Subgroups — C₁₉×#23628 #18029587

Order 4864.

1226 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#418	C₂	1	2	yes	maximal
2	C₂×C₁₉×#390	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2285	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#2073	C₂	1	4	yes	maximal, complemented
5	C₁₉×#2071	C₂	1	4	yes	maximal, complemented
6	C₁₉×#1736	C₂	1	2	yes	maximal, complemented
7	C₁₉×#1728	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₄×C₁₉	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
10	C₂×C₁₉×#117	C₂²	1	2	yes	—
11	C₂²×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
12	C₂²×C₂²⋊C₄×C₁₉	—	2	2	no	—
13	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
14	C₁₉×#418	—	2	8	no	—
15	C₂×C₁₉×#102	C₂²	1	8	yes	—
16	C₁₉×#390	—	2	2	no	—
17	C₁₉×#390	C₂²	1	2	yes	—
18	C₁₉×#390	C₂²	1	2	yes	—
19	C₁₉×#384	C₂²	1	2	yes	complemented
20	C₁₉×#382	C₂²	1	2	yes	complemented
21	C₁₉×#380	—	2	2	no	—
22	C₄×C₄⋊C₄×C₁₉	—	2	2	no	—
23	C₁₉×#340	C₂²	1	8	yes	complemented
24	C₁₉×#336	C₂²	1	2	yes	complemented
25	C₁₉×#336	C₂²	1	2	yes	complemented
26	C₂⁵×C₁₉	—	2	1	no	—
27	C₂²×D₄×C₁₉	—	2	1	no	—
28	C₂²×D₄×C₁₉	—	2	4	no	—
29	C₂²×D₄×C₁₉	—	2	4	no	—
30	C₂²×D₄×C₁₉	—	2	2	no	—
31	C₂²×D₄×C₁₉	D₄	1	4	yes	—
32	C₂²×D₄×C₁₉	C₂³	1	2	yes	—
33	C₂³×C₄×C₁₉	—	2	1	no	—
34	C₂³×C₄×C₁₉	D₄	1	2	yes	—
35	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₁₉	—	2	4	no	—
37	C₂×M₁₆×C₁₉	—	2	4	no	—
38	C₂×M₁₆×C₁₉	—	2	4	no	—
39	C₂²×C₈×C₁₉	C₂³	1	4	yes	—
40	C₁₉×#117	—	2	4	no	—
41	C₁₉×#117	—	2	4	no	—
42	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
43	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
45	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
46	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
48	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₁₉	—	2	4	no	—
50	C₂×C₂²⋊C₄×C₁₉	—	2	4	no	—
51	C₂×C₂²⋊C₄×C₁₉	—	4	8	no	—
52	C₂×C₄²×C₁₉	—	2	4	no	—
53	C₂×C₄²×C₁₉	—	2	2	no	—
54	C₂×C₄²×C₁₉	D₄	1	2	yes	—
55	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
57	C₁₉×#102	—	2	16	no	—
58	C₁₉×#95	—	2	4	no	—
59	C₂⁴×C₁₉	—	2	4	no	—
60	C₂⁴×C₁₉	—	2	2	no	—
61	C₂⁴×C₁₉	—	4	4	no	—
62	C₂⁴×C₁₉	—	4	4	no	—
63	C₂⁴×C₁₉	—	4	2	no	—
64	C₂⁴×C₁₉	—	4	2	no	—
65	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
66	C₂×D₄×C₁₉	—	2	4	no	—
67	C₂×D₄×C₁₉	—	2	16	no	—
68	C₂×D₄×C₁₉	—	2	4	no	—
69	C₂×D₄×C₁₉	—	2	4	no	—
