Subgroups — #18082721

Order 4864.

592 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14571987	C₂	1	1	yes	maximal, characteristic
2	#14563615	C₂	1	1	yes	maximal, characteristic
3	#14561514	C₂	1	4	yes	maximal
4	C₁₉×#1434	C₂	1	1	yes	maximal, characteristic
5	C₂×#10597315	C₄	1	1	yes	characteristic, complemented
6	C₂×#10597370	—	2	1	no	—
7	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
8	C₂×C₂²⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
9	#11926450	—	2	1	no	—
10	#11926442	—	2	1	no	—
11	#11926442	C₄	1	2	yes	complemented
12	#11926414	C₄	1	1	yes	characteristic, complemented
13	#11926404	—	2	1	no	—
14	#11926404	C₂²	1	2	yes	—
15	#11926299	—	2	1	no	—
16	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₉×#95	—	2	1	no	—
18	C₂×C₁₉×#95	—	2	1	no	—
19	C₂×C₁₉×#95	C₂²	1	2	yes	—
20	C₂³×C₄×C₁₉	—	2	2	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	C₂²×Dic₃₈	—	2	1	no	—
25	C₂×C₄²×C₁₉	—	4	2	no	—
26	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
30	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
31	C₂×C₄⋊Dic₁₉	—	2	1	no	—
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×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
36	#10597370	—	4	4	no	—
37	C₂×Dic₁₉⋊C₄	—	2	2	no	—
38	C₂²×C₁₉⋊C₈	—	2	2	no	—
39	#10597315	—	4	2	no	—
40	C₁₉×#95	—	2	2	no	—
41	C₁₉×#95	—	4	2	no	—
42	C₁₉×#95	—	4	2	no	—
43	C₁₉×#95	Q₈	1	4	yes	complemented
44	C₁₉⋊#95	—	2	2	no	—
45	C₁₉⋊#95	—	2	2	no	—
46	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
47	C₄⋊(C₁₉⋊C₈)	—	4	2	no	—
48	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	—	2	2	no	—
50	C₂²×C₄×C₁₉	—	2	2	no	—
51	C₂²×C₄×C₁₉	—	2	4	no	—
52	C₂²×C₄×C₁₉	—	4	1	no	—
53	C₂²×C₄×C₁₉	—	4	1	no	—
54	C₂²×C₄×C₁₉	—	4	1	no	—
55	C₂²×C₄×C₁₉	—	4	1	no	—
56	C₂²×C₄×C₁₉	—	4	1	no	—
57	C₂²×C₄×C₁₉	—	4	2	no	—
58	C₂²×C₄×C₁₉	—	4	2	no	—
59	C₂²×C₄×C₁₉	—	4	2	no	—
60	C₂²×C₄×C₁₉	—	4	2	no	—
61	C₂²×C₄×C₁₉	—	4	2	no	—
62	C₂²×C₄×C₁₉	—	4	2	no	—
63	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
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	SD₁₆	1	2	yes	—
68	C₂²×C₄×C₁₉	Q₁₆	1	2	yes	—
69	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₁₉	#41	1	2	yes	—
73	C₂²×Dic₁₉	—	2	2	no	—
74	C₂²×Dic₁₉	—	2	2	no	—
75	C₂×Dic₃₈	—	4	2	no	—
76	C₂×Dic₃₈	—	4	2	no	—
77	C₂×Dic₃₈	—	4	2	no	—
78	C₄²×C₁₉	—	2	2	no	—
79	C₄²×C₁₉	—	2	2	no	—
80	C₄²×C₁₉	—	4	1	no	—
81	C₄²×C₁₉	—	4	1	no	—
82	C₂²⋊Dic₁₉	—	4	4	no	—
83	C₄⋊Dic₁₉	—	4	2	no	—
84	C₄⋊Dic₁₉	—	4	2	no	—
85	C₄⋊Dic₁₉	—	4	2	no	—
86	C₄⋊Dic₁₉	—	4	2	no	—
87	C₄⋊Dic₁₉	—	4	2	no	—
88	Dic₁₉⋊C₄	—	4	4	no	—
89	C₂×C₁₉⋊C₈	—	4	2	no	—
90	C₂×C₁₉⋊C₈	—	4	2	no	—
91	C₂×C₁₉⋊C₈	—	4	2	no	—
92	#9911	—	19	1	no	maximal, Sylow
93	C₂³×C₁₉	—	2	1	no	—
94	C₂³×C₁₉	—	2	1	no	—
95	C₂³×C₁₉	—	2	1	no	—
96	C₂³×C₁₉	—	2	1	no	—
97	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂³×C₁₉	C₂²⋊Q₈	1	2	yes	—
99	C₂³×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
100	C₂³×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
101	C₂³×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
102	C₂×C₄×C₁₉	—	2	1	no	—
103	C₂×C₄×C₁₉	—	2	1	no	—
104	C₂×C₄×C₁₉	—	2	1	no	—
105	C₂×C₄×C₁₉	—	2	1	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	2	1	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	—	4	2	no	—
116	C₂×C₄×C₁₉	—	4	2	no	—
117	C₂×C₄×C₁₉	—	4	2	no	—
118	C₂×C₄×C₁₉	—	4	2	no	—
