Subgroups — #18078179

Order 4864.

588 conjugacy classes of subgroups.

The subgroup lattice has 344 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×#11926292	C₂	1	2	yes	maximal, complemented
4	#14561519	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₉×#431	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597315	C₂²	1	1	yes	characteristic
7	C₂×#10597370	C₄	1	2	yes	—
8	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
10	#11926450	C₄	1	2	yes	—
11	#11926442	—	2	1	no	—
12	#11926442	—	2	1	no	—
13	#11926414	C₂²	1	1	yes	characteristic
14	#11926404	—	2	1	no	—
15	#11926404	—	2	1	no	—
16	#11926299	—	2	1	no	—
17	#11926292	—	2	4	no	—
18	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
19	C₁₉×#431	—	2	4	no	—
20	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₁₉×C₂²⋊C₈	—	2	1	no	—
22	C₂×C₄×C₈×C₁₉	—	2	1	no	—
23	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
26	C₂²×Dic₃₈	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₁₉	—	2	2	no	—
28	C₂×C₄²×C₁₉	—	2	2	no	—
29	C₂×C₄²×C₁₉	—	2	2	no	—
30	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
34	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
35	C₂×C₄⋊Dic₁₉	—	2	1	no	—
36	C₂×C₄⋊Dic₁₉	—	2	1	no	—
37	C₂×C₄⋊Dic₁₉	—	2	1	no	—
38	C₂×C₄⋊Dic₁₉	—	2	1	no	—
39	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
40	#10597370	—	4	4	no	—
41	C₂×Dic₁₉⋊C₄	—	2	2	no	—
42	C₂²×C₁₉⋊C₈	—	2	2	no	—
43	#10597315	—	2	2	no	—
44	#10597315	—	2	2	no	—
45	C₁₉×C₄⋊C₈	—	2	2	no	—
46	C₁₉×C₄⋊C₈	D₄	1	4	yes	—
47	C₁₉×C₂²⋊C₈	—	2	4	no	—
48	C₄×C₈×C₁₉	—	2	4	no	—
49	C₁₉⋊#95	—	2	2	no	—
50	C₁₉⋊#95	—	2	2	no	—
51	C₂²⋊(C₁₉⋊C₈)	—	4	2	no	—
52	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
53	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
54	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
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	1	no	—
60	C₂²×C₄×C₁₉	—	2	1	no	—
61	C₂²×C₄×C₁₉	—	2	2	no	—
62	C₂²×C₄×C₁₉	—	2	2	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	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×D₄	1	1	yes	characteristic
72	C₂²×Dic₁₉	—	2	2	no	—
73	C₂²×Dic₁₉	—	2	2	no	—
74	C₂×Dic₃₈	—	2	4	no	—
75	C₂×Dic₃₈	—	4	2	no	—
76	C₂×Dic₃₈	—	4	2	no	—
77	C₂×C₈×C₁₉	—	2	4	no	—
78	C₂×C₈×C₁₉	—	4	2	no	—
79	C₂×C₈×C₁₉	—	4	2	no	—
80	C₄²×C₁₉	—	2	2	no	—
81	C₄²×C₁₉	—	2	1	no	—
82	C₄²×C₁₉	—	2	2	no	—
83	C₄²×C₁₉	—	2	1	no	—
84	C₄²×C₁₉	C₂×D₄	1	2	yes	—
85	C₄²×C₁₉	#41	1	2	yes	—
86	C₂²⋊Dic₁₉	—	4	4	no	—
87	C₄⋊Dic₁₉	—	4	1	no	—
88	C₄⋊Dic₁₉	—	4	2	no	—
89	C₄⋊Dic₁₉	—	4	1	no	—
90	C₄⋊Dic₁₉	—	4	2	no	—
91	C₄⋊Dic₁₉	—	4	2	no	—
92	C₄⋊Dic₁₉	—	4	2	no	—
93	Dic₁₉⋊C₄	—	4	4	no	—
94	C₂×C₁₉⋊C₈	—	4	2	no	—
95	C₂×C₁₉⋊C₈	—	4	2	no	—
96	C₂×C₁₉⋊C₈	—	4	2	no	—
97	#9128	—	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	—	2	1	no	—
103	C₂³×C₁₉	—	2	1	no	—
104	C₂³×C₁₉	#103	1	2	yes	—
105	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	4	no	—
113	C₂×C₄×C₁₉	—	2	4	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	—	4	4	no	—
118	C₂×C₄×C₁₉	—	4	2	no	—
119	C₂×C₄×C₁₉	—	4	2	no	—
120	C₂×C₄×C₁₉	#100	1	2	yes	—