70	C₂×D₄×C₁₉	—	2	2	no	—
71	C₂×D₄×C₁₉	—	4	2	no	—
72	C₂×D₄×C₁₉	—	4	2	no	—
73	C₂×D₄×C₁₉	—	4	8	no	—
74	C₂×D₄×C₁₉	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₁₉	—	2	4	no	—
76	C₂²×C₄×C₁₉	—	2	4	no	—
77	C₂²×C₄×C₁₉	—	2	4	no	—
78	C₂²×C₄×C₁₉	—	2	2	no	—
79	C₂²×C₄×C₁₉	—	4	2	no	—
80	C₂²×C₄×C₁₉	—	4	4	no	—
81	C₂²×C₄×C₁₉	—	4	2	no	—
82	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₁₉	#41	1	2	yes	—
87	C₂²×C₄×C₁₉	#41	1	4	yes	—
88	C₂²×C₄×C₁₉	#41	1	4	yes	—
89	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
90	M₁₆×C₁₉	—	4	8	no	—
91	C₂×C₈×C₁₉	—	2	4	no	—
92	C₂×C₈×C₁₉	—	2	4	no	—
93	C₂×C₈×C₁₉	—	2	4	no	—
94	C₄⋊C₄×C₁₉	—	2	4	no	—
95	C₄⋊C₄×C₁₉	—	2	4	no	—
96	C₄⋊C₄×C₁₉	—	2	4	no	—
97	C₄⋊C₄×C₁₉	—	4	4	no	—
98	C₄⋊C₄×C₁₉	—	4	2	no	—
99	C₄⋊C₄×C₁₉	—	4	4	no	—
100	C₂²⋊C₄×C₁₉	—	4	8	no	—
101	C₂²⋊C₄×C₁₉	—	4	4	no	—
102	C₂²⋊C₄×C₁₉	—	4	8	no	—
103	C₄²×C₁₉	—	2	4	no	—
104	C₄²×C₁₉	—	2	2	no	—
105	C₄²×C₁₉	—	2	2	no	—
106	C₄²×C₁₉	—	4	4	no	—
107	#23628	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂³×C₁₉	—	2	2	no	—
109	C₂³×C₁₉	—	2	2	no	—
110	C₂³×C₁₉	—	2	4	no	—
111	C₂³×C₁₉	—	4	16	no	—
112	C₂³×C₁₉	—	4	2	no	—
113	C₂³×C₁₉	—	4	2	no	—
114	C₂³×C₁₉	—	4	4	no	—
115	C₂³×C₁₉	—	4	4	no	—
116	C₂³×C₁₉	—	4	8	no	—
117	C₂³×C₁₉	—	8	8	no	—
118	C₂³×C₁₉	—	8	4	no	—
119	C₂³×C₁₉	—	8	4	no	—
120	C₂³×C₁₉	#121	1	2	yes	—
121	C₂³×C₁₉	#123	1	2	yes	—
122	C₂³×C₁₉	#124	1	2	yes	—
123	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
124	D₄×C₁₉	—	2	8	no	—
125	D₄×C₁₉	—	4	4	no	—
126	D₄×C₁₉	—	4	8	no	—
127	C₂×C₄×C₁₉	—	2	2	no	—
128	C₂×C₄×C₁₉	—	2	2	no	—
129	C₂×C₄×C₁₉	—	2	2	no	—
130	C₂×C₄×C₁₉	—	2	2	no	—
131	C₂×C₄×C₁₉	—	2	2	no	—
132	C₂×C₄×C₁₉	—	4	8	no	—
133	C₂×C₄×C₁₉	—	4	4	no	—
134	C₂×C₄×C₁₉	—	4	2	no	—
135	C₂×C₄×C₁₉	—	4	4	no	—
136	C₂×C₄×C₁₉	—	4	4	no	—
137	C₂×C₄×C₁₉	—	4	4	no	—
138	C₂×C₄×C₁₉	—	8	4	no	—
139	C₂×C₄×C₁₉	#121	1	2	yes	—
140	C₂×C₄×C₁₉	#123	1	8	yes	—
141	C₂×C₄×C₁₉	#123	1	2	yes	—
142	C₂×C₄×C₁₉	#124	1	2	yes	—
143	C₂×C₄×C₁₉	#126	1	8	yes	—
144	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	4	yes	—
145	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
147	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
148	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
149	C₈×C₁₉	—	4	4	no	—
150	C₂×#418	C₂×C₁₉	1	2	yes	—
151	C₂×#390	C₂×C₁₉	1	1	yes	characteristic
152	#2285	C₂×C₁₉	1	1	yes	characteristic, complemented
