Subgroups — C₁₉×#12813 #18018772

Order 4864.

494 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₉×#357	C₂	1	2	yes	maximal
4	C₁₉×#1846	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1380	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₁₉	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₄×C₁₉	—	2	1	no	—
8	C₂²×C₁₆×C₁₉	C₂²	1	2	yes	—
9	C₄×D₈×C₁₉	—	2	2	no	—
10	C₄×D₈×C₁₉	C₄	1	4	yes	complemented
11	C₂×C₁₉×#107	C₂²	1	2	yes	—
12	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
13	C₂×C₁₉×#106	—	2	1	no	—
14	C₂×C₁₉×#102	—	2	1	no	—
15	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₁₉×#382	—	2	1	no	—
17	C₁₉×C₄⋊C₁₆	—	2	2	no	—
18	C₁₉×#357	—	2	4	no	—
19	C₂²×D₄×C₁₉	—	2	1	no	—
20	C₂³×C₄×C₁₉	—	4	1	no	—
21	C₂×D₈×C₁₉	—	2	1	no	—
22	C₂×D₈×C₁₉	—	2	1	no	—
23	C₂×D₈×C₁₉	—	2	1	no	—
24	C₂×D₈×C₁₉	C₂×C₄	1	2	yes	—
25	C₂×D₈×C₁₉	D₄	1	2	yes	—
26	C₂×D₈×C₁₉	Q₈	1	2	yes	—
27	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
29	C₄×D₄×C₁₉	—	2	4	no	—
30	C₄×D₄×C₁₉	—	4	2	no	—
31	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
33	C₂×C₂²⋊C₄×C₁₉	—	4	1	no	—
34	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₁₉	—	2	2	no	—
36	C₂×C₁₆×C₁₉	—	2	2	no	—
37	C₂×C₁₆×C₁₉	—	2	2	no	—
38	C₁₉×#107	—	2	1	no	—
39	C₁₉×#107	—	2	2	no	—
40	C₁₉×#107	—	2	2	no	—
41	C₁₉×#107	C₂×C₄	1	2	yes	—
42	C₁₉×#106	—	4	2	no	—
43	C₁₉×#102	—	2	4	no	—
44	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
45	C₄×C₈×C₁₉	D₄	1	2	yes	—
46	C₁₉×#95	—	4	1	no	—
47	C₁₉×#95	—	4	1	no	—
48	C₂⁴×C₁₉	—	4	1	no	—
49	C₂×D₄×C₁₉	—	2	2	no	—
50	C₂×D₄×C₁₉	—	2	2	no	—
51	C₂×D₄×C₁₉	—	2	2	no	—
52	C₂×D₄×C₁₉	—	4	1	no	—
53	C₂×D₄×C₁₉	—	4	1	no	—
54	C₂×D₄×C₁₉	—	4	1	no	—
55	C₂²×C₄×C₁₉	—	4	1	no	—
56	C₂²×C₄×C₁₉	—	4	4	no	—
57	C₂²×C₄×C₁₉	—	4	2	no	—
58	C₂²×C₄×C₁₉	—	8	4	no	—
59	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
62	D₈×C₁₉	—	2	2	no	—
63	D₈×C₁₉	—	2	2	no	—
64	D₈×C₁₉	—	2	2	no	—
65	D₈×C₁₉	C₄⋊C₄	1	4	yes	—
66	C₂×C₈×C₁₉	—	2	1	no	—
67	C₂×C₈×C₁₉	—	2	1	no	—
68	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
69	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
74	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
75	C₄⋊C₄×C₁₉	—	2	2	no	—
76	C₄⋊C₄×C₁₉	—	4	1	no	—
77	C₄⋊C₄×C₁₉	—	4	2	no	—
78	C₄⋊C₄×C₁₉	—	4	2	no	—
79	C₂²⋊C₄×C₁₉	—	4	4	no	—
80	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
81	C₄²×C₁₉	SD₁₆	1	2	yes	—
82	C₁₆×C₁₉	—	4	2	no	—
83	#12813	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂³×C₁₉	—	4	2	no	—
85	C₂³×C₁₉	—	4	2	no	—
86	C₂³×C₁₉	—	4	2	no	—
87	C₂³×C₁₉	—	8	2	no	—
88	C₂³×C₁₉	—	8	2	no	—
89	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
90	D₄×C₁₉	—	2	4	no	—
91	D₄×C₁₉	—	4	2	no	—
92	D₄×C₁₉	—	4	2	no	—
93	D₄×C₁₉	—	4	2	no	—
94	C₂×C₄×C₁₉	—	2	2	no	—
95	C₂×C₄×C₁₉	—	2	2	no	—
96	C₂×C₄×C₁₉	—	4	2	no	—
97	C₂×C₄×C₁₉	—	8	8	no	—
98	C₂×C₄×C₁₉	—	8	2	no	—
99	C₂×C₄×C₁₉	—	8	2	no	—
100	C₂×C₄×C₁₉	—	8	2	no	—
101	C₂×C₄×C₁₉	—	8	2	no	—
102	C₂×C₄×C₁₉	#102	1	2	yes	—
103	C₂×C₄×C₁₉	#102	1	2	yes	—
