Subgroups — #18072004

Order 4864.

758 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	C₂×#11926568	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#14561469	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₉×#423	C₂	1	1	yes	maximal, characteristic
5	C₂²×#64520	—	2	1	no	—
6	C₂³×C₁₉⋊C₈	C₄	1	2	yes	—
7	#11926568	—	2	2	no	—
8	#11926568	C₄	1	4	yes	—
9	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
10	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄×C₁₉⋊C₈	—	2	2	no	—
12	C₂×C₄×C₁₉⋊C₈	—	2	2	no	—
13	C₂×C₄×C₁₉⋊C₈	C₄	1	2	yes	—
14	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
15	C₂²×M₁₆×C₁₉	—	2	1	no	—
16	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#423	—	2	2	no	—
18	C₁₉×#423	C₄	1	4	yes	—
19	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
20	C₂×C₁₉×C₄⋊C₈	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	—	4	2	no	—
26	C₂×M₁₆×C₁₉	—	4	2	no	—
27	C₂²×C₈×C₁₉	—	2	2	no	—
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	C₂×#64520	—	2	4	no	—
37	C₂×#64520	—	4	2	no	—
38	C₂×#64520	—	4	2	no	—
39	C₂²×C₁₉⋊C₈	—	2	4	no	—
40	C₂²×C₁₉⋊C₈	—	2	8	no	—
41	C₂²×C₁₉⋊C₈	—	2	2	no	—
42	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
43	C₁₉×C₄⋊C₈	—	2	4	no	—
44	C₁₉×C₄⋊C₈	—	2	2	no	—
45	C₁₉×C₄⋊C₈	C₂×C₄	1	4	yes	—
46	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
47	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
48	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	4	yes	—
49	C₄×C₁₉⋊C₈	—	2	4	no	—
50	C₄×C₁₉⋊C₈	—	2	2	no	—
51	C₄×C₁₉⋊C₈	—	2	2	no	—
52	C₂⁴×C₁₉	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	—	2	4	no	—
54	C₂²×C₄×C₁₉	—	2	4	no	—
55	C₂²×C₄×C₁₉	C₄²	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	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
61	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₁₉	D₈	1	2	yes	—
65	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
66	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
67	C₂²×C₄×C₁₉	Q₁₆	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×Q₈	1	1	yes	characteristic
71	M₁₆×C₁₉	—	4	8	no	—
72	C₂×C₈×C₁₉	—	2	4	no	—
73	C₂×C₈×C₁₉	—	4	2	no	—
74	C₂×C₈×C₁₉	—	4	2	no	—
75	C₄²×C₁₉	C₄²	1	2	yes	—
76	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
79	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
80	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
81	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
85	#64520	—	4	8	no	—
86	C₂×C₁₉⋊C₈	—	2	16	no	—
87	C₂×C₁₉⋊C₈	—	2	4	no	—
88	C₂×C₁₉⋊C₈	—	2	4	no	—
89	C₂×C₁₉⋊C₈	—	2	4	no	—
90	C₂×C₁₉⋊C₈	—	4	2	no	—
91	C₂×C₁₉⋊C₈	—	4	2	no	—
92	C₂×C₁₉⋊C₈	M₁₆	1	8	yes	complemented
93	#7874	—	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	#106	1	2	yes	—
98	C₂³×C₁₉	#107	1	2	yes	—
99	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₉	—	2	2	no	—
101	C₂×C₄×C₁₉	—	2	2	no	—
102	C₂×C₄×C₁₉	—	2	4	no	—
103	C₂×C₄×C₁₉	—	2	4	no	—
104	C₂×C₄×C₁₉	—	2	4	no	—
105	C₂×C₄×C₁₉	—	2	4	no	—
106	C₂×C₄×C₁₉	#95	1	2	yes	—
107	C₂×C₄×C₁₉	#95	1	2	yes	—
108	C₂×C₄×C₁₉	#95	1	2	yes	—
109	C₂×C₄×C₁₉	#102	1	4	yes	—
110	C₂×C₄×C₁₉	#102	1	4	yes	—
111	C₂×C₄×C₁₉	#103	1	4	yes	—
112	C₂×C₄×C₁₉	#103	1	4	yes	—
113	C₂×C₄×C₁₉	#106	1	2	yes	—
114	C₂×C₄×C₁₉	#107	1	2	yes	—
