Subgroups — #18088346

Order 4864.

762 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926568	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#64520	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉⋊#338	C₂	1	4	yes	maximal
4	C₂×C₁₉×#423	C₂	1	1	yes	maximal, characteristic
5	C₂²×#64520	C₂²	1	2	yes	—
6	C₂²×#64520	C₂²	1	2	yes	—
7	#11926568	—	2	1	no	—
8	#11926568	—	2	1	no	—
9	#11926568	—	2	2	no	—
10	C₂×C₄⋊(C₁₉⋊C₈)	C₄	1	4	yes	—
11	C₄×#64520	—	2	1	no	—
12	C₄×#64520	—	2	1	no	—
13	C₄×#64520	—	2	1	no	—
14	C₄×#64520	—	2	1	no	—
15	C₂×#10597260	C₄	1	2	yes	—
16	C₂×C₄×C₁₉⋊C₈	C₄	1	2	yes	—
17	C₁₉⋊#338	—	2	8	no	—
18	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
19	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
20	C₁₉×#423	—	2	2	no	—
21	C₁₉×#423	—	2	1	no	—
22	C₁₉×#423	—	2	1	no	—
23	C₂×C₁₉×C₄⋊C₈	C₄	1	4	yes	—
24	C₂³×C₄×C₁₉	Q₈	1	2	yes	—
25	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₁₉	—	2	4	no	—
27	C₂×M₁₆×C₁₉	—	2	2	no	—
28	C₂×M₁₆×C₁₉	—	2	2	no	—
29	C₂×M₁₆×C₁₉	—	2	4	no	—
30	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₉	D₄	1	2	yes	—
38	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
40	C₂×#64520	—	2	2	no	—
41	C₂×#64520	—	2	2	no	—
42	C₂×#64520	—	2	2	no	—
43	C₂×#64520	—	2	2	no	—
44	C₂×#64520	—	2	4	no	—
45	C₂×#64520	—	2	2	no	—
46	C₂×#64520	—	2	2	no	—
47	C₂×#64520	—	2	4	no	—
48	C₂×#64520	D₄	1	4	yes	—
49	C₂×#64520	Q₈	1	4	yes	—
50	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
51	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
52	C₁₉×C₄⋊C₈	—	2	4	no	—
53	C₁₉×C₄⋊C₈	—	2	4	no	—
54	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
55	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
56	#10597260	—	2	2	no	—
57	#10597260	—	2	2	no	—
58	C₄×C₁₉⋊C₈	—	2	2	no	—
59	C₄×C₁₉⋊C₈	—	2	2	no	—
60	C₂⁴×C₁₉	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	—	4	4	no	—
62	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₁₉	#41	1	2	yes	—
77	C₂²×C₄×C₁₉	#41	1	2	yes	—
78	M₁₆×C₁₉	—	4	4	no	—
79	M₁₆×C₁₉	—	4	2	no	—
80	M₁₆×C₁₉	—	4	2	no	—
81	C₂×C₈×C₁₉	—	2	4	no	—
82	C₂×C₈×C₁₉	—	2	8	no	—
83	C₄²×C₁₉	C₄²	1	2	yes	—
84	C₄²×C₁₉	C₄²	1	2	yes	—
85	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
88	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
89	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
90	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
91	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
92	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
93	#64520	—	2	4	no	—
94	#64520	—	2	4	no	—
95	#64520	—	2	4	no	—
96	#64520	—	2	4	no	—
97	#64520	—	4	4	no	—
98	#64520	—	4	2	no	—
99	#64520	—	4	2	no	—
100	C₂×C₁₉⋊C₈	—	2	4	no	—
101	C₂×C₁₉⋊C₈	—	2	4	no	—
102	C₂×C₁₉⋊C₈	—	2	4	no	—
103	C₂×C₁₉⋊C₈	—	2	8	no	—
104	C₂×C₁₉⋊C₈	C₄⋊C₄	1	8	yes	—
105	C₄⋊#338	—	19	1	no	maximal, Sylow
106	C₂³×C₁₉	—	4	2	no	—
107	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂³×C₁₉	#125	1	2	yes	—
111	C₂³×C₁₉	C₄⋊Q₈	1	2	yes	—
112	C₂×C₄×C₁₉	—	2	4	no	—
113	C₂×C₄×C₁₉	—	2	4	no	—
114	C₂×C₄×C₁₉	—	2	4	no	—
115	C₂×C₄×C₁₉	—	4	2	no	—
116	C₂×C₄×C₁₉	—	4	2	no	—
117	C₂×C₄×C₁₉	—	4	2	no	—
118	C₂×C₄×C₁₉	#95	1	2	yes	—
119	C₂×C₄×C₁₉	#95	1	2	yes	—
120	C₂×C₄×C₁₉	#95	1	2	yes	—
121	C₂×C₄×C₁₉	#95	1	2	yes	—
122	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₁₉	#117	1	2	yes	—
128	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
130	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