104	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
106	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
110	C₈×C₁₉	—	2	2	no	—
111	C₈×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₈×C₁₉	#117	1	1	yes	characteristic
113	C₈×C₁₉	#121	1	1	yes	characteristic
114	C₈×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₄×D₈	C₂×C₁₉	1	1	yes	characteristic
116	C₂×C₄⋊C₁₆	C₂×C₁₉	1	1	yes	characteristic, complemented
117	C₂×#357	C₂×C₁₉	1	2	yes	—
118	#1846	C₂×C₁₉	1	1	yes	characteristic, complemented
119	#1380	C₂×C₁₉	1	2	yes	complemented
120	C₂²×C₁₉	—	4	4	no	—
121	C₂²×C₁₉	—	8	2	no	—
122	C₂²×C₁₉	—	8	2	no	—
123	C₂²×C₁₉	—	8	4	no	—
124	C₂²×C₁₉	—	8	4	no	—
125	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
126	C₂²×C₁₉	#459	1	1	yes	characteristic
127	C₂²×C₁₉	#474	1	1	yes	characteristic
128	C₂²×C₁₉	#508	1	2	yes	—
129	C₂²×C₁₉	#509	1	2	yes	—
130	C₄×C₁₉	—	2	2	no	—
131	C₄×C₁₉	—	2	2	no	—
132	C₄×C₁₉	—	8	2	no	—
133	C₄×C₁₉	—	16	2	no	—
134	C₄×C₁₉	#380	1	1	yes	characteristic
135	C₄×C₁₉	#417	1	1	yes	characteristic
136	C₄×C₁₉	#461	1	1	yes	characteristic
137	C₄×C₁₉	#476	1	1	yes	characteristic
138	C₂²×D₈	C₂²×C₁₉	1	1	yes	characteristic
139	C₂×C₄×D₄	—	2	1	no	—
140	C₂²×C₁₆	C₂²×C₁₉	1	2	yes	—
141	C₄×D₈	—	2	2	no	—
142	C₄×D₈	C₄×C₁₉	1	4	yes	complemented
143	C₂×#107	C₂²×C₁₉	1	2	yes	—
144	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
145	C₂×#106	—	2	1	no	—
146	C₂×#102	—	2	1	no	—
147	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
148	#382	—	2	1	no	—
149	C₄⋊C₁₆	—	2	2	no	—
150	#357	—	2	4	no	—
151	C₂×C₁₉	—	8	4	no	—
152	C₂×C₁₉	#1802	1	1	yes	characteristic
153	C₂×C₁₉	#2138	1	1	yes	characteristic
154	C₂×C₁₉	#2140	1	1	yes	characteristic
155	C₂×C₁₉	#2210	1	2	yes	—
156	C₂×C₁₉	#2228	1	2	yes	—
157	C₂²×D₄	—	2	1	no	—
158	C₂³×C₄	—	4	1	no	—
159	C₂×D₈	—	2	1	no	—
160	C₂×D₈	—	2	1	no	—
161	C₂×D₈	—	2	1	no	—
162	C₂×D₈	C₂×C₄×C₁₉	1	2	yes	—
163	C₂×D₈	D₄×C₁₉	1	2	yes	—
164	C₂×D₈	Q₈×C₁₉	1	2	yes	—
165	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
166	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
167	C₄×D₄	—	2	4	no	—
168	C₄×D₄	—	4	2	no	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₂²⋊C₄	—	4	1	no	—
172	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	—	2	2	no	—
175	C₂×C₁₆	—	2	2	no	—
176	#107	—	2	1	no	—
177	#107	—	2	2	no	—
178	#107	—	2	2	no	—
179	#107	C₂×C₄×C₁₉	1	2	yes	—
180	#106	—	4	2	no	—
181	#102	—	2	4	no	—
182	C₄×C₈	C₂×C₄×C₁₉	1	2	yes	—
183	C₄×C₈	D₄×C₁₉	1	2	yes	—
184	#95	—	4	1	no	—
185	#95	—	4	1	no	—
186	C₁₉	#12813	1	1	yes	characteristic, direct product factor, Sylow
187	C₂⁴	—	4	1	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	4	1	no	—
192	C₂×D₄	—	4	1	no	—
193	C₂×D₄	—	4	1	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	4	4	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	—	8	4	no	—
198	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
199	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
201	D₈	—	2	2	no	—
