Subgroups — #18078184

Order 4864.

644 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14571987	C₂	1	1	yes	maximal, characteristic
2	#14563615	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926289	C₂	1	2	yes	maximal, complemented
4	#14561519	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597315	C₂²	1	1	yes	characteristic
7	C₂×#10597370	—	2	1	no	—
8	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
10	#11926450	—	2	1	no	—
11	#11926442	—	2	1	no	—
12	#11926442	C₄	1	2	yes	—
13	#11926414	C₂²	1	1	yes	characteristic
14	#11926404	—	2	1	no	—
15	#11926404	C₄	1	2	yes	—
16	#11926299	—	2	1	no	—
17	#11926289	—	2	4	no	—
18	C₂³×C₈×C₁₉	—	2	1	no	—
19	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
20	C₂×C₁₉×C₄⋊C₈	—	2	4	no	—
21	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
22	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
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×Dic₃₈	—	2	1	no	—
27	C₂²×C₈×C₁₉	—	2	4	no	—
28	C₂²×C₈×C₁₉	—	2	2	no	—
29	C₂²×C₈×C₁₉	—	4	2	no	—
30	C₂²×C₈×C₁₉	—	4	2	no	—
31	C₂×C₄²×C₁₉	—	2	2	no	—
32	C₂×C₄²×C₁₉	—	2	2	no	—
33	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
37	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
38	C₂×C₄⋊Dic₁₉	—	2	1	no	—
39	C₂×C₄⋊Dic₁₉	—	2	1	no	—
40	C₂×C₄⋊Dic₁₉	—	2	1	no	—
41	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
42	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
43	#10597370	—	4	2	no	—
44	#10597370	—	4	2	no	—
45	C₂×Dic₁₉⋊C₄	—	2	2	no	—
46	C₂²×C₁₉⋊C₈	—	2	2	no	—
47	#10597315	—	2	2	no	—
48	#10597315	—	2	2	no	—
49	C₁₉×C₄⋊C₈	—	2	4	no	—
50	C₁₉×C₄⋊C₈	—	2	2	no	—
51	C₁₉×C₄⋊C₈	D₄	1	4	yes	—
52	C₁₉⋊#95	—	2	2	no	—
53	C₁₉⋊#95	—	2	2	no	—
54	C₂²⋊(C₁₉⋊C₈)	—	4	2	no	—
55	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
56	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
57	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	—	2	1	no	—
59	C₂²×C₄×C₁₉	—	2	2	no	—
60	C₂²×C₄×C₁₉	—	2	1	no	—
61	C₂²×C₄×C₁₉	—	2	2	no	—
62	C₂²×C₄×C₁₉	—	2	1	no	—
63	C₂²×C₄×C₁₉	—	2	1	no	—
64	C₂²×C₄×C₁₉	—	2	2	no	—
65	C₂²×C₄×C₁₉	—	2	2	no	—
66	C₂²×C₄×C₁₉	—	4	1	no	—
67	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	D₈	1	2	yes	—
71	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
72	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
73	C₂²×C₄×C₁₉	Q₁₆	1	2	yes	—
74	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×Dic₁₉	—	2	2	no	—
78	C₂²×Dic₁₉	—	2	2	no	—
79	C₂×Dic₃₈	—	4	2	no	—
80	C₂×Dic₃₈	—	4	1	no	—
81	C₂×Dic₃₈	—	4	1	no	—
82	C₂×Dic₃₈	—	4	2	no	—
83	C₂×C₈×C₁₉	—	2	4	no	—
84	C₂×C₈×C₁₉	—	4	8	no	—
85	C₂×C₈×C₁₉	—	4	2	no	—
86	C₂×C₈×C₁₉	—	4	2	no	—
87	C₄²×C₁₉	—	2	2	no	—
88	C₄²×C₁₉	—	2	1	no	—
89	C₄²×C₁₉	—	2	2	no	—
90	C₄²×C₁₉	—	2	1	no	—
91	C₄²×C₁₉	C₂×D₄	1	2	yes	—
92	C₄²×C₁₉	#41	1	2	yes	—
93	C₂²⋊Dic₁₉	—	4	4	no	—
