Subgroups — #18077975

Order 4864.

504 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572675	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926642	C₂	1	1	yes	maximal, characteristic
3	#14562027	C₂	1	1	yes	maximal, characteristic
4	#14561959	C₂	1	1	yes	maximal, characteristic, complemented
5	#14561479	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561465	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1479	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#64527	C₄	1	1	yes	characteristic
9	C₂×C₄×D₇₆	C₂²	1	1	yes	characteristic
10	#11926642	—	2	2	no	—
11	#11926642	—	2	2	no	—
12	#11926623	—	2	2	no	—
13	#11926621	—	2	2	no	—
14	#11926597	C₂²	1	1	yes	characteristic
15	#11926597	C₂²	1	1	yes	characteristic
16	C₂×#10597264	C₄	1	1	yes	characteristic
17	C₂×#10597292	C₄	1	1	yes	characteristic
18	C₂×#10597266	C₄	1	1	yes	characteristic
19	#11926506	—	2	2	no	—
20	C₂×#10597260	C₂²	1	1	yes	characteristic
21	#11926273	—	2	2	no	—
22	#11926270	—	2	2	no	—
23	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
24	C₁₉×#431	C₂²	1	1	yes	characteristic
25	C₁₉×#431	C₂²	1	1	yes	characteristic
26	C₁₉×Q₈⋊C₈	—	2	2	no	—
27	C₂²×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×D₇₆	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×D₁₉	—	2	1	no	—
30	C₄×Q₈×C₁₉	—	2	1	no	—
31	C₄×Q₈×C₁₉	—	2	1	no	—
32	C₄×Q₈×C₁₉	—	2	1	no	—
33	C₄×Q₈×C₁₉	—	2	1	no	—
34	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
35	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₉	—	2	1	no	—
37	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
38	C₂×#64527	—	2	2	no	—
39	C₂×#64527	—	2	2	no	—
40	C₂×#64527	—	2	2	no	—
41	C₂×(C₂×D₁₉)⋊C₄	—	2	1	no	—
42	C₂×C₄⋊Dic₁₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₂×#64515	—	2	4	no	—
46	C₄×D₇₆	—	2	1	no	—
47	C₄×D₇₆	—	2	1	no	—
48	C₄×D₇₆	D₄	1	2	yes	—
49	C₄×D₇₆	D₄	1	2	yes	—
50	C₁₉×C₄⋊C₈	—	2	1	no	—
51	C₁₉×C₄⋊C₈	—	2	1	no	—
52	C₁₉×C₂²⋊C₈	—	2	1	no	—
53	C₁₉×C₂²⋊C₈	—	2	1	no	—
54	C₄×C₈×C₁₉	—	2	1	no	—
55	C₄×C₈×C₁₉	D₄	1	2	yes	—
56	#10597292	—	2	2	no	—
57	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
58	C₂²⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
59	(C₂×D₁₉)⋊C₈	—	2	4	no	—
60	#10597266	—	2	2	no	—
61	#10597264	—	2	2	no	—
62	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
63	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
64	#10597260	—	2	1	no	—
65	#10597260	—	2	1	no	—
66	C₄×C₁₉⋊C₈	—	2	1	no	—
67	C₄×C₁₉⋊C₈	—	2	1	no	—
68	C₂³×D₁₉	—	2	1	no	—
69	C₂×Q₈×C₁₉	—	2	1	no	—
70	C₂×Q₈×C₁₉	—	2	1	no	—
71	C₂×Q₈×C₁₉	—	2	1	no	—
72	C₂×Q₈×C₁₉	—	2	1	no	—
73	C₂×Q₈×C₁₉	—	2	1	no	—
74	C₂×Q₈×C₁₉	M₁₆	1	2	yes	complemented
75	C₂²×C₄×C₁₉	—	2	1	no	—
76	C₂²×C₄×C₁₉	—	2	1	no	—
77	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
80	C₂²×Dic₁₉	—	2	1	no	—
81	C₂×D₇₆	—	2	1	no	—
82	C₂×D₇₆	—	2	1	no	—
83	C₂×D₇₆	—	2	1	no	—
