Subgroups — C₁₉×#29545 #18035504

Order 4864.

746 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2319	C₂	1	2	yes	maximal, complemented
2	C₁₉×#2086	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#2065	C₂	1	2	yes	maximal, complemented
4	C₁₉×#2054	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1925	C₂	1	2	yes	maximal, complemented
6	C₁₉×#1919	C₂	1	2	yes	maximal, complemented
7	C₁₉×#1879	C₂	1	2	yes	maximal
8	C₁₉×#1864	C₂	1	2	yes	maximal, complemented
9	C₁₉×#1832	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₈×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₄×C₁₉	C₂²	1	2	yes	—
12	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
13	C₂×C₁₉×#106	C₂²	1	2	yes	—
14	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₁₉×#102	C₂²	1	2	yes	—
17	C₂×C₁₉×#102	C₂²	1	2	yes	—
18	C₂×C₁₉×#102	C₂²	1	2	yes	—
19	C₁₉×#390	C₂²	1	2	yes	—
20	C₁₉×#390	C₂²	1	2	yes	—
21	C₁₉×#389	C₂²	1	2	yes	—
22	C₁₉×#389	C₂²	1	2	yes	—
23	C₁₉×#388	—	2	2	no	—
24	C₁₉×#388	—	2	2	no	—
25	C₁₉×#381	—	2	2	no	—
26	C₁₉×#380	—	2	2	no	—
27	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
28	C₁₉×#376	C₂²	1	1	yes	characteristic
29	C₁₉×#340	C₂²	1	1	yes	characteristic
30	C₁₉×#340	C₂²	1	2	yes	—
31	C₁₉×#340	C₂²	1	2	yes	—
32	C₁₉×#340	C₂²	1	1	yes	characteristic
33	C₁₉×#336	C₂²	1	2	yes	—
34	C₁₉×#336	C₂²	1	2	yes	—
35	C₂²×D₄×C₁₉	C₂³	1	2	yes	—
36	C₂³×C₄×C₁₉	—	2	2	no	—
37	C₂×D₈×C₁₉	—	2	4	no	—
38	C₂×D₈×C₁₉	—	4	1	no	—
39	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
42	C₄×D₄×C₁₉	—	2	4	no	—
43	C₄×D₄×C₁₉	—	2	4	no	—
44	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
45	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
49	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
51	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
53	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
54	C₂×C₄²×C₁₉	—	2	2	no	—
55	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
56	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
57	C₁₉×#107	—	4	1	no	—
58	C₁₉×#106	—	2	4	no	—
59	C₁₉×C₄⋊C₈	—	4	1	no	—
60	C₁₉×C₄⋊C₈	—	4	1	no	—
61	C₁₉×#102	—	2	4	no	—
62	C₁₉×#102	—	2	4	no	—
63	C₁₉×#102	—	4	2	no	—
64	C₁₉×#95	—	2	2	no	—
65	C₁₉×#95	—	2	2	no	—
66	C₁₉×#95	—	2	2	no	—
67	C₁₉×#95	D₄	1	2	yes	—
68	C₁₉×#95	D₄	1	1	yes	characteristic
69	C₁₉×#95	D₄	1	1	yes	characteristic
70	C₂⁴×C₁₉	—	2	2	no	—
71	C₂×D₄×C₁₉	—	2	2	no	—
72	C₂×D₄×C₁₉	—	2	2	no	—
73	C₂×D₄×C₁₉	—	2	2	no	—
74	C₂×D₄×C₁₉	—	2	2	no	—
75	C₂×D₄×C₁₉	—	2	2	no	—
76	C₂×D₄×C₁₉	#41	1	4	yes	—
77	C₂²×C₄×C₁₉	—	2	2	no	—
78	C₂²×C₄×C₁₉	—	2	8	no	—
79	C₂²×C₄×C₁₉	—	2	2	no	—
80	C₂²×C₄×C₁₉	—	2	2	no	—
81	C₂²×C₄×C₁₉	—	8	2	no	—
82	C₂²×C₄×C₁₉	—	8	2	no	—
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	1	yes	characteristic
