Subgroups — #18087243

Order 4864.

418 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14564428	C₂	1	1	yes	maximal, characteristic
2	#14563794	C₂	1	1	yes	maximal, characteristic, complemented
3	#14563636	C₂	1	1	yes	maximal, characteristic
4	#14563632	C₂	1	1	yes	maximal, characteristic
5	#14561519	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561469	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1296	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₇₆	C₂²	1	1	yes	characteristic
9	C₂×#10597264	—	2	1	no	—
10	C₂×#10597264	—	2	1	no	—
11	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×#10597278	—	2	1	no	—
13	C₂×#10597266	—	2	1	no	—
14	C₂×(C₂×D₁₉)⋊C₈	—	2	1	no	—
15	C₂×(C₂×D₁₉)⋊C₈	—	2	1	no	—
16	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
17	#11926414	C₂²	1	1	yes	characteristic
18	#11926307	—	2	1	no	—
19	#11926306	—	2	1	no	—
20	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₁₉×#384	C₂²	1	1	yes	characteristic
23	C₂²×D₇₆	D₄	1	1	yes	characteristic
24	C₂²×C₄×D₁₉	—	2	1	no	—
25	C₂²×C₈×C₁₉	—	2	1	no	—
26	C₂²×C₈×C₁₉	—	2	1	no	—
27	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
28	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
30	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
31	C₂×(C₂×D₁₉)⋊C₄	—	2	1	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	D₄	1	1	yes	characteristic
36	C₂²×C₁₉⋊C₈	—	2	1	no	—
37	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
38	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
39	C₄×D₇₆	—	2	1	no	—
40	C₄×D₇₆	—	2	1	no	—
41	C₄×D₇₆	—	2	1	no	—
42	C₄×D₇₆	—	2	1	no	—
43	C₁₉×C₄⋊C₈	—	2	1	no	—
44	C₁₉×C₄⋊C₈	—	2	1	no	—
45	C₁₉×C₄⋊C₈	—	2	1	no	—
46	C₁₉×C₄⋊C₈	—	2	1	no	—
47	#10597278	—	4	2	no	—
48	(C₂×D₁₉)⋊C₈	—	4	2	no	—
49	(C₂×D₁₉)⋊C₈	—	4	2	no	—
50	#10597266	—	4	2	no	—
51	#10597264	—	4	2	no	—
52	#10597264	—	4	2	no	—
53	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
54	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
55	C₄×C₁₉⋊C₈	—	2	1	no	—
56	C₄×C₁₉⋊C₈	—	2	1	no	—
57	C₂³×D₁₉	—	2	1	no	—
58	C₂²×C₄×C₁₉	—	4	1	no	—
59	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
62	C₂²×Dic₁₉	—	2	1	no	—
63	C₂²×Dic₁₉	—	4	1	no	—
64	C₂×D₇₆	—	2	1	no	—
65	C₂×D₇₆	—	2	1	no	—
66	C₂×D₇₆	—	2	1	no	—
67	C₂×D₇₆	—	2	1	no	—
68	C₂×D₇₆	—	2	1	no	—
69	C₂×D₇₆	D₈	1	2	yes	—
70	C₂×C₄×D₁₉	—	4	1	no	—
71	C₂×C₄×D₁₉	—	4	1	no	—
72	C₂×C₄×D₁₉	—	4	1	no	—
