Subgroups — #18072648

Order 4864.

686 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	C₂²×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	#14566042	C₂	1	1	yes	maximal, characteristic
3	#14561519	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₉×#432	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₁₉⋊C₈	—	2	1	no	—
6	C₂×#10597372	C₄	1	2	yes	—
7	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
8	C₂×C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
9	C₂×C₄⋊(C₁₉⋊C₈)	C₄	1	4	yes	complemented
10	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
11	#11926414	C₂²	1	2	yes	—
12	#11926410	C₄	1	2	yes	—
13	#11926404	—	2	1	no	—
14	#11926404	—	2	1	no	—
15	C₄×C₄⋊Dic₁₉	—	2	1	no	—
16	C₄×C₄⋊Dic₁₉	—	2	1	no	—
17	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#432	—	2	2	no	—
19	C₁₉×#432	C₄	1	4	yes	complemented
20	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₁₉×C₂²⋊C₈	—	2	1	no	—
22	C₂×C₁₉×#97	—	2	1	no	—
23	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₉	—	2	2	no	—
27	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₉	D₄	1	2	yes	—
33	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
34	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
35	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
36	#10597372	—	4	4	no	—
37	C₂×C₄⋊Dic₁₉	—	2	2	no	—
38	C₂×C₄⋊Dic₁₉	—	2	2	no	—
39	C₂×C₄⋊Dic₁₉	C₂×C₄	1	4	yes	—
40	C₂×C₄×Dic₁₉	—	2	2	no	—
41	C₂²×C₁₉⋊C₈	—	2	4	no	—
42	C₂²×C₁₉⋊C₈	—	2	2	no	—
43	C₂²×C₁₉⋊C₈	—	4	4	no	—
44	C₁₉×C₄⋊C₈	—	2	2	no	—
45	C₁₉×C₄⋊C₈	C₂×C₄	1	4	yes	—
46	C₁₉×C₂²⋊C₈	—	2	4	no	—
47	C₁₉×#97	—	2	4	no	—
48	C₁₉⋊#95	—	2	2	no	—
49	C₁₉⋊#95	—	2	2	no	—
50	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
51	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
52	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	4	yes	—
53	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	—	2	2	no	—
55	C₂²×C₄×C₁₉	—	2	2	no	—
56	C₂²×C₄×C₁₉	—	2	4	no	—
57	C₂²×C₄×C₁₉	—	2	2	no	—
58	C₂²×C₄×C₁₉	—	2	2	no	—
59	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
60	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×Dic₁₉	—	2	4	no	—
72	C₂×C₈×C₁₉	—	2	4	no	—
73	C₂×C₈×C₁₉	—	4	4	no	—
74	C₄²×C₁₉	C₄²	1	2	yes	—
75	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
76	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
78	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
79	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
80	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
84	C₂²⋊Dic₁₉	—	4	4	no	—
85	C₄⋊Dic₁₉	—	4	4	no	—
86	C₄⋊Dic₁₉	—	4	2	no	—
87	C₄⋊Dic₁₉	—	4	2	no	—
88	C₄⋊Dic₁₉	—	4	4	no	—
89	C₄×Dic₁₉	—	4	4	no	—
90	C₂×C₁₉⋊C₈	—	2	4	no	—
91	C₂×C₁₉⋊C₈	—	4	8	no	—
92	C₂×C₁₉⋊C₈	—	4	4	no	—
93	#8044	—	19	1	no	maximal, Sylow
94	C₂³×C₁₉	—	2	2	no	—
95	C₂³×C₁₉	—	2	2	no	—
96	C₂³×C₁₉	#95	1	2	yes	—
97	C₂³×C₁₉	#100	1	2	yes	—
98	C₂³×C₁₉	#101	1	2	yes	—
99	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₉	—	2	4	no	—
101	C₂×C₄×C₁₉	—	2	4	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	2	no	—
107	C₂×C₄×C₁₉	—	2	2	no	—
108	C₂×C₄×C₁₉	—	2	4	no	—
109	C₂×C₄×C₁₉	—	2	2	no	—
110	C₂×C₄×C₁₉	—	2	2	no	—
