Subgroups — #18078407

Order 4864.

478 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	#14573444	C₂	1	1	yes	maximal, characteristic
2	#14572200	C₂	1	1	yes	maximal, characteristic
3	#14562320	C₂	1	1	yes	maximal, characteristic, complemented
4	#14562020	C₂	1	1	yes	maximal, characteristic
5	#14561532	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561510	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₉×#333	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10597347	—	2	1	no	—
9	C₄×#64541	C₂²	1	1	yes	characteristic
10	#11926836	—	2	1	no	—
11	#11926831	—	2	1	no	—
12	#11926826	C₄	1	2	yes	—
13	#11926821	C₄	1	2	yes	—
14	#11926597	C₂²	1	1	yes	characteristic
15	#11926584	—	2	1	no	—
16	#11926570	—	2	1	no	—
17	#11926569	—	2	2	no	—
18	#11926568	C₂²	1	1	yes	characteristic
19	#11926568	C₂²	1	1	yes	characteristic
20	#11926566	—	2	1	no	—
21	C₄⋊C₈×D₁₉	—	2	1	no	—
22	#11926543	—	2	1	no	—
23	#11926290	—	2	2	no	—
24	#11926274	—	2	1	no	—
25	#11926271	—	2	1	no	—
26	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
27	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
29	C₁₉×#333	—	2	1	no	—
30	C₁₉×#333	—	2	1	no	—
31	C₂²×Q₈×C₁₉	—	2	1	no	—
32	C₂×#64541	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₉	—	2	1	no	—
34	C₂²×C₈×C₁₉	—	2	1	no	—
35	C₁₉×C₄⋊Q₈	—	2	1	no	—
36	C₁₉×C₄⋊Q₈	—	2	1	no	—
37	C₁₉×C₄⋊Q₈	—	2	1	no	—
38	C₁₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
40	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
41	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
42	C₄×C₂²⋊D₁₉	—	2	1	no	—
43	C₄×C₂²⋊D₁₉	—	2	1	no	—
44	#10597372	D₄	1	1	yes	characteristic
45	C₂×#64520	—	2	1	no	—
46	C₂×#64520	—	2	1	no	—
47	#10597347	—	4	2	no	—
48	#10597347	—	4	2	no	—
49	C₂×#64515	—	2	2	no	—
50	C₂×C₈×D₁₉	—	2	2	no	—
51	C₄×D₇₆	C₂×C₄	1	1	yes	characteristic
52	C₄×D₇₆	D₄	1	1	yes	characteristic
53	#10597318	—	2	1	no	—
54	C₄²×D₁₉	—	2	1	no	—
55	C₄×Dic₃₈	C₂×C₄	1	1	yes	characteristic
56	C₄×Dic₃₈	D₄	1	1	yes	characteristic
57	C₁₉×C₄⋊C₈	—	2	1	no	—
58	C₁₉×C₄⋊C₈	—	2	1	no	—
59	C₁₉×C₄⋊C₈	—	2	1	no	—
60	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
61	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
62	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
63	Dic₁₉⋊C₈	—	2	2	no	—
64	Dic₁₉⋊C₈	—	2	2	no	—
65	#10597267	—	2	2	no	—
66	#10597266	—	2	2	no	—
67	#10597265	—	2	2	no	—
68	#10597264	—	2	2	no	—
69	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
70	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
71	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
72	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
73	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
74	C₄×C₁₉⋊C₈	—	2	1	no	—
75	C₂×Q₈×C₁₉	—	2	2	no	—
76	C₂×Q₈×C₁₉	—	4	1	no	—
77	C₂×Q₈×C₁₉	—	8	2	no	—
78	C₂²×C₄×C₁₉	—	4	1	no	—
79	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
81	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
