Subgroups — C₁₉×#13156 #18019115

Order 4864.

306 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2148	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#360	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2110	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#1976	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1951	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#1389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#1386	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₉×#135	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×#428	—	2	1	no	—
13	C₁₉×#427	C₂²	1	1	yes	characteristic
14	C₂×C₁₉×#106	C₂²	1	1	yes	characteristic
15	C₁₉×#421	—	2	1	no	—
16	C₁₉×#410	—	2	1	no	—
17	C₁₉×#405	C₂²	1	1	yes	characteristic
18	C₁₉×#362	C₄	1	2	yes	complemented
19	C₁₉×#361	C₄	1	2	yes	complemented
20	C₁₉×#360	—	2	2	no	—
21	C₁₉×#347	—	2	2	no	—
22	C₁₉×#343	—	2	1	no	—
23	C₂×#41×C₁₉	—	2	1	no	—
24	C₁₉×#135	—	2	1	no	—
25	C₁₉×#135	—	2	1	no	—
26	C₁₉×#135	—	2	1	no	—
27	C₁₉×#135	—	2	1	no	—
28	C₂×Q₁₆×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂×SD₁₆×C₁₉	—	2	1	no	—
30	C₂×D₈×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂×M₁₆×C₁₉	—	2	1	no	—
32	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
34	C₁₉×#117	—	2	1	no	—
35	C₁₉×#117	D₄	1	1	yes	characteristic
36	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
37	C₁₉×M₃₂	—	2	1	no	—
38	C₁₉×M₃₂	—	2	1	no	—
39	C₁₉×M₃₂	C₂×C₄	1	2	yes	—
40	C₁₉×M₃₂	D₄	1	2	yes	—
41	C₂×C₁₆×C₁₉	—	2	1	no	—
42	C₂×C₁₆×C₁₉	—	2	1	no	—
43	C₁₉×#108	—	2	1	no	—
44	C₁₉×#108	C₂×C₄	1	2	yes	—
45	C₁₉×#107	—	2	1	no	—
46	C₁₉×#107	—	4	1	no	—
47	C₁₉×#106	—	2	1	no	—
48	C₁₉×#106	—	2	1	no	—
49	C₁₉×#106	—	2	1	no	—
50	C₁₉×#106	—	4	1	no	—
51	C₁₉×#104	—	4	2	no	—
52	C₁₉×#99	—	4	2	no	—
53	C₁₉×#97	—	2	1	no	—
54	C₄×C₈×C₁₉	—	2	1	no	—
55	#41×C₁₉	—	4	1	no	—
56	#41×C₁₉	—	4	1	no	—
57	#41×C₁₉	—	4	1	no	—
58	#41×C₁₉	—	4	1	no	—
59	C₂×Q₈×C₁₉	—	2	1	no	—
60	C₂×D₄×C₁₉	—	2	1	no	—
61	C₂×D₄×C₁₉	—	4	1	no	—
62	C₂²×C₄×C₁₉	—	4	1	no	—
63	C₂²×C₄×C₁₉	—	4	1	no	—
64	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
65	Q₁₆×C₁₉	—	2	1	no	—
66	Q₁₆×C₁₉	—	2	1	no	—
67	SD₁₆×C₁₉	—	4	1	no	—
68	SD₁₆×C₁₉	—	4	1	no	—
69	D₈×C₁₉	—	2	1	no	—
70	D₈×C₁₉	—	2	1	no	—
71	M₁₆×C₁₉	—	2	2	no	—
72	M₁₆×C₁₉	—	2	1	no	—
73	M₁₆×C₁₉	—	2	1	no	—
74	M₁₆×C₁₉	—	4	1	no	—
75	C₂×C₈×C₁₉	—	2	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	#41	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	C₄⋊C₄×C₁₉	—	2	1	no	—
84	C₄⋊C₄×C₁₉	—	4	1	no	—
85	C₄⋊C₄×C₁₉	—	4	1	no	—
86	C₄⋊C₄×C₁₉	—	4	1	no	—
87	C₂²⋊C₄×C₁₉	—	2	1	no	—
88	C₂²⋊C₄×C₁₉	—	4	1	no	—
89	C₄²×C₁₉	—	2	1	no	—
90	C₄²×C₁₉	—	4	1	no	—
91	C₁₆×C₁₉	—	2	2	no	—
92	C₁₆×C₁₉	—	2	2	no	—
93	#13156	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₁₉	—	4	1	no	—
95	C₂³×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
96	Q₈×C₁₉	—	4	1	no	—
97	Q₈×C₁₉	—	4	1	no	—
98	D₄×C₁₉	—	4	1	no	—
99	D₄×C₁₉	—	4	1	no	—
100	D₄×C₁₉	—	8	1	no	—
101	D₄×C₁₉	—	8	1	no	—
102	C₂×C₄×C₁₉	—	4	1	no	—
103	C₂×C₄×C₁₉	—	4	1	no	—
