Subgroups — C₁₉×#34644 #18040603

Order 4864.

1570 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2667	C₂	1	4	yes	maximal
2	C₁₉×#2666	C₂	1	8	yes	maximal, complemented
3	C₁₉×#2665	C₂	1	4	yes	maximal
4	C₁₉×#2402	C₂	1	2	yes	maximal, complemented
5	C₁₉×#2399	C₂	1	2	yes	maximal, complemented
6	C₁₉×#2396	C₂	1	4	yes	maximal
7	C₁₉×#2295	C₂	1	4	yes	maximal, complemented
8	C₁₉×#2283	C₂	1	2	yes	maximal
9	C₄×C₁₉×#117	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Q₈×C₁₉	C₂²	1	4	yes	—
11	C₂×C₁₉×#117	C₂²	1	2	yes	—
12	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
13	C₁₉×#400	C₂²	1	8	yes	—
14	C₁₉×#400	C₂²	1	4	yes	—
15	C₁₉×#399	C₂²	1	4	yes	—
16	C₁₉×#398	C₂²	1	8	yes	—
17	C₁₉×#398	C₂²	1	8	yes	—
18	C₁₉×#398	C₂²	1	8	yes	—
19	C₁₉×#396	C₂²	1	8	yes	—
20	C₁₉×#395	C₂²	1	8	yes	—
21	C₁₉×#395	C₂²	1	4	yes	—
22	C₁₉×#393	C₂²	1	4	yes	—
23	C₁₉×#391	C₂²	1	8	yes	—
24	C₁₉×#389	C₂²	1	8	yes	—
25	C₁₉×#387	C₂²	1	8	yes	—
26	C₁₉×#387	C₂²	1	8	yes	—
27	C₁₉×#383	C₂²	1	4	yes	—
28	C₁₉×#382	C₂²	1	4	yes	—
29	C₁₉×#381	C₂²	1	2	yes	—
30	C₁₉×#381	C₂²	1	2	yes	—
31	C₁₉×#380	C₂²	1	4	yes	—
32	C₄×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
33	C₄×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
34	C₄×C₄⋊C₄×C₁₉	C₂²	1	8	yes	—
35	C₄×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
36	C₄×C₂²⋊C₄×C₁₉	C₂²	1	4	yes	—
37	C₄×C₂²⋊C₄×C₁₉	C₂²	1	4	yes	—
38	C₁₉×#376	C₂²	1	1	yes	characteristic
39	C₁₉×#376	C₂²	1	1	yes	characteristic
40	C₁₉×#376	C₂²	1	4	yes	—
41	C₄³×C₁₉	C₂²	1	2	yes	—
42	C₂²×Q₈×C₁₉	C₂³	1	4	yes	—
43	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
45	C₄×Q₈×C₁₉	—	2	16	no	—
46	C₁₉×#117	—	2	8	no	—
47	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
48	C₂×C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
49	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
50	C₂×C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
51	C₂×C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
52	C₂×C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
53	C₂×C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
54	C₂×C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
55	C₂×C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
56	C₂×C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
57	C₂×C₄²×C₁₉	Q₈	1	8	yes	complemented
58	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
59	C₂×C₄²×C₁₉	C₂³	1	8	yes	—
60	C₂×C₄²×C₁₉	C₂³	1	8	yes	—
61	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
62	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
65	C₁₉×#95	C₂³	1	4	yes	—
66	C₁₉×#95	C₂³	1	4	yes	—
67	C₁₉×#95	C₂³	1	8	yes	—
