Subgroups — #18075661

Order 4864.

482 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	C₂×D₄×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#14563806	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562381	C₂	1	1	yes	maximal, characteristic, complemented
4	#14562060	C₂	1	1	yes	maximal, characteristic
5	#14561531	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561530	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1486	C₂	1	1	yes	maximal, characteristic
8	C₂³×C₁₉⋊C₈	—	2	1	no	—
9	D₄×C₁₉⋊C₈	—	2	1	no	—
10	D₄×C₁₉⋊C₈	—	2	1	no	—
11	D₄×C₁₉⋊C₈	C₄	1	2	yes	—
12	D₄×C₁₉⋊C₈	C₄	1	2	yes	—
13	#11926597	C₂²	1	1	yes	characteristic
14	C₂×#10597279	—	2	1	no	—
15	C₂×#10597279	—	2	1	no	—
16	#11926568	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
19	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
20	#11926296	—	2	1	no	—
21	#11926294	—	2	1	no	—
22	D₄⋊(C₁₉⋊C₈)	—	2	2	no	—
23	C₂×C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
24	C₁₉×#431	C₂²	1	1	yes	characteristic
25	C₁₉×#423	C₂²	1	1	yes	characteristic
26	C₁₉×D₄⋊C₈	—	2	2	no	—
27	C₂²×D₄×C₁₉	Q₈	1	1	yes	characteristic
28	C₂³×C₄×C₁₉	—	2	1	no	—
29	C₂×M₁₆×C₁₉	—	2	1	no	—
30	C₄×D₄×C₁₉	—	2	1	no	—
31	C₄×D₄×C₁₉	—	2	1	no	—
32	C₄×D₄×C₁₉	C₂×C₄	1	2	yes	—
33	C₄×D₄×C₁₉	D₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₁₉	Q₈	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
36	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
37	C₂×#64520	—	2	1	no	—
38	C₂²×C₁₉⋊C₈	—	2	1	no	—
39	C₂²×C₁₉⋊C₈	—	2	2	no	—
40	C₂²×C₁₉⋊C₈	—	2	2	no	—
41	C₂²×C₁₉⋊C₈	—	4	2	no	—
42	C₂²×C₁₉⋊C₈	—	4	2	no	—
43	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
44	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
45	C₁₉×C₄⋊C₈	—	2	1	no	—
46	C₁₉×C₄⋊C₈	—	2	1	no	—
47	C₁₉×C₄⋊C₈	—	2	1	no	—
48	C₁₉×C₂²⋊C₈	—	2	1	no	—
49	C₄×C₈×C₁₉	—	2	1	no	—
50	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
51	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
52	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
53	#10597279	—	4	2	no	—
54	#10597279	—	4	2	no	—
55	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
56	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
57	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
58	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
59	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
60	C₄×C₁₉⋊C₈	—	2	1	no	—
61	C₄×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
62	C₄×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
63	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
64	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
65	C₂⁴×C₁₉	—	2	1	no	—
