Subgroups — C₁₉×#23157 #18029116

Order 4864.

806 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#500	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2365	C₂	1	2	yes	maximal
3	C₁₉×#1988	C₂	1	2	yes	maximal
4	C₁₉×#1942	C₂	1	2	yes	maximal
5	C₁₉×#1918	C₂	1	4	yes	maximal
6	C₄×C₁₉×#107	C₂	1	1	yes	maximal, characteristic
7	C₄×C₁₉×#107	C₂	1	2	yes	maximal
8	C₈×C₄⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₁₉	—	2	2	no	—
11	C₁₉×#500	C₄	1	8	yes	complemented
12	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
13	C₂×C₁₉×#107	C₂²	1	2	yes	—
14	C₂×C₁₉×#107	C₂²	1	2	yes	—
15	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
16	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
19	C₁₉×#391	—	2	2	no	—
20	C₁₉×#389	C₂²	1	4	yes	—
21	C₁₉×#387	—	2	4	no	—
22	C₄×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
23	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
25	C₁₉×#340	C₂²	1	2	yes	—
26	C₁₉×#340	C₂²	1	4	yes	—
27	C₁₉×#340	C₂²	1	4	yes	—
28	C₁₉×#336	C₂²	1	2	yes	—
29	C₂²×Q₈×C₁₉	—	2	2	no	—
30	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
31	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
33	C₁₉×C₄⋊Q₈	—	2	4	no	—
34	C₁₉×C₄⋊Q₈	C₂×C₄	1	8	yes	—
35	C₄×Q₈×C₁₉	—	4	8	no	—
36	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
37	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
38	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
42	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
43	C₂×C₄²×C₁₉	—	2	4	no	—
44	C₂×C₄²×C₁₉	—	2	2	no	—
45	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
46	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
47	C₁₉×#107	—	2	2	no	—
48	C₁₉×#107	C₂×C₄	1	8	yes	—
49	C₁₉×#107	C₂×C₄	1	8	yes	—
50	C₁₉×#107	Q₈	1	4	yes	complemented
51	C₁₉×C₄⋊C₈	—	2	2	no	—
52	C₄×C₈×C₁₉	—	2	4	no	—
53	C₄×C₈×C₁₉	C₂×C₄	1	4	yes	—
54	C₁₉×#95	—	2	2	no	—
55	C₁₉×#95	—	2	4	no	—
56	C₁₉×#95	—	2	4	no	—
57	C₁₉×#95	D₄	1	2	yes	—
58	C₂×Q₈×C₁₉	—	2	8	no	—
59	C₂×Q₈×C₁₉	—	4	4	no	—
60	C₂×Q₈×C₁₉	—	4	4	no	—
61	C₂²×C₄×C₁₉	—	2	2	no	—
62	C₂²×C₄×C₁₉	—	2	4	no	—
63	C₂²×C₄×C₁₉	—	2	2	no	—
64	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
69	C₂×C₈×C₁₉	—	2	1	no	—
70	C₂×C₈×C₁₉	—	2	1	no	—
71	C₂×C₈×C₁₉	—	2	2	no	—
72	C₂×C₈×C₁₉	—	2	2	no	—
73	C₂×C₈×C₁₉	—	2	2	no	—
74	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₁₉	C₂×Q₈	1	2	yes	—
77	C₂×C₈×C₁₉	#41	1	2	yes	—
78	C₄⋊C₄×C₁₉	—	2	8	no	—
79	C₄⋊C₄×C₁₉	—	2	2	no	—
80	C₄⋊C₄×C₁₉	—	2	4	no	—
81	C₄⋊C₄×C₁₉	—	2	2	no	—
82	C₄⋊C₄×C₁₉	—	2	2	no	—
83	C₄⋊C₄×C₁₉	—	2	2	no	—
84	C₄⋊C₄×C₁₉	—	4	8	no	—
85	C₄⋊C₄×C₁₉	D₈	1	4	yes	—
86	C₄⋊C₄×C₁₉	C₂²×C₄	1	8	yes	—
87	C₄⋊C₄×C₁₉	C₂×Q₈	1	4	yes	—
88	C₄²×C₁₉	—	2	2	no	—
89	C₄²×C₁₉	—	2	2	no	—
90	C₄²×C₁₉	—	4	8	no	—
91	C₄²×C₁₉	—	4	4	no	—
92	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
93	C₄²×C₁₉	#41	1	2	yes	—
94	#23157	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
96	Q₈×C₁₉	—	4	16	no	—
97	C₂×C₄×C₁₉	—	2	2	no	—
98	C₂×C₄×C₁₉	—	2	2	no	—
99	C₂×C₄×C₁₉	—	2	1	no	—
100	C₂×C₄×C₁₉	—	2	1	no	—
101	C₂×C₄×C₁₉	—	2	8	no	—
102	C₂×C₄×C₁₉	—	2	2	no	—
103	C₂×C₄×C₁₉	—	2	4	no	—
104	C₂×C₄×C₁₉	—	2	2	no	—
105	C₂×C₄×C₁₉	—	2	2	no	—
106	C₂×C₄×C₁₉	—	4	2	no	—
107	C₂×C₄×C₁₉	—	4	4	no	—
108	C₂×C₄×C₁₉	—	4	4	no	—
109	C₂×C₄×C₁₉	—	4	2	no	—
110	C₂×C₄×C₁₉	—	4	2	no	—
