Subgroups — #18082607

Order 4864.

654 conjugacy classes of subgroups.

The subgroup lattice has 384 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	C₄×C₂²⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#14561514	C₂	1	4	yes	maximal
4	C₁₉×C₄⋊#95	C₂	1	1	yes	maximal, characteristic
5	C₂×#10597315	C₄	1	1	yes	characteristic, complemented
6	C₂×#10597370	—	2	1	no	—
7	C₂×C₂²⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
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×C₄⋊C₄×C₁₉	—	2	1	no	—
17	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
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×C₄⋊C₄×C₁₉	—	2	4	no	—
26	C₂×C₄⋊C₄×C₁₉	—	4	2	no	—
27	C₂×C₄⋊C₄×C₁₉	—	4	2	no	—
28	C₂×C₄²×C₁₉	—	4	2	no	—
29	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
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×C₂²⋊Dic₁₉	—	2	2	no	—
34	C₂×C₄⋊Dic₁₉	—	2	1	no	—
35	C₂×C₄⋊Dic₁₉	—	2	1	no	—
36	C₂×C₄⋊Dic₁₉	—	2	1	no	—
37	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
39	#10597370	—	4	2	no	—
40	#10597370	—	4	2	no	—
41	C₂×Dic₁₉⋊C₄	—	2	2	no	—
42	C₂²×C₁₉⋊C₈	—	2	2	no	—
43	#10597315	—	4	2	no	—
44	C₁₉×#95	—	2	4	no	—
45	C₁₉×#95	—	4	2	no	—
46	C₁₉⋊#95	—	2	2	no	—
47	C₁₉⋊#95	—	2	2	no	—
48	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
49	C₂²⋊(C₁₉⋊C₈)	Q₈	1	4	yes	complemented
50	C₄×C₁₉⋊C₈	—	4	2	no	—
51	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	—	2	2	no	—
53	C₂²×C₄×C₁₉	—	2	2	no	—
54	C₂²×C₄×C₁₉	—	2	4	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	D₈	1	2	yes	—
70	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
71	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
72	C₂²×C₄×C₁₉	Q₁₆	1	2	yes	—
73	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₁₉	#41	1	2	yes	—
77	C₂²×Dic₁₉	—	2	2	no	—
78	C₂²×Dic₁₉	—	2	2	no	—
79	C₂×Dic₃₈	—	4	2	no	—
80	C₂×Dic₃₈	—	4	2	no	—
81	C₂×Dic₃₈	—	4	2	no	—
82	C₄⋊C₄×C₁₉	—	4	8	no	—
83	C₄²×C₁₉	—	2	2	no	—
84	C₄²×C₁₉	—	2	2	no	—
85	C₄²×C₁₉	—	4	1	no	—
86	C₄²×C₁₉	—	4	1	no	—
87	C₂²⋊Dic₁₉	—	4	2	no	—
88	C₂²⋊Dic₁₉	—	4	2	no	—
89	C₄⋊Dic₁₉	—	4	2	no	—
90	C₄⋊Dic₁₉	—	4	2	no	—
91	C₄⋊Dic₁₉	—	4	2	no	—
92	C₄⋊Dic₁₉	—	4	2	no	—
93	C₄⋊Dic₁₉	—	4	2	no	—
94	Dic₁₉⋊C₄	—	4	4	no	—
95	C₂×C₁₉⋊C₈	—	2	4	no	—
96	C₂×C₁₉⋊C₈	—	4	2	no	—
97	C₂×C₁₉⋊C₈	—	4	2	no	—
98	#9885	—	19	1	no	maximal, Sylow
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	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂³×C₁₉	#124	1	1	yes	characteristic
105	C₂³×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
106	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
107	C₂³×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
