Subgroups — #18081514

Order 4864.

720 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563488	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄⋊(C₁₉⋊C₈)	C₂	1	4	yes	maximal, complemented
4	C₄×C₁₉×#127	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
6	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×#10597260	—	2	1	no	—
8	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
9	#11926299	—	2	1	no	—
10	#11926299	—	2	1	no	—
11	D₄⋊(C₁₉⋊C₈)	—	2	8	no	—
12	C₂×C₁₉×#127	C₄	1	1	yes	characteristic
13	C₂×C₄×D₄×C₁₉	—	2	1	no	—
14	C₂×C₄×D₄×C₁₉	—	2	1	no	—
15	C₂×C₄×D₄×C₁₉	C₂²	1	2	yes	—
16	C₁₉×#390	—	2	1	no	—
17	C₁₉×#390	—	2	1	no	—
18	C₁₉×#390	C₄	1	2	yes	—
19	C₁₉×#384	C₄	1	1	yes	characteristic
20	C₄³×C₁₉	C₂²	1	1	yes	characteristic
21	C₂²×D₄×C₁₉	—	2	1	no	—
22	C₂²×D₄×C₁₉	—	2	1	no	—
23	C₂²×D₄×C₁₉	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₁₉	—	2	2	no	—
25	C₁₉×#127	—	2	4	no	—
26	C₄×D₄×C₁₉	—	2	4	no	—
27	C₄×D₄×C₁₉	—	2	4	no	—
28	C₄×D₄×C₁₉	—	4	4	no	—
29	C₄×D₄×C₁₉	—	4	2	no	—
30	C₄×D₄×C₁₉	—	4	2	no	—
31	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
33	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
34	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
35	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
36	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
37	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₉	C₂×C₄	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	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
44	C₂²×C₁₉⋊C₈	—	2	2	no	—
45	C₂²×C₁₉⋊C₈	—	2	2	no	—
46	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
47	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
48	C₄⋊(C₁₉⋊C₈)	D₄	1	4	yes	complemented
49	#10597260	—	4	2	no	—
50	C₄×C₁₉⋊C₈	—	2	4	no	—
51	C₂⁴×C₁₉	—	2	2	no	—
52	C₂×D₄×C₁₉	—	2	4	no	—
53	C₂×D₄×C₁₉	—	2	4	no	—
54	C₂×D₄×C₁₉	—	2	4	no	—
55	C₂×D₄×C₁₉	—	2	2	no	—
56	C₂×D₄×C₁₉	—	2	4	no	—
57	C₂×D₄×C₁₉	—	4	2	no	—
58	C₂×D₄×C₁₉	—	4	2	no	—
59	C₂×D₄×C₁₉	—	4	2	no	—
60	C₂×D₄×C₁₉	—	4	2	no	—
61	C₂×D₄×C₁₉	M₁₆	1	4	yes	—
62	C₂²×C₄×C₁₉	—	2	2	no	—
63	C₂²×C₄×C₁₉	—	4	4	no	—
64	C₂²×C₄×C₁₉	—	4	2	no	—
65	C₂²×C₄×C₁₉	—	4	2	no	—
66	C₂²×C₄×C₁₉	—	4	4	no	—
67	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
74	C₄⋊C₄×C₁₉	—	2	4	no	—
75	C₄⋊C₄×C₁₉	—	4	2	no	—
76	C₄⋊C₄×C₁₉	—	4	2	no	—
77	C₂²⋊C₄×C₁₉	—	4	4	no	—
78	C₂²⋊C₄×C₁₉	—	4	4	no	—
79	C₂²⋊C₄×C₁₉	—	4	2	no	—
80	C₂²⋊C₄×C₁₉	—	4	2	no	—
81	C₄²×C₁₉	—	2	2	no	—
82	C₄²×C₁₉	—	2	2	no	—
83	C₄²×C₁₉	—	2	2	no	—
84	C₄²×C₁₉	—	2	2	no	—
85	C₄²×C₁₉	—	4	1	no	—
86	C₄²×C₁₉	D₈	1	2	yes	—
87	C₄²×C₁₉	SD₁₆	1	2	yes	—
88	C₄²×C₁₉	C₂×D₄	1	2	yes	—
89	C₄²×C₁₉	#41	1	2	yes	—
90	C₂×C₁₉⋊C₈	—	2	4	no	—
91	C₂×C₁₉⋊C₈	—	2	4	no	—
92	C₂×C₁₉⋊C₈	—	4	2	no	—
93	C₂×C₁₉⋊C₈	—	4	2	no	—
94	C₂×C₁₉⋊C₈	—	4	2	no	—
95	C₂×C₁₉⋊C₈	—	4	2	no	—
96	#9678	—	19	1	no	maximal, Sylow
97	C₂³×C₁₉	—	2	4	no	—
98	C₂³×C₁₉	—	4	2	no	—
99	C₂³×C₁₉	—	4	2	no	—
100	C₂³×C₁₉	—	4	2	no	—
101	C₂³×C₁₉	—	4	2	no	—
102	C₂³×C₁₉	—	4	2	no	—
103	C₂³×C₁₉	—	4	2	no	—
104	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	D₄×C₁₉	—	2	8	no	—
106	D₄×C₁₉	—	4	8	no	—
107	D₄×C₁₉	—	4	8	no	—
108	D₄×C₁₉	—	4	4	no	—
109	C₂×C₄×C₁₉	—	2	1	no	—
110	C₂×C₄×C₁₉	—	2	1	no	—
111	C₂×C₄×C₁₉	—	2	1	no	—
112	C₂×C₄×C₁₉	—	2	1	no	—
113	C₂×C₄×C₁₉	—	2	1	no	—
114	C₂×C₄×C₁₉	—	2	1	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	2	1	no	—
