Subgroups — #18075383

Order 4864.

546 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#64515	C₂	1	1	yes	maximal, characteristic
2	#14563612	C₂	1	2	yes	maximal
3	C₂×#11926295	C₂	1	1	yes	maximal, characteristic, complemented
4	#14561469	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₉×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#64515	C₄	1	1	yes	characteristic
7	C₂²×#64515	C₄	1	1	yes	characteristic
8	C₂×C₄²×D₁₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
10	C₂×(C₂×D₁₉)⋊C₈	—	2	2	no	—
11	C₄×#64515	—	2	2	no	—
12	C₄×#64515	—	2	2	no	—
13	C₂×#10597272	C₄	1	1	yes	characteristic
14	C₂×#10597272	C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
16	#11926298	—	2	2	no	—
17	#11926295	—	2	2	no	—
18	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
20	C₁₉×#335	—	2	2	no	—
21	C₂²×C₄×D₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×D₁₉	D₄	1	1	yes	characteristic
23	C₂²×C₄×D₁₉	Q₈	1	1	yes	characteristic
24	C₂²×C₈×C₁₉	—	2	2	no	—
25	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
28	C₂×C₄×Dic₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
30	C₂×C₄×Dic₁₉	Q₈	1	1	yes	characteristic
31	C₂²×C₁₉⋊C₈	—	2	2	no	—
32	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×#64515	—	2	2	no	—
35	C₂×#64515	—	2	2	no	—
36	C₂×#64515	—	2	2	no	—
37	C₂×#64515	—	2	2	no	—
38	C₂×#64515	—	2	2	no	—
39	C₂×#64515	C₈	1	4	yes	complemented
40	C₄²×D₁₉	—	2	1	no	—
41	C₄²×D₁₉	—	2	1	no	—
42	C₄²×D₁₉	—	2	1	no	—
43	C₄²×D₁₉	—	2	1	no	—
44	C₁₉×C₄⋊C₈	—	2	2	no	—
45	C₁₉×C₄⋊C₈	—	2	2	no	—
46	C₄×C₈×C₁₉	D₄	1	2	yes	—
47	C₄×C₈×C₁₉	D₄	1	2	yes	—
48	(C₂×D₁₉)⋊C₈	—	4	4	no	—
49	#10597272	—	2	2	no	—
50	#10597272	—	2	2	no	—
51	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
52	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
53	C₄×C₁₉⋊C₈	—	2	1	no	—
54	C₄×C₁₉⋊C₈	—	2	1	no	—
55	C₂³×D₁₉	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
59	C₂²×Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×Dic₁₉	C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×D₁₉	—	2	1	no	—
63	C₂×C₄×D₁₉	—	2	1	no	—
64	C₂×C₄×D₁₉	—	2	1	no	—
65	C₂×C₄×D₁₉	—	2	1	no	—
66	C₂×C₄×D₁₉	—	2	1	no	—
67	C₂×C₄×D₁₉	—	2	1	no	—
68	C₂×C₄×D₁₉	—	2	1	no	—
69	C₂×C₄×D₁₉	—	2	1	no	—
70	C₂×C₄×D₁₉	—	2	2	no	—
71	C₂×C₄×D₁₉	—	2	2	no	—
72	C₂×C₄×D₁₉	—	4	2	no	—
73	C₂×C₄×D₁₉	C₂×C₈	1	2	yes	—
74	C₂×C₄×D₁₉	M₁₆	1	2	yes	—
75	C₂×C₈×C₁₉	—	2	1	no	—
76	C₂×C₈×C₁₉	—	2	1	no	—
77	C₂×C₈×C₁₉	—	4	2	no	—
78	C₂×C₈×C₁₉	—	4	2	no	—
79	C₂×C₈×C₁₉	—	4	2	no	—
80	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₁₉	C₂×C₈	1	2	yes	—
83	C₂×C₈×C₁₉	M₁₆	1	2	yes	—
84	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₁₉	#41	1	1	yes	characteristic
87	C₄²×C₁₉	#41	1	1	yes	characteristic
