Subgroups — #18072001

Order 4864.

722 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926597	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#14561470	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₁₉⋊C₈	—	2	1	no	—
6	#11926597	—	2	2	no	—
7	#11926597	C₄	1	4	yes	—
8	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
9	C₂×C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
10	C₂×C₄⋊(C₁₉⋊C₈)	C₄	1	4	yes	—
11	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
14	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
15	C₂²×M₁₆×C₁₉	C₄	1	2	yes	—
16	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₄×M₁₆×C₁₉	—	2	2	no	—
18	C₄×M₁₆×C₁₉	—	2	2	no	—
19	C₂×C₁₉×#97	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
21	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₉	Q₈	1	1	yes	characteristic
24	C₂×M₁₆×C₁₉	—	2	4	no	—
25	C₂×M₁₆×C₁₉	—	2	4	no	—
26	C₂×M₁₆×C₁₉	C₈	1	8	yes	complemented
27	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₉	D₄	1	2	yes	—
33	C₂×C₄²×C₁₉	D₄	1	2	yes	—
34	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
35	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
36	C₂²×C₁₉⋊C₈	—	2	4	no	—
37	C₂²×C₁₉⋊C₈	—	2	2	no	—
38	C₂²×C₁₉⋊C₈	—	2	2	no	—
39	C₂²×C₁₉⋊C₈	—	4	4	no	—
40	C₁₉×#97	—	2	4	no	—
41	C₄×C₈×C₁₉	—	2	2	no	—
42	C₄×C₈×C₁₉	—	2	2	no	—
43	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
44	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
45	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
46	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
47	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	4	yes	—
48	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	4	yes	—
49	C₄×C₁₉⋊C₈	—	2	4	no	—
50	C₂⁴×C₁₉	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	—	2	2	no	—
52	C₂²×C₄×C₁₉	—	2	2	no	—
53	C₂²×C₄×C₁₉	—	4	2	no	—
54	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
55	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂×C₈	1	4	yes	—
64	C₂²×C₄×C₁₉	M₁₆	1	4	yes	—
65	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
68	M₁₆×C₁₉	—	2	16	no	—
69	C₂×C₈×C₁₉	—	2	4	no	—
70	C₂×C₈×C₁₉	—	2	4	no	—
71	C₂×C₈×C₁₉	C₂×C₈	1	8	yes	complemented
72	C₄²×C₁₉	C₄²	1	2	yes	—
73	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
76	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
77	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
78	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₁₉⋊C₈	—	2	4	no	—
83	C₂×C₁₉⋊C₈	—	2	4	no	—
84	C₂×C₁₉⋊C₈	—	4	8	no	—
85	C₂×C₁₉⋊C₈	—	4	2	no	—
86	C₂×C₁₉⋊C₈	—	4	4	no	—
87	C₂×C₁₉⋊C₈	—	4	2	no	—
88	#7873	—	19	1	no	maximal, Sylow
89	C₂³×C₁₉	—	2	2	no	—
90	C₂³×C₁₉	—	2	2	no	—
91	C₂³×C₁₉	#95	1	2	yes	—
92	C₂³×C₁₉	C₄⋊C₈	1	4	yes	—
93	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₉	—	2	2	no	—
95	C₂×C₄×C₁₉	—	2	2	no	—
96	C₂×C₄×C₁₉	—	2	2	no	—
97	C₂×C₄×C₁₉	—	2	4	no	—
98	C₂×C₄×C₁₉	—	2	2	no	—
99	C₂×C₄×C₁₉	—	2	2	no	—
100	C₂×C₄×C₁₉	—	4	2	no	—
101	C₂×C₄×C₁₉	—	4	2	no	—
102	C₂×C₄×C₁₉	#95	1	2	yes	—
103	C₂×C₄×C₁₉	#95	1	2	yes	—
104	C₂×C₄×C₁₉	#95	1	2	yes	—
105	C₂×C₄×C₁₉	C₄×C₈	1	4	yes	—
106	C₂×C₄×C₁₉	#97	1	4	yes	—
107	C₂×C₄×C₁₉	C₂²⋊C₈	1	8	yes	—
108	C₂×C₄×C₁₉	C₄⋊C₈	1	4	yes	—