94	C₄⋊Dic₁₉	—	2	4	no	—
95	C₄⋊Dic₁₉	—	4	1	no	—
96	C₄⋊Dic₁₉	—	4	1	no	—
97	C₄⋊Dic₁₉	—	4	1	no	—
98	C₄⋊Dic₁₉	—	4	1	no	—
99	C₄⋊Dic₁₉	—	4	1	no	—
100	C₄⋊Dic₁₉	—	4	1	no	—
101	C₄⋊Dic₁₉	—	4	2	no	—
102	Dic₁₉⋊C₄	—	4	4	no	—
103	C₂×C₁₉⋊C₈	—	4	2	no	—
104	C₂×C₁₉⋊C₈	—	4	2	no	—
105	C₂×C₁₉⋊C₈	—	4	2	no	—
106	#9129	—	19	1	no	maximal, Sylow
107	C₂³×C₁₉	—	2	1	no	—
108	C₂³×C₁₉	—	2	1	no	—
109	C₂³×C₁₉	—	2	1	no	—
110	C₂³×C₁₉	—	2	1	no	—
111	C₂³×C₁₉	#101	1	2	yes	—
112	C₂³×C₁₉	#102	1	2	yes	—
113	C₂³×C₁₉	#103	1	2	yes	—
114	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	2	1	no	—
118	C₂×C₄×C₁₉	—	2	1	no	—
119	C₂×C₄×C₁₉	—	2	1	no	—
120	C₂×C₄×C₁₉	—	2	1	no	—
121	C₂×C₄×C₁₉	—	2	2	no	—
122	C₂×C₄×C₁₉	—	2	2	no	—
123	C₂×C₄×C₁₉	—	2	4	no	—
124	C₂×C₄×C₁₉	—	2	4	no	—
125	C₂×C₄×C₁₉	—	2	2	no	—
126	C₂×C₄×C₁₉	—	2	2	no	—
127	C₂×C₄×C₁₉	—	2	1	no	—
128	C₂×C₄×C₁₉	—	2	1	no	—
129	C₂×C₄×C₁₉	—	2	1	no	—
130	C₂×C₄×C₁₉	—	4	4	no	—
131	C₂×C₄×C₁₉	#102	1	2	yes	—
132	C₂×C₄×C₁₉	#103	1	2	yes	—
133	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
135	C₂×C₄×C₁₉	#121	1	2	yes	—
136	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
137	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
139	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
140	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
141	C₂×C₄×C₁₉	#135	1	2	yes	—
142	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
144	C₂×Dic₁₉	—	4	2	no	—
145	C₂×Dic₁₉	—	4	2	no	—
146	C₂×Dic₁₉	—	4	2	no	—
147	C₂×Dic₁₉	—	4	2	no	—
148	C₂×Dic₁₉	—	4	2	no	—
149	C₂×Dic₁₉	—	4	2	no	—
150	Dic₃₈	—	8	2	no	—
151	Dic₃₈	—	8	2	no	—
152	C₈×C₁₉	—	4	4	no	—
153	C₁₉⋊C₈	—	8	2	no	—
154	C₂²×C₄⋊C₈	D₁₉	1	1	yes	characteristic
155	#2857	—	19	1	no	—
156	#1797	—	19	1	no	—
157	C₂×#332	—	19	2	no	—
158	#1296	—	19	2	no	—
159	C₂²×C₁₉	—	2	2	no	—
160	C₂²×C₁₉	—	2	2	no	—
161	C₂²×C₁₉	—	2	2	no	—
162	C₂²×C₁₉	—	2	2	no	—
163	C₂²×C₁₉	—	2	2	no	—
164	C₂²×C₁₉	—	2	2	no	—
165	C₂²×C₁₉	#332	1	4	yes	complemented
166	C₂²×C₁₉	#386	1	1	yes	characteristic
167	C₂²×C₁₉	C₂×#101	1	1	yes	characteristic
168	C₂²×C₁₉	#413	1	1	yes	characteristic
169	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
170	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
171	C₂²×C₁₉	#416	1	1	yes	characteristic
172	C₂²×C₁₉	#417	1	1	yes	characteristic
173	C₄×C₁₉	—	2	2	no	—
174	C₄×C₁₉	—	2	2	no	—
175	C₄×C₁₉	—	2	2	no	—
176	C₄×C₁₉	—	4	1	no	—
177	C₄×C₁₉	#450	1	1	yes	characteristic
178	C₄×C₁₉	#451	1	1	yes	characteristic
179	C₄×C₁₉	#465	1	2	yes	—
180	C₄×C₁₉	#466	1	2	yes	—
181	C₄×C₁₉	#483	1	1	yes	characteristic
182	C₄×C₁₉	#484	1	1	yes	characteristic
183	Dic₁₉	—	8	2	no	—
184	Dic₁₉	—	8	2	no	—
185	C₂³×C₈	—	2	1	no	—
186	C₂×C₄⋊Q₈	—	19	1	no	—
187	C₂×C₂²⋊Q₈	—	38	1	no	—
188	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
189	C₂×C₄⋊C₈	—	2	1	no	—
190	C₂×C₄⋊C₈	—	2	4	no	—
