Subgroups — #18075418

Order 4864.

462 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	#14562383	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11926297	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562022	C₂	1	2	yes	maximal
4	#14561486	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₉×#131	C₂	1	1	yes	maximal, characteristic
6	#11926597	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
8	#11926297	—	2	1	no	—
9	#11926297	—	2	1	no	—
10	#11926297	—	2	1	no	—
11	#11926295	—	2	1	no	—
12	Q₈⋊(C₁₉⋊C₈)	—	2	2	no	—
13	D₄⋊(C₁₉⋊C₈)	—	2	2	no	—
14	C₂×C₁₉×#131	C₄	1	1	yes	characteristic
15	C₂×C₁₉×#131	C₄	1	1	yes	characteristic
16	C₄×#41×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#405	—	2	1	no	—
18	C₁₉×#405	—	2	1	no	—
19	C₁₉×#405	C₄	1	1	yes	characteristic
20	C₁₉×#405	C₄	1	1	yes	characteristic
21	C₄×M₁₆×C₁₉	—	2	1	no	—
22	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₁₉	—	2	1	no	—
24	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
25	C₂×#41×C₁₉	C₂×C₄	1	1	yes	characteristic
26	C₁₉×#131	—	2	2	no	—
27	C₁₉×#131	—	2	2	no	—
28	C₁₉×#131	—	2	2	no	—
29	C₁₉×#131	C₈	1	4	yes	complemented
30	C₂×M₁₆×C₁₉	—	2	1	no	—
31	C₂×M₁₆×C₁₉	—	2	1	no	—
32	C₂×M₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂×M₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₉	—	2	1	no	—
35	C₂²×C₈×C₁₉	—	2	1	no	—
36	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
38	C₄×Q₈×C₁₉	D₄	1	1	yes	characteristic
39	C₄×Q₈×C₁₉	Q₈	1	1	yes	characteristic
40	C₄×D₄×C₁₉	—	2	1	no	—
41	C₄×D₄×C₁₉	—	2	1	no	—
42	C₄×D₄×C₁₉	D₄	1	1	yes	characteristic
43	C₄×D₄×C₁₉	Q₈	1	1	yes	characteristic
44	C₁₉×#117	—	2	1	no	—
45	C₁₉×#117	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₁₉	—	2	1	no	—
47	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
48	C₂²×C₁₉⋊C₈	—	2	2	no	—
49	C₁₉×#97	—	2	1	no	—
50	C₁₉×#97	—	2	1	no	—
51	C₁₉×#97	—	2	1	no	—
52	C₄×C₈×C₁₉	—	2	1	no	—
53	C₄×C₈×C₁₉	—	2	1	no	—
54	C₄×C₈×C₁₉	—	2	1	no	—
55	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
56	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
57	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
58	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
59	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
60	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
61	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
62	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
63	C₄×C₁₉⋊C₈	—	2	2	no	—
64	#41×C₁₉	—	2	1	no	—
65	#41×C₁₉	—	2	1	no	—
66	#41×C₁₉	C₂×C₈	1	2	yes	—
67	#41×C₁₉	M₁₆	1	2	yes	—
68	C₂×Q₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
69	C₂×D₄×C₁₉	—	2	1	no	—
70	C₂×D₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	—	2	1	no	—
72	C₂²×C₄×C₁₉	—	2	1	no	—
73	C₂²×C₄×C₁₉	—	2	1	no	—
74	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
