄	—	19	2	no	—
310	C₂×C₄	—	19	2	no	—
311	C₂×C₄	—	19	2	no	—
312	C₂×C₄	—	38	1	no	—
313	C₂×C₄	—	38	1	no	—
314	C₂×C₄	—	38	1	no	—
315	C₂×C₄	—	38	1	no	—
316	C₂×C₄	—	76	2	no	—
317	C₂×C₄	—	76	2	no	—
318	C₂×C₄	—	76	2	no	—
319	C₂×C₄	C₄×Dic₃₈	1	1	yes	characteristic
320	C₂×C₄	#10597324	1	1	yes	characteristic
321	C₂×C₄	#10597331	1	1	yes	characteristic
322	C₂×C₄	#10597333	1	1	yes	characteristic
323	C₂×C₄	#10597334	1	1	yes	characteristic
324	C₂×C₄	#10597342	1	1	yes	characteristic
325	C₂×C₄	D₈×D₁₉	1	1	yes	characteristic
326	C₂×C₄	#10597357	1	1	yes	characteristic
327	C₂×C₄	SD₁₆×D₁₉	1	1	yes	characteristic
328	C₂×C₄	SD₁₆×D₁₉	1	1	yes	characteristic
329	C₂×C₄	SD₁₆×D₁₉	1	1	yes	characteristic
330	C₂×C₄	#10597360	1	1	yes	characteristic
331	C₂×C₄	Q₁₆×D₁₉	1	1	yes	characteristic
332	C₂×C₄	Q₁₆×D₁₉	1	1	yes	characteristic
333	C₈	—	4	2	no	—
334	C₈	—	8	1	no	—
335	C₈	—	76	2	no	—
336	C₈	—	152	1	no	—
337	C₂²	#11926448	1	1	yes	characteristic
338	C₂²	#100×D₁₉	1	1	yes	characteristic
339	C₂²	#101×D₁₉	1	1	yes	characteristic
340	C₂²	#102×D₁₉	1	1	yes	characteristic
341	C₂²	#103×D₁₉	1	1	yes	characteristic
342	C₂²	#103×D₁₉	1	1	yes	characteristic
343	C₂²	#11926547	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	8	2	no	—
347	C₄	—	19	4	no	—
348	C₄	—	19	4	no	—
349	C₄	—	38	2	no	—
350	C₄	—	38	2	no	—
351	C₄	—	152	2	no	—
352	C₄	#11926832	1	1	yes	characteristic
353	C₄	#11926833	1	1	yes	characteristic
354	C₄	#11926837	1	1	yes	characteristic
355	C₄	#11926838	1	1	yes	characteristic
356	C₄	#11926889	1	1	yes	characteristic
357	C₄	#11926893	1	1	yes	characteristic
358	C₄	#11926941	1	1	yes	characteristic
359	C₄	#11926948	1	1	yes	characteristic
360	C₂	D₁₉×#332	1	1	yes	characteristic
361	C₂	D₁₉×#333	1	1	yes	characteristic
362	C₂	#14562301	1	1	yes	characteristic
363	C₂	#14562318	1	1	yes	characteristic
364	C₂	#14564581	1	1	yes	characteristic
365	C₂	#14564594	1	1	yes	characteristic
366	C₂	#14564622	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊C₄×D₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1296	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1296	D₁₉
hypocenter	C₁₉	#9896
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

Complements

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

N	H ≅ G/N	type
54. C₁₉×C₄⋊C₈	Q₈	semidirect
61. C₄⋊(C₁₉⋊C₈)	Q₈	semidirect
243. C₄⋊C₈	Dic₃₈	semidirect
245. C₁₉	#9896	semidirect

Maximal subgroups

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

subgroup	index
#14572032	2
#14563786	2
C₄⋊C₈×Dic₁₉	2
#14561519	2
#14561518	2
C₁₉×#1296	2
#9896	19