Subgroups — C₁₉×#19698 #18025657

Order 4864.

896 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3208	C₂	1	2	yes	maximal, complemented
2	C₁₉×#3205	C₂	1	2	yes	maximal, complemented
3	C₂×C₁₉×#493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#3131	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#1721	C₂	1	4	yes	maximal, complemented
6	C₁₉×#1717	C₂	1	4	yes	maximal, complemented
7	C₁₉×#1452	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₁₉	—	2	2	no	—
9	C₁₉×#533	C₂²	1	1	yes	characteristic
10	C₁₉×#532	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×#127	C₂²	1	2	yes	—
12	C₁₉×#493	—	2	2	no	—
13	C₁₉×#493	C₂²	1	4	yes	—
14	C₁₉×#493	C₂²	1	4	yes	complemented
15	C₁₉×#492	C₂²	1	4	yes	complemented
16	C₁₉×#488	—	2	2	no	—
17	C₁₉×#487	C₂²	1	2	yes	complemented
18	C₁₉×#486	C₂²	1	4	yes	complemented
19	C₁₉×#486	C₂²	1	4	yes	complemented
20	C₁₉×#486	C₂²	1	2	yes	complemented
21	C₁₉×#460	—	2	4	no	—
22	C₁₉×#459	—	2	4	no	—
23	C₁₉×#450	—	2	4	no	—
24	C₁₉×#447	—	2	4	no	—
25	C₁₉×#431	C₂²	1	2	yes	—
26	C₁₉×#416	—	2	2	no	—
27	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
28	C₁₉×#322	C₂²	1	4	yes	complemented
29	C₂×#41×C₁₉	—	2	1	no	—
30	C₂²×D₄×C₁₉	—	2	2	no	—
31	C₂²×D₄×C₁₉	—	2	2	no	—
32	C₂²×D₄×C₁₉	—	2	2	no	—
33	C₂×SD₁₆×C₁₉	—	2	8	no	—
34	C₂×D₈×C₁₉	—	2	8	no	—
35	C₂×D₈×C₁₉	—	2	8	no	—
36	C₂×D₈×C₁₉	—	4	8	no	—
37	C₂²×C₈×C₁₉	D₄	1	2	yes	—
38	C₁₉×C₄⋊Q₈	C₂³	1	1	yes	characteristic
39	C₁₉×C₄⋊Q₈	C₂³	1	2	yes	—
40	C₁₉×#127	—	2	2	no	—
41	C₁₉×#127	—	2	2	no	—
42	C₁₉×#127	—	2	2	no	—
43	C₁₉×#127	C₂³	1	4	yes	—
44	C₁₉×#127	C₂³	1	1	yes	characteristic
45	C₁₉×#125	—	2	1	no	—
46	C₁₉×#124	—	2	1	no	—
47	C₁₉×C₂²⋊Q₈	—	2	2	no	—
48	C₁₉×#121	—	2	2	no	—
49	C₄×Q₈×C₁₉	—	2	2	no	—
50	C₄×D₄×C₁₉	—	2	2	no	—
51	C₁₉×#117	—	2	1	no	—
52	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
53	C₁₉×C₄⋊C₈	D₄	1	4	yes	—
54	C₁₉×#103	—	2	4	no	—
55	C₁₉×#102	—	2	4	no	—
56	C₁₉×#102	—	2	8	no	—
57	C₁₉×#102	—	2	8	no	—
58	C₁₉×C₂²⋊C₈	D₄	1	4	yes	—
59	C₄×C₈×C₁₉	D₄	1	2	yes	—
60	C₄×C₈×C₁₉	C₂³	1	2	yes	—
61	C₄×C₈×C₁₉	C₂³	1	4	yes	—
62	C₂⁴×C₁₉	—	4	2	no	—
63	#41×C₁₉	—	4	4	no	—
64	C₂×Q₈×C₁₉	—	2	2	no	—
65	C₂×Q₈×C₁₉	—	2	1	no	—
66	C₂×D₄×C₁₉	—	2	2	no	—
67	C₂×D₄×C₁₉	—	2	1	no	—
68	C₂×D₄×C₁₉	—	2	4	no	—
69	C₂×D₄×C₁₉	—	2	4	no	—
70	C₂×D₄×C₁₉	—	2	4	no	—
71	C₂×D₄×C₁₉	—	4	1	no	—
72	C₂×D₄×C₁₉	—	4	2	no	—
73	C₂×D₄×C₁₉	—	4	2	no	—
74	C₂×D₄×C₁₉	—	4	2	no	—
75	C₂×D₄×C₁₉	—	4	8	no	—
