Subgroups — C₁₉×#12637 #18018596

Order 4864.

526 conjugacy classes of subgroups.

The subgroup lattice has 254 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×#357	C₂	1	2	yes	maximal
3	C₂×C₁₉×#346	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₁₉×#107	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	1	yes	characteristic
12	C₂×C₁₉×#107	C₂²	1	2	yes	—
13	C₂×C₁₉×#102	—	2	1	no	—
14	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₁₉	—	2	1	no	—
16	C₁₉×#357	C₄	1	8	yes	complemented
17	C₁₉×#346	—	2	2	no	—
18	C₁₉×#340	—	2	1	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	2	no	—
23	C₂×D₈×C₁₉	C₂×C₄	1	4	yes	—
24	C₂×D₈×C₁₉	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
27	C₄×D₄×C₁₉	—	2	4	no	—
28	C₄×D₄×C₁₉	—	4	2	no	—
29	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
31	C₂×C₂²⋊C₄×C₁₉	—	4	1	no	—
32	C₂×C₄²×C₁₉	—	4	1	no	—
33	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₁₉	—	2	2	no	—
35	C₂×C₁₆×C₁₉	—	2	2	no	—
36	C₂×C₁₆×C₁₉	C₂×C₄	1	4	yes	complemented
37	C₁₉×#107	—	2	1	no	—
38	C₁₉×#107	—	2	2	no	—
39	C₁₉×#107	C₂×C₄	1	2	yes	—
40	C₁₉×#107	C₂×C₄	1	4	yes	complemented
41	C₁₉×#102	—	2	4	no	—
42	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₁₉	D₄	1	2	yes	—
44	C₁₉×#95	—	4	1	no	—
45	C₂⁴×C₁₉	—	4	1	no	—
46	C₂×D₄×C₁₉	—	2	2	no	—
47	C₂×D₄×C₁₉	—	2	4	no	—
48	C₂×D₄×C₁₉	—	4	1	no	—
49	C₂×D₄×C₁₉	—	4	2	no	—
50	C₂²×C₄×C₁₉	—	4	2	no	—
51	C₂²×C₄×C₁₉	—	4	1	no	—
52	C₂²×C₄×C₁₉	—	4	4	no	—
53	C₂²×C₄×C₁₉	—	8	4	no	—
54	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
57	D₈×C₁₉	—	2	2	no	—
58	D₈×C₁₉	—	2	4	no	—
59	D₈×C₁₉	C₄²	1	4	yes	complemented
60	C₂×C₈×C₁₉	—	2	2	no	—
61	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₁₉	#41	1	2	yes	—
66	C₄⋊C₄×C₁₉	—	2	2	no	—
67	C₄⋊C₄×C₁₉	—	2	4	no	—
68	C₄⋊C₄×C₁₉	—	4	1	no	—
69	C₄⋊C₄×C₁₉	—	4	2	no	—
70	C₂²⋊C₄×C₁₉	—	4	4	no	—
71	C₄²×C₁₉	—	8	2	no	—
72	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
73	C₄²×C₁₉	SD₁₆	1	2	yes	—
74	C₁₆×C₁₉	—	2	4	no	—
75	#12637	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₁₉	—	4	2	no	—
77	C₂³×C₁₉	—	4	4	no	—
78	C₂³×C₁₉	—	8	2	no	—
79	C₂³×C₁₉	—	8	1	no	—
80	C₂³×C₁₉	—	8	1	no	—
81	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
82	D₄×C₁₉	—	2	4	no	—
83	D₄×C₁₉	—	4	2	no	—
84	D₄×C₁₉	—	4	4	no	—
85	C₂×C₄×C₁₉	—	2	1	no	—
86	C₂×C₄×C₁₉	—	2	2	no	—
87	C₂×C₄×C₁₉	—	2	1	no	—
88	C₂×C₄×C₁₉	—	4	2	no	—
89	C₂×C₄×C₁₉	—	4	4	no	—
90	C₂×C₄×C₁₉	—	8	8	no	—
91	C₂×C₄×C₁₉	—	8	2	no	—
92	C₂×C₄×C₁₉	—	8	2	no	—
93	C₂×C₄×C₁₉	—	8	2	no	—
94	C₂×C₄×C₁₉	#102	1	2	yes	—
95	C₂×C₄×C₁₉	#102	1	2	yes	—
96	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
99	C₂×C₄×C₁₉	#135	1	2	yes	—
100	C₈×C₁₉	—	2	2	no	—
101	C₈×C₁₉	C₂×C₄²	1	1	yes	characteristic
102	C₈×C₁₉	#117	1	1	yes	characteristic
103	C₈×C₁₉	C₄×D₄	1	2	yes	—
104	C₂×C₄×D₈	C₂×C₁₉	1	1	yes	characteristic