77	M₁₆×C₁₉	—	2	4	no	—
78	M₁₆×C₁₉	—	2	1	no	—
79	M₁₆×C₁₉	—	2	1	no	—
80	M₁₆×C₁₉	—	2	4	no	—
81	M₁₆×C₁₉	C₂×C₈	1	2	yes	—
82	M₁₆×C₁₉	M₁₆	1	2	yes	—
83	C₂×C₈×C₁₉	—	2	4	no	—
84	C₂×C₈×C₁₉	—	2	1	no	—
85	C₂×C₈×C₁₉	—	2	1	no	—
86	C₂×C₈×C₁₉	—	2	1	no	—
87	C₂×C₈×C₁₉	—	2	1	no	—
88	C₂×C₈×C₁₉	—	2	4	no	—
89	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₉	C₂×C₈	1	2	yes	—
94	C₂×C₈×C₁₉	M₁₆	1	2	yes	—
95	C₄⋊C₄×C₁₉	—	2	1	no	—
96	C₄⋊C₄×C₁₉	—	2	1	no	—
97	C₄⋊C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
98	C₄⋊C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
99	C₂²⋊C₄×C₁₉	—	2	1	no	—
100	C₂²⋊C₄×C₁₉	—	2	1	no	—
101	C₂²⋊C₄×C₁₉	—	2	1	no	—
102	C₄²×C₁₉	—	2	1	no	—
103	C₄²×C₁₉	—	2	1	no	—
104	C₄²×C₁₉	—	2	1	no	—
105	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
109	C₂×C₁₉⋊C₈	—	4	2	no	—
110	C₂×C₁₉⋊C₈	—	4	2	no	—
111	C₂×C₁₉⋊C₈	—	8	2	no	—
112	#8605	—	19	1	no	maximal, Sylow
113	C₂³×C₁₉	—	2	1	no	—
114	C₂³×C₁₉	#117	1	1	yes	characteristic
115	Q₈×C₁₉	—	2	1	no	—
116	Q₈×C₁₉	C₄⋊C₈	1	2	yes	—
117	D₄×C₁₉	—	2	2	no	—
118	D₄×C₁₉	—	2	2	no	—
119	D₄×C₁₉	—	2	1	no	—
120	D₄×C₁₉	C₄⋊C₈	1	2	yes	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	2	2	no	—
123	C₂×C₄×C₁₉	—	2	2	no	—
124	C₂×C₄×C₁₉	—	2	1	no	—
125	C₂×C₄×C₁₉	—	2	1	no	—
126	C₂×C₄×C₁₉	—	2	1	no	—
127	C₂×C₄×C₁₉	—	2	1	no	—
128	C₂×C₄×C₁₉	—	4	1	no	—
129	C₂×C₄×C₁₉	—	4	2	no	—
130	C₂×C₄×C₁₉	—	4	1	no	—
131	C₂×C₄×C₁₉	—	4	1	no	—
132	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
135	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
136	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
137	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
138	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₄×C₁₉	C₂×M₁₆	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	C₈×C₁₉	—	2	2	no	—
143	C₈×C₁₉	—	2	1	no	—
144	C₈×C₁₉	—	2	2	no	—
145	C₈×C₁₉	—	2	1	no	—
146	C₈×C₁₉	C₂²⋊C₈	1	2	yes	—
147	C₈×C₁₉	C₂²⋊C₈	1	2	yes	—
148	C₁₉⋊C₈	—	8	2	no	—
149	C₁₉⋊C₈	—	8	2	no	—
150	C₄×#131	D₁₉	1	1	yes	characteristic
151	#1564	—	19	1	no	—
152	C₂×#335	—	19	1	no	—
153	#1477	—	19	2	no	—
154	#1283	—	19	2	no	—
155	C₂²×C₁₉	—	2	1	no	—
156	C₂²×C₁₉	—	2	1	no	—
157	C₂²×C₁₉	—	2	2	no	—
158	C₂²×C₁₉	—	4	1	no	—
159	C₂²×C₁₉	—	4	1	no	—
160	C₂²×C₁₉	#380	1	1	yes	characteristic
161	C₂²×C₁₉	#431	1	1	yes	characteristic
162	C₂²×C₁₉	#432	1	1	yes	characteristic
163	C₄×C₁₉	—	2	2	no	—
164	C₄×C₁₉	—	2	1	no	—
165	C₄×C₁₉	—	2	1	no	—
166	C₄×C₁₉	—	2	1	no	—
167	C₄×C₁₉	—	4	1	no	—
168	C₄×C₁₉	—	4	1	no	—
169	C₄×C₁₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
170	C₄×C₁₉	#407	1	1	yes	characteristic
171	C₄×C₁₉	C₂×C₄⋊C₈	1	1	yes	characteristic
172	C₄×C₁₉	#423	1	1	yes	characteristic
173	C₄×C₁₉	#472	1	1	yes	characteristic
174	C₄×C₁₉	#473	1	1	yes	characteristic
175	C₂×#131	Dic₁₉	1	1	yes	characteristic
176	C₂×#131	Dic₁₉	1	1	yes	characteristic