118	C₂×C₄×C₁₉	—	2	1	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	4	2	no	—
121	C₂×C₄×C₁₉	—	4	2	no	—
122	C₂×C₄×C₁₉	—	4	2	no	—
123	C₂×C₄×C₁₉	—	8	4	no	—
124	C₂×C₄×C₁₉	—	8	4	no	—
125	C₂×C₄×C₁₉	#102	1	2	yes	—
126	C₂×C₄×C₁₉	#102	1	2	yes	—
127	C₂×C₄×C₁₉	#104	1	2	yes	—
128	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₁₉	#121	1	2	yes	—
130	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
131	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
133	C₂×C₄×C₁₉	#131	1	2	yes	—
134	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
137	C₁₉⋊C₈	—	4	4	no	—
138	C₁₉⋊C₈	—	4	4	no	—
139	C₄×#127	D₁₉	1	1	yes	characteristic
140	#1767	—	19	1	no	—
141	C₄×C₄⋊C₈	—	19	1	no	—
142	C₂×D₄⋊C₈	—	19	4	no	—
143	C₂²×C₁₉	—	4	4	no	—
144	C₂²×C₁₉	—	4	4	no	—
145	C₂²×C₁₉	—	4	4	no	—
146	C₂²×C₁₉	—	8	2	no	—
147	C₂²×C₁₉	—	8	2	no	—
148	C₂²×C₁₉	—	8	2	no	—
149	C₂²×C₁₉	—	8	2	no	—
150	C₂²×C₁₉	#390	1	1	yes	characteristic
151	C₂²×C₁₉	#407	1	1	yes	characteristic
152	C₂²×C₁₉	#408	1	1	yes	characteristic
153	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
154	C₂²×C₁₉	#418	1	1	yes	characteristic
155	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
156	C₂²×C₁₉	#421	1	1	yes	characteristic
157	C₄×C₁₉	—	2	1	no	—
158	C₄×C₁₉	—	2	1	no	—
159	C₄×C₁₉	—	2	2	no	—
160	C₄×C₁₉	—	2	2	no	—
161	C₄×C₁₉	—	2	2	no	—
162	C₄×C₁₉	—	4	1	no	—
163	C₄×C₁₉	—	8	2	no	—
164	C₄×C₁₉	#435	1	4	yes	—
165	C₄×C₁₉	#459	1	2	yes	—
166	C₄×C₁₉	#460	1	2	yes	—
167	C₂×#127	Dic₁₉	1	1	yes	characteristic
168	C₂×C₄×D₄	—	2	1	no	—
169	C₂×C₄×D₄	—	2	1	no	—
170	C₂×C₄×D₄	C₂×D₁₉	1	2	yes	—
171	C₂×C₄⋊C₈	—	19	2	no	—
172	C₂×C₄⋊C₈	—	38	1	no	—
173	C₂×#97	—	38	1	no	—
174	C₂×C₄×C₈	—	19	2	no	—
175	#390	—	2	1	no	—
176	#390	—	2	1	no	—
177	#390	Dic₁₉	1	2	yes	—
178	#384	Dic₁₉	1	1	yes	characteristic
179	C₄³	C₂×D₁₉	1	1	yes	characteristic
180	#336	—	38	1	no	—
181	#336	—	38	1	no	—
182	D₄⋊C₈	—	38	8	no	—
183	C₂×C₁₉	—	8	4	no	—
184	C₂×C₁₉	#1486	1	1	yes	characteristic
185	C₂×C₁₉	#1490	1	1	yes	characteristic
186	C₂×C₁₉	#1492	1	1	yes	characteristic
187	C₂×C₁₉	#1496	1	1	yes	characteristic
188	C₂×C₁₉	#1955	1	1	yes	characteristic
189	C₂×C₁₉	#1980	1	1	yes	characteristic
190	C₂×C₁₉	#1982	1	1	yes	characteristic
191	C₂²×D₄	—	2	1	no	—
192	C₂²×D₄	—	2	1	no	—
193	C₂²×D₄	C₂×Dic₁₉	1	2	yes	—
194	C₂³×C₄	—	2	2	no	—
195	C₂²×C₈	—	38	2	no	—
196	C₂²×C₈	—	38	2	no	—
197	#127	—	2	4	no	—
198	C₄×D₄	—	2	4	no	—
199	C₄×D₄	—	2	4	no	—
200	C₄×D₄	—	4	4	no	—
201	C₄×D₄	—	4	2	no	—
202	C₄×D₄	—	4	2	no	—
203	C₂×C₄⋊C₄	—	2	1	no	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
206	C₂×C₂²⋊C₄	—	2	2	no	—
207	C₂×C₂²⋊C₄	—	2	2	no	—
208	C₂×C₂²⋊C₄	—	2	2	no	—
209	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
210	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
211	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
212	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
213	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
214	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
215	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
216	C₄⋊C₈	—	19	4	no	—
217	C₄⋊C₈	—	38	4	no	—
218	C₄⋊C₈	—	38	2	no	—
219	#97	—	76	2	no	—
220	C₄×C₈	—	38	4	no	—
221	C₁₉	#9678	1	1	yes	characteristic, complemented, Sylow
222	C₂⁴	—	2	2	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	2	4	no	—
225	C₂×D₄	—	2	4	no	—