66	C₂×D₄×C₁₉	—	2	4	no	—
67	C₂×D₄×C₁₉	—	2	1	no	—
68	C₂×D₄×C₁₉	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₁₉	—	2	1	no	—
70	C₂²×C₄×C₁₉	—	2	2	no	—
71	C₂²×C₄×C₁₉	—	2	2	no	—
72	C₂²×C₄×C₁₉	—	4	1	no	—
73	C₂²×C₄×C₁₉	—	4	1	no	—
74	C₂²×C₄×C₁₉	—	4	1	no	—
75	C₂²×C₄×C₁₉	—	4	1	no	—
76	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
78	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
79	M₁₆×C₁₉	—	4	1	no	—
80	M₁₆×C₁₉	—	4	1	no	—
81	C₂×C₈×C₁₉	—	4	1	no	—
82	C₂×C₈×C₁₉	—	4	1	no	—
83	C₄⋊C₄×C₁₉	—	2	1	no	—
84	C₄⋊C₄×C₁₉	C₄⋊C₄	1	2	yes	—
85	C₂²⋊C₄×C₁₉	—	2	2	no	—
86	C₂²⋊C₄×C₁₉	—	2	2	no	—
87	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₉	#41	1	1	yes	characteristic
91	#64520	—	4	1	no	—
92	#64520	—	4	1	no	—
93	C₂×C₁₉⋊C₈	—	2	1	no	—
94	C₂×C₁₉⋊C₈	—	2	1	no	—
95	C₂×C₁₉⋊C₈	—	2	2	no	—
96	C₂×C₁₉⋊C₈	—	4	1	no	—
97	C₂×C₁₉⋊C₈	—	4	1	no	—
98	C₂×C₁₉⋊C₈	—	4	4	no	—
99	C₂×C₁₉⋊C₈	—	4	4	no	—
100	C₂×C₁₉⋊C₈	—	4	1	no	—
101	C₂×C₁₉⋊C₈	—	4	1	no	—
102	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₁₉⋊C₈	D₈	1	2	yes	—
107	C₂×C₁₉⋊C₈	SD₁₆	1	2	yes	—
108	#8658	—	19	1	no	maximal, Sylow
109	C₂³×C₁₉	—	2	4	no	—
110	C₂³×C₁₉	—	2	2	no	—
111	C₂³×C₁₉	—	4	2	no	—
112	C₂³×C₁₉	—	4	1	no	—
113	C₂³×C₁₉	—	4	1	no	—
114	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
115	D₄×C₁₉	—	2	4	no	—
116	D₄×C₁₉	—	2	2	no	—
117	D₄×C₁₉	#108	1	4	yes	complemented
118	C₂×C₄×C₁₉	—	2	2	no	—
119	C₂×C₄×C₁₉	—	2	2	no	—
120	C₂×C₄×C₁₉	—	4	1	no	—
121	C₂×C₄×C₁₉	—	4	1	no	—
122	C₂×C₄×C₁₉	—	4	1	no	—
123	C₂×C₄×C₁₉	—	4	2	no	—
124	C₂×C₄×C₁₉	—	4	2	no	—
125	C₂×C₄×C₁₉	—	4	2	no	—
126	C₂×C₄×C₁₉	—	4	2	no	—
127	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
131	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
133	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
135	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
137	C₈×C₁₉	—	8	1	no	—
138	C₈×C₁₉	—	8	1	no	—
139	C₁₉⋊C₈	—	2	1	no	—
140	C₁₉⋊C₈	—	2	1	no	—
141	C₁₉⋊C₈	—	4	2	no	—
142	C₁₉⋊C₈	—	8	1	no	—
143	C₁₉⋊C₈	—	8	1	no	—
144	C₁₉⋊C₈	#102	1	2	yes	—
145	C₁₉⋊C₈	#102	1	2	yes	—
146	C₂×C₈×D₄	—	19	1	no	—
147	#1848	—	19	1	no	—
148	#1564	—	19	1	no	—
149	#1486	—	19	1	no	—
150	#1486	D₁₉	1	1	yes	characteristic
151	#1301	—	19	1	no	—
152	#1301	—	19	1	no	—
153	C₂²×C₁₉	—	2	1	no	—
154	C₂²×C₁₉	—	2	1	no	—
155	C₂²×C₁₉	—	2	4	no	—
156	C₂²×C₁₉	—	4	4	no	—
157	C₂²×C₁₉	—	4	4	no	—
158	C₂²×C₁₉	—	4	2	no	—
159	C₂²×C₁₉	—	4	2	no	—
160	C₂²×C₁₉	#380	1	1	yes	characteristic
161	C₂²×C₁₉	#474	1	1	yes	characteristic
162	C₂²×C₁₉	#476	1	1	yes	characteristic
163	C₄×C₁₉	—	2	1	no	—
164	C₄×C₁₉	—	2	1	no	—
165	C₄×C₁₉	—	2	1	no	—
166	C₄×C₁₉	—	4	2	no	—
167	C₄×C₁₉	C₂×#102	1	1	yes	characteristic
168	C₄×C₁₉	#417	1	1	yes	characteristic