73	C₂×C₄×D₁₉	—	4	1	no	—
74	C₂×C₄×D₁₉	—	4	1	no	—
75	C₂×C₄×D₁₉	—	4	1	no	—
76	C₂×C₈×C₁₉	—	4	1	no	—
77	C₂×C₈×C₁₉	—	4	1	no	—
78	C₂×C₈×C₁₉	—	4	1	no	—
79	C₂×C₈×C₁₉	—	4	1	no	—
80	C₂×C₈×C₁₉	—	4	1	no	—
81	C₂×C₈×C₁₉	—	4	1	no	—
82	C₄⋊C₄×C₁₉	—	4	1	no	—
83	C₄⋊C₄×C₁₉	—	4	1	no	—
84	C₄⋊C₄×C₁₉	—	4	1	no	—
85	C₄⋊C₄×C₁₉	—	4	1	no	—
86	C₄⋊C₄×C₁₉	—	8	1	no	—
87	C₄²×C₁₉	#41	1	1	yes	characteristic
88	C₄²×C₁₉	#41	1	1	yes	characteristic
89	C₄²×C₁₉	#41	1	1	yes	characteristic
90	C₄²×C₁₉	#41	1	1	yes	characteristic
91	(C₂×D₁₉)⋊C₄	—	4	1	no	—
92	(C₂×D₁₉)⋊C₄	—	4	1	no	—
93	C₄⋊Dic₁₉	—	2	1	no	—
94	C₄⋊Dic₁₉	—	2	1	no	—
95	C₄⋊Dic₁₉	—	4	1	no	—
96	C₄⋊Dic₁₉	—	4	1	no	—
97	C₄⋊Dic₁₉	—	4	1	no	—
98	C₄⋊Dic₁₉	—	4	1	no	—
99	C₄⋊Dic₁₉	—	8	1	no	—
100	C₂×C₁₉⋊C₈	—	2	1	no	—
101	C₂×C₁₉⋊C₈	—	2	1	no	—
102	C₂×C₁₉⋊C₈	—	2	1	no	—
103	C₂×C₁₉⋊C₈	—	2	1	no	—
104	C₂×C₁₉⋊C₈	—	4	1	no	—
105	C₂×C₁₉⋊C₈	—	4	1	no	—
106	C₂×C₁₉⋊C₈	—	4	1	no	—
107	C₂×C₁₉⋊C₈	D₈	1	2	yes	—
108	C₂×C₁₉⋊C₈	SD₁₆	1	2	yes	—
109	#10615	—	19	1	no	maximal, Sylow
110	C₂³×C₁₉	#121	1	1	yes	characteristic
111	C₂²×D₁₉	—	2	2	no	—
112	C₂²×D₁₉	—	4	1	no	—
113	C₂²×D₁₉	—	4	1	no	—
114	C₂²×D₁₉	—	4	1	no	—
115	C₂²×D₁₉	—	4	1	no	—
116	C₂²×D₁₉	—	4	1	no	—
117	C₂²×D₁₉	—	4	1	no	—
118	C₂×C₄×C₁₉	—	2	1	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	8	1	no	—
121	C₂×C₄×C₁₉	—	8	1	no	—
122	C₂×C₄×C₁₉	—	8	1	no	—
123	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
124	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
125	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
126	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
127	C₂×C₄×C₁₉	#126	1	1	yes	characteristic
128	C₂×C₄×C₁₉	#126	1	1	yes	characteristic
129	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
133	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
134	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
135	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
136	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
137	C₂×Dic₁₉	—	4	1	no	—
138	C₂×Dic₁₉	—	4	1	no	—
139	C₂×Dic₁₉	—	4	1	no	—
140	C₂×Dic₁₉	—	8	1	no	—
141	C₂×Dic₁₉	—	8	1	no	—
142	C₂×Dic₁₉	—	8	1	no	—
143	D₇₆	—	2	2	no	—
144	D₇₆	—	2	2	no	—
145	D₇₆	—	4	1	no	—
146	D₇₆	—	4	1	no	—
147	C₄×D₁₉	—	8	1	no	—
148	C₄×D₁₉	—	8	1	no	—
149	C₄×D₁₉	—	8	1	no	—
150	C₄×D₁₉	—	8	1	no	—
151	C₈×C₁₉	—	8	1	no	—
152	C₈×C₁₉	—	8	1	no	—
153	C₁₉⋊C₈	—	2	2	no	—
154	C₁₉⋊C₈	—	2	2	no	—
155	C₁₉⋊C₈	—	8	1	no	—