111	C₂×C₄×C₁₉	—	2	4	no	—
112	C₂×C₄×C₁₉	—	2	4	no	—
113	C₂×C₄×C₁₉	#95	1	2	yes	—
114	C₂×C₄×C₁₉	#95	1	2	yes	—
115	C₂×C₄×C₁₉	#95	1	2	yes	—
116	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	#135	1	4	yes	—
123	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
124	C₂×C₄×C₁₉	#137	1	2	yes	—
125	C₂×Dic₁₉	—	4	4	no	—
126	C₂×Dic₁₉	—	4	4	no	—
127	C₂×Dic₁₉	—	4	4	no	—
128	C₈×C₁₉	—	4	4	no	—
129	C₁₉⋊C₈	—	4	4	no	—
130	C₂×#432	D₁₉	1	1	yes	characteristic
131	C₂²×C₄⋊C₈	—	19	1	no	—
132	#2285	—	19	1	no	—
133	#1296	—	19	4	no	—
134	C₂²×C₁₉	—	2	4	no	—
135	C₂²×C₁₉	—	2	4	no	—
136	C₂²×C₁₉	—	2	2	no	—
137	C₂²×C₁₉	—	2	2	no	—
138	C₂²×C₁₉	#343	1	4	yes	—
139	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
140	C₂²×C₁₉	C₂×#100	1	1	yes	characteristic
141	C₂²×C₁₉	C₂×#101	1	1	yes	characteristic
142	C₂²×C₁₉	#416	1	2	yes	—
143	C₂²×C₁₉	#417	1	2	yes	—
144	C₄×C₁₉	—	2	2	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	#416	1	2	yes	—
150	C₄×C₁₉	#418	1	1	yes	characteristic
151	C₄×C₁₉	#419	1	1	yes	characteristic
152	C₄×C₁₉	#427	1	2	yes	—
153	C₄×C₁₉	#428	1	2	yes	—
154	Dic₁₉	—	8	4	no	—
155	C₂³×C₈	—	38	1	no	—
156	C₂×#117	—	19	2	no	—
157	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
158	#432	—	2	2	no	—
159	#432	Dic₁₉	1	4	yes	complemented
160	C₂×C₄⋊C₈	—	19	4	no	—
161	C₂×C₄⋊C₈	—	19	2	no	—
162	C₂×C₄⋊C₈	—	38	1	no	—
163	C₂×C₄⋊C₈	—	38	2	no	—
164	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
165	C₂×C₂²⋊C₈	—	2	1	no	—
166	C₂×#97	—	2	1	no	—
167	#384	—	19	2	no	—
168	#382	—	19	2	no	—
169	#380	—	38	1	no	—
170	#380	—	38	1	no	—
171	C₄×C₄⋊C₄	—	38	1	no	—
172	C₄×C₄⋊C₄	—	38	1	no	—
173	C₂×C₁₉	—	2	4	no	—
174	C₂×C₁₉	#1551	1	2	yes	—
175	C₂×C₁₉	#1552	1	2	yes	—
176	C₂×C₁₉	#1735	1	1	yes	characteristic
177	C₂×C₁₉	#1736	1	1	yes	characteristic
178	C₂×C₁₉	C₂×#343	1	1	yes	characteristic
179	C₂³×C₄	C₄×D₁₉	1	1	yes	characteristic
180	C₂³×C₄	D₇₆	1	1	yes	characteristic
181	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	38	4	no	—
184	C₂²×C₈	—	38	2	no	—
185	C₂²×C₈	—	76	4	no	—
186	#117	—	76	4	no	—
187	C₂×C₄⋊C₄	—	19	4	no	—
188	C₂×C₄⋊C₄	—	38	2	no	—
189	C₂×C₄⋊C₄	—	38	2	no	—
190	C₂×C₂²⋊C₄	—	38	2	no	—
191	C₂×C₄²	—	38	2	no	—
192	C₂×C₄²	Dic₃₈	1	2	yes	—
193	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
194	C₂×C₄²	C₄×D₁₉	1	2	yes	—
195	C₂×C₄²	D₇₆	1	1	yes	characteristic
196	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
197	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
198	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
199	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
200	C₄⋊C₈	—	2	2	no	—
201	C₄⋊C₈	—	19	4	no	—
202	C₄⋊C₈	—	38	4	no	—
203	C₄⋊C₈	—	38	2	no	—
204	C₄⋊C₈	C₂×Dic₁₉	1	4	yes	—
205	C₂²⋊C₈	—	2	4	no	—
206	#97	—	2	4	no	—
207	#95	—	38	2	no	—
208	#95	—	38	2	no	—
209	C₁₉	#8044	1	1	yes	characteristic, complemented, Sylow
210	C₂⁴	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	38	4	no	—
217	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
218	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
219	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
220	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
