Subgroups — #18082713

Order 4864.

684 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	#14563615	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₉⋊#95	C₂	1	1	yes	maximal, characteristic
3	#14561513	C₂	1	4	yes	maximal
4	C₁₉×#2857	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄×Dic₁₉	—	2	1	no	—
6	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
7	C₂×C₂²⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₁₉⋊#95	—	2	1	no	—
9	C₂×C₁₉⋊#95	C₂²	1	2	yes	—
10	#11926299	—	2	1	no	—
11	C₂×C₁₉×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
12	C₂×C₁₉×C₂²⋊Q₈	—	2	1	no	—
13	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
14	C₁₉×#391	—	2	1	no	—
15	C₁₉×#389	—	2	1	no	—
16	C₁₉×#389	C₄	1	2	yes	complemented
17	C₁₉×#384	C₄	1	1	yes	characteristic, complemented
18	C₁₉×#380	—	2	1	no	—
19	C₁₉×#380	C₂²	1	2	yes	—
20	C₂²×Q₈×C₁₉	—	2	1	no	—
21	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
24	C₂³×Dic₁₉	—	2	2	no	—
25	C₁₉×C₄⋊Q₈	—	4	2	no	—
26	C₁₉×C₂²⋊Q₈	—	4	2	no	—
27	C₁₉×C₂²⋊Q₈	—	4	2	no	—
28	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
32	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
34	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
35	C₂×C₄²×C₁₉	—	4	2	no	—
36	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
40	C₂×C₄×Dic₁₉	—	2	4	no	—
41	C₂×C₄×Dic₁₉	—	4	2	no	—
42	C₂×C₄×Dic₁₉	—	4	2	no	—
43	C₂²×C₁₉⋊C₈	—	2	2	no	—
44	C₁₉×#95	—	2	2	no	—
45	C₁₉×#95	—	2	2	no	—
46	C₁₉⋊#95	—	2	4	no	—
47	C₁₉⋊#95	—	2	2	no	—
48	C₁₉⋊#95	Q₈	1	4	yes	complemented
49	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
50	C₄⋊(C₁₉⋊C₈)	—	4	2	no	—
51	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
52	C₂×Q₈×C₁₉	—	4	2	no	—
53	C₂×Q₈×C₁₉	—	4	2	no	—
54	C₂×Q₈×C₁₉	—	4	2	no	—
55	C₂²×C₄×C₁₉	—	2	2	no	—
56	C₂²×C₄×C₁₉	—	2	2	no	—
57	C₂²×C₄×C₁₉	—	2	2	no	—
58	C₂²×C₄×C₁₉	—	2	2	no	—
59	C₂²×C₄×C₁₉	—	2	2	no	—
60	C₂²×C₄×C₁₉	—	2	2	no	—
61	C₂²×C₄×C₁₉	—	4	1	no	—
62	C₂²×C₄×C₁₉	—	4	1	no	—
63	C₂²×C₄×C₁₉	—	4	1	no	—
64	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	D₈	1	2	yes	—
69	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
70	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₁₉	#41	1	2	yes	—
74	C₂²×Dic₁₉	—	2	4	no	—
75	C₂²×Dic₁₉	—	2	4	no	—
76	C₂²×Dic₁₉	—	2	4	no	—
77	C₂²×Dic₁₉	—	4	2	no	—
78	C₂²×Dic₁₉	—	4	2	no	—
79	C₂²×Dic₁₉	—	4	2	no	—
80	C₂²×Dic₁₉	—	4	2	no	—
81	C₄⋊C₄×C₁₉	—	4	2	no	—
82	C₄⋊C₄×C₁₉	—	4	2	no	—
83	C₄⋊C₄×C₁₉	—	4	2	no	—
84	C₄⋊C₄×C₁₉	—	4	2	no	—
85	C₄⋊C₄×C₁₉	—	4	4	no	—
86	C₄⋊C₄×C₁₉	—	4	2	no	—
87	C₂²⋊C₄×C₁₉	—	4	2	no	—
88	C₂²⋊C₄×C₁₉	—	4	2	no	—
89	C₄²×C₁₉	—	2	2	no	—
90	C₄²×C₁₉	—	2	2	no	—
91	C₄²×C₁₉	—	4	1	no	—
