Subgroups — #18079338

Order 4864.

518 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14572676	C₂	1	1	yes	maximal, characteristic, complemented
2	#14563645	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562195	C₂	1	1	yes	maximal, characteristic, complemented
4	#14562034	C₂	1	1	yes	maximal, characteristic, complemented
5	#14561530	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561495	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#3036	C₂	1	1	yes	maximal, characteristic
8	#11926622	—	2	2	no	—
9	#11926622	—	2	2	no	—
10	#11926597	C₂²	1	1	yes	characteristic
11	#11926597	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×#10597289	—	2	1	no	—
14	#11926508	—	2	2	no	—
15	#11926410	C₂²	1	1	yes	characteristic
16	#11926306	—	2	1	no	—
17	#11926278	—	2	2	no	—
18	D₄⋊(C₁₉⋊C₈)	—	2	2	no	—
19	C₂×C₁₉×AGL(1,ℤ/8)	—	2	1	no	—
20	C₁₉×#532	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
22	C₁₉×#463	—	2	2	no	—
23	C₁₉×#461	C₄	1	2	yes	complemented
24	C₁₉×#459	C₄	1	2	yes	complemented
25	C₁₉×#423	C₂²	1	1	yes	characteristic
26	C₁₉×#417	—	2	1	no	—
27	C₂×#41×C₁₉	—	2	1	no	—
28	C₂²×D₄×C₁₉	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₁₉	—	2	1	no	—
30	C₁₉×AGL(1,ℤ/8)	—	4	2	no	—
31	C₁₉×AGL(1,ℤ/8)	—	4	2	no	—
32	C₂×SD₁₆×C₁₉	—	2	2	no	—
33	C₂×D₈×C₁₉	—	2	2	no	—
34	C₂×M₁₆×C₁₉	—	2	1	no	—
35	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
36	C₁₉×#127	C₂×C₄	1	1	yes	characteristic
37	C₁₉×#124	—	2	1	no	—
38	C₁₉×#121	—	2	1	no	—
39	C₁₉×#121	—	2	1	no	—
40	C₄×D₄×C₁₉	—	2	1	no	—
41	C₄×D₄×C₁₉	—	2	1	no	—
42	C₄×D₄×C₁₉	—	2	1	no	—
43	C₄×D₄×C₁₉	—	2	1	no	—
44	C₄×D₄×C₁₉	C₂×C₄	1	2	yes	—
45	C₄×D₄×C₁₉	D₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
47	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
48	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
49	C₂×C₄⋊Dic₁₉	—	2	1	no	—
50	C₂×#64520	—	2	4	no	—
51	C₂²×C₁₉⋊C₈	—	2	1	no	—
52	C₁₉×C₄⋊C₈	—	2	1	no	—
53	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
54	C₁₉×#103	—	2	2	no	—
55	C₁₉×#102	—	2	2	no	—
56	#10597289	—	4	2	no	—
57	C₁₉⋊#95	—	2	1	no	—
58	C₁₉⋊#95	—	2	1	no	—
59	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
60	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
61	C₂²⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
62	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
63	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