82	C₂×C₂²⋊D₁₉	—	2	1	no	—
83	#64541	—	2	1	no	—
84	#64541	—	2	1	no	—
85	#64541	SD₁₆	1	2	yes	—
86	#64541	Q₁₆	1	2	yes	—
87	C₂×D₇₆	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×D₁₉	—	2	1	no	—
89	C₂×C₄×D₁₉	—	2	1	no	—
90	C₂×C₄×D₁₉	—	2	1	no	—
91	C₂×Dic₃₈	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₉	—	2	1	no	—
93	C₂×C₈×C₁₉	—	2	1	no	—
94	C₂×C₈×C₁₉	—	4	1	no	—
95	C₂×C₈×C₁₉	—	4	1	no	—
96	C₂×C₈×C₁₉	—	4	1	no	—
97	C₂×C₈×C₁₉	—	8	1	no	—
98	C₄⋊C₄×C₁₉	—	2	2	no	—
99	C₄⋊C₄×C₁₉	—	2	2	no	—
100	C₄⋊C₄×C₁₉	—	4	1	no	—
101	C₄⋊C₄×C₁₉	—	4	1	no	—
102	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
106	C₂²⋊Dic₁₉	—	2	1	no	—
107	(C₂×D₁₉)⋊C₄	—	2	1	no	—
108	(C₂×D₁₉)⋊C₄	—	2	1	no	—
109	C₄⋊Dic₁₉	C₂²⋊C₄	1	1	yes	characteristic
110	C₄⋊Dic₁₉	C₂²⋊C₄	1	1	yes	characteristic
111	Dic₁₉⋊C₄	—	2	1	no	—
112	Dic₁₉⋊C₄	—	2	1	no	—
113	C₄×Dic₁₉	—	2	1	no	—
114	C₄×Dic₁₉	—	2	1	no	—
115	C₄×Dic₁₉	—	2	1	no	—
116	#64520	—	4	1	no	—
117	#64520	—	4	1	no	—
118	#64520	—	4	1	no	—
119	#64520	—	4	1	no	—
120	C₂×C₁₉⋊C₈	—	2	2	no	—
121	C₂×C₁₉⋊C₈	—	2	2	no	—
122	C₂×C₁₉⋊C₈	—	4	1	no	—
123	#64515	—	4	4	no	—
124	C₈×D₁₉	—	4	4	no	—
125	#9169	—	19	1	no	maximal, Sylow
126	C₂³×C₁₉	#123	1	1	yes	characteristic
127	C₂²×D₁₉	—	2	1	no	—
128	Q₈×C₁₉	—	8	2	no	—
129	Q₈×C₁₉	—	8	2	no	—
130	C₂×C₄×C₁₉	—	2	1	no	—
131	C₂×C₄×C₁₉	—	2	1	no	—
132	C₂×C₄×C₁₉	—	4	1	no	—
133	C₂×C₄×C₁₉	—	4	2	no	—
134	C₂×C₄×C₁₉	—	8	2	no	—
135	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
139	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
140	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
141	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
142	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
143	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
145	C₂²⋊D₁₉	—	2	2	no	—
146	C₂²⋊D₁₉	—	2	2	no	—
147	C₂×Dic₁₉	—	2	1	no	—
148	C₂×Dic₁₉	—	2	1	no	—
149	C₂×Dic₁₉	—	2	1	no	—
150	D₇₆	—	2	1	no	—
151	D₇₆	#103	1	2	yes	—
152	C₄×D₁₉	—	2	2	no	—
153	C₄×D₁₉	—	2	2	no	—
154	C₄×D₁₉	—	4	1	no	—
155	C₄×D₁₉	—	4	1	no	—
156	C₄×D₁₉	—	8	1	no	—
157	Dic₃₈	—	2	1	no	—
158	Dic₃₈	#103	1	2	yes	—
159	C₈×C₁₉	—	4	1	no	—
160	C₈×C₁₉	—	4	1	no	—
161	C₈×C₁₉	—	8	1	no	—
162	C₁₉⋊C₈	—	4	2	no	—
163	C₁₉⋊C₈	—	4	2	no	—
164	C₁₉⋊C₈	—	8	1	no	—
165	#3079	—	19	1	no	—
166	#2906	—	19	1	no	—
167	#1549	—	19	1	no	—
168	C₂×#333	D₁₉	1	1	yes	characteristic, complemented
169	#1477	—	19	1	no	—
170	#1301	—	19	1	no	—
171	#1292	—	19	1	no	—
172	C₂²×C₁₉	—	2	1	no	—
173	C₂²×C₁₉	—	2	1	no	—
174	C₂²×C₁₉	#386	1	1	yes	characteristic
175	C₂²×C₁₉	#483	1	1	yes	characteristic
176	C₂²×C₁₉	#484	1	1	yes	characteristic
177	C₂×D₁₉	—	2	2	no	—
178	C₂×D₁₉	—	4	1	no	—
179	C₂×D₁₉	—	4	1	no	—
180	C₄×C₁₉	—	2	1	no	—
181	C₄×C₁₉	—	2	1	no	—
182	C₄×C₁₉	—	2	1	no	—
183	C₄×C₁₉	—	8	2	no	—
184	C₄×C₁₉	C₂×#103	1	1	yes	characteristic
185	C₄×C₁₉	#417	1	1	yes	characteristic
186	C₄×C₁₉	#450	1	1	yes	characteristic
187	C₄×C₁₉	#451	1	1	yes	characteristic
188	C₄×C₁₉	#471	1	1	yes	characteristic
