Subgroups — #18077772

Order 4864.

500 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14574190	C₂	1	1	yes	maximal, characteristic
2	#14572379	C₂	1	1	yes	maximal, characteristic
3	#14564578	C₂	1	1	yes	maximal, characteristic, complemented
4	#14562039	C₂	1	1	yes	maximal, characteristic, complemented
5	#14561476	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561471	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1502	C₂	1	1	yes	maximal, characteristic
8	C₂×#10597394	—	2	1	no	—
9	#11927101	C₂²	1	1	yes	characteristic
10	#11926940	C₄	1	1	yes	characteristic, complemented
11	#11926916	C₄	1	1	yes	characteristic, complemented
12	#11926896	C₄	1	1	yes	characteristic, complemented
13	#11926890	C₄	1	1	yes	characteristic, complemented
14	#11926771	—	2	2	no	—
15	C₄×#64526	—	2	2	no	—
16	#11926622	—	2	1	no	—
17	#11926616	—	2	2	no	—
18	D₄×C₁₉⋊C₈	—	2	1	no	—
19	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
20	#11926511	—	2	1	no	—
21	C₄×#64520	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
23	#11926304	—	2	1	no	—
24	#11926304	—	2	1	no	—
25	#11926276	—	2	1	no	—
26	#11926276	—	2	1	no	—
27	C₁₉×#532	C₂²	1	1	yes	characteristic
28	C₁₉×#431	C₂²	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
30	C₁₉×#327	—	2	2	no	—
31	C₂×#41×C₁₉	—	2	1	no	—
32	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
33	C₁₉×#127	C₂×C₄	1	1	yes	characteristic
34	C₁₉×#124	—	2	1	no	—
35	C₁₉×#121	—	2	1	no	—
36	C₁₉×#121	D₄	1	2	yes	—
37	C₄×D₄×C₁₉	—	2	1	no	—
38	C₄×D₄×C₁₉	D₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
40	C₂×C₄²×C₁₉	—	2	1	no	—
41	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
42	#10597394	—	4	2	no	—
43	#10597394	—	4	2	no	—
44	C₂×#64526	—	2	2	no	—
45	#10597372	—	2	1	no	—
46	#10597370	—	2	1	no	—
47	#10597370	—	2	1	no	—
48	C₂×#64520	—	2	2	no	—
49	C₂×#64520	D₄	1	1	yes	characteristic
50	C₂×#64520	D₄	1	1	yes	characteristic
51	C₂²×C₁₉⋊C₈	—	2	2	no	—
52	C₂²×C₁₉⋊C₈	—	2	1	no	—
53	C₂²×C₁₉⋊C₈	—	2	1	no	—
54	#10597316	—	2	1	no	—
55	#10597315	C₂×C₄	1	1	yes	characteristic
56	#10597315	C₂×C₄	1	1	yes	characteristic
57	C₄×Dic₃₈	—	2	1	no	—
58	C₄×Dic₃₈	—	2	1	no	—
59	C₁₉×C₄⋊C₈	—	2	1	no	—
60	C₁₉×C₂²⋊C₈	—	2	1	no	—
61	C₄×C₈×C₁₉	—	2	1	no	—
62	C₁₉×#95	—	2	1	no	—
63	#10597289	—	2	2	no	—
64	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
65	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
66	#10597267	—	2	2	no	—
67	#10597265	—	2	2	no	—
