Subgroups — #18070586

Order 4864.

426 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563778	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#14561458	C₂	1	2	yes	maximal
4	#14561458	C₂	1	2	yes	maximal
5	C₄×C₁₉×#125	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
7	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
8	#11926299	—	2	1	no	—
9	#11926299	—	2	1	no	—
10	#11926299	C₂²	1	2	yes	—
11	#11926299	C₂²	1	2	yes	—
12	C₂×C₁₉×#125	C₄	1	1	yes	characteristic
13	C₁₉×#389	C₂²	1	1	yes	characteristic
14	C₁₉×#389	C₂²	1	1	yes	characteristic
15	C₁₉×#387	—	2	1	no	—
16	C₁₉×#387	—	2	1	no	—
17	C₁₉×#382	C₄	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₁₉	—	2	1	no	—
19	C₄×C₄⋊C₄×C₁₉	—	2	1	no	—
20	C₄×C₄⋊C₄×C₁₉	C₄	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₁₉	C₄	1	1	yes	characteristic
22	C₄³×C₁₉	C₂²	1	1	yes	characteristic
23	C₁₉×#125	—	2	2	no	—
24	C₁₉×#125	C₈	1	4	yes	complemented
25	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
26	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
27	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
28	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₉	—	2	1	no	—
34	C₂×C₄²×C₁₉	—	2	1	no	—
35	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
42	C₂²×C₁₉⋊C₈	—	2	2	no	—
43	C₂²×C₁₉⋊C₈	—	2	2	no	—
44	C₁₉×#95	—	2	1	no	—
45	C₁₉×#95	—	2	1	no	—
46	C₁₉×#95	—	2	1	no	—
47	C₁₉×#95	—	2	1	no	—
48	C₁₉×#95	—	2	1	no	—
49	C₁₉×#95	—	2	1	no	—
50	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
51	C₄×C₁₉⋊C₈	—	4	2	no	—
52	C₂²×C₄×C₁₉	—	2	1	no	—
53	C₂²×C₄×C₁₉	—	2	1	no	—
54	C₂²×C₄×C₁₉	—	2	1	no	—
55	C₂²×C₄×C₁₉	—	2	1	no	—
56	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
63	C₄⋊C₄×C₁₉	—	2	4	no	—
64	C₄⋊C₄×C₁₉	—	2	4	no	—
65	C₄⋊C₄×C₁₉	—	2	1	no	—
66	C₄⋊C₄×C₁₉	—	2	1	no	—
67	C₄⋊C₄×C₁₉	—	4	1	no	—
68	C₄⋊C₄×C₁₉	—	4	1	no	—
69	C₄⋊C₄×C₁₉	—	4	2	no	—
70	C₄⋊C₄×C₁₉	—	4	1	no	—
71	C₄⋊C₄×C₁₉	—	4	1	no	—
72	C₄⋊C₄×C₁₉	C₂×C₈	1	2	yes	—
73	C₄⋊C₄×C₁₉	C₂×C₈	1	2	yes	—
74	C₄²×C₁₉	—	2	2	no	—
75	C₄²×C₁₉	—	2	2	no	—
76	C₄²×C₁₉	—	2	2	no	—
77	C₄²×C₁₉	—	2	1	no	—
78	C₄²×C₁₉	—	2	1	no	—
79	C₄²×C₁₉	—	4	1	no	—
80	C₄²×C₁₉	—	4	1	no	—
81	C₄²×C₁₉	—	4	2	no	—
82	C₄²×C₁₉	—	4	2	no	—
83	C₄²×C₁₉	C₂×C₈	1	2	yes	—
84	C₄²×C₁₉	M₁₆	1	2	yes	—
85	C₂×C₁₉⋊C₈	—	2	4	no	—
86	C₂×C₁₉⋊C₈	—	4	2	no	—
87	C₂×C₁₉⋊C₈	—	4	2	no	—
88	C₂×C₁₉⋊C₈	—	4	2	no	—
89	C₂×C₁₉⋊C₈	—	4	2	no	—
90	C₂×C₁₉⋊C₈	—	4	2	no	—
91	#7585	—	19	1	no	maximal, Sylow
92	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₉	—	2	1	no	—
94	C₂×C₄×C₁₉	—	2	1	no	—
95	C₂×C₄×C₁₉	—	2	1	no	—
96	C₂×C₄×C₁₉	—	2	1	no	—
97	C₂×C₄×C₁₉	—	2	1	no	—
98	C₂×C₄×C₁₉	—	2	1	no	—
99	C₂×C₄×C₁₉	—	2	1	no	—
100	C₂×C₄×C₁₉	—	2	1	no	—
101	C₂×C₄×C₁₉	—	2	1	no	—
102	C₂×C₄×C₁₉	—	2	1	no	—
103	C₂×C₄×C₁₉	—	2	1	no	—
104	C₂×C₄×C₁₉	—	2	1	no	—
105	C₂×C₄×C₁₉	—	2	1	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	2	2	no	—
108	C₂×C₄×C₁₉	—	2	2	no	—
109	C₂×C₄×C₁₉	—	4	1	no	—
110	C₂×C₄×C₁₉	—	4	1	no	—
