Subgroups — C₁₉×#12954 #18018913

Order 4864.

380 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2950	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2155	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#362	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₉×#362	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1953	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1389	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1378	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
9	C₁₉×#519	—	2	1	no	—
10	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×M₃₂	C₂²	1	1	yes	characteristic
12	C₁₉×#441	—	2	2	no	—
13	C₄×Q₁₆×C₁₉	—	2	2	no	—
14	C₁₉×#430	—	2	1	no	—
15	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
16	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
17	C₁₉×#427	C₂²	1	1	yes	characteristic
18	C₁₉×#419	—	2	1	no	—
19	C₁₉×#405	C₂²	1	1	yes	characteristic
20	C₁₉×#365	—	2	2	no	—
21	C₁₉×#362	—	2	2	no	—
22	C₁₉×#362	C₄	1	4	yes	complemented
23	C₁₉×#344	—	2	1	no	—
24	C₂²×Q₈×C₁₉	—	2	1	no	—
25	C₂×Q₁₆×C₁₉	—	2	1	no	—
26	C₂×Q₁₆×C₁₉	—	2	1	no	—
27	C₂×Q₁₆×C₁₉	—	2	1	no	—
28	C₂×Q₁₆×C₁₉	C₂×C₄	1	2	yes	—
29	C₂×Q₁₆×C₁₉	D₄	1	2	yes	—
30	C₂×Q₁₆×C₁₉	Q₈	1	2	yes	—
31	C₂×M₁₆×C₁₉	—	2	1	no	—
32	C₂×M₁₆×C₁₉	—	2	1	no	—
33	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
35	C₄×Q₈×C₁₉	—	2	4	no	—
36	C₄×Q₈×C₁₉	—	4	2	no	—
37	C₁₉×#117	—	2	1	no	—
38	C₁₉×#117	—	4	2	no	—
39	C₁₉×#117	D₄	1	1	yes	characteristic
40	C₁₉×M₃₂	—	2	1	no	—
41	C₁₉×M₃₂	—	2	1	no	—
42	C₁₉×M₃₂	—	2	1	no	—
43	C₁₉×M₃₂	C₂×C₄	1	2	yes	—
44	C₂×C₁₆×C₁₉	—	2	1	no	—
45	C₂×C₁₆×C₁₉	—	2	1	no	—
46	C₁₉×#108	—	2	1	no	—
47	C₁₉×#108	—	2	1	no	—
48	C₁₉×#108	—	2	1	no	—
49	C₁₉×#108	—	4	2	no	—
50	C₁₉×#108	C₂×C₄	1	2	yes	—
51	C₁₉×#107	—	2	1	no	—
52	C₁₉×#106	D₄	1	2	yes	—
53	C₁₉×#103	—	2	4	no	—
54	C₁₉×#97	—	2	1	no	—
55	C₄×C₈×C₁₉	—	2	1	no	—
56	C₂×Q₈×C₁₉	—	2	2	no	—
57	C₂×Q₈×C₁₉	—	2	2	no	—
58	C₂×Q₈×C₁₉	—	2	2	no	—
59	C₂×Q₈×C₁₉	—	4	1	no	—
60	C₂×Q₈×C₁₉	—	4	1	no	—
61	C₂×Q₈×C₁₉	—	4	1	no	—
62	C₂²×C₄×C₁₉	—	4	1	no	—
63	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
64	Q₁₆×C₁₉	—	2	2	no	—
65	Q₁₆×C₁₉	—	2	2	no	—
66	Q₁₆×C₁₉	—	2	2	no	—
67	Q₁₆×C₁₉	C₄⋊C₄	1	4	yes	—
68	M₁₆×C₁₉	—	2	2	no	—
69	M₁₆×C₁₉	—	2	1	no	—
70	M₁₆×C₁₉	—	2	1	no	—
71	M₁₆×C₁₉	—	4	1	no	—
72	M₁₆×C₁₉	—	4	1	no	—
73	M₁₆×C₁₉	—	4	1	no	—
74	C₂×C₈×C₁₉	—	2	1	no	—
75	C₂×C₈×C₁₉	—	4	1	no	—
76	C₂×C₈×C₁₉	—	4	1	no	—
77	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
81	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
82	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
83	C₄⋊C₄×C₁₉	—	2	1	no	—
84	C₄⋊C₄×C₁₉	—	2	2	no	—
85	C₄⋊C₄×C₁₉	—	4	4	no	—
86	C₂²⋊C₄×C₁₉	—	4	1	no	—
87	C₂²⋊C₄×C₁₉	SD₁₆	1	2	yes	—
88	C₄²×C₁₉	—	2	1	no	—
89	C₄²×C₁₉	—	4	4	no	—
90	C₄²×C₁₉	—	4	1	no	—
91	C₁₆×C₁₉	—	2	2	no	—
92	C₁₆×C₁₉	—	4	1	no	—
93	#12954	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
