Subgroups — #18088144

Order 4864.

1062 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926299	C₂	1	2	yes	maximal
2	C₂×#11926299	C₂	1	2	yes	maximal
3	C₂×#11926299	C₂	1	2	yes	maximal
4	C₂×C₄³×C₁₉	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₁₉⋊C₈	C₂²	1	4	yes	—
6	#11926299	—	2	4	no	—
7	#11926299	—	2	4	no	—
8	#11926299	C₄	1	8	yes	complemented
9	C₂²×C₄²×C₁₉	C₄	1	2	yes	—
10	C₂²×C₄²×C₁₉	C₄	1	2	yes	—
11	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
14	C₄³×C₁₉	—	2	4	no	—
15	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
17	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₁₉	—	2	4	no	—
21	C₂×C₄²×C₁₉	—	2	4	no	—
22	C₂×C₄²×C₁₉	—	2	4	no	—
23	C₂×C₄²×C₁₉	—	2	4	no	—
24	C₂×C₄²×C₁₉	—	2	4	no	—
25	C₂×C₄²×C₁₉	—	2	2	no	—
26	C₂×C₄²×C₁₉	—	2	2	no	—
27	C₂×C₄²×C₁₉	—	2	2	no	—
28	C₂×C₄²×C₁₉	—	2	2	no	—
29	C₂×C₄²×C₁₉	—	2	2	no	—
30	C₂×C₄²×C₁₉	—	2	2	no	—
31	C₂×C₄²×C₁₉	C₈	1	8	yes	complemented
32	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₁₉	D₄	1	4	yes	—
34	C₂×C₄²×C₁₉	D₄	1	2	yes	—
35	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
36	C₂²×C₁₉⋊C₈	—	2	4	no	—
37	C₂²×C₁₉⋊C₈	—	2	4	no	—
38	C₂²×C₁₉⋊C₈	—	2	4	no	—
39	C₂²×C₁₉⋊C₈	—	2	4	no	—
40	C₂²×C₁₉⋊C₈	—	2	4	no	—
41	C₂²×C₁₉⋊C₈	—	2	4	no	—
42	C₂²×C₁₉⋊C₈	C₂×C₄	1	8	yes	—
43	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	—	2	1	no	—
45	C₂²×C₄×C₁₉	—	2	1	no	—
46	C₂²×C₄×C₁₉	—	2	2	no	—
47	C₂²×C₄×C₁₉	—	2	1	no	—
48	C₂²×C₄×C₁₉	—	2	1	no	—
49	C₂²×C₄×C₁₉	—	2	2	no	—
50	C₂²×C₄×C₁₉	—	2	2	no	—
51	C₂²×C₄×C₁₉	—	2	2	no	—
52	C₂²×C₄×C₁₉	—	2	2	no	—
53	C₂²×C₄×C₁₉	—	2	2	no	—
54	C₂²×C₄×C₁₉	—	2	2	no	—
55	C₂²×C₄×C₁₉	—	2	2	no	—
56	C₂²×C₄×C₁₉	—	2	2	no	—
57	C₂²×C₄×C₁₉	—	2	2	no	—
58	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
59	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
60	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
61	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₁₉	C₂×C₈	1	4	yes	—
63	C₂²×C₄×C₁₉	C₂×C₈	1	4	yes	—
64	C₂²×C₄×C₁₉	C₂×C₈	1	4	yes	—
65	C₂²×C₄×C₁₉	M₁₆	1	4	yes	—
66	C₂²×C₄×C₁₉	M₁₆	1	4	yes	—
67	C₂²×C₄×C₁₉	M₁₆	1	4	yes	—
68	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	#41	1	2	yes	—
73	C₂²×C₄×C₁₉	#41	1	2	yes	—
74	C₂²×C₄×C₁₉	#41	1	2	yes	—
75	C₂²×C₄×C₁₉	#41	1	2	yes	—
76	C₄²×C₁₉	—	2	16	no	—
77	C₄²×C₁₉	—	2	16	no	—
78	C₄²×C₁₉	—	2	8	no	—
79	C₄²×C₁₉	—	2	4	no	—
80	C₄²×C₁₉	—	2	4	no	—
81	C₄²×C₁₉	—	2	8	no	—
82	C₂×C₁₉⋊C₈	—	2	8	no	—
83	C₂×C₁₉⋊C₈	—	2	8	no	—
84	C₂×C₁₉⋊C₈	—	2	8	no	—
85	C₂×C₁₉⋊C₈	—	4	4	no	—
86	C₂×C₁₉⋊C₈	—	4	4	no	—
87	C₂×C₁₉⋊C₈	—	4	4	no	—
88	C₂×C₁₉⋊C₈	—	4	4	no	—
89	#10795	—	19	1	no	maximal, Sylow
90	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