156	#1954	—	19	1	no	—
157	#1845	—	19	1	no	—
158	#1802	—	19	1	no	—
159	#1802	—	19	1	no	—
160	#1296	—	19	1	no	—
161	#1296	D₁₉	1	1	yes	characteristic, complemented
162	#1277	—	19	1	no	—
163	C₂²×C₁₉	#397	1	1	yes	characteristic
164	C₂²×C₁₉	#459	1	1	yes	characteristic
165	C₂²×C₁₉	#461	1	1	yes	characteristic
166	C₂²×C₁₉	#466	1	1	yes	characteristic
167	C₂²×C₁₉	#470	1	1	yes	characteristic
168	C₂²×C₁₉	#472	1	1	yes	characteristic
169	C₂²×C₁₉	#473	1	1	yes	characteristic
170	C₂×D₁₉	—	4	2	no	—
171	C₂×D₁₉	—	4	2	no	—
172	C₂×D₁₉	—	4	2	no	—
173	C₂×D₁₉	—	8	1	no	—
174	C₂×D₁₉	—	8	1	no	—
175	C₂×D₁₉	—	8	1	no	—
176	C₂×D₁₉	—	8	1	no	—
177	C₄×C₁₉	—	2	1	no	—
178	C₄×C₁₉	—	2	1	no	—
179	C₄×C₁₉	—	16	1	no	—
180	C₄×C₁₉	#480	1	1	yes	characteristic
181	C₄×C₁₉	#480	1	1	yes	characteristic
182	C₄×C₁₉	#480	1	1	yes	characteristic
183	C₄×C₁₉	#481	1	1	yes	characteristic
184	C₄×C₁₉	#485	1	1	yes	characteristic
185	C₄×C₁₉	#485	1	1	yes	characteristic
186	C₄×C₁₉	#486	1	1	yes	characteristic
187	C₄×C₁₉	#489	1	1	yes	characteristic
188	Dic₁₉	—	8	1	no	—
189	Dic₁₉	—	16	1	no	—
190	C₂×C₄×D₄	—	19	1	no	—
191	C₂×#107	—	38	1	no	—
192	C₂×#107	—	38	1	no	—
193	C₂×C₄⋊C₈	—	19	1	no	—
194	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
195	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
196	C₂×#102	—	38	1	no	—
197	C₂×#102	—	38	1	no	—
198	C₂×C₂²⋊C₈	—	38	1	no	—
199	C₂×C₂²⋊C₈	—	38	1	no	—
200	C₂×C₄×C₈	—	19	1	no	—
201	#384	—	19	1	no	—
202	#384	C₂×D₁₉	1	1	yes	characteristic
203	#340	—	38	1	no	—
204	#340	—	38	1	no	—
205	C₂×C₁₉	#1667	1	1	yes	characteristic
206	C₂×C₁₉	#1673	1	1	yes	characteristic
207	C₂×C₁₉	#1689	1	1	yes	characteristic
208	C₂×C₁₉	#1695	1	1	yes	characteristic
209	C₂×C₁₉	#2071	1	1	yes	characteristic
210	C₂×C₁₉	#2075	1	1	yes	characteristic
211	C₂×C₁₉	#2078	1	1	yes	characteristic
212	D₁₉	—	8	2	no	—
213	C₂²×D₄	—	19	1	no	—
214	C₂³×C₄	—	38	1	no	—
215	C₂²×C₈	—	2	1	no	—
216	C₂²×C₈	—	2	1	no	—
217	C₂²×C₈	—	19	1	no	—
218	C₂²×C₈	—	19	1	no	—
219	C₂²×C₈	—	38	1	no	—
220	C₄×D₄	—	38	1	no	—
221	C₄×D₄	—	38	1	no	—
222	C₄×D₄	—	38	1	no	—
223	C₄×D₄	—	38	1	no	—
224	C₂×C₄⋊C₄	—	2	1	no	—
225	C₂×C₄⋊C₄	—	2	1	no	—
226	C₂×C₄⋊C₄	—	2	1	no	—
227	C₂×C₄⋊C₄	—	19	1	no	—
228	C₂×C₄⋊C₄	—	38	1	no	—
229	C₂×C₄⋊C₄	—	38	1	no	—
230	C₂×C₄⋊C₄	—	38	1	no	—
231	C₂×C₂²⋊C₄	—	38	1	no	—
232	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
233	#107	—	76	2	no	—
234	#107	—	76	2	no	—