108	C₂³×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
109	C₂³×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
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	—	2	1	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	2	4	no	—
121	C₂×C₄×C₁₉	—	2	4	no	—
122	C₂×C₄×C₁₉	—	4	2	no	—
123	C₂×C₄×C₁₉	—	4	2	no	—
124	C₂×C₄×C₁₉	—	4	2	no	—
125	C₂×C₄×C₁₉	—	4	2	no	—
126	C₂×C₄×C₁₉	—	4	4	no	—
127	C₂×C₄×C₁₉	—	4	4	no	—
128	C₂×C₄×C₁₉	—	4	4	no	—
129	C₂×C₄×C₁₉	—	8	2	no	—
130	C₂×C₄×C₁₉	—	8	2	no	—
131	C₂×C₄×C₁₉	—	8	2	no	—
132	C₂×C₄×C₁₉	—	8	2	no	—
133	C₂×C₄×C₁₉	#99	1	2	yes	—
134	C₂×C₄×C₁₉	#102	1	2	yes	—
135	C₂×C₄×C₁₉	#102	1	2	yes	—
136	C₂×C₄×C₁₉	#103	1	2	yes	—
137	C₂×C₄×C₁₉	#103	1	2	yes	—
138	C₂×C₄×C₁₉	#104	1	2	yes	—
139	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
140	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
141	C₂×Dic₁₉	—	4	2	no	—
142	C₂×Dic₁₉	—	4	2	no	—
143	C₂×Dic₁₉	—	4	2	no	—
144	C₂×Dic₁₉	—	4	2	no	—
145	C₂×Dic₁₉	—	4	2	no	—
146	C₂×Dic₁₉	—	4	2	no	—
147	Dic₃₈	—	8	4	no	—
148	C₁₉⋊C₈	—	4	4	no	—
149	#2857	—	19	1	no	—
150	C₄×C₂²⋊C₈	—	19	1	no	—
151	C₄⋊#95	D₁₉	1	1	yes	characteristic
152	#1294	—	19	4	no	—
153	C₂²×C₁₉	—	2	1	no	—
154	C₂²×C₁₉	—	2	1	no	—
155	C₂²×C₁₉	—	2	1	no	—
156	C₂²×C₁₉	—	2	1	no	—
157	C₂²×C₁₉	—	2	1	no	—
158	C₂²×C₁₉	—	2	1	no	—
159	C₂²×C₁₉	—	2	1	no	—
160	C₂²×C₁₉	—	2	1	no	—
161	C₂²×C₁₉	—	2	1	no	—
162	C₂²×C₁₉	—	2	1	no	—
163	C₂²×C₁₉	—	2	1	no	—
164	C₂²×C₁₉	—	2	1	no	—
165	C₂²×C₁₉	#391	1	1	yes	characteristic
166	C₂²×C₁₉	C₂×#99	1	1	yes	characteristic
167	C₂²×C₁₉	#410	1	1	yes	characteristic
168	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
169	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
170	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
171	C₂²×C₁₉	#421	1	1	yes	characteristic
172	C₂²×C₁₉	#486	1	1	yes	characteristic
173	C₂²×C₁₉	#487	1	1	yes	characteristic
174	C₂²×C₁₉	#498	1	1	yes	characteristic
175	C₂²×C₁₉	#500	1	1	yes	characteristic
176	C₄×C₁₉	—	2	2	no	—
177	C₄×C₁₉	—	2	2	no	—
178	C₄×C₁₉	—	2	2	no	—
179	C₄×C₁₉	—	2	2	no	—
180	C₄×C₁₉	—	2	2	no	—
181	C₄×C₁₉	—	8	4	no	—
182	C₄×C₁₉	#327	1	4	yes	—
183	Dic₁₉	—	8	2	no	—
184	Dic₁₉	—	8	2	no	—
185	C₂×C₄⋊Q₈	—	19	1	no	—
186	C₂×C₂²⋊Q₈	—	38	1	no	—
187	C₂²×C₄⋊C₄	—	2	1	no	—
188	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
189	C₂×C₂²⋊C₈	—	19	2	no	—
190	C₂×C₄×C₈	—	38	1	no	—
191	#391	—	38	1	no	—
192	#389	—	19	2	no	—
193	#389	—	38	1	no	—
194	#384	—	19	1	no	—
195	#380	—	19	2	no	—
196	#380	—	38	1	no	—
197	C₂×#95	—	2	1	no	—
198	C₂×#95	C₂×D₁₉	1	2	yes	—
199	#336	—	38	1	no	—
200	C₂×C₁₉	—	2	1	no	—
201	C₂×C₁₉	—	2	1	no	—