111	C₂×C₄×C₁₉	—	4	2	no	—
112	C₂×C₄×C₁₉	—	4	2	no	—
113	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₁₉	C₂×D₈	1	4	yes	—
116	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
117	C₂×C₄×C₁₉	#135	1	2	yes	—
118	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
123	C₂×C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
124	C₈×C₁₉	—	2	2	no	—
125	C₈×C₁₉	—	4	2	no	—
126	C₈×C₁₉	C₄×Q₈	1	4	yes	complemented
127	C₂×#500	C₂×C₁₉	1	1	yes	characteristic
128	#2365	C₂×C₁₉	1	2	yes	—
129	#1988	C₂×C₁₉	1	2	yes	—
130	#1942	C₂×C₁₉	1	2	yes	—
131	#1918	C₂×C₁₉	1	4	yes	—
132	C₄×#107	C₂×C₁₉	1	1	yes	characteristic
133	C₄×#107	C₂×C₁₉	1	2	yes	—
134	C₈×C₄⋊C₄	C₂×C₁₉	1	1	yes	characteristic
135	C₂²×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂²×C₁₉	#547	1	1	yes	characteristic
137	C₂²×C₁₉	D₄×Q₈	1	1	yes	characteristic
138	C₂²×C₁₉	C₂²×D₈	1	1	yes	characteristic
139	C₂²×C₁₉	C₂×#135	1	1	yes	characteristic
140	C₂²×C₁₉	#578	1	1	yes	characteristic
141	C₂²×C₁₉	#578	1	1	yes	characteristic
142	C₄×C₁₉	—	2	4	no	—
143	C₄×C₁₉	—	2	2	no	—
144	C₄×C₁₉	—	4	2	no	—
145	C₄×C₁₉	—	4	4	no	—
146	C₄×C₁₉	—	4	8	no	—
147	C₄×C₁₉	—	8	2	no	—
148	C₄×C₁₉	C₄×D₈	1	4	yes	—
149	C₄×C₁₉	C₂×C₄×Q₈	1	1	yes	characteristic
150	C₄×C₁₉	#519	1	1	yes	characteristic
151	C₄×C₁₉	#556	1	1	yes	characteristic
152	C₄×C₁₉	#557	1	1	yes	characteristic
153	C₂×C₄⋊Q₈	C₂²×C₁₉	1	2	yes	—
154	C₂×C₄×Q₈	—	2	2	no	—
155	#500	C₄×C₁₉	1	8	yes	complemented
156	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
157	C₂×#107	C₂²×C₁₉	1	2	yes	—
158	C₂×#107	C₂²×C₁₉	1	2	yes	—
159	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
160	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
161	C₂×C₄×C₈	C₂²×C₁₉	1	2	yes	—
162	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
163	#391	—	2	2	no	—
164	#389	C₂²×C₁₉	1	4	yes	—
165	#387	—	2	4	no	—
166	C₄×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
167	C₄×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
168	C₄×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
169	#340	C₂²×C₁₉	1	2	yes	—
170	#340	C₂²×C₁₉	1	4	yes	—
171	#340	C₂²×C₁₉	1	4	yes	—
172	#336	C₂²×C₁₉	1	2	yes	—
173	C₂×C₁₉	#2365	1	1	yes	characteristic
174	C₂×C₁₉	C₂×C₄×D₈	1	1	yes	characteristic
175	C₂×C₁₉	#2950	1	1	yes	characteristic
176	C₂×C₁₉	#3334	1	1	yes	characteristic
177	C₂×C₁₉	#3342	1	1	yes	characteristic
178	C₂×C₁₉	Q₈×D₈	1	1	yes	characteristic
179	C₂×C₁₉	#3383	1	1	yes	characteristic
180	C₂²×Q₈	—	2	2	no	—
181	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
182	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
183	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
184	C₄⋊Q₈	—	2	4	no	—
185	C₄⋊Q₈	C₂×C₄×C₁₉	1	8	yes	—
186	C₄×Q₈	—	4	8	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	2	4	no	—
189	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
190	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
191	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
192	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
193	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
194	C₂×C₄²	—	2	4	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
197	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
198	#107	—	2	2	no	—
199	#107	C₂×C₄×C₁₉	1	8	yes	—
200	#107	C₂×C₄×C₁₉	1	8	yes	—
