Subgroups — #18087694

Order 4864.

1174 conjugacy classes of subgroups.

The subgroup lattice has 322 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	C₂²×C₈×Dic₁₉	C₂	1	2	yes	maximal
3	C₂×#11926298	C₂	1	2	yes	maximal
4	C₄×C₁₉⋊#95	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₉×#336	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
7	C₂³×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₁₉	C₂²	1	2	yes	—
9	C₂×C₄×C₁₉⋊C₈	—	2	4	no	—
10	C₂×C₄×C₁₉⋊C₈	C₄	1	4	yes	complemented
11	C₂×C₈×Dic₁₉	—	2	4	no	—
12	C₂×C₈×Dic₁₉	—	2	4	no	—
13	C₂×C₈×Dic₁₉	C₄	1	8	yes	complemented
14	C₂×C₁₉⋊#95	C₄	1	4	yes	—
15	#11926298	C₄	1	8	yes	complemented
16	C₂³×C₈×C₁₉	C₂²	1	2	yes	—
17	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#336	C₄	1	4	yes	complemented
19	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
21	C₂³×Dic₁₉	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₁₉	—	2	2	no	—
23	C₂²×C₈×C₁₉	—	2	2	no	—
24	C₂²×C₈×C₁₉	—	2	2	no	—
25	C₂²×C₈×C₁₉	—	2	2	no	—
26	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₁₉	—	2	2	no	—
30	C₂×C₄²×C₁₉	—	2	2	no	—
31	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
33	C₂×C₄×Dic₁₉	—	2	2	no	—
34	C₂×C₄×Dic₁₉	—	2	2	no	—
35	C₂×C₄×Dic₁₉	—	2	2	no	—
36	C₂×C₄×Dic₁₉	—	2	2	no	—
37	C₂×C₄×Dic₁₉	C₂×C₄	1	4	yes	—
38	C₂×C₄×Dic₁₉	C₂×C₄	1	4	yes	—
39	C₂²×C₁₉⋊C₈	—	2	2	no	—
40	C₂²×C₁₉⋊C₈	—	2	1	no	—
41	C₂²×C₁₉⋊C₈	—	2	1	no	—
42	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
43	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
44	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
45	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
46	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
47	C₂²×C₁₉⋊C₈	D₄	1	4	yes	—
48	C₂²×C₁₉⋊C₈	Q₈	1	2	yes	—
49	C₁₉⋊#95	C₈	1	16	yes	complemented
50	C₈×Dic₁₉	—	2	16	no	—
51	C₄×C₁₉⋊C₈	—	2	8	no	—
52	C₂⁴×C₁₉	C₂²×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	4	yes	—
56	C₂²×C₄×C₁₉	C₂×C₈	1	8	yes	—
57	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	#41	1	2	yes	—
66	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
67	C₂²×Dic₁₉	—	2	4	no	—
68	C₂²×Dic₁₉	—	2	4	no	—
69	C₂²×Dic₁₉	—	2	8	no	—
70	C₂²×Dic₁₉	C₄²	1	8	yes	—
71	C₂²×Dic₁₉	C₂×C₈	1	16	yes	—
72	C₂×C₈×C₁₉	—	2	4	no	—
73	C₂×C₈×C₁₉	—	2	4	no	—
74	C₂×C₈×C₁₉	—	2	4	no	—
75	C₂×C₈×C₁₉	—	2	4	no	—
76	C₂×C₈×C₁₉	—	2	4	no	—
77	C₂×C₈×C₁₉	—	2	4	no	—
78	C₂×C₈×C₁₉	C₄²	1	8	yes	—
79	C₄²×C₁₉	—	2	2	no	—
80	C₄²×C₁₉	—	2	2	no	—
81	C₄²×C₁₉	—	2	4	no	—
82	C₄×Dic₁₉	—	2	4	no	—
83	C₄×Dic₁₉	—	2	4	no	—
84	C₄×Dic₁₉	—	2	4	no	—
85	C₄×Dic₁₉	—	2	4	no	—
86	C₂×C₁₉⋊C₈	—	2	2	no	—
87	C₂×C₁₉⋊C₈	—	2	2	no	—
88	C₂×C₁₉⋊C₈	—	2	4	no	—