169	C₄×C₁₉	C₂×#108	1	2	yes	—
170	C₄×C₁₉	#459	1	1	yes	characteristic
171	C₄×C₁₉	#461	1	1	yes	characteristic
172	C₂³×C₈	—	38	1	no	—
173	C₂×C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
174	C₈×D₄	—	19	2	no	—
175	C₈×D₄	—	19	2	no	—
176	C₈×D₄	—	38	1	no	—
177	C₈×D₄	—	38	1	no	—
178	#431	—	19	1	no	—
179	#431	C₂×D₁₉	1	1	yes	characteristic
180	C₂×#108	—	38	1	no	—
181	C₂×#108	—	38	1	no	—
182	#423	—	19	1	no	—
183	#423	C₂×D₁₉	1	1	yes	characteristic
184	C₂×C₄⋊C₈	—	19	1	no	—
185	C₂×C₂²⋊C₈	—	38	1	no	—
186	C₂×C₄×C₈	—	19	1	no	—
187	#335	—	38	1	no	—
188	#334	—	38	1	no	—
189	D₄⋊C₈	—	2	2	no	—
190	D₄⋊C₈	—	38	2	no	—
191	C₂×C₁₉	—	2	1	no	—
192	C₂×C₁₉	—	2	1	no	—
193	C₂×C₁₉	—	4	4	no	—
194	C₂×C₁₉	#1802	1	1	yes	characteristic
195	C₂×C₁₉	#1810	1	1	yes	characteristic
196	C₂×C₁₉	#1814	1	1	yes	characteristic
197	C₂²×D₄	Dic₃₈	1	1	yes	characteristic
198	C₂³×C₄	—	2	1	no	—
199	C₂×M₁₆	—	2	1	no	—
200	C₂×M₁₆	—	38	1	no	—
201	C₂²×C₈	—	19	1	no	—
202	C₂²×C₈	—	19	1	no	—
203	C₂²×C₈	—	38	2	no	—
204	C₂²×C₈	—	38	2	no	—
205	C₂²×C₈	—	38	1	no	—
206	C₂²×C₈	—	76	2	no	—
207	C₂²×C₈	—	76	2	no	—
208	C₄×D₄	—	2	1	no	—
209	C₄×D₄	—	2	1	no	—
210	C₄×D₄	C₄×D₁₉	1	2	yes	—
211	C₄×D₄	C₂²⋊D₁₉	1	2	yes	—
212	C₂×C₄⋊C₄	Dic₃₈	1	1	yes	characteristic
213	C₂×C₂²⋊C₄	—	2	1	no	—
214	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
215	#108	—	76	2	no	—
216	#108	—	76	2	no	—
217	C₄⋊C₈	—	2	1	no	—
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	19	2	no	—
221	C₄⋊C₈	—	38	1	no	—
222	C₄⋊C₈	—	38	1	no	—
223	C₄⋊C₈	—	38	1	no	—
224	C₄⋊C₈	—	38	1	no	—
225	C₂²⋊C₈	—	2	1	no	—
226	C₂²⋊C₈	—	38	2	no	—
227	C₂²⋊C₈	—	38	2	no	—
228	C₂²⋊C₈	—	38	1	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	19	1	no	—
231	C₄×C₈	—	19	1	no	—
232	C₄×C₈	—	19	1	no	—
233	C₄×C₈	—	19	1	no	—
234	C₄×C₈	—	38	1	no	—
235	C₁₉	#8658	1	1	yes	characteristic, complemented, Sylow
236	C₂⁴	—	2	1	no	—
237	C₂×D₄	—	2	4	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	Dic₁₉⋊C₄	1	2	yes	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	—	4	1	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	—	4	1	no	—
247	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
248	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
249	C₂²×C₄	#64543	1	1	yes	characteristic
250	M₁₆	—	4	1	no	—
251	M₁₆	—	4	1	no	—
252	M₁₆	—	76	1	no	—
253	M₁₆	—	76	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	19	1	no	—
257	C₂×C₈	—	19	1	no	—
258	C₂×C₈	—	19	2	no	—
259	C₂×C₈	—	19	2	no	—
260	C₂×C₈	—	19	1	no	—
261	C₂×C₈	—	19	1	no	—
262	C₂×C₈	—	38	1	no	—
263	C₂×C₈	—	38	1	no	—
264	C₂×C₈	—	38	2	no	—
265	C₂×C₈	—	76	4	no	—
266	C₂×C₈	—	76	4	no	—
267	C₂×C₈	—	76	1	no	—
268	C₂×C₈	—	76	1	no	—
