Subgroups — #18090643

Order 4864.

682 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10597276	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926570	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926313	C₂	1	2	yes	maximal
4	#14563354	C₂	1	2	yes	maximal
5	C₂×C₁₉×#340	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	1	yes	characteristic
7	C₂²×#64520	C₂²	1	2	yes	—
8	C₂×#10597372	C₂²	1	1	yes	characteristic
9	C₂×#10597276	—	2	4	no	—
10	C₂×#10597276	—	2	2	no	—
11	#11926570	—	2	2	no	—
12	#11926570	C₄	1	4	yes	complemented
13	C₂×Dic₁₉⋊C₈	—	2	2	no	—
14	#11926313	—	2	4	no	—
15	#11926306	—	2	2	no	—
16	#11926306	—	2	2	no	—
17	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
19	C₁₉×#340	—	2	2	no	—
20	C₂³×C₄×C₁₉	—	2	2	no	—
21	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
22	C₂²×C₈×C₁₉	—	2	2	no	—
23	C₂²×C₈×C₁₉	—	2	2	no	—
24	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₉	D₄	1	2	yes	—
26	C₂²×C₈×C₁₉	D₄	1	2	yes	—
27	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
28	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
31	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
32	C₂×C₄⋊C₄×C₁₉	Q₈	1	4	yes	—
33	C₂×C₂²⋊Dic₁₉	—	2	1	no	—
34	#10597372	—	2	1	no	—
35	#10597372	—	2	1	no	—
36	#10597372	C₂×C₄	1	2	yes	—
37	#10597372	Q₈	1	2	yes	—
38	C₂×C₄⋊Dic₁₉	D₄	1	1	yes	characteristic
39	C₂×C₄⋊Dic₁₉	D₄	1	1	yes	characteristic
40	C₂×C₄×Dic₁₉	—	2	1	no	—
41	C₂×#64520	—	2	1	no	—
42	C₂×#64520	—	2	2	no	—
43	C₂×#64520	—	2	2	no	—
44	C₂×#64520	—	2	2	no	—
45	C₂×#64520	—	2	2	no	—
46	C₂×#64520	—	2	2	no	—
47	C₂×#64520	—	2	2	no	—
48	C₂×#64520	—	2	2	no	—
49	C₂×#64520	C₂×C₄	1	2	yes	—
50	C₂×#64520	D₄	1	4	yes	—
51	C₂²×C₁₉⋊C₈	—	2	2	no	—
52	C₂²×C₁₉⋊C₈	—	2	1	no	—
53	#10597276	—	2	8	no	—
54	Dic₁₉⋊C₈	—	2	8	no	—
55	C₂⁴×C₁₉	#41	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	—	2	4	no	—
57	C₂²×C₄×C₁₉	—	4	2	no	—
58	C₂²×C₄×C₁₉	—	4	2	no	—
59	C₂²×C₄×C₁₉	—	8	2	no	—
60	C₂²×C₄×C₁₉	—	8	2	no	—
61	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
67	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	#41	1	2	yes	—
69	C₂²×C₄×C₁₉	#41	1	2	yes	—
70	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
71	C₂²×Dic₁₉	—	2	1	no	—
72	C₂²×Dic₁₉	—	2	1	no	—
73	C₂×C₈×C₁₉	—	2	4	no	—
74	C₂×C₈×C₁₉	—	2	2	no	—
75	C₂×C₈×C₁₉	—	2	4	no	—
76	C₂×C₈×C₁₉	—	2	2	no	—
77	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
78	C₄⋊C₄×C₁₉	—	2	4	no	—
79	C₄⋊C₄×C₁₉	—	2	4	no	—
80	C₄⋊C₄×C₁₉	—	4	2	no	—
81	C₄⋊C₄×C₁₉	—	4	2	no	—
82	C₂²⋊Dic₁₉	—	2	2	no	—
83	C₂²⋊Dic₁₉	—	2	2	no	—
84	C₄⋊Dic₁₉	—	2	1	no	—
85	C₄⋊Dic₁₉	—	2	1	no	—
86	C₄⋊Dic₁₉	C₄⋊C₄	1	2	yes	—
