Subgroups — #18075753

Order 4864.

434 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563799	C₂	1	1	yes	maximal, characteristic, complemented
2	#14562383	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562195	C₂	1	2	yes	maximal
4	#14561495	C₂	1	2	yes	maximal, complemented
5	C₁₉×#2931	C₂	1	1	yes	maximal, characteristic
6	#11926597	C₂²	1	2	yes	—
7	C₂×#10597275	—	2	1	no	—
8	C₂×#10597274	—	2	1	no	—
9	#11926410	C₂²	1	2	yes	—
10	#11926297	—	2	1	no	—
11	#11926295	—	2	1	no	—
12	#11926278	—	2	4	no	—
13	C₂×C₁₉×#131	—	2	1	no	—
14	C₁₉×#532	C₂²	1	1	yes	characteristic
15	C₁₉×#436	—	2	1	no	—
16	C₁₉×#436	C₄	1	2	yes	complemented
17	C₈×D₄×C₁₉	—	2	1	no	—
18	C₈×D₄×C₁₉	C₄	1	2	yes	complemented
19	C₁₉×#423	C₂²	1	1	yes	characteristic
20	C₁₉×#408	—	2	1	no	—
21	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
22	C₂×#41×C₁₉	—	2	1	no	—
23	C₁₉×#131	—	4	2	no	—
24	C₁₉×#131	—	4	2	no	—
25	C₂×M₁₆×C₁₉	—	2	1	no	—
26	C₂×M₁₆×C₁₉	—	2	2	no	—
27	C₂²×C₈×C₁₉	—	2	2	no	—
28	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
30	C₁₉×C₄⋊Q₈	Q₈	1	1	yes	characteristic
31	C₁₉×#127	Q₈	1	1	yes	characteristic
32	C₁₉×#124	—	2	1	no	—
33	C₁₉×#121	—	2	1	no	—
34	C₁₉×#121	D₄	1	2	yes	—
35	C₄×D₄×C₁₉	—	2	1	no	—
36	C₄×D₄×C₁₉	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊Dic₁₉	—	2	2	no	—
39	C₁₉×C₄⋊C₈	—	2	1	no	—
40	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
41	C₁₉×C₂²⋊C₈	—	2	2	no	—
42	C₁₉×C₂²⋊C₈	—	2	2	no	—
43	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
46	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
47	C₁₉⋊#95	—	2	2	no	—
48	C₁₉⋊#95	—	2	2	no	—
49	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
50	#10597275	—	4	2	no	—
51	#10597274	—	4	2	no	—
52	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
53	C₄×C₁₉⋊C₈	—	2	2	no	—
54	#41×C₁₉	—	4	1	no	—
55	#41×C₁₉	—	4	1	no	—
56	#41×C₁₉	—	4	1	no	—
57	#41×C₁₉	—	4	1	no	—
58	C₂×Q₈×C₁₉	—	2	1	no	—
59	C₂×D₄×C₁₉	—	2	2	no	—
60	C₂×D₄×C₁₉	—	2	1	no	—
61	C₂×D₄×C₁₉	—	2	1	no	—
62	C₂×D₄×C₁₉	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₉	—	2	2	no	—
64	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
67	C₂²×Dic₁₉	—	4	2	no	—
68	M₁₆×C₁₉	—	4	1	no	—
69	M₁₆×C₁₉	—	4	1	no	—
70	M₁₆×C₁₉	—	4	4	no	—
71	C₂×C₈×C₁₉	—	2	1	no	—
72	C₂×C₈×C₁₉	—	2	1	no	—
73	C₂×C₈×C₁₉	—	2	2	no	—
74	C₂×C₈×C₁₉	—	4	4	no	—
75	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₉	D₈	1	2	yes	—
80	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
81	C₄⋊C₄×C₁₉	—	2	1	no	—
82	C₄⋊C₄×C₁₉	C₄⋊C₄	1	2	yes	—
83	C₂²⋊C₄×C₁₉	—	2	2	no	—
84	C₂²⋊C₄×C₁₉	—	2	2	no	—
85	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
88	C₄²×C₁₉	#41	1	1	yes	characteristic
89	C₄⋊Dic₁₉	—	4	2	no	—
90	C₄⋊Dic₁₉	—	4	2	no	—
91	C₂×C₁₉⋊C₈	—	4	2	no	—
92	#8676	—	19	1	no	maximal, Sylow
93	C₂³×C₁₉	—	2	2	no	—
94	C₂³×C₁₉	#121	1	1	yes	characteristic
95	Q₈×C₁₉	—	4	1	no	—
96	Q₈×C₁₉	—	4	1	no	—
97	D₄×C₁₉	—	4	2	no	—
98	D₄×C₁₉	—	4	2	no	—
99	D₄×C₁₉	—	4	2	no	—
100	D₄×C₁₉	—	4	2	no	—
101	D₄×C₁₉	—	4	1	no	—
102	D₄×C₁₉	—	4	1	no	—
103	C₂×C₄×C₁₉	—	2	2	no	—
104	C₂×C₄×C₁₉	—	4	1	no	—
105	C₂×C₄×C₁₉	—	4	1	no	—
106	C₂×C₄×C₁₉	—	4	2	no	—
107	C₂×C₄×C₁₉	—	4	2	no	—
