Subgroups — C₁₉×#13136 #18019095

Order 4864.

360 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2159	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₉×#361	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2140	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1970	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1772	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#1387	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₁₉	C₂²	1	1	yes	characteristic
9	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
12	C₁₉×#427	C₂²	1	1	yes	characteristic
13	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
14	C₂×C₁₉×#102	—	2	1	no	—
15	C₁₉×#405	—	2	1	no	—
16	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#390	—	2	1	no	—
18	C₁₉×#367	—	2	1	no	—
19	C₁₉×#366	—	2	1	no	—
20	C₁₉×#361	C₄	1	4	yes	complemented
21	C₁₉×#357	—	2	1	no	—
22	C₁₉×#357	—	2	1	no	—
23	C₁₉×#339	—	2	1	no	—
24	C₂²×D₄×C₁₉	—	2	1	no	—
25	C₂×D₈×C₁₉	—	2	1	no	—
26	C₂×D₈×C₁₉	—	2	1	no	—
27	C₂×D₈×C₁₉	—	2	1	no	—
28	C₂×D₈×C₁₉	—	2	1	no	—
29	C₂×D₈×C₁₉	—	2	1	no	—
30	C₂×D₈×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×M₁₆×C₁₉	—	2	1	no	—
32	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
34	C₁₉×#117	—	2	1	no	—
35	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
36	C₂×C₂²⋊C₄×C₁₉	—	4	2	no	—
37	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
38	C₁₉×M₃₂	—	2	1	no	—
39	C₁₉×M₃₂	—	2	1	no	—
40	C₁₉×M₃₂	—	2	1	no	—
41	C₁₉×M₃₂	C₂×C₄	1	2	yes	—
42	C₂×C₁₆×C₁₉	—	2	1	no	—
43	C₂×C₁₆×C₁₉	—	2	1	no	—
44	C₁₉×#108	—	2	1	no	—
45	C₁₉×#108	C₂×C₄	1	2	yes	—
46	C₁₉×#107	—	2	1	no	—
47	C₁₉×#107	—	2	1	no	—
48	C₁₉×#107	D₄	1	1	yes	characteristic
49	C₁₉×#107	D₄	1	1	yes	characteristic
50	C₁₉×#106	—	2	1	no	—
51	C₁₉×#102	—	4	1	no	—
52	C₁₉×#102	—	4	1	no	—
53	C₁₉×#97	—	4	1	no	—
54	C₄×C₈×C₁₉	—	2	1	no	—
55	C₄×C₈×C₁₉	—	2	1	no	—
56	C₄×C₈×C₁₉	—	4	1	no	—
57	C₂⁴×C₁₉	—	4	1	no	—
58	C₂×D₄×C₁₉	—	2	2	no	—
59	C₂×D₄×C₁₉	—	4	1	no	—
60	C₂×D₄×C₁₉	—	4	1	no	—
61	C₂×D₄×C₁₉	—	4	1	no	—
62	C₂×D₄×C₁₉	—	4	1	no	—
63	C₂×D₄×C₁₉	—	4	1	no	—
64	C₂²×C₄×C₁₉	—	2	1	no	—
65	C₂²×C₄×C₁₉	—	4	1	no	—
66	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
67	D₈×C₁₉	—	2	2	no	—
68	D₈×C₁₉	—	2	2	no	—
69	D₈×C₁₉	—	4	1	no	—
70	D₈×C₁₉	—	4	1	no	—
71	M₁₆×C₁₉	—	2	2	no	—
72	M₁₆×C₁₉	—	4	1	no	—
73	C₂×C₈×C₁₉	—	2	1	no	—
74	C₂×C₈×C₁₉	—	2	1	no	—
75	C₂×C₈×C₁₉	—	4	1	no	—
76	C₂×C₈×C₁₉	D₈	1	1	yes	characteristic
77	C₂×C₈×C₁₉	D₈	1	1	yes	characteristic
78	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
81	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
82	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
83	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
84	C₄⋊C₄×C₁₉	—	2	1	no	—
85	C₄⋊C₄×C₁₉	—	2	1	no	—
86	C₄⋊C₄×C₁₉	—	4	1	no	—
87	C₄⋊C₄×C₁₉	—	4	1	no	—
88	C₂²⋊C₄×C₁₉	—	4	1	no	—
89	C₂²⋊C₄×C₁₉	—	8	4	no	—
90	C₄²×C₁₉	—	2	1	no	—
91	C₄²×C₁₉	—	2	1	no	—
92	C₄²×C₁₉	—	4	1	no	—
93	C₁₆×C₁₉	—	2	2	no	—
94	C₁₆×C₁₉	—	4	1	no	—
95	#13136	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂³×C₁₉	—	4	2	no	—
97	C₂³×C₁₉	—	8	2	no	—
98	C₂³×C₁₉	—	8	2	no	—
99	C₂³×C₁₉	—	8	1	no	—
100	C₂³×C₁₉	—	8	1	no	—
101	C₂³×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
102	D₄×C₁₉	—	4	2	no	—
103	D₄×C₁₉	—	4	2	no	—
104	D₄×C₁₉	—	8	1	no	—
105	D₄×C₁₉	—	8	1	no	—
