Subgroups — C₁₉×#13113 #18019072

Order 4864.

262 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2158	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1945	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1772	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1382	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1382	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1381	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1374	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
9	C₂²×C₁₆×C₁₉	C₂²	1	1	yes	characteristic
10	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
12	C₁₉×#427	C₂²	1	1	yes	characteristic
13	C₁₉×#427	C₂²	1	1	yes	characteristic
14	C₁₉×#424	—	2	1	no	—
15	C₁₉×#405	—	2	1	no	—
16	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#367	C₄	1	1	yes	characteristic, complemented
18	C₁₉×#367	C₄	1	1	yes	characteristic, complemented
19	C₁₉×#366	C₄	1	1	yes	characteristic, complemented
20	C₁₉×#366	C₄	1	1	yes	characteristic, complemented
21	C₁₉×#339	—	2	1	no	—
22	C₁₉×#339	—	2	1	no	—
23	C₂×M₁₆×C₁₉	—	2	1	no	—
24	C₂×M₁₆×C₁₉	—	2	1	no	—
25	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
27	C₁₉×#117	—	2	1	no	—
28	C₁₉×#117	—	2	1	no	—
29	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
30	C₂×C₁₆×C₁₉	—	2	1	no	—
31	C₂×C₁₆×C₁₉	—	2	1	no	—
32	C₂×C₁₆×C₁₉	—	2	1	no	—
33	C₂×C₁₆×C₁₉	—	2	1	no	—
34	C₂×C₁₆×C₁₉	—	2	1	no	—
35	C₂×C₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
37	C₁₉×#108	—	2	1	no	—
38	C₁₉×#108	—	2	1	no	—
39	C₁₉×#108	—	2	1	no	—
40	C₁₉×#108	Q₈	1	2	yes	—
41	C₁₉×#107	C₂×C₄	1	1	yes	characteristic
42	C₁₉×#107	C₂×C₄	1	1	yes	characteristic
43	C₁₉×#107	C₂×C₄	1	1	yes	characteristic
44	C₁₉×#107	C₂×C₄	1	1	yes	characteristic
45	C₁₉×#106	—	2	1	no	—
46	C₁₉×#106	—	2	1	no	—
47	C₁₉×C₄⋊C₈	—	4	1	no	—
48	C₁₉×C₄⋊C₈	—	4	1	no	—
49	C₁₉×#97	—	4	1	no	—
50	C₄×C₈×C₁₉	—	2	1	no	—
51	C₄×C₈×C₁₉	—	2	1	no	—
52	C₄×C₈×C₁₉	—	4	1	no	—
53	C₂²×C₄×C₁₉	—	2	1	no	—
54	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
55	M₁₆×C₁₉	—	2	2	no	—
56	M₁₆×C₁₉	—	4	1	no	—
57	M₁₆×C₁₉	—	4	1	no	—
58	M₁₆×C₁₉	—	4	1	no	—
59	C₂×C₈×C₁₉	—	2	1	no	—
60	C₂×C₈×C₁₉	—	2	1	no	—
61	C₂×C₈×C₁₉	—	4	1	no	—
62	C₂×C₈×C₁₉	—	4	1	no	—
63	C₂×C₈×C₁₉	D₈	1	1	yes	characteristic
64	C₂×C₈×C₁₉	D₈	1	1	yes	characteristic
65	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
68	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
69	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
70	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
71	C₄⋊C₄×C₁₉	—	2	1	no	—
72	C₄⋊C₄×C₁₉	—	2	1	no	—
73	C₄⋊C₄×C₁₉	—	2	1	no	—
74	C₄⋊C₄×C₁₉	—	2	1	no	—
75	C₂²⋊C₄×C₁₉	—	4	1	no	—
76	C₂²⋊C₄×C₁₉	—	4	1	no	—
77	C₄²×C₁₉	—	2	1	no	—
78	C₄²×C₁₉	—	2	1	no	—
79	C₄²×C₁₉	—	4	1	no	—
80	C₄²×C₁₉	—	4	1	no	—
81	C₁₆×C₁₉	—	2	1	no	—
82	C₁₆×C₁₉	—	2	1	no	—
83	C₁₆×C₁₉	—	4	1	no	—
84	#13113	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂³×C₁₉	#135	1	1	yes	characteristic
86	C₂×C₄×C₁₉	—	2	2	no	—
87	C₂×C₄×C₁₉	—	4	1	no	—
88	C₂×C₄×C₁₉	—	4	1	no	—
89	C₂×C₄×C₁₉	—	4	1	no	—