109	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₂²×C₈	1	2	yes	—
116	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
117	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
118	C₂×C₄×C₁₉	#137	1	2	yes	—
119	C₈×C₁₉	—	2	8	no	—
120	C₁₉⋊C₈	—	4	4	no	—
121	C₁₉⋊C₈	—	4	4	no	—
122	C₂×#431	—	19	1	no	—
123	C₂²×C₄⋊C₈	—	19	1	no	—
124	C₂×C₄×M₁₆	D₁₉	1	1	yes	characteristic
125	#1277	—	19	4	no	—
126	C₂²×C₁₉	—	2	8	no	—
127	C₂²×C₁₉	—	2	2	no	—
128	C₂²×C₁₉	—	2	2	no	—
129	C₂²×C₁₉	#336	1	4	yes	—
130	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
131	C₂²×C₁₉	C₂×C₄⋊C₈	1	2	yes	—
132	C₂²×C₁₉	#428	1	2	yes	—
133	C₂²×C₁₉	#430	1	2	yes	—
134	C₄×C₁₉	—	2	2	no	—
135	C₄×C₁₉	—	2	2	no	—
136	C₄×C₁₉	—	2	1	no	—
137	C₄×C₁₉	—	2	1	no	—
138	C₄×C₁₉	#336	1	4	yes	—
139	C₄×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
140	C₄×C₁₉	C₂×#97	1	1	yes	characteristic
141	C₄×C₁₉	C₂×C₂²⋊C₈	1	2	yes	—
142	C₄×C₁₉	#417	1	2	yes	—
143	C₄×C₁₉	#418	1	1	yes	characteristic
144	C₄×C₁₉	#419	1	1	yes	characteristic
145	C₂²×M₁₆	Dic₁₉	1	2	yes	—
146	C₂³×C₈	—	38	1	no	—
147	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
148	#431	—	19	4	no	—
149	#431	—	38	2	no	—
150	C₂×C₄⋊C₈	—	19	4	no	—
151	C₂×C₄⋊C₈	—	19	2	no	—
152	C₂×C₄⋊C₈	—	19	2	no	—
153	C₂×C₄⋊C₈	—	38	1	no	—
154	C₂×C₄⋊C₈	—	38	2	no	—
155	C₂×C₂²⋊C₈	—	38	1	no	—
156	C₄×M₁₆	—	2	2	no	—
157	C₄×M₁₆	—	2	2	no	—
158	C₂×#97	Dic₁₉	1	2	yes	—
159	C₂×C₄×C₈	—	38	1	no	—
160	C₂×C₄×C₈	C₂×D₁₉	1	2	yes	—
161	C₂×C₁₉	—	2	4	no	—
162	C₂×C₁₉	#1565	1	4	yes	—
163	C₂×C₁₉	C₂×#336	1	1	yes	characteristic
164	C₂×C₁₉	#1736	1	1	yes	characteristic
165	C₂×C₁₉	#1738	1	1	yes	characteristic
166	C₂³×C₄	Dic₃₈	1	1	yes	characteristic
167	C₂³×C₄	D₇₆	1	1	yes	characteristic
168	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
169	C₂×M₁₆	—	2	4	no	—
170	C₂×M₁₆	—	2	4	no	—
171	C₂×M₁₆	C₁₉⋊C₈	1	8	yes	complemented
172	C₂²×C₈	—	38	4	no	—
173	C₂²×C₈	—	38	2	no	—
174	C₂²×C₈	—	38	2	no	—
175	C₂²×C₈	—	76	4	no	—
176	C₂²×C₈	C₂×Dic₁₉	1	4	yes	—
177	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
178	C₂×C₄²	C₄×D₁₉	1	2	yes	—
179	C₂×C₄²	C₄×D₁₉	1	2	yes	—
180	C₂×C₄²	D₇₆	1	1	yes	characteristic
181	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
182	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
183	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
184	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
185	C₄⋊C₈	—	19	4	no	—
186	C₄⋊C₈	—	19	4	no	—
187	C₄⋊C₈	—	38	4	no	—
188	C₄⋊C₈	—	38	2	no	—
189	C₄⋊C₈	—	38	2	no	—
190	C₂²⋊C₈	—	38	4	no	—
191	#97	—	2	4	no	—
192	C₄×C₈	—	2	2	no	—
193	C₄×C₈	—	2	2	no	—
194	C₄×C₈	—	38	4	no	—
195	C₁₉	#7873	1	1	yes	characteristic, complemented, Sylow
196	C₂⁴	C₄⋊Dic₁₉	1	1	yes	characteristic
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	C₂×C₁₉⋊C₈	1	4	yes	—
201	C₂²×C₄	#64520	1	4	yes	—
202	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
203	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
204	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
205	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
206	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
207	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
208	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
