Subgroups — #18087794

Order 4864.

1422 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926508	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926308	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₉×#407	C₂	1	1	yes	maximal, characteristic
5	C₂²×#64520	C₂²	1	2	yes	—
6	C₂³×C₁₉⋊C₈	—	2	8	no	—
7	C₂³×C₁₉⋊C₈	—	2	2	no	—
8	C₂³×C₁₉⋊C₈	—	2	2	no	—
9	C₂³×C₁₉⋊C₈	C₄	1	4	yes	—
10	C₂³×C₁₉⋊C₈	C₂²	1	2	yes	—
11	#11926508	—	2	2	no	—
12	#11926508	C₄	1	4	yes	—
13	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
14	#11926308	—	2	8	no	—
15	C₂⁴×C₄×C₁₉	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
17	C₁₉×#407	—	2	2	no	—
18	C₁₉×#407	C₄	1	4	yes	—
19	C₂×C₁₉×C₂²⋊C₈	—	2	2	no	—
20	C₂⁵×C₁₉	Q₈	1	1	yes	characteristic
21	C₂³×C₄×C₁₉	—	2	4	no	—
22	C₂³×C₄×C₁₉	—	2	4	no	—
23	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₁₉	D₄	1	2	yes	—
27	C₂³×C₄×C₁₉	D₄	1	2	yes	—
28	C₂³×C₄×C₁₉	D₄	1	2	yes	—
29	C₂³×C₄×C₁₉	Q₈	1	1	yes	characteristic
30	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
31	C₂×M₁₆×C₁₉	—	2	2	no	—
32	C₂×M₁₆×C₁₉	—	2	4	no	—
33	C₂×M₁₆×C₁₉	—	2	4	no	—
34	C₂×M₁₆×C₁₉	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₁₉	—	2	2	no	—
36	C₂×#64520	—	2	2	no	—
37	C₂×#64520	—	2	4	no	—
38	C₂×#64520	—	2	4	no	—
39	C₂×#64520	C₂×C₄	1	4	yes	—
40	C₂²×C₁₉⋊C₈	—	2	16	no	—
41	C₂²×C₁₉⋊C₈	—	2	16	no	—
42	C₂²×C₁₉⋊C₈	—	2	8	no	—
43	C₂²×C₁₉⋊C₈	—	2	2	no	—
44	C₂²×C₁₉⋊C₈	—	2	4	no	—
45	C₂²×C₁₉⋊C₈	—	2	2	no	—
46	C₂²×C₁₉⋊C₈	—	2	16	no	—
47	C₂²×C₁₉⋊C₈	—	2	2	no	—
48	C₂²×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
49	C₂²×C₁₉⋊C₈	D₄	1	8	yes	—
50	C₁₉×C₂²⋊C₈	—	2	8	no	—
51	C₂²⋊(C₁₉⋊C₈)	—	2	8	no	—
52	C₂⁴×C₁₉	—	2	8	no	—
53	C₂⁴×C₁₉	—	2	4	no	—
54	C₂⁴×C₁₉	C₄⋊C₄	1	2	yes	—
55	C₂⁴×C₁₉	C₄⋊C₄	1	2	yes	—
56	C₂⁴×C₁₉	C₄⋊C₄	1	2	yes	—
57	C₂⁴×C₁₉	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	—	2	4	no	—
59	C₂²×C₄×C₁₉	—	2	4	no	—
60	C₂²×C₄×C₁₉	—	2	4	no	—
61	C₂²×C₄×C₁₉	—	2	8	no	—
62	C₂²×C₄×C₁₉	—	2	8	no	—
63	C₂²×C₄×C₁₉	—	2	4	no	—
64	C₂²×C₄×C₁₉	—	2	4	no	—
65	C₂²×C₄×C₁₉	—	2	8	no	—
66	C₂²×C₄×C₁₉	—	2	4	no	—
67	C₂²×C₄×C₁₉	—	2	4	no	—
68	C₂²×C₄×C₁₉	C₄²	1	4	yes	—
69	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
70	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
71	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
72	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
74	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
75	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
