Subgroups — #18081674

Order 4864.

624 conjugacy classes of subgroups.

The subgroup lattice has 360 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	C₂×#11926280	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	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
8	C₂×C₂²⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
9	#11926450	C₄	1	2	yes	complemented
10	#11926442	—	2	1	no	—
11	#11926442	—	2	1	no	—
12	#11926414	C₄	1	1	yes	characteristic, complemented
13	#11926404	—	2	1	no	—
14	#11926404	—	2	1	no	—
15	#11926299	—	2	1	no	—
16	#11926280	—	2	8	no	—
17	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
18	C₂×C₁₉×#95	—	2	1	no	—
19	C₂×C₁₉×#95	—	2	1	no	—
20	C₂×C₁₉×#95	C₂²	1	2	yes	—
21	C₂³×C₄×C₁₉	—	2	2	no	—
22	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
25	C₂²×Dic₃₈	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₁₉	—	4	2	no	—
27	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
31	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
32	C₂×C₄⋊Dic₁₉	—	2	1	no	—
33	C₂×C₄⋊Dic₁₉	—	2	1	no	—
34	C₂×C₄⋊Dic₁₉	—	2	1	no	—
35	C₂×C₄⋊Dic₁₉	—	2	1	no	—
36	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
37	#10597370	—	2	4	no	—
38	#10597370	—	2	4	no	—
39	C₂×Dic₁₉⋊C₄	—	2	2	no	—
40	C₂²×C₁₉⋊C₈	—	2	2	no	—
41	#10597315	—	4	2	no	—
42	C₁₉×#95	—	2	2	no	—
43	C₁₉×#95	—	4	2	no	—
44	C₁₉×#95	—	4	2	no	—
45	C₁₉×#95	D₄	1	4	yes	—
46	C₁₉⋊#95	—	2	2	no	—
47	C₁₉⋊#95	—	2	2	no	—
48	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
49	C₄⋊(C₁₉⋊C₈)	—	4	2	no	—
50	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	—	2	2	no	—
52	C₂²×C₄×C₁₉	—	2	2	no	—
53	C₂²×C₄×C₁₉	—	2	4	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	1	no	—
58	C₂²×C₄×C₁₉	—	4	1	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	—	4	2	no	—
64	C₂²×C₄×C₁₉	—	4	2	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
70	C₂²×C₄×C₁₉	Q₁₆	1	2	yes	—
71	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₉	#41	1	2	yes	—
75	C₂²×Dic₁₉	—	2	2	no	—
76	C₂²×Dic₁₉	—	2	2	no	—
77	C₂×Dic₃₈	—	2	4	no	—
78	C₂×Dic₃₈	—	4	2	no	—
79	C₂×Dic₃₈	—	4	2	no	—
80	C₄²×C₁₉	—	2	2	no	—
81	C₄²×C₁₉	—	2	2	no	—
82	C₄²×C₁₉	—	4	1	no	—
83	C₄²×C₁₉	—	4	1	no	—
84	C₂²⋊Dic₁₉	—	4	4	no	—
85	C₄⋊Dic₁₉	—	2	4	no	—
86	C₄⋊Dic₁₉	—	4	2	no	—
87	C₄⋊Dic₁₉	—	4	2	no	—
88	C₄⋊Dic₁₉	—	4	2	no	—
89	C₄⋊Dic₁₉	—	4	2	no	—
90	Dic₁₉⋊C₄	—	4	2	no	—
91	Dic₁₉⋊C₄	—	4	2	no	—
92	C₂×C₁₉⋊C₈	—	4	2	no	—
93	C₂×C₁₉⋊C₈	—	4	2	no	—
94	C₂×C₁₉⋊C₈	—	4	2	no	—
95	#9714	—	19	1	no	maximal, Sylow
96	C₂³×C₁₉	—	2	1	no	—
97	C₂³×C₁₉	—	2	1	no	—
98	C₂³×C₁₉	—	2	1	no	—
99	C₂³×C₁₉	—	2	1	no	—
100	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂³×C₁₉	#121	1	2	yes	—
102	C₂³×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
103	C₂³×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
104	C₂³×C₁₉	#137	1	2	yes	—
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	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	2	1	no	—
118	C₂×C₄×C₁₉	—	4	2	no	—
119	C₂×C₄×C₁₉	—	4	2	no	—
120	C₂×C₄×C₁₉	—	4	2	no	—
121	C₂×C₄×C₁₉	—	4	2	no	—
122	C₂×C₄×C₁₉	—	4	4	no	—
123	C₂×C₄×C₁₉	—	4	4	no	—
124	C₂×C₄×C₁₉	—	4	4	no	—
125	C₂×C₄×C₁₉	—	4	4	no	—
