Subgroups — #18087750

Order 4864.

842 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926500	C₂	1	2	yes	maximal
2	C₂×#11926500	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926314	C₂	1	2	yes	maximal
4	#14563374	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1740	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	2	yes	—
7	C₂²×#64520	C₂²	1	1	yes	characteristic
8	C₂³×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
9	#11926500	—	2	4	no	—
10	#11926500	—	2	2	no	—
11	#11926500	—	2	2	no	—
12	#11926500	C₄	1	8	yes	complemented
13	C₂×#10597260	—	2	2	no	—
14	C₂×#10597260	C₄	1	2	yes	complemented
15	C₂×C₄×C₁₉⋊C₈	—	2	2	no	—
16	C₂×C₄×C₁₉⋊C₈	C₄	1	2	yes	complemented
17	#11926314	C₄	1	8	yes	complemented
18	#11926313	C₄	1	4	yes	complemented
19	C₂×C₁₉×#117	C₂²	1	2	yes	—
20	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
21	C₁₉×#342	C₄	1	4	yes	—
22	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
23	C₁₉×#117	—	2	2	no	—
24	C₁₉×#117	—	2	2	no	—
25	C₁₉×#117	—	2	2	no	—
26	C₁₉×#117	—	2	2	no	—
27	C₁₉×#117	C₂×C₄	1	4	yes	—
28	C₁₉×#117	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
32	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₁₉	—	2	2	no	—
34	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
35	C₂×#64520	—	2	2	no	—
36	C₂×#64520	—	2	2	no	—
37	C₂×#64520	—	2	1	no	—
38	C₂×#64520	—	2	1	no	—
39	C₂×#64520	—	2	2	no	—
40	C₂×#64520	—	2	2	no	—
41	C₂×#64520	C₂×C₄	1	4	yes	—
42	C₂×#64520	C₂×C₄	1	4	yes	—
43	C₂×#64520	C₂×C₄	1	4	yes	—
44	C₂×#64520	C₂×C₄	1	4	yes	—
45	C₂²×C₁₉⋊C₈	—	2	2	no	—
46	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
47	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
50	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
51	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
52	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
53	C₂²×C₁₉⋊C₈	Q₈	1	2	yes	—
54	#10597260	—	2	8	no	—
55	#10597260	—	2	4	no	—
56	C₄×C₁₉⋊C₈	—	2	8	no	—
57	C₄×C₁₉⋊C₈	—	2	2	no	—
58	C₄×C₁₉⋊C₈	—	2	2	no	—
59	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	—	2	4	no	—
61	C₂²×C₄×C₁₉	C₄²	1	4	yes	—
62	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	#41	1	2	yes	—
70	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
72	C₄⋊C₄×C₁₉	—	2	4	no	—
73	C₄⋊C₄×C₁₉	—	2	4	no	—
74	C₄⋊C₄×C₁₉	—	2	4	no	—
75	C₂²⋊C₄×C₁₉	—	2	4	no	—
76	C₂²⋊C₄×C₁₉	—	2	4	no	—
77	C₂²⋊C₄×C₁₉	C₄²	1	8	yes	—
78	C₄²×C₁₉	—	2	4	no	—
79	C₄²×C₁₉	—	2	2	no	—
80	C₄²×C₁₉	—	2	2	no	—
81	C₄²×C₁₉	—	2	4	no	—
82	#64520	—	2	4	no	—
83	#64520	—	2	4	no	—
84	#64520	—	2	4	no	—
85	#64520	—	2	4	no	—
86	#64520	—	2	4	no	—
87	#64520	C₄²	1	8	yes	—
88	C₂×C₁₉⋊C₈	—	2	4	no	—
89	C₂×C₁₉⋊C₈	—	2	2	no	—
90	C₂×C₁₉⋊C₈	—	2	2	no	—
91	C₂×C₁₉⋊C₈	—	2	4	no	—
92	C₂×C₁₉⋊C₈	—	2	4	no	—
93	C₂×C₁₉⋊C₈	C₄²	1	4	yes	—
94	C₂×C₁₉⋊C₈	C₄²	1	2	yes	—
95	C₂×C₁₉⋊C₈	C₄²	1	2	yes	—
96	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	4	yes	—
97	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	4	yes	—
100	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	2	yes	—
101	C₂×C₁₉⋊C₈	C₄⋊C₄	1	2	yes	—
102	C₂×C₁₉⋊C₈	C₄⋊C₄	1	2	yes	—
103	C₂×C₁₉⋊C₈	C₄⋊C₄	1	4	yes	—
104	C₂×C₁₉⋊C₈	C₄⋊C₄	1	4	yes	—
105	#10711	—	19	1	no	maximal, Sylow
106	C₂³×C₁₉	—	4	2	no	—
107	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
108	C₂³×C₁₉	C₂×C₄²	1	2	yes	—
109	C₂³×C₁₉	C₄×D₄	1	2	yes	—
110	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