177	C₄×#41	C₂×D₁₉	1	1	yes	characteristic
178	#431	—	19	2	no	—
179	C₂×C₄⋊C₈	—	19	2	no	—
180	#405	—	2	1	no	—
181	#405	—	2	1	no	—
182	#405	Dic₁₉	1	1	yes	characteristic
183	#405	Dic₁₉	1	1	yes	characteristic
184	C₄×M₁₆	—	2	1	no	—
185	C₄×M₁₆	C₂×D₁₉	1	1	yes	characteristic
186	C₂×C₄×C₈	—	2	1	no	—
187	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
188	#335	—	38	1	no	—
189	#335	—	38	1	no	—
190	#335	—	38	1	no	—
191	#334	—	38	1	no	—
192	Q₈⋊C₈	—	38	2	no	—
193	D₄⋊C₈	—	38	2	no	—
194	C₂×C₁₉	—	2	1	no	—
195	C₂×C₁₉	—	2	1	no	—
196	C₂×C₁₉	—	4	2	no	—
197	C₂×C₁₉	#1797	1	1	yes	characteristic
198	C₂×C₁₉	#1814	1	1	yes	characteristic
199	C₂×C₁₉	#1816	1	1	yes	characteristic
200	C₂×#41	C₂×Dic₁₉	1	1	yes	characteristic
201	#131	—	2	2	no	—
202	#131	—	2	2	no	—
203	#131	—	2	2	no	—
204	#131	C₁₉⋊C₈	1	4	yes	complemented
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	—	2	1	no	—
207	C₂×M₁₆	C₂×Dic₁₉	1	1	yes	characteristic
208	C₂×M₁₆	C₂×Dic₁₉	1	1	yes	characteristic
209	C₂²×C₈	—	2	1	no	—
210	C₂²×C₈	—	2	1	no	—
211	C₂²×C₈	—	38	2	no	—
212	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
213	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
214	C₄×Q₈	Dic₃₈	1	1	yes	characteristic
215	C₄×Q₈	D₇₆	1	1	yes	characteristic
216	C₄×D₄	—	2	1	no	—
217	C₄×D₄	—	2	1	no	—
218	C₄×D₄	Dic₃₈	1	1	yes	characteristic
219	C₄×D₄	D₇₆	1	1	yes	characteristic
220	#117	—	2	1	no	—
221	#117	C₂×Dic₁₉	1	1	yes	characteristic
222	C₂×C₄²	—	2	1	no	—
223	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
224	C₄⋊C₈	—	38	2	no	—
225	C₄⋊C₈	—	38	2	no	—
226	C₄⋊C₈	—	38	2	no	—
227	C₂²⋊C₈	—	38	2	no	—
228	#97	—	2	1	no	—
229	#97	—	2	1	no	—
230	#97	—	2	1	no	—
231	C₄×C₈	—	2	1	no	—
232	C₄×C₈	—	2	1	no	—
233	C₄×C₈	—	2	1	no	—
234	C₄×C₈	—	38	2	no	—
235	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
236	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
237	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
238	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
239	C₁₉	#8605	1	1	yes	characteristic, complemented, Sylow
240	#41	—	2	1	no	—
241	#41	—	2	1	no	—
242	#41	C₂×C₁₉⋊C₈	1	2	yes	—
243	#41	#64520	1	2	yes	—
244	C₂×Q₈	C₄⋊Dic₁₉	1	1	yes	characteristic
245	C₂×D₄	—	2	1	no	—
246	C₂×D₄	C₄⋊Dic₁₉	1	1	yes	characteristic
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	#64541	1	1	yes	characteristic
251	C₂²×C₄	#64541	1	1	yes	characteristic
252	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
253	M₁₆	—	2	4	no	—
254	M₁₆	—	2	1	no	—
255	M₁₆	—	2	1	no	—
256	M₁₆	—	2	4	no	—
257	M₁₆	C₂×C₁₉⋊C₈	1	2	yes	—
258	M₁₆	#64520	1	2	yes	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	76	2	no	—
266	C₂×C₈	—	76	2	no	—
267	C₂×C₈	—	152	2	no	—
268	C₂×C₈	C₂×C₁₉⋊C₈	1	2	yes	—
269	C₂×C₈	#64520	1	2	yes	—
270	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
271	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