87	C₄⋊Dic₁₉	C₄⋊C₄	1	2	yes	—
88	C₄×Dic₁₉	—	2	2	no	—
89	C₄×Dic₁₉	—	2	2	no	—
90	#64520	—	2	4	no	—
91	#64520	—	2	4	no	—
92	#64520	—	2	4	no	—
93	#64520	—	4	1	no	—
94	#64520	—	4	1	no	—
95	#64520	—	4	2	no	—
96	#64520	—	4	2	no	—
97	C₂×C₁₉⋊C₈	—	2	2	no	—
98	C₂×C₁₉⋊C₈	—	2	4	no	—
99	C₂×C₁₉⋊C₈	—	4	2	no	—
100	C₂×C₁₉⋊C₈	—	4	2	no	—
101	C₂×C₁₉⋊C₈	—	4	2	no	—
102	#11242	—	19	1	no	maximal, Sylow
103	C₂³×C₁₉	—	2	2	no	—
104	C₂³×C₁₉	—	2	1	no	—
105	C₂³×C₁₉	—	2	1	no	—
106	C₂³×C₁₉	#117	1	1	yes	characteristic
107	C₂³×C₁₉	#121	1	1	yes	characteristic
108	C₂³×C₁₉	C₂²⋊Q₈	1	2	yes	—
109	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂³×C₁₉	#126	1	2	yes	—
111	C₂×C₄×C₁₉	—	4	4	no	—
112	C₂×C₄×C₁₉	—	8	4	no	—
113	C₂×C₄×C₁₉	—	8	2	no	—
114	C₂×C₄×C₁₉	—	8	2	no	—
115	C₂×C₄×C₁₉	—	8	4	no	—
116	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
120	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
121	C₂×C₄×C₁₉	#121	1	2	yes	—
122	C₂×C₄×C₁₉	#121	1	2	yes	—
123	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
126	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
128	C₂×C₄×C₁₉	#123	1	2	yes	—
129	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#125	1	2	yes	—
131	C₂×C₄×C₁₉	#125	1	2	yes	—
132	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
133	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
134	C₂×Dic₁₉	—	2	2	no	—
135	C₂×Dic₁₉	—	2	2	no	—
136	C₂×Dic₁₉	—	4	2	no	—
137	C₂×Dic₁₉	—	4	1	no	—
138	C₂×Dic₁₉	—	4	1	no	—
139	C₈×C₁₉	—	2	4	no	—
140	C₁₉⋊C₈	—	4	2	no	—
141	C₁₉⋊C₈	—	4	4	no	—
142	C₂²×#108	—	19	1	no	—
143	C₂×#424	—	19	1	no	—
144	C₂×#343	—	19	2	no	—
145	#1736	—	19	2	no	—
146	C₂×#340	D₁₉	1	1	yes	characteristic
147	C₂²×C₁₉	—	2	2	no	—
148	C₂²×C₁₉	—	2	2	no	—
149	C₂²×C₁₉	—	2	2	no	—
150	C₂²×C₁₉	—	2	2	no	—
151	C₂²×C₁₉	—	2	1	no	—
152	C₂²×C₁₉	—	2	1	no	—
153	C₂²×C₁₉	—	2	1	no	—
154	C₂²×C₁₉	—	2	1	no	—
155	C₂²×C₁₉	#380	1	1	yes	characteristic
156	C₂²×C₁₉	#387	1	2	yes	—
157	C₂²×C₁₉	#395	1	1	yes	characteristic
158	C₂²×C₁₉	#396	1	2	yes	—
159	C₂²×C₁₉	#400	1	1	yes	characteristic
160	C₂²×C₁₉	#471	1	2	yes	—
161	C₂²×C₁₉	#472	1	1	yes	characteristic
162	C₂²×C₁₉	#473	1	1	yes	characteristic
163	C₄×C₁₉	—	8	4	no	—
164	C₄×C₁₉	#386	1	1	yes	characteristic
165	C₄×C₁₉	#389	1	2	yes	—
166	C₄×C₁₉	#391	1	1	yes	characteristic
167	C₄×C₁₉	#394	1	1	yes	characteristic
168	C₄×C₁₉	#398	1	1	yes	characteristic
169	C₄×C₁₉	#398	1	2	yes	—
170	Dic₁₉	—	4	2	no	—
171	Dic₁₉	—	4	2	no	—
172	C₂²×M₁₆	—	19	1	no	—
173	C₂²×M₁₆	—	19	2	no	—
174	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
175	C₂×#117	—	19	1	no	—
176	C₂²×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