80	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₉	#41	1	4	yes	—
83	M₁₆×C₁₉	—	2	8	no	—
84	M₁₆×C₁₉	—	4	2	no	—
85	M₁₆×C₁₉	—	4	2	no	—
86	C₂×C₈×C₁₉	—	2	4	no	—
87	C₂×C₈×C₁₉	—	4	4	no	—
88	#64520	—	2	8	no	—
89	#64520	—	4	2	no	—
90	#64520	—	4	2	no	—
91	C₂×C₁₉⋊C₈	—	2	32	no	—
92	C₂×C₁₉⋊C₈	—	2	4	no	—
93	C₂×C₁₉⋊C₈	—	2	4	no	—
94	C₂×C₁₉⋊C₈	—	2	8	no	—
95	C₂×C₁₉⋊C₈	—	2	8	no	—
96	C₂×C₁₉⋊C₈	—	2	4	no	—
97	C₂×C₁₉⋊C₈	—	4	4	no	—
98	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	8	yes	—
99	#10723	—	19	1	no	maximal, Sylow
100	C₂³×C₁₉	—	2	32	no	—
101	C₂³×C₁₉	—	2	8	no	—
102	C₂³×C₁₉	—	2	8	no	—
103	C₂³×C₁₉	—	2	4	no	—
104	C₂³×C₁₉	—	2	4	no	—
105	C₂³×C₁₉	—	2	4	no	—
106	C₂³×C₁₉	—	2	4	no	—
107	C₂³×C₁₉	—	2	4	no	—
108	C₂³×C₁₉	#95	1	4	yes	—
109	C₂³×C₁₉	#108	1	8	yes	—
110	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂³×C₁₉	C₂²⋊Q₈	1	4	yes	—
114	C₂×C₄×C₁₉	—	2	4	no	—
115	C₂×C₄×C₁₉	—	2	4	no	—
116	C₂×C₄×C₁₉	—	2	4	no	—
117	C₂×C₄×C₁₉	—	2	8	no	—
118	C₂×C₄×C₁₉	—	2	4	no	—
119	C₂×C₄×C₁₉	—	2	4	no	—
120	C₂×C₄×C₁₉	—	2	8	no	—
121	C₂×C₄×C₁₉	—	2	8	no	—
122	C₂×C₄×C₁₉	#95	1	4	yes	—
123	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
126	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₉	#117	1	2	yes	—
129	C₂×C₄×C₁₉	C₄×D₄	1	4	yes	—
130	C₂×C₄×C₁₉	C₄×D₄	1	4	yes	—
131	C₂×C₄×C₁₉	#120	1	2	yes	—
132	C₂×C₄×C₁₉	#120	1	2	yes	—
133	C₂×C₄×C₁₉	#121	1	4	yes	—
134	C₂×C₄×C₁₉	#123	1	2	yes	—
135	C₂×C₄×C₁₉	#123	1	2	yes	—
136	C₈×C₁₉	—	4	4	no	—
137	C₁₉⋊C₈	—	2	8	no	—
138	C₁₉⋊C₈	—	4	4	no	—
139	C₂⁴×C₈	—	19	1	no	—
140	C₂×#407	—	19	1	no	—
141	C₂×#407	D₁₉	1	1	yes	characteristic
142	C₂×#341	—	19	4	no	—
143	C₂²×C₁₉	—	2	16	no	—
144	C₂²×C₁₉	—	2	16	no	—
145	C₂²×C₁₉	—	2	16	no	—
146	C₂²×C₁₉	—	2	8	no	—
147	C₂²×C₁₉	—	2	4	no	—
148	C₂²×C₁₉	—	2	4	no	—
149	C₂²×C₁₉	—	2	2	no	—
150	C₂²×C₁₉	—	2	2	no	—
151	C₂²×C₁₉	#341	1	8	yes	complemented
152	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
153	C₂²×C₁₉	#380	1	2	yes	—
154	C₂²×C₁₉	#385	1	2	yes	—
155	C₂²×C₁₉	#385	1	2	yes	—
156	C₂²×C₁₉	C₂×#108	1	4	yes	—
157	C₄×C₁₉	—	2	4	no	—
158	C₄×C₁₉	C₄×C₂²⋊C₄	1	2	yes	—
159	C₄×C₁₉	#379	1	1	yes	characteristic
160	C₄×C₁₉	#381	1	1	yes	characteristic
161	C₄×C₁₉	#386	1	2	yes	—
162	C₄×C₁₉	#386	1	2	yes	—
163	C₂⁴×C₄	C₂×D₁₉	1	1	yes	characteristic
164	C₂²×M₁₆	—	19	2	no	—
165	C₂²×M₁₆	C₂×D₁₉	1	2	yes	—
166	C₂³×C₈	—	19	2	no	—
167	C₂³×C₈	—	19	4	no	—
168	C₂³×C₈	—	38	8	no	—
169	C₂³×C₈	—	38	2	no	—
170	C₂³×C₈	—	38	2	no	—