91	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂³×C₁₉	#117	1	1	yes	characteristic
94	C₂³×C₁₉	#117	1	1	yes	characteristic
95	C₂³×C₁₉	#117	1	1	yes	characteristic
96	C₂³×C₁₉	#117	1	1	yes	characteristic
97	C₂³×C₁₉	C₂²×C₈	1	2	yes	—
98	C₂³×C₁₉	C₂×M₁₆	1	2	yes	—
99	C₂³×C₁₉	C₂×M₁₆	1	2	yes	—
100	C₂³×C₁₉	C₂×M₁₆	1	2	yes	—
101	C₂×C₄×C₁₉	—	2	2	no	—
102	C₂×C₄×C₁₉	—	2	2	no	—
103	C₂×C₄×C₁₉	—	2	4	no	—
104	C₂×C₄×C₁₉	—	2	4	no	—
105	C₂×C₄×C₁₉	—	2	2	no	—
106	C₂×C₄×C₁₉	—	2	2	no	—
107	C₂×C₄×C₁₉	—	2	4	no	—
108	C₂×C₄×C₁₉	—	2	4	no	—
109	C₂×C₄×C₁₉	—	2	4	no	—
110	C₂×C₄×C₁₉	—	2	4	no	—
111	C₂×C₄×C₁₉	—	2	4	no	—
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	—	2	4	no	—
116	C₂×C₄×C₁₉	—	2	4	no	—
117	C₂×C₄×C₁₉	—	2	4	no	—
118	C₂×C₄×C₁₉	—	2	4	no	—
119	C₂×C₄×C₁₉	—	2	4	no	—
120	C₂×C₄×C₁₉	—	2	4	no	—
121	C₂×C₄×C₁₉	C₄×C₈	1	4	yes	—
122	C₂×C₄×C₁₉	#97	1	4	yes	—
123	C₂×C₄×C₁₉	C₂²⋊C₈	1	8	yes	—
124	C₂×C₄×C₁₉	C₄⋊C₈	1	8	yes	—
125	C₂×C₄×C₁₉	#117	1	4	yes	—
126	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₁₉	C₄×D₄	1	4	yes	—
128	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
129	C₂×C₄×C₁₉	#121	1	2	yes	—
130	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
131	C₂×C₄×C₁₉	#123	1	4	yes	—
132	C₂×C₄×C₁₉	#124	1	2	yes	—
133	C₂×C₄×C₁₉	#125	1	2	yes	—
134	C₂×C₄×C₁₉	#126	1	4	yes	—
135	C₁₉⋊C₈	—	4	8	no	—
136	C₂×C₄³	D₁₉	1	1	yes	characteristic
137	C₂×#336	—	19	2	no	—
138	C₂×#336	—	19	2	no	—
139	C₂×#336	—	19	2	no	—
140	C₂²×C₁₉	#376	1	1	yes	characteristic
141	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
143	C₂²×C₁₉	#380	1	1	yes	characteristic
144	C₂²×C₁₉	#381	1	1	yes	characteristic
145	C₂²×C₁₉	#382	1	1	yes	characteristic
146	C₂²×C₁₉	#383	1	1	yes	characteristic
147	C₂²×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
148	C₂²×C₁₉	C₂×#97	1	1	yes	characteristic
149	C₂²×C₁₉	C₄×M₁₆	1	1	yes	characteristic
150	C₂²×C₁₉	C₄×M₁₆	1	1	yes	characteristic
151	C₂²×C₁₉	C₂×C₂²⋊C₈	1	2	yes	—
152	C₂²×C₁₉	#407	1	2	yes	—
153	C₂²×C₁₉	C₂×C₄⋊C₈	1	2	yes	—
154	C₂²×C₁₉	#423	1	2	yes	—
155	C₂²×C₁₉	#431	1	2	yes	—
156	C₂²×C₁₉	#431	1	2	yes	—
157	C₂²×C₁₉	#431	1	2	yes	—
158	C₂²×C₁₉	#431	1	2	yes	—
159	C₂²×C₁₉	#432	1	2	yes	—
160	C₂²×C₁₉	#432	1	2	yes	—
161	C₂²×C₁₉	#432	1	2	yes	—
162	C₂²×C₁₉	#432	1	2	yes	—
163	C₄×C₁₉	—	2	8	no	—
164	C₄×C₁₉	—	2	8	no	—
165	C₄×C₁₉	—	2	4	no	—
166	C₄×C₁₉	—	2	4	no	—
167	C₄×C₁₉	#387	1	4	yes	—
168	C₄×C₁₉	#388	1	4	yes	—
169	C₂³×C₈	—	19	4	no	—
170	C₂²×C₄²	Dic₁₉	1	2	yes	—
171	C₂²×C₄²	Dic₁₉	1	2	yes	—
172	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
173	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
174	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