126	C₂×C₄×C₁₉	—	8	2	no	—
127	C₂×C₄×C₁₉	—	8	2	no	—
128	C₂×C₄×C₁₉	—	8	2	no	—
129	C₂×C₄×C₁₉	—	8	2	no	—
130	C₂×C₄×C₁₉	#99	1	2	yes	—
131	C₂×C₄×C₁₉	#103	1	2	yes	—
132	C₂×C₄×C₁₉	#103	1	2	yes	—
133	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
135	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
136	C₂×Dic₁₉	—	4	2	no	—
137	C₂×Dic₁₉	—	4	2	no	—
138	C₂×Dic₁₉	—	4	2	no	—
139	C₂×Dic₁₉	—	4	2	no	—
140	C₂×Dic₁₉	—	4	2	no	—
141	C₂×Dic₁₉	—	4	2	no	—
142	Dic₃₈	—	8	4	no	—
143	C₁₉⋊C₈	—	8	2	no	—
144	#2857	—	19	1	no	—
145	#1797	—	19	1	no	—
146	C₂×#328	—	19	4	no	—
147	#1434	D₁₉	1	1	yes	characteristic
148	C₂²×C₁₉	—	2	2	no	—
149	C₂²×C₁₉	—	2	2	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	#390	1	1	yes	characteristic
157	C₂²×C₁₉	C₂×#99	1	1	yes	characteristic
158	C₂²×C₁₉	#410	1	1	yes	characteristic
159	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
160	C₂²×C₁₉	#419	1	1	yes	characteristic
161	C₂²×C₁₉	#421	1	1	yes	characteristic
162	C₂²×C₁₉	#421	1	1	yes	characteristic
163	C₂²×C₁₉	#461	1	2	yes	—
164	C₂²×C₁₉	#462	1	2	yes	—
165	C₄×C₁₉	—	2	1	no	—
166	C₄×C₁₉	—	2	2	no	—
167	C₄×C₁₉	—	2	1	no	—
168	C₄×C₁₉	—	2	2	no	—
169	C₄×C₁₉	—	2	2	no	—
170	C₄×C₁₉	—	2	2	no	—
171	C₄×C₁₉	—	4	1	no	—
172	C₄×C₁₉	—	8	4	no	—
173	Dic₁₉	—	8	2	no	—
174	Dic₁₉	—	8	2	no	—
175	C₂×C₄⋊Q₈	—	19	1	no	—
176	C₂×C₂²⋊Q₈	—	19	2	no	—
177	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
178	C₂×C₄⋊C₈	—	38	1	no	—
179	C₂×C₂²⋊C₈	—	19	2	no	—
180	#391	—	19	2	no	—
181	#389	—	38	1	no	—
182	#389	—	38	1	no	—
183	#384	—	19	1	no	—
184	#380	—	38	1	no	—
185	#380	—	38	1	no	—
186	C₂×#95	—	2	1	no	—
187	C₂×#95	—	2	1	no	—
188	C₂×#95	C₂×D₁₉	1	2	yes	—
189	#336	—	38	1	no	—
190	#328	—	38	8	no	—
191	C₂×C₁₉	—	2	2	no	—
192	C₂×C₁₉	—	2	2	no	—
193	C₂×C₁₉	#1511	1	1	yes	characteristic
194	C₂×C₁₉	#1515	1	1	yes	characteristic
195	C₂×C₁₉	#1518	1	1	yes	characteristic
196	C₂×C₁₉	#1520	1	1	yes	characteristic
197	C₂×C₁₉	#1963	1	1	yes	characteristic
198	C₂×C₁₉	#1964	1	1	yes	characteristic
199	C₂×C₁₉	#1981	1	1	yes	characteristic
200	C₂²×Q₈	—	19	2	no	—
201	C₂³×C₄	—	2	2	no	—
202	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
203	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
204	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
205	C₂²×C₈	—	38	2	no	—
206	C₄⋊Q₈	—	76	2	no	—
207	C₂²⋊Q₈	—	38	4	no	—
208	C₂²⋊Q₈	—	38	4	no	—
209	C₂×C₄⋊C₄	—	19	2	no	—
210	C₂×C₄⋊C₄	—	38	1	no	—
211	C₂×C₄⋊C₄	—	38	1	no	—
212	C₂×C₄⋊C₄	—	38	1	no	—
213	C₂×C₄⋊C₄	—	38	1	no	—
214	C₂×C₄⋊C₄	—	38	2	no	—
215	C₂×C₂²⋊C₄	—	38	2	no	—
216	C₂×C₄²	—	4	2	no	—
217	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
218	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
219	C₂×C₄²	D₇₆	1	1	yes	characteristic
220	C₂×C₄²	D₇₆	1	1	yes	characteristic
221	C₄⋊C₈	—	76	2	no	—
222	C₂²⋊C₈	—	38	4	no	—
223	#95	—	2	2	no	—
224	#95	—	4	2	no	—
225	#95	—	4	2	no	—
226	#95	—	38	2	no	—
227	#95	—	38	2	no	—
228	#95	D₇₆	1	4	yes	—
229	C₁₉	#9714	1	1	yes	characteristic, complemented, Sylow
230	C₂⁴	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
231	C₂×Q₈	—	38	4	no	—
232	C₂×Q₈	—	76	2	no	—
