Subgroups — #18073208

Order 4864.

574 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14566100	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₈×Dic₁₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#14563380	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	#14561475	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561474	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₉×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×#10597372	—	2	1	no	—
9	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₁₉	—	2	1	no	—
12	#11926447	C₄	1	2	yes	—
13	#11926443	C₄	1	2	yes	—
14	#11926437	—	2	2	no	—
15	C₄×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
16	C₄×C₂²⋊Dic₁₉	—	2	1	no	—
17	C₄²×Dic₁₉	C₂²	1	1	yes	characteristic
18	#11926298	—	2	2	no	—
19	#11926298	—	2	1	no	—
20	D₄⋊(C₁₉⋊C₈)	C₄	1	4	yes	complemented
21	C₂×C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
22	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
24	C₁₉×D₄⋊C₈	C₄	1	4	yes	complemented
25	C₂²×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₁₉	—	2	1	no	—
27	C₂²×C₈×C₁₉	—	2	1	no	—
28	C₂²×C₈×C₁₉	—	2	1	no	—
29	C₄×D₄×C₁₉	—	2	1	no	—
30	C₄×D₄×C₁₉	—	2	1	no	—
31	C₄×D₄×C₁₉	C₂×C₄	1	2	yes	—
32	C₄×D₄×C₁₉	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
35	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
36	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
37	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
38	#10597372	—	4	2	no	—
39	#10597372	—	4	2	no	—
40	C₂×C₄⋊Dic₁₉	—	2	2	no	—
41	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
42	C₂×C₄×Dic₁₉	—	2	1	no	—
43	C₂×C₄×Dic₁₉	C₂×C₄	1	2	yes	—
44	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
45	C₂×C₄×Dic₁₉	D₄	1	2	yes	—
46	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
47	C₂²×C₁₉⋊C₈	—	2	1	no	—
48	C₂²×C₁₉⋊C₈	—	2	1	no	—
49	C₁₉×C₄⋊C₈	—	2	1	no	—
50	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
51	C₄×C₈×C₁₉	—	2	1	no	—
52	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
53	C₁₉⋊#95	—	2	1	no	—
54	C₁₉⋊#95	—	2	2	no	—
55	C₈×Dic₁₉	—	4	2	no	—
56	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
57	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
58	C₄×C₁₉⋊C₈	—	2	1	no	—
59	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
60	C₂⁴×C₁₉	—	2	1	no	—
61	C₂×D₄×C₁₉	—	2	1	no	—
62	C₂×D₄×C₁₉	—	2	2	no	—
63	C₂×D₄×C₁₉	—	2	2	no	—
64	C₂×D₄×C₁₉	C₄²	1	2	yes	—
65	C₂²×C₄×C₁₉	—	2	1	no	—
66	C₂²×C₄×C₁₉	—	2	2	no	—
67	C₂²×C₄×C₁₉	—	2	2	no	—
68	C₂²×C₄×C₁₉	—	4	1	no	—
69	C₂²×C₄×C₁₉	—	4	1	no	—
70	C₂²×C₄×C₁₉	—	4	1	no	—
71	C₂²×C₄×C₁₉	—	4	1	no	—
72	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×Dic₁₉	—	2	2	no	—
76	C₂²×Dic₁₉	C₂²⋊C₄	1	2	yes	—
77	C₂²×Dic₁₉	C₂²⋊C₄	1	2	yes	—
78	C₂×C₈×C₁₉	—	2	2	no	—
79	C₂×C₈×C₁₉	—	2	2	no	—
80	C₂×C₈×C₁₉	—	4	2	no	—
81	C₂×C₈×C₁₉	—	4	1	no	—
