Subgroups — #18073117

Order 4864.

494 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	C₄×#10597290	C₂	1	1	yes	maximal, characteristic
2	#14561956	C₂	1	1	yes	maximal, characteristic
3	#14561924	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×#64520	C₂	1	1	yes	maximal, characteristic, complemented
5	#14561485	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561485	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₉×#436	C₂	1	1	yes	maximal, characteristic
8	#11926597	C₂²	1	1	yes	characteristic
9	#11926597	C₂²	1	1	yes	characteristic
10	C₂×#10597290	C₄	1	2	yes	—
11	C₄×#64520	C₂²	1	1	yes	characteristic
12	C₄×#64520	C₂²	1	1	yes	characteristic
13	#11926313	C₄	1	2	yes	—
14	#11926268	C₄	1	4	yes	—
15	#11926266	—	2	1	no	—
16	#11926266	—	2	1	no	—
17	#11926265	—	2	1	no	—
18	C₈×C₁₉⋊C₈	—	2	1	no	—
19	C₂²×M₁₆×C₁₉	C₄	1	2	yes	—
20	C₂×C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
21	C₁₉×#436	—	2	1	no	—
22	C₁₉×#436	—	2	1	no	—
23	C₁₉×#436	—	2	1	no	—
24	C₁₉×#436	—	2	1	no	—
25	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₁₉×C₂²⋊C₈	C₄	1	2	yes	—
27	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
28	C₂²×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₁₉	—	2	2	no	—
31	C₂×M₁₆×C₁₉	—	2	2	no	—
32	C₂×M₁₆×C₁₉	—	4	4	no	—
33	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
36	C₄×D₄×C₁₉	—	2	1	no	—
37	C₄×D₄×C₁₉	—	2	1	no	—
38	C₄×D₄×C₁₉	—	2	1	no	—
39	C₄×D₄×C₁₉	—	2	1	no	—
40	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
43	C₂×#64520	C₂×C₄	1	2	yes	—
44	C₂×#64520	C₂×C₄	1	2	yes	—
45	C₁₉×C₄⋊C₈	—	2	1	no	—
46	C₁₉×C₄⋊C₈	—	2	1	no	—
47	C₁₉×C₂²⋊C₈	—	2	2	no	—
48	C₁₉×C₂²⋊C₈	—	2	2	no	—
49	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
50	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
51	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
52	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
53	#10597290	—	2	4	no	—
54	C₂²⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
55	C₂²⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
56	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
57	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
58	#10597260	—	2	1	no	—
59	#10597260	—	2	1	no	—
60	C₄×C₁₉⋊C₈	—	2	1	no	—
61	C₄×C₁₉⋊C₈	—	2	1	no	—
62	C₄×C₁₉⋊C₈	—	2	1	no	—
63	C₄×C₁₉⋊C₈	—	2	1	no	—
64	C₂⁴×C₁₉	C₄²	1	2	yes	—
65	C₂×D₄×C₁₉	—	2	1	no	—
66	C₂×D₄×C₁₉	—	2	1	no	—
67	C₂×D₄×C₁₉	—	2	2	no	—
68	C₂×D₄×C₁₉	—	2	2	no	—
69	C₂²×C₄×C₁₉	—	2	2	no	—
70	C₂²×C₄×C₁₉	—	2	2	no	—
71	C₂²×C₄×C₁₉	—	4	2	no	—
72	C₂²×C₄×C₁₉	—	4	2	no	—
73	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
74	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
77	M₁₆×C₁₉	—	4	4	no	—
78	M₁₆×C₁₉	—	4	4	no	—
79	C₂×C₈×C₁₉	—	2	2	no	—
80	C₂×C₈×C₁₉	—	2	2	no	—
81	C₂×C₈×C₁₉	—	4	1	no	—
82	C₂×C₈×C₁₉	—	4	2	no	—
83	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₄⋊C₄×C₁₉	—	2	1	no	—
88	C₄⋊C₄×C₁₉	—	2	1	no	—
89	C₂²⋊C₄×C₁₉	—	2	2	no	—
90	C₂²⋊C₄×C₁₉	—	2	2	no	—
91	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₁₉	#41	1	1	yes	characteristic
94	C₄²×C₁₉	#41	1	1	yes	characteristic
95	#64520	—	2	2	no	—
96	#64520	—	2	2	no	—
97	#64520	—	2	2	no	—
98	#64520	M₁₆	1	4	yes	complemented
99	C₂×C₁₉⋊C₈	—	2	2	no	—
100	C₂×C₁₉⋊C₈	—	2	2	no	—
101	C₂×C₁₉⋊C₈	—	2	2	no	—
102	C₂×C₁₉⋊C₈	—	2	2	no	—
103	#8166	—	19	1	no	maximal, Sylow
104	C₂³×C₁₉	—	2	2	no	—