90	C₂×C₄×C₁₉	—	4	1	no	—
91	C₂×C₄×C₁₉	—	4	1	no	—
92	C₂×C₄×C₁₉	—	4	1	no	—
93	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
94	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
96	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
98	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
99	C₈×C₁₉	—	2	1	no	—
100	C₈×C₁₉	—	2	1	no	—
101	C₈×C₁₉	—	4	2	no	—
102	C₈×C₁₉	—	8	1	no	—
103	C₈×C₁₉	#117	1	1	yes	characteristic
104	C₈×C₁₉	C₄×Q₈	1	1	yes	characteristic
105	C₈×C₁₉	#125	1	1	yes	characteristic
106	C₈×C₁₉	#126	1	1	yes	characteristic
107	#2158	C₂×C₁₉	1	1	yes	characteristic
108	#1945	C₂×C₁₉	1	1	yes	characteristic
109	#1772	C₂×C₁₉	1	1	yes	characteristic
110	#1382	C₂×C₁₉	1	1	yes	characteristic
111	#1382	C₂×C₁₉	1	1	yes	characteristic
112	#1381	C₂×C₁₉	1	1	yes	characteristic
113	#1374	C₂×C₁₉	1	1	yes	characteristic
114	C₂²×C₁₉	—	2	1	no	—
115	C₂²×C₁₉	—	2	1	no	—
116	C₂²×C₁₉	C₄×D₈	1	1	yes	characteristic
117	C₂²×C₁₉	#479	1	1	yes	characteristic
118	C₂²×C₁₉	#485	1	1	yes	characteristic
119	C₄×C₁₉	—	4	2	no	—
120	C₄×C₁₉	—	8	1	no	—
121	C₄×C₁₉	—	8	1	no	—
122	C₄×C₁₉	—	8	1	no	—
123	C₄×C₁₉	—	8	1	no	—
124	C₄×C₁₉	#387	1	1	yes	characteristic
125	C₄×C₁₉	#441	1	1	yes	characteristic
126	C₄×C₁₉	#474	1	1	yes	characteristic
127	C₄×C₁₉	#480	1	1	yes	characteristic
128	C₂²×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
129	C₂²×C₁₆	C₂²×C₁₉	1	1	yes	characteristic
130	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
131	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
132	#427	C₂²×C₁₉	1	1	yes	characteristic
133	#427	C₂²×C₁₉	1	1	yes	characteristic
134	#424	—	2	1	no	—
135	#405	—	2	1	no	—
136	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
137	#367	C₄×C₁₉	1	1	yes	characteristic, complemented
138	#367	C₄×C₁₉	1	1	yes	characteristic, complemented
139	#366	C₄×C₁₉	1	1	yes	characteristic, complemented
140	#366	C₄×C₁₉	1	1	yes	characteristic, complemented
141	#339	—	2	1	no	—
142	#339	—	2	1	no	—
143	C₂×C₁₉	—	2	1	no	—
144	C₂×C₁₉	—	2	1	no	—
145	C₂×C₁₉	#1918	1	1	yes	characteristic
146	C₂×C₁₉	#2178	1	1	yes	characteristic
147	C₂×C₁₉	#2179	1	1	yes	characteristic
148	C₂×M₁₆	—	2	1	no	—
149	C₂×M₁₆	—	2	1	no	—
150	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
151	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
152	#117	—	2	1	no	—
153	#117	—	2	1	no	—
154	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
155	C₂×C₁₆	—	2	1	no	—
156	C₂×C₁₆	—	2	1	no	—
157	C₂×C₁₆	—	2	1	no	—
158	C₂×C₁₆	—	2	1	no	—
159	C₂×C₁₆	—	2	1	no	—
160	C₂×C₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
161	C₂×C₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
162	#108	—	2	1	no	—
163	#108	—	2	1	no	—
164	#108	—	2	1	no	—
165	#108	Q₈×C₁₉	1	2	yes	—
166	#107	C₂×C₄×C₁₉	1	1	yes	characteristic
167	#107	C₂×C₄×C₁₉	1	1	yes	characteristic
168	#107	C₂×C₄×C₁₉	1	1	yes	characteristic
169	#107	C₂×C₄×C₁₉	1	1	yes	characteristic
170	#106	—	2	1	no	—
171	#106	—	2	1	no	—
172	C₄⋊C₈	—	4	1	no	—
173	C₄⋊C₈	—	4	1	no	—
174	#97	—	4	1	no	—
175	C₄×C₈	—	2	1	no	—
176	C₄×C₈	—	2	1	no	—