272	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
273	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
277	C₄⋊C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	2	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	2	1	no	—
284	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
285	C₄²	C₂×D₇₆	1	1	yes	characteristic
286	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
287	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
288	C₂³	—	2	1	no	—
289	C₂³	#10597372	1	1	yes	characteristic
290	Q₈	—	2	1	no	—
291	Q₈	C₄⋊(C₁₉⋊C₈)	1	2	yes	—
292	D₄	—	2	2	no	—
293	D₄	—	2	2	no	—
294	D₄	—	2	1	no	—
295	D₄	C₄⋊(C₁₉⋊C₈)	1	2	yes	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	C₂²×C₁₉⋊C₈	1	1	yes	characteristic
308	C₂×C₄	C₂×#64520	1	1	yes	characteristic
309	C₂×C₄	C₂×#64520	1	1	yes	characteristic
310	C₂×C₄	C₂×#64520	1	1	yes	characteristic
311	C₂×C₄	#10597370	1	1	yes	characteristic
312	C₂×C₄	#10597370	1	1	yes	characteristic
313	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
314	C₂×C₄	#10597376	1	1	yes	characteristic
315	C₂×C₄	#10597376	1	1	yes	characteristic
316	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
317	C₈	—	2	2	no	—
318	C₈	—	2	1	no	—
319	C₈	—	2	2	no	—
320	C₈	—	2	1	no	—
321	C₈	—	152	2	no	—
322	C₈	—	152	2	no	—
323	C₈	C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
324	C₈	C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	#11926404	1	1	yes	characteristic
331	C₂²	#11926597	1	1	yes	characteristic
332	C₂²	#11926601	1	1	yes	characteristic
333	C₄	—	2	2	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	C₂×C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
340	C₄	#11926508	1	1	yes	characteristic
341	C₄	C₂×C₄⋊(C₁₉⋊C₈)	1	1	yes	characteristic
342	C₄	#11926568	1	1	yes	characteristic
343	C₄	#11926814	1	1	yes	characteristic
344	C₄	#11926819	1	1	yes	characteristic
345	C₂	—	2	1	no	—
346	C₂	—	2	1	no	—
347	C₂	—	4	2	no	—
348	C₂	#14563615	1	1	yes	characteristic
349	C₂	#14563689	1	1	yes	characteristic
350	C₂	#14563699	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597376
commutator	C₂×C₄×C₁₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₄×C₁₉×#131	C₂
socle	C₂²×C₁₉	#380
prime radical	C₂×#41×C₁₉	C₂×C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄×#131	D₁₉
hypocenter	C₁₉	#8605
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#131

Complements

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

N	H ≅ G/N	type
1. #14562383	C₂	semidirect
2. C₂×#11926297	C₂	semidirect
4. #14561486	C₂	semidirect
29. C₁₉×#131	C₈	semidirect
204. #131	C₁₉⋊C₈	semidirect
239. C₁₉	#8605	semidirect

Maximal subgroups

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

subgroup	index
#14562383	2
C₂×#11926297	2
#14562022	2
#14561486	2
C₄×C₁₉×#131	2
#8605	19