131	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
132	C₂×C₄×C₁₉	#121	1	2	yes	—
133	C₂×C₄×C₁₉	#121	1	2	yes	—
134	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
135	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
136	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
137	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
138	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
139	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
140	C₈×C₁₉	—	4	4	no	—
141	C₁₉⋊C₈	—	2	8	no	—
142	C₁₉⋊C₈	—	4	4	no	—
143	C₂×#423	—	19	1	no	—
144	C₂×#423	D₁₉	1	1	yes	characteristic
145	C₂×C₄×M₁₆	—	19	1	no	—
146	C₂×#338	—	19	4	no	—
147	C₂²×C₁₉	—	2	2	no	—
148	C₂²×C₁₉	—	2	2	no	—
149	C₂²×C₁₉	—	2	2	no	—
150	C₂²×C₁₉	—	4	2	no	—
151	C₂²×C₁₉	—	4	2	no	—
152	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
153	C₂²×C₁₉	#382	1	1	yes	characteristic
154	C₂²×C₁₉	#384	1	1	yes	characteristic
155	C₂²×C₁₉	#389	1	2	yes	—
156	C₂²×C₁₉	#389	1	2	yes	—
157	C₄×C₁₉	—	2	2	no	—
158	C₄×C₁₉	—	2	2	no	—
159	C₄×C₁₉	—	2	2	no	—
160	C₄×C₁₉	#338	1	4	yes	complemented
161	C₄×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
162	C₄×C₁₉	#380	1	2	yes	—
163	C₄×C₁₉	#390	1	1	yes	characteristic
164	C₄×C₁₉	#390	1	1	yes	characteristic
165	C₄×C₁₉	#391	1	1	yes	characteristic
166	C₄×C₁₉	#391	1	1	yes	characteristic
167	C₂²×M₁₆	—	19	2	no	—
168	C₂²×M₁₆	—	19	2	no	—
169	C₂²×M₁₆	C₂×D₁₉	1	2	yes	—
170	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
171	#423	—	2	1	no	—
172	#423	—	2	1	no	—
173	#423	—	2	2	no	—
174	#423	—	38	1	no	—
175	#423	—	38	1	no	—
176	#423	—	38	2	no	—
177	C₂×C₄⋊C₈	—	19	4	no	—
178	C₂×C₄⋊C₈	Dic₁₉	1	4	yes	—
179	C₄×M₁₆	—	38	1	no	—
180	C₄×M₁₆	—	38	1	no	—
181	C₄×M₁₆	—	38	1	no	—
182	C₄×M₁₆	—	38	1	no	—
183	C₂×#97	—	19	2	no	—
184	C₂×C₄×C₈	—	19	2	no	—
185	#338	—	38	8	no	—
186	C₂×C₁₉	—	2	1	no	—
187	C₂×C₁₉	—	2	1	no	—
188	C₂×C₁₉	—	2	2	no	—
189	C₂×C₁₉	—	2	2	no	—
190	C₂×C₁₉	C₄⋊#95	1	1	yes	characteristic
191	C₂×C₁₉	C₂×#338	1	2	yes	—
192	C₂³×C₄	Dic₃₈	1	2	yes	—
193	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
194	C₂×M₁₆	—	2	4	no	—
195	C₂×M₁₆	—	2	4	no	—
196	C₂×M₁₆	—	2	2	no	—
197	C₂×M₁₆	—	2	2	no	—
198	C₂×M₁₆	—	19	4	no	—
199	C₂×M₁₆	—	19	4	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	C₂×M₁₆	—	38	2	no	—
205	C₂×M₁₆	—	38	4	no	—
206	C₂×M₁₆	—	38	4	no	—
207	C₂×M₁₆	—	38	2	no	—
208	C₂²×C₈	—	19	4	no	—
209	C₂²×C₈	—	19	4	no	—
210	C₂²×C₈	C₂×Dic₁₉	1	4	yes	—
211	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
212	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
213	C₂×C₄²	C₄×D₁₉	1	2	yes	—
214	C₂×C₄²	D₇₆	1	1	yes	characteristic
215	C₂×C₄²	D₇₆	1	1	yes	characteristic
216	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
217	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
218	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
219	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
220	C₄⋊C₈	—	2	4	no	—
221	C₄⋊C₈	—	2	4	no	—
222	C₄⋊C₈	—	38	4	no	—
223	C₄⋊C₈	—	38	4	no	—
224	#97	—	38	2	no	—
225	#97	—	38	2	no	—
226	C₄×C₈	—	38	2	no	—
227	C₄×C₈	—	38	2	no	—
228	C₁₉	C₄⋊#338	1	1	yes	characteristic, complemented, Sylow
229	C₂⁴	C₂×Dic₃₈	1	1	yes	characteristic
230	C₂²×C₄	—	4	4	no	—
231	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
232	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
233	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
234	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
235	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