111	C₂×C₄×C₁₉	—	4	1	no	—
112	C₂×C₄×C₁₉	—	4	1	no	—
113	C₂×C₄×C₁₉	—	4	1	no	—
114	C₂×C₄×C₁₉	—	4	1	no	—
115	C₂×C₄×C₁₉	—	4	1	no	—
116	C₂×C₄×C₁₉	—	4	1	no	—
117	C₂×C₄×C₁₉	—	4	1	no	—
118	C₂×C₄×C₁₉	—	4	1	no	—
119	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
120	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
121	C₂×C₄×C₁₉	#101	1	2	yes	—
122	C₂×C₄×C₁₉	#104	1	2	yes	—
123	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
125	C₂×C₄×C₁₉	#135	1	2	yes	—
126	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
128	C₁₉⋊C₈	—	4	4	no	—
129	C₁₉⋊C₈	—	8	2	no	—
130	C₄×#125	D₁₉	1	1	yes	characteristic
131	#1839	—	19	1	no	—
132	C₄×C₄⋊C₈	—	19	1	no	—
133	#1271	—	19	2	no	—
134	#1271	—	19	2	no	—
135	C₂²×C₁₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
136	C₂²×C₁₉	#410	1	1	yes	characteristic
137	C₂²×C₁₉	C₂×#101	1	1	yes	characteristic
138	C₂²×C₁₉	#416	1	1	yes	characteristic
139	C₂²×C₁₉	#417	1	1	yes	characteristic
140	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
141	C₂²×C₁₉	#421	1	1	yes	characteristic
142	C₄×C₁₉	—	2	2	no	—
143	C₄×C₁₉	—	2	1	no	—
144	C₄×C₁₉	—	2	1	no	—
145	C₄×C₁₉	—	2	2	no	—
146	C₄×C₁₉	—	2	2	no	—
147	C₄×C₁₉	—	2	1	no	—
148	C₄×C₁₉	—	2	1	no	—
149	C₄×C₁₉	—	2	2	no	—
150	C₄×C₁₉	—	2	2	no	—
151	C₄×C₁₉	—	4	2	no	—
152	C₄×C₁₉	—	4	2	no	—
153	C₄×C₁₉	—	8	1	no	—
154	C₄×C₁₉	—	8	1	no	—
155	C₂×#125	Dic₁₉	1	1	yes	characteristic
156	C₂×C₄⋊C₈	—	38	1	no	—
157	C₂×C₄×C₈	—	38	1	no	—
158	#389	C₂×D₁₉	1	1	yes	characteristic
159	#389	C₂×D₁₉	1	1	yes	characteristic
160	#387	—	2	1	no	—
161	#387	—	2	1	no	—
162	#382	Dic₁₉	1	1	yes	characteristic
163	C₄×C₄⋊C₄	—	2	1	no	—
164	C₄×C₄⋊C₄	—	2	1	no	—
165	C₄×C₄⋊C₄	Dic₁₉	1	1	yes	characteristic
166	C₄×C₄⋊C₄	Dic₁₉	1	1	yes	characteristic
167	C₄³	C₂×D₁₉	1	1	yes	characteristic
168	#336	—	19	2	no	—
169	#336	—	19	2	no	—
170	#336	—	38	1	no	—
171	#336	—	38	1	no	—
172	C₂×C₁₉	#1461	1	1	yes	characteristic
173	C₂×C₁₉	#1476	1	1	yes	characteristic
174	C₂×C₁₉	#1477	1	1	yes	characteristic
175	C₂×C₁₉	#1509	1	1	yes	characteristic
176	C₂×C₁₉	#1521	1	1	yes	characteristic
177	C₂×C₁₉	#1528	1	1	yes	characteristic
178	C₂×C₁₉	#1552	1	1	yes	characteristic
179	C₂²×C₈	—	38	2	no	—
180	C₂²×C₈	—	38	2	no	—
181	#125	—	2	2	no	—
182	#125	C₁₉⋊C₈	1	4	yes	complemented
183	C₂×C₄⋊C₄	—	2	1	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	—	2	1	no	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
188	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
189	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
190	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
191	C₂×C₄²	—	2	1	no	—
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
194	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
195	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
196	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
197	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
198	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