68	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
69	C₄⋊(C₁₉⋊C₈)	D₄	1	1	yes	characteristic
70	C₄⋊(C₁₉⋊C₈)	D₄	1	1	yes	characteristic
71	#10597260	—	2	1	no	—
72	C₄×C₁₉⋊C₈	—	2	1	no	—
73	C₄×C₁₉⋊C₈	—	2	1	no	—
74	C₄×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
75	C₄×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
76	#41×C₁₉	—	4	1	no	—
77	#41×C₁₉	—	4	1	no	—
78	#41×C₁₉	—	4	1	no	—
79	#41×C₁₉	—	4	1	no	—
80	C₂×Q₈×C₁₉	—	2	1	no	—
81	C₂×D₄×C₁₉	—	2	2	no	—
82	C₂×D₄×C₁₉	—	2	1	no	—
83	C₂×D₄×C₁₉	—	2	1	no	—
84	C₂×D₄×C₁₉	C₂²⋊C₄	1	2	yes	—
85	C₂²×C₄×C₁₉	—	2	2	no	—
86	C₂²×C₄×C₁₉	—	4	1	no	—
87	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
90	C₂×Dic₃₈	—	2	1	no	—
91	C₂×Dic₃₈	—	2	1	no	—
92	C₂×C₈×C₁₉	—	4	1	no	—
93	C₄⋊C₄×C₁₉	—	2	1	no	—
94	C₄⋊C₄×C₁₉	—	2	2	no	—
95	C₄⋊C₄×C₁₉	—	4	1	no	—
96	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
97	C₂²⋊C₄×C₁₉	—	2	2	no	—
98	C₂²⋊C₄×C₁₉	—	2	2	no	—
99	C₄²×C₁₉	—	4	1	no	—
100	C₄²×C₁₉	—	4	1	no	—
101	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₁₉	#41	1	1	yes	characteristic
105	C₂²⋊Dic₁₉	—	4	1	no	—
106	#64526	—	4	4	no	—
107	C₄⋊Dic₁₉	—	2	1	no	—
108	C₄⋊Dic₁₉	—	2	1	no	—
109	C₄⋊Dic₁₉	—	2	1	no	—
110	C₄⋊Dic₁₉	—	2	1	no	—
111	Dic₁₉⋊C₄	—	4	1	no	—
112	Dic₁₉⋊C₄	—	4	1	no	—
113	C₄×Dic₁₉	—	4	1	no	—
114	#64520	—	2	1	no	—
115	#64520	—	2	1	no	—
116	#64520	—	2	1	no	—
117	#64520	—	4	4	no	—
118	#64520	SD₁₆	1	2	yes	complemented
119	C₂×C₁₉⋊C₈	—	2	1	no	—
120	C₂×C₁₉⋊C₈	—	2	1	no	—
121	C₂×C₁₉⋊C₈	—	2	1	no	—
122	C₂×C₁₉⋊C₈	—	2	1	no	—
123	C₂×C₁₉⋊C₈	—	2	1	no	—
124	C₂×C₁₉⋊C₈	—	2	1	no	—
125	C₂×C₁₉⋊C₈	—	4	4	no	—
126	C₂×C₁₉⋊C₈	—	4	2	no	—
127	C₂×C₁₉⋊C₈	—	4	1	no	—
128	C₂×C₁₉⋊C₈	—	4	1	no	—
129	#9055	—	19	1	no	maximal, Sylow
130	C₂³×C₁₉	—	2	2	no	—
131	C₂³×C₁₉	#121	1	1	yes	characteristic
132	Q₈×C₁₉	—	4	1	no	—
133	Q₈×C₁₉	—	4	1	no	—
134	D₄×C₁₉	—	4	2	no	—
135	D₄×C₁₉	—	4	2	no	—
136	D₄×C₁₉	—	4	2	no	—
137	D₄×C₁₉	—	4	1	no	—
138	D₄×C₁₉	—	4	1	no	—
139	D₄×C₁₉	—	4	1	no	—
140	D₄×C₁₉	—	4	1	no	—
141	C₂×C₄×C₁₉	—	2	1	no	—
142	C₂×C₄×C₁₉	—	2	1	no	—
143	C₂×C₄×C₁₉	—	2	2	no	—
144	C₂×C₄×C₁₉	—	4	1	no	—
145	C₂×C₄×C₁₉	—	4	2	no	—
146	C₂×C₄×C₁₉	—	4	2	no	—
147	C₂×C₄×C₁₉	—	4	2	no	—
148	C₂×C₄×C₁₉	—	8	1	no	—
149	C₂×C₄×C₁₉	—	8	1	no	—
150	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
154	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
155	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
156	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