209	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
210	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
211	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
213	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
214	M₁₆	—	2	16	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	38	4	no	—
218	C₂×C₈	—	38	4	no	—
219	C₂×C₈	—	76	8	no	—
220	C₂×C₈	—	76	2	no	—
221	C₂×C₈	—	76	4	no	—
222	C₂×C₈	—	76	2	no	—
223	C₂×C₈	C₂×C₁₉⋊C₈	1	8	yes	complemented
224	C₄²	C₄×Dic₁₉	1	2	yes	—
225	C₄²	Dic₁₉⋊C₄	1	2	yes	—
226	C₄²	Dic₁₉⋊C₄	1	2	yes	—
227	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
228	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
229	C₄²	C₂²⋊Dic₁₉	1	2	yes	—
230	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
231	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
232	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
233	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	C₄⋊(C₁₉⋊C₈)	1	4	yes	—
237	C₂³	C₁₉⋊#95	1	2	yes	—
238	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	C₄×C₁₉⋊C₈	1	4	yes	—
248	C₂×C₄	#10597260	1	4	yes	—
249	C₂×C₄	C₄⋊(C₁₉⋊C₈)	1	4	yes	—
250	C₂×C₄	C₂²⋊(C₁₉⋊C₈)	1	8	yes	—
251	C₂×C₄	C₁₉⋊#95	1	2	yes	—
252	C₂×C₄	C₁₉⋊#95	1	2	yes	—
253	C₂×C₄	C₁₉⋊#95	1	2	yes	—
254	C₂×C₄	#10597354	1	2	yes	—
255	C₂×C₄	#10597355	1	2	yes	—
256	C₂×C₄	C₂²×C₁₉⋊C₈	1	2	yes	—
257	C₂×C₄	C₂×#64520	1	2	yes	—
258	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
259	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
260	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
261	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
262	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
263	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
264	C₈	—	2	8	no	—
265	C₈	—	76	4	no	—
266	C₈	—	76	4	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	8	no	—
270	C₂²	#11926299	1	4	yes	—
271	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
272	C₂²	C₂×C₄⋊(C₁₉⋊C₈)	1	2	yes	—
273	C₂²	#11926586	1	2	yes	—
274	C₂²	#11926593	1	2	yes	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	#11926299	1	4	yes	—
280	C₄	C₂×C₄×C₁₉⋊C₈	1	1	yes	characteristic
281	C₄	C₂×#10597260	1	1	yes	characteristic
282	C₄	C₂×C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
283	C₄	#11926546	1	2	yes	—
284	C₄	#11926550	1	1	yes	characteristic
285	C₄	#11926554	1	1	yes	characteristic
286	C₂	—	2	4	no	—
287	C₂	#14562387	1	4	yes	—
288	C₂	C₂×#11926299	1	1	yes	characteristic
289	C₂	#14563355	1	1	yes	characteristic
290	C₂	#14563362	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₇₆
commutator	C₄×C₁₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₄×M₁₆×C₁₉	C₂
socle	C₂³×C₁₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₉	C₄⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×C₄×M₁₆	D₁₉
hypocenter	C₁₉	#7873
LCS	G ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #14561470	C₂	semidirect
26. C₂×M₁₆×C₁₉	C₈	semidirect
71. C₂×C₈×C₁₉	C₂×C₈	semidirect
171. C₂×M₁₆	C₁₉⋊C₈	semidirect
195. C₁₉	#7873	semidirect
223. C₂×C₈	C₂×C₁₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926597	2
C₂²×C₄⋊(C₁₉⋊C₈)	2
#14561470	2
C₂×C₄×M₁₆×C₁₉	2
#7873	19