189	C₄×C₁₉	#473	1	1	yes	characteristic
190	Dic₁₉	—	2	2	no	—
191	Dic₁₉	—	4	1	no	—
192	Dic₁₉	—	4	1	no	—
193	C₂×#131	—	38	1	no	—
194	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
195	C₄×#41	—	19	1	no	—
196	#477	—	38	1	no	—
197	#476	—	38	1	no	—
198	#475	—	19	2	no	—
199	#474	—	19	2	no	—
200	#431	—	19	1	no	—
201	#428	—	38	1	no	—
202	#424	—	38	2	no	—
203	#424	—	38	1	no	—
204	#423	—	19	1	no	—
205	#423	—	19	1	no	—
206	#423	—	38	1	no	—
207	C₂×C₄⋊C₈	—	38	1	no	—
208	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
209	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
210	#417	—	38	1	no	—
211	#333	—	2	1	no	—
212	#333	—	2	1	no	—
213	#332	—	38	2	no	—
214	Q₈⋊C₈	—	38	1	no	—
215	D₄⋊C₈	—	38	1	no	—
216	C₂×C₁₉	—	2	1	no	—
217	C₂×C₁₉	—	2	1	no	—
218	C₂×C₁₉	#1872	1	1	yes	characteristic
219	C₂×C₁₉	#1903	1	1	yes	characteristic
220	C₂×C₁₉	#1908	1	1	yes	characteristic
221	D₁₉	—	4	2	no	—
222	C₂×#41	—	19	1	no	—
223	C₂²×Q₈	—	2	1	no	—
224	#131	—	76	2	no	—
225	#131	—	76	2	no	—
226	C₂×M₁₆	—	38	2	no	—
227	C₂×M₁₆	—	38	1	no	—
228	C₂×M₁₆	—	38	1	no	—
229	C₂²×C₈	—	2	1	no	—
230	C₂²×C₈	—	2	1	no	—
231	C₂²×C₈	—	38	2	no	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	2	1	no	—
234	C₄⋊Q₈	—	2	1	no	—
235	C₄⋊Q₈	C₄×D₁₉	1	2	yes	—
236	C₄×Q₈	—	19	1	no	—
237	C₄×Q₈	—	19	1	no	—
238	C₄×D₄	—	19	1	no	—
239	C₄×D₄	—	19	1	no	—
240	C₄×D₄	—	38	1	no	—
241	C₄×D₄	—	38	1	no	—
242	#117	—	19	1	no	—
243	#117	—	38	1	no	—
244	C₂×C₄⋊C₄	—	2	1	no	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	C₂×C₄²	—	38	1	no	—
247	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
248	#107	—	38	2	no	—
249	#106	—	38	2	no	—
250	C₄⋊C₈	—	2	1	no	—
251	C₄⋊C₈	—	2	1	no	—
252	C₄⋊C₈	—	2	1	no	—
253	C₄⋊C₈	—	19	2	no	—
254	C₄⋊C₈	—	19	2	no	—
255	C₄⋊C₈	—	38	2	no	—
256	C₄⋊C₈	—	38	2	no	—
257	C₄⋊C₈	—	38	1	no	—
258	C₄⋊C₈	—	38	1	no	—
259	C₄⋊C₈	—	38	1	no	—
260	C₄⋊C₈	C₂²⋊D₁₉	1	1	yes	characteristic
261	C₄⋊C₈	C₂²⋊D₁₉	1	1	yes	characteristic
262	#103	—	38	2	no	—
263	#102	—	38	2	no	—
264	C₂²⋊C₈	—	38	1	no	—
265	C₄×C₈	—	38	1	no	—
266	C₁₉	#9169	1	1	yes	characteristic, complemented, Sylow
267	#41	—	19	2	no	—
268	#41	—	19	2	no	—
269	#41	—	38	1	no	—
270	#41	—	38	1	no	—
271	C₂×Q₈	—	2	2	no	—
272	C₂×Q₈	—	4	1	no	—
273	C₂×Q₈	—	8	2	no	—
274	C₂×Q₈	—	19	1	no	—
275	C₂×D₄	—	19	1	no	—
276	C₂×D₄	—	38	1	no	—
277	C₂²×C₄	—	4	1	no	—
278	C₂²×C₄	—	38	1	no	—
279	C₂²×C₄	—	38	1	no	—
280	C₂²×C₄	—	38	1	no	—
281	C₂²×C₄	#64541	1	1	yes	characteristic
282	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
283	C₂²×C₄	#64543	1	1	yes	characteristic
284	M₁₆	—	76	4	no	—
285	M₁₆	—	76	1	no	—
286	M₁₆	—	76	1	no	—
287	M₁₆	—	76	1	no	—
288	M₁₆	—	76	1	no	—
289	C₂×C₈	—	2	1	no	—
290	C₂×C₈	—	2	1	no	—
291	C₂×C₈	—	4	1	no	—
292	C₂×C₈	—	4	1	no	—
293	C₂×C₈	—	4	1	no	—
294	C₂×C₈	—	8	1	no	—
295	C₂×C₈	—	38	2	no	—
296	C₂×C₈	—	38	2	no	—
297	C₂×C₈	—	76	4	no	—
298	C₂×C₈	—	76	1	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	4	1	no	—
303	C₄⋊C₄	—	19	1	no	—