171	#407	—	2	2	no	—
172	#407	—	19	4	no	—
173	#407	—	38	2	no	—
174	#407	Dic₁₉	1	4	yes	—
175	C₂×C₂²⋊C₈	—	2	2	no	—
176	C₂×C₂²⋊C₈	—	38	2	no	—
177	#341	—	38	8	no	—
178	C₂×C₁₉	—	2	8	no	—
179	C₂×C₁₉	—	2	2	no	—
180	C₂×C₁₉	—	2	2	no	—
181	C₂×C₁₉	C₂²⋊#95	1	1	yes	characteristic
182	C₂×C₁₉	C₂×#341	1	2	yes	—
183	C₂×C₁₉	#1810	1	4	yes	—
184	C₂⁵	Dic₃₈	1	1	yes	characteristic
185	C₂³×C₄	—	2	4	no	—
186	C₂³×C₄	—	2	4	no	—
187	C₂³×C₄	Dic₃₈	1	1	yes	characteristic
188	C₂³×C₄	C₄×D₁₉	1	2	yes	—
189	C₂³×C₄	C₄×D₁₉	1	2	yes	—
190	C₂³×C₄	D₇₆	1	2	yes	—
191	C₂³×C₄	C₂×Dic₁₉	1	2	yes	—
192	C₂³×C₄	C₂²⋊D₁₉	1	2	yes	—
193	C₂³×C₄	C₂²⋊D₁₉	1	2	yes	—
194	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
195	C₂×M₁₆	—	2	2	no	—
196	C₂×M₁₆	—	2	4	no	—
197	C₂×M₁₆	—	2	4	no	—
198	C₂×M₁₆	—	19	4	no	—
199	C₂×M₁₆	—	38	2	no	—
200	C₂×M₁₆	—	38	4	no	—
201	C₂×M₁₆	—	38	4	no	—
202	C₂×M₁₆	C₂×Dic₁₉	1	4	yes	—
203	C₂²×C₈	—	2	2	no	—
204	C₂²×C₈	—	19	4	no	—
205	C₂²×C₈	—	19	8	no	—
206	C₂²×C₈	—	38	16	no	—
207	C₂²×C₈	—	38	16	no	—
208	C₂²×C₈	—	38	16	no	—
209	C₂²×C₈	—	38	8	no	—
210	C₂²×C₈	—	38	2	no	—
211	C₂²×C₈	—	38	4	no	—
212	C₂²×C₈	—	38	2	no	—
213	C₂²×C₈	—	38	2	no	—
214	C₂²⋊C₈	—	2	8	no	—
215	C₂²⋊C₈	—	38	8	no	—
216	C₁₉	#10723	1	1	yes	characteristic, complemented, Sylow
217	C₂⁴	—	2	8	no	—
218	C₂⁴	—	2	4	no	—
219	C₂⁴	Dic₁₉⋊C₄	1	2	yes	—
220	C₂⁴	Dic₁₉⋊C₄	1	2	yes	—
221	C₂⁴	C₄⋊Dic₁₉	1	2	yes	—
222	C₂⁴	C₂×Dic₃₈	1	1	yes	characteristic
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	8	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	8	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	8	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	C₄×Dic₁₉	1	4	yes	—
234	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
235	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
236	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
237	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	4	yes	—
238	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	4	yes	—
239	C₂²×C₄	C₂²⋊Dic₁₉	1	4	yes	—
240	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
243	C₂²×C₄	D₄×D₁₉	1	4	yes	—
244	C₂²×C₄	#64543	1	4	yes	—
245	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
248	M₁₆	—	2	8	no	—
249	M₁₆	—	4	2	no	—
250	M₁₆	—	4	2	no	—
251	M₁₆	—	38	8	no	—
252	M₁₆	—	76	2	no	—
253	M₁₆	—	76	2	no	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	4	4	no	—
256	C₂×C₈	—	19	8	no	—
257	C₂×C₈	—	38	32	no	—
258	C₂×C₈	—	38	8	no	—
259	C₂×C₈	—	38	4	no	—