235	C₄⋊C₈	—	2	1	no	—
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	—	2	1	no	—
238	C₄⋊C₈	—	2	1	no	—
239	C₄⋊C₈	—	38	1	no	—
240	C₄⋊C₈	—	38	1	no	—
241	#102	—	76	2	no	—
242	#102	—	76	2	no	—
243	C₂²⋊C₈	—	76	2	no	—
244	C₂²⋊C₈	—	76	2	no	—
245	C₄×C₈	—	38	1	no	—
246	C₄×C₈	—	38	1	no	—
247	C₁₉	#10615	1	1	yes	characteristic, complemented, Sylow
248	C₂⁴	—	38	1	no	—
249	C₂×D₄	—	19	2	no	—
250	C₂×D₄	—	38	1	no	—
251	C₂×D₄	—	38	1	no	—
252	C₂×D₄	—	38	1	no	—
253	C₂×D₄	—	38	1	no	—
254	C₂×D₄	—	38	1	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	—	38	1	no	—
257	C₂²×C₄	—	76	1	no	—
258	C₂²×C₄	—	76	1	no	—
259	C₂²×C₄	—	76	1	no	—
260	C₂²×C₄	—	76	1	no	—
261	C₂²×C₄	—	76	1	no	—
262	C₂²×C₄	—	76	1	no	—
263	C₂²×C₄	—	76	1	no	—
264	C₂²×C₄	#64541	1	1	yes	characteristic
265	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
266	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	4	1	no	—
271	C₂×C₈	—	4	1	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	—	19	2	no	—
274	C₂×C₈	—	19	2	no	—
275	C₂×C₈	—	38	1	no	—
276	C₂×C₈	—	38	1	no	—
277	C₂×C₈	—	38	1	no	—
278	C₂×C₈	—	38	1	no	—
279	C₂×C₈	—	76	1	no	—
280	C₂×C₈	—	76	1	no	—
281	C₂×C₈	—	76	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	—	8	1	no	—
287	C₄⋊C₄	—	38	1	no	—
288	C₄⋊C₄	—	38	1	no	—
289	C₄⋊C₄	—	76	1	no	—
290	C₄⋊C₄	—	76	1	no	—
291	C₄⋊C₄	—	76	1	no	—
292	C₄⋊C₄	—	76	1	no	—
293	C₄⋊C₄	—	152	1	no	—
294	C₂²⋊C₄	—	76	1	no	—
295	C₂²⋊C₄	—	76	1	no	—
296	C₄²	#64541	1	1	yes	characteristic
297	C₄²	#64541	1	1	yes	characteristic
298	C₄²	#64543	1	1	yes	characteristic
299	C₄²	#64545	1	1	yes	characteristic
300	C₂³	—	38	2	no	—
301	C₂³	—	76	1	no	—
302	C₂³	—	76	1	no	—
303	C₂³	—	76	1	no	—
304	C₂³	—	76	1	no	—
305	C₂³	—	76	1	no	—
306	C₂³	—	76	1	no	—
307	C₂³	#10597330	1	1	yes	characteristic
308	D₄	—	38	2	no	—
309	D₄	—	38	2	no	—
310	D₄	—	76	1	no	—
311	D₄	—	76	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	8	1	no	—
315	C₂×C₄	—	8	1	no	—
316	C₂×C₄	—	8	1	no	—
317	C₂×C₄	—	76	1	no	—
318	C₂×C₄	—	76	1	no	—
319	C₂×C₄	—	76	1	no	—
320	C₂×C₄	—	152	1	no	—
321	C₂×C₄	—	152	1	no	—
322	C₂×C₄	—	152	1	no	—
323	C₂×C₄	—	152	1	no	—
324	C₂×C₄	—	152	1	no	—
325	C₂×C₄	—	152	1	no	—
326	C₂×C₄	—	152	1	no	—
327	C₂×C₄	#10597322	1	1	yes	characteristic
328	C₂×C₄	#10597329	1	1	yes	characteristic
329	C₂×C₄	#10597331	1	1	yes	characteristic
330	C₂×C₄	#10597340	1	1	yes	characteristic