201	#107	Q₈×C₁₉	1	4	yes	complemented
202	C₄⋊C₈	—	2	2	no	—
203	C₄×C₈	—	2	4	no	—
204	C₄×C₈	C₂×C₄×C₁₉	1	4	yes	—
205	#95	—	2	2	no	—
206	#95	—	2	4	no	—
207	#95	—	2	4	no	—
208	#95	D₄×C₁₉	1	2	yes	—
209	C₁₉	#23157	1	1	yes	characteristic, direct product factor, Sylow
210	C₂×Q₈	—	2	8	no	—
211	C₂×Q₈	—	4	4	no	—
212	C₂×Q₈	—	4	4	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
217	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
219	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
220	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
227	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
228	C₂×C₈	C₂×Q₈×C₁₉	1	2	yes	—
229	C₂×C₈	#41×C₁₉	1	2	yes	—
230	C₄⋊C₄	—	2	8	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	4	8	no	—
237	C₄⋊C₄	D₈×C₁₉	1	4	yes	—
238	C₄⋊C₄	C₂²×C₄×C₁₉	1	8	yes	—
239	C₄⋊C₄	C₂×Q₈×C₁₉	1	4	yes	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	4	8	no	—
243	C₄²	—	4	4	no	—
244	C₄²	C₂²×C₄×C₁₉	1	2	yes	—
245	C₄²	#41×C₁₉	1	2	yes	—
246	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
247	Q₈	—	4	16	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
265	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
266	C₂×C₄	C₂×D₈×C₁₉	1	4	yes	—
267	C₂×C₄	C₂×D₈×C₁₉	1	2	yes	—
268	C₂×C₄	C₁₉×#135	1	2	yes	—
269	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
270	C₂×C₄	C₂²×Q₈×C₁₉	1	1	yes	characteristic
271	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
272	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
273	C₂×C₄	C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
274	C₂×C₄	C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
275	C₈	—	2	2	no	—
276	C₈	—	4	2	no	—
277	C₈	C₄×Q₈×C₁₉	1	4	yes	complemented
278	C₂²	C₂×C₄×D₄×C₁₉	1	1	yes	characteristic
279	C₂²	C₁₉×#547	1	1	yes	characteristic
280	C₂²	D₄×Q₈×C₁₉	1	1	yes	characteristic
281	C₂²	C₂²×D₈×C₁₉	1	1	yes	characteristic
282	C₂²	C₂×C₁₉×#135	1	1	yes	characteristic
283	C₂²	C₁₉×#578	1	1	yes	characteristic
284	C₂²	C₁₉×#578	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	2	4	no	—
287	C₄	—	4	8	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	4	no	—
290	C₄	—	8	2	no	—
291	C₄	C₄×D₈×C₁₉	1	4	yes	—
292	C₄	C₂×C₄×Q₈×C₁₉	1	1	yes	characteristic
293	C₄	C₁₉×#519	1	1	yes	characteristic
294	C₄	C₁₉×#556	1	1	yes	characteristic
295	C₄	C₁₉×#557	1	1	yes	characteristic
296	C₂	C₁₉×#2365	1	1	yes	characteristic
297	C₂	C₂×C₄×D₈×C₁₉	1	1	yes	characteristic
298	C₂	C₁₉×#2950	1	1	yes	characteristic
299	C₂	C₁₉×#3334	1	1	yes	characteristic
300	C₂	C₁₉×#3342	1	1	yes	characteristic
301	C₂	Q₈×D₈×C₁₉	1	1	yes	characteristic
302	C₂	C₁₉×#3383	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₉	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₄×C₄⋊C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₁₉×#500	C₄	semidirect
50. C₁₉×#107	Q₈	semidirect
94. #23157	C₁₉	direct (N × H)
126. C₈×C₁₉	C₄×Q₈	semidirect
155. #500	C₄×C₁₉	semidirect
201. #107	Q₈×C₁₉	semidirect
209. C₁₉	#23157	direct (N × H)
277. C₈	C₄×Q₈×C₁₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₁₉×#500	2
C₁₉×#2365	2
C₁₉×#1988	2
C₁₉×#1942	2
C₁₉×#1918	2
C₄×C₁₉×#107	2
C₈×C₄⋊C₄×C₁₉	2
#23157	19

Subgroups — C19×#23157 #18029116

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#23157 #18029116