89	C₂×C₁₉⋊C₈	—	2	4	no	—
90	C₂×C₁₉⋊C₈	C₄²	1	4	yes	—
91	C₂×C₁₉⋊C₈	C₄²	1	4	yes	—
92	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	4	yes	—
93	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	8	yes	—
94	C₂×C₁₉⋊C₈	C₄⋊C₄	1	8	yes	—
95	C₂×C₁₉⋊C₈	C₄⋊C₄	1	4	yes	—
96	C₈×#95	—	19	1	no	maximal, Sylow
97	C₂³×C₁₉	C₂×C₄²	1	2	yes	—
98	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
99	C₂³×C₁₉	C₄×D₄	1	2	yes	—
100	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
101	C₂³×C₁₉	C₄×Q₈	1	1	yes	characteristic
102	C₂³×C₁₉	C₂²×C₈	1	4	yes	—
103	C₂³×C₁₉	#131	1	4	yes	—
104	C₂×C₄×C₁₉	—	2	4	no	—
105	C₂×C₄×C₁₉	—	2	4	no	—
106	C₂×C₄×C₁₉	—	2	16	no	—
107	C₂×C₄×C₁₉	C₄×C₈	1	8	yes	complemented
108	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₁₉	C₂×C₄²	1	2	yes	—
111	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
113	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₁₉	#117	1	2	yes	—
116	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
117	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
118	C₂×C₄×C₁₉	#117	1	2	yes	—
119	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
126	C₂×Dic₁₉	—	2	8	no	—
127	C₂×Dic₁₉	—	2	8	no	—
128	C₂×Dic₁₉	—	2	16	no	—
129	C₂×Dic₁₉	—	2	16	no	—
130	C₂×Dic₁₉	C₄×C₈	1	16	yes	complemented
131	C₈×C₁₉	—	2	8	no	—
132	C₁₉⋊C₈	—	2	4	no	—
133	C₁₉⋊C₈	#95	1	8	yes	complemented
134	C₂²×C₄×C₈	—	19	1	no	—
135	C₂²×C₄×C₈	—	19	2	no	—
136	C₂×#336	—	19	2	no	—
137	C₂×#336	D₁₉	1	1	yes	characteristic
138	C₄×#95	—	19	1	no	—
139	C₂²×C₁₉	C₄³	1	1	yes	characteristic
140	C₂²×C₁₉	C₄×C₂²⋊C₄	1	2	yes	—
141	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
143	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
144	C₂²×C₁₉	C₂×C₄×C₈	1	4	yes	—
145	C₂²×C₁₉	C₂×C₄×C₈	1	2	yes	—
146	C₂²×C₁₉	#405	1	4	yes	—
147	C₂²×C₁₉	#405	1	2	yes	—
148	C₂²×C₁₉	C₈×D₄	1	8	yes	—
149	C₂²×C₁₉	C₈×D₄	1	4	yes	—
150	C₂²×C₁₉	C₈×Q₈	1	4	yes	—
151	C₄×C₁₉	—	2	8	no	—
152	C₄×C₁₉	C₂×#95	1	1	yes	characteristic
153	C₄×C₁₉	#376	1	1	yes	characteristic
154	C₄×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
155	C₄×C₁₉	C₄×C₂²⋊C₄	1	2	yes	—
156	C₄×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
157	C₄×C₁₉	C₄×C₄⋊C₄	1	2	yes	—
158	Dic₁₉	—	2	16	no	—
159	C₂³×C₈	—	19	1	no	—
160	C₂³×C₈	—	19	1	no	—
161	C₂³×C₈	C₂×D₁₉	1	2	yes	—
162	C₂²×C₄²	—	19	2	no	—
163	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
164	C₂×C₄×C₈	—	19	8	no	—
165	C₂×C₄×C₈	—	19	4	no	—
166	C₂×C₄×C₈	—	38	4	no	—
167	C₂×C₄×C₈	—	38	4	no	—
168	C₂×C₄×C₈	—	38	4	no	—
169	C₂×#95	—	19	4	no	—
170	#336	—	19	8	no	—
171	#336	Dic₁₉	1	4	yes	complemented
172	C₂×C₁₉	C₄×#95	1	1	yes	characteristic