221	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
222	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
223	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
224	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
225	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
226	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	4	4	no	—
231	C₂×C₈	—	38	4	no	—
232	C₂×C₈	—	76	8	no	—
233	C₂×C₈	—	76	4	no	—
234	C₄⋊C₄	—	76	4	no	—
235	C₄⋊C₄	—	76	4	no	—
236	C₄⋊C₄	—	76	2	no	—
237	C₄⋊C₄	—	76	2	no	—
238	C₂²⋊C₄	—	76	4	no	—
239	C₄²	—	76	4	no	—
240	C₄²	C₄×Dic₁₉	1	2	yes	—
241	C₄²	Dic₁₉⋊C₄	1	2	yes	—
242	C₄²	Dic₁₉⋊C₄	1	2	yes	—
243	C₄²	C₄⋊Dic₁₉	1	2	yes	—
244	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
245	C₄²	C₂²⋊Dic₁₉	1	2	yes	—
246	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
247	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
248	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
249	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
250	C₂³	—	2	2	no	—
251	C₂³	—	2	2	no	—
252	C₂³	#10597280	1	2	yes	—
253	C₂³	#10597281	1	2	yes	—
254	C₂³	C₁₉⋊#95	1	2	yes	—
255	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	76	4	no	—
270	C₂×C₄	—	76	4	no	—
271	C₂×C₄	—	76	4	no	—
272	C₂×C₄	C₁₉⋊#95	1	2	yes	—
273	C₂×C₄	C₁₉⋊#95	1	2	yes	—
274	C₂×C₄	C₁₉⋊#95	1	2	yes	—
275	C₂×C₄	#10597354	1	2	yes	—
276	C₂×C₄	#10597355	1	2	yes	—
277	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
278	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
279	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
280	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
281	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
282	C₂×C₄	#10597396	1	4	yes	—
283	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
284	C₈	—	4	4	no	—
285	C₈	—	76	4	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	4	no	—
290	C₂²	#11926312	1	4	yes	—
291	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
292	C₂²	C₂×#10597280	1	1	yes	characteristic
293	C₂²	C₂×#10597281	1	1	yes	characteristic
294	C₂²	#11926538	1	2	yes	—
295	C₂²	#11926546	1	2	yes	—
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	152	4	no	—
302	C₄	#11926540	1	2	yes	—
303	C₄	#11926550	1	1	yes	characteristic
304	C₄	#11926554	1	1	yes	characteristic
305	C₄	#11926580	1	2	yes	—
306	C₄	#11926586	1	2	yes	—
307	C₂	—	2	4	no	—
308	C₂	#14562331	1	2	yes	—
309	C₂	#14562336	1	2	yes	—
310	C₂	#14563351	1	1	yes	characteristic
311	C₂	#14563355	1	1	yes	characteristic
312	C₂	C₂×#11926312	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₁₉)⋊C₄
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#432	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#432	D₁₉
hypocenter	C₁₉	#8044
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#432

Complements

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

N	H ≅ G/N	type
3. #14561519	C₂	semidirect
9. C₂×C₄⋊(C₁₉⋊C₈)	C₄	semidirect
19. C₁₉×#432	C₄	semidirect
159. #432	Dic₁₉	semidirect
209. C₁₉	#8044	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊(C₁₉⋊C₈)	2
#14566042	2
#14561519	2
C₂×C₁₉×#432	2
#8044	19