236	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
237	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
240	C₂²×C₄	D₄×D₁₉	1	2	yes	—
241	C₂²×C₄	#64543	1	2	yes	—
242	C₂²×C₄	Q₈×D₁₉	1	2	yes	—
243	C₂²×C₄	#64545	1	2	yes	—
244	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
247	M₁₆	—	4	2	no	—
248	M₁₆	—	4	4	no	—
249	M₁₆	—	4	2	no	—
250	M₁₆	—	38	4	no	—
251	M₁₆	—	38	4	no	—
252	M₁₆	—	38	4	no	—
253	M₁₆	—	38	4	no	—
254	M₁₆	—	76	2	no	—
255	M₁₆	—	76	2	no	—
256	M₁₆	—	76	4	no	—
257	C₂×C₈	—	2	8	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	19	8	no	—
260	C₂×C₈	—	38	4	no	—
261	C₂×C₈	—	38	4	no	—
262	C₂×C₈	—	38	8	no	—
263	C₂×C₈	—	38	4	no	—
264	C₄²	C₄×Dic₁₉	1	2	yes	—
265	C₄²	C₄×Dic₁₉	1	2	yes	—
266	C₄²	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
267	C₄²	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
268	C₄²	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
269	C₄²	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
270	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
271	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
272	C₄²	C₂²⋊Dic₁₉	1	2	yes	—
273	C₄²	C₂²⋊Dic₁₉	1	2	yes	—
274	C₂³	—	4	2	no	—
275	C₂³	#10597334	1	2	yes	—
276	C₂³	#10597336	1	2	yes	—
277	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
278	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
279	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	C₁₉⋊#95	1	2	yes	—
287	C₂×C₄	C₁₉⋊#95	1	2	yes	—
288	C₂×C₄	C₁₉⋊#95	1	2	yes	—
289	C₂×C₄	C₁₉⋊#95	1	2	yes	—
290	C₂×C₄	#10597333	1	2	yes	—
291	C₂×C₄	C₄⋊C₄×D₁₉	1	2	yes	—
292	C₂×C₄	#10597338	1	2	yes	—
293	C₂×C₄	#10597339	1	2	yes	—
294	C₂×C₄	#10597341	1	2	yes	—
295	C₂×C₄	#10597343	1	2	yes	—
296	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
297	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
298	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
299	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
300	C₂×C₄	#10597382	1	1	yes	characteristic
301	C₂×C₄	#10597384	1	2	yes	—
302	C₂×C₄	#10597386	1	1	yes	characteristic
303	C₂×C₄	#10597390	1	1	yes	characteristic
304	C₂×C₄	Q₈×Dic₁₉	1	2	yes	—
305	C₂×C₄	#10597392	1	2	yes	—
306	C₂×C₄	#10597393	1	1	yes	characteristic
307	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
308	C₈	—	4	4	no	—
309	C₈	—	38	8	no	—
310	C₈	—	76	4	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
317	C₂²	#11926409	1	1	yes	characteristic
318	C₂²	#11926413	1	1	yes	characteristic
319	C₂²	#11926440	1	2	yes	—
320	C₂²	#11926443	1	2	yes	—
321	C₄	—	2	2	no	—
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	C₁₉⋊#338	1	4	yes	complemented
325	C₄	C₄⋊C₄×Dic₁₉	1	2	yes	—
326	C₄	#11926401	1	2	yes	—
327	C₄	#11926445	1	1	yes	characteristic
328	C₄	#11926447	1	1	yes	characteristic
329	C₄	#11926449	1	1	yes	characteristic
330	C₄	#11926451	1	1	yes	characteristic
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	2	2	no	—
334	C₂	—	2	2	no	—
335	C₂	#14561864	1	1	yes	characteristic
336	C₂	C₂×C₁₉⋊#338	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×#423	C₂
socle	C₂²×C₁₉	C₂×#95
prime radical	C₂⁴×C₁₉	C₂×Q₈
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#423	D₁₉
hypocenter	C₁₉	C₄⋊#338
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
160. C₄×C₁₉	#338	semidirect
228. C₁₉	C₄⋊#338	semidirect
324. C₄	C₁₉⋊#338	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926568	2
C₂×C₄×#64520	2
C₂×C₁₉⋊#338	2
C₂×C₁₉×#423	2
C₄⋊#338	19