119	C₂×C₄×C₁₉	—	4	4	no	—
120	C₂×C₄×C₁₉	—	4	4	no	—
121	C₂×C₄×C₁₉	—	4	4	no	—
122	C₂×C₄×C₁₉	—	4	4	no	—
123	C₂×C₄×C₁₉	—	8	2	no	—
124	C₂×C₄×C₁₉	—	8	2	no	—
125	C₂×C₄×C₁₉	—	8	2	no	—
126	C₂×C₄×C₁₉	—	8	2	no	—
127	C₂×C₄×C₁₉	#99	1	2	yes	—
128	C₂×C₄×C₁₉	#103	1	2	yes	—
129	C₂×C₄×C₁₉	#103	1	2	yes	—
130	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
131	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
133	C₂×Dic₁₉	—	4	2	no	—
134	C₂×Dic₁₉	—	4	2	no	—
135	C₂×Dic₁₉	—	4	2	no	—
136	C₂×Dic₁₉	—	4	2	no	—
137	C₂×Dic₁₉	—	4	2	no	—
138	C₂×Dic₁₉	—	4	2	no	—
139	Dic₃₈	—	8	4	no	—
140	C₁₉⋊C₈	—	8	2	no	—
141	#2857	—	19	1	no	—
142	#1797	—	19	1	no	—
143	#1434	D₁₉	1	1	yes	characteristic
144	#1294	—	19	4	no	—
145	C₂²×C₁₉	—	2	2	no	—
146	C₂²×C₁₉	—	2	2	no	—
147	C₂²×C₁₉	—	2	2	no	—
148	C₂²×C₁₉	—	2	2	no	—
149	C₂²×C₁₉	—	2	1	no	—
150	C₂²×C₁₉	—	2	1	no	—
151	C₂²×C₁₉	—	2	1	no	—
152	C₂²×C₁₉	—	2	1	no	—
153	C₂²×C₁₉	#391	1	1	yes	characteristic
154	C₂²×C₁₉	C₂×#99	1	1	yes	characteristic
155	C₂²×C₁₉	#410	1	1	yes	characteristic
156	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
157	C₂²×C₁₉	#418	1	1	yes	characteristic
158	C₂²×C₁₉	#421	1	1	yes	characteristic
159	C₂²×C₁₉	#421	1	1	yes	characteristic
160	C₂²×C₁₉	#476	1	2	yes	—
161	C₂²×C₁₉	#477	1	2	yes	—
162	C₄×C₁₉	—	2	1	no	—
163	C₄×C₁₉	—	2	2	no	—
164	C₄×C₁₉	—	2	1	no	—
165	C₄×C₁₉	—	2	2	no	—
166	C₄×C₁₉	—	2	2	no	—
167	C₄×C₁₉	—	2	2	no	—
168	C₄×C₁₉	—	4	1	no	—
169	C₄×C₁₉	—	8	4	no	—
170	Dic₁₉	—	8	2	no	—
171	Dic₁₉	—	8	2	no	—
172	C₂×C₄⋊Q₈	—	19	1	no	—
173	C₂×C₂²⋊Q₈	—	38	1	no	—
174	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
175	C₂×C₄⋊C₈	—	38	1	no	—
176	C₂×C₂²⋊C₈	—	19	2	no	—
177	#391	—	38	1	no	—
178	#389	—	19	2	no	—
179	#389	—	38	1	no	—
180	#384	—	19	1	no	—
181	#380	—	19	2	no	—
182	#380	—	38	1	no	—
183	C₂×#95	—	2	1	no	—
184	C₂×#95	—	2	1	no	—
185	C₂×#95	C₂×D₁₉	1	2	yes	—
186	#336	—	38	1	no	—
187	C₂×C₁₉	—	2	2	no	—
188	C₂×C₁₉	—	2	2	no	—
189	C₂×C₁₉	#1510	1	1	yes	characteristic
190	C₂×C₁₉	#1514	1	1	yes	characteristic
191	C₂×C₁₉	#1518	1	1	yes	characteristic
192	C₂×C₁₉	#1520	1	1	yes	characteristic
193	C₂×C₁₉	#1987	1	1	yes	characteristic
194	C₂×C₁₉	#1995	1	1	yes	characteristic
195	C₂×C₁₉	#1996	1	1	yes	characteristic
196	C₂²×Q₈	—	38	1	no	—
197	C₂³×C₄	—	2	2	no	—
198	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
199	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
200	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
201	C₂²×C₈	—	38	2	no	—
202	C₄⋊Q₈	—	76	2	no	—
203	C₂²⋊Q₈	—	76	4	no	—
204	C₂×C₄⋊C₄	—	19	2	no	—
205	C₂×C₄⋊C₄	—	19	2	no	—
206	C₂×C₄⋊C₄	—	38	1	no	—
207	C₂×C₄⋊C₄	—	38	2	no	—
208	C₂×C₄⋊C₄	—	38	1	no	—
209	C₂×C₄⋊C₄	—	38	1	no	—
210	C₂×C₂²⋊C₄	—	38	2	no	—
211	C₂×C₄²	—	4	2	no	—
212	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
213	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
214	C₂×C₄²	D₇₆	1	1	yes	characteristic
215	C₂×C₄²	D₇₆	1	1	yes	characteristic
216	C₄⋊C₈	—	76	2	no	—
217	C₂²⋊C₈	—	38	4	no	—
218	#95	—	2	2	no	—
219	#95	—	4	2	no	—
220	#95	—	4	2	no	—
221	#95	—	38	2	no	—
222	#95	—	38	2	no	—
223	#95	Dic₃₈	1	4	yes	complemented
224	C₁₉	#9911	1	1	yes	characteristic, complemented, Sylow
225	C₂⁴	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
226	C₂×Q₈	—	76	2	no	—
227	C₂×Q₈	—	76	2	no	—