86	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
88	D₈×C₁₉	—	4	4	no	—
89	C₂×C₈×C₁₉	—	2	2	no	—
90	C₂×C₈×C₁₉	—	2	2	no	—
91	C₂×C₈×C₁₉	—	2	2	no	—
92	C₂×C₈×C₁₉	—	2	2	no	—
93	C₂×C₈×C₁₉	—	2	1	no	—
94	C₂×C₈×C₁₉	—	2	2	no	—
95	C₂×C₈×C₁₉	—	2	1	no	—
96	C₄⋊C₄×C₁₉	—	2	2	no	—
97	C₄⋊C₄×C₁₉	—	2	2	no	—
98	C₄⋊C₄×C₁₉	—	2	2	no	—
99	C₄⋊C₄×C₁₉	—	2	2	no	—
100	C₄⋊C₄×C₁₉	—	2	2	no	—
101	C₄⋊C₄×C₁₉	—	8	2	no	—
102	C₄⋊C₄×C₁₉	—	8	2	no	—
103	C₄⋊C₄×C₁₉	#41	1	4	yes	—
104	C₂²⋊C₄×C₁₉	—	2	8	no	—
105	C₂²⋊C₄×C₁₉	—	8	2	no	—
106	C₂²⋊C₄×C₁₉	—	8	2	no	—
107	C₂²⋊C₄×C₁₉	—	8	2	no	—
108	C₂²⋊C₄×C₁₉	—	8	2	no	—
109	C₄²×C₁₉	—	2	2	no	—
110	C₄²×C₁₉	—	2	2	no	—
111	C₄²×C₁₉	—	2	1	no	—
112	C₄²×C₁₉	—	2	1	no	—
113	C₄²×C₁₉	—	8	2	no	—
114	#29545	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂³×C₁₉	—	2	4	no	—
116	C₂³×C₁₉	—	4	2	no	—
117	C₂³×C₁₉	—	4	2	no	—
118	C₂³×C₁₉	—	4	2	no	—
119	C₂³×C₁₉	—	4	2	no	—
120	C₂³×C₁₉	—	4	2	no	—
121	C₂³×C₁₉	—	4	2	no	—
122	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
123	D₄×C₁₉	—	2	4	no	—
124	D₄×C₁₉	—	2	4	no	—
125	D₄×C₁₉	—	4	2	no	—
126	D₄×C₁₉	—	4	2	no	—
127	C₂×C₄×C₁₉	—	2	2	no	—
128	C₂×C₄×C₁₉	—	2	2	no	—
129	C₂×C₄×C₁₉	—	2	4	no	—
130	C₂×C₄×C₁₉	—	2	1	no	—
131	C₂×C₄×C₁₉	—	2	1	no	—
132	C₂×C₄×C₁₉	—	2	2	no	—
133	C₂×C₄×C₁₉	—	2	1	no	—
134	C₂×C₄×C₁₉	—	2	2	no	—
135	C₂×C₄×C₁₉	—	2	1	no	—
136	C₂×C₄×C₁₉	—	2	2	no	—
137	C₂×C₄×C₁₉	—	2	2	no	—
138	C₂×C₄×C₁₉	—	2	2	no	—
139	C₂×C₄×C₁₉	—	4	2	no	—
140	C₂×C₄×C₁₉	—	4	2	no	—
141	C₂×C₄×C₁₉	—	4	2	no	—
142	C₂×C₄×C₁₉	—	4	2	no	—
143	C₂×C₄×C₁₉	—	4	2	no	—
144	C₂×C₄×C₁₉	—	4	2	no	—
145	C₂×C₄×C₁₉	—	4	2	no	—
146	C₂×C₄×C₁₉	—	4	2	no	—
147	C₂×C₄×C₁₉	—	8	8	no	—
148	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
149	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
150	C₂×C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
151	C₂×C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
152	C₈×C₁₉	—	4	1	no	—
153	C₈×C₁₉	—	4	1	no	—
154	C₈×C₁₉	—	4	2	no	—
155	#2319	C₂×C₁₉	1	2	yes	complemented
156	#2086	C₂×C₁₉	1	1	yes	characteristic, complemented
157	#2065	C₂×C₁₉	1	2	yes	complemented
158	#2054	C₂×C₁₉	1	1	yes	characteristic
159	#1925	C₂×C₁₉	1	2	yes	complemented
160	#1919	C₂×C₁₉	1	2	yes	complemented
161	#1879	C₂×C₁₉	1	2	yes	—
162	#1864	C₂×C₁₉	1	2	yes	complemented
163	#1832	C₂×C₁₉	1	1	yes	characteristic, complemented
164	C₂²×C₁₉	—	4	4	no	—
165	C₂²×C₁₉	—	4	4	no	—
166	C₂²×C₁₉	—	4	4	no	—
167	C₂²×C₁₉	—	8	2	no	—
168	C₂²×C₁₉	—	8	2	no	—
169	C₂²×C₁₉	—	8	2	no	—
170	C₂²×C₁₉	—	8	2	no	—
171	C₂²×C₁₉	#525	1	2	yes	—
172	C₂²×C₁₉	#537	1	1	yes	characteristic
173	C₂²×C₁₉	#576	1	1	yes	characteristic
174	C₂²×C₁₉	#576	1	1	yes	characteristic
175	C₂²×C₁₉	#577	1	2	yes	—
176	C₄×C₁₉	—	4	2	no	—
177	C₄×C₁₉	—	4	4	no	—
178	C₄×C₁₉	—	4	2	no	—
179	C₄×C₁₉	—	4	1	no	—