121	C₂×C₄×C₁₉	#103	1	2	yes	—
122	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
124	C₂×C₄×C₁₉	#121	1	2	yes	—
125	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
128	C₂×C₄×C₁₉	#135	1	2	yes	—
129	C₂×C₄×C₁₉	#135	1	2	yes	—
130	C₂×C₄×C₁₉	#137	1	2	yes	—
131	C₂×Dic₁₉	—	4	2	no	—
132	C₂×Dic₁₉	—	4	2	no	—
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	Dic₃₈	—	8	4	no	—
138	C₈×C₁₉	—	4	4	no	—
139	C₁₉⋊C₈	—	8	2	no	—
140	C₂×#431	D₁₉	1	1	yes	characteristic
141	#2857	—	19	1	no	—
142	#1797	—	19	1	no	—
143	C₂×#333	—	19	2	no	—
144	#1296	—	19	2	no	—
145	C₂²×C₁₉	—	2	2	no	—
146	C₂²×C₁₉	—	2	2	no	—
147	C₂²×C₁₉	—	2	2	no	—
148	C₂²×C₁₉	—	4	1	no	—
149	C₂²×C₁₉	—	4	1	no	—
150	C₂²×C₁₉	—	4	1	no	—
151	C₂²×C₁₉	—	4	1	no	—
152	C₂²×C₁₉	#386	1	1	yes	characteristic
153	C₂²×C₁₉	C₂×#100	1	1	yes	characteristic
154	C₂²×C₁₉	#413	1	1	yes	characteristic
155	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
156	C₂²×C₁₉	#416	1	1	yes	characteristic
157	C₂²×C₁₉	#416	1	1	yes	characteristic
158	C₂²×C₁₉	#419	1	1	yes	characteristic
159	C₄×C₁₉	—	2	2	no	—
160	C₄×C₁₉	—	2	2	no	—
161	C₄×C₁₉	—	2	2	no	—
162	C₄×C₁₉	—	4	1	no	—
163	C₄×C₁₉	#452	1	2	yes	—
164	C₄×C₁₉	#465	1	2	yes	—
165	C₄×C₁₉	#467	1	2	yes	—
166	C₄×C₁₉	#485	1	2	yes	—
167	Dic₁₉	—	8	2	no	—
168	Dic₁₉	—	8	2	no	—
169	C₂×C₄⋊Q₈	—	19	1	no	—
170	C₂×C₂²⋊Q₈	—	19	2	no	—
171	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
172	#431	—	2	4	no	—
173	C₂×C₄⋊C₈	—	19	2	no	—
174	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
175	C₂×C₂²⋊C₈	—	2	1	no	—
176	C₂×C₂²⋊C₈	—	38	1	no	—
177	C₂×C₄×C₈	—	2	1	no	—
178	#391	—	19	2	no	—
179	#389	—	38	1	no	—
180	#389	—	38	1	no	—
181	#384	—	19	1	no	—
182	#380	—	38	1	no	—
183	#380	—	38	1	no	—
184	#336	—	38	1	no	—
185	#333	—	38	4	no	—
186	C₂×C₁₉	—	4	2	no	—
187	C₂×C₁₉	#1542	1	1	yes	characteristic
188	C₂×C₁₉	#1543	1	1	yes	characteristic
189	C₂×C₁₉	#1548	1	1	yes	characteristic
190	C₂×C₁₉	#1554	1	1	yes	characteristic
191	C₂×C₁₉	#1873	1	1	yes	characteristic
192	C₂×C₁₉	#1892	1	1	yes	characteristic
193	C₂×C₁₉	#1900	1	1	yes	characteristic
194	C₂²×Q₈	—	19	2	no	—
195	C₂³×C₄	C₄×D₁₉	1	1	yes	characteristic
196	C₂³×C₄	D₇₆	1	1	yes	characteristic
197	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	—	38	2	no	—
200	C₄⋊Q₈	—	38	2	no	—
201	C₄⋊Q₈	—	38	2	no	—
202	C₂²⋊Q₈	—	76	4	no	—
203	C₂×C₄⋊C₄	—	19	2	no	—
204	C₂×C₄⋊C₄	—	38	2	no	—
205	C₂×C₄⋊C₄	—	38	1	no	—
206	C₂×C₄⋊C₄	—	38	1	no	—
207	C₂×C₄⋊C₄	—	38	1	no	—
208	C₂×C₄⋊C₄	—	38	1	no	—
209	C₂×C₂²⋊C₄	—	38	2	no	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	—	2	2	no	—
212	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
213	C₂×C₄²	D₇₆	1	1	yes	characteristic
214	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
215	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
216	C₄⋊C₈	—	2	2	no	—
217	C₄⋊C₈	—	38	2	no	—
218	C₄⋊C₈	—	38	2	no	—
219	C₄⋊C₈	C₂²⋊D₁₉	1	4	yes	—
220	C₂²⋊C₈	—	2	4	no	—
221	C₂²⋊C₈	—	76	2	no	—
222	C₄×C₈	—	2	4	no	—
223	#95	—	38	2	no	—
224	#95	—	38	2	no	—
225	C₁₉	#9128	1	1	yes	characteristic, complemented, Sylow