157	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
158	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
159	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
160	C₂×Dic₁₉	—	4	1	no	—
161	C₂×Dic₁₉	—	4	1	no	—
162	Dic₃₈	—	4	2	no	—
163	Dic₃₈	—	4	1	no	—
164	Dic₃₈	—	4	1	no	—
165	C₈×C₁₉	—	8	1	no	—
166	C₁₉⋊C₈	—	2	2	no	—
167	C₁₉⋊C₈	—	2	2	no	—
168	C₁₉⋊C₈	—	4	1	no	—
169	C₁₉⋊C₈	—	4	1	no	—
170	C₁₉⋊C₈	—	4	1	no	—
171	C₁₉⋊C₈	—	4	1	no	—
172	#3167	—	19	1	no	—
173	#2935	—	19	1	no	—
174	#1986	—	19	1	no	—
175	#1502	D₁₉	1	1	yes	characteristic
176	#1481	—	19	1	no	—
177	#1279	—	19	1	no	—
178	#1278	—	19	1	no	—
179	C₂²×C₁₉	—	2	1	no	—
180	C₂²×C₁₉	—	2	1	no	—
181	C₂²×C₁₉	—	4	2	no	—
182	C₂²×C₁₉	—	4	2	no	—
183	C₂²×C₁₉	—	4	2	no	—
184	C₂²×C₁₉	#386	1	1	yes	characteristic
185	C₂²×C₁₉	#461	1	1	yes	characteristic
186	C₂²×C₁₉	#463	1	1	yes	characteristic
187	C₄×C₁₉	—	2	1	no	—
188	C₄×C₁₉	—	2	1	no	—
189	C₄×C₁₉	—	2	1	no	—
190	C₄×C₁₉	—	4	2	no	—
191	C₄×C₁₉	—	8	2	no	—
192	C₄×C₁₉	C₄×SD₁₆	1	1	yes	characteristic
193	C₄×C₁₉	#440	1	1	yes	characteristic
194	C₄×C₁₉	#443	1	1	yes	characteristic
195	C₄×C₁₉	#444	1	1	yes	characteristic
196	C₄×C₁₉	#450	1	1	yes	characteristic
197	C₄×C₁₉	#452	1	1	yes	characteristic
198	Dic₁₉	—	8	1	no	—
199	Dic₁₉	—	8	1	no	—
200	C₂×#131	—	38	1	no	—
201	#533	—	19	1	no	—
202	#532	C₂×D₁₉	1	1	yes	characteristic
203	#498	—	19	1	no	—
204	#492	—	19	1	no	—
205	#487	—	19	1	no	—
206	#486	—	19	1	no	—
207	#465	—	38	2	no	—
208	C₄×SD₁₆	—	38	2	no	—
209	#436	—	38	1	no	—
210	#435	—	38	2	no	—
211	C₈×D₄	—	38	1	no	—
212	#431	C₂×D₁₉	1	1	yes	characteristic
213	C₂×C₄⋊C₈	—	19	1	no	—
214	#408	—	38	1	no	—
215	C₄×M₁₆	—	19	1	no	—
216	C₂×C₄×C₈	—	19	1	no	—
217	C₄×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
218	#339	—	38	1	no	—
219	#339	—	38	1	no	—
220	#327	—	2	2	no	—
221	Q₈⋊C₈	—	38	1	no	—
222	Q₈⋊C₈	—	38	1	no	—
223	C₂×C₁₉	—	2	1	no	—
224	C₂×C₁₉	—	2	1	no	—
225	C₂×C₁₉	—	8	2	no	—
226	C₂×C₁₉	#1876	1	1	yes	characteristic
227	C₂×C₁₉	#1877	1	1	yes	characteristic
228	C₂×C₁₉	#1884	1	1	yes	characteristic
229	C₂×#41	—	2	1	no	—
230	C₂×SD₁₆	—	38	2	no	—
231	#131	—	76	2	no	—
232	#131	—	76	2	no	—
233	C₂×M₁₆	—	19	1	no	—
234	C₂×M₁₆	—	19	1	no	—
235	C₂×M₁₆	—	38	2	no	—
236	C₂²×C₈	—	38	2	no	—
237	C₂²×C₈	—	38	1	no	—
238	C₂²×C₈	—	38	1	no	—
239	C₄⋊Q₈	—	19	1	no	—
240	C₄⋊Q₈	—	19	1	no	—
241	C₄⋊Q₈	C₄×D₁₉	1	1	yes	characteristic
242	#127	C₄×D₁₉	1	1	yes	characteristic
243	#125	—	38	1	no	—
244	#124	—	2	1	no	—