64	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
65	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
66	C₄×C₁₉⋊C₈	—	2	1	no	—
67	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
68	C₂⁴×C₁₉	—	2	1	no	—
69	#41×C₁₉	—	4	1	no	—
70	#41×C₁₉	—	4	1	no	—
71	#41×C₁₉	—	4	1	no	—
72	#41×C₁₉	—	4	1	no	—
73	C₂×Q₈×C₁₉	—	2	1	no	—
74	C₂×D₄×C₁₉	—	2	1	no	—
75	C₂×D₄×C₁₉	—	2	1	no	—
76	C₂×D₄×C₁₉	—	2	1	no	—
77	C₂×D₄×C₁₉	—	2	1	no	—
78	C₂×D₄×C₁₉	—	2	1	no	—
79	C₂×D₄×C₁₉	—	2	1	no	—
80	C₂×D₄×C₁₉	—	2	1	no	—
81	C₂×D₄×C₁₉	—	2	1	no	—
82	C₂×D₄×C₁₉	—	4	1	no	—
83	C₂×D₄×C₁₉	C₂²⋊C₄	1	2	yes	—
84	C₂²×C₄×C₁₉	—	2	1	no	—
85	C₂²×C₄×C₁₉	—	2	2	no	—
86	C₂²×C₄×C₁₉	—	2	2	no	—
87	C₂²×C₄×C₁₉	—	4	1	no	—
88	C₂²×C₄×C₁₉	—	8	2	no	—
89	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
92	C₂²×Dic₁₉	—	4	1	no	—
93	SD₁₆×C₁₉	—	4	4	no	—
94	D₈×C₁₉	—	4	4	no	—
95	M₁₆×C₁₉	—	4	1	no	—
96	M₁₆×C₁₉	—	4	1	no	—
97	C₂×C₈×C₁₉	—	2	2	no	—
98	C₄⋊C₄×C₁₉	—	2	1	no	—
99	C₄⋊C₄×C₁₉	—	2	1	no	—
100	C₄⋊C₄×C₁₉	—	2	1	no	—
101	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
102	C₂²⋊C₄×C₁₉	—	2	2	no	—
103	C₂²⋊C₄×C₁₉	—	2	2	no	—
104	C₂²⋊C₄×C₁₉	—	4	1	no	—
105	C₂²⋊C₄×C₁₉	—	4	1	no	—
106	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₁₉	#41	1	1	yes	characteristic
110	C₄⋊Dic₁₉	—	4	1	no	—
111	C₄⋊Dic₁₉	—	4	1	no	—
112	#64520	—	8	4	no	—
113	C₂×C₁₉⋊C₈	—	2	2	no	—
114	C₂×C₁₉⋊C₈	—	2	2	no	—
115	C₂×C₁₉⋊C₈	—	4	1	no	—
116	C₂×C₁₉⋊C₈	—	4	1	no	—
117	C₂×C₁₉⋊C₈	—	4	1	no	—
118	#9309	—	19	1	no	maximal, Sylow
119	C₂³×C₁₉	—	2	2	no	—
120	C₂³×C₁₉	—	4	1	no	—
121	C₂³×C₁₉	—	4	1	no	—
122	C₂³×C₁₉	—	4	1	no	—
123	C₂³×C₁₉	—	8	1	no	—
124	C₂³×C₁₉	—	8	1	no	—
125	C₂³×C₁₉	#121	1	1	yes	characteristic
126	Q₈×C₁₉	—	4	1	no	—
127	Q₈×C₁₉	—	4	1	no	—
128	D₄×C₁₉	—	2	2	no	—
129	D₄×C₁₉	—	2	2	no	—
130	D₄×C₁₉	—	4	2	no	—
131	D₄×C₁₉	—	4	1	no	—
132	D₄×C₁₉	—	4	1	no	—
133	D₄×C₁₉	—	4	1	no	—
134	D₄×C₁₉	—	4	1	no	—
135	D₄×C₁₉	—	8	1	no	—
136	D₄×C₁₉	—	8	1	no	—
137	D₄×C₁₉	—	8	1	no	—
138	C₂×C₄×C₁₉	—	2	2	no	—
139	C₂×C₄×C₁₉	—	4	1	no	—
140	C₂×C₄×C₁₉	—	4	1	no	—
141	C₂×C₄×C₁₉	—	4	4	no	—
142	C₂×C₄×C₁₉	—	4	1	no	—
143	C₂×C₄×C₁₉	—	8	2	no	—
144	C₂×C₄×C₁₉	—	8	1	no	—
145	C₂×C₄×C₁₉	—	8	1	no	—
146	C₂×C₄×C₁₉	—	8	1	no	—
147	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
149	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
150	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
151	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