82	C₂×C₈×C₁₉	—	4	1	no	—
83	C₄⋊C₄×C₁₉	—	2	1	no	—
84	C₄⋊C₄×C₁₉	C₄²	1	2	yes	—
85	C₂²⋊C₄×C₁₉	—	2	2	no	—
86	C₂²⋊C₄×C₁₉	—	2	2	no	—
87	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₁₉	#41	1	1	yes	characteristic
90	C₄²×C₁₉	#41	1	1	yes	characteristic
91	C₂²⋊Dic₁₉	—	4	4	no	—
92	C₂²⋊Dic₁₉	—	4	4	no	—
93	C₄⋊Dic₁₉	—	2	2	no	—
94	C₄⋊Dic₁₉	—	2	2	no	—
95	C₄⋊Dic₁₉	—	4	4	no	—
96	C₄×Dic₁₉	—	2	2	no	—
97	C₄×Dic₁₉	—	2	2	no	—
98	C₄×Dic₁₉	—	2	1	no	—
99	C₄×Dic₁₉	—	2	1	no	—
100	C₄×Dic₁₉	—	4	2	no	—
101	C₄×Dic₁₉	—	4	1	no	—
102	C₄×Dic₁₉	—	4	1	no	—
103	C₄×Dic₁₉	D₈	1	2	yes	—
104	C₄×Dic₁₉	SD₁₆	1	2	yes	—
105	C₂×C₁₉⋊C₈	—	2	2	no	—
106	C₂×C₁₉⋊C₈	—	2	2	no	—
107	C₂×C₁₉⋊C₈	—	4	2	no	—
108	C₂×C₁₉⋊C₈	—	4	1	no	—
109	C₂×C₁₉⋊C₈	—	4	1	no	—
110	#8183	—	19	1	no	maximal, Sylow
111	C₂³×C₁₉	—	2	2	no	—
112	C₂³×C₁₉	—	4	2	no	—
113	C₂³×C₁₉	—	4	1	no	—
114	C₂³×C₁₉	—	4	1	no	—
115	C₂³×C₁₉	—	4	2	no	—
116	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	D₄×C₁₉	—	2	2	no	—
118	D₄×C₁₉	—	2	2	no	—
119	D₄×C₁₉	—	4	2	no	—
120	C₂×C₄×C₁₉	—	2	2	no	—
121	C₂×C₄×C₁₉	—	2	2	no	—
122	C₂×C₄×C₁₉	—	4	1	no	—
123	C₂×C₄×C₁₉	—	4	1	no	—
124	C₂×C₄×C₁₉	—	4	2	no	—
125	C₂×C₄×C₁₉	—	4	2	no	—
126	C₂×C₄×C₁₉	—	4	2	no	—
127	C₂×C₄×C₁₉	—	4	2	no	—
128	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
129	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
130	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₉	#131	1	2	yes	—
135	C₂×C₄×C₁₉	#131	1	2	yes	—
136	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
138	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
139	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
140	C₂×Dic₁₉	—	2	2	no	—
141	C₂×Dic₁₉	—	2	2	no	—
142	C₂×Dic₁₉	—	2	2	no	—
143	C₂×Dic₁₉	—	2	2	no	—
144	C₂×Dic₁₉	—	2	2	no	—
145	C₂×Dic₁₉	—	4	2	no	—
146	C₂×Dic₁₉	—	4	2	no	—
147	C₂×Dic₁₉	—	4	2	no	—
148	C₂×Dic₁₉	#102	1	4	yes	—
149	C₈×C₁₉	—	4	2	no	—
150	C₈×C₁₉	—	4	2	no	—
151	C₁₉⋊C₈	—	4	2	no	—
152	C₁₉⋊C₈	—	4	2	no	—
153	#2293	—	19	1	no	—
154	C₄×C₄⋊C₈	—	19	1	no	—
155	#1744	—	19	1	no	—
156	C₂×D₄⋊C₈	—	19	1	no	—
157	C₂×D₄⋊C₈	D₁₉	1	1	yes	characteristic
158	#1279	—	19	1	no	—
159	#1279	—	19	1	no	—
160	C₂²×C₁₉	—	4	2	no	—
161	C₂²×C₁₉	—	4	2	no	—
162	C₂²×C₁₉	—	4	2	no	—
163	C₂²×C₁₉	—	8	2	no	—
164	C₂²×C₁₉	—	8	2	no	—
165	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
166	C₂²×C₁₉	#408	1	2	yes	—
167	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
168	C₂²×C₁₉	#416	1	1	yes	characteristic
169	C₂²×C₁₉	#421	1	2	yes	—
170	C₄×C₁₉	—	2	1	no	—
171	C₄×C₁₉	—	2	1	no	—
172	C₄×C₁₉	—	4	2	no	—
173	C₄×C₁₉	#405	1	1	yes	characteristic
174	C₄×C₁₉	#405	1	1	yes	characteristic
175	C₄×C₁₉	#433	1	2	yes	—
176	C₄×C₁₉	C₄×D₈	1	1	yes	characteristic
177	C₄×C₁₉	C₄×D₈	1	1	yes	characteristic
178	C₄×C₁₉	C₄×SD₁₆	1	1	yes	characteristic
179	C₄×C₁₉	C₄×SD₁₆	1	1	yes	characteristic
180	Dic₁₉	—	2	4	no	—
181	Dic₁₉	—	4	2	no	—