84	C₂×D₇₆	—	2	1	no	—
85	C₂×D₇₆	—	2	1	no	—
86	C₂×D₇₆	C₂²⋊C₄	1	2	yes	—
87	C₂×C₄×D₁₉	—	2	2	no	—
88	C₂×C₄×D₁₉	—	2	2	no	—
89	C₂×C₄×D₁₉	—	8	2	no	—
90	C₂×C₈×C₁₉	—	2	2	no	—
91	C₂×C₈×C₁₉	—	4	1	no	—
92	C₄⋊C₄×C₁₉	—	2	1	no	—
93	C₄⋊C₄×C₁₉	—	2	1	no	—
94	C₄⋊C₄×C₁₉	—	4	1	no	—
95	C₄⋊C₄×C₁₉	—	4	1	no	—
96	C₄²×C₁₉	—	4	1	no	—
97	C₄²×C₁₉	—	4	1	no	—
98	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₁₉	#41	1	1	yes	characteristic
102	#64527	—	2	8	no	—
103	(C₂×D₁₉)⋊C₄	—	2	2	no	—
104	(C₂×D₁₉)⋊C₄	—	2	2	no	—
105	C₄⋊Dic₁₉	—	2	1	no	—
106	C₄⋊Dic₁₉	C₂²⋊C₄	1	2	yes	—
107	C₂×C₁₉⋊C₈	—	2	1	no	—
108	C₂×C₁₉⋊C₈	—	2	1	no	—
109	C₂×C₁₉⋊C₈	—	2	1	no	—
110	C₂×C₁₉⋊C₈	—	2	2	no	—
111	C₂×C₁₉⋊C₈	—	2	1	no	—
112	C₂×C₁₉⋊C₈	—	2	1	no	—
113	C₂×C₁₉⋊C₈	—	4	1	no	—
114	C₂×C₁₉⋊C₈	M₁₆	1	2	yes	complemented
115	#64515	—	8	4	no	—
116	#9091	—	19	1	no	maximal, Sylow
117	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
118	C₂²×D₁₉	—	2	2	no	—
119	C₂²×D₁₉	—	4	1	no	—
120	C₂²×D₁₉	—	4	1	no	—
121	C₂²×D₁₉	—	8	1	no	—
122	C₂²×D₁₉	—	8	1	no	—
123	Q₈×C₁₉	—	2	2	no	—
124	Q₈×C₁₉	—	2	2	no	—
125	Q₈×C₁₉	—	4	1	no	—
126	Q₈×C₁₉	—	4	1	no	—
127	C₂×C₄×C₁₉	—	2	2	no	—
128	C₂×C₄×C₁₉	—	4	1	no	—
129	C₂×C₄×C₁₉	—	4	1	no	—
130	C₂×C₄×C₁₉	—	4	1	no	—
131	C₂×C₄×C₁₉	—	4	1	no	—
132	C₂×C₄×C₁₉	—	4	1	no	—
133	C₂×C₄×C₁₉	—	8	1	no	—
134	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
137	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
138	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
139	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
140	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
141	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
142	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
143	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
144	C₂×Dic₁₉	—	2	2	no	—
145	C₂×Dic₁₉	—	8	1	no	—
146	D₇₆	—	2	2	no	—
147	D₇₆	—	2	2	no	—
148	D₇₆	—	4	1	no	—
149	D₇₆	—	4	1	no	—
150	C₄×D₁₉	—	4	4	no	—
151	C₄×D₁₉	—	8	2	no	—
152	C₈×C₁₉	—	4	2	no	—
153	C₈×C₁₉	—	8	1	no	—
154	C₁₉⋊C₈	—	2	2	no	—
155	C₁₉⋊C₈	—	4	2	no	—
156	C₁₉⋊C₈	—	4	1	no	—
157	C₁₉⋊C₈	—	8	1	no	—
158	#2975	—	19	1	no	—
159	C₂×#440	—	19	1	no	—
160	#1479	D₁₉	1	1	yes	characteristic, complemented
161	#1478	—	19	1	no	—
162	#1461	—	19	1	no	—
163	#1281	—	19	1	no	—
164	#1275	—	19	1	no	—
165	C₂²×C₁₉	—	2	1	no	—
166	C₂²×C₁₉	—	2	1	no	—
167	C₂²×C₁₉	#386	1	1	yes	characteristic
168	C₂²×C₁₉	#435	1	1	yes	characteristic
169	C₂²×C₁₉	#436	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	2	no	—
174	C₂×D₁₉	—	8	2	no	—
175	C₄×C₁₉	—	2	1	no	—
176	C₄×C₁₉	—	2	1	no	—
177	C₄×C₁₉	—	2	1	no	—
178	C₄×C₁₉	—	2	1	no	—
179	C₄×C₁₉	—	4	2	no	—
180	C₄×C₁₉	—	8	1	no	—
181	C₄×C₁₉	#407	1	1	yes	characteristic
182	C₄×C₁₉	#408	1	1	yes	characteristic
183	C₄×C₁₉	#435	1	1	yes	characteristic
184	C₄×C₁₉	#436	1	1	yes	characteristic
185	Dic₁₉	—	4	2	no	—
186	C₂²×SD₁₆	—	19	1	no	—