105	C₂³×C₁₉	—	2	2	no	—
106	C₂³×C₁₉	—	2	2	no	—
107	C₂³×C₁₉	—	4	2	no	—
108	C₂³×C₁₉	—	4	2	no	—
109	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
110	D₄×C₁₉	—	4	2	no	—
111	D₄×C₁₉	—	4	2	no	—
112	C₂×C₄×C₁₉	—	2	1	no	—
113	C₂×C₄×C₁₉	—	2	1	no	—
114	C₂×C₄×C₁₉	—	2	2	no	—
115	C₂×C₄×C₁₉	—	4	1	no	—
116	C₂×C₄×C₁₉	—	4	4	no	—
117	C₂×C₄×C₁₉	—	4	2	no	—
118	C₂×C₄×C₁₉	—	4	2	no	—
119	C₂×C₄×C₁₉	—	4	2	no	—
120	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
127	C₂×C₄×C₁₉	#131	1	2	yes	—
128	C₈×C₁₉	—	2	1	no	—
129	C₈×C₁₉	—	2	1	no	—
130	C₈×C₁₉	—	2	1	no	—
131	C₈×C₁₉	—	4	2	no	—
132	C₈×C₁₉	#100	1	2	yes	complemented
133	C₁₉⋊C₈	—	2	4	no	—
134	C₁₉⋊C₈	—	4	2	no	—
135	C₁₉⋊C₈	—	4	2	no	—
136	C₁₉⋊C₈	—	4	2	no	—
137	C₂×#436	D₁₉	1	1	yes	characteristic
138	C₄×#100	—	19	1	no	—
139	#1460	—	19	1	no	—
140	#1451	—	19	1	no	—
141	C₈×M₁₆	—	19	1	no	—
142	#1283	—	19	1	no	—
143	#1283	—	19	1	no	—
144	C₂²×C₁₉	—	2	1	no	—
145	C₂²×C₁₉	—	2	1	no	—
146	C₂²×C₁₉	—	4	2	no	—
147	C₂²×C₁₉	—	4	2	no	—
148	C₂²×C₁₉	—	4	2	no	—
149	C₂²×C₁₉	—	8	2	no	—
150	C₂²×C₁₉	—	8	2	no	—
151	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₁₉	C₄×M₁₆	1	1	yes	characteristic
153	C₂²×C₁₉	#405	1	1	yes	characteristic
154	C₄×C₁₉	—	2	1	no	—
155	C₄×C₁₉	—	2	1	no	—
156	C₄×C₁₉	—	2	1	no	—
157	C₄×C₁₉	—	4	2	no	—
158	C₄×C₁₉	C₂×#100	1	1	yes	characteristic
159	C₄×C₁₉	#413	1	1	yes	characteristic
160	C₄×C₁₉	#435	1	2	yes	—
161	C₄×C₁₉	#436	1	2	yes	—
162	C₂²×M₁₆	Dic₁₉	1	2	yes	—
163	C₂×C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
164	#436	—	2	1	no	—
165	#436	—	2	1	no	—
166	#436	—	2	1	no	—
167	#436	—	2	1	no	—
168	#431	—	19	1	no	—
169	#431	—	19	1	no	—
170	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
171	C₂×#100	—	19	2	no	—
172	C₂×C₂²⋊C₈	Dic₁₉	1	2	yes	—
173	C₄×M₁₆	—	19	1	no	—
174	C₄×M₁₆	—	19	1	no	—
175	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
176	#343	—	19	2	no	—
177	#323	—	19	4	no	—
178	#322	—	38	1	no	—
179	#322	—	38	1	no	—
180	#322	—	38	1	no	—
181	C₈²	—	38	1	no	—
182	C₂×C₁₉	—	2	1	no	—
183	C₂×C₁₉	—	2	1	no	—
184	C₂×C₁₉	—	8	2	no	—
185	C₂×C₁₉	#1753	1	1	yes	characteristic
186	C₂×C₁₉	C₄×#100	1	1	yes	characteristic
187	C₂×C₁₉	#1757	1	1	yes	characteristic
188	C₂²×D₄	C₄×D₁₉	1	1	yes	characteristic
189	C₂³×C₄	C₂×Dic₁₉	1	2	yes	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	4	4	no	—
193	C₂×M₁₆	—	19	2	no	—
194	C₂×M₁₆	—	19	2	no	—
195	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
196	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
197	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
198	C₄×D₄	—	2	1	no	—
199	C₄×D₄	—	2	1	no	—
200	C₄×D₄	—	2	1	no	—
201	C₄×D₄	—	2	1	no	—
202	C₂×C₄⋊C₄	C₄×D₁₉	1	1	yes	characteristic
203	C₂×C₂²⋊C₄	C₄×D₁₉	1	2	yes	—
204	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
205	C₄⋊C₈	—	2	1	no	—
206	C₄⋊C₈	—	2	1	no	—
207	C₄⋊C₈	—	38	1	no	—
208	C₄⋊C₈	—	38	1	no	—
209	#100	—	38	4	no	—
210	C₂²⋊C₈	—	2	2	no	—
211	C₂²⋊C₈	—	2	2	no	—
212	C₂²⋊C₈	—	19	2	no	—
213	C₂²⋊C₈	—	19	2	no	—
214	#97	—	38	1	no	—
215	#97	—	38	1	no	—
216	C₄×C₈	—	38	1	no	—
217	C₄×C₈	—	38	1	no	—
218	C₄×C₈	—	38	1	no	—
219	C₄×C₈	—	38	1	no	—
220	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
221	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
222	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
223	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