191	C₂×C₄⋊C₈	—	19	2	no	—
192	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
193	C₂×C₂²⋊C₈	—	38	1	no	—
194	#391	—	38	1	no	—
195	#389	—	19	2	no	—
196	#389	—	38	1	no	—
197	#384	—	19	1	no	—
198	#380	—	19	2	no	—
199	#380	—	38	1	no	—
200	#336	—	38	1	no	—
201	#332	—	38	4	no	—
202	C₂×C₁₉	—	2	4	no	—
203	C₂×C₁₉	C₂×#332	1	1	yes	characteristic
204	C₂×C₁₉	#1542	1	1	yes	characteristic
205	C₂×C₁₉	#1547	1	1	yes	characteristic
206	C₂×C₁₉	#1552	1	1	yes	characteristic
207	C₂×C₁₉	#1872	1	1	yes	characteristic
208	C₂×C₁₉	#1893	1	1	yes	characteristic
209	C₂×C₁₉	#1901	1	1	yes	characteristic
210	C₂²×Q₈	—	38	1	no	—
211	C₂³×C₄	C₄×D₁₉	1	1	yes	characteristic
212	C₂³×C₄	D₇₆	1	1	yes	characteristic
213	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
214	C₂²×C₈	—	2	4	no	—
215	C₂²×C₈	—	2	2	no	—
216	C₂²×C₈	—	4	2	no	—
217	C₂²×C₈	—	4	2	no	—
218	C₂²×C₈	—	38	2	no	—
219	C₄⋊Q₈	—	38	2	no	—
220	C₄⋊Q₈	—	38	2	no	—
221	C₂²⋊Q₈	—	76	2	no	—
222	C₂²⋊Q₈	—	76	2	no	—
223	C₂×C₄⋊C₄	—	19	2	no	—
224	C₂×C₄⋊C₄	—	19	2	no	—
225	C₂×C₄⋊C₄	—	38	2	no	—
226	C₂×C₄⋊C₄	—	38	1	no	—
227	C₂×C₄⋊C₄	—	38	1	no	—
228	C₂×C₄⋊C₄	—	38	1	no	—
229	C₂×C₂²⋊C₄	—	38	2	no	—
230	C₂×C₄²	—	2	2	no	—
231	C₂×C₄²	—	2	2	no	—
232	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
233	C₂×C₄²	D₇₆	1	1	yes	characteristic
234	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
235	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
236	C₄⋊C₈	—	2	4	no	—
237	C₄⋊C₈	—	2	2	no	—
238	C₄⋊C₈	—	38	2	no	—
239	C₄⋊C₈	—	38	2	no	—
240	C₄⋊C₈	C₂²⋊D₁₉	1	4	yes	—
241	C₂²⋊C₈	—	76	2	no	—
242	#95	—	38	2	no	—
243	#95	—	38	2	no	—
244	C₁₉	#9129	1	1	yes	characteristic, complemented, Sylow
245	C₂⁴	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
246	C₂×Q₈	—	76	2	no	—
247	C₂×Q₈	—	76	1	no	—
248	C₂×Q₈	—	76	1	no	—
249	C₂×Q₈	—	76	2	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	1	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	4	1	no	—
259	C₂²×C₄	—	38	2	no	—
260	C₂²×C₄	—	38	2	no	—
261	C₂²×C₄	#64516	1	2	yes	—
262	C₂²×C₄	D₁₅₂	1	2	yes	—
263	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
264	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
265	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
266	C₂²×C₄	#64526	1	2	yes	—
267	C₂²×C₄	#64528	1	2	yes	—
268	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
270	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
271	C₂×C₈	—	2	4	no	—
272	C₂×C₈	—	4	8	no	—
273	C₂×C₈	—	4	2	no	—
274	C₂×C₈	—	4	2	no	—
275	C₂×C₈	—	76	2	no	—
276	C₂×C₈	—	76	2	no	—
277	C₂×C₈	—	76	2	no	—
278	C₄⋊C₄	—	38	4	no	—
279	C₄⋊C₄	—	76	2	no	—
280	C₄⋊C₄	—	76	1	no	—
281	C₄⋊C₄	—	76	1	no	—
282	C₄⋊C₄	—	76	1	no	—
283	C₄⋊C₄	—	76	4	no	—
284	C₄⋊C₄	—	76	1	no	—
285	C₄⋊C₄	—	76	1	no	—
286	C₄⋊C₄	—	76	1	no	—
287	C₂²⋊C₄	—	76	4	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	1	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	1	no	—