105	C₂×#357	C₂×C₁₉	1	2	yes	—
106	C₂×#346	C₂×C₁₉	1	1	yes	characteristic, complemented
107	C₄×#107	C₂×C₁₉	1	1	yes	characteristic, complemented
108	#1380	C₂×C₁₉	1	2	yes	complemented
109	C₂²×C₁₉	—	4	4	no	—
110	C₂²×C₁₉	—	8	2	no	—
111	C₂²×C₁₉	—	8	2	no	—
112	C₂²×C₁₉	—	8	4	no	—
113	C₂²×C₁₉	—	8	4	no	—
114	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
115	C₂²×C₁₉	C₄×D₈	1	2	yes	—
116	C₂²×C₁₉	#509	1	1	yes	characteristic
117	C₂²×C₁₉	#509	1	2	yes	—
118	C₂²×C₁₉	#510	1	1	yes	characteristic
119	C₄×C₁₉	—	2	2	no	—
120	C₄×C₁₉	—	2	2	no	—
121	C₄×C₁₉	—	8	2	no	—
122	C₄×C₁₉	—	8	4	no	—
123	C₄×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
124	C₄×C₁₉	#416	1	1	yes	characteristic
125	C₄×C₁₉	C₄×SD₁₆	1	2	yes	—
126	C₂²×D₈	C₂²×C₁₉	1	1	yes	characteristic
127	C₂×C₄×D₄	—	2	1	no	—
128	C₂²×C₁₆	C₂²×C₁₉	1	2	yes	—
129	C₄×D₈	—	2	2	no	—
130	C₄×D₈	C₄×C₁₉	1	4	yes	complemented
131	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
132	C₂×#107	C₂²×C₁₉	1	2	yes	—
133	C₂×#102	—	2	1	no	—
134	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
135	C₄×C₄⋊C₄	—	2	1	no	—
136	#357	C₄×C₁₉	1	8	yes	complemented
137	#346	—	2	2	no	—
138	#340	—	2	1	no	—
139	C₂×C₁₉	—	8	4	no	—
140	C₂×C₁₉	C₄×#102	1	1	yes	characteristic
141	C₂×C₁₉	#2139	1	1	yes	characteristic
142	C₂×C₁₉	#2140	1	1	yes	characteristic
143	C₂×C₁₉	#2175	1	2	yes	—
144	C₂×C₁₉	#2177	1	2	yes	—
145	C₂²×D₄	—	2	1	no	—
146	C₂³×C₄	—	4	1	no	—
147	C₂×D₈	—	2	1	no	—
148	C₂×D₈	—	2	2	no	—
149	C₂×D₈	C₂×C₄×C₁₉	1	4	yes	—
150	C₂×D₈	C₂×C₄×C₁₉	1	2	yes	—
151	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
152	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
153	C₄×D₄	—	2	4	no	—
154	C₄×D₄	—	4	2	no	—
155	C₂×C₄⋊C₄	—	2	1	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₂²⋊C₄	—	4	1	no	—
158	C₂×C₄²	—	4	1	no	—
159	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
160	C₂×C₁₆	—	2	2	no	—
161	C₂×C₁₆	—	2	2	no	—
162	C₂×C₁₆	C₂×C₄×C₁₉	1	4	yes	complemented
163	#107	—	2	1	no	—
164	#107	—	2	2	no	—
165	#107	C₂×C₄×C₁₉	1	2	yes	—
166	#107	C₂×C₄×C₁₉	1	4	yes	complemented
167	#102	—	2	4	no	—
168	C₄×C₈	C₂×C₄×C₁₉	1	2	yes	—
169	C₄×C₈	D₄×C₁₉	1	2	yes	—
170	#95	—	4	1	no	—
171	C₁₉	#12637	1	1	yes	characteristic, direct product factor, Sylow
172	C₂⁴	—	4	1	no	—
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	—	2	4	no	—
175	C₂×D₄	—	4	1	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	—	4	1	no	—
179	C₂²×C₄	—	4	4	no	—
180	C₂²×C₄	—	8	4	no	—
181	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
182	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
183	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
184	D₈	—	2	2	no	—
185	D₈	—	2	4	no	—
186	D₈	C₄²×C₁₉	1	4	yes	complemented
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	C₂²⋊C₄×C₁₉	1	2	yes	—