202	C₂×C₁₉	—	2	1	no	—
203	C₂×C₁₉	—	2	1	no	—
204	C₂×C₁₉	C₂×#327	1	1	yes	characteristic
205	C₂×C₁₉	#1503	1	1	yes	characteristic
206	C₂×C₁₉	#1517	1	1	yes	characteristic
207	C₂×C₁₉	#1520	1	1	yes	characteristic
208	C₂×C₁₉	#1986	1	1	yes	characteristic
209	C₂×C₁₉	#1990	1	1	yes	characteristic
210	C₂×C₁₉	#1996	1	1	yes	characteristic
211	C₂²×Q₈	—	38	1	no	—
212	C₂³×C₄	—	2	2	no	—
213	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
214	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
215	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
216	C₂²×C₈	—	38	2	no	—
217	C₄⋊Q₈	—	76	2	no	—
218	C₂²⋊Q₈	—	76	2	no	—
219	C₂²⋊Q₈	—	76	2	no	—
220	C₂×C₄⋊C₄	—	2	4	no	—
221	C₂×C₄⋊C₄	—	4	2	no	—
222	C₂×C₄⋊C₄	—	4	2	no	—
223	C₂×C₄⋊C₄	—	19	2	no	—
224	C₂×C₄⋊C₄	—	19	2	no	—
225	C₂×C₄⋊C₄	—	38	1	no	—
226	C₂×C₄⋊C₄	—	38	2	no	—
227	C₂×C₄⋊C₄	—	38	1	no	—
228	C₂×C₄⋊C₄	—	38	1	no	—
229	C₂×C₂²⋊C₄	—	38	2	no	—
230	C₂×C₄²	—	4	2	no	—
231	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
232	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
233	C₂×C₄²	D₇₆	1	1	yes	characteristic
234	C₂×C₄²	D₇₆	1	1	yes	characteristic
235	C₂²⋊C₈	—	19	4	no	—
236	C₂²⋊C₈	—	38	2	no	—
237	C₄×C₈	—	76	2	no	—
238	#95	—	2	4	no	—
239	#95	—	4	2	no	—
240	#95	—	38	2	no	—
241	#95	—	38	2	no	—
242	C₁₉	#9885	1	1	yes	characteristic, complemented, Sylow
243	C₂⁴	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
244	C₂×Q₈	—	76	2	no	—
245	C₂×Q₈	—	76	2	no	—
246	C₂×Q₈	—	76	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	—	4	1	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	4	2	no	—
255	C₂²×C₄	—	4	2	no	—
256	C₂²×C₄	—	4	2	no	—
257	C₂²×C₄	—	4	2	no	—
258	C₂²×C₄	—	4	2	no	—
259	C₂²×C₄	—	4	2	no	—
260	C₂²×C₄	—	38	2	no	—
261	C₂²×C₄	—	38	2	no	—
262	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
263	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
264	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
265	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
266	C₂²×C₄	#64525	1	2	yes	—
267	C₂²×C₄	#64526	1	2	yes	—
268	C₂²×C₄	#64527	1	2	yes	—
269	C₂²×C₄	#64528	1	2	yes	—
270	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
271	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
272	C₂²×C₄	#64543	1	2	yes	—
273	C₂²×C₄	Q₈×D₁₉	1	2	yes	—
274	C₂×C₈	—	38	4	no	—
275	C₂×C₈	—	76	2	no	—
276	C₂×C₈	—	76	2	no	—
277	C₄⋊C₄	—	4	8	no	—
278	C₄⋊C₄	—	76	2	no	—
279	C₄⋊C₄	—	76	2	no	—
280	C₄⋊C₄	—	76	2	no	—
281	C₄⋊C₄	—	76	2	no	—
282	C₄⋊C₄	—	76	4	no	—
283	C₄⋊C₄	—	76	2	no	—
284	C₂²⋊C₄	—	76	2	no	—
285	C₂²⋊C₄	—	76	2	no	—
286	C₄²	—	2	2	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	4	1	no	—