88	C₄×Dic₁₉	—	2	1	no	—
89	C₄×Dic₁₉	—	2	1	no	—
90	C₄×Dic₁₉	—	2	1	no	—
91	C₄×Dic₁₉	—	2	1	no	—
92	C₄×Dic₁₉	—	2	1	no	—
93	C₄×Dic₁₉	—	2	1	no	—
94	C₂×C₁₉⋊C₈	—	2	1	no	—
95	C₂×C₁₉⋊C₈	—	2	1	no	—
96	C₂×C₁₉⋊C₈	—	2	1	no	—
97	C₂×C₁₉⋊C₈	—	2	1	no	—
98	C₂×C₁₉⋊C₈	—	4	2	no	—
99	C₂×C₁₉⋊C₈	—	4	2	no	—
100	C₂×C₁₉⋊C₈	—	4	2	no	—
101	C₂×C₁₉⋊C₈	C₂×C₈	1	2	yes	—
102	C₂×C₁₉⋊C₈	M₁₆	1	2	yes	—
103	#64515	—	2	8	no	—
104	#64515	—	2	8	no	—
105	#8597	—	19	1	no	maximal, Sylow
106	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×D₁₉	—	2	1	no	—
108	C₂²×D₁₉	—	2	1	no	—
109	C₂²×D₁₉	—	2	2	no	—
110	C₂²×D₁₉	—	2	2	no	—
111	C₂²×D₁₉	C₄⋊C₈	1	2	yes	—
112	C₂×C₄×C₁₉	—	2	2	no	—
113	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
115	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
116	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
123	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
124	C₂×C₄×C₁₉	#137	1	2	yes	—
125	C₂×Dic₁₉	—	2	1	no	—
126	C₂×Dic₁₉	—	2	1	no	—
127	C₂×Dic₁₉	—	2	1	no	—
128	C₂×Dic₁₉	—	2	1	no	—
129	C₂×Dic₁₉	—	2	2	no	—
130	C₂×Dic₁₉	—	2	2	no	—
131	C₂×Dic₁₉	C₄⋊C₈	1	2	yes	—
132	C₄×D₁₉	—	2	2	no	—
133	C₄×D₁₉	—	2	2	no	—
134	C₄×D₁₉	—	2	2	no	—
135	C₄×D₁₉	—	2	2	no	—
136	C₄×D₁₉	—	4	2	no	—
137	C₄×D₁₉	—	4	2	no	—
138	C₄×D₁₉	—	4	2	no	—
139	C₄×D₁₉	—	4	1	no	—
140	C₄×D₁₉	—	4	1	no	—
141	C₈×C₁₉	—	2	2	no	—
142	C₈×C₁₉	—	8	2	no	—
143	C₈×C₁₉	C₂²⋊C₈	1	4	yes	complemented
144	C₁₉⋊C₈	—	2	2	no	—
145	C₁₉⋊C₈	—	2	2	no	—
146	C₁₉⋊C₈	—	8	2	no	—
147	C₂×C₄×M₁₆	—	19	1	no	—
148	#1797	—	19	2	no	—
149	C₂×#335	D₁₉	1	1	yes	characteristic, complemented
150	C₂×#334	—	19	1	no	—
151	#1277	—	19	2	no	—
152	C₂²×C₁₉	#380	1	1	yes	characteristic
153	C₂²×C₁₉	C₂×C₄⋊C₈	1	1	yes	characteristic
154	C₂²×C₁₉	#423	1	1	yes	characteristic
155	C₂²×C₁₉	#428	1	2	yes	—
156	C₂²×C₁₉	#430	1	2	yes	—
157	C₂×D₁₉	—	2	2	no	—
158	C₂×D₁₉	—	2	4	no	—
159	C₂×D₁₉	—	4	1	no	—
160	C₂×D₁₉	—	4	1	no	—
161	C₂×D₁₉	—	4	2	no	—
162	C₄×C₁₉	—	2	1	no	—
163	C₄×C₁₉	—	2	1	no	—
164	C₄×C₁₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
165	C₄×C₁₉	#407	1	1	yes	characteristic
166	C₄×C₁₉	#431	1	1	yes	characteristic
167	C₄×C₁₉	#432	1	1	yes	characteristic
168	C₄×C₁₉	#468	1	2	yes	—
169	C₄×C₁₉	#469	1	1	yes	characteristic
170	C₄×C₁₉	#470	1	1	yes	characteristic
171	Dic₁₉	—	2	2	no	—
172	Dic₁₉	—	4	1	no	—
173	Dic₁₉	—	4	1	no	—
174	C₂²×M₁₆	—	19	1	no	—
175	C₂²×M₁₆	—	19	1	no	—
176	C₂²×C₄²	—	19	1	no	—
177	C₂×C₄⋊C₈	—	19	2	no	—
178	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
179	C₂×C₂²⋊C₈	—	38	2	no	—
180	C₄×M₁₆	—	38	2	no	—
181	C₄×M₁₆	—	38	2	no	—
182	C₂×#97	—	19	1	no	—
183	C₂×#97	—	19	1	no	—
184	C₂×C₄×C₈	—	19	1	no	—
185	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