187	C₂×C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
188	C₂×C₄×D₄	—	19	1	no	—
189	#440	—	38	2	no	—
190	#440	—	38	2	no	—
191	#436	—	38	2	no	—
192	#436	—	38	2	no	—
193	#431	—	19	1	no	—
194	#431	—	19	1	no	—
195	#431	C₂×D₁₉	1	1	yes	characteristic
196	#431	C₂×D₁₉	1	1	yes	characteristic
197	C₂×#107	—	19	1	no	—
198	C₂×#103	—	19	1	no	—
199	C₂×#102	—	19	1	no	—
200	#407	—	38	2	no	—
201	C₂×#97	—	19	1	no	—
202	Q₈⋊C₈	—	2	2	no	—
203	D₄⋊C₈	—	38	2	no	—
204	#324	—	38	2	no	—
205	C₂×C₁₉	—	2	1	no	—
206	C₂×C₁₉	—	2	1	no	—
207	C₂×C₁₉	#1870	1	1	yes	characteristic
208	C₂×C₁₉	#1902	1	1	yes	characteristic
209	C₂×C₁₉	#1904	1	1	yes	characteristic
210	D₁₉	—	8	2	no	—
211	C₂²×Q₈	C₄×D₁₉	1	1	yes	characteristic
212	C₂²×D₄	—	19	1	no	—
213	C₂³×C₄	—	38	1	no	—
214	C₂×SD₁₆	—	38	2	no	—
215	C₂×SD₁₆	—	38	2	no	—
216	C₂×SD₁₆	—	38	2	no	—
217	C₂×M₁₆	—	38	4	no	—
218	C₂²×C₈	—	19	1	no	—
219	C₂²×C₈	—	19	1	no	—
220	C₄×Q₈	—	2	1	no	—
221	C₄×Q₈	—	2	1	no	—
222	C₄×Q₈	—	2	1	no	—
223	C₄×Q₈	—	2	1	no	—
224	C₄×D₄	—	19	2	no	—
225	C₄×D₄	—	19	2	no	—
226	C₄×D₄	—	38	1	no	—
227	C₄×D₄	—	38	1	no	—
228	C₂×C₄⋊C₄	—	2	1	no	—
229	C₂×C₄⋊C₄	—	19	1	no	—
230	C₂×C₄⋊C₄	C₄×D₁₉	1	1	yes	characteristic
231	C₂×C₂²⋊C₄	—	38	1	no	—
232	C₂×C₄²	—	2	1	no	—
233	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
234	#107	—	38	2	no	—
235	C₄⋊C₈	—	2	1	no	—
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	—	38	1	no	—
238	C₄⋊C₈	—	38	1	no	—
239	#103	—	38	2	no	—
240	#102	—	38	2	no	—
241	C₂²⋊C₈	—	2	1	no	—
242	C₂²⋊C₈	—	2	1	no	—
243	C₂²⋊C₈	—	19	2	no	—
244	C₂²⋊C₈	—	38	4	no	—
245	C₂²⋊C₈	—	38	1	no	—
246	#97	—	38	1	no	—
247	#97	—	38	1	no	—
248	C₄×C₈	—	2	1	no	—
249	C₄×C₈	—	38	1	no	—
250	C₄×C₈	—	38	1	no	—
251	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
252	C₁₉	#9091	1	1	yes	characteristic, complemented, Sylow
253	C₂⁴	—	38	1	no	—
254	C₂×Q₈	—	2	1	no	—
255	C₂×Q₈	—	2	1	no	—
256	C₂×Q₈	—	2	1	no	—
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	—	2	1	no	—
259	C₂×Q₈	#64515	1	2	yes	complemented
260	C₂×D₄	—	19	2	no	—
261	C₂×D₄	—	38	1	no	—
262	C₂×D₄	—	38	1	no	—
263	C₂×D₄	—	38	1	no	—
264	C₂×D₄	—	38	1	no	—
265	C₂×D₄	—	38	1	no	—
266	C₂²×C₄	—	2	1	no	—
267	C₂²×C₄	—	2	1	no	—
268	C₂²×C₄	—	38	1	no	—
269	C₂²×C₄	—	38	2	no	—
270	C₂²×C₄	—	38	2	no	—
271	C₂²×C₄	—	152	2	no	—
272	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
273	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
274	C₂²×C₄	#64543	1	1	yes	characteristic
275	SD₁₆	—	38	8	no	—
276	M₁₆	—	152	4	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	4	1	no	—
279	C₂×C₈	—	19	2	no	—
280	C₂×C₈	—	38	1	no	—
281	C₂×C₈	—	38	1	no	—
282	C₂×C₈	—	38	2	no	—
283	C₂×C₈	—	38	1	no	—
284	C₂×C₈	—	38	1	no	—
285	C₂×C₈	—	38	1	no	—
286	C₂×C₈	—	76	1	no	—
287	C₄⋊C₄	—	2	1	no	—
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	19	2	no	—
292	C₄⋊C₄	—	38	1	no	—
293	C₂²⋊C₄	—	38	2	no	—
