Subgroups — C₁₉×#21739 #18027698

Order 4864.

1550 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#525	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₉×C₂²⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₁₉×#1869	C₂	1	8	yes	maximal, complemented
4	C₁₉×#1818	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1780	C₂	1	2	yes	maximal, complemented
6	C₂²×#41×C₁₉	C₂²	1	1	yes	characteristic
7	C₂⁴×C₄×C₁₉	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₁₉	C₂²	1	4	yes	—
9	C₁₉×#525	—	2	8	no	—
10	C₂×C₁₉×#123	C₄	1	2	yes	—
11	C₂×C₁₉×C₂²⋊Q₈	C₄	1	2	yes	—
12	C₂×C₁₉×#121	C₄	1	2	yes	—
13	C₂×C₁₉×#120	C₄	1	2	yes	—
14	C₂×C₄×D₄×C₁₉	C₂²	1	4	yes	—
15	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
16	C₂×C₁₉×C₂²⋊C₈	—	2	8	no	—
17	C₂×C₁₉×C₂²⋊C₈	C₂²	1	4	yes	—
18	C₂×C₁₉×C₂²⋊C₈	C₂²	1	4	yes	—
19	C₂×C₁₉×C₂²⋊C₈	C₂²	1	8	yes	complemented
20	C₁₉×#336	C₂²	1	8	yes	complemented
21	C₂⁵×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂×#41×C₁₉	—	2	4	no	—
23	C₂×#41×C₁₉	—	4	4	no	—
24	C₂²×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×D₄×C₁₉	C₂×C₄	1	4	yes	—
26	C₂²×D₄×C₁₉	C₂×C₄	1	2	yes	—
27	C₂²×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₉	—	4	4	no	—
29	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₁₉	D₄	1	8	yes	—
32	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
33	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₉	—	2	8	no	—
36	C₂²×C₈×C₁₉	—	4	8	no	—
37	C₂²×C₈×C₁₉	C₂³	1	8	yes	—
38	C₁₉×#123	—	2	8	no	—
39	C₁₉×C₂²⋊Q₈	—	2	8	no	—
40	C₁₉×#121	—	2	8	no	—
41	C₁₉×#120	—	2	8	no	—
42	C₄×D₄×C₁₉	—	2	8	no	—
43	C₄×D₄×C₁₉	—	2	8	no	—
44	C₁₉×#117	—	2	4	no	—
45	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
47	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
49	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
51	C₁₉×C₂²⋊C₈	—	2	8	no	—
52	C₁₉×C₂²⋊C₈	—	2	8	no	—
53	C₁₉×C₂²⋊C₈	—	4	8	no	—
54	C₂⁴×C₁₉	—	2	8	no	—
55	C₂⁴×C₁₉	—	4	2	no	—
56	C₂⁴×C₁₉	C₂²⋊C₄	1	4	yes	—
57	C₂⁴×C₁₉	C₂²×C₄	1	2	yes	—
58	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
59	C₂⁴×C₁₉	C₂²×C₄	1	2	yes	—
60	#41×C₁₉	—	4	16	no	—
61	C₂×Q₈×C₁₉	—	2	2	no	—
62	C₂×Q₈×C₁₉	—	4	2	no	—
63	C₂×D₄×C₁₉	—	2	8	no	—
64	C₂×D₄×C₁₉	—	2	4	no	—
65	C₂×D₄×C₁₉	—	2	16	no	—
66	C₂×D₄×C₁₉	—	2	2	no	—
67	C₂×D₄×C₁₉	—	4	2	no	—
68	C₂×D₄×C₁₉	—	4	4	no	—
69	C₂²×C₄×C₁₉	—	2	8	no	—
70	C₂²×C₄×C₁₉	—	2	4	no	—
71	C₂²×C₄×C₁₉	—	2	4	no	—
72	C₂²×C₄×C₁₉	—	2	8	no	—
73	C₂²×C₄×C₁₉	—	4	16	no	—
74	C₂²×C₄×C₁₉	—	4	2	no	—
75	C₂²×C₄×C₁₉	—	4	4	no	—
76	C₂²×C₄×C₁₉	C₂²⋊C₄	1	8	yes	—
77	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
78	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
83	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
84	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
85	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