245	C₂²⋊Q₈	—	38	1	no	—
246	C₂²⋊Q₈	—	38	1	no	—
247	#121	—	2	1	no	—
248	#121	C₂²⋊D₁₉	1	2	yes	—
249	C₄×Q₈	—	38	1	no	—
250	C₄×Q₈	—	38	1	no	—
251	C₄×D₄	—	2	1	no	—
252	C₄×D₄	D₇₆	1	2	yes	—
253	#117	—	38	1	no	—
254	C₂×C₄⋊C₄	—	2	1	no	—
255	C₂×C₄²	—	2	1	no	—
256	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
257	#106	—	38	2	no	—
258	C₄⋊C₈	—	2	1	no	—
259	C₄⋊C₈	—	19	1	no	—
260	C₄⋊C₈	—	19	1	no	—
261	C₄⋊C₈	—	38	1	no	—
262	#103	—	38	2	no	—
263	#102	—	38	2	no	—
264	C₂²⋊C₈	—	2	1	no	—
265	C₂²⋊C₈	—	38	2	no	—
266	C₂²⋊C₈	—	38	2	no	—
267	#97	—	38	1	no	—
268	C₄×C₈	—	2	1	no	—
269	C₄×C₈	—	19	1	no	—
270	C₄×C₈	—	19	1	no	—
271	C₄×C₈	—	38	1	no	—
272	C₄×C₈	—	38	1	no	—
273	#95	—	2	1	no	—
274	C₁₉	#9055	1	1	yes	characteristic, complemented, Sylow
275	#41	—	4	1	no	—
276	#41	—	4	1	no	—
277	#41	—	4	1	no	—
278	#41	—	4	1	no	—
279	C₂×Q₈	—	2	1	no	—
280	C₂×Q₈	—	38	1	no	—
281	C₂×Q₈	—	38	1	no	—
282	C₂×D₄	—	2	2	no	—
283	C₂×D₄	—	2	1	no	—
284	C₂×D₄	—	2	1	no	—
285	C₂×D₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	4	1	no	—
288	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
289	C₂²×C₄	#64543	1	1	yes	characteristic
290	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
291	SD₁₆	—	76	4	no	—
292	M₁₆	—	19	2	no	—
293	M₁₆	—	38	1	no	—
294	M₁₆	—	38	1	no	—
295	M₁₆	—	38	1	no	—
296	M₁₆	—	76	4	no	—
297	C₂×C₈	—	4	1	no	—
298	C₂×C₈	—	38	1	no	—
299	C₂×C₈	—	38	1	no	—
300	C₂×C₈	—	38	1	no	—
301	C₂×C₈	—	38	1	no	—
302	C₂×C₈	—	38	1	no	—
303	C₂×C₈	—	38	1	no	—
304	C₂×C₈	—	76	2	no	—
305	C₂×C₈	—	76	4	no	—
306	C₂×C₈	—	76	1	no	—
307	C₂×C₈	—	76	1	no	—
308	C₄⋊C₄	—	2	1	no	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	4	1	no	—
311	C₄⋊C₄	—	38	1	no	—
312	C₄⋊C₄	—	38	1	no	—
313	C₄⋊C₄	—	38	1	no	—
314	C₄⋊C₄	—	38	1	no	—
315	C₄⋊C₄	—	76	1	no	—
316	C₄⋊C₄	—	76	1	no	—
317	C₄⋊C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
318	C₂²⋊C₄	—	2	2	no	—
319	C₂²⋊C₄	—	2	2	no	—
320	C₂²⋊C₄	—	76	1	no	—
321	C₄²	—	4	1	no	—
322	C₄²	—	4	1	no	—
323	C₄²	—	76	1	no	—
324	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
325	C₄²	C₂×D₇₆	1	1	yes	characteristic
326	C₄²	D₄×D₁₉	1	1	yes	characteristic
327	C₄²	#64543	1	1	yes	characteristic
328	C₂³	—	2	2	no	—
329	C₂³	#10597385	1	1	yes	characteristic
330	Q₈	—	4	1	no	—
331	Q₈	—	4	1	no	—
332	Q₈	—	76	2	no	—
333	Q₈	—	76	1	no	—
334	Q₈	—	76	1	no	—
335	D₄	—	4	2	no	—
336	D₄	—	4	2	no	—
337	D₄	—	4	2	no	—
338	D₄	—	4	1	no	—
339	D₄	—	4	1	no	—
340	D₄	—	4	1	no	—