228	C₂×Q₈	—	76	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	4	1	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	—	4	1	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	4	2	no	—
238	C₂²×C₄	—	4	2	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	—	38	2	no	—
244	C₂²×C₄	—	38	2	no	—
245	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
246	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
247	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
248	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
249	C₂²×C₄	#64526	1	2	yes	—
250	C₂²×C₄	#64528	1	2	yes	—
251	C₂²×C₄	C₂×Dic₃₈	1	2	yes	—
252	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
254	C₂²×C₄	#64541	1	2	yes	—
255	C₂×C₈	—	76	2	no	—
256	C₂×C₈	—	76	2	no	—
257	C₂×C₈	—	76	2	no	—
258	C₄⋊C₄	—	76	2	no	—
259	C₄⋊C₄	—	76	2	no	—
260	C₄⋊C₄	—	76	4	no	—
261	C₄⋊C₄	—	76	2	no	—
262	C₄⋊C₄	—	76	2	no	—
263	C₄⋊C₄	—	76	2	no	—
264	C₂²⋊C₄	—	76	4	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	4	1	no	—
268	C₄²	—	4	1	no	—
269	C₂³	—	2	1	no	—
270	C₂³	—	2	1	no	—
271	C₂³	—	2	1	no	—
272	C₂³	—	2	1	no	—
273	C₂³	#10597370	1	2	yes	—
274	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
275	C₂³	C₂×#64526	1	1	yes	characteristic
276	C₂³	C₂×#64528	1	1	yes	characteristic
277	C₂³	#10597395	1	2	yes	—
278	Q₈	—	152	4	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	—	8	2	no	—
302	C₂×C₄	—	8	2	no	—
303	C₂×C₄	—	8	2	no	—
304	C₂×C₄	—	76	2	no	—
305	C₂×C₄	—	76	2	no	—
306	C₂×C₄	—	76	2	no	—
307	C₂×C₄	—	76	2	no	—
308	C₂×C₄	—	76	2	no	—
309	C₂×C₄	—	76	2	no	—
310	C₂×C₄	#10597263	1	2	yes	—
311	C₂×C₄	#10597267	1	2	yes	—
312	C₂×C₄	#10597267	1	2	yes	—
313	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
314	C₂×C₄	#10597315	1	1	yes	characteristic
315	C₂×C₄	#10597321	1	1	yes	characteristic
316	C₈	—	152	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	#11926450	1	1	yes	characteristic
326	C₂²	C₂×#10597263	1	1	yes	characteristic
327	C₂²	#11926517	1	1	yes	characteristic
328	C₂²	C₂×#10597267	1	1	yes	characteristic
329	C₂²	#11926548	1	1	yes	characteristic
330	C₂²	#11926562	1	1	yes	characteristic
331	C₂²	#11926562	1	1	yes	characteristic
332	C₂²	#11926834	1	2	yes	—
333	C₂²	#11926839	1	2	yes	—
334	C₄	—	2	1	no	—
335	C₄	—	2	2	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	2	no	—
338	C₄	—	2	2	no	—
339	C₄	—	2	2	no	—
340	C₄	—	4	1	no	—
341	C₄	—	8	4	no	—
342	C₄	—	152	2	no	—
343	C₄	—	152	2	no	—
344	C₂	—	2	2	no	—
345	C₂	—	2	2	no	—
346	C₂	#14562163	1	1	yes	characteristic
347	C₂	#14562181	1	1	yes	characteristic
348	C₂	#14562198	1	1	yes	characteristic
349	C₂	#14562206	1	1	yes	characteristic
350	C₂	#14564583	1	1	yes	characteristic
351	C₂	#14564616	1	1	yes	characteristic
352	C₂	#14564620	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₁₉)⋊C₄
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#1434	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#1434	D₁₉
hypocenter	C₁₉	#9911
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1434

Complements

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

N	H ≅ G/N	type
5. C₂×#10597315	C₄	semidirect
11. #11926442	C₄	semidirect
12. #11926414	C₄	semidirect
43. C₁₉×#95	Q₈	semidirect
223. #95	Dic₃₈	semidirect
224. C₁₉	#9911	semidirect

Maximal subgroups

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

subgroup	index
#14571987	2
#14563615	2
#14561514	2
C₁₉×#1434	2
#9911	19