202	D₈	—	2	2	no	—
203	D₈	—	2	2	no	—
204	D₈	C₄⋊C₄×C₁₉	1	4	yes	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	C₂²⋊C₄×C₁₉	1	2	yes	—
208	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
209	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
210	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
211	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
212	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
213	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	4	1	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₂²⋊C₄	—	4	4	no	—
219	C₄²	C₂²⋊C₄×C₁₉	1	2	yes	—
220	C₄²	SD₁₆×C₁₉	1	2	yes	—
221	C₁₆	—	4	2	no	—
222	C₂³	—	4	2	no	—
223	C₂³	—	4	2	no	—
224	C₂³	—	4	2	no	—
225	C₂³	—	8	2	no	—
226	C₂³	—	8	2	no	—
227	C₂³	C₂×D₈×C₁₉	1	1	yes	characteristic
228	D₄	—	2	4	no	—
229	D₄	—	4	2	no	—
230	D₄	—	4	2	no	—
231	D₄	—	4	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	8	8	no	—
236	C₂×C₄	—	8	2	no	—
237	C₂×C₄	—	8	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	C₁₉×#102	1	2	yes	—
241	C₂×C₄	C₁₉×#102	1	2	yes	—
242	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
243	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
244	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
245	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
246	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
247	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
250	C₈	C₁₉×#117	1	1	yes	characteristic
251	C₈	C₁₉×#121	1	1	yes	characteristic
252	C₈	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
253	C₂²	—	4	4	no	—
254	C₂²	—	8	4	no	—
255	C₂²	—	8	4	no	—
256	C₂²	—	8	2	no	—
257	C₂²	—	8	2	no	—
258	C₂²	C₂×C₁₉×#102	1	1	yes	characteristic
259	C₂²	C₁₉×#459	1	1	yes	characteristic
260	C₂²	C₁₉×#474	1	1	yes	characteristic
261	C₂²	C₁₉×#508	1	2	yes	—
262	C₂²	C₁₉×#509	1	2	yes	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	8	2	no	—
266	C₄	—	16	2	no	—
267	C₄	C₁₉×#380	1	1	yes	characteristic
268	C₄	C₁₉×#417	1	1	yes	characteristic
269	C₄	C₁₉×#461	1	1	yes	characteristic
270	C₄	C₁₉×#476	1	1	yes	characteristic
271	C₂	—	8	4	no	—
272	C₂	C₁₉×#1802	1	1	yes	characteristic
273	C₂	C₁₉×#2138	1	1	yes	characteristic
274	C₂	C₁₉×#2140	1	1	yes	characteristic
275	C₂	C₁₉×#2210	1	2	yes	—
276	C₂	C₁₉×#2228	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂×D₈
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 ◁ C₂³×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₉×C₄⋊C₁₆	C₂	semidirect
4. C₁₉×#1846	C₂	semidirect
5. C₁₉×#1380	C₂	semidirect
10. C₄×D₈×C₁₉	C₄	semidirect
83. #12813	C₁₉	direct (N × H)
116. C₂×C₄⋊C₁₆	C₂×C₁₉	semidirect
118. #1846	C₂×C₁₉	semidirect
119. #1380	C₂×C₁₉	semidirect
142. C₄×D₈	C₄×C₁₉	semidirect
186. C₁₉	#12813	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×D₈×C₁₉	2
C₂×C₁₉×C₄⋊C₁₆	2
C₂×C₁₉×#357	2
C₁₉×#1846	2
C₁₉×#1380	2
#12813	19

Subgroups — C19×#12813 #18018772

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12813 #18018772