304	C₄⋊C₄	—	19	1	no	—
305	C₄⋊C₄	—	38	1	no	—
306	C₄⋊C₄	—	38	1	no	—
307	C₂²⋊C₄	—	38	1	no	—
308	C₂²⋊C₄	—	38	1	no	—
309	C₂²⋊C₄	—	38	1	no	—
310	C₄²	—	38	1	no	—
311	C₄²	—	38	1	no	—
312	C₄²	—	38	1	no	—
313	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
314	C₄²	D₄×D₁₉	1	1	yes	characteristic
315	C₄²	D₄×D₁₉	1	1	yes	characteristic
316	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
317	C₂³	—	38	1	no	—
318	C₂³	#10597329	1	1	yes	characteristic
319	Q₈	—	8	2	no	—
320	Q₈	—	8	2	no	—
321	Q₈	—	19	2	no	—
322	Q₈	—	38	1	no	—
323	D₄	—	19	2	no	—
324	D₄	—	38	2	no	—
325	D₄	—	38	2	no	—
326	D₄	—	38	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	8	2	no	—
332	C₂×C₄	—	38	1	no	—
333	C₂×C₄	—	38	2	no	—
334	C₂×C₄	—	38	2	no	—
335	C₂×C₄	—	38	1	no	—
336	C₂×C₄	—	38	1	no	—
337	C₂×C₄	—	76	1	no	—
338	C₂×C₄	—	76	1	no	—
339	C₂×C₄	—	152	1	no	—
340	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
341	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
342	C₂×C₄	#10597330	1	1	yes	characteristic
343	C₂×C₄	SD₁₆×D₁₉	1	1	yes	characteristic
344	C₂×C₄	#10597360	1	1	yes	characteristic
345	C₂×C₄	Q₁₆×D₁₉	1	1	yes	characteristic
346	C₂×C₄	#10597364	1	1	yes	characteristic
347	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
348	C₂×C₄	#10597383	1	1	yes	characteristic
349	C₂×C₄	#10597384	1	1	yes	characteristic
350	C₈	—	4	1	no	—
351	C₈	—	4	1	no	—
352	C₈	—	8	1	no	—
353	C₈	—	76	2	no	—
354	C₈	—	76	2	no	—
355	C₈	—	152	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	38	2	no	—
359	C₂²	—	76	1	no	—
360	C₂²	—	76	1	no	—
361	C₂²	#11926420	1	1	yes	characteristic
362	C₂²	#11926873	1	1	yes	characteristic
363	C₂²	#11926878	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	8	2	no	—
368	C₄	—	38	2	no	—
369	C₄	—	76	1	no	—
370	C₄	—	76	1	no	—
371	C₄	#103×D₁₉	1	1	yes	characteristic
372	C₄	#11926544	1	1	yes	characteristic
373	C₄	#11926693	1	1	yes	characteristic
374	C₄	#11926698	1	1	yes	characteristic
375	C₄	#11926804	1	1	yes	characteristic
376	C₄	#11926816	1	1	yes	characteristic
377	C₂	—	2	1	no	—
378	C₂	—	2	1	no	—
379	C₂	—	76	2	no	—
380	C₂	#14563939	1	1	yes	characteristic
381	C₂	#14564118	1	1	yes	characteristic
382	C₂	#14564153	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926420
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#333	C₂
socle	C₂²×C₁₉	#386
prime radical	C₂×#64541	D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#333	D₁₉
hypocenter	C₁₉	#9169
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#333

Complements

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

N	H ≅ G/N	type
3. #14562320	C₂	semidirect
5. #14561532	C₂	semidirect
6. #14561510	C₂	semidirect
7. C₂×C₁₉×#333	C₂	semidirect
168. C₂×#333	D₁₉	semidirect
266. C₁₉	#9169	semidirect

Maximal subgroups

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

subgroup	index
#14573444	2
#14572200	2
#14562320	2
#14562020	2
#14561532	2
#14561510	2
C₂×C₁₉×#333	2
#9169	19