180	C₄×C₁₉	—	4	1	no	—
181	C₄×C₁₉	—	8	2	no	—
182	C₄×C₁₉	—	8	2	no	—
183	C₄×C₁₉	#551	1	1	yes	characteristic
184	C₄×C₁₉	#552	1	2	yes	—
185	C₄×C₁₉	#555	1	1	yes	characteristic
186	C₂²×D₈	C₂²×C₁₉	1	1	yes	characteristic
187	C₂×C₄×D₄	C₂²×C₁₉	1	2	yes	—
188	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
189	C₂×#106	C₂²×C₁₉	1	2	yes	—
190	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
191	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
192	C₂×#102	C₂²×C₁₉	1	2	yes	—
193	C₂×#102	C₂²×C₁₉	1	2	yes	—
194	C₂×#102	C₂²×C₁₉	1	2	yes	—
195	#390	C₂²×C₁₉	1	2	yes	—
196	#390	C₂²×C₁₉	1	2	yes	—
197	#389	C₂²×C₁₉	1	2	yes	—
198	#389	C₂²×C₁₉	1	2	yes	—
199	#388	—	2	2	no	—
200	#388	—	2	2	no	—
201	#381	—	2	2	no	—
202	#380	—	2	2	no	—
203	C₄×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
204	#376	C₂²×C₁₉	1	1	yes	characteristic
205	#340	C₂²×C₁₉	1	1	yes	characteristic
206	#340	C₂²×C₁₉	1	2	yes	—
207	#340	C₂²×C₁₉	1	1	yes	characteristic
208	#340	C₂²×C₁₉	1	2	yes	—
209	#336	C₂²×C₁₉	1	2	yes	—
210	#336	C₂²×C₁₉	1	2	yes	—
211	C₂×C₁₉	—	8	4	no	—
212	C₂×C₁₉	#2609	1	1	yes	characteristic
213	C₂×C₁₉	#3118	1	2	yes	—
214	C₂×C₁₉	#3119	1	2	yes	—
215	C₂×C₁₉	#3259	1	1	yes	characteristic
216	C₂×C₁₉	#3263	1	1	yes	characteristic
217	C₂²×D₄	C₂³×C₁₉	1	2	yes	—
218	C₂³×C₄	—	2	2	no	—
219	C₂×D₈	—	2	4	no	—
220	C₂×D₈	—	4	1	no	—
221	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
222	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
223	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
224	C₄×D₄	—	2	4	no	—
225	C₄×D₄	—	2	4	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
229	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
230	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
231	C₂×C₂²⋊C₄	—	2	2	no	—
232	C₂×C₂²⋊C₄	—	2	2	no	—
233	C₂×C₂²⋊C₄	—	2	2	no	—
234	C₂×C₂²⋊C₄	—	2	2	no	—
235	C₂×C₂²⋊C₄	—	2	2	no	—
236	C₂×C₄²	—	2	2	no	—
237	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
238	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
239	#107	—	4	1	no	—
240	#106	—	2	4	no	—
241	C₄⋊C₈	—	4	1	no	—
242	C₄⋊C₈	—	4	1	no	—
243	#102	—	2	4	no	—
244	#102	—	2	4	no	—
245	#102	—	4	2	no	—
246	#95	—	2	2	no	—
247	#95	—	2	2	no	—
248	#95	—	2	2	no	—
249	#95	D₄×C₁₉	1	2	yes	—
250	#95	D₄×C₁₉	1	1	yes	characteristic
251	#95	D₄×C₁₉	1	1	yes	characteristic
252	C₁₉	#29545	1	1	yes	characteristic, direct product factor, Sylow
253	C₂⁴	—	2	2	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	#41×C₁₉	1	4	yes	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	2	8	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	8	2	no	—
265	C₂²×C₄	—	8	2	no	—
266	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
268	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