95	Q₈×C₁₉	—	2	4	no	—
96	Q₈×C₁₉	—	4	2	no	—
97	Q₈×C₁₉	—	4	2	no	—
98	Q₈×C₁₉	—	4	2	no	—
99	C₂×C₄×C₁₉	—	2	2	no	—
100	C₂×C₄×C₁₉	—	4	2	no	—
101	C₂×C₄×C₁₉	—	4	2	no	—
102	C₂×C₄×C₁₉	—	4	2	no	—
103	C₂×C₄×C₁₉	—	4	2	no	—
104	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
106	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
108	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₈×C₁₉	—	4	2	no	—
111	C₈×C₁₉	—	4	1	no	—
112	C₈×C₁₉	—	8	1	no	—
113	C₈×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₈×C₁₉	C₄×D₄	1	1	yes	characteristic
115	C₈×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
116	C₈×C₁₉	#123	1	1	yes	characteristic
117	#2950	C₂×C₁₉	1	1	yes	characteristic
118	#2155	C₂×C₁₉	1	1	yes	characteristic
119	C₂×#362	C₂×C₁₉	1	1	yes	characteristic
120	C₂×#362	C₂×C₁₉	1	1	yes	characteristic
121	#1953	C₂×C₁₉	1	1	yes	characteristic
122	#1389	C₂×C₁₉	1	1	yes	characteristic
123	#1378	C₂×C₁₉	1	1	yes	characteristic
124	C₂²×C₁₉	—	4	1	no	—
125	C₂²×C₁₉	C₄×SD₁₆	1	1	yes	characteristic
126	C₂²×C₁₉	#450	1	1	yes	characteristic
127	C₂²×C₁₉	#476	1	1	yes	characteristic
128	C₄×C₁₉	—	4	2	no	—
129	C₄×C₁₉	—	4	4	no	—
130	C₄×C₁₉	—	8	2	no	—
131	C₄×C₁₉	#385	1	1	yes	characteristic
132	C₄×C₁₉	#442	1	1	yes	characteristic
133	C₄×C₁₉	#449	1	1	yes	characteristic
134	C₄×C₁₉	#478	1	1	yes	characteristic
135	C₂²×Q₁₆	C₂²×C₁₉	1	1	yes	characteristic
136	#519	—	2	1	no	—
137	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
138	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
139	#441	—	2	2	no	—
140	C₄×Q₁₆	—	2	2	no	—
141	#430	—	2	1	no	—
142	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
143	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
144	#427	C₂²×C₁₉	1	1	yes	characteristic
145	#419	—	2	1	no	—
146	#405	C₂²×C₁₉	1	1	yes	characteristic
147	#365	—	2	2	no	—
148	#362	—	2	2	no	—
149	#362	C₄×C₁₉	1	4	yes	complemented
150	#344	—	2	1	no	—
151	C₂×C₁₉	—	2	1	no	—
152	C₂×C₁₉	—	2	1	no	—
153	C₂×C₁₉	—	2	1	no	—
154	C₂×C₁₉	#1864	1	1	yes	characteristic
155	C₂²×Q₈	—	2	1	no	—
156	C₂×Q₁₆	—	2	1	no	—
157	C₂×Q₁₆	—	2	1	no	—
158	C₂×Q₁₆	—	2	1	no	—
159	C₂×Q₁₆	C₂×C₄×C₁₉	1	2	yes	—
160	C₂×Q₁₆	D₄×C₁₉	1	2	yes	—
161	C₂×Q₁₆	Q₈×C₁₉	1	2	yes	—
162	C₂×M₁₆	—	2	1	no	—
163	C₂×M₁₆	—	2	1	no	—
164	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
165	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
166	C₄×Q₈	—	2	4	no	—
167	C₄×Q₈	—	4	2	no	—
168	#117	—	2	1	no	—
169	#117	—	4	2	no	—
170	#117	D₄×C₁₉	1	1	yes	characteristic
171	M₃₂	—	2	1	no	—
172	M₃₂	—	2	1	no	—
173	M₃₂	—	2	1	no	—
174	M₃₂	C₂×C₄×C₁₉	1	2	yes	—
175	C₂×C₁₆	—	2	1	no	—
176	C₂×C₁₆	—	2	1	no	—
177	#108	—	2	1	no	—
178	#108	—	2	1	no	—
179	#108	—	2	1	no	—
180	#108	—	4	2	no	—
181	#108	C₂×C₄×C₁₉	1	2	yes	—
182	#107	—	2	1	no	—
183	#106	D₄×C₁₉	1	2	yes	—
184	#103	—	2	4	no	—
185	#97	—	2	1	no	—
186	C₄×C₈	—	2	1	no	—
187	C₁₉	#12954	1	1	yes	characteristic, direct product factor, Sylow