108	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
111	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
116	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
118	C₂×Dic₁₉	—	8	2	no	—
119	C₂×Dic₁₉	—	8	2	no	—
120	C₂×Dic₁₉	—	8	2	no	—
121	C₈×C₁₉	—	2	1	no	—
122	C₈×C₁₉	—	2	1	no	—
123	C₈×C₁₉	—	4	2	no	—
124	C₈×C₁₉	#102	1	2	yes	—
125	C₈×C₁₉	#102	1	2	yes	—
126	C₁₉⋊C₈	—	8	2	no	—
127	#2931	D₁₉	1	1	yes	characteristic
128	#1846	—	19	1	no	—
129	#1564	—	19	1	no	—
130	#1517	—	19	2	no	—
131	#1286	—	19	2	no	—
132	C₂²×C₁₉	—	2	1	no	—
133	C₂²×C₁₉	—	2	1	no	—
134	C₂²×C₁₉	—	4	2	no	—
135	C₂²×C₁₉	—	4	2	no	—
136	C₂²×C₁₉	—	4	2	no	—
137	C₂²×C₁₉	#380	1	1	yes	characteristic
138	C₂²×C₁₉	#459	1	1	yes	characteristic
139	C₂²×C₁₉	#461	1	1	yes	characteristic
140	C₄×C₁₉	—	2	1	no	—
141	C₄×C₁₉	—	2	1	no	—
142	C₄×C₁₉	—	2	1	no	—
143	C₄×C₁₉	—	2	1	no	—
144	C₄×C₁₉	—	4	2	no	—
145	C₄×C₁₉	C₂×#102	1	1	yes	characteristic
146	C₄×C₁₉	#417	1	1	yes	characteristic
147	C₄×C₁₉	#474	1	1	yes	characteristic
148	C₄×C₁₉	#476	1	1	yes	characteristic
149	Dic₁₉	—	16	2	no	—
150	C₂×#131	—	2	1	no	—
151	#532	C₂×D₁₉	1	1	yes	characteristic
152	#436	—	2	1	no	—
153	#436	Dic₁₉	1	2	yes	complemented
154	C₈×D₄	—	2	1	no	—
155	C₈×D₄	Dic₁₉	1	2	yes	complemented
156	#431	—	19	2	no	—
157	C₂×#107	—	38	1	no	—
158	C₂×#106	—	38	1	no	—
159	#423	C₂×D₁₉	1	1	yes	characteristic
160	#408	—	2	1	no	—
161	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
162	#382	—	19	2	no	—
163	#335	—	38	1	no	—
164	#334	—	38	1	no	—
165	#327	—	38	4	no	—
166	C₂×C₁₉	—	2	1	no	—
167	C₂×C₁₉	—	2	1	no	—
168	C₂×C₁₉	—	8	2	no	—
169	C₂×C₁₉	#1802	1	1	yes	characteristic
170	C₂×C₁₉	#1812	1	1	yes	characteristic
171	C₂×C₁₉	#1814	1	1	yes	characteristic
172	C₂×#41	—	2	1	no	—
173	#131	—	4	2	no	—
174	#131	—	4	2	no	—
175	C₂×M₁₆	—	2	1	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
179	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
180	C₄⋊Q₈	Dic₃₈	1	1	yes	characteristic
181	#127	Dic₃₈	1	1	yes	characteristic
182	#124	—	2	1	no	—
183	#121	—	2	1	no	—
184	#121	D₇₆	1	2	yes	—
185	C₄×D₄	—	2	1	no	—
186	C₄×D₄	C₂×Dic₁₉	1	2	yes	—
187	C₂×C₄⋊C₄	—	38	2	no	—
188	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
189	#107	—	76	2	no	—
190	#106	—	76	2	no	—
191	C₄⋊C₈	—	2	1	no	—
192	C₄⋊C₈	—	38	2	no	—
193	C₄⋊C₈	C₂×Dic₁₉	1	2	yes	—
194	C₂²⋊C₈	—	2	2	no	—
195	C₂²⋊C₈	—	2	2	no	—
196	C₂²⋊C₈	—	38	2	no	—
197	C₄×C₈	—	38	2	no	—
198	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
199	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
200	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
201	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
202	#95	—	38	2	no	—
203	#95	—	38	2	no	—
204	C₁₉	#8676	1	1	yes	characteristic, complemented, Sylow
205	#41	—	4	1	no	—
206	#41	—	4	1	no	—
207	#41	—	4	1	no	—
208	#41	—	4	1	no	—
209	C₂×Q₈	—	2	1	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	1	no	—
212	C₂×D₄	—	2	1	no	—
213	C₂×D₄	C₄⋊Dic₁₉	1	2	yes	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	76	2	no	—
216	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