92	C₄²×C₁₉	—	4	1	no	—
93	C₄×Dic₁₉	—	4	8	no	—
94	C₂×C₁₉⋊C₈	—	4	2	no	—
95	C₂×C₁₉⋊C₈	—	4	2	no	—
96	C₂×C₁₉⋊C₈	—	4	2	no	—
97	#9909	—	19	1	no	maximal, Sylow
98	C₂³×C₁₉	—	2	1	no	—
99	C₂³×C₁₉	—	2	1	no	—
100	C₂³×C₁₉	—	2	1	no	—
101	C₂³×C₁₉	—	2	1	no	—
102	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂³×C₁₉	C₂²⋊Q₈	1	2	yes	—
104	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
105	C₂³×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
106	C₂³×C₁₉	#137	1	2	yes	—
107	Q₈×C₁₉	—	8	4	no	—
108	C₂×C₄×C₁₉	—	2	1	no	—
109	C₂×C₄×C₁₉	—	2	1	no	—
110	C₂×C₄×C₁₉	—	2	1	no	—
111	C₂×C₄×C₁₉	—	2	1	no	—
112	C₂×C₄×C₁₉	—	2	1	no	—
113	C₂×C₄×C₁₉	—	2	1	no	—
114	C₂×C₄×C₁₉	—	2	1	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	4	no	—
117	C₂×C₄×C₁₉	—	2	4	no	—
118	C₂×C₄×C₁₉	—	2	1	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	4	2	no	—
122	C₂×C₄×C₁₉	—	4	2	no	—
123	C₂×C₄×C₁₉	—	4	4	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	—	4	2	no	—
129	C₂×C₄×C₁₉	—	4	2	no	—
130	C₂×C₄×C₁₉	#99	1	2	yes	—
131	C₂×C₄×C₁₉	#102	1	2	yes	—
132	C₂×C₄×C₁₉	#102	1	2	yes	—
133	C₂×C₄×C₁₉	#104	1	2	yes	—
134	C₂×C₄×C₁₉	#104	1	2	yes	—
135	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
136	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
137	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
138	C₂×Dic₁₉	—	2	8	no	—
139	C₂×Dic₁₉	—	4	4	no	—
140	C₂×Dic₁₉	—	4	4	no	—
141	C₂×Dic₁₉	—	4	4	no	—
142	C₂×Dic₁₉	—	4	4	no	—
143	C₂×Dic₁₉	—	4	4	no	—
144	C₂×Dic₁₉	—	4	4	no	—
145	C₁₉⋊C₈	—	8	2	no	—
146	#2857	D₁₉	1	1	yes	characteristic
147	#1797	—	19	1	no	—
148	C₄×#95	—	19	1	no	—
149	#1294	—	19	4	no	—
150	C₂²×C₁₉	—	2	2	no	—
151	C₂²×C₁₉	—	2	2	no	—
152	C₂²×C₁₉	—	2	1	no	—
153	C₂²×C₁₉	—	2	1	no	—
154	C₂²×C₁₉	—	2	1	no	—
155	C₂²×C₁₉	—	2	1	no	—
156	C₂²×C₁₉	—	2	2	no	—
157	C₂²×C₁₉	—	2	2	no	—
158	C₂²×C₁₉	#391	1	1	yes	characteristic
159	C₂²×C₁₉	C₂×#99	1	1	yes	characteristic
160	C₂²×C₁₉	#410	1	1	yes	characteristic
161	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
162	C₂²×C₁₉	#419	1	1	yes	characteristic
163	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
164	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
165	C₂²×C₁₉	#474	1	2	yes	—
166	C₂²×C₁₉	#475	1	2	yes	—
167	C₄×C₁₉	—	2	1	no	—
168	C₄×C₁₉	—	2	2	no	—
169	C₄×C₁₉	—	2	1	no	—
170	C₄×C₁₉	—	2	2	no	—
171	C₄×C₁₉	—	2	2	no	—
172	C₄×C₁₉	—	4	1	no	—
173	C₄×C₁₉	—	8	2	no	—
174	C₄×C₁₉	—	8	2	no	—
175	C₄×C₁₉	#327	1	4	yes	—
176	Dic₁₉	—	4	8	no	—
177	C₂×C₄⋊Q₈	Dic₁₉	1	1	yes	characteristic, complemented
178	C₂×C₂²⋊Q₈	—	2	1	no	—
179	C₂²×C₄²	—	38	1	no	—
180	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
181	C₂×C₄⋊C₈	—	38	1	no	—
