Subgroups — #18073069

Order 4864.

506 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926500	C₂	1	1	yes	maximal, characteristic
2	#14561949	C₂	1	1	yes	maximal, characteristic
3	#14561919	C₂	1	2	yes	maximal, complemented
4	#14561479	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1458	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₄	1	1	yes	characteristic
7	C₂³×C₁₉⋊C₈	C₄	1	1	yes	characteristic
8	#11926597	C₂²	1	2	yes	—
9	#11926500	—	2	2	no	—
10	#11926500	—	2	2	no	—
11	C₂×#10597260	C₄	1	1	yes	characteristic
12	C₂×#10597260	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₁₉⋊C₈	C₄	1	1	yes	characteristic
14	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
15	#11926270	C₄	1	2	yes	—
16	#11926268	C₄	1	2	yes	—
17	#11926266	—	2	2	no	—
18	C₈×C₁₉⋊C₈	—	2	2	no	—
19	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
20	C₁₉×#431	C₂²	1	2	yes	—
21	C₁₉×#324	C₄	1	2	yes	—
22	C₁₉×#323	C₄	1	2	yes	—
23	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₁₉×#117	—	2	2	no	—
25	C₁₉×#117	—	2	2	no	—
26	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
32	C₂×#64520	—	2	2	no	—
33	C₂×#64520	—	2	2	no	—
34	C₂×#64520	C₈	1	4	yes	complemented
35	C₂²×C₁₉⋊C₈	—	2	1	no	—
36	C₂²×C₁₉⋊C₈	—	2	1	no	—
37	C₂²×C₁₉⋊C₈	—	2	2	no	—
38	C₂²×C₁₉⋊C₈	C₈	1	4	yes	complemented
39	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
43	C₁₉×C₄⋊C₈	—	2	2	no	—
44	C₁₉×C₂²⋊C₈	C₂×C₄	1	4	yes	—
45	C₄×C₈×C₁₉	—	2	2	no	—
46	C₂²⋊(C₁₉⋊C₈)	C₂×C₄	1	4	yes	—
47	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
48	#10597260	—	2	2	no	—
49	#10597260	—	2	2	no	—
50	C₄×C₁₉⋊C₈	—	2	2	no	—
51	C₄×C₁₉⋊C₈	—	2	1	no	—
52	C₄×C₁₉⋊C₈	—	2	1	no	—
53	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
54	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
55	C₂⁴×C₁₉	C₄²	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	—	2	1	no	—
57	C₂²×C₄×C₁₉	—	2	1	no	—
58	C₂²×C₄×C₁₉	—	2	1	no	—
59	C₂²×C₄×C₁₉	—	2	1	no	—
60	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂×C₈	1	2	yes	—
64	C₂²×C₄×C₁₉	C₂×C₈	1	2	yes	—
65	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₉	—	2	4	no	—
69	C₄⋊C₄×C₁₉	—	2	2	no	—
70	C₄⋊C₄×C₁₉	—	2	2	no	—
71	C₂²⋊C₄×C₁₉	—	2	2	no	—
72	C₂²⋊C₄×C₁₉	—	2	2	no	—
73	C₄²×C₁₉	—	2	1	no	—
74	C₄²×C₁₉	—	2	1	no	—
75	C₄²×C₁₉	C₂×D₄	1	2	yes	—
76	C₄²×C₁₉	#41	1	2	yes	—
77	#64520	—	2	8	no	—
78	C₂×C₁₉⋊C₈	—	2	1	no	—
79	C₂×C₁₉⋊C₈	—	2	1	no	—
80	C₂×C₁₉⋊C₈	—	2	1	no	—
81	C₂×C₁₉⋊C₈	—	2	1	no	—
82	C₂×C₁₉⋊C₈	—	2	4	no	—
83	C₂×C₁₉⋊C₈	—	2	4	no	—
84	C₂×C₁₉⋊C₈	—	2	4	no	—
85	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	2	yes	—
88	C₂×C₁₉⋊C₈	C₂×C₈	1	2	yes	—
89	C₂×C₁₉⋊C₈	C₂×C₈	1	2	yes	—
90	C₂×C₁₉⋊C₈	C₂×C₈	1	2	yes	—
91	C₂×C₁₉⋊C₈	C₂×C₈	1	2	yes	—
92	C₂×C₁₉⋊C₈	M₁₆	1	2	yes	—
93	C₂×C₁₉⋊C₈	M₁₆	1	2	yes	—
94	#8156	—	19	1	no	maximal, Sylow
95	C₂³×C₁₉	—	2	1	no	—
96	C₂³×C₁₉	—	2	1	no	—
97	C₂³×C₁₉	—	4	1	no	—
98	C₂³×C₁₉	—	4	1	no	—
99	C₂³×C₁₉	C₄×C₈	1	2	yes	—
100	C₂³×C₁₉	C₂×C₄²	1	1	yes	characteristic
101	C₂×C₄×C₁₉	—	2	1	no	—
102	C₂×C₄×C₁₉	—	2	1	no	—