260	C₂×C₈	—	38	8	no	—
261	C₂×C₈	—	38	4	no	—
262	C₂×C₈	—	38	4	no	—
263	C₂×C₈	—	76	4	no	—
264	C₂³	—	2	32	no	—
265	C₂³	—	2	8	no	—
266	C₂³	—	2	8	no	—
267	C₂³	—	2	4	no	—
268	C₂³	—	2	4	no	—
269	C₂³	—	2	4	no	—
270	C₂³	—	2	4	no	—
271	C₂³	—	2	4	no	—
272	C₂³	#10597279	1	8	yes	—
273	C₂³	C₁₉⋊#95	1	4	yes	—
274	C₂³	#10597324	1	4	yes	—
275	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
276	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
277	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	8	no	—
285	C₂×C₄	—	2	8	no	—
286	C₂×C₄	C₁₉⋊#95	1	4	yes	—
287	C₂×C₄	#10597323	1	2	yes	—
288	C₂×C₄	C₂²⋊C₄×D₁₉	1	2	yes	—
289	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	4	yes	—
290	C₂×C₄	#10597328	1	2	yes	—
291	C₂×C₄	#10597332	1	2	yes	—
292	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
293	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
294	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
295	C₂×C₄	D₄×Dic₁₉	1	4	yes	—
296	C₂×C₄	#10597381	1	2	yes	—
297	C₂×C₄	#10597383	1	2	yes	—
298	C₂×C₄	#10597385	1	4	yes	—
299	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
300	C₈	—	4	4	no	—
301	C₈	—	38	8	no	—
302	C₈	—	76	4	no	—
303	C₂²	—	2	16	no	—
304	C₂²	—	2	16	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	4	no	—
309	C₂²	—	2	16	no	—
310	C₂²	—	2	8	no	—
311	C₂²	#11926308	1	8	yes	complemented
312	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
313	C₂²	#11926402	1	2	yes	—
314	C₂²	#11926416	1	2	yes	—
315	C₂²	#11926419	1	2	yes	—
316	C₂²	C₂×#10597279	1	4	yes	—
317	C₄	—	2	4	no	—
318	C₄	C₂²⋊C₄×Dic₁₉	1	2	yes	—
319	C₄	#11926399	1	1	yes	characteristic
320	C₄	#11926405	1	1	yes	characteristic
321	C₄	#11926421	1	2	yes	—
322	C₄	#11926424	1	2	yes	—
323	C₂	—	2	8	no	—
324	C₂	—	2	2	no	—
325	C₂	—	2	2	no	—
326	C₂	#14561857	1	1	yes	characteristic
327	C₂	C₂×#11926308	1	2	yes	—
328	C₂	#14563671	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×#407	C₂
socle	C₂³×C₁₉	C₂×C₄⋊C₄
prime radical	C₂⁵×C₁₉	Q₈
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#407	D₁₉
hypocenter	C₁₉	#10723
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ C₂×#407

Complements

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

N	H ≅ G/N	type
3. C₂×#11926308	C₂	semidirect
151. C₂²×C₁₉	#341	semidirect
216. C₁₉	#10723	semidirect
311. C₂²	#11926308	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×C₁₉⋊C₈	2
C₂×#11926508	2
C₂×#11926308	2
C₂×C₁₉×#407	2
#10723	19