153	#2073	C₂×C₁₉	1	4	yes	complemented
154	#2071	C₂×C₁₉	1	4	yes	complemented
155	#1736	C₂×C₁₉	1	2	yes	complemented
156	#1728	C₂×C₁₉	1	1	yes	characteristic, complemented
157	C₂²×C₁₉	—	2	4	no	—
158	C₂²×C₁₉	—	2	2	no	—
159	C₂²×C₁₉	—	2	2	no	—
160	C₂²×C₁₉	—	2	4	no	—
161	C₂²×C₁₉	—	4	4	no	—
162	C₂²×C₁₉	—	4	8	no	—
163	C₂²×C₁₉	—	8	16	no	—
164	C₂²×C₁₉	—	8	4	no	—
165	C₂²×C₁₉	—	8	2	no	—
166	C₂²×C₁₉	—	8	2	no	—
167	C₂²×C₁₉	#397	1	4	yes	—
168	C₂²×C₁₉	C₂×#121	1	1	yes	characteristic
169	C₂²×C₁₉	C₂×#123	1	1	yes	characteristic
170	C₂²×C₁₉	C₂×#124	1	1	yes	characteristic
171	C₂²×C₁₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
172	C₂²×C₁₉	#575	1	2	yes	—
173	C₄×C₁₉	—	2	4	no	—
174	C₄×C₁₉	—	4	2	no	—
175	C₄×C₁₉	—	8	2	no	—
176	C₄×C₁₉	—	8	4	no	—
177	C₄×C₁₉	#397	1	4	yes	—
178	C₄×C₁₉	C₂×#123	1	2	yes	—
179	C₄×C₁₉	C₂×#126	1	2	yes	—
180	C₂³×D₄	C₂²×C₁₉	1	1	yes	characteristic
181	C₂²×M₁₆	C₂²×C₁₉	1	2	yes	—
182	C₂×#117	C₂²×C₁₉	1	2	yes	—
183	C₂²×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
184	C₂²×C₂²⋊C₄	—	2	2	no	—
185	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
186	#418	—	2	8	no	—
187	C₂×#102	C₂²×C₁₉	1	8	yes	—
188	#390	—	2	2	no	—
189	#390	C₂²×C₁₉	1	2	yes	—
190	#390	C₂²×C₁₉	1	2	yes	—
191	#384	C₂²×C₁₉	1	2	yes	complemented
192	#382	C₂²×C₁₉	1	2	yes	complemented
193	#380	—	2	2	no	—
194	C₄×C₄⋊C₄	—	2	2	no	—
195	#340	C₂²×C₁₉	1	8	yes	complemented
196	#336	C₂²×C₁₉	1	2	yes	complemented
197	#336	C₂²×C₁₉	1	2	yes	complemented
198	C₂×C₁₉	—	2	4	no	—
199	C₂×C₁₉	—	8	4	no	—
200	C₂×C₁₉	C₂×#397	1	1	yes	characteristic
201	C₂×C₁₉	#3036	1	2	yes	—
202	C₂×C₁₉	#3094	1	2	yes	—
203	C₂×C₁₉	#3138	1	1	yes	characteristic
204	C₂×C₁₉	#3140	1	1	yes	characteristic
205	C₂⁵	—	2	1	no	—
206	C₂²×D₄	—	2	1	no	—
207	C₂²×D₄	—	2	4	no	—
208	C₂²×D₄	—	2	4	no	—
209	C₂²×D₄	—	2	2	no	—
210	C₂²×D₄	D₄×C₁₉	1	4	yes	—
211	C₂²×D₄	C₂³×C₁₉	1	2	yes	—
212	C₂³×C₄	—	2	1	no	—
213	C₂³×C₄	D₄×C₁₉	1	2	yes	—
214	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
215	C₂×M₁₆	—	2	4	no	—
216	C₂×M₁₆	—	2	4	no	—
217	C₂×M₁₆	—	2	4	no	—
218	C₂²×C₈	C₂³×C₁₉	1	4	yes	—
219	#117	—	2	4	no	—
220	#117	—	2	4	no	—
221	C₂×C₄⋊C₄	—	2	4	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	—	2	2	no	—
224	C₂×C₄⋊C₄	—	2	2	no	—
225	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
226	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