226	C₂⁴	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
227	C₂×Q₈	—	38	4	no	—
228	C₂×Q₈	—	76	2	no	—
229	C₂×Q₈	—	76	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	—	38	2	no	—
240	C₂²×C₄	—	38	2	no	—
241	C₂²×C₄	#64516	1	2	yes	—
242	C₂²×C₄	Dic₇₆	1	2	yes	—
243	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
244	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
245	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	4	2	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	—	76	2	no	—
253	C₂×C₈	—	76	2	no	—
254	C₂×C₈	—	76	2	no	—
255	C₄⋊C₄	—	76	2	no	—
256	C₄⋊C₄	—	76	2	no	—
257	C₄⋊C₄	—	76	2	no	—
258	C₄⋊C₄	—	76	1	no	—
259	C₄⋊C₄	—	76	4	no	—
260	C₄⋊C₄	—	76	2	no	—
261	C₄⋊C₄	—	76	1	no	—
262	C₂²⋊C₄	—	76	4	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	1	no	—
267	C₄²	#64541	1	2	yes	—
268	C₄²	C₂×C₂²⋊D₁₉	1	2	yes	—
269	C₂³	—	2	1	no	—
270	C₂³	—	2	1	no	—
271	C₂³	—	2	1	no	—
272	C₂³	—	2	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	#10597273	1	2	yes	—
276	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
277	Q₈	—	152	4	no	—
278	C₂×C₄	—	2	1	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	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	76	2	no	—
293	C₂×C₄	—	76	2	no	—
294	C₂×C₄	—	76	2	no	—
295	C₂×C₄	—	76	2	no	—
296	C₂×C₄	—	76	2	no	—
297	C₂×C₄	—	76	2	no	—
298	C₂×C₄	#10597273	1	2	yes	—
299	C₂×C₄	#10597280	1	2	yes	—
300	C₂×C₄	C₂×#64516	1	1	yes	characteristic
301	C₂×C₄	#10597350	1	2	yes	—
302	C₂×C₄	C₂×Dic₇₆	1	1	yes	characteristic
303	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
304	C₂×C₄	#10597375	1	1	yes	characteristic
305	C₂×C₄	#10597376	1	2	yes	—
306	C₂×C₄	#10597396	1	2	yes	—
307	C₂×C₄	#10597397	1	2	yes	—
308	C₂×C₄	#10597399	1	1	yes	characteristic
309	C₈	—	4	4	no	—
310	C₈	—	152	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	#11926423	1	1	yes	characteristic
319	C₂²	C₂×#10597280	1	1	yes	characteristic
320	C₂²	#11926526	1	1	yes	characteristic
321	C₂²	C₂×#10597273	1	1	yes	characteristic
322	C₂²	#11926538	1	1	yes	characteristic
323	C₂²	#11926541	1	1	yes	characteristic
324	C₂²	#11926552	1	1	yes	characteristic
325	C₄	—	2	2	no	—
326	C₄	—	2	2	no	—
327	C₄	—	2	2	no	—
328	C₄	—	4	1	no	—
329	C₄	—	152	2	no	—
330	C₄	—	152	2	no	—
331	C₄	#11926706	1	2	yes	—
332	C₄	#11926773	1	2	yes	—
333	C₄	#11926785	1	2	yes	—
334	C₄	#11926885	1	2	yes	—
335	C₂	—	4	2	no	—
336	C₂	#14562290	1	1	yes	characteristic
337	C₂	#14562295	1	1	yes	characteristic
338	C₂	#14562315	1	1	yes	characteristic
339	C₂	#14562343	1	1	yes	characteristic
340	C₂	#14563947	1	1	yes	characteristic
341	C₂	#14564058	1	1	yes	characteristic
342	C₂	#14564098	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₁₉)⋊C₄
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#431	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#431	D₁₉
hypocenter	C₁₉	#9128
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. C₂×#11926292	C₂	semidirect
4. #14561519	C₂	semidirect
225. C₁₉	#9128	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×#11926292	2
#14561519	2
C₂×C₁₉×#431	2
#9128	19