152	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
153	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
156	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
157	C₂×Dic₁₉	—	8	1	no	—
158	C₂×Dic₁₉	—	8	1	no	—
159	C₂×Dic₁₉	—	8	1	no	—
160	C₈×C₁₉	—	4	2	no	—
161	C₁₉⋊C₈	—	4	2	no	—
162	C₁₉⋊C₈	—	4	2	no	—
163	C₁₉⋊C₈	—	8	1	no	—
164	#3036	D₁₉	1	1	yes	characteristic
165	#2975	—	19	1	no	—
166	#1804	—	19	1	no	—
167	#1517	—	19	1	no	—
168	#1480	—	19	1	no	—
169	#1301	—	19	1	no	—
170	#1286	—	19	1	no	—
171	C₂²×C₁₉	—	2	1	no	—
172	C₂²×C₁₉	—	2	1	no	—
173	C₂²×C₁₉	—	4	2	no	—
174	C₂²×C₁₉	—	4	2	no	—
175	C₂²×C₁₉	—	4	2	no	—
176	C₂²×C₁₉	—	8	2	no	—
177	C₂²×C₁₉	—	8	2	no	—
178	C₂²×C₁₉	—	8	1	no	—
179	C₂²×C₁₉	—	8	1	no	—
180	C₂²×C₁₉	—	8	1	no	—
181	C₂²×C₁₉	#386	1	1	yes	characteristic
182	C₂²×C₁₉	#468	1	1	yes	characteristic
183	C₂²×C₁₉	#469	1	1	yes	characteristic
184	C₄×C₁₉	—	2	1	no	—
185	C₄×C₁₉	—	2	1	no	—
186	C₄×C₁₉	—	2	1	no	—
187	C₄×C₁₉	—	2	1	no	—
188	C₄×C₁₉	—	4	2	no	—
189	C₄×C₁₉	—	8	1	no	—
190	C₄×C₁₉	#417	1	1	yes	characteristic
191	C₄×C₁₉	#418	1	1	yes	characteristic
192	C₄×C₁₉	#468	1	1	yes	characteristic
193	C₄×C₁₉	#469	1	1	yes	characteristic
194	Dic₁₉	—	16	1	no	—
195	C₂×AGL(1,ℤ/8)	—	2	1	no	—
196	#532	C₂×D₁₉	1	1	yes	characteristic
197	C₂×C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
198	#463	—	2	2	no	—
199	#461	Dic₁₉	1	2	yes	complemented
200	#459	Dic₁₉	1	2	yes	complemented
201	#436	—	38	2	no	—
202	#436	—	38	2	no	—
203	#431	—	19	1	no	—
204	#431	—	19	1	no	—
205	#423	C₂×D₁₉	1	1	yes	characteristic
206	C₂×C₄⋊C₈	—	19	1	no	—
207	#417	—	2	1	no	—
208	C₂×#102	—	38	1	no	—
209	#407	—	38	2	no	—
210	#382	—	19	1	no	—
211	#340	—	38	1	no	—
212	#327	—	38	2	no	—
213	D₄⋊C₈	—	38	2	no	—
214	C₂×C₁₉	—	2	1	no	—
215	C₂×C₁₉	—	2	1	no	—
216	C₂×C₁₉	—	8	2	no	—
217	C₂×C₁₉	—	16	1	no	—
218	C₂×C₁₉	#1881	1	1	yes	characteristic
219	C₂×C₁₉	#1895	1	1	yes	characteristic
220	C₂×C₁₉	#1904	1	1	yes	characteristic
221	C₂×#41	—	2	1	no	—
222	C₂²×D₄	C₂²⋊D₁₉	1	1	yes	characteristic
223	C₂³×C₄	—	2	1	no	—
224	AGL(1,ℤ/8)	—	4	2	no	—
225	AGL(1,ℤ/8)	—	4	2	no	—
226	C₂×SD₁₆	—	2	2	no	—
227	C₂×D₈	—	2	2	no	—
228	C₂×M₁₆	—	2	1	no	—
229	C₂×M₁₆	—	38	4	no	—
230	C₂²×C₈	—	38	1	no	—
231	C₄⋊Q₈	C₂×Dic₁₉	1	1	yes	characteristic
232	#127	C₂×Dic₁₉	1	1	yes	characteristic
233	#124	—	2	1	no	—
234	#121	—	2	1	no	—
235	#121	—	2	1	no	—
236	C₄×D₄	—	2	1	no	—
237	C₄×D₄	—	2	1	no	—
238	C₄×D₄	—	2	1	no	—
239	C₄×D₄	—	2	1	no	—