226	C₂×D₄	—	2	4	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂×D₄	—	4	2	no	—
230	C₂×D₄	—	4	2	no	—
231	C₂×D₄	—	4	2	no	—
232	C₂×D₄	#64520	1	4	yes	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	—	4	4	no	—
236	C₂²×C₄	—	4	2	no	—
237	C₂²×C₄	—	4	4	no	—
238	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
239	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
240	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
241	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
242	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
245	C₂×C₈	—	38	4	no	—
246	C₂×C₈	—	38	4	no	—
247	C₂×C₈	—	76	2	no	—
248	C₂×C₈	—	76	2	no	—
249	C₂×C₈	—	76	2	no	—
250	C₂×C₈	—	76	2	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₂²⋊C₄	—	4	4	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	—	4	4	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	4	1	no	—
263	C₄²	#64525	1	2	yes	—
264	C₄²	#64526	1	2	yes	—
265	C₄²	D₄×D₁₉	1	2	yes	—
266	C₄²	#64543	1	2	yes	—
267	C₂³	—	2	4	no	—
268	C₂³	—	4	2	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	4	2	no	—
271	C₂³	—	4	2	no	—
272	C₂³	—	4	2	no	—
273	C₂³	—	4	2	no	—
274	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
275	D₄	—	2	8	no	—
276	D₄	—	4	8	no	—
277	D₄	—	4	8	no	—
278	D₄	—	4	4	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	8	4	no	—
294	C₂×C₄	—	8	4	no	—
295	C₂×C₄	#10597289	1	2	yes	—
296	C₂×C₄	#10597289	1	2	yes	—
297	C₂×C₄	#10597294	1	2	yes	—
298	C₂×C₄	C₂×#64520	1	2	yes	—
299	C₂×C₄	C₂×#64525	1	1	yes	characteristic
300	C₂×C₄	#10597378	1	2	yes	—
301	C₂×C₄	C₂×#64526	1	1	yes	characteristic
302	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
303	C₂×C₄	#10597382	1	1	yes	characteristic
304	C₂×C₄	#10597384	1	2	yes	—
305	C₂×C₄	#10597386	1	1	yes	characteristic
306	C₂×C₄	#10597394	1	2	yes	—
307	C₈	—	76	4	no	—
308	C₈	—	76	4	no	—
309	C₂²	—	4	4	no	—
310	C₂²	—	4	4	no	—
311	C₂²	—	4	4	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	#11926447	1	1	yes	characteristic
317	C₂²	#11926508	1	1	yes	characteristic
318	C₂²	#11926511	1	1	yes	characteristic
319	C₂²	C₂×#10597289	1	1	yes	characteristic
320	C₂²	#11926549	1	1	yes	characteristic
321	C₂²	C₂×#10597294	1	1	yes	characteristic
322	C₂²	#11926561	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	8	2	no	—
330	C₄	#11926616	1	4	yes	—
331	C₄	#11926736	1	2	yes	—
332	C₄	#11926741	1	2	yes	—
333	C₂	—	8	4	no	—
334	C₂	#14562060	1	1	yes	characteristic
335	C₂	#14562078	1	1	yes	characteristic
336	C₂	#14562086	1	1	yes	characteristic
337	C₂	#14562104	1	1	yes	characteristic
338	C₂	#14564434	1	1	yes	characteristic
339	C₂	#14564555	1	1	yes	characteristic
340	C₂	#14564563	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₄×C₁₉×#127	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂×C₁₉×#127	C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄×#127	D₁₉
hypocenter	C₁₉	#9678
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×#127

Complements

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

N	H ≅ G/N	type
1. #14563488	C₂	semidirect
2. C₄×C₄⋊(C₁₉⋊C₈)	C₂	semidirect
3. C₂×D₄⋊(C₁₉⋊C₈)	C₂	semidirect
48. C₄⋊(C₁₉⋊C₈)	D₄	semidirect
221. C₁₉	#9678	semidirect

Maximal subgroups

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

subgroup	index
#14563488	2
C₄×C₄⋊(C₁₉⋊C₈)	2
C₂×D₄⋊(C₁₉⋊C₈)	2
C₄×C₁₉×#127	2
#9678	19