76	C₂×D₄×C₁₉	—	8	4	no	—
77	C₂×D₄×C₁₉	—	8	4	no	—
78	C₂²×C₄×C₁₉	—	4	1	no	—
79	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
81	SD₁₆×C₁₉	—	8	8	no	—
82	D₈×C₁₉	—	4	16	no	—
83	D₈×C₁₉	—	8	8	no	—
84	C₂×C₈×C₁₉	—	2	4	no	—
85	C₂×C₈×C₁₉	C₂×D₄	1	4	yes	—
86	C₂×C₈×C₁₉	C₂×D₄	1	8	yes	—
87	C₄⋊C₄×C₁₉	—	2	2	no	—
88	C₄⋊C₄×C₁₉	—	2	2	no	—
89	C₄⋊C₄×C₁₉	—	2	2	no	—
90	C₄⋊C₄×C₁₉	—	4	2	no	—
91	C₂²⋊C₄×C₁₉	—	4	1	no	—
92	C₂²⋊C₄×C₁₉	—	4	2	no	—
93	C₄²×C₁₉	—	4	1	no	—
94	C₄²×C₁₉	C₂×D₄	1	2	yes	—
95	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
97	#19698	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂³×C₁₉	—	4	1	no	—
99	C₂³×C₁₉	—	4	4	no	—
100	C₂³×C₁₉	—	8	4	no	—
101	C₂³×C₁₉	—	8	4	no	—
102	C₂³×C₁₉	—	8	4	no	—
103	C₂³×C₁₉	#127	1	1	yes	characteristic
104	Q₈×C₁₉	—	4	2	no	—
105	Q₈×C₁₉	—	4	4	no	—
106	D₄×C₁₉	—	4	8	no	—
107	D₄×C₁₉	—	4	2	no	—
108	D₄×C₁₉	—	4	4	no	—
109	D₄×C₁₉	—	4	8	no	—
110	D₄×C₁₉	—	4	8	no	—
111	D₄×C₁₉	—	8	4	no	—
112	D₄×C₁₉	—	8	4	no	—
113	D₄×C₁₉	—	8	2	no	—
114	D₄×C₁₉	—	8	4	no	—
115	C₂×C₄×C₁₉	—	2	4	no	—
116	C₂×C₄×C₁₉	—	4	2	no	—
117	C₂×C₄×C₁₉	—	4	1	no	—
118	C₂×C₄×C₁₉	—	8	2	no	—
119	C₂×C₄×C₁₉	#127	1	2	yes	—
120	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
121	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	4	yes	—
122	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
124	C₈×C₁₉	—	2	4	no	—
125	C₈×C₁₉	—	2	8	no	—
126	#3208	C₂×C₁₉	1	2	yes	complemented
127	#3205	C₂×C₁₉	1	2	yes	complemented
128	C₂×#493	C₂×C₁₉	1	1	yes	characteristic, complemented
129	#3131	C₂×C₁₉	1	1	yes	characteristic, complemented
130	#1721	C₂×C₁₉	1	4	yes	complemented
131	#1717	C₂×C₁₉	1	4	yes	complemented
132	#1452	C₂×C₁₉	1	1	yes	characteristic, complemented
133	C₂²×C₁₉	—	2	2	no	—
134	C₂²×C₁₉	—	8	1	no	—
135	C₂²×C₁₉	—	8	4	no	—
136	C₂²×C₁₉	—	8	4	no	—
137	C₂²×C₁₉	—	8	4	no	—
138	C₂²×C₁₉	—	8	2	no	—
139	C₂²×C₁₉	—	16	8	no	—
140	C₂²×C₁₉	#496	1	2	yes	—
141	C₂²×C₁₉	C₂×#127	1	1	yes	characteristic
142	C₄×C₁₉	—	2	2	no	—
143	C₄×C₁₉	—	8	1	no	—
144	C₄×C₁₉	—	8	2	no	—
145	C₄×C₁₉	#496	1	2	yes	—
146	C₄×C₁₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
147	C₄×C₁₉	#576	1	4	yes	—
148	C₂²×D₈	—	2	2	no	—
149	#533	C₂²×C₁₉	1	1	yes	characteristic
150	#532	C₂²×C₁₉	1	1	yes	characteristic
151	C₂×#127	C₂²×C₁₉	1	2	yes	—
152	#493	—	2	2	no	—
153	#493	C₂²×C₁₉	1	4	yes	—
154	#493	C₂²×C₁₉	1	4	yes	complemented
155	#492	C₂²×C₁₉	1	4	yes	complemented
156	#488	—	2	2	no	—