186	#336	—	38	2	no	—
187	#335	—	2	2	no	—
188	#334	—	38	2	no	—
189	C₂×C₁₉	#1565	1	2	yes	—
190	C₂×C₁₉	#1569	1	2	yes	—
191	C₂×C₁₉	#1797	1	1	yes	characteristic
192	C₂×C₁₉	#1809	1	1	yes	characteristic
193	C₂×C₁₉	#1811	1	1	yes	characteristic
194	D₁₉	—	4	2	no	—
195	C₂³×C₄	—	19	1	no	—
196	C₂³×C₄	—	19	1	no	—
197	C₂³×C₄	—	19	1	no	—
198	C₂×M₁₆	—	19	4	no	—
199	C₂×M₁₆	—	38	2	no	—
200	C₂×M₁₆	—	38	2	no	—
201	C₂×M₁₆	—	38	2	no	—
202	C₂×M₁₆	—	38	2	no	—
203	C₂×M₁₆	—	38	2	no	—
204	C₂²×C₈	—	2	2	no	—
205	C₂²×C₈	—	19	1	no	—
206	C₂²×C₈	—	19	1	no	—
207	C₂²×C₈	—	38	2	no	—
208	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
209	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
210	C₂×C₄²	—	19	1	no	—
211	C₂×C₄²	—	19	1	no	—
212	C₂×C₄²	—	19	1	no	—
213	C₂×C₄²	—	38	1	no	—
214	C₂×C₄²	—	38	1	no	—
215	C₂×C₄²	—	38	1	no	—
216	C₂×C₄²	—	38	1	no	—
217	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
218	C₄⋊C₈	—	2	2	no	—
219	C₄⋊C₈	—	2	2	no	—
220	C₄⋊C₈	—	38	2	no	—
221	C₄⋊C₈	—	38	2	no	—
222	C₂²⋊C₈	—	76	4	no	—
223	#97	—	38	2	no	—
224	#97	—	38	2	no	—
225	C₄×C₈	—	38	1	no	—
226	C₄×C₈	—	38	1	no	—
227	C₄×C₈	D₇₆	1	2	yes	—
228	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
229	C₁₉	#8597	1	1	yes	characteristic, complemented, Sylow
230	C₂⁴	—	19	1	no	—
231	C₂²×C₄	—	19	1	no	—
232	C₂²×C₄	—	19	2	no	—
233	C₂²×C₄	—	19	2	no	—
234	C₂²×C₄	—	19	1	no	—
235	C₂²×C₄	—	19	1	no	—
236	C₂²×C₄	—	38	1	no	—
237	C₂²×C₄	—	38	1	no	—
238	C₂²×C₄	—	38	1	no	—
239	C₂²×C₄	—	38	1	no	—
240	C₂²×C₄	—	38	1	no	—
241	C₂²×C₄	—	38	1	no	—
242	C₂²×C₄	—	38	1	no	—
243	C₂²×C₄	—	38	1	no	—
244	C₂²×C₄	—	38	2	no	—
245	C₂²×C₄	—	38	2	no	—
246	C₂²×C₄	—	76	2	no	—
247	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
248	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
249	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
250	M₁₆	—	38	8	no	—
251	M₁₆	—	38	8	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	4	2	no	—
255	C₂×C₈	—	4	2	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	19	2	no	—
258	C₂×C₈	—	19	2	no	—
259	C₂×C₈	—	38	1	no	—
260	C₂×C₈	—	38	1	no	—
261	C₂×C₈	—	38	1	no	—
262	C₂×C₈	—	38	1	no	—
263	C₂×C₈	—	76	2	no	—
264	C₂×C₈	—	76	2	no	—
265	C₂×C₈	—	76	2	no	—
266	C₂×C₈	C₈×D₁₉	1	2	yes	—
267	C₂×C₈	#64515	1	2	yes	—
268	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
269	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
270	C₄²	—	38	1	no	—
271	C₄²	—	38	1	no	—
272	C₄²	—	38	1	no	—
273	C₄²	—	38	1	no	—
274	C₄²	—	38	1	no	—
275	C₄²	—	38	1	no	—
276	C₄²	C₂×D₇₆	1	1	yes	characteristic
277	C₄²	#64543	1	1	yes	characteristic
278	C₄²	#64545	1	1	yes	characteristic
279	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
280	C₂³	—	19	2	no	—
281	C₂³	—	38	1	no	—