233	C₂×Q₈	—	76	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	—	4	2	no	—
244	C₂²×C₄	—	4	2	no	—
245	C₂²×C₄	—	4	2	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	—	38	2	no	—
249	C₂²×C₄	—	38	2	no	—
250	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
251	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
252	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
253	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
254	C₂²×C₄	#64526	1	2	yes	—
255	C₂²×C₄	#64528	1	2	yes	—
256	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
258	C₂²×C₄	C₂×D₇₆	1	2	yes	—
259	C₂²×C₄	#64541	1	2	yes	—
260	C₂×C₈	—	76	2	no	—
261	C₂×C₈	—	76	2	no	—
262	C₂×C₈	—	76	2	no	—
263	C₄⋊C₄	—	38	4	no	—
264	C₄⋊C₄	—	76	2	no	—
265	C₄⋊C₄	—	76	2	no	—
266	C₄⋊C₄	—	76	2	no	—
267	C₄⋊C₄	—	76	2	no	—
268	C₄⋊C₄	—	76	2	no	—
269	C₄⋊C₄	—	76	2	no	—
270	C₂²⋊C₄	—	76	4	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	4	1	no	—
274	C₄²	—	4	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	—	2	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	—	2	1	no	—
279	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
280	C₂³	#10597376	1	2	yes	—
281	C₂³	C₂×#64526	1	1	yes	characteristic
282	C₂³	C₂×#64528	1	1	yes	characteristic
283	C₂³	#10597397	1	2	yes	—
284	Q₈	—	152	4	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	—	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	1	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	—	4	4	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	—	8	2	no	—
307	C₂×C₄	—	8	2	no	—
308	C₂×C₄	—	8	2	no	—
309	C₂×C₄	—	8	2	no	—
310	C₂×C₄	—	76	2	no	—
311	C₂×C₄	—	76	2	no	—
312	C₂×C₄	—	76	2	no	—
313	C₂×C₄	—	76	2	no	—
314	C₂×C₄	—	76	2	no	—
315	C₂×C₄	—	76	2	no	—
316	C₂×C₄	#10597263	1	2	yes	—
317	C₂×C₄	#10597267	1	2	yes	—
318	C₂×C₄	#10597267	1	2	yes	—
319	C₂×C₄	C₄×D₇₆	1	2	yes	—
320	C₂×C₄	#10597320	1	1	yes	characteristic
321	C₂×C₄	#10597321	1	1	yes	characteristic
322	C₈	—	152	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	#11926446	1	1	yes	characteristic
332	C₂²	C₂×#10597263	1	1	yes	characteristic
333	C₂²	#11926517	1	1	yes	characteristic
334	C₂²	C₂×#10597267	1	1	yes	characteristic
335	C₂²	#11926552	1	1	yes	characteristic
336	C₂²	#11926562	1	1	yes	characteristic
337	C₂²	#11926562	1	1	yes	characteristic
338	C₂²	#11926749	1	2	yes	—
339	C₂²	#11926754	1	2	yes	—
340	C₄	—	2	2	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	2	no	—
343	C₄	—	2	1	no	—
344	C₄	—	2	2	no	—
345	C₄	—	2	2	no	—
346	C₄	—	4	1	no	—
347	C₄	—	8	4	no	—
348	C₄	—	152	2	no	—
349	C₄	—	152	2	no	—
350	C₂	—	2	2	no	—
351	C₂	—	2	2	no	—
352	C₂	#14562167	1	1	yes	characteristic
353	C₂	#14562185	1	1	yes	characteristic
354	C₂	#14562198	1	1	yes	characteristic
355	C₂	#14562206	1	1	yes	characteristic
356	C₂	#14564467	1	1	yes	characteristic
357	C₂	#14564471	1	1	yes	characteristic
358	C₂	#14564558	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	#9714
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
9. #11926450	C₄	semidirect
12. #11926414	C₄	semidirect
229. C₁₉	#9714	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
C₂×#11926280	2
C₁₉×#1434	2
#9714	19