157	#487	C₂²×C₁₉	1	2	yes	complemented
158	#486	C₂²×C₁₉	1	2	yes	complemented
159	#486	C₂²×C₁₉	1	4	yes	complemented
160	#486	C₂²×C₁₉	1	4	yes	complemented
161	#460	—	2	4	no	—
162	#459	—	2	4	no	—
163	#450	—	2	4	no	—
164	#447	—	2	4	no	—
165	#431	C₂²×C₁₉	1	2	yes	—
166	#416	—	2	2	no	—
167	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
168	#322	C₂²×C₁₉	1	4	yes	complemented
169	C₂×C₁₉	—	2	2	no	—
170	C₂×C₁₉	—	16	1	no	—
171	C₂×C₁₉	—	16	4	no	—
172	C₂×C₁₉	C₂×#496	1	1	yes	characteristic
173	C₂×C₁₉	#3155	1	2	yes	—
174	C₂×#41	—	2	1	no	—
175	C₂²×D₄	—	2	2	no	—
176	C₂²×D₄	—	2	2	no	—
177	C₂²×D₄	—	2	2	no	—
178	C₂×SD₁₆	—	2	8	no	—
179	C₂×D₈	—	2	8	no	—
180	C₂×D₈	—	2	8	no	—
181	C₂×D₈	—	4	8	no	—
182	C₂²×C₈	D₄×C₁₉	1	2	yes	—
183	C₄⋊Q₈	C₂³×C₁₉	1	1	yes	characteristic
184	C₄⋊Q₈	C₂³×C₁₉	1	2	yes	—
185	#127	—	2	2	no	—
186	#127	—	2	2	no	—
187	#127	—	2	2	no	—
188	#127	C₂³×C₁₉	1	4	yes	—
189	#127	C₂³×C₁₉	1	1	yes	characteristic
190	#125	—	2	1	no	—
191	#124	—	2	1	no	—
192	C₂²⋊Q₈	—	2	2	no	—
193	#121	—	2	2	no	—
194	C₄×Q₈	—	2	2	no	—
195	C₄×D₄	—	2	2	no	—
196	#117	—	2	1	no	—
197	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
198	C₄⋊C₈	D₄×C₁₉	1	4	yes	—
199	#103	—	2	4	no	—
200	#102	—	2	4	no	—
201	#102	—	2	8	no	—
202	#102	—	2	8	no	—
203	C₂²⋊C₈	D₄×C₁₉	1	4	yes	—
204	C₄×C₈	D₄×C₁₉	1	2	yes	—
205	C₄×C₈	C₂³×C₁₉	1	2	yes	—
206	C₄×C₈	C₂³×C₁₉	1	4	yes	—
207	C₁₉	#19698	1	1	yes	characteristic, direct product factor, Sylow
208	C₂⁴	—	4	2	no	—
209	#41	—	4	4	no	—
210	C₂×Q₈	—	2	1	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×D₄	—	2	1	no	—
213	C₂×D₄	—	2	4	no	—
214	C₂×D₄	—	2	4	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	2	4	no	—
217	C₂×D₄	—	4	1	no	—
218	C₂×D₄	—	4	2	no	—
219	C₂×D₄	—	4	2	no	—
220	C₂×D₄	—	4	2	no	—
221	C₂×D₄	—	4	8	no	—
222	C₂×D₄	—	8	4	no	—
223	C₂×D₄	—	8	4	no	—
224	C₂²×C₄	—	4	1	no	—
225	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
226	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
227	SD₁₆	—	8	8	no	—
228	D₈	—	4	16	no	—
229	D₈	—	8	8	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	C₂×D₄×C₁₉	1	4	yes	—
232	C₂×C₈	C₂×D₄×C₁₉	1	8	yes	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₂²⋊C₄	—	4	1	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₄²	—	4	1	no	—
240	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
241	C₄²	C₂×D₄×C₁₉	1	2	yes	—
242	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
243	C₂³	—	4	1	no	—