282	C₂³	—	38	2	no	—
283	C₂³	—	38	2	no	—
284	C₂³	—	38	1	no	—
285	C₂³	C₄⋊C₄×D₁₉	1	1	yes	characteristic
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	19	2	no	—
288	C₂×C₄	—	38	1	no	—
289	C₂×C₄	—	38	1	no	—
290	C₂×C₄	—	38	2	no	—
291	C₂×C₄	—	38	2	no	—
292	C₂×C₄	—	38	2	no	—
293	C₂×C₄	—	38	2	no	—
294	C₂×C₄	—	38	2	no	—
295	C₂×C₄	—	38	2	no	—
296	C₂×C₄	—	38	1	no	—
297	C₂×C₄	—	38	1	no	—
298	C₂×C₄	—	76	2	no	—
299	C₂×C₄	—	76	2	no	—
300	C₂×C₄	—	76	2	no	—
301	C₂×C₄	—	76	1	no	—
302	C₂×C₄	—	76	1	no	—
303	C₂×C₄	#10597338	1	1	yes	characteristic
304	C₂×C₄	#10597341	1	1	yes	characteristic
305	C₂×C₄	#10597343	1	1	yes	characteristic
306	C₂×C₄	C₂×C₈×D₁₉	1	1	yes	characteristic
307	C₂×C₄	C₂×#64515	1	1	yes	characteristic
308	C₂×C₄	M₁₆×D₁₉	1	1	yes	characteristic
309	C₂×C₄	M₁₆×D₁₉	1	1	yes	characteristic
310	C₂×C₄	#10597357	1	2	yes	—
311	C₂×C₄	#10597364	1	2	yes	—
312	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
313	C₂×C₄	#10597384	1	1	yes	characteristic
314	C₂×C₄	#10597392	1	1	yes	characteristic
315	C₈	—	2	2	no	—
316	C₈	—	8	2	no	—
317	C₈	—	38	2	no	—
318	C₈	—	38	2	no	—
319	C₈	—	152	2	no	—
320	C₈	(C₂×D₁₉)⋊C₈	1	4	yes	complemented
321	C₂²	—	38	2	no	—
322	C₂²	—	38	4	no	—
323	C₂²	—	76	1	no	—
324	C₂²	—	76	1	no	—
325	C₂²	—	76	2	no	—
326	C₂²	#11926401	1	1	yes	characteristic
327	C₂²	C₄⋊C₈×D₁₉	1	1	yes	characteristic
328	C₂²	#11926566	1	1	yes	characteristic
329	C₂²	#11926585	1	2	yes	—
330	C₂²	#11926591	1	2	yes	—
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	38	2	no	—
334	C₄	—	76	1	no	—
335	C₄	—	76	1	no	—
336	C₄	C₂×(C₂×D₁₉)⋊C₈	1	1	yes	characteristic
337	C₄	#11926507	1	1	yes	characteristic
338	C₄	#11926596	1	1	yes	characteristic
339	C₄	#11926600	1	1	yes	characteristic
340	C₄	#11926791	1	2	yes	—
341	C₄	#11926794	1	1	yes	characteristic
342	C₄	#11926799	1	1	yes	characteristic
343	C₂	—	76	2	no	—
344	C₂	#14562386	1	2	yes	—
345	C₂	#14562401	1	2	yes	—
346	C₂	#14563612	1	1	yes	characteristic
347	C₂	#14563669	1	1	yes	characteristic
348	C₂	#14563674	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₉
commutator	C₂×C₄×C₁₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#335	C₂
socle	C₂³×C₁₉	C₂×C₄⋊C₄
prime radical	C₂³×D₁₉	C₄⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#335	D₁₉
hypocenter	C₁₉	#8597
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
3. C₂×#11926295	C₂	semidirect
4. #14561469	C₂	semidirect
5. C₂×C₁₉×#335	C₂	semidirect
39. C₂×#64515	C₈	semidirect
143. C₈×C₁₉	C₂²⋊C₈	semidirect
149. C₂×#335	D₁₉	semidirect
229. C₁₉	#8597	semidirect
320. C₈	(C₂×D₁₉)⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#64515	2
#14563612	2
C₂×#11926295	2
#14561469	2
C₂×C₁₉×#335	2
#8597	19