240	C₄×D₄	C₂×Dic₁₉	1	2	yes	—
241	C₄×D₄	C₂²⋊D₁₉	1	2	yes	—
242	C₂×C₄⋊C₄	—	38	1	no	—
243	C₂×C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
244	C₂×C₂²⋊C₄	—	2	1	no	—
245	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
246	C₄⋊C₈	—	2	1	no	—
247	C₄⋊C₈	—	38	1	no	—
248	C₄⋊C₈	—	38	1	no	—
249	C₄⋊C₈	—	38	1	no	—
250	C₄⋊C₈	—	38	1	no	—
251	C₄⋊C₈	C₂×Dic₁₉	1	2	yes	—
252	#103	—	2	2	no	—
253	#102	—	2	2	no	—
254	#102	—	76	2	no	—
255	C₂²⋊C₈	—	19	2	no	—
256	C₂²⋊C₈	—	38	4	no	—
257	C₂²⋊C₈	—	38	1	no	—
258	C₄×C₈	—	19	2	no	—
259	C₄×C₈	—	38	1	no	—
260	#95	—	38	1	no	—
261	#95	—	38	1	no	—
262	C₁₉	#9309	1	1	yes	characteristic, complemented, Sylow
263	C₂⁴	—	2	1	no	—
264	#41	—	4	1	no	—
265	#41	—	4	1	no	—
266	#41	—	4	1	no	—
267	#41	—	4	1	no	—
268	C₂×Q₈	—	2	1	no	—
269	C₂×D₄	—	2	1	no	—
270	C₂×D₄	—	2	1	no	—
271	C₂×D₄	—	2	1	no	—
272	C₂×D₄	—	2	1	no	—
273	C₂×D₄	—	2	1	no	—
274	C₂×D₄	—	2	1	no	—
275	C₂×D₄	—	2	1	no	—
276	C₂×D₄	—	2	1	no	—
277	C₂×D₄	—	4	1	no	—
278	C₂×D₄	C₂²⋊Dic₁₉	1	2	yes	—
279	C₂²×C₄	—	2	1	no	—
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	4	1	no	—
283	C₂²×C₄	—	8	2	no	—
284	C₂²×C₄	—	76	1	no	—
285	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
286	C₂²×C₄	#64543	1	1	yes	characteristic
287	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
288	SD₁₆	—	4	4	no	—
289	D₈	—	4	4	no	—
290	M₁₆	—	4	1	no	—
291	M₁₆	—	4	1	no	—
292	M₁₆	—	152	4	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	38	2	no	—
295	C₂×C₈	—	38	2	no	—
296	C₂×C₈	—	76	1	no	—
297	C₂×C₈	—	76	1	no	—
298	C₂×C₈	—	76	1	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	1	no	—
302	C₄⋊C₄	—	76	1	no	—
303	C₄⋊C₄	—	76	1	no	—
304	C₄⋊C₄	C₂²⋊Dic₁₉	1	2	yes	—
305	C₂²⋊C₄	—	2	2	no	—
306	C₂²⋊C₄	—	2	2	no	—
307	C₂²⋊C₄	—	4	1	no	—
308	C₂²⋊C₄	—	4	1	no	—
309	C₄²	D₄×D₁₉	1	1	yes	characteristic
310	C₄²	#64543	1	1	yes	characteristic
311	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
312	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
313	C₂³	—	2	2	no	—
314	C₂³	—	4	1	no	—
315	C₂³	—	4	1	no	—
316	C₂³	—	4	1	no	—
317	C₂³	—	8	1	no	—
318	C₂³	—	8	1	no	—
319	C₂³	#10597385	1	1	yes	characteristic
320	Q₈	—	4	1	no	—
321	Q₈	—	4	1	no	—
322	D₄	—	2	2	no	—
323	D₄	—	2	2	no	—
324	D₄	—	4	2	no	—
325	D₄	—	4	1	no	—
326	D₄	—	4	1	no	—
327	D₄	—	4	1	no	—
328	D₄	—	4	1	no	—
329	D₄	—	8	1	no	—
330	D₄	—	8	1	no	—
331	D₄	—	8	1	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	4	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	8	2	no	—