175	C₄³	—	2	4	no	—
176	#336	—	19	8	no	—
177	#336	—	38	4	no	—
178	#336	—	38	4	no	—
179	C₂×C₁₉	#1434	1	1	yes	characteristic
180	C₂×C₁₉	#1744	1	1	yes	characteristic
181	C₂×C₁₉	#1745	1	1	yes	characteristic
182	C₂×C₁₉	C₄×C₂²⋊C₈	1	1	yes	characteristic
183	C₂×C₁₉	#1749	1	1	yes	characteristic
184	C₂×C₁₉	C₄×C₄⋊C₈	1	1	yes	characteristic
185	C₂×C₁₉	#1767	1	1	yes	characteristic
186	C₂×C₁₉	#1797	1	2	yes	—
187	C₂×C₁₉	#1817	1	2	yes	—
188	C₂×C₁₉	#1831	1	2	yes	—
189	C₂×C₁₉	#1839	1	2	yes	—
190	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
191	C₂³×C₄	C₂×Dic₁₉	1	2	yes	—
192	C₂³×C₄	C₂×Dic₁₉	1	2	yes	—
193	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
194	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
195	C₂²×C₈	—	19	8	no	—
196	C₂²×C₈	—	38	4	no	—
197	C₂²×C₈	—	38	4	no	—
198	C₂²×C₈	—	38	4	no	—
199	C₂²×C₈	—	38	4	no	—
200	C₂²×C₈	—	38	4	no	—
201	C₂²×C₈	—	38	4	no	—
202	C₂×C₄²	—	2	4	no	—
203	C₂×C₄²	—	2	4	no	—
204	C₂×C₄²	—	2	4	no	—
205	C₂×C₄²	—	2	4	no	—
206	C₂×C₄²	—	2	4	no	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	—	2	2	no	—
209	C₂×C₄²	—	2	2	no	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	—	2	2	no	—
212	C₂×C₄²	—	2	2	no	—
213	C₂×C₄²	C₁₉⋊C₈	1	8	yes	complemented
214	C₂×C₄²	Dic₃₈	1	2	yes	—
215	C₂×C₄²	C₄×D₁₉	1	4	yes	—
216	C₂×C₄²	D₇₆	1	2	yes	—
217	C₂×C₄²	C₂²⋊D₁₉	1	4	yes	—
218	C₁₉	#10795	1	1	yes	characteristic, complemented, Sylow
219	C₂⁴	C₂²×Dic₁₉	1	1	yes	characteristic
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	C₂×C₁₉⋊C₈	1	4	yes	—
235	C₂²×C₄	C₂×C₁₉⋊C₈	1	4	yes	—
236	C₂²×C₄	C₂×C₁₉⋊C₈	1	4	yes	—
237	C₂²×C₄	#64520	1	4	yes	—
238	C₂²×C₄	#64520	1	4	yes	—
239	C₂²×C₄	#64520	1	4	yes	—
240	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
241	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
242	C₂²×C₄	C₄⋊Dic₁₉	1	4	yes	—
243	C₂²×C₄	C₂²⋊Dic₁₉	1	4	yes	—
244	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
246	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
247	C₂²×C₄	#64541	1	2	yes	—
248	C₂²×C₄	#64541	1	2	yes	—
249	C₂²×C₄	#64541	1	2	yes	—
250	C₂²×C₄	#64541	1	2	yes	—
251	C₂²×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
252	C₂×C₈	—	38	8	no	—
253	C₂×C₈	—	38	8	no	—
254	C₂×C₈	—	38	8	no	—
255	C₂×C₈	—	76	4	no	—
256	C₂×C₈	—	76	4	no	—
257	C₂×C₈	—	76	4	no	—
258	C₂×C₈	—	76	4	no	—
259	C₄²	—	2	16	no	—
260	C₄²	—	2	4	no	—
261	C₄²	—	2	16	no	—
262	C₄²	—	2	4	no	—
263	C₄²	—	2	8	no	—
264	C₄²	—	2	8	no	—
265	C₂³	C₂²×C₁₉⋊C₈	1	2	yes	—
266	C₂³	C₂×#64520	1	2	yes	—
267	C₂³	C₂×#64520	1	2	yes	—
268	C₂³	C₂×#64520	1	2	yes	—
269	C₂³	C₂×C₄×Dic₁₉	1	1	yes	characteristic
270	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