106	C₂×C₄×C₁₉	—	2	2	no	—
107	C₂×C₄×C₁₉	—	4	1	no	—
108	C₂×C₄×C₁₉	—	4	1	no	—
109	C₂×C₄×C₁₉	—	4	1	no	—
110	C₂×C₄×C₁₉	—	4	1	no	—
111	C₂×C₄×C₁₉	—	8	2	no	—
112	C₂×C₄×C₁₉	—	8	1	no	—
113	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
115	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
118	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
119	C₈×C₁₉	—	2	1	no	—
120	C₈×C₁₉	—	2	1	no	—
121	C₈×C₁₉	—	4	2	no	—
122	C₈×C₁₉	#117	1	1	yes	characteristic
123	C₈×C₁₉	C₄×D₄	1	1	yes	characteristic
124	C₈×C₁₉	#124	1	1	yes	characteristic
125	C₈×C₁₉	#126	1	1	yes	characteristic
126	#2159	C₂×C₁₉	1	1	yes	characteristic, complemented
127	C₂×#361	C₂×C₁₉	1	1	yes	characteristic
128	#2140	C₂×C₁₉	1	1	yes	characteristic
129	#1970	C₂×C₁₉	1	1	yes	characteristic
130	#1772	C₂×C₁₉	1	1	yes	characteristic, complemented
131	#1387	C₂×C₁₉	1	1	yes	characteristic, complemented
132	#1376	C₂×C₁₉	1	1	yes	characteristic, complemented
133	C₂²×C₁₉	—	2	1	no	—
134	C₂²×C₁₉	—	2	1	no	—
135	C₂²×C₁₉	—	8	4	no	—
136	C₂²×C₁₉	—	8	2	no	—
137	C₂²×C₁₉	—	16	2	no	—
138	C₂²×C₁₉	—	16	1	no	—
139	C₂²×C₁₉	—	16	1	no	—
140	C₂²×C₁₉	#440	1	1	yes	characteristic
141	C₂²×C₁₉	#459	1	1	yes	characteristic
142	C₂²×C₁₉	#482	1	1	yes	characteristic
143	C₄×C₁₉	—	4	2	no	—
144	C₄×C₁₉	—	8	1	no	—
145	C₄×C₁₉	—	8	1	no	—
146	C₄×C₁₉	—	16	1	no	—
147	C₄×C₁₉	#388	1	1	yes	characteristic
148	C₄×C₁₉	C₄×D₈	1	1	yes	characteristic
149	C₄×C₁₉	#464	1	1	yes	characteristic
150	C₄×C₁₉	#480	1	1	yes	characteristic
151	C₂²×D₈	C₂²×C₁₉	1	1	yes	characteristic
152	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
153	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
154	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
155	#427	C₂²×C₁₉	1	1	yes	characteristic
156	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
157	C₂×#102	—	2	1	no	—
158	#405	—	2	1	no	—
159	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
160	#390	—	2	1	no	—
161	#367	—	2	1	no	—
162	#366	—	2	1	no	—
163	#361	C₄×C₁₉	1	4	yes	complemented
164	#357	—	2	1	no	—
165	#357	—	2	1	no	—
166	#339	—	2	1	no	—
167	C₂×C₁₉	—	2	1	no	—
168	C₂×C₁₉	—	2	1	no	—
169	C₂×C₁₉	—	16	2	no	—
170	C₂×C₁₉	#1925	1	1	yes	characteristic
171	C₂×C₁₉	#2212	1	1	yes	characteristic
172	C₂×C₁₉	#2213	1	1	yes	characteristic
173	C₂²×D₄	—	2	1	no	—
174	C₂×D₈	—	2	1	no	—
175	C₂×D₈	—	2	1	no	—
176	C₂×D₈	—	2	1	no	—
177	C₂×D₈	—	2	1	no	—
178	C₂×D₈	—	2	1	no	—
179	C₂×D₈	C₂×C₄×C₁₉	1	2	yes	—
180	C₂×M₁₆	—	2	1	no	—
181	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
182	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
183	#117	—	2	1	no	—
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₂²⋊C₄	—	4	2	no	—
186	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
187	M₃₂	—	2	1	no	—
188	M₃₂	—	2	1	no	—
189	M₃₂	—	2	1	no	—
190	M₃₂	C₂×C₄×C₁₉	1	2	yes	—
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	—	2	1	no	—
193	#108	—	2	1	no	—
194	#108	C₂×C₄×C₁₉	1	2	yes	—
195	#107	—	2	1	no	—
196	#107	—	2	1	no	—
197	#107	D₄×C₁₉	1	1	yes	characteristic
198	#107	D₄×C₁₉	1	1	yes	characteristic
199	#106	—	2	1	no	—
200	#102	—	4	1	no	—
201	#102	—	4	1	no	—
202	#97	—	4	1	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	4	1	no	—
206	C₁₉	#13136	1	1	yes	characteristic, direct product factor, Sylow
207	C₂⁴	—	4	1	no	—