338	C₂×C₄	—	8	1	no	—
339	C₂×C₄	—	8	1	no	—
340	C₂×C₄	—	8	1	no	—
341	C₂×C₄	—	152	1	no	—
342	C₂×C₄	—	152	1	no	—
343	C₂×C₄	—	152	1	no	—
344	C₂×C₄	#10597378	1	1	yes	characteristic
345	C₂×C₄	#10597378	1	1	yes	characteristic
346	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
347	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
348	C₂×C₄	#10597381	1	1	yes	characteristic
349	C₂×C₄	#10597383	1	1	yes	characteristic
350	C₂×C₄	#10597385	1	1	yes	characteristic
351	C₂×C₄	#10597395	1	1	yes	characteristic
352	C₂×C₄	#10597397	1	1	yes	characteristic
353	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
354	C₈	—	4	2	no	—
355	C₈	—	76	2	no	—
356	C₈	—	76	2	no	—
357	C₈	—	152	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	4	2	no	—
361	C₂²	—	4	2	no	—
362	C₂²	—	4	2	no	—
363	C₂²	—	8	2	no	—
364	C₂²	—	8	2	no	—
365	C₂²	—	8	1	no	—
366	C₂²	—	8	1	no	—
367	C₂²	—	8	1	no	—
368	C₂²	#11926424	1	1	yes	characteristic
369	C₂²	#11926788	1	1	yes	characteristic
370	C₂²	#11926795	1	1	yes	characteristic
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	4	2	no	—
376	C₄	—	8	1	no	—
377	C₄	—	304	1	no	—
378	C₄	#11926545	1	1	yes	characteristic
379	C₄	#11926549	1	1	yes	characteristic
380	C₄	#11926788	1	1	yes	characteristic
381	C₄	#11926795	1	1	yes	characteristic
382	C₂	—	2	1	no	—
383	C₂	—	2	1	no	—
384	C₂	—	8	2	no	—
385	C₂	—	16	1	no	—
386	C₂	#14563999	1	1	yes	characteristic
387	C₂	#14564074	1	1	yes	characteristic
388	C₂	#14564124	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926424
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting, prime radical	C₁₉×#3036	C₂
socle	C₂²×C₁₉	#386
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#3036	D₁₉
hypocenter	C₁₉	#9309
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3036

Complements

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

N	H ≅ G/N	type
1. #14572676	C₂	semidirect
2. #14563645	C₂	semidirect
3. #14562195	C₂	semidirect
4. #14562034	C₂	semidirect
5. #14561530	C₂	semidirect
6. #14561495	C₂	semidirect
23. C₁₉×#461	C₄	semidirect
24. C₁₉×#459	C₄	semidirect
199. #461	Dic₁₉	semidirect
200. #459	Dic₁₉	semidirect
262. C₁₉	#9309	semidirect

Maximal subgroups

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

subgroup	index
#14572676	2
#14563645	2
#14562195	2
#14562034	2
#14561530	2
#14561495	2
C₁₉×#3036	2
#9309	19