292	C₄²	#64541	1	2	yes	—
293	C₄²	C₂×C₂²⋊D₁₉	1	2	yes	—
294	C₂³	—	2	1	no	—
295	C₂³	—	2	1	no	—
296	C₂³	—	2	1	no	—
297	C₂³	—	2	1	no	—
298	C₂³	#10597267	1	2	yes	—
299	C₂³	#10597278	1	2	yes	—
300	C₂³	#10597281	1	2	yes	—
301	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
302	Q₈	—	152	2	no	—
303	Q₈	—	152	2	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	4	4	no	—
320	C₂×C₄	—	76	2	no	—
321	C₂×C₄	—	76	2	no	—
322	C₂×C₄	—	76	2	no	—
323	C₂×C₄	—	76	2	no	—
324	C₂×C₄	—	76	2	no	—
325	C₂×C₄	—	76	2	no	—
326	C₂×C₄	#10597267	1	2	yes	—
327	C₂×C₄	#10597278	1	2	yes	—
328	C₂×C₄	C₂×#64516	1	1	yes	characteristic
329	C₂×C₄	C₂×D₁₅₂	1	1	yes	characteristic
330	C₂×C₄	#10597350	1	2	yes	—
331	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
332	C₂×C₄	#10597375	1	1	yes	characteristic
333	C₂×C₄	#10597376	1	2	yes	—
334	C₂×C₄	#10597378	1	1	yes	characteristic
335	C₂×C₄	C₂×#64526	1	1	yes	characteristic
336	C₂×C₄	#10597388	1	1	yes	characteristic
337	C₂×C₄	C₂×#64528	1	1	yes	characteristic
338	C₂×C₄	#10597399	1	1	yes	characteristic
339	C₈	—	4	4	no	—
340	C₈	—	152	2	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	2	2	no	—
345	C₂²	—	2	2	no	—
346	C₂²	—	2	2	no	—
347	C₂²	#11926289	1	4	yes	complemented
348	C₂²	#11926423	1	1	yes	characteristic
349	C₂²	C₂×#10597281	1	1	yes	characteristic
350	C₂²	#11926526	1	1	yes	characteristic
351	C₂²	C₂×#10597278	1	1	yes	characteristic
352	C₂²	C₂×#10597267	1	1	yes	characteristic
353	C₂²	#11926541	1	1	yes	characteristic
354	C₂²	#11926543	1	1	yes	characteristic
355	C₄	—	2	2	no	—
356	C₄	—	2	2	no	—
357	C₄	—	2	2	no	—
358	C₄	—	4	1	no	—
359	C₄	—	152	2	no	—
360	C₄	—	152	2	no	—
361	C₄	#11926695	1	1	yes	characteristic
362	C₄	#11926700	1	1	yes	characteristic
363	C₄	#11926773	1	2	yes	—
364	C₄	#11926780	1	2	yes	—
365	C₄	#11926874	1	1	yes	characteristic
366	C₄	#11926879	1	1	yes	characteristic
367	C₂	—	2	4	no	—
368	C₂	C₂×#11926289	1	1	yes	characteristic
369	C₂	#14562293	1	1	yes	characteristic
370	C₂	#14562310	1	1	yes	characteristic
371	C₂	#14562333	1	1	yes	characteristic
372	C₂	#14563942	1	1	yes	characteristic
373	C₂	#14564063	1	1	yes	characteristic
374	C₂	#14564103	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₁₉)⋊C₄
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂²×C₁₉×C₄⋊C₈	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂²×C₄⋊C₈	D₁₉
hypocenter	C₁₉	#9129
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#11926289	C₂	semidirect
4. #14561519	C₂	semidirect
165. C₂²×C₁₉	#332	semidirect
244. C₁₉	#9129	semidirect
347. C₂²	#11926289	semidirect

Maximal subgroups

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

subgroup	index
#14571987	2
#14563615	2
C₂×#11926289	2
#14561519	2
C₂²×C₁₉×C₄⋊C₈	2
#9129	19