115	C₂×C₄×C₁₉	#108	1	4	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×C₂²⋊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	C₂×M₁₆	1	4	yes	—
123	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
127	C₈×C₁₉	—	4	4	no	—
128	C₁₉⋊C₈	—	2	8	no	—
129	C₁₉⋊C₈	—	4	4	no	—
130	C₂×#423	—	19	1	no	—
131	C₂×#423	D₁₉	1	1	yes	characteristic
132	C₂²×C₄×C₈	—	19	1	no	—
133	#1277	—	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	#340	1	4	yes	—
139	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
140	C₂²×C₁₉	#423	1	2	yes	—
141	C₂²×C₁₉	C₂×#106	1	1	yes	characteristic
142	C₂²×C₁₉	C₂×#107	1	1	yes	characteristic
143	C₂²×C₁₉	C₂×#108	1	2	yes	—
144	C₄×C₁₉	—	2	2	no	—
145	C₄×C₁₉	—	2	1	no	—
146	C₄×C₁₉	—	2	1	no	—
147	C₄×C₁₉	#340	1	4	yes	—
148	C₄×C₁₉	#341	1	4	yes	—
149	C₄×C₁₉	C₄×M₁₆	1	2	yes	—
150	C₄×C₁₉	#407	1	2	yes	—
151	C₄×C₁₉	C₂×#102	1	1	yes	characteristic
152	C₄×C₁₉	C₂×#102	1	1	yes	characteristic
153	C₄×C₁₉	C₂×#103	1	1	yes	characteristic
154	C₄×C₁₉	C₂×#103	1	1	yes	characteristic
155	C₂²×M₁₆	—	2	1	no	—
156	C₂²×M₁₆	—	38	1	no	—
157	C₂³×C₈	—	19	2	no	—
158	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
159	#423	—	2	2	no	—
160	#423	—	19	4	no	—
161	#423	—	38	2	no	—
162	#423	Dic₁₉	1	4	yes	—
163	C₂×C₄⋊C₈	—	2	1	no	—
164	C₂×C₄⋊C₈	—	19	2	no	—
165	C₂×C₄⋊C₈	—	38	1	no	—
166	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
167	C₂×C₄×C₈	—	19	2	no	—
168	C₂×C₄×C₈	—	19	2	no	—
169	C₂×C₄×C₈	—	38	2	no	—
170	C₂×C₄×C₈	—	38	2	no	—
171	C₂×C₁₉	—	2	4	no	—
172	C₂×C₁₉	#1566	1	2	yes	—
173	C₂×C₁₉	#1567	1	2	yes	—
174	C₂×C₁₉	#1728	1	1	yes	characteristic
175	C₂×C₁₉	C₂×#340	1	1	yes	characteristic
176	C₂×C₁₉	C₂×#341	1	1	yes	characteristic
177	C₂³×C₄	Dic₃₈	1	1	yes	characteristic
178	C₂³×C₄	C₄×D₁₉	1	1	yes	characteristic
179	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
180	C₂×M₁₆	—	2	4	no	—
181	C₂×M₁₆	—	4	2	no	—
182	C₂×M₁₆	—	4	2	no	—
183	C₂×M₁₆	—	38	4	no	—
184	C₂×M₁₆	—	76	2	no	—
185	C₂×M₁₆	—	76	2	no	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	19	4	no	—
188	C₂²×C₈	—	38	8	no	—
189	C₂²×C₈	—	38	4	no	—
190	C₂²×C₈	—	38	2	no	—
191	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
192	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
193	C₂×C₄²	C₄×D₁₉	1	2	yes	—
194	C₂×C₄²	D₇₆	1	2	yes	—
195	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
196	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
197	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
198	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
199	C₄⋊C₈	—	2	4	no	—
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	C₄×C₈	—	38	4	no	—
206	C₄×C₈	—	38	2	no	—
207	C₄×C₈	—	38	2	no	—
208	C₁₉	#7874	1	1	yes	characteristic, complemented, Sylow
209	C₂⁴	Dic₁₉⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
213	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
214	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
215	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
216	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
217	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
218	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
219	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
220	C₂²×C₄	#64525	1	2	yes	—