68	C₁₉×#95	C₂³	1	4	yes	—
69	C₁₉×#95	C₂³	1	4	yes	—
70	C₁₉×#95	C₂³	1	8	yes	—
71	C₁₉×#95	C₂³	1	4	yes	—
72	C₁₉×#95	C₂³	1	4	yes	—
73	C₁₉×#95	C₂³	1	4	yes	—
74	C₁₉×#95	C₂³	1	4	yes	—
75	C₁₉×#95	C₂³	1	8	yes	—
76	C₁₉×#95	C₂³	1	8	yes	—
77	C₂⁴×C₁₉	C₂⁴	1	1	yes	characteristic
78	C₂×Q₈×C₁₉	—	2	8	no	—
79	C₂×Q₈×C₁₉	—	4	8	no	—
80	C₂²×C₄×C₁₉	—	2	4	no	—
81	C₂²×C₄×C₁₉	—	4	4	no	—
82	C₂²×C₄×C₁₉	C₂×Q₈	1	4	yes	—
83	C₂²×C₄×C₁₉	C₂×Q₈	1	8	yes	—
84	C₂²×C₄×C₁₉	C₂⁴	1	4	yes	—
85	C₂²×C₄×C₁₉	C₂⁴	1	8	yes	—
86	C₂²×C₄×C₁₉	C₂⁴	1	8	yes	—
87	C₂²×C₄×C₁₉	C₂⁴	1	2	yes	—
88	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
89	C₂²×C₄×C₁₉	C₂⁴	1	2	yes	—
90	C₂²×C₄×C₁₉	C₂⁴	1	4	yes	—
91	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
92	C₄⋊C₄×C₁₉	—	2	8	no	—
93	C₄⋊C₄×C₁₉	—	2	8	no	—
94	C₄⋊C₄×C₁₉	—	2	16	no	—
95	C₄⋊C₄×C₁₉	—	4	8	no	—
96	C₄⋊C₄×C₁₉	—	4	8	no	—
97	C₄⋊C₄×C₁₉	—	4	8	no	—
98	C₄⋊C₄×C₁₉	—	4	8	no	—
99	C₂²⋊C₄×C₁₉	—	2	8	no	—
100	C₂²⋊C₄×C₁₉	—	4	4	no	—
101	C₂²⋊C₄×C₁₉	—	4	4	no	—
102	C₄²×C₁₉	—	2	4	no	—
103	C₄²×C₁₉	—	2	16	no	—
104	C₄²×C₁₉	—	4	8	no	—
105	C₄²×C₁₉	C₂×Q₈	1	8	yes	—
106	C₄²×C₁₉	C₂×Q₈	1	4	yes	—
107	C₄²×C₁₉	C₂×Q₈	1	4	yes	—
108	C₄²×C₁₉	#41	1	8	yes	—
109	#34644	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂³×C₁₉	—	2	2	no	—
111	C₂³×C₁₉	—	2	4	no	—
112	C₂³×C₁₉	C₂²×Q₈	1	2	yes	—
113	C₂³×C₁₉	C₂⁵	1	1	yes	characteristic
114	Q₈×C₁₉	—	4	16	no	—
115	C₂×C₄×C₁₉	—	2	2	no	—
116	C₂×C₄×C₁₉	—	2	4	no	—
117	C₂×C₄×C₁₉	—	2	8	no	—
118	C₂×C₄×C₁₉	—	2	4	no	—
119	C₂×C₄×C₁₉	—	2	8	no	—
120	C₂×C₄×C₁₉	—	2	4	no	—
121	C₂×C₄×C₁₉	—	2	8	no	—
122	C₂×C₄×C₁₉	—	2	8	no	—
123	C₂×C₄×C₁₉	—	2	8	no	—
124	C₂×C₄×C₁₉	—	2	8	no	—
125	C₂×C₄×C₁₉	—	2	8	no	—
126	C₂×C₄×C₁₉	—	2	8	no	—
127	C₂×C₄×C₁₉	—	2	8	no	—
128	C₂×C₄×C₁₉	—	4	8	no	—
129	C₂×C₄×C₁₉	C₂²×Q₈	1	2	yes	—
130	C₂×C₄×C₁₉	C₂²×Q₈	1	4	yes	—
131	C₂×C₄×C₁₉	C₂²×Q₈	1	4	yes	—
132	C₂×C₄×C₁₉	C₂²×Q₈	1	2	yes	—
133	C₂×C₄×C₁₉	C₂×#41	1	8	yes	—
134	C₂×C₄×C₁₉	C₂×#41	1	4	yes	—
135	#2667	C₂×C₁₉	1	4	yes	—
136	#2666	C₂×C₁₉	1	8	yes	complemented
137	#2665	C₂×C₁₉	1	4	yes	—
138	#2402	C₂×C₁₉	1	2	yes	complemented
139	#2399	C₂×C₁₉	1	2	yes	complemented
140	#2396	C₂×C₁₉	1	4	yes	—
141	#2295	C₂×C₁₉	1	4	yes	complemented
142	#2283	C₂×C₁₉	1	2	yes	—
143	C₄×#117	C₂×C₁₉	1	1	yes	characteristic
144	C₂²×C₁₉	—	2	2	no	—
145	C₂²×C₁₉	—	2	4	no	—
146	C₂²×C₁₉	—	4	4	no	—
147	C₂²×C₁₉	C₂³×Q₈	1	1	yes	characteristic
148	C₂²×C₁₉	C₂²×#41	1	2	yes	—
149	C₂²×C₁₉	#585	1	4	yes	—
150	C₄×C₁₉	—	2	2	no	—
151	C₄×C₁₉	—	2	8	no	—
152	C₄×C₁₉	—	2	4	no	—
153	C₄×C₁₉	—	2	4	no	—
154	C₄×C₁₉	—	4	4	no	—
155	C₄×C₁₉	—	4	8	no	—
156	C₄×C₁₉	—	4	8	no	—
157	C₄×C₁₉	#533	1	4	yes	—
