Subgroups — #18073120

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	C₂×#11926622	C₂	1	1	yes	maximal, characteristic
2	#14561956	C₂	1	1	yes	maximal, characteristic
3	#14561922	C₂	1	1	yes	maximal, characteristic, complemented
4	M₁₆×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#14561486	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₁₉×#100	C₂	1	1	yes	maximal, characteristic
8	C₂²×#64520	C₄	1	2	yes	—
9	#11926622	—	2	1	no	—
10	#11926622	—	2	1	no	—
11	#11926622	—	2	1	no	—
12	#11926622	—	2	1	no	—
13	#11926597	C₂²	1	1	yes	characteristic
14	#11926597	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊(C₁₉⋊C₈)	C₄	1	2	yes	—
17	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
18	#11926268	C₄	1	4	yes	—
19	#11926266	—	2	1	no	—
20	#11926266	—	2	1	no	—
21	#11926265	—	2	1	no	—
22	C₈×C₁₉⋊C₈	—	2	1	no	—
23	C₂×C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
24	C₂×C₁₉×#100	C₄	1	2	yes	—
25	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
26	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
27	C₁₉×#343	C₄	1	2	yes	—
28	C₂²×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
32	C₄×D₄×C₁₉	—	2	1	no	—
33	C₄×D₄×C₁₉	—	2	1	no	—
34	C₄×D₄×C₁₉	—	2	1	no	—
35	C₄×D₄×C₁₉	—	2	1	no	—
36	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
39	C₂×#64520	—	2	2	no	—
40	C₂×#64520	—	2	2	no	—
41	C₂×#64520	—	4	4	no	—
42	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
44	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₁₉×#100	—	2	4	no	—
46	C₁₉×#97	—	2	1	no	—
47	C₁₉×#97	—	2	1	no	—
48	C₄×C₈×C₁₉	—	2	1	no	—
49	C₄×C₈×C₁₉	—	2	1	no	—
50	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
51	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
52	C₂²⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
53	C₂²⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
54	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
55	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
56	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
57	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
58	C₄×C₁₉⋊C₈	—	2	1	no	—
59	C₄×C₁₉⋊C₈	—	2	1	no	—
60	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
61	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
62	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
63	C₄×C₁₉⋊C₈	D₄	1	1	yes	characteristic
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	—	2	2	no	—
78	M₁₆×C₁₉	—	2	2	no	—
79	M₁₆×C₁₉	—	2	2	no	—
80	M₁₆×C₁₉	M₁₆	1	4	yes	complemented
81	C₂×C₈×C₁₉	—	2	2	no	—
82	C₂×C₈×C₁₉	—	2	2	no	—
83	C₄⋊C₄×C₁₉	—	2	1	no	—
84	C₄⋊C₄×C₁₉	—	2	1	no	—
85	C₂²⋊C₄×C₁₉	—	2	2	no	—
86	C₂²⋊C₄×C₁₉	—	2	2	no	—
87	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₉	#41	1	1	yes	characteristic
90	C₄²×C₁₉	#41	1	1	yes	characteristic
91	#64520	—	4	4	no	—
92	#64520	—	4	4	no	—
93	C₂×C₁₉⋊C₈	—	2	2	no	—
94	C₂×C₁₉⋊C₈	—	2	2	no	—
95	C₂×C₁₉⋊C₈	—	2	2	no	—
96	C₂×C₁₉⋊C₈	—	2	2	no	—
97	C₂×C₁₉⋊C₈	—	4	1	no	—
98	C₂×C₁₉⋊C₈	—	4	2	no	—
99	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
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	4	no	—
129	C₈×C₁₉	—	4	2	no	—
130	C₁₉⋊C₈	—	2	1	no	—
131	C₁₉⋊C₈	—	2	1	no	—
132	C₁₉⋊C₈	—	2	1	no	—
133	C₁₉⋊C₈	—	4	2	no	—
134	C₁₉⋊C₈	—	4	2	no	—
135	C₁₉⋊C₈	—	4	2	no	—
136	C₁₉⋊C₈	#100	1	2	yes	complemented
137	C₂×#436	—	19	1	no	—
138	C₄×#100	D₁₉	1	1	yes	characteristic
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	—	19	2	no	—
163	C₂×C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
164	#436	—	38	1	no	—
165	#436	—	38	1	no	—
166	#436	—	38	1	no	—
167	#436	—	38	1	no	—
168	#431	—	19	1	no	—
169	#431	—	19	1	no	—
170	C₂×C₄⋊C₈	—	19	1	no	—
171	C₂×#100	Dic₁₉	1	2	yes	—
172	C₂×C₂²⋊C₈	—	19	2	no	—