199	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
200	C₄⋊C₈	—	38	4	no	—
201	C₄×C₈	—	76	2	no	—
202	#95	—	2	1	no	—
203	#95	—	2	1	no	—
204	#95	—	2	1	no	—
205	#95	—	2	1	no	—
206	#95	—	2	1	no	—
207	#95	—	2	1	no	—
208	C₁₉	#7585	1	1	yes	characteristic, complemented, Sylow
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
214	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
215	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
216	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
217	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
220	C₂×C₈	—	38	4	no	—
221	C₂×C₈	—	76	2	no	—
222	C₂×C₈	—	76	2	no	—
223	C₂×C₈	—	76	2	no	—
224	C₂×C₈	—	76	2	no	—
225	C₂×C₈	—	76	2	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	4	1	no	—
231	C₄⋊C₄	—	4	1	no	—
232	C₄⋊C₄	—	4	1	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	1	no	—
235	C₄⋊C₄	C₂×C₁₉⋊C₈	1	2	yes	—
236	C₄⋊C₄	C₂×C₁₉⋊C₈	1	2	yes	—
237	C₄²	—	2	1	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	1	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	4	1	no	—
243	C₄²	—	4	1	no	—
244	C₄²	—	4	2	no	—
245	C₄²	—	4	2	no	—
246	C₄²	C₂×C₁₉⋊C₈	1	2	yes	—
247	C₄²	#64520	1	2	yes	—
248	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
276	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
277	C₂×C₄	#10597293	1	2	yes	—
278	C₂×C₄	#10597294	1	2	yes	—
279	C₂×C₄	C₂²×C₁₉⋊C₈	1	1	yes	characteristic
280	C₂×C₄	C₂×#64520	1	1	yes	characteristic
281	C₂×C₄	#10597395	1	1	yes	characteristic
282	C₂×C₄	#10597396	1	2	yes	—
283	C₂×C₄	#10597397	1	1	yes	characteristic
284	C₈	—	76	4	no	—
285	C₈	—	152	2	no	—
286	C₂²	C₂×C₂²⋊(C₁₉⋊C₈)	1	1	yes	characteristic
287	C₂²	#11926518	1	1	yes	characteristic
288	C₂²	C₂×#10597293	1	1	yes	characteristic
289	C₂²	#11926540	1	1	yes	characteristic
290	C₂²	#11926545	1	1	yes	characteristic
291	C₂²	C₂×#10597294	1	1	yes	characteristic
292	C₂²	#11926561	1	1	yes	characteristic
293	C₄	—	2	2	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	4	2	no	—
303	C₄	—	4	2	no	—
304	C₄	—	8	1	no	—
305	C₄	—	8	1	no	—
306	C₂	#14561959	1	1	yes	characteristic
307	C₂	#14562017	1	1	yes	characteristic
308	C₂	#14562022	1	1	yes	characteristic
309	C₂	#14562160	1	1	yes	characteristic
310	C₂	#14562212	1	1	yes	characteristic
311	C₂	#14562237	1	1	yes	characteristic
312	C₂	#14562335	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₉
commutator	C₂×C₄×C₁₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₄×C₁₉×#125	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄×#125	D₁₉
hypocenter	C₁₉	#7585
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×#125

Complements

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

N	H ≅ G/N	type
24. C₁₉×#125	C₈	semidirect
182. #125	C₁₉⋊C₈	semidirect
208. C₁₉	#7585	semidirect

Maximal subgroups

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

subgroup	index
#14563778	2
C₄×C₄⋊(C₁₉⋊C₈)	2
#14561458	2
C₄×C₁₉×#125	2
#7585	19