103	C₂×C₄×C₁₉	—	2	2	no	—
104	C₂×C₄×C₁₉	—	2	2	no	—
105	C₂×C₄×C₁₉	—	2	2	no	—
106	C₂×C₄×C₁₉	—	2	2	no	—
107	C₂×C₄×C₁₉	—	2	1	no	—
108	C₂×C₄×C₁₉	—	2	1	no	—
109	C₂×C₄×C₁₉	—	4	1	no	—
110	C₂×C₄×C₁₉	—	4	1	no	—
111	C₂×C₄×C₁₉	—	4	1	no	—
112	C₂×C₄×C₁₉	—	4	1	no	—
113	C₂×C₄×C₁₉	C₄×C₈	1	2	yes	—
114	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
122	C₈×C₁₉	—	4	4	no	—
123	C₁₉⋊C₈	—	2	2	no	—
124	C₁₉⋊C₈	—	2	2	no	—
125	C₁₉⋊C₈	—	4	4	no	—
126	C₁₉⋊C₈	C₂²⋊C₈	1	2	yes	—
127	C₁₉⋊C₈	C₂²⋊C₈	1	2	yes	—
128	C₁₉⋊C₈	C₂²⋊C₈	1	4	yes	—
129	C₂×#405	—	19	1	no	—
130	#1458	—	19	1	no	—
131	#1458	D₁₉	1	1	yes	characteristic
132	#1450	—	19	2	no	—
133	#1281	—	19	2	no	—
134	C₂²×C₁₉	—	2	1	no	—
135	C₂²×C₁₉	—	2	1	no	—
136	C₂²×C₁₉	—	2	2	no	—
137	C₂²×C₁₉	—	4	2	no	—
138	C₂²×C₁₉	—	4	2	no	—
139	C₂²×C₁₉	—	4	1	no	—
140	C₂²×C₁₉	—	4	1	no	—
141	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
143	C₂²×C₁₉	C₄×M₁₆	1	1	yes	characteristic
144	C₄×C₁₉	—	2	2	no	—
145	C₄×C₁₉	—	2	2	no	—
146	C₄×C₁₉	—	2	2	no	—
147	C₄×C₁₉	—	4	1	no	—
148	C₄×C₁₉	—	4	1	no	—
149	C₄×C₁₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
150	C₄×C₁₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
151	C₄×C₁₉	#431	1	1	yes	characteristic
152	C₄×C₁₉	#431	1	1	yes	characteristic
153	C₂²×M₁₆	—	19	1	no	—
154	C₂³×C₈	—	19	1	no	—
155	C₂×#117	C₂×D₁₉	1	1	yes	characteristic
156	#431	—	19	2	no	—
157	#431	C₂×D₁₉	1	2	yes	—
158	#405	—	38	2	no	—
159	#405	—	38	2	no	—
160	C₂×#97	—	19	1	no	—
161	C₂×#97	—	19	1	no	—
162	C₂×C₄×C₈	—	19	1	no	—
163	C₂×C₄×C₈	—	19	1	no	—
164	#324	—	19	2	no	—
165	#324	Dic₁₉	1	2	yes	—
166	#323	—	19	2	no	—
167	#323	Dic₁₉	1	2	yes	—
168	#322	—	38	2	no	—
169	C₈²	—	38	2	no	—
170	C₂×C₁₉	—	2	1	no	—
171	C₂×C₁₉	—	2	1	no	—
172	C₂×C₁₉	—	4	2	no	—
173	C₂×C₁₉	C₄×C₂²⋊C₈	1	1	yes	characteristic
174	C₂×C₁₉	#1757	1	1	yes	characteristic
175	C₂×C₁₉	#1757	1	1	yes	characteristic
176	C₂³×C₄	C₄×D₁₉	1	1	yes	characteristic
177	C₂×M₁₆	—	19	4	no	—
178	C₂×M₁₆	—	38	2	no	—
179	C₂×M₁₆	—	38	2	no	—
180	C₂²×C₈	—	19	1	no	—
181	C₂²×C₈	—	19	1	no	—
182	C₂²×C₈	—	19	4	no	—
183	C₂²×C₈	—	19	1	no	—
184	C₂²×C₈	—	19	1	no	—
185	C₂²×C₈	—	38	1	no	—
186	C₂²×C₈	—	38	1	no	—
187	C₂²×C₈	—	38	2	no	—
188	#117	—	2	2	no	—
189	#117	—	2	2	no	—
190	C₂×C₄⋊C₄	C₄×D₁₉	1	1	yes	characteristic
191	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
192	C₂×C₂²⋊C₄	C₄×D₁₉	1	1	yes	characteristic
193	C₂×C₂²⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
194	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
195	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
196	C₄⋊C₈	—	2	2	no	—
197	C₄⋊C₈	—	38	2	no	—
198	C₂²⋊C₈	—	19	4	no	—
199	C₂²⋊C₈	C₂×Dic₁₉	1	4	yes	—
200	#97	—	38	2	no	—
201	#97	—	38	2	no	—
202	C₄×C₈	—	2	2	no	—
203	C₄×C₈	—	19	2	no	—
204	C₄×C₈	—	19	2	no	—
205	C₄×C₈	—	38	1	no	—
206	C₄×C₈	—	38	1	no	—
207	C₄×C₈	—	38	2	no	—
208	C₁₉	#8156	1	1	yes	characteristic, complemented, Sylow
209	C₂⁴	C₄×Dic₁₉	1	1	yes	characteristic
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	2	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	2	1	no	—