173	C₂×C₁₉	C₄²×C₈	1	1	yes	characteristic
174	C₂×C₁₉	#1726	1	1	yes	characteristic
175	C₂×C₁₉	C₈×C₂²⋊C₄	1	4	yes	—
176	C₂×C₁₉	C₈×C₂²⋊C₄	1	2	yes	—
177	C₂×C₁₉	C₈×C₄⋊C₄	1	4	yes	—
178	C₂×C₁₉	C₈×C₄⋊C₄	1	2	yes	—
179	C₂³×C₄	—	19	4	no	—
180	C₂³×C₄	C₄×D₁₉	1	2	yes	—
181	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	19	2	no	—
187	C₂²×C₈	—	19	4	no	—
188	C₂²×C₈	—	19	2	no	—
189	C₂²×C₈	—	19	4	no	—
190	C₂²×C₈	—	19	2	no	—
191	C₂²×C₈	—	19	4	no	—
192	C₂²×C₈	—	19	2	no	—
193	C₂²×C₈	—	38	2	no	—
194	C₂²×C₈	—	38	1	no	—
195	C₂²×C₈	—	38	1	no	—
196	C₂²×C₈	C₄×D₁₉	1	4	yes	—
197	C₂²×C₈	C₄×D₁₉	1	4	yes	—
198	C₂²×C₈	C₂×Dic₁₉	1	4	yes	—
199	C₂×C₄²	—	2	2	no	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	—	19	4	no	—
202	C₂×C₄²	—	19	4	no	—
203	C₂×C₄²	—	38	2	no	—
204	C₂×C₄²	—	38	2	no	—
205	C₂×C₄²	—	38	2	no	—
206	C₂×C₄²	—	38	2	no	—
207	C₂×C₄²	C₄×D₁₉	1	2	yes	—
208	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
209	C₄×C₈	—	38	8	no	—
210	C₄×C₈	—	38	16	no	—
211	#95	—	19	16	no	—
212	C₁₉	C₈×#95	1	1	yes	characteristic, complemented, Sylow
213	C₂⁴	C₂×C₄×D₁₉	1	1	yes	characteristic
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	19	8	no	—
217	C₂²×C₄	—	19	16	no	—
218	C₂²×C₄	—	38	4	no	—
219	C₂²×C₄	—	38	4	no	—
220	C₂²×C₄	—	38	8	no	—
221	C₂²×C₄	C₈×D₁₉	1	8	yes	—
222	C₂²×C₄	C₄×Dic₁₉	1	4	yes	—
223	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
225	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
226	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
227	C₂²×C₄	D₄×D₁₉	1	2	yes	—
228	C₂²×C₄	#64543	1	1	yes	characteristic
229	C₂²×C₄	#64543	1	2	yes	—
230	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
231	C₂²×C₄	#64545	1	1	yes	characteristic
232	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	19	4	no	—
240	C₂×C₈	—	19	4	no	—
241	C₂×C₈	—	19	8	no	—
242	C₂×C₈	—	19	4	no	—
243	C₂×C₈	—	19	8	no	—
244	C₂×C₈	—	19	4	no	—
245	C₂×C₈	—	38	2	no	—
246	C₂×C₈	—	38	4	no	—
247	C₂×C₈	—	38	2	no	—
248	C₂×C₈	—	38	4	no	—
249	C₂×C₈	C₄×Dic₁₉	1	8	yes	—
250	C₄²	—	2	4	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	38	4	no	—
254	C₄²	—	38	4	no	—
255	C₄²	—	38	4	no	—
256	C₄²	—	38	4	no	—
257	C₂³	C₄²×D₁₉	1	2	yes	—
258	C₂³	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
259	C₂³	#10597333	1	1	yes	characteristic
260	C₂³	#10597339	1	1	yes	characteristic
261	C₂³	C₂×C₈×D₁₉	1	4	yes	—
262	C₂³	#10597353	1	4	yes	—
263	C₂³	C₂×C₄×Dic₁₉	1	1	yes	characteristic
264	C₂×C₄	—	2	16	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	19	16	no	—
268	C₂×C₄	—	38	16	no	—
269	C₂×C₄	—	38	16	no	—
270	C₂×C₄	—	38	8	no	—
271	C₂×C₄	—	38	8	no	—