289	C₄²	—	4	1	no	—
290	C₂³	—	2	1	no	—
291	C₂³	—	2	1	no	—
292	C₂³	—	2	1	no	—
293	C₂³	—	2	1	no	—
294	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
295	C₂³	C₂×#64525	1	1	yes	characteristic
296	C₂³	C₂×#64526	1	1	yes	characteristic
297	C₂³	#10597382	1	1	yes	characteristic
298	C₂³	C₂×#64527	1	1	yes	characteristic
299	C₂³	C₂×#64528	1	1	yes	characteristic
300	C₂³	#10597390	1	1	yes	characteristic
301	Q₈	—	152	4	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	4	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	4	no	—
319	C₂×C₄	—	4	4	no	—
320	C₂×C₄	—	4	4	no	—
321	C₂×C₄	—	8	2	no	—
322	C₂×C₄	—	8	2	no	—
323	C₂×C₄	—	8	2	no	—
324	C₂×C₄	—	8	2	no	—
325	C₂×C₄	—	76	2	no	—
326	C₂×C₄	—	76	2	no	—
327	C₂×C₄	—	76	2	no	—
328	C₂×C₄	—	76	2	no	—
329	C₂×C₄	—	76	2	no	—
330	C₂×C₄	—	76	2	no	—
331	C₂×C₄	#10597262	1	2	yes	—
332	C₂×C₄	#10597263	1	2	yes	—
333	C₂×C₄	#10597266	1	2	yes	—
334	C₂×C₄	#10597266	1	2	yes	—
335	C₂×C₄	#10597267	1	2	yes	—
336	C₂×C₄	#10597267	1	2	yes	—
337	C₂×C₄	#10597333	1	2	yes	—
338	C₂×C₄	#10597343	1	2	yes	—
339	C₈	—	76	4	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	#11926449	1	1	yes	characteristic
353	C₂²	C₂×#10597263	1	1	yes	characteristic
354	C₂²	#11926517	1	1	yes	characteristic
355	C₂²	C₂×#10597266	1	1	yes	characteristic
356	C₂²	C₂×#10597267	1	1	yes	characteristic
357	C₂²	C₂×#10597262	1	1	yes	characteristic
358	C₂²	#11926562	1	1	yes	characteristic
359	C₂²	#11926890	1	1	yes	characteristic
360	C₂²	#11926894	1	1	yes	characteristic
361	C₂²	#11926940	1	1	yes	characteristic
362	C₂²	#11926947	1	1	yes	characteristic
363	C₄	—	2	2	no	—
364	C₄	—	2	2	no	—
365	C₄	—	2	2	no	—
366	C₄	—	2	2	no	—
367	C₄	—	2	2	no	—
368	C₄	—	8	4	no	—
369	C₄	—	152	2	no	—
370	C₄	—	152	2	no	—
371	C₄	#11926277	1	4	yes	—
372	C₂	—	2	1	no	—
373	C₂	—	2	1	no	—
374	C₂	—	2	1	no	—
375	C₂	—	2	1	no	—
376	C₂	C₂×#11926277	1	1	yes	characteristic
377	C₂	#14562133	1	1	yes	characteristic
378	C₂	#14562194	1	1	yes	characteristic
379	C₂	#14562206	1	1	yes	characteristic
380	C₂	#14564578	1	1	yes	characteristic
381	C₂	#14564595	1	1	yes	characteristic
382	C₂	#14564619	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×C₄⋊#95	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₄⋊#95	D₁₉
hypocenter	C₁₉	#9885
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₄⋊#95

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
49. C₂²⋊(C₁₉⋊C₈)	Q₈	semidirect
242. C₁₉	#9885	semidirect

Maximal subgroups

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

subgroup	index
#14571987	2
C₄×C₂²⋊(C₁₉⋊C₈)	2
#14561514	2
C₁₉×C₄⋊#95	2
#9885	19