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	—	4	1	no	—
210	C₂×D₄	—	4	1	no	—
211	C₂×D₄	—	4	1	no	—
212	C₂×D₄	—	4	1	no	—
213	C₂×D₄	—	4	1	no	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
217	D₈	—	2	2	no	—
218	D₈	—	2	2	no	—
219	D₈	—	4	1	no	—
220	D₈	—	4	1	no	—
221	M₁₆	—	2	2	no	—
222	M₁₆	—	4	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	D₈×C₁₉	1	1	yes	characteristic
227	C₂×C₈	D₈×C₁₉	1	1	yes	characteristic
228	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
229	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
230	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
231	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
232	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
233	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	4	1	no	—
237	C₄⋊C₄	—	4	1	no	—
238	C₂²⋊C₄	—	4	1	no	—
239	C₂²⋊C₄	—	8	4	no	—
240	C₄²	—	2	1	no	—
241	C₄²	—	2	1	no	—
242	C₄²	—	4	1	no	—
243	C₁₆	—	2	2	no	—
244	C₁₆	—	4	1	no	—
245	C₂³	—	4	2	no	—
246	C₂³	—	8	2	no	—
247	C₂³	—	8	2	no	—
248	C₂³	—	8	1	no	—
249	C₂³	—	8	1	no	—
250	C₂³	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
251	D₄	—	4	2	no	—
252	D₄	—	4	2	no	—
253	D₄	—	8	1	no	—
254	D₄	—	8	1	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	8	2	no	—
261	C₂×C₄	—	8	1	no	—
262	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
263	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
264	C₂×C₄	C₁₉×#123	1	1	yes	characteristic
265	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
266	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
267	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
268	C₈	—	2	1	no	—
269	C₈	—	2	1	no	—
270	C₈	—	4	2	no	—
271	C₈	C₁₉×#117	1	1	yes	characteristic
272	C₈	C₄×D₄×C₁₉	1	1	yes	characteristic
273	C₈	C₁₉×#124	1	1	yes	characteristic
274	C₈	C₁₉×#126	1	1	yes	characteristic
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	8	4	no	—
278	C₂²	—	8	2	no	—
279	C₂²	—	16	2	no	—
280	C₂²	—	16	1	no	—
281	C₂²	—	16	1	no	—
282	C₂²	C₁₉×#440	1	1	yes	characteristic
283	C₂²	C₁₉×#459	1	1	yes	characteristic
284	C₂²	C₁₉×#482	1	1	yes	characteristic
285	C₄	—	4	2	no	—
286	C₄	—	8	1	no	—
287	C₄	—	8	1	no	—
288	C₄	—	16	1	no	—
289	C₄	C₁₉×#388	1	1	yes	characteristic
290	C₄	C₄×D₈×C₁₉	1	1	yes	characteristic
291	C₄	C₁₉×#464	1	1	yes	characteristic
292	C₄	C₁₉×#480	1	1	yes	characteristic
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	16	2	no	—
296	C₂	C₁₉×#1925	1	1	yes	characteristic
297	C₂	C₁₉×#2212	1	1	yes	characteristic
298	C₂	C₁₉×#2213	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#459
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂²×D₈×C₁₉	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₉×#2159	C₂	semidirect
5. C₁₉×#1772	C₂	semidirect
6. C₁₉×#1387	C₂	semidirect
7. C₁₉×#1376	C₂	semidirect
20. C₁₉×#361	C₄	semidirect
95. #13136	C₁₉	direct (N × H)
126. #2159	C₂×C₁₉	semidirect
130. #1772	C₂×C₁₉	semidirect
131. #1387	C₂×C₁₉	semidirect
132. #1376	C₂×C₁₉	semidirect
163. #361	C₄×C₁₉	semidirect
206. C₁₉	#13136	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₉×#2159	2
C₂×C₁₉×#361	2
C₁₉×#2140	2
C₁₉×#1970	2
C₁₉×#1772	2
C₁₉×#1387	2
C₁₉×#1376	2
#13136	19

Subgroups — C19×#13136 #18019095

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13136 #18019095