217	C₂²×C₄	#64541	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
219	M₁₆	—	4	1	no	—
220	M₁₆	—	4	1	no	—
221	M₁₆	—	4	4	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	4	4	no	—
226	C₂×C₈	—	76	2	no	—
227	C₂×C₈	#64525	1	2	yes	—
228	C₂×C₈	#64526	1	2	yes	—
229	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
230	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
231	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
232	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	76	2	no	—
235	C₄⋊C₄	—	76	2	no	—
236	C₄⋊C₄	C₄⋊Dic₁₉	1	2	yes	—
237	C₂²⋊C₄	—	2	2	no	—
238	C₂²⋊C₄	—	2	2	no	—
239	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
240	C₄²	#64541	1	1	yes	characteristic
241	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
242	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
243	C₂³	—	2	2	no	—
244	C₂³	#10597376	1	1	yes	characteristic
245	Q₈	—	4	1	no	—
246	Q₈	—	4	1	no	—
247	D₄	—	4	2	no	—
248	D₄	—	4	2	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	2	no	—
251	D₄	—	4	1	no	—
252	D₄	—	4	1	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	152	2	no	—
259	C₂×C₄	—	152	2	no	—
260	C₂×C₄	—	152	2	no	—
261	C₂×C₄	#10597370	1	1	yes	characteristic
262	C₂×C₄	#10597370	1	1	yes	characteristic
263	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
264	C₂×C₄	#10597372	1	1	yes	characteristic
265	C₂×C₄	#10597376	1	1	yes	characteristic
266	C₂×C₄	C₂×#64525	1	1	yes	characteristic
267	C₂×C₄	C₂×#64526	1	1	yes	characteristic
268	C₂×C₄	#10597395	1	1	yes	characteristic
269	C₂×C₄	#10597397	1	1	yes	characteristic
270	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
271	C₈	—	2	1	no	—
272	C₈	—	2	1	no	—
273	C₈	—	4	2	no	—
274	C₈	—	152	2	no	—
275	C₈	#10597289	1	2	yes	—
276	C₈	#10597289	1	2	yes	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	#11926404	1	1	yes	characteristic
283	C₂²	#11926739	1	1	yes	characteristic
284	C₂²	#11926749	1	1	yes	characteristic
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	4	2	no	—
290	C₄	—	304	2	no	—
291	C₄	C₂×#10597289	1	1	yes	characteristic
292	C₄	#11926545	1	1	yes	characteristic
293	C₄	#11926824	1	1	yes	characteristic
294	C₄	#11926834	1	1	yes	characteristic
295	C₂	—	2	1	no	—
296	C₂	—	2	1	no	—
297	C₂	—	8	2	no	—
298	C₂	#14563635	1	1	yes	characteristic
299	C₂	#14563681	1	1	yes	characteristic
300	C₂	#14563689	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926404
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2931	C₂
socle	C₂²×C₁₉	#380
prime radical	C₁₉×#532	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2931	D₁₉
hypocenter	C₁₉	#8676
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931

Complements

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

N	H ≅ G/N	type
1. #14563799	C₂	semidirect
2. #14562383	C₂	semidirect
4. #14561495	C₂	semidirect
16. C₁₉×#436	C₄	semidirect
18. C₈×D₄×C₁₉	C₄	semidirect
153. #436	Dic₁₉	semidirect
155. C₈×D₄	Dic₁₉	semidirect
204. C₁₉	#8676	semidirect

Maximal subgroups

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

subgroup	index
#14563799	2
#14562383	2
#14562195	2
#14561495	2
C₁₉×#2931	2
#8676	19