224	C₁₉	#8166	1	1	yes	characteristic, complemented, Sylow
225	C₂⁴	C₄×Dic₁₉	1	2	yes	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	1	no	—
229	C₂×D₄	—	2	1	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
235	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
237	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
238	M₁₆	—	4	4	no	—
239	M₁₆	—	4	4	no	—
240	M₁₆	—	19	4	no	—
241	M₁₆	—	38	2	no	—
242	M₁₆	—	38	2	no	—
243	M₁₆	—	38	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	4	1	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	—	38	2	no	—
249	C₂×C₈	—	38	2	no	—
250	C₂×C₈	—	38	2	no	—
251	C₂×C₈	—	38	2	no	—
252	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
253	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
254	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
255	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₂²⋊C₄	—	2	2	no	—
260	C₄²	#64541	1	1	yes	characteristic
261	C₄²	#64541	1	1	yes	characteristic
262	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
263	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
264	C₂³	—	2	2	no	—
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	4	2	no	—
268	C₂³	—	4	2	no	—
269	C₂³	C₂×C₄×Dic₁₉	1	1	yes	characteristic
270	D₄	—	4	2	no	—
271	D₄	—	4	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	M₁₆×D₁₉	1	2	yes	—
281	C₂×C₄	#10597353	1	2	yes	—
282	C₂×C₄	#10597372	1	1	yes	characteristic
283	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
284	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
285	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
286	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
287	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
288	C₈	—	2	1	no	—
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	4	2	no	—
292	C₈	—	38	4	no	—
293	C₈	—	76	2	no	—
294	C₈	—	76	2	no	—
295	C₈	—	76	2	no	—
296	C₈	#10597290	1	2	yes	complemented
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	4	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	—	8	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	C₄×C₂²⋊Dic₁₉	1	1	yes	characteristic
305	C₂²	M₁₆×Dic₁₉	1	1	yes	characteristic
306	C₂²	#11926502	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	4	2	no	—
311	C₄	C₂×#10597290	1	1	yes	characteristic
312	C₄	#11926528	1	1	yes	characteristic
313	C₄	#11926613	1	2	yes	—
314	C₄	#11926620	1	2	yes	—
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	8	2	no	—
318	C₂	#14563416	1	1	yes	characteristic
319	C₂	C₄×#10597290	1	1	yes	characteristic
320	C₂	#14563435	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₁₉
commutator	C₂³×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#436	C₂
socle	C₂²×C₁₉	C₄×C₂²⋊C₄
prime radical	C₂²×D₄×C₁₉	C₂×C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#436	D₁₉
hypocenter	C₁₉	#8166
LCS	G ▷ C₂³×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#436

Complements

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

N	H ≅ G/N	type
3. #14561924	C₂	semidirect
4. C₈×#64520	C₂	semidirect
5. #14561485	C₂	semidirect
6. #14561485	C₂	semidirect
98. #64520	M₁₆	semidirect
132. C₈×C₁₉	#100	semidirect
224. C₁₉	#8166	semidirect
296. C₈	#10597290	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10597290	2
#14561956	2
#14561924	2
C₈×#64520	2
#14561485	2
C₂×C₁₉×#436	2
#8166	19