111	C₂³×C₁₉	C₄×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₁₉	—	2	4	no	—
113	C₂×C₄×C₁₉	—	2	8	no	—
114	C₂×C₄×C₁₉	—	4	8	no	—
115	C₂×C₄×C₁₉	—	4	4	no	—
116	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₂×C₄²	1	2	yes	—
119	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
120	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
124	C₂×C₄×C₁₉	#117	1	2	yes	—
125	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
126	C₂×C₄×C₁₉	#117	1	2	yes	—
127	C₂×C₄×C₁₉	C₄×D₄	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×D₄	1	2	yes	—
131	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
134	C₁₉⋊C₈	—	2	8	no	—
135	C₁₉⋊C₈	—	2	4	no	—
136	C₁₉⋊C₈	#95	1	2	yes	—
137	C₁₉⋊C₈	#95	1	4	yes	—
138	C₁₉⋊C₈	#95	1	2	yes	—
139	C₂×#405	—	19	1	no	—
140	C₂×#405	—	19	2	no	—
141	C₂×#344	—	19	2	no	—
142	#1742	—	19	1	no	—
143	#1740	D₁₉	1	1	yes	characteristic
144	C₂²×C₁₉	—	2	2	no	—
145	C₂²×C₁₉	—	2	4	no	—
146	C₂²×C₁₉	—	4	2	no	—
147	C₂²×C₁₉	—	4	1	no	—
148	C₂²×C₁₉	—	4	1	no	—
149	C₂²×C₁₉	C₄³	1	1	yes	characteristic
150	C₂²×C₁₉	C₄×C₂²⋊C₄	1	2	yes	—
151	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
153	C₂²×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
154	C₄×C₁₉	—	4	8	no	—
155	C₄×C₁₉	—	4	4	no	—
156	C₄×C₁₉	C₂×#95	1	1	yes	characteristic
157	C₄×C₁₉	#376	1	1	yes	characteristic
158	C₄×C₁₉	C₄×C₂²⋊C₄	1	2	yes	—
159	C₄×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
160	C₄×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
161	C₄×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
162	C₂²×M₁₆	—	19	1	no	—
163	C₂²×M₁₆	—	19	2	no	—
164	C₂³×C₈	—	19	1	no	—
165	C₂×#117	C₂×D₁₉	1	2	yes	—
166	C₂×#117	C₂×D₁₉	1	1	yes	characteristic
167	#405	—	19	8	no	—
168	#405	—	38	2	no	—
169	#405	—	38	2	no	—
170	#405	—	38	4	no	—
171	C₂×#97	—	19	2	no	—
172	C₂×#97	—	38	2	no	—
173	C₂×C₄×C₈	—	19	2	no	—
174	C₂×C₄×C₈	—	38	2	no	—
175	#344	—	19	8	no	—
176	#343	—	19	4	no	—
177	#342	Dic₁₉	1	4	yes	—
178	C₂×C₁₉	—	2	2	no	—
179	C₂×C₁₉	—	2	1	no	—
180	C₂×C₁₉	—	2	1	no	—
181	C₂×C₁₉	—	2	2	no	—
182	C₂×C₁₉	C₄×#95	1	1	yes	characteristic
183	C₂×C₁₉	#1757	1	2	yes	—
184	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
185	C₂×M₁₆	—	19	4	no	—
186	C₂×M₁₆	—	19	4	no	—
187	C₂×M₁₆	—	19	4	no	—
188	C₂×M₁₆	—	19	4	no	—
189	C₂×M₁₆	—	38	2	no	—
190	C₂×M₁₆	—	38	2	no	—
191	C₂×M₁₆	—	38	2	no	—
192	C₂×M₁₆	—	38	1	no	—
193	C₂×M₁₆	—	38	1	no	—
194	C₂×M₁₆	—	38	2	no	—
195	C₂²×C₈	—	19	1	no	—
196	C₂²×C₈	—	19	1	no	—
197	C₂²×C₈	—	19	4	no	—
198	C₂²×C₈	—	19	2	no	—
199	C₂²×C₈	—	19	2	no	—
200	C₂²×C₈	—	19	2	no	—
201	C₂²×C₈	—	19	2	no	—
202	C₂²×C₈	—	19	2	no	—
203	C₂²×C₈	—	38	2	no	—
204	#117	—	2	2	no	—
205	#117	—	2	2	no	—
206	#117	—	2	2	no	—
207	#117	—	2	2	no	—
208	#117	C₄×D₁₉	1	4	yes	—
209	#117	C₄×D₁₉	1	4	yes	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	C₄×D₁₉	1	2	yes	—
212	C₂×C₂²⋊C₄	C₂×Dic₁₉	1	2	yes	—
213	C₂×C₂²⋊C₄	C₂×Dic₁₉	1	4	yes	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	C₄×D₁₉	1	2	yes	—
216	#97	—	38	4	no	—
217	#97	—	38	8	no	—
218	C₄×C₈	—	38	2	no	—
219	C₄×C₈	—	38	2	no	—
220	C₄×C₈	—	38	8	no	—
221	C₁₉	#10711	1	1	yes	characteristic, complemented, Sylow
222	C₂⁴	C₂²×Dic₁₉	1	1	yes	characteristic
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	C₄×Dic₁₉	1	4	yes	—
225	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
226	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
229	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
230	C₂²×C₄	D₄×D₁₉	1	2	yes	—
231	C₂²×C₄	#64543	1	2	yes	—