341	D₄	—	4	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	4	1	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	8	1	no	—
350	C₂×C₄	—	8	1	no	—
351	C₂×C₄	—	76	1	no	—
352	C₂×C₄	—	76	1	no	—
353	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
354	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
355	C₂×C₄	#10597328	1	1	yes	characteristic
356	C₂×C₄	#10597332	1	1	yes	characteristic
357	C₂×C₄	#10597357	1	1	yes	characteristic
358	C₂×C₄	#10597358	1	1	yes	characteristic
359	C₂×C₄	SD₁₆×D₁₉	1	1	yes	characteristic
360	C₂×C₄	#10597361	1	1	yes	characteristic
361	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
362	C₂×C₄	#10597385	1	1	yes	characteristic
363	C₈	—	8	1	no	—
364	C₈	—	38	2	no	—
365	C₈	—	38	2	no	—
366	C₈	—	76	1	no	—
367	C₈	—	76	1	no	—
368	C₈	—	76	1	no	—
369	C₈	—	76	1	no	—
370	C₂²	—	2	1	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	4	2	no	—
373	C₂²	—	4	2	no	—
374	C₂²	—	4	2	no	—
375	C₂²	#11926421	1	1	yes	characteristic
376	C₂²	#11926747	1	1	yes	characteristic
377	C₂²	#11926757	1	1	yes	characteristic
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	4	2	no	—
382	C₄	—	8	2	no	—
383	C₄	—	152	1	no	—
384	C₄	—	152	1	no	—
385	C₄	#11926635	1	1	yes	characteristic
386	C₄	#11926641	1	1	yes	characteristic
387	C₄	#11926658	1	1	yes	characteristic
388	C₄	#11926663	1	1	yes	characteristic
389	C₄	#11926696	1	1	yes	characteristic
390	C₄	#11926707	1	1	yes	characteristic
391	C₂	—	2	1	no	—
392	C₂	—	2	1	no	—
393	C₂	—	8	2	no	—
394	C₂	#14563960	1	1	yes	characteristic
395	C₂	#14563967	1	1	yes	characteristic
396	C₂	#14564016	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597328
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1502	C₂
socle	C₂²×C₁₉	#386
prime radical	C₁₉×#532	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1502	D₁₉
hypocenter	C₁₉	#9055
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1502

Complements

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

N	H ≅ G/N	type
3. #14564578	C₂	semidirect
4. #14562039	C₂	semidirect
5. #14561476	C₂	semidirect
6. #14561471	C₂	semidirect
10. #11926940	C₄	semidirect
11. #11926916	C₄	semidirect
12. #11926896	C₄	semidirect
13. #11926890	C₄	semidirect
118. #64520	SD₁₆	semidirect
274. C₁₉	#9055	semidirect

Maximal subgroups

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

subgroup	index
#14574190	2
#14572379	2
#14564578	2
#14562039	2
#14561476	2
#14561471	2
C₁₉×#1502	2
#9055	19