272	C₂×C₄	C₈×Dic₁₉	1	8	yes	complemented
273	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
274	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
275	C₂×C₄	#10597323	1	2	yes	—
276	C₂×C₄	#10597323	1	1	yes	characteristic
277	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
278	C₂×C₄	C₂²⋊C₄×D₁₉	1	2	yes	—
279	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
280	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
281	C₂×C₄	#10597333	1	2	yes	—
282	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
283	C₂×C₄	C₄⋊C₄×D₁₉	1	2	yes	—
284	C₂×C₄	#10597338	1	1	yes	characteristic
285	C₂×C₄	#10597338	1	2	yes	—
286	C₂×C₄	#10597339	1	2	yes	—
287	C₂×C₄	C₂×C₄×Dic₁₉	1	2	yes	—
288	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
289	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
290	C₂×C₄	Q₈×Dic₁₉	1	1	yes	characteristic
291	C₈	—	2	8	no	—
292	C₈	—	19	8	no	—
293	C₈	—	38	4	no	—
294	C₂²	C₄²×Dic₁₉	1	1	yes	characteristic
295	C₂²	#11926391	1	2	yes	—
296	C₂²	C₂²⋊C₄×Dic₁₉	1	1	yes	characteristic
297	C₂²	#11926395	1	2	yes	—
298	C₂²	C₄⋊C₄×Dic₁₉	1	1	yes	characteristic
299	C₂²	C₄×C₈×D₁₉	1	4	yes	—
300	C₂²	C₂×C₈×Dic₁₉	1	2	yes	—
301	C₂²	#11926501	1	4	yes	—
302	C₂²	#11926502	1	2	yes	—
303	C₂²	(C₂²⋊D₁₉)⋊C₈	1	8	yes	—
304	C₂²	#11926609	1	4	yes	—
305	C₂²	#11926668	1	4	yes	—
306	C₄	—	2	8	no	—
307	C₄	—	38	16	no	—
308	C₄	#95×D₁₉	1	1	yes	characteristic
309	C₄	#11926389	1	1	yes	characteristic
310	C₄	#11926391	1	1	yes	characteristic
311	C₄	C₂²⋊C₄×Dic₁₉	1	2	yes	—
312	C₄	#11926395	1	1	yes	characteristic
313	C₄	C₄⋊C₄×Dic₁₉	1	2	yes	—
314	C₂	#95×Dic₁₉	1	1	yes	characteristic
315	C₂	C₄×C₈×Dic₁₉	1	1	yes	characteristic
316	C₂	#14563323	1	1	yes	characteristic
317	C₂	#14563405	1	4	yes	—
318	C₂	#14563409	1	2	yes	—
319	C₂	#14563496	1	4	yes	—
320	C₂	#14563500	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³×C₄	C₂²×D₁₉
commutator	C₂×C₁₉	C₄²×C₈
Fitting	C₂×C₁₉×#336	C₂
socle, prime radical	C₂⁴×C₁₉	C₂²×C₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#336	D₁₉
hypocenter	C₁₉	C₈×#95
LCS	G ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×#336

Complements

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

N	H ≅ G/N	type
10. C₂×C₄×C₁₉⋊C₈	C₄	semidirect
13. C₂×C₈×Dic₁₉	C₄	semidirect
15. #11926298	C₄	semidirect
18. C₁₉×#336	C₄	semidirect
49. C₁₉⋊#95	C₈	semidirect
107. C₂×C₄×C₁₉	C₄×C₈	semidirect
130. C₂×Dic₁₉	C₄×C₈	semidirect
133. C₁₉⋊C₈	#95	semidirect
171. #336	Dic₁₉	semidirect
212. C₁₉	C₈×#95	semidirect
272. C₂×C₄	C₈×Dic₁₉	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
C₂²×C₈×Dic₁₉	2
C₂×#11926298	2
C₄×C₁₉⋊#95	2
C₂×C₁₉×#336	2
C₈×#95	19