214	C₂²×C₄	C₈×D₁₉	1	2	yes	—
215	C₂²×C₄	C₈×D₁₉	1	2	yes	—
216	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
217	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
218	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
221	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
222	M₁₆	—	38	8	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	19	1	no	—
225	C₂×C₈	—	19	1	no	—
226	C₂×C₈	—	19	2	no	—
227	C₂×C₈	—	19	2	no	—
228	C₂×C₈	—	19	2	no	—
229	C₂×C₈	—	19	2	no	—
230	C₂×C₈	—	19	2	no	—
231	C₂×C₈	—	19	2	no	—
232	C₂×C₈	—	19	2	no	—
233	C₂×C₈	—	38	4	no	—
234	C₂×C₈	—	38	4	no	—
235	C₂×C₈	—	38	1	no	—
236	C₂×C₈	—	38	1	no	—
237	C₂×C₈	—	38	1	no	—
238	C₂×C₈	—	38	1	no	—
239	C₂×C₈	—	38	4	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₂²⋊C₄	—	2	2	no	—
243	C₂²⋊C₄	—	2	2	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	2	1	no	—
246	C₄²	D₄×D₁₉	1	2	yes	—
247	C₄²	#64543	1	2	yes	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	1	no	—
250	C₂³	—	4	1	no	—
251	C₂³	—	4	1	no	—
252	C₂³	C₈×Dic₁₉	1	2	yes	—
253	C₂³	C₂×C₄×Dic₁₉	1	1	yes	characteristic
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	C₈×Dic₁₉	1	2	yes	—
267	C₂×C₄	#10597323	1	1	yes	characteristic
268	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
269	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
270	C₂×C₄	C₂×C₈×D₁₉	1	1	yes	characteristic
271	C₂×C₄	C₂×C₈×D₁₉	1	1	yes	characteristic
272	C₂×C₄	M₁₆×D₁₉	1	1	yes	characteristic
273	C₂×C₄	M₁₆×D₁₉	1	1	yes	characteristic
274	C₂×C₄	D₄×Dic₁₉	1	2	yes	—
275	C₈	—	4	4	no	—
276	C₈	—	19	2	no	—
277	C₈	—	19	2	no	—
278	C₈	—	19	4	no	—
279	C₈	—	38	2	no	—
280	C₈	—	38	2	no	—
281	C₈	—	76	4	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	C₂²⋊C₄×Dic₁₉	1	1	yes	characteristic
290	C₂²	C₂×C₈×Dic₁₉	1	1	yes	characteristic
291	C₂²	M₁₆×Dic₁₉	1	1	yes	characteristic
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	1	no	—
297	C₄	C₂²⋊C₈×D₁₉	1	1	yes	characteristic
298	C₄	C₂²⋊C₈×D₁₉	1	1	yes	characteristic
299	C₄	#11926594	1	1	yes	characteristic
300	C₄	#11926594	1	1	yes	characteristic
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	4	2	no	—
304	C₂	C₂²⋊C₈×Dic₁₉	1	1	yes	characteristic
305	C₂	#14563437	1	1	yes	characteristic
306	C₂	#14563438	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₁₉
commutator	C₂²×C₁₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1458	C₂
socle	C₂²×C₁₉	C₄×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₄²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1458	D₁₉
hypocenter	C₁₉	#8156
LCS	G ▷ C₂²×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1458

Complements

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

N	H ≅ G/N	type
3. #14561919	C₂	semidirect
4. #14561479	C₂	semidirect
34. C₂×#64520	C₈	semidirect
38. C₂²×C₁₉⋊C₈	C₈	semidirect
208. C₁₉	#8156	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926500	2
#14561949	2
#14561919	2
#14561479	2
C₁₉×#1458	2
#8156	19