221	C₂²×C₄	#64526	1	2	yes	—
222	C₂²×C₄	#64527	1	2	yes	—
223	C₂²×C₄	#64528	1	2	yes	—
224	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
225	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
228	M₁₆	—	4	8	no	—
229	M₁₆	—	76	8	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	19	8	no	—
234	C₂×C₈	—	38	16	no	—
235	C₂×C₈	—	38	4	no	—
236	C₂×C₈	—	38	4	no	—
237	C₂×C₈	—	38	4	no	—
238	C₂×C₈	—	76	2	no	—
239	C₂×C₈	—	76	2	no	—
240	C₄²	C₄×Dic₁₉	1	2	yes	—
241	C₄²	Dic₁₉⋊C₄	1	2	yes	—
242	C₄²	C₄⋊Dic₁₉	1	2	yes	—
243	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
244	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
245	C₄²	C₂²⋊Dic₁₉	1	2	yes	—
246	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
247	C₄²	C₂×D₇₆	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	#10597264	1	2	yes	—
253	C₂³	#10597265	1	2	yes	—
254	C₂³	C₁₉⋊#95	1	2	yes	—
255	C₂³	C₂×Dic₁₉⋊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	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	#10597264	1	2	yes	—
263	C₂×C₄	#10597265	1	2	yes	—
264	C₂×C₄	#10597266	1	4	yes	—
265	C₂×C₄	#10597267	1	4	yes	—
266	C₂×C₄	#10597279	1	4	yes	—
267	C₂×C₄	C₁₉⋊#95	1	2	yes	—
268	C₂×C₄	C₁₉⋊#95	1	2	yes	—
269	C₂×C₄	C₁₉⋊#95	1	2	yes	—
270	C₂×C₄	#10597289	1	4	yes	—
271	C₂×C₄	#10597292	1	4	yes	—
272	C₂×C₄	M₁₆×D₁₉	1	4	yes	—
273	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
274	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
275	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
276	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
277	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
278	C₂×C₄	C₂×#64525	1	1	yes	characteristic
279	C₂×C₄	C₂×#64526	1	1	yes	characteristic
280	C₂×C₄	C₂×#64527	1	1	yes	characteristic
281	C₂×C₄	C₂×#64528	1	1	yes	characteristic
282	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
283	C₈	—	4	4	no	—
284	C₈	—	38	8	no	—
285	C₈	—	76	4	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	4	no	—
290	C₂²	#11926306	1	4	yes	—
291	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
292	C₂²	#11926567	1	2	yes	—
293	C₂²	C₂×#10597265	1	1	yes	characteristic
294	C₂²	C₂×#10597264	1	1	yes	characteristic
295	C₂²	C₂×#10597279	1	2	yes	—
296	C₄	—	2	2	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	#11926306	1	4	yes	—
300	C₄	#11926308	1	4	yes	—
301	C₄	M₁₆×Dic₁₉	1	2	yes	—
302	C₄	#11926507	1	2	yes	—
303	C₄	C₂×#10597266	1	1	yes	characteristic
304	C₄	C₂×#10597289	1	1	yes	characteristic
305	C₄	C₂×#10597267	1	1	yes	characteristic
306	C₄	C₂×#10597292	1	1	yes	characteristic
307	C₂	—	2	4	no	—
308	C₂	#14562390	1	2	yes	—
309	C₂	#14562393	1	2	yes	—
310	C₂	#14563328	1	1	yes	characteristic
311	C₂	C₂×#11926306	1	1	yes	characteristic
312	C₂	C₂×#11926308	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#423	C₂
socle	C₂³×C₁₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₉	C₄⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#423	D₁₉
hypocenter	C₁₉	#7874
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
3. #14561469	C₂	semidirect
92. C₂×C₁₉⋊C₈	M₁₆	semidirect
208. C₁₉	#7874	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926568	2
C₂²×C₄×C₁₉⋊C₈	2
#14561469	2
C₂×C₁₉×#423	2
#7874	19