Subgroups — #18075413

Order 4864.

462 conjugacy classes of subgroups.

The subgroup lattice has 346 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×#11926295	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	#11926295	—	2	1	no	—
10	#11926295	—	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	#8604	—	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	2	no	—
125	C₂×C₄×C₁₉	—	2	1	no	—
126	C₂×C₄×C₁₉	—	2	1	no	—
127	C₂×C₄×C₁₉	—	4	1	no	—
128	C₂×C₄×C₁₉	—	4	2	no	—
129	C₂×C₄×C₁₉	—	4	2	no	—
130	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
133	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
134	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
136	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
138	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
139	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
140	C₈×C₁₉	—	2	2	no	—
141	C₈×C₁₉	—	2	1	no	—
142	C₈×C₁₉	—	2	2	no	—
143	C₈×C₁₉	—	2	1	no	—
144	C₈×C₁₉	C₂²⋊C₈	1	2	yes	—
145	C₈×C₁₉	C₂²⋊C₈	1	2	yes	—
146	C₁₉⋊C₈	—	8	2	no	—
147	C₁₉⋊C₈	—	8	2	no	—
148	C₄×#131	D₁₉	1	1	yes	characteristic
149	#1564	—	19	1	no	—
150	C₂×#334	—	19	1	no	—
151	#1477	—	19	2	no	—
152	#1283	—	19	2	no	—
153	C₂²×C₁₉	—	2	1	no	—
154	C₂²×C₁₉	—	2	1	no	—
155	C₂²×C₁₉	—	2	2	no	—
156	C₂²×C₁₉	—	4	1	no	—
157	C₂²×C₁₉	—	4	1	no	—
158	C₂²×C₁₉	#380	1	1	yes	characteristic
159	C₂²×C₁₉	#431	1	1	yes	characteristic
160	C₂²×C₁₉	#432	1	1	yes	characteristic
161	C₄×C₁₉	—	2	2	no	—
162	C₄×C₁₉	—	2	1	no	—
163	C₄×C₁₉	—	2	1	no	—
164	C₄×C₁₉	—	2	1	no	—
165	C₄×C₁₉	—	4	1	no	—
166	C₄×C₁₉	—	4	1	no	—
167	C₄×C₁₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
168	C₄×C₁₉	#407	1	1	yes	characteristic
169	C₄×C₁₉	C₂×C₄⋊C₈	1	1	yes	characteristic
170	C₄×C₁₉	#423	1	1	yes	characteristic
171	C₄×C₁₉	#471	1	2	yes	—
172	C₂×#131	Dic₁₉	1	1	yes	characteristic
173	C₂×#131	Dic₁₉	1	1	yes	characteristic
174	C₄×#41	C₂×D₁₉	1	1	yes	characteristic
175	#431	—	19	2	no	—
176	C₂×C₄⋊C₈	—	19	2	no	—
177	#405	—	2	1	no	—
178	#405	—	2	1	no	—
179	#405	Dic₁₉	1	1	yes	characteristic
180	#405	Dic₁₉	1	1	yes	characteristic
181	C₄×M₁₆	—	2	1	no	—
182	C₄×M₁₆	C₂×D₁₉	1	1	yes	characteristic
183	C₂×C₄×C₈	—	2	1	no	—
184	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
185	#335	—	38	1	no	—
186	#334	—	38	1	no	—
187	#334	—	38	1	no	—
188	#334	—	38	1	no	—
189	Q₈⋊C₈	—	38	2	no	—
190	D₄⋊C₈	—	38	2	no	—
191	C₂×C₁₉	—	2	1	no	—
192	C₂×C₁₉	—	2	1	no	—
193	C₂×C₁₉	—	4	2	no	—
194	C₂×C₁₉	#1797	1	1	yes	characteristic
195	C₂×C₁₉	#1814	1	1	yes	characteristic
196	C₂×C₁₉	#1815	1	1	yes	characteristic
197	C₂×#41	C₂×Dic₁₉	1	1	yes	characteristic
198	#131	—	2	2	no	—
199	#131	—	2	2	no	—
200	#131	—	2	2	no	—
201	#131	C₁₉⋊C₈	1	4	yes	complemented
202	C₂×M₁₆	—	2	1	no	—
203	C₂×M₁₆	—	2	1	no	—
204	C₂×M₁₆	C₂×Dic₁₉	1	1	yes	characteristic
205	C₂×M₁₆	C₂×Dic₁₉	1	1	yes	characteristic
206	C₂²×C₈	—	2	1	no	—
207	C₂²×C₈	—	2	1	no	—
208	C₂²×C₈	—	38	2	no	—
209	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
210	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
211	C₄×Q₈	Dic₃₈	1	1	yes	characteristic
212	C₄×Q₈	D₇₆	1	1	yes	characteristic
213	C₄×D₄	—	2	1	no	—
214	C₄×D₄	—	2	1	no	—
215	C₄×D₄	Dic₃₈	1	1	yes	characteristic
216	C₄×D₄	D₇₆	1	1	yes	characteristic
217	#117	—	2	1	no	—
218	#117	C₂×Dic₁₉	1	1	yes	characteristic
219	C₂×C₄²	—	2	1	no	—
220	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
221	C₄⋊C₈	—	38	2	no	—
222	C₄⋊C₈	—	38	2	no	—
223	C₄⋊C₈	—	38	2	no	—
224	C₂²⋊C₈	—	38	2	no	—
225	#97	—	2	1	no	—
226	#97	—	2	1	no	—
227	#97	—	2	1	no	—