227	C₂×C₂²⋊C₄	—	2	2	no	—
228	C₂×C₂²⋊C₄	—	2	4	no	—
229	C₂×C₂²⋊C₄	—	2	4	no	—
230	C₂×C₂²⋊C₄	—	4	8	no	—
231	C₂×C₄²	—	2	4	no	—
232	C₂×C₄²	—	2	2	no	—
233	C₂×C₄²	D₄×C₁₉	1	2	yes	—
234	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
235	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
236	#102	—	2	16	no	—
237	#95	—	2	4	no	—
238	C₁₉	#23628	1	1	yes	characteristic, direct product factor, Sylow
239	C₂⁴	—	2	4	no	—
240	C₂⁴	—	2	2	no	—
241	C₂⁴	—	4	4	no	—
242	C₂⁴	—	4	4	no	—
243	C₂⁴	—	4	2	no	—
244	C₂⁴	—	4	2	no	—
245	C₂⁴	C₂×D₄×C₁₉	1	1	yes	characteristic
246	C₂×D₄	—	2	4	no	—
247	C₂×D₄	—	2	16	no	—
248	C₂×D₄	—	2	4	no	—
249	C₂×D₄	—	2	4	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	4	2	no	—
252	C₂×D₄	—	4	2	no	—
253	C₂×D₄	—	4	8	no	—
254	C₂×D₄	C₂×D₄×C₁₉	1	4	yes	—
255	C₂²×C₄	—	2	4	no	—
256	C₂²×C₄	—	2	4	no	—
257	C₂²×C₄	—	2	4	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	4	2	no	—
260	C₂²×C₄	—	4	4	no	—
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
263	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
264	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
265	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
266	C₂²×C₄	#41×C₁₉	1	2	yes	—
267	C₂²×C₄	#41×C₁₉	1	4	yes	—
268	C₂²×C₄	#41×C₁₉	1	4	yes	—
269	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
270	M₁₆	—	4	8	no	—
271	C₂×C₈	—	2	4	no	—
272	C₂×C₈	—	2	4	no	—
273	C₂×C₈	—	2	4	no	—
274	C₄⋊C₄	—	2	4	no	—
275	C₄⋊C₄	—	2	4	no	—
276	C₄⋊C₄	—	2	4	no	—
277	C₄⋊C₄	—	4	4	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	4	no	—
280	C₂²⋊C₄	—	4	8	no	—
281	C₂²⋊C₄	—	4	4	no	—
282	C₂²⋊C₄	—	4	8	no	—
283	C₄²	—	2	4	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	4	4	no	—
287	C₂³	—	2	2	no	—
288	C₂³	—	2	2	no	—
289	C₂³	—	2	4	no	—
290	C₂³	—	4	16	no	—
291	C₂³	—	4	8	no	—
292	C₂³	—	4	2	no	—
293	C₂³	—	4	2	no	—
294	C₂³	—	4	4	no	—
295	C₂³	—	4	4	no	—
296	C₂³	—	8	8	no	—
297	C₂³	—	8	4	no	—
298	C₂³	—	8	4	no	—
299	C₂³	C₁₉×#121	1	2	yes	—
300	C₂³	C₁₉×#123	1	2	yes	—
301	C₂³	C₁₉×#124	1	2	yes	—
302	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
303	D₄	—	2	8	no	—
304	D₄	—	4	8	no	—
305	D₄	—	4	4	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	8	no	—
312	C₂×C₄	—	4	4	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	4	no	—
315	C₂×C₄	—	4	4	no	—
316	C₂×C₄	—	4	4	no	—
317	C₂×C₄	—	8	4	no	—