104	C₂×C₄×C₁₉	—	8	1	no	—
105	C₂×C₄×C₁₉	—	8	1	no	—
106	C₂×C₄×C₁₉	—	8	1	no	—
107	C₂×C₄×C₁₉	—	8	1	no	—
108	C₂×C₄×C₁₉	—	16	1	no	—
109	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
111	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
112	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
115	C₈×C₁₉	—	4	2	no	—
116	C₈×C₁₉	—	4	1	no	—
117	C₈×C₁₉	#117	1	1	yes	characteristic
118	C₈×C₁₉	C₄×D₄	1	1	yes	characteristic
119	C₈×C₁₉	#123	1	1	yes	characteristic
120	C₈×C₁₉	#126	1	1	yes	characteristic
121	#2148	C₂×C₁₉	1	1	yes	characteristic
122	C₂×#360	C₂×C₁₉	1	1	yes	characteristic
123	#2110	C₂×C₁₉	1	1	yes	characteristic, complemented
124	#1976	C₂×C₁₉	1	1	yes	characteristic
125	#1951	C₂×C₁₉	1	1	yes	characteristic, complemented
126	#1389	C₂×C₁₉	1	1	yes	characteristic, complemented
127	#1386	C₂×C₁₉	1	1	yes	characteristic, complemented
128	C₂²×C₁₉	—	4	1	no	—
129	C₂²×C₁₉	—	8	1	no	—
130	C₂²×C₁₉	—	16	1	no	—
131	C₂²×C₁₉	#440	1	1	yes	characteristic
132	C₂²×C₁₉	#469	1	1	yes	characteristic
133	C₂²×C₁₉	#490	1	1	yes	characteristic
134	C₄×C₁₉	—	8	1	no	—
135	C₄×C₁₉	—	8	1	no	—
136	C₄×C₁₉	—	16	1	no	—
137	C₄×C₁₉	—	16	1	no	—
138	C₄×C₁₉	#388	1	1	yes	characteristic
139	C₄×C₁₉	#442	1	1	yes	characteristic
140	C₄×C₁₉	#468	1	1	yes	characteristic
141	C₄×C₁₉	#489	1	1	yes	characteristic
142	C₂×#135	C₂²×C₁₉	1	1	yes	characteristic
143	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
144	C₂×M₃₂	C₂²×C₁₉	1	1	yes	characteristic
145	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
146	#428	—	2	1	no	—
147	#427	C₂²×C₁₉	1	1	yes	characteristic
148	C₂×#106	C₂²×C₁₉	1	1	yes	characteristic
149	#421	—	2	1	no	—
150	#410	—	2	1	no	—
151	#405	C₂²×C₁₉	1	1	yes	characteristic
152	#362	C₄×C₁₉	1	2	yes	complemented
153	#361	C₄×C₁₉	1	2	yes	complemented
154	#360	—	2	2	no	—
155	#347	—	2	2	no	—
156	#343	—	2	1	no	—
157	C₂×C₁₉	—	2	1	no	—
158	C₂×C₁₉	—	2	1	no	—
159	C₂×C₁₉	—	2	1	no	—
160	C₂×C₁₉	—	16	1	no	—
161	C₂×C₁₉	#1936	1	1	yes	characteristic
162	C₂×#41	—	2	1	no	—
163	#135	—	2	1	no	—
164	#135	—	2	1	no	—
165	#135	—	2	1	no	—
166	#135	—	2	1	no	—
167	C₂×Q₁₆	C₂×C₄×C₁₉	1	1	yes	characteristic
168	C₂×SD₁₆	—	2	1	no	—
169	C₂×D₈	C₂×C₄×C₁₉	1	1	yes	characteristic
170	C₂×M₁₆	—	2	1	no	—
171	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
172	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
173	#117	—	2	1	no	—
174	#117	D₄×C₁₉	1	1	yes	characteristic
175	C₂×C₄⋊C₄	—	2	1	no	—
176	M₃₂	—	2	1	no	—
177	M₃₂	—	2	1	no	—
178	M₃₂	C₂×C₄×C₁₉	1	2	yes	—
179	M₃₂	D₄×C₁₉	1	2	yes	—
180	C₂×C₁₆	—	2	1	no	—
181	C₂×C₁₆	—	2	1	no	—
182	#108	—	2	1	no	—
183	#108	C₂×C₄×C₁₉	1	2	yes	—
184	#107	—	2	1	no	—
185	#107	—	4	1	no	—
186	#106	—	2	1	no	—
187	#106	—	2	1	no	—
188	#106	—	2	1	no	—
189	#106	—	4	1	no	—
190	#104	—	4	2	no	—
191	#99	—	4	2	no	—
192	#97	—	2	1	no	—
193	C₄×C₈	—	2	1	no	—
194	C₁₉	#13156	1	1	yes	characteristic, direct product factor, Sylow
195	#41	—	4	1	no	—
196	#41	—	4	1	no	—
197	#41	—	4	1	no	—
198	#41	—	4	1	no	—
199	C₂×Q₈	—	2	1	no	—
200	C₂×D₄	—	2	1	no	—
201	C₂×D₄	—	4	1	no	—
202	C₂²×C₄	—	4	1	no	—