294	C₂²⋊C₄	—	38	2	no	—
295	C₄²	—	4	1	no	—
296	C₄²	—	4	1	no	—
297	C₄²	#64541	1	1	yes	characteristic
298	C₄²	D₄×D₁₉	1	1	yes	characteristic
299	C₄²	D₄×D₁₉	1	1	yes	characteristic
300	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
301	C₂³	—	38	2	no	—
302	C₂³	—	76	1	no	—
303	C₂³	—	76	1	no	—
304	C₂³	—	152	1	no	—
305	C₂³	—	152	1	no	—
306	C₂³	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
307	Q₈	—	2	2	no	—
308	Q₈	—	2	2	no	—
309	Q₈	—	4	1	no	—
310	Q₈	—	4	1	no	—
311	D₄	—	38	2	no	—
312	D₄	—	38	2	no	—
313	D₄	—	76	1	no	—
314	D₄	—	76	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	8	1	no	—
322	C₂×C₄	—	38	2	no	—
323	C₂×C₄	—	76	4	no	—
324	C₂×C₄	—	152	2	no	—
325	C₂×C₄	—	152	1	no	—
326	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
327	C₂×C₄	#10597329	1	1	yes	characteristic
328	C₂×C₄	#10597330	1	1	yes	characteristic
329	C₂×C₄	C₂×#64515	1	1	yes	characteristic
330	C₂×C₄	#10597347	1	1	yes	characteristic
331	C₂×C₄	M₁₆×D₁₉	1	1	yes	characteristic
332	C₂×C₄	#10597353	1	1	yes	characteristic
333	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
334	C₂×C₄	#10597383	1	1	yes	characteristic
335	C₂×C₄	#10597384	1	1	yes	characteristic
336	C₈	—	4	2	no	—
337	C₈	—	8	1	no	—
338	C₈	—	38	2	no	—
339	C₈	—	76	2	no	—
340	C₈	—	76	1	no	—
341	C₈	—	152	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	76	2	no	—
345	C₂²	—	76	2	no	—
346	C₂²	—	76	2	no	—
347	C₂²	—	152	2	no	—
348	C₂²	—	152	2	no	—
349	C₂²	#11926420	1	1	yes	characteristic
350	C₂²	#11926612	1	1	yes	characteristic
351	C₂²	#11926619	1	1	yes	characteristic
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	4	2	no	—
357	C₄	—	8	1	no	—
358	C₄	—	76	2	no	—
359	C₄	#11926506	1	1	yes	characteristic
360	C₄	#11926509	1	1	yes	characteristic
361	C₄	#11926615	1	1	yes	characteristic
362	C₄	#11926620	1	1	yes	characteristic
363	C₂	—	2	1	no	—
364	C₂	—	2	1	no	—
365	C₂	—	152	2	no	—
366	C₂	#14563929	1	1	yes	characteristic
367	C₂	#14564105	1	1	yes	characteristic
368	C₂	#14564119	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926420
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1479	C₂
socle	C₂²×C₁₉	#386
prime radical	C₂²×D₇₆	C₂×C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1479	D₁₉
hypocenter	C₁₉	#9091
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1479

Complements

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

N	H ≅ G/N	type
4. #14561959	C₂	semidirect
5. #14561479	C₂	semidirect
6. #14561465	C₂	semidirect
7. C₁₉×#1479	C₂	semidirect
74. C₂×Q₈×C₁₉	M₁₆	semidirect
114. C₂×C₁₉⋊C₈	M₁₆	semidirect
160. #1479	D₁₉	semidirect
252. C₁₉	#9091	semidirect
259. C₂×Q₈	#64515	semidirect

Maximal subgroups

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

subgroup	index
#14572675	2
C₂×#11926642	2
#14562027	2
#14561959	2
#14561479	2
#14561465	2
C₁₉×#1479	2
#9091	19