158	C₂×C₄×Q₈	C₂²×C₁₉	1	4	yes	—
159	C₂×#117	C₂²×C₁₉	1	2	yes	—
160	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
161	#400	C₂²×C₁₉	1	4	yes	—
162	#400	C₂²×C₁₉	1	8	yes	—
163	#399	C₂²×C₁₉	1	4	yes	—
164	#398	C₂²×C₁₉	1	8	yes	—
165	#398	C₂²×C₁₉	1	8	yes	—
166	#398	C₂²×C₁₉	1	8	yes	—
167	#396	C₂²×C₁₉	1	8	yes	—
168	#395	C₂²×C₁₉	1	4	yes	—
169	#395	C₂²×C₁₉	1	8	yes	—
170	#393	C₂²×C₁₉	1	4	yes	—
171	#391	C₂²×C₁₉	1	8	yes	—
172	#389	C₂²×C₁₉	1	8	yes	—
173	#387	C₂²×C₁₉	1	8	yes	—
174	#387	C₂²×C₁₉	1	8	yes	—
175	#383	C₂²×C₁₉	1	4	yes	—
176	#382	C₂²×C₁₉	1	4	yes	—
177	#381	C₂²×C₁₉	1	2	yes	—
178	#381	C₂²×C₁₉	1	2	yes	—
179	#380	C₂²×C₁₉	1	4	yes	—
180	C₄×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
181	C₄×C₄⋊C₄	C₂²×C₁₉	1	8	yes	—
182	C₄×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
183	C₄×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
184	C₄×C₂²⋊C₄	C₂²×C₁₉	1	4	yes	—
185	C₄×C₂²⋊C₄	C₂²×C₁₉	1	4	yes	—
186	#376	C₂²×C₁₉	1	1	yes	characteristic
187	#376	C₂²×C₁₉	1	1	yes	characteristic
188	#376	C₂²×C₁₉	1	4	yes	—
189	C₄³	C₂²×C₁₉	1	2	yes	—
190	C₂×C₁₉	—	4	2	no	—
191	C₂×C₁₉	C₂×#533	1	1	yes	characteristic
192	C₂×C₁₉	#3455	1	4	yes	—
193	C₂×C₁₉	#3458	1	2	yes	—
194	C₂²×Q₈	C₂³×C₁₉	1	4	yes	—
195	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
196	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
197	C₄×Q₈	—	2	16	no	—
198	#117	—	2	8	no	—
199	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
200	C₂×C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
201	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
202	C₂×C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
203	C₂×C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
204	C₂×C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
205	C₂×C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
206	C₂×C₂²⋊C₄	C₂³×C₁₉	1	4	yes	—
207	C₂×C₂²⋊C₄	C₂³×C₁₉	1	4	yes	—
208	C₂×C₂²⋊C₄	C₂³×C₁₉	1	4	yes	—
209	C₂×C₄²	Q₈×C₁₉	1	8	yes	complemented
210	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
211	C₂×C₄²	C₂³×C₁₉	1	8	yes	—
212	C₂×C₄²	C₂³×C₁₉	1	8	yes	—
213	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
214	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
215	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
216	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
217	#95	C₂³×C₁₉	1	4	yes	—
218	#95	C₂³×C₁₉	1	4	yes	—
219	#95	C₂³×C₁₉	1	8	yes	—
220	#95	C₂³×C₁₉	1	4	yes	—
221	#95	C₂³×C₁₉	1	4	yes	—
222	#95	C₂³×C₁₉	1	8	yes	—
223	#95	C₂³×C₁₉	1	4	yes	—
224	#95	C₂³×C₁₉	1	4	yes	—
225	#95	C₂³×C₁₉	1	4	yes	—
226	#95	C₂³×C₁₉	1	4	yes	—
227	#95	C₂³×C₁₉	1	8	yes	—