331	C₂×C₄	#10597340	1	1	yes	characteristic
332	C₂×C₄	#10597344	1	1	yes	characteristic
333	C₂×C₄	D₈×D₁₉	1	1	yes	characteristic
334	C₂×C₄	D₈×D₁₉	1	1	yes	characteristic
335	C₂×C₄	SD₁₆×D₁₉	1	1	yes	characteristic
336	C₂×C₄	#10597360	1	1	yes	characteristic
337	C₂×C₄	#10597361	1	1	yes	characteristic
338	C₂×C₄	#10597362	1	1	yes	characteristic
339	C₂×C₄	#10597364	1	1	yes	characteristic
340	C₂×C₄	#10597364	1	1	yes	characteristic
341	C₈	—	8	1	no	—
342	C₈	—	8	1	no	—
343	C₈	—	38	2	no	—
344	C₈	—	38	2	no	—
345	C₈	—	152	1	no	—
346	C₂²	—	76	2	no	—
347	C₂²	—	76	2	no	—
348	C₂²	—	76	2	no	—
349	C₂²	—	152	1	no	—
350	C₂²	—	152	1	no	—
351	C₂²	—	152	1	no	—
352	C₂²	—	152	1	no	—
353	C₂²	#11926473	1	1	yes	characteristic
354	C₂²	#11926738	1	1	yes	characteristic
355	C₂²	#11926748	1	1	yes	characteristic
356	C₂²	#11926779	1	1	yes	characteristic
357	C₂²	#11926801	1	1	yes	characteristic
358	C₂²	#11926813	1	1	yes	characteristic
359	C₂²	#11926818	1	1	yes	characteristic
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	16	1	no	—
363	C₄	—	304	1	no	—
364	C₄	—	152	1	no	—
365	C₄	#11926856	1	1	yes	characteristic
366	C₄	#11926859	1	1	yes	characteristic
367	C₄	#11926859	1	1	yes	characteristic
368	C₄	#11926864	1	1	yes	characteristic
369	C₄	#11926883	1	1	yes	characteristic
370	C₄	#11926888	1	1	yes	characteristic
371	C₄	#11926889	1	1	yes	characteristic
372	C₄	#11926905	1	1	yes	characteristic
373	C₂	—	152	2	no	—
374	C₂	#14562989	1	1	yes	characteristic
375	C₂	#14563023	1	1	yes	characteristic
376	C₂	#14563123	1	1	yes	characteristic
377	C₂	#14563157	1	1	yes	characteristic
378	C₂	#14565042	1	1	yes	characteristic
379	C₂	#14565062	1	1	yes	characteristic
380	C₂	#14565079	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597330
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1296	C₂
socle	C₂³×C₁₉	#121
prime radical	C₂²×D₇₆	D₄
hypercenter	#1296	D₁₉
hypocenter	C₁₉	#10615
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

Complements

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

N	H ≅ G/N	type
2. #14563794	C₂	semidirect
5. #14561519	C₂	semidirect
6. #14561469	C₂	semidirect
7. C₁₉×#1296	C₂	semidirect
161. #1296	D₁₉	semidirect
247. C₁₉	#10615	semidirect

Maximal subgroups

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

subgroup	index
#14564428	2
#14563794	2
#14563636	2
#14563632	2
#14561519	2
#14561469	2
C₁₉×#1296	2
#10615	19