182	Dic₁₉	—	8	2	no	—
183	C₂×C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
184	C₂×#117	—	38	1	no	—
185	C₂×C₄⋊C₈	—	19	1	no	—
186	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
187	C₂×C₄×C₈	—	19	1	no	—
188	C₂×C₄×C₈	—	38	1	no	—
189	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
190	#390	—	19	2	no	—
191	#389	—	19	2	no	—
192	#388	—	38	2	no	—
193	C₄×C₄⋊C₄	—	19	1	no	—
194	C₄×C₂²⋊C₄	—	38	1	no	—
195	C₄³	—	19	1	no	—
196	#336	—	38	2	no	—
197	#336	—	38	1	no	—
198	D₄⋊C₈	—	19	4	no	—
199	D₄⋊C₈	Dic₁₉	1	4	yes	complemented
200	C₂×C₁₉	—	8	2	no	—
201	C₂×C₁₉	#1480	1	2	yes	—
202	C₂×C₁₉	#1482	1	2	yes	—
203	C₂×C₁₉	#1750	1	1	yes	characteristic
204	C₂×C₁₉	#1759	1	1	yes	characteristic
205	C₂×C₁₉	C₄×#102	1	1	yes	characteristic
206	C₂²×D₄	C₄×D₁₉	1	1	yes	characteristic
207	C₂³×C₄	—	2	1	no	—
208	C₂²×C₈	—	2	1	no	—
209	C₂²×C₈	—	2	1	no	—
210	C₂²×C₈	—	38	1	no	—
211	C₂²×C₈	—	38	1	no	—
212	C₄×D₄	—	2	1	no	—
213	C₄×D₄	—	2	1	no	—
214	C₄×D₄	C₄×D₁₉	1	2	yes	—
215	C₄×D₄	C₂×Dic₁₉	1	2	yes	—
216	#117	—	76	2	no	—
217	#117	—	76	2	no	—
218	C₂×C₄⋊C₄	—	19	2	no	—
219	C₂×C₄⋊C₄	—	38	2	no	—
220	C₂×C₄⋊C₄	C₄×D₁₉	1	1	yes	characteristic
221	C₂×C₂²⋊C₄	—	2	1	no	—
222	C₂×C₂²⋊C₄	—	38	2	no	—
223	C₂×C₂²⋊C₄	—	38	2	no	—
224	C₂×C₄²	—	19	1	no	—
225	C₂×C₄²	—	19	2	no	—
226	C₂×C₄²	—	19	1	no	—
227	C₂×C₄²	—	19	2	no	—
228	C₂×C₄²	—	38	1	no	—
229	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
230	C₄⋊C₈	—	2	1	no	—
231	C₄⋊C₈	—	19	2	no	—
232	C₄⋊C₈	—	38	1	no	—
233	C₄⋊C₈	C₂×Dic₁₉	1	2	yes	—
234	C₄×C₈	—	2	1	no	—
235	C₄×C₈	—	19	2	no	—
236	C₄×C₈	—	38	1	no	—
237	C₄×C₈	—	76	2	no	—
238	C₄×C₈	C₂×Dic₁₉	1	2	yes	—
239	#95	—	38	2	no	—
240	#95	—	38	1	no	—
241	C₁₉	#8183	1	1	yes	characteristic, complemented, Sylow
242	C₂⁴	—	2	1	no	—
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	C₄×Dic₁₉	1	2	yes	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	—	4	1	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	19	2	no	—
255	C₂²×C₄	—	19	2	no	—
256	C₂²×C₄	—	38	2	no	—
257	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
258	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
259	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	4	2	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	4	1	no	—
265	C₂×C₈	—	38	2	no	—
266	C₂×C₈	—	38	2	no	—
267	C₂×C₈	—	76	2	no	—
268	C₂×C₈	—	76	1	no	—
269	C₂×C₈	—	76	1	no	—
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	38	2	no	—
272	C₄⋊C₄	—	38	2	no	—
273	C₄⋊C₄	—	76	4	no	—
274	C₄⋊C₄	C₄×Dic₁₉	1	2	yes	—
275	C₂²⋊C₄	—	2	2	no	—
276	C₂²⋊C₄	—	2	2	no	—
277	C₂²⋊C₄	—	76	4	no	—
278	C₂²⋊C₄	—	76	4	no	—
279	C₄²	—	19	2	no	—
280	C₄²	—	19	2	no	—
281	C₄²	—	38	2	no	—
282	C₄²	—	38	2	no	—
283	C₄²	—	38	1	no	—
284	C₄²	—	38	1	no	—
285	C₄²	—	76	2	no	—
286	C₄²	—	76	1	no	—
287	C₄²	—	76	1	no	—
288	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