318	C₂×C₄	C₁₉×#121	1	2	yes	—
319	C₂×C₄	C₁₉×#123	1	2	yes	—
320	C₂×C₄	C₁₉×#123	1	8	yes	—
321	C₂×C₄	C₁₉×#124	1	2	yes	—
322	C₂×C₄	C₁₉×#126	1	8	yes	—
323	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	4	yes	—
324	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
325	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
326	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
327	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
328	C₈	—	4	4	no	—
329	C₂²	—	2	4	no	—
330	C₂²	—	2	4	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	2	2	no	—
333	C₂²	—	4	8	no	—
334	C₂²	—	4	4	no	—
335	C₂²	—	8	16	no	—
336	C₂²	—	8	4	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	C₁₉×#397	1	4	yes	—
340	C₂²	C₂×C₁₉×#121	1	1	yes	characteristic
341	C₂²	C₂×C₁₉×#123	1	1	yes	characteristic
342	C₂²	C₂×C₁₉×#124	1	1	yes	characteristic
343	C₂²	C₂×C₁₉×AGL(1,ℤ/8)	1	2	yes	—
344	C₂²	C₁₉×#575	1	2	yes	—
345	C₄	—	2	4	no	—
346	C₄	—	4	2	no	—
347	C₄	—	8	4	no	—
348	C₄	—	8	2	no	—
349	C₄	C₁₉×#397	1	4	yes	—
350	C₄	C₂×C₁₉×#123	1	2	yes	—
351	C₄	C₂×C₁₉×#126	1	2	yes	—
352	C₂	—	2	4	no	—
353	C₂	—	8	4	no	—
354	C₂	C₂×C₁₉×#397	1	1	yes	characteristic
355	C₂	C₁₉×#3036	1	2	yes	—
356	C₂	C₁₉×#3094	1	2	yes	—
357	C₂	C₁₉×#3138	1	1	yes	characteristic
358	C₂	C₁₉×#3140	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
prime radical	C₂³×D₄×C₁₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂²×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₉×#2285	C₂	semidirect
4. C₁₉×#2073	C₂	semidirect
5. C₁₉×#2071	C₂	semidirect
6. C₁₉×#1736	C₂	semidirect
7. C₁₉×#1728	C₂	semidirect
19. C₁₉×#384	C₂²	semidirect
20. C₁₉×#382	C₂²	semidirect
23. C₁₉×#340	C₂²	semidirect
24. C₁₉×#336	C₂²	semidirect
25. C₁₉×#336	C₂²	semidirect
107. #23628	C₁₉	direct (N × H)
152. #2285	C₂×C₁₉	semidirect
153. #2073	C₂×C₁₉	semidirect
154. #2071	C₂×C₁₉	semidirect
155. #1736	C₂×C₁₉	semidirect
156. #1728	C₂×C₁₉	semidirect
191. #384	C₂²×C₁₉	semidirect
192. #382	C₂²×C₁₉	semidirect
195. #340	C₂²×C₁₉	semidirect
196. #336	C₂²×C₁₉	semidirect
197. #336	C₂²×C₁₉	semidirect
238. C₁₉	#23628	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#418	2
C₂×C₁₉×#390	2
C₁₉×#2285	2
C₁₉×#2073	2
C₁₉×#2071	2
C₁₉×#1736	2
C₁₉×#1728	2
#23628	19

Subgroups — C19×#23628 #18029587

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#23628 #18029587