177	C₂×#108	—	38	4	no	—
178	C₂×#108	—	38	2	no	—
179	#424	—	19	4	no	—
180	#424	—	38	2	no	—
181	C₂×C₄⋊C₈	—	38	2	no	—
182	#343	—	38	4	no	—
183	#340	—	2	2	no	—
184	#340	—	38	2	no	—
185	#340	—	38	2	no	—
186	C₂×C₁₉	—	2	2	no	—
187	C₂×C₁₉	—	2	1	no	—
188	C₂×C₁₉	—	2	1	no	—
189	C₂×C₁₉	#1441	1	1	yes	characteristic
190	C₂×C₁₉	#1815	1	1	yes	characteristic
191	C₂×C₁₉	#1816	1	1	yes	characteristic
192	C₂×C₁₉	#2063	1	2	yes	—
193	C₂×C₁₉	#2064	1	2	yes	—
194	C₂³×C₄	—	2	2	no	—
195	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
196	C₂×M₁₆	—	19	2	no	—
197	C₂×M₁₆	—	19	4	no	—
198	C₂×M₁₆	—	38	1	no	—
199	C₂×M₁₆	—	38	2	no	—
200	C₂×M₁₆	—	38	2	no	—
201	C₂×M₁₆	—	38	2	no	—
202	C₂×M₁₆	—	38	2	no	—
203	C₂×M₁₆	—	38	2	no	—
204	C₂×M₁₆	—	38	2	no	—
205	C₂×M₁₆	—	38	2	no	—
206	C₂²×C₈	—	2	2	no	—
207	C₂²×C₈	—	2	2	no	—
208	C₂²×C₈	—	38	2	no	—
209	C₂²×C₈	—	38	1	no	—
210	C₂²×C₈	C₄×D₁₉	1	2	yes	—
211	C₂²×C₈	D₇₆	1	2	yes	—
212	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
213	#117	—	19	2	no	—
214	#117	—	19	2	no	—
215	#117	—	38	1	no	—
216	#117	—	38	1	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	19	1	no	—
223	C₂×C₄⋊C₄	—	19	1	no	—
224	C₂×C₄⋊C₄	Dic₃₈	1	4	yes	—
225	C₂×C₂²⋊C₄	—	38	1	no	—
226	C₂×C₄²	—	38	1	no	—
227	#108	—	38	8	no	—
228	C₄⋊C₈	—	38	8	no	—
229	C₁₉	#11242	1	1	yes	characteristic, complemented, Sylow
230	C₂⁴	#64543	1	1	yes	characteristic
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	—	8	2	no	—
235	C₂²×C₄	—	8	2	no	—
236	C₂²×C₄	—	38	1	no	—
237	C₂²×C₄	—	38	1	no	—
238	C₂²×C₄	C₂×Dic₃₈	1	2	yes	—
239	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
241	C₂²×C₄	#64541	1	2	yes	—
242	C₂²×C₄	D₄×D₁₉	1	2	yes	—
243	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
244	C₂²×C₄	#64543	1	2	yes	—
245	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
246	C₂²×C₄	#64545	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
248	M₁₆	—	38	4	no	—
249	M₁₆	—	38	4	no	—
250	M₁₆	—	38	4	no	—
251	M₁₆	—	76	2	no	—
252	M₁₆	—	76	1	no	—
253	M₁₆	—	76	1	no	—
254	M₁₆	—	76	2	no	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	38	4	no	—
260	C₂×C₈	—	38	2	no	—
261	C₂×C₈	—	76	2	no	—
262	C₂×C₈	—	76	2	no	—
263	C₂×C₈	—	76	2	no	—
264	C₂×C₈	(C₂×D₁₉)⋊C₄	1	4	yes	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	2	4	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	19	2	no	—
270	C₄⋊C₄	—	19	2	no	—
271	C₄⋊C₄	—	38	1	no	—
272	C₄⋊C₄	—	38	1	no	—
273	C₂²⋊C₄	—	38	2	no	—
274	C₂²⋊C₄	—	38	2	no	—
275	C₄²	—	38	2	no	—
276	C₄²	—	38	2	no	—
277	C₂³	—	2	2	no	—
278	C₂³	—	2	1	no	—
279	C₂³	—	2	1	no	—
280	C₂³	#10597324	1	2	yes	—