271	C₂³	#10597372	1	1	yes	characteristic
272	C₂³	#10597372	1	1	yes	characteristic
273	C₂³	#10597372	1	1	yes	characteristic
274	C₂³	#10597372	1	1	yes	characteristic
275	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	C₄×C₁₉⋊C₈	1	4	yes	—
297	C₂×C₄	#10597260	1	4	yes	—
298	C₂×C₄	C₄⋊(C₁₉⋊C₈)	1	8	yes	—
299	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	8	yes	—
300	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
301	C₂×C₄	#10597316	1	2	yes	—
302	C₂×C₄	#10597318	1	4	yes	—
303	C₂×C₄	C₄×D₇₆	1	2	yes	—
304	C₂×C₄	#10597321	1	2	yes	—
305	C₂×C₄	#10597322	1	4	yes	—
306	C₂×C₄	#10597370	1	2	yes	—
307	C₂×C₄	C₄×C₂²⋊D₁₉	1	4	yes	—
308	C₂×C₄	#10597375	1	4	yes	—
309	C₂×C₄	#10597376	1	2	yes	—
310	C₈	—	76	8	no	—
311	C₂²	#11926390	1	1	yes	characteristic
312	C₂²	C₄×C₂²⋊Dic₁₉	1	1	yes	characteristic
313	C₂²	C₄×C₄⋊Dic₁₉	1	1	yes	characteristic
314	C₂²	#11926404	1	1	yes	characteristic
315	C₂²	#11926407	1	1	yes	characteristic
316	C₂²	#11926410	1	1	yes	characteristic
317	C₂²	#11926412	1	1	yes	characteristic
318	C₂²	C₂×C₄×C₁₉⋊C₈	1	1	yes	characteristic
319	C₂²	C₂×#10597260	1	1	yes	characteristic
320	C₂²	C₄×#64520	1	1	yes	characteristic
321	C₂²	C₄×#64520	1	1	yes	characteristic
322	C₂²	C₂×C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
323	C₂²	#11926508	1	2	yes	—
324	C₂²	C₂×C₄⋊(C₁₉⋊C₈)	1	2	yes	—
325	C₂²	#11926568	1	2	yes	—
326	C₂²	#11926597	1	2	yes	—
327	C₂²	#11926597	1	2	yes	—
328	C₂²	#11926597	1	2	yes	—
329	C₂²	#11926597	1	2	yes	—
330	C₂²	#11926601	1	2	yes	—
331	C₂²	#11926601	1	2	yes	—
332	C₂²	#11926601	1	2	yes	—
333	C₂²	#11926601	1	2	yes	—
334	C₄	—	2	8	no	—
335	C₄	—	2	4	no	—
336	C₄	—	2	4	no	—
337	C₄	—	2	8	no	—
338	C₄	#11926428	1	4	yes	—
339	C₄	#11926435	1	4	yes	—
340	C₂	#14561862	1	1	yes	characteristic
341	C₂	#14563381	1	1	yes	characteristic
342	C₂	#14563384	1	1	yes	characteristic
343	C₂	C₄×C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
344	C₂	#14563397	1	1	yes	characteristic
345	C₂	C₄×C₄⋊(C₁₉⋊C₈)	1	1	yes	characteristic
346	C₂	#14563488	1	1	yes	characteristic
347	C₂	#14563615	1	2	yes	—
348	C₂	#14563703	1	2	yes	—
349	C₂	#14563752	1	2	yes	—
350	C₂	#14563778	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₁₉
commutator	C₂²×C₁₉	C₂×C₄×C₈
Fitting	C₂×C₄³×C₁₉	C₂
socle, prime radical	C₂⁴×C₁₉	C₂²×C₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×C₄³	D₁₉
hypocenter	C₁₉	#10795
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
8. #11926299	C₄	semidirect
31. C₂×C₄²×C₁₉	C₈	semidirect
213. C₂×C₄²	C₁₉⋊C₈	semidirect
218. C₁₉	#10795	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926299	2
C₂×C₄³×C₁₉	2
#10795	19