203	C₂²×C₄	—	4	1	no	—
204	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
205	Q₁₆	—	2	1	no	—
206	Q₁₆	—	2	1	no	—
207	SD₁₆	—	4	1	no	—
208	SD₁₆	—	4	1	no	—
209	D₈	—	2	1	no	—
210	D₈	—	2	1	no	—
211	M₁₆	—	2	2	no	—
212	M₁₆	—	2	1	no	—
213	M₁₆	—	2	1	no	—
214	M₁₆	—	4	1	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
218	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
219	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
220	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
221	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
222	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	4	1	no	—
225	C₄⋊C₄	—	4	1	no	—
226	C₄⋊C₄	—	4	1	no	—
227	C₂²⋊C₄	—	2	1	no	—
228	C₂²⋊C₄	—	4	1	no	—
229	C₄²	—	2	1	no	—
230	C₄²	—	4	1	no	—
231	C₁₆	—	2	2	no	—
232	C₁₆	—	2	2	no	—
233	C₂³	—	4	1	no	—
234	C₂³	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
235	Q₈	—	4	1	no	—
236	Q₈	—	4	1	no	—
237	D₄	—	4	1	no	—
238	D₄	—	4	1	no	—
239	D₄	—	8	1	no	—
240	D₄	—	8	1	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	8	1	no	—
244	C₂×C₄	—	8	1	no	—
245	C₂×C₄	—	8	1	no	—
246	C₂×C₄	—	8	1	no	—
247	C₂×C₄	—	16	1	no	—
248	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
249	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
250	C₂×C₄	C₁₉×#124	1	1	yes	characteristic
251	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
252	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
253	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
254	C₈	—	4	2	no	—
255	C₈	—	4	1	no	—
256	C₈	C₁₉×#117	1	1	yes	characteristic
257	C₈	C₄×D₄×C₁₉	1	1	yes	characteristic
258	C₈	C₁₉×#123	1	1	yes	characteristic
259	C₈	C₁₉×#126	1	1	yes	characteristic
260	C₂²	—	4	1	no	—
261	C₂²	—	8	1	no	—
262	C₂²	—	16	1	no	—
263	C₂²	C₁₉×#440	1	1	yes	characteristic
264	C₂²	C₁₉×#469	1	1	yes	characteristic
265	C₂²	C₁₉×#490	1	1	yes	characteristic
266	C₄	—	8	1	no	—
267	C₄	—	8	1	no	—
268	C₄	—	16	1	no	—
269	C₄	—	16	1	no	—
270	C₄	C₁₉×#388	1	1	yes	characteristic
271	C₄	C₁₉×#442	1	1	yes	characteristic
272	C₄	C₁₉×#468	1	1	yes	characteristic
273	C₄	C₁₉×#489	1	1	yes	characteristic
274	C₂	—	2	1	no	—
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	—	16	1	no	—
278	C₂	C₁₉×#1936	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₉	#1936
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂×C₁₉×#135	C₂²
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
3. C₁₉×#2110	C₂	semidirect
5. C₁₉×#1951	C₂	semidirect
6. C₁₉×#1389	C₂	semidirect
7. C₁₉×#1386	C₂	semidirect
18. C₁₉×#362	C₄	semidirect
19. C₁₉×#361	C₄	semidirect
93. #13156	C₁₉	direct (N × H)
123. #2110	C₂×C₁₉	semidirect
125. #1951	C₂×C₁₉	semidirect
126. #1389	C₂×C₁₉	semidirect
127. #1386	C₂×C₁₉	semidirect
152. #362	C₄×C₁₉	semidirect
153. #361	C₄×C₁₉	semidirect
194. C₁₉	#13156	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×#2148	2
C₂×C₁₉×#360	2
C₁₉×#2110	2
C₁₉×#1976	2
C₁₉×#1951	2
C₁₉×#1389	2
C₁₉×#1386	2
#13156	19

Subgroups — C19×#13156 #18019115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13156 #18019115