281	C₂³	#10597325	1	2	yes	—
282	C₂³	#10597338	1	1	yes	characteristic
283	C₂³	#10597343	1	1	yes	characteristic
284	C₂³	#10597384	1	1	yes	characteristic
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	8	4	no	—
287	C₂×C₄	—	8	2	no	—
288	C₂×C₄	—	8	2	no	—
289	C₂×C₄	—	8	4	no	—
290	C₂×C₄	—	38	2	no	—
291	C₂×C₄	—	38	2	no	—
292	C₂×C₄	—	76	2	no	—
293	C₂×C₄	—	76	1	no	—
294	C₂×C₄	—	76	1	no	—
295	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
296	C₂×C₄	#10597315	1	1	yes	characteristic
297	C₂×C₄	#10597321	1	1	yes	characteristic
298	C₂×C₄	#10597324	1	2	yes	—
299	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
300	C₂×C₄	#10597328	1	1	yes	characteristic
301	C₂×C₄	#10597330	1	2	yes	—
302	C₂×C₄	#10597332	1	1	yes	characteristic
303	C₂×C₄	#10597334	1	2	yes	—
304	C₂×C₄	#10597335	1	2	yes	—
305	C₂×C₄	#10597336	1	2	yes	—
306	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
307	C₂×C₄	#10597340	1	2	yes	—
308	C₂×C₄	#10597341	1	1	yes	characteristic
309	C₂×C₄	#10597370	1	2	yes	—
310	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
311	C₂×C₄	#10597385	1	2	yes	—
312	C₂×C₄	#10597392	1	1	yes	characteristic
313	C₈	—	2	4	no	—
314	C₈	—	76	2	no	—
315	C₈	—	76	4	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	#11926401	1	1	yes	characteristic
325	C₂²	#11926431	1	2	yes	—
326	C₂²	#11926465	1	1	yes	characteristic
327	C₂²	#11926467	1	2	yes	—
328	C₂²	#11926484	1	1	yes	characteristic
329	C₂²	#11926804	1	2	yes	—
330	C₂²	#11926811	1	1	yes	characteristic
331	C₂²	#11926816	1	1	yes	characteristic
332	C₄	—	8	4	no	—
333	C₄	—	76	2	no	—
334	C₄	—	76	2	no	—
335	C₄	#11926421	1	1	yes	characteristic
336	C₄	#11926441	1	2	yes	—
337	C₄	#11926450	1	1	yes	characteristic
338	C₄	#11926462	1	1	yes	characteristic
339	C₄	#11926476	1	1	yes	characteristic
340	C₄	#11926477	1	2	yes	—
341	C₂	—	2	2	no	—
342	C₂	—	2	1	no	—
343	C₂	—	2	1	no	—
344	C₂	#14561889	1	1	yes	characteristic
345	C₂	#14563695	1	1	yes	characteristic
346	C₂	#14563700	1	1	yes	characteristic
347	C₂	#14564993	1	2	yes	—
348	C₂	#14565000	1	2	yes	—

	subgroup	G/H
center	C₂³	#10597384
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×#340	C₂
socle	C₂³×C₁₉	#121
prime radical	C₂⁴×C₁₉	#41
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#340	D₁₉
hypocenter	C₁₉	#11242
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#340

Complements

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

N	H ≅ G/N	type
12. #11926570	C₄	semidirect
229. C₁₉	#11242	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#10597276	2
C₂×#11926570	2
C₂×#11926313	2
#14563354	2
C₂×C₁₉×#340	2
#11242	19