188	C₂×Q₈	—	2	2	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	—	4	1	no	—
192	C₂×Q₈	—	4	1	no	—
193	C₂×Q₈	—	4	1	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
196	Q₁₆	—	2	2	no	—
197	Q₁₆	—	2	2	no	—
198	Q₁₆	—	2	2	no	—
199	Q₁₆	C₄⋊C₄×C₁₉	1	4	yes	—
200	M₁₆	—	2	2	no	—
201	M₁₆	—	2	1	no	—
202	M₁₆	—	2	1	no	—
203	M₁₆	—	4	1	no	—
204	M₁₆	—	4	1	no	—
205	M₁₆	—	4	1	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	4	1	no	—
208	C₂×C₈	—	4	1	no	—
209	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
210	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
211	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
212	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
213	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
214	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
215	C₄⋊C₄	—	2	1	no	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	4	4	no	—
218	C₂²⋊C₄	—	4	1	no	—
219	C₂²⋊C₄	SD₁₆×C₁₉	1	2	yes	—
220	C₄²	—	2	1	no	—
221	C₄²	—	4	4	no	—
222	C₄²	—	4	1	no	—
223	C₁₆	—	2	2	no	—
224	C₁₆	—	4	1	no	—
225	C₂³	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
226	Q₈	—	2	4	no	—
227	Q₈	—	4	2	no	—
228	Q₈	—	4	2	no	—
229	Q₈	—	4	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
236	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
237	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
238	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
239	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
240	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
241	C₈	—	4	2	no	—
242	C₈	—	4	1	no	—
243	C₈	—	8	1	no	—
244	C₈	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
245	C₈	C₄×D₄×C₁₉	1	1	yes	characteristic
246	C₈	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
247	C₈	C₁₉×#123	1	1	yes	characteristic
248	C₂²	—	4	1	no	—
249	C₂²	C₄×SD₁₆×C₁₉	1	1	yes	characteristic
250	C₂²	C₁₉×#450	1	1	yes	characteristic
251	C₂²	C₁₉×#476	1	1	yes	characteristic
252	C₄	—	4	2	no	—
253	C₄	—	4	4	no	—
254	C₄	—	8	2	no	—
255	C₄	C₁₉×#385	1	1	yes	characteristic
256	C₄	C₁₉×#442	1	1	yes	characteristic
257	C₄	C₁₉×#449	1	1	yes	characteristic
258	C₄	C₁₉×#478	1	1	yes	characteristic
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	C₁₉×#1864	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₉	#1864
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂³×C₁₉	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#405 ◁ G

Complements

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

N	H ≅ G/N	type
22. C₁₉×#362	C₄	semidirect
93. #12954	C₁₉	direct (N × H)
149. #362	C₄×C₁₉	semidirect
187. C₁₉	#12954	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×#2950	2
C₁₉×#2155	2
C₂×C₁₉×#362	2
C₁₉×#1953	2
C₁₉×#1389	2
C₁₉×#1378	2
#12954	19

Subgroups — C19×#12954 #18018913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12954 #18018913