269	C₂×C₈	—	76	1	no	—
270	C₂×C₈	—	76	1	no	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	Dic₁₉⋊C₄	1	2	yes	—
273	C₂²⋊C₄	—	2	2	no	—
274	C₂²⋊C₄	—	2	2	no	—
275	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
276	C₄²	D₄×D₁₉	1	1	yes	characteristic
277	C₄²	#64543	1	1	yes	characteristic
278	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
279	C₂³	—	2	4	no	—
280	C₂³	—	2	2	no	—
281	C₂³	—	4	2	no	—
282	C₂³	—	4	1	no	—
283	C₂³	—	4	1	no	—
284	C₂³	#10597324	1	1	yes	characteristic
285	D₄	—	2	4	no	—
286	D₄	—	2	2	no	—
287	D₄	#10597279	1	4	yes	complemented
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	#10597323	1	1	yes	characteristic
298	C₂×C₄	#10597324	1	1	yes	characteristic
299	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
300	C₂×C₄	D₈×D₁₉	1	1	yes	characteristic
301	C₂×C₄	#10597357	1	1	yes	characteristic
302	C₂×C₄	SD₁₆×D₁₉	1	1	yes	characteristic
303	C₂×C₄	#10597361	1	1	yes	characteristic
304	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
305	C₂×C₄	#10597385	1	1	yes	characteristic
306	C₂×C₄	#10597385	1	1	yes	characteristic
307	C₈	—	8	1	no	—
308	C₈	—	8	1	no	—
309	C₈	—	19	2	no	—
310	C₈	—	19	2	no	—
311	C₈	—	38	1	no	—
312	C₈	—	38	1	no	—
313	C₈	—	76	2	no	—
314	C₈	—	152	1	no	—
315	C₈	—	152	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	4	no	—
319	C₂²	—	4	4	no	—
320	C₂²	—	4	4	no	—
321	C₂²	—	4	2	no	—
322	C₂²	—	4	2	no	—
323	C₂²	#11926402	1	1	yes	characteristic
324	C₂²	#11926822	1	1	yes	characteristic
325	C₂²	#11926832	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	4	2	no	—
330	C₄	#102×D₁₉	1	1	yes	characteristic
331	C₄	#11926542	1	1	yes	characteristic
332	C₄	C₂×#10597279	1	2	yes	—
333	C₄	#11926737	1	1	yes	characteristic
334	C₄	#11926747	1	1	yes	characteristic
335	C₂	—	2	1	no	—
336	C₂	—	2	1	no	—
337	C₂	—	4	4	no	—
338	C₂	#14563633	1	1	yes	characteristic
339	C₂	#14563671	1	1	yes	characteristic
340	C₂	#14563687	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597385
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1486	C₂
socle	C₂²×C₁₉	#380
prime radical	C₂²×D₄×C₁₉	Q₈
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1486	D₁₉
hypocenter	C₁₉	#8658
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1486

Complements

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

N	H ≅ G/N	type
2. #14563806	C₂	semidirect
3. #14562381	C₂	semidirect
5. #14561531	C₂	semidirect
6. #14561530	C₂	semidirect
117. D₄×C₁₉	#108	semidirect
235. C₁₉	#8658	semidirect
287. D₄	#10597279	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×C₁₉⋊C₈	2
#14563806	2
#14562381	2
#14562060	2
#14561531	2
#14561530	2
C₁₉×#1486	2
#8658	19