173	C₄×M₁₆	C₂×D₁₉	1	1	yes	characteristic
174	C₄×M₁₆	C₂×D₁₉	1	1	yes	characteristic
175	C₂×C₄×C₈	—	19	1	no	—
176	#343	Dic₁₉	1	2	yes	—
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	C₂×Dic₁₉	1	1	yes	characteristic
189	C₂³×C₄	C₄×D₁₉	1	2	yes	—
190	C₂×M₁₆	—	38	2	no	—
191	C₂×M₁₆	—	38	2	no	—
192	C₂×M₁₆	—	76	4	no	—
193	C₂×M₁₆	C₂×Dic₁₉	1	2	yes	—
194	C₂×M₁₆	C₂×Dic₁₉	1	2	yes	—
195	C₂²×C₈	—	19	1	no	—
196	C₂²×C₈	—	19	1	no	—
197	C₂²×C₈	—	19	2	no	—
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	C₂×Dic₁₉	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	—	38	1	no	—
206	C₄⋊C₈	—	38	1	no	—
207	C₄⋊C₈	—	38	1	no	—
208	C₄⋊C₈	—	38	1	no	—
209	#100	—	2	4	no	—
210	C₂²⋊C₈	—	19	2	no	—
211	C₂²⋊C₈	—	19	2	no	—
212	C₂²⋊C₈	—	38	2	no	—
213	C₂²⋊C₈	—	38	2	no	—
214	#97	—	2	1	no	—
215	#97	—	2	1	no	—
216	C₄×C₈	—	2	1	no	—
217	C₄×C₈	—	2	1	no	—
218	C₄×C₈	—	19	1	no	—
219	C₄×C₈	—	19	1	no	—
220	C₄×C₈	—	19	1	no	—
221	C₄×C₈	—	19	1	no	—
222	C₄×C₈	—	38	1	no	—
223	C₄×C₈	—	38	1	no	—
224	C₁₉	#8166	1	1	yes	characteristic, complemented, Sylow
225	C₂⁴	C₄×Dic₁₉	1	2	yes	—
226	C₂×D₄	—	2	1	no	—
227	C₂×D₄	—	2	1	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	2	2	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	—	2	2	no	—
239	M₁₆	—	2	2	no	—
240	M₁₆	—	2	2	no	—
241	M₁₆	—	76	4	no	—
242	M₁₆	—	76	4	no	—
243	M₁₆	#64520	1	4	yes	complemented
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	19	1	no	—
247	C₂×C₈	—	19	1	no	—
248	C₂×C₈	—	19	1	no	—
249	C₂×C₈	—	19	1	no	—
250	C₂×C₈	—	38	2	no	—
251	C₂×C₈	—	38	2	no	—
252	C₂×C₈	—	38	2	no	—
253	C₂×C₈	—	38	2	no	—
254	C₂×C₈	—	76	1	no	—
255	C₂×C₈	—	76	2	no	—
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	D₄×D₁₉	1	1	yes	characteristic
261	C₄²	D₄×D₁₉	1	1	yes	characteristic
262	C₄²	#64543	1	1	yes	characteristic
263	C₄²	#64543	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	#10597323	1	1	yes	characteristic
281	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
282	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
283	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
284	C₂×C₄	C₂×#64520	1	2	yes	—
285	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
286	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
287	C₂×C₄	#10597394	1	2	yes	—
288	C₈	—	2	4	no	—
289	C₈	—	4	2	no	—
290	C₈	—	19	2	no	—
291	C₈	—	38	1	no	—
292	C₈	—	38	1	no	—
293	C₈	—	38	1	no	—
294	C₈	—	76	2	no	—
295	C₈	—	76	2	no	—
296	C₈	—	76	2	no	—
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	C₂²⋊C₄×Dic₁₉	1	1	yes	characteristic
305	C₂²	C₄×#64520	1	1	yes	characteristic
306	C₂²	#11926500	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	#100×D₁₉	1	1	yes	characteristic
312	C₄	#11926525	1	1	yes	characteristic
313	C₄	#11926616	1	2	yes	—
314	C₄	#11926622	1	2	yes	—
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	8	2	no	—
318	C₂	#14563419	1	1	yes	characteristic
319	C₂	#100×Dic₁₉	1	1	yes	characteristic
320	C₂	#14563438	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₁₉
commutator	C₂³×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₄×C₁₉×#100	C₂
socle	C₂²×C₁₉	C₄×C₂²⋊C₄
prime radical	C₂²×D₄×C₁₉	C₂×C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄×#100	D₁₉
hypocenter	C₁₉	#8166
LCS	G ▷ C₂³×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#100

Complements

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

N	H ≅ G/N	type
3. #14561922	C₂	semidirect
4. M₁₆×C₁₉⋊C₈	C₂	semidirect
5. #14561486	C₂	semidirect
6. #14561486	C₂	semidirect
80. M₁₆×C₁₉	M₁₆	semidirect
136. C₁₉⋊C₈	#100	semidirect
224. C₁₉	#8166	semidirect
243. M₁₆	#64520	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926622	2
#14561956	2
#14561922	2
M₁₆×C₁₉⋊C₈	2
#14561486	2
C₄×C₁₉×#100	2
#8166	19