232	C₂²×C₄	#64543	1	1	yes	characteristic
233	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
234	C₂²×C₄	#64545	1	1	yes	characteristic
235	M₁₆	—	19	8	no	—
236	M₁₆	—	38	4	no	—
237	M₁₆	—	38	4	no	—
238	M₁₆	—	38	4	no	—
239	M₁₆	—	38	4	no	—
240	M₁₆	—	38	4	no	—
241	C₂×C₈	—	19	2	no	—
242	C₂×C₈	—	19	2	no	—
243	C₂×C₈	—	19	4	no	—
244	C₂×C₈	—	19	4	no	—
245	C₂×C₈	—	19	1	no	—
246	C₂×C₈	—	19	1	no	—
247	C₂×C₈	—	19	2	no	—
248	C₂×C₈	—	19	2	no	—
249	C₂×C₈	—	19	2	no	—
250	C₂×C₈	—	19	4	no	—
251	C₂×C₈	—	19	4	no	—
252	C₂×C₈	—	19	4	no	—
253	C₂×C₈	—	38	4	no	—
254	C₂×C₈	—	38	2	no	—
255	C₂×C₈	—	38	4	no	—
256	C₂×C₈	—	38	2	no	—
257	C₂×C₈	—	38	4	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₂²⋊C₄	—	2	4	no	—
262	C₂²⋊C₄	—	2	4	no	—
263	C₂²⋊C₄	C₄×Dic₁₉	1	8	yes	—
264	C₄²	—	2	2	no	—
265	C₄²	—	2	4	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	4	no	—
268	C₂³	—	4	2	no	—
269	C₂³	C₂×C₄×Dic₁₉	1	1	yes	characteristic
270	C₂³	C₂×C₄×Dic₁₉	1	2	yes	—
271	C₂³	D₄×Dic₁₉	1	2	yes	—
272	C₂³	D₄×Dic₁₉	1	1	yes	characteristic
273	C₂³	Q₈×Dic₁₉	1	1	yes	characteristic
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	4	8	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
279	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
280	C₂×C₄	C₄²×D₁₉	1	2	yes	—
281	C₂×C₄	#10597323	1	1	yes	characteristic
282	C₂×C₄	#10597323	1	2	yes	—
283	C₂×C₄	C₂²⋊C₄×D₁₉	1	2	yes	—
284	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
285	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
286	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
287	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
288	C₂×C₄	#10597333	1	1	yes	characteristic
289	C₂×C₄	#10597333	1	2	yes	—
290	C₂×C₄	C₄⋊C₄×D₁₉	1	2	yes	—
291	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
292	C₂×C₄	#10597338	1	1	yes	characteristic
293	C₂×C₄	#10597338	1	2	yes	—
294	C₂×C₄	#10597339	1	2	yes	—
295	C₂×C₄	#10597339	1	1	yes	characteristic
296	C₈	—	19	2	no	—
297	C₈	—	19	4	no	—
298	C₈	—	19	2	no	—
299	C₈	—	38	4	no	—
300	C₈	—	38	8	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	2	no	—
306	C₂²	C₄²×Dic₁₉	1	1	yes	characteristic
307	C₂²	C₂²⋊C₄×Dic₁₉	1	2	yes	—
308	C₂²	C₂²⋊C₄×Dic₁₉	1	1	yes	characteristic
309	C₂²	C₄⋊C₄×Dic₁₉	1	1	yes	characteristic
310	C₂²	C₄⋊C₄×Dic₁₉	1	2	yes	—
311	C₄	—	4	8	no	—
312	C₄	—	4	4	no	—
313	C₄	#95×D₁₉	1	1	yes	characteristic
314	C₄	#11926389	1	1	yes	characteristic
315	C₄	#11926391	1	2	yes	—
316	C₄	#11926391	1	1	yes	characteristic
317	C₄	#11926395	1	1	yes	characteristic
318	C₄	#11926395	1	2	yes	—
319	C₂	—	2	2	no	—
320	C₂	—	2	1	no	—
321	C₂	—	2	2	no	—
322	C₂	—	2	1	no	—
323	C₂	#95×Dic₁₉	1	1	yes	characteristic
324	C₂	#14563437	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₁₉
commutator	C₂²×C₁₉	C₄³
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#1740	C₂
socle	C₂²×C₁₉	C₄×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂²×C₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#1740	D₁₉
hypocenter	C₁₉	#10711
LCS	G ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1740

Complements

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

N	H ≅ G/N	type
12. #11926500	C₄	semidirect
14. C₂×#10597260	C₄	semidirect
16. C₂×C₄×C₁₉⋊C₈	C₄	semidirect
17. #11926314	C₄	semidirect
18. #11926313	C₄	semidirect
221. C₁₉	#10711	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926500	2
C₂×#11926314	2
#14563374	2
C₁₉×#1740	2
#10711	19