177	C₄×C₈	—	4	1	no	—
178	C₁₉	#13113	1	1	yes	characteristic, direct product factor, Sylow
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
181	M₁₆	—	2	2	no	—
182	M₁₆	—	4	1	no	—
183	M₁₆	—	4	1	no	—
184	M₁₆	—	4	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	4	1	no	—
188	C₂×C₈	—	4	1	no	—
189	C₂×C₈	D₈×C₁₉	1	1	yes	characteristic
190	C₂×C₈	D₈×C₁₉	1	1	yes	characteristic
191	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
192	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
193	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
194	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
195	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
196	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	4	1	no	—
202	C₂²⋊C₄	—	4	1	no	—
203	C₄²	—	2	1	no	—
204	C₄²	—	2	1	no	—
205	C₄²	—	4	1	no	—
206	C₄²	—	4	1	no	—
207	C₁₆	—	2	1	no	—
208	C₁₆	—	2	1	no	—
209	C₁₆	—	4	1	no	—
210	C₂³	C₁₉×#135	1	1	yes	characteristic
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	1	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	—	4	1	no	—
215	C₂×C₄	—	4	1	no	—
216	C₂×C₄	—	4	1	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
219	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
220	C₂×C₄	C₁₉×#123	1	1	yes	characteristic
221	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
222	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
223	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
224	C₈	—	2	1	no	—
225	C₈	—	2	1	no	—
226	C₈	—	4	2	no	—
227	C₈	—	8	1	no	—
228	C₈	C₁₉×#117	1	1	yes	characteristic
229	C₈	C₄×Q₈×C₁₉	1	1	yes	characteristic
230	C₈	C₁₉×#125	1	1	yes	characteristic
231	C₈	C₁₉×#126	1	1	yes	characteristic
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	C₄×D₈×C₁₉	1	1	yes	characteristic
235	C₂²	C₁₉×#479	1	1	yes	characteristic
236	C₂²	C₁₉×#485	1	1	yes	characteristic
237	C₄	—	4	2	no	—
238	C₄	—	8	1	no	—
239	C₄	—	8	1	no	—
240	C₄	—	8	1	no	—
241	C₄	—	8	1	no	—
242	C₄	C₁₉×#387	1	1	yes	characteristic
243	C₄	C₁₉×#441	1	1	yes	characteristic
244	C₄	C₁₉×#474	1	1	yes	characteristic
245	C₄	C₁₉×#480	1	1	yes	characteristic
246	C₂	—	2	1	no	—
247	C₂	—	2	1	no	—
248	C₂	C₁₉×#1918	1	1	yes	characteristic
249	C₂	C₁₉×#2178	1	1	yes	characteristic
250	C₂	C₁₉×#2179	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₉	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
socle	C₂²×C₁₉	C₄×D₈
prime radical	C₂³×C₁₉	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₈×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
17. C₁₉×#367	C₄	semidirect
18. C₁₉×#367	C₄	semidirect
19. C₁₉×#366	C₄	semidirect
20. C₁₉×#366	C₄	semidirect
84. #13113	C₁₉	direct (N × H)
137. #367	C₄×C₁₉	semidirect
138. #367	C₄×C₁₉	semidirect
139. #366	C₄×C₁₉	semidirect
140. #366	C₄×C₁₉	semidirect
178. C₁₉	#13113	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×#2158	2
C₁₉×#1945	2
C₁₉×#1772	2
C₁₉×#1382	2
C₁₉×#1381	2
C₁₉×#1374	2
#13113	19

Subgroups — C19×#13113 #18019072

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13113 #18019072