228	#95	C₂³×C₁₉	1	8	yes	—
229	C₁₉	#34644	1	1	yes	characteristic, direct product factor, Sylow
230	C₂⁴	C₂⁴×C₁₉	1	1	yes	characteristic
231	C₂×Q₈	—	2	8	no	—
232	C₂×Q₈	—	4	8	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	4	4	no	—
235	C₂²×C₄	C₂×Q₈×C₁₉	1	4	yes	—
236	C₂²×C₄	C₂×Q₈×C₁₉	1	8	yes	—
237	C₂²×C₄	C₂⁴×C₁₉	1	8	yes	—
238	C₂²×C₄	C₂⁴×C₁₉	1	4	yes	—
239	C₂²×C₄	C₂⁴×C₁₉	1	8	yes	—
240	C₂²×C₄	C₂⁴×C₁₉	1	2	yes	—
241	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
242	C₂²×C₄	C₂⁴×C₁₉	1	2	yes	—
243	C₂²×C₄	C₂⁴×C₁₉	1	4	yes	—
244	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
245	C₄⋊C₄	—	2	8	no	—
246	C₄⋊C₄	—	2	8	no	—
247	C₄⋊C₄	—	2	16	no	—
248	C₄⋊C₄	—	4	8	no	—
249	C₄⋊C₄	—	4	8	no	—
250	C₄⋊C₄	—	4	8	no	—
251	C₄⋊C₄	—	4	8	no	—
252	C₂²⋊C₄	—	2	8	no	—
253	C₂²⋊C₄	—	4	4	no	—
254	C₂²⋊C₄	—	4	4	no	—
255	C₄²	—	2	4	no	—
256	C₄²	—	2	16	no	—
257	C₄²	—	4	8	no	—
258	C₄²	C₂×Q₈×C₁₉	1	8	yes	—
259	C₄²	C₂×Q₈×C₁₉	1	4	yes	—
260	C₄²	C₂×Q₈×C₁₉	1	4	yes	—
261	C₄²	#41×C₁₉	1	8	yes	—
262	C₂³	—	2	2	no	—
263	C₂³	—	2	4	no	—
264	C₂³	C₂²×Q₈×C₁₉	1	2	yes	—
265	C₂³	C₂⁵×C₁₉	1	1	yes	characteristic
266	Q₈	—	4	16	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	8	no	—
279	C₂×C₄	—	2	8	no	—
280	C₂×C₄	—	4	8	no	—
281	C₂×C₄	C₂²×Q₈×C₁₉	1	2	yes	—
282	C₂×C₄	C₂²×Q₈×C₁₉	1	4	yes	—
283	C₂×C₄	C₂²×Q₈×C₁₉	1	4	yes	—
284	C₂×C₄	C₂²×Q₈×C₁₉	1	2	yes	—
285	C₂×C₄	C₂×#41×C₁₉	1	8	yes	—
286	C₂×C₄	C₂×#41×C₁₉	1	4	yes	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	4	4	no	—
290	C₂²	C₂³×Q₈×C₁₉	1	1	yes	characteristic
291	C₂²	C₂²×#41×C₁₉	1	2	yes	—
292	C₂²	C₁₉×#585	1	4	yes	—
293	C₄	—	2	2	no	—
294	C₄	—	2	8	no	—
295	C₄	—	2	4	no	—
296	C₄	—	2	4	no	—
297	C₄	—	4	8	no	—
298	C₄	—	4	4	no	—
299	C₄	—	4	8	no	—
300	C₄	C₁₉×#533	1	4	yes	—
301	C₂	—	4	2	no	—
302	C₂	C₂×C₁₉×#533	1	1	yes	characteristic
303	C₂	C₁₉×#3455	1	4	yes	—
304	C₂	C₁₉×#3458	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₉
socle	C₂³×C₁₉	C₂⁵
prime radical	C₂⁴×C₁₉	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#2666	C₂	semidirect
4. C₁₉×#2402	C₂	semidirect
5. C₁₉×#2399	C₂	semidirect
7. C₁₉×#2295	C₂	semidirect
57. C₂×C₄²×C₁₉	Q₈	semidirect
109. #34644	C₁₉	direct (N × H)
136. #2666	C₂×C₁₉	semidirect
138. #2402	C₂×C₁₉	semidirect
139. #2399	C₂×C₁₉	semidirect
141. #2295	C₂×C₁₉	semidirect
209. C₂×C₄²	Q₈×C₁₉	semidirect
229. C₁₉	#34644	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×#2667	2
C₁₉×#2666	2
C₁₉×#2665	2
C₁₉×#2402	2
C₁₉×#2399	2
C₁₉×#2396	2
C₁₉×#2295	2
C₁₉×#2283	2
C₄×C₁₉×#117	2
#34644	19

Subgroups — C19×#34644 #18040603

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#34644 #18040603