228	C₄×C₈	—	2	1	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	—	38	2	no	—
232	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
233	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
234	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
235	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
236	C₁₉	#8604	1	1	yes	characteristic, complemented, Sylow
237	#41	—	2	1	no	—
238	#41	—	2	1	no	—
239	#41	C₂×C₁₉⋊C₈	1	2	yes	—
240	#41	#64520	1	2	yes	—
241	C₂×Q₈	C₄⋊Dic₁₉	1	1	yes	characteristic
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	C₄⋊Dic₁₉	1	1	yes	characteristic
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	#64541	1	1	yes	characteristic
248	C₂²×C₄	#64541	1	1	yes	characteristic
249	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
250	M₁₆	—	2	4	no	—
251	M₁₆	—	2	1	no	—
252	M₁₆	—	2	1	no	—
253	M₁₆	—	2	4	no	—
254	M₁₆	C₂×C₁₉⋊C₈	1	2	yes	—
255	M₁₆	#64520	1	2	yes	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	4	no	—
262	C₂×C₈	—	76	2	no	—
263	C₂×C₈	—	76	2	no	—
264	C₂×C₈	—	152	2	no	—
265	C₂×C₈	C₂×C₁₉⋊C₈	1	2	yes	—
266	C₂×C₈	#64520	1	2	yes	—
267	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
268	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
269	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
270	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
274	C₄⋊C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	2	1	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	2	1	no	—
280	C₄²	—	2	1	no	—
281	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
282	C₄²	C₂×D₇₆	1	1	yes	characteristic
283	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
284	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
285	C₂³	—	2	1	no	—
286	C₂³	#10597372	1	1	yes	characteristic
287	Q₈	—	2	1	no	—
288	Q₈	C₄⋊(C₁₉⋊C₈)	1	2	yes	—
289	D₄	—	2	2	no	—
290	D₄	—	2	2	no	—
291	D₄	—	2	1	no	—
292	D₄	C₄⋊(C₁₉⋊C₈)	1	2	yes	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	C₂²×C₁₉⋊C₈	1	1	yes	characteristic
303	C₂×C₄	C₂×#64520	1	1	yes	characteristic
304	C₂×C₄	C₂×#64520	1	1	yes	characteristic
305	C₂×C₄	C₂×#64520	1	1	yes	characteristic
306	C₂×C₄	#10597370	1	1	yes	characteristic
307	C₂×C₄	#10597370	1	1	yes	characteristic
308	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
309	C₂×C₄	#10597376	1	1	yes	characteristic
310	C₂×C₄	#10597376	1	1	yes	characteristic
311	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
312	C₈	—	2	2	no	—
313	C₈	—	2	1	no	—
314	C₈	—	2	2	no	—
315	C₈	—	2	1	no	—
316	C₈	—	152	2	no	—
317	C₈	—	152	2	no	—
318	C₈	C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
319	C₈	C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	#11926404	1	1	yes	characteristic
326	C₂²	#11926597	1	1	yes	characteristic
327	C₂²	#11926601	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	C₂×C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
335	C₄	#11926508	1	1	yes	characteristic
336	C₄	C₂×C₄⋊(C₁₉⋊C₈)	1	1	yes	characteristic
337	C₄	#11926568	1	1	yes	characteristic
338	C₄	#11926807	1	2	yes	—
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	4	2	no	—
342	C₂	#14563615	1	1	yes	characteristic
343	C₂	#14563689	1	1	yes	characteristic
344	C₂	#14563694	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	#8604
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×#11926295	C₂	semidirect
4. #14561486	C₂	semidirect
29. C₁₉×#131	C₈	semidirect
201. #131	C₁₉⋊C₈	semidirect
236. C₁₉	#8604	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×#11926295	2
#14562022	2
#14561486	2
C₄×C₁₉×#131	2
#8604	19