269	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
270	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
271	D₈	—	4	4	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	2	1	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	8	2	no	—
285	C₄⋊C₄	—	8	2	no	—
286	C₄⋊C₄	#41×C₁₉	1	4	yes	—
287	C₂²⋊C₄	—	2	8	no	—
288	C₂²⋊C₄	—	8	2	no	—
289	C₂²⋊C₄	—	8	2	no	—
290	C₂²⋊C₄	—	8	2	no	—
291	C₂²⋊C₄	—	8	2	no	—
292	C₄²	—	2	2	no	—
293	C₄²	—	2	2	no	—
294	C₄²	—	2	1	no	—
295	C₄²	—	2	1	no	—
296	C₄²	—	8	2	no	—
297	C₂³	—	2	4	no	—
298	C₂³	—	4	2	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	4	2	no	—
302	C₂³	—	4	2	no	—
303	C₂³	—	4	2	no	—
304	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
305	D₄	—	2	4	no	—
306	D₄	—	2	4	no	—
307	D₄	—	4	2	no	—
308	D₄	—	4	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	8	8	no	—
330	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
331	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
332	C₂×C₄	C₁₉×H(5,𝔽₂)	1	1	yes	characteristic
333	C₂×C₄	C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
334	C₈	—	4	1	no	—
335	C₈	—	4	1	no	—
336	C₈	—	4	2	no	—
337	C₂²	—	4	4	no	—
338	C₂²	—	4	4	no	—
339	C₂²	—	4	4	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	C₁₉×#525	1	2	yes	—
345	C₂²	C₁₉×#537	1	1	yes	characteristic
346	C₂²	C₁₉×#576	1	1	yes	characteristic
347	C₂²	C₁₉×#576	1	1	yes	characteristic
348	C₂²	C₁₉×#577	1	2	yes	—
349	C₄	—	4	1	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	4	no	—
353	C₄	—	4	2	no	—
354	C₄	—	8	2	no	—
355	C₄	—	8	2	no	—
356	C₄	C₁₉×#551	1	1	yes	characteristic
357	C₄	C₁₉×#552	1	2	yes	—
358	C₄	C₁₉×#555	1	1	yes	characteristic
359	C₂	—	8	4	no	—
360	C₂	C₁₉×#2609	1	1	yes	characteristic
361	C₂	C₁₉×#3118	1	2	yes	—
362	C₂	C₁₉×#3119	1	2	yes	—
363	C₂	C₁₉×#3259	1	1	yes	characteristic
364	C₂	C₁₉×#3263	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
prime radical	C₂²×D₈×C₁₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₁₉×#376 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₉×#2319	C₂	semidirect
2. C₁₉×#2086	C₂	semidirect
3. C₁₉×#2065	C₂	semidirect
5. C₁₉×#1925	C₂	semidirect
6. C₁₉×#1919	C₂	semidirect
8. C₁₉×#1864	C₂	semidirect
9. C₁₉×#1832	C₂	semidirect
114. #29545	C₁₉	direct (N × H)
155. #2319	C₂×C₁₉	semidirect
156. #2086	C₂×C₁₉	semidirect
157. #2065	C₂×C₁₉	semidirect
159. #1925	C₂×C₁₉	semidirect
160. #1919	C₂×C₁₉	semidirect
162. #1864	C₂×C₁₉	semidirect
163. #1832	C₂×C₁₉	semidirect
252. C₁₉	#29545	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×#2319	2
C₁₉×#2086	2
C₁₉×#2065	2
C₁₉×#2054	2
C₁₉×#1925	2
C₁₉×#1919	2
C₁₉×#1879	2
C₁₉×#1864	2
C₁₉×#1832	2
#29545	19

Subgroups — C19×#29545 #18035504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#29545 #18035504