289	C₄²	#64543	1	1	yes	characteristic
290	C₄²	#64543	1	1	yes	characteristic
291	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
292	C₂³	—	2	2	no	—
293	C₂³	—	4	1	no	—
294	C₂³	—	4	1	no	—
295	C₂³	—	4	2	no	—
296	C₂³	—	4	2	no	—
297	C₂³	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
298	D₄	—	2	2	no	—
299	D₄	—	2	2	no	—
300	D₄	—	4	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	19	4	no	—
310	C₂×C₄	—	38	2	no	—
311	C₂×C₄	—	38	2	no	—
312	C₂×C₄	—	38	2	no	—
313	C₂×C₄	—	38	2	no	—
314	C₂×C₄	—	38	2	no	—
315	C₂×C₄	—	76	2	no	—
316	C₂×C₄	—	76	2	no	—
317	C₂×C₄	—	76	2	no	—
318	C₂×C₄	#10597323	1	1	yes	characteristic
319	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
320	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
321	C₂×C₄	#10597347	1	2	yes	—
322	C₂×C₄	#10597353	1	2	yes	—
323	C₂×C₄	D₈×D₁₉	1	1	yes	characteristic
324	C₂×C₄	#10597358	1	1	yes	characteristic
325	C₂×C₄	SD₁₆×D₁₉	1	1	yes	characteristic
326	C₂×C₄	#10597362	1	1	yes	characteristic
327	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
328	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
329	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
330	C₈	—	4	2	no	—
331	C₈	—	4	2	no	—
332	C₈	—	76	2	no	—
333	C₈	—	76	2	no	—
334	C₂²	—	4	2	no	—
335	C₂²	—	4	2	no	—
336	C₂²	—	4	2	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	C₂²⋊C₄×Dic₁₉	1	1	yes	characteristic
340	C₂²	#11926509	1	2	yes	—
341	C₂²	#102×D₁₉	1	1	yes	characteristic
342	C₂²	#11926537	1	1	yes	characteristic
343	C₂²	#11926559	1	2	yes	—
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	4	2	no	—
347	C₄	—	38	4	no	—
348	C₄	—	76	2	no	—
349	C₄	—	152	2	no	—
350	C₄	#11926499	1	1	yes	characteristic
351	C₄	#11926502	1	1	yes	characteristic
352	C₄	#11926602	1	2	yes	—
353	C₄	#11926624	1	1	yes	characteristic
354	C₄	D₈×Dic₁₉	1	1	yes	characteristic
355	C₄	SD₁₆×Dic₁₉	1	1	yes	characteristic
356	C₄	#11926635	1	1	yes	characteristic
357	C₂	—	8	2	no	—
358	C₂	#14562032	1	2	yes	—
359	C₂	#14562040	1	2	yes	—
360	C₂	#14563403	1	1	yes	characteristic
361	C₂	#14563447	1	1	yes	characteristic
362	C₂	#102×Dic₁₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×D₄⋊C₈	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₁₉	C₂×C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×D₄⋊C₈	D₁₉
hypocenter	C₁₉	#8183
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×D₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₄⋊C₈×Dic₁₉	C₂	semidirect
3. #14563380	C₂	semidirect
5. #14561475	C₂	semidirect
6. #14561474	C₂	semidirect
20. D₄⋊(C₁₉⋊C₈)	C₄	semidirect
24. C₁₉×D₄⋊C₈	C₄	semidirect
199. D₄⋊C₈	Dic₁₉	semidirect
241. C₁₉	#8183	semidirect

Maximal subgroups

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

subgroup	index
#14566100	2
C₄⋊C₈×Dic₁₉	2
#14563380	2
C₂×D₄⋊(C₁₉⋊C₈)	2
#14561475	2
#14561474	2
C₂×C₁₉×D₄⋊C₈	2
#8183	19