182	C₂×C₂²⋊C₈	—	19	2	no	—
183	#391	—	2	1	no	—
184	#389	—	2	1	no	—
185	#389	Dic₁₉	1	2	yes	complemented
186	#384	Dic₁₉	1	1	yes	characteristic, complemented
187	#380	—	2	1	no	—
188	#380	C₂×D₁₉	1	2	yes	—
189	C₂×#95	—	19	2	no	—
190	C₂×#95	—	38	1	no	—
191	#336	—	38	1	no	—
192	C₂×C₁₉	—	2	2	no	—
193	C₂×C₁₉	—	2	2	no	—
194	C₂×C₁₉	C₂×#327	1	1	yes	characteristic
195	C₂×C₁₉	#1502	1	1	yes	characteristic
196	C₂×C₁₉	#1506	1	1	yes	characteristic
197	C₂×C₁₉	#1508	1	1	yes	characteristic
198	C₂×C₁₉	#1988	1	1	yes	characteristic
199	C₂×C₁₉	#1994	1	1	yes	characteristic
200	C₂×C₁₉	#1996	1	1	yes	characteristic
201	C₂²×Q₈	—	2	1	no	—
202	C₂³×C₄	—	38	2	no	—
203	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
204	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
205	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
206	C₂²×C₈	—	38	2	no	—
207	C₄⋊Q₈	—	4	2	no	—
208	C₂²⋊Q₈	—	4	2	no	—
209	C₂²⋊Q₈	—	4	2	no	—
210	C₂×C₄⋊C₄	—	2	1	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
215	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
216	C₂×C₂²⋊C₄	—	2	2	no	—
217	C₂×C₄²	—	4	2	no	—
218	C₂×C₄²	—	38	4	no	—
219	C₂×C₄²	—	76	2	no	—
220	C₂×C₄²	—	76	2	no	—
221	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
222	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
223	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
224	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
225	C₄⋊C₈	—	76	2	no	—
226	C₂²⋊C₈	—	38	4	no	—
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	#95	—	19	4	no	—
230	#95	—	38	4	no	—
231	#95	—	38	2	no	—
232	C₁₉	#9909	1	1	yes	characteristic, complemented, Sylow
233	C₂⁴	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
234	C₂×Q₈	—	4	2	no	—
235	C₂×Q₈	—	4	2	no	—
236	C₂×Q₈	—	4	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	—	4	1	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	—	38	4	no	—
247	C₂²×C₄	—	38	4	no	—
248	C₂²×C₄	—	38	4	no	—
249	C₂²×C₄	—	76	2	no	—
250	C₂²×C₄	—	76	2	no	—
251	C₂²×C₄	—	76	2	no	—
252	C₂²×C₄	—	76	2	no	—
253	C₂²×C₄	#64525	1	2	yes	—
254	C₂²×C₄	#64526	1	2	yes	—
255	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
256	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
257	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
258	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
259	C₂²×C₄	Q₈×D₁₉	1	2	yes	—
260	C₂²×C₄	#64545	1	2	yes	—
261	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
262	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
263	C₂×C₈	—	76	2	no	—
264	C₂×C₈	—	76	2	no	—
265	C₂×C₈	—	76	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	4	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	4	1	no	—