244	C₂³	—	4	4	no	—
245	C₂³	—	8	4	no	—
246	C₂³	—	8	4	no	—
247	C₂³	—	8	4	no	—
248	C₂³	C₁₉×#127	1	1	yes	characteristic
249	Q₈	—	4	2	no	—
250	Q₈	—	4	4	no	—
251	D₄	—	4	8	no	—
252	D₄	—	4	2	no	—
253	D₄	—	4	4	no	—
254	D₄	—	4	8	no	—
255	D₄	—	4	8	no	—
256	D₄	—	8	4	no	—
257	D₄	—	8	4	no	—
258	D₄	—	8	2	no	—
259	D₄	—	8	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	8	2	no	—
264	C₂×C₄	C₁₉×#127	1	2	yes	—
265	C₂×C₄	C₁₉×#127	1	1	yes	characteristic
266	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	4	yes	—
267	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
268	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
269	C₈	—	2	4	no	—
270	C₈	—	2	8	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	8	1	no	—
273	C₂²	—	8	4	no	—
274	C₂²	—	8	4	no	—
275	C₂²	—	8	4	no	—
276	C₂²	—	8	2	no	—
277	C₂²	—	16	8	no	—
278	C₂²	C₁₉×#496	1	2	yes	—
279	C₂²	C₂×C₁₉×#127	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	8	1	no	—
282	C₄	—	8	2	no	—
283	C₄	C₁₉×#496	1	2	yes	—
284	C₄	C₂×C₁₉×AGL(1,ℤ/8)	1	2	yes	—
285	C₄	C₁₉×#576	1	4	yes	—
286	C₂	—	2	2	no	—
287	C₂	—	16	4	no	—
288	C₂	—	16	1	no	—
289	C₂	C₂×C₁₉×#496	1	1	yes	characteristic
290	C₂	C₁₉×#3155	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₉
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₉×#3208	C₂	semidirect
2. C₁₉×#3205	C₂	semidirect
3. C₂×C₁₉×#493	C₂	semidirect
4. C₁₉×#3131	C₂	semidirect
5. C₁₉×#1721	C₂	semidirect
6. C₁₉×#1717	C₂	semidirect
7. C₁₉×#1452	C₂	semidirect
14. C₁₉×#493	C₂²	semidirect
15. C₁₉×#492	C₂²	semidirect
17. C₁₉×#487	C₂²	semidirect
18. C₁₉×#486	C₂²	semidirect
19. C₁₉×#486	C₂²	semidirect
20. C₁₉×#486	C₂²	semidirect
28. C₁₉×#322	C₂²	semidirect
97. #19698	C₁₉	direct (N × H)
126. #3208	C₂×C₁₉	semidirect
127. #3205	C₂×C₁₉	semidirect
128. C₂×#493	C₂×C₁₉	semidirect
129. #3131	C₂×C₁₉	semidirect
130. #1721	C₂×C₁₉	semidirect
131. #1717	C₂×C₁₉	semidirect
132. #1452	C₂×C₁₉	semidirect
154. #493	C₂²×C₁₉	semidirect
155. #492	C₂²×C₁₉	semidirect
157. #487	C₂²×C₁₉	semidirect
158. #486	C₂²×C₁₉	semidirect
159. #486	C₂²×C₁₉	semidirect
160. #486	C₂²×C₁₉	semidirect
168. #322	C₂²×C₁₉	semidirect
207. C₁₉	#19698	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×#3208	2
C₁₉×#3205	2
C₂×C₁₉×#493	2
C₁₉×#3131	2
C₁₉×#1721	2
C₁₉×#1717	2
C₁₉×#1452	2
#19698	19

Subgroups — C19×#19698 #18025657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#19698 #18025657