189	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
190	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
191	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
192	C₂×C₈	#41×C₁₉	1	2	yes	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	4	1	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₂²⋊C₄	—	4	4	no	—
198	C₄²	—	8	2	no	—
199	C₄²	C₂²⋊C₄×C₁₉	1	2	yes	—
200	C₄²	SD₁₆×C₁₉	1	2	yes	—
201	C₁₆	—	2	4	no	—
202	C₂³	—	4	2	no	—
203	C₂³	—	4	4	no	—
204	C₂³	—	8	2	no	—
205	C₂³	—	8	1	no	—
206	C₂³	—	8	1	no	—
207	C₂³	C₂×D₈×C₁₉	1	1	yes	characteristic
208	D₄	—	2	4	no	—
209	D₄	—	4	2	no	—
210	D₄	—	4	4	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	8	8	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	—	8	2	no	—
219	C₂×C₄	—	8	2	no	—
220	C₂×C₄	C₁₉×#102	1	2	yes	—
221	C₂×C₄	C₁₉×#102	1	2	yes	—
222	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
223	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
224	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
225	C₂×C₄	C₁₉×#135	1	2	yes	—
226	C₈	—	2	2	no	—
227	C₈	C₂×C₄²×C₁₉	1	1	yes	characteristic
228	C₈	C₁₉×#117	1	1	yes	characteristic
229	C₈	C₄×D₄×C₁₉	1	2	yes	—
230	C₂²	—	4	4	no	—
231	C₂²	—	8	4	no	—
232	C₂²	—	8	4	no	—
233	C₂²	—	8	2	no	—
234	C₂²	—	8	2	no	—
235	C₂²	C₂×C₁₉×#102	1	1	yes	characteristic
236	C₂²	C₄×D₈×C₁₉	1	2	yes	—
237	C₂²	C₁₉×#509	1	2	yes	—
238	C₂²	C₁₉×#509	1	1	yes	characteristic
239	C₂²	C₁₉×#510	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	8	4	no	—
243	C₄	—	8	2	no	—
244	C₄	C₄×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
245	C₄	C₁₉×#416	1	1	yes	characteristic
246	C₄	C₄×SD₁₆×C₁₉	1	2	yes	—
247	C₂	—	8	4	no	—
248	C₂	C₄×C₁₉×#102	1	1	yes	characteristic
249	C₂	C₁₉×#2139	1	1	yes	characteristic
250	C₂	C₁₉×#2140	1	1	yes	characteristic
251	C₂	C₁₉×#2175	1	2	yes	—
252	C₂	C₁₉×#2177	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂×D₈
commutator	C₈	C₂×C₄²×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂²×D₈×C₁₉	C₂²
LCS	G ▷ 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
3. C₂×C₁₉×#346	C₂	semidirect
4. C₄×C₁₉×#107	C₂	semidirect
5. C₁₉×#1380	C₂	semidirect
10. C₄×D₈×C₁₉	C₄	semidirect
16. C₁₉×#357	C₄	semidirect
36. C₂×C₁₆×C₁₉	C₂×C₄	semidirect
40. C₁₉×#107	C₂×C₄	semidirect
59. D₈×C₁₉	C₄²	semidirect
75. #12637	C₁₉	direct (N × H)
106. C₂×#346	C₂×C₁₉	semidirect
107. C₄×#107	C₂×C₁₉	semidirect
108. #1380	C₂×C₁₉	semidirect
130. C₄×D₈	C₄×C₁₉	semidirect
136. #357	C₄×C₁₉	semidirect
162. C₂×C₁₆	C₂×C₄×C₁₉	semidirect
166. #107	C₂×C₄×C₁₉	semidirect
171. C₁₉	#12637	direct (N × H)
186. D₈	C₄²×C₁₉	semidirect

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×#357	2
C₂×C₁₉×#346	2
C₄×C₁₉×#107	2
C₁₉×#1380	2
#12637	19

Subgroups — C19×#12637 #18018596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12637 #18018596