277	C₄²	—	4	1	no	—
278	C₄²	—	76	8	no	—
279	C₂³	—	2	1	no	—
280	C₂³	—	2	1	no	—
281	C₂³	—	2	1	no	—
282	C₂³	—	2	1	no	—
283	C₂³	C₂×#64525	1	1	yes	characteristic
284	C₂³	C₂×#64526	1	1	yes	characteristic
285	C₂³	#10597388	1	2	yes	—
286	C₂³	#10597392	1	2	yes	—
287	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
288	Q₈	—	8	4	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	4	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	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	38	8	no	—
312	C₂×C₄	—	76	4	no	—
313	C₂×C₄	—	76	4	no	—
314	C₂×C₄	—	76	4	no	—
315	C₂×C₄	—	76	4	no	—
316	C₂×C₄	—	76	4	no	—
317	C₂×C₄	—	76	4	no	—
318	C₂×C₄	#10597289	1	2	yes	—
319	C₂×C₄	#10597289	1	2	yes	—
320	C₂×C₄	#10597291	1	2	yes	—
321	C₂×C₄	#10597294	1	2	yes	—
322	C₂×C₄	#10597294	1	2	yes	—
323	C₂×C₄	#10597390	1	1	yes	characteristic
324	C₂×C₄	Q₈×Dic₁₉	1	2	yes	—
325	C₂×C₄	#10597393	1	1	yes	characteristic
326	C₈	—	152	2	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	2	no	—
334	C₂²	—	2	2	no	—
335	C₂²	#11926451	1	1	yes	characteristic
336	C₂²	C₂×#10597291	1	1	yes	characteristic
337	C₂²	#11926518	1	1	yes	characteristic
338	C₂²	C₂×#10597289	1	1	yes	characteristic
339	C₂²	#11926553	1	1	yes	characteristic
340	C₂²	C₂×#10597294	1	1	yes	characteristic
341	C₂²	C₂×#10597294	1	1	yes	characteristic
342	C₂²	#11926821	1	2	yes	—
343	C₂²	#11926826	1	2	yes	—
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	2	no	—
347	C₄	—	2	2	no	—
348	C₄	—	2	2	no	—
349	C₄	—	4	1	no	—
350	C₄	—	8	2	no	—
351	C₄	—	8	2	no	—
352	C₄	—	76	8	no	—
353	C₄	#11926278	1	4	yes	—
354	C₂	—	2	2	no	—
355	C₂	—	2	2	no	—
356	C₂	C₂×#11926278	1	1	yes	characteristic
357	C₂	#14562130	1	1	yes	characteristic
358	C₂	#14562147	1	1	yes	characteristic
359	C₂	#14562155	1	1	yes	characteristic
360	C₂	#14564588	1	1	yes	characteristic
361	C₂	#14564613	1	1	yes	characteristic
362	C₂	#14564621	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#2857	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#2857	D₁₉
hypocenter	C₁₉	#9909
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #2857

Complements

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

N	H ≅ G/N	type
11. C₂×C₁₉×C₄⋊Q₈	C₄	semidirect
16. C₁₉×#389	C₄	semidirect
17. C₁₉×#384	C₄	semidirect
48. C₁₉⋊#95	Q₈	semidirect
177. C₂×C₄⋊Q₈	Dic₁₉	semidirect
185. #389	Dic₁₉	semidirect
186. #384	Dic₁₉	semidirect
232. C₁₉	#9909	semidirect

Maximal subgroups

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

subgroup	index
#14563615	2
C₄×C₁₉⋊#95	2
#14561513	2
C₁₉×#2857	2
#9909	19