86	C₂×C₈×C₁₉	—	2	8	no	—
87	C₂×C₈×C₁₉	—	2	16	no	—
88	C₂×C₈×C₁₉	—	4	16	no	—
89	C₄⋊C₄×C₁₉	—	2	4	no	—
90	C₄⋊C₄×C₁₉	—	2	8	no	—
91	C₂²⋊C₄×C₁₉	—	2	4	no	—
92	C₂²⋊C₄×C₁₉	—	2	8	no	—
93	C₂²⋊C₄×C₁₉	—	2	8	no	—
94	C₄²×C₁₉	—	2	4	no	—
95	#21739	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂³×C₁₉	—	2	4	no	—
97	C₂³×C₁₉	—	2	4	no	—
98	C₂³×C₁₉	—	2	2	no	—
99	C₂³×C₁₉	—	2	4	no	—
100	C₂³×C₁₉	—	2	16	no	—
101	C₂³×C₁₉	—	2	2	no	—
102	C₂³×C₁₉	—	2	4	no	—
103	C₂³×C₁₉	—	2	8	no	—
104	C₂³×C₁₉	—	4	4	no	—
105	C₂³×C₁₉	—	4	16	no	—
106	C₂³×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
107	C₂³×C₁₉	C₂×C₂²⋊C₄	1	4	yes	—
108	C₂³×C₁₉	#131	1	8	yes	—
109	C₂³×C₁₉	C₂³×C₄	1	1	yes	characteristic
110	Q₈×C₁₉	—	4	4	no	—
111	D₄×C₁₉	—	4	4	no	—
112	D₄×C₁₉	—	4	8	no	—
113	D₄×C₁₉	—	4	16	no	—
114	C₂×C₄×C₁₉	—	2	4	no	—
115	C₂×C₄×C₁₉	—	2	2	no	—
116	C₂×C₄×C₁₉	—	2	8	no	—
117	C₂×C₄×C₁₉	—	2	8	no	—
118	C₂×C₄×C₁₉	—	2	8	no	—
119	C₂×C₄×C₁₉	—	2	4	no	—
120	C₂×C₄×C₁₉	—	2	2	no	—
121	C₂×C₄×C₁₉	—	2	4	no	—
122	C₂×C₄×C₁₉	—	2	8	no	—
123	C₂×C₄×C₁₉	—	4	4	no	—
124	C₂×C₄×C₁₉	—	4	8	no	—
125	C₂×C₄×C₁₉	—	4	8	no	—
126	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	4	yes	—
127	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
128	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
129	C₂×C₄×C₁₉	H(5,𝔽₂)	1	4	yes	—
130	C₈×C₁₉	—	4	8	no	—
131	C₂×#525	C₂×C₁₉	1	1	yes	characteristic
132	C₂²×C₂²⋊C₈	C₂×C₁₉	1	2	yes	complemented
133	#1869	C₂×C₁₉	1	8	yes	complemented
134	#1818	C₂×C₁₉	1	2	yes	complemented
135	#1780	C₂×C₁₉	1	2	yes	complemented
136	C₂²×C₁₉	—	2	4	no	—
137	C₂²×C₁₉	—	2	4	no	—
138	C₂²×C₁₉	—	2	16	no	—
139	C₂²×C₁₉	—	2	2	no	—
140	C₂²×C₁₉	—	2	4	no	—
141	C₂²×C₁₉	—	4	16	no	—
142	C₂²×C₁₉	—	4	4	no	—
143	C₂²×C₁₉	—	4	2	no	—
144	C₂²×C₁₉	—	4	4	no	—
145	C₂²×C₁₉	—	4	8	no	—
146	C₂²×C₁₉	#408	1	8	yes	—
147	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₁₉	#518	1	2	yes	—
149	C₂²×C₁₉	C₂×#131	1	4	yes	—
150	C₄×C₁₉	—	2	4	no	—
151	C₄×C₁₉	—	2	2	no	—
152	C₄×C₁₉	—	4	2	no	—
153	C₄×C₁₉	—	4	4	no	—
154	C₄×C₁₉	#534	1	4	yes	—
155	C₂²×#41	C₂²×C₁₉	1	1	yes	characteristic
156	C₂⁴×C₄	C₂²×C₁₉	1	1	yes	characteristic
157	C₂³×C₈	C₂²×C₁₉	1	4	yes	—
158	#525	—	2	8	no	—
159	C₂×#123	C₄×C₁₉	1	2	yes	—
160	C₂×C₂²⋊Q₈	C₄×C₁₉	1	2	yes	—
161	C₂×#121	C₄×C₁₉	1	2	yes	—
162	C₂×#120	C₄×C₁₉	1	2	yes	—
163	C₂×C₄×D₄	C₂²×C₁₉	1	4	yes	—
164	C₂×#117	C₂²×C₁₉	1	1	yes	characteristic
165	C₂×C₂²⋊C₈	—	2	8	no	—
166	C₂×C₂²⋊C₈	C₂²×C₁₉	1	4	yes	—
167	C₂×C₂²⋊C₈	C₂²×C₁₉	1	4	yes	—
168	C₂×C₂²⋊C₈	C₂²×C₁₉	1	8	yes	complemented
169	#336	C₂²×C₁₉	1	8	yes	complemented
170	C₂×C₁₉	—	2	8	no	—
171	C₂×C₁₉	—	4	2	no	—
172	C₂×C₁₉	—	8	2	no	—
173	C₂×C₁₉	#2308	1	1	yes	characteristic
174	C₂×C₁₉	C₂×#408	1	2	yes	—
175	C₂×C₁₉	#2977	1	4	yes	—
176	C₂⁵	C₂×C₄×C₁₉	1	1	yes	characteristic
177	C₂×#41	—	2	4	no	—
178	C₂×#41	—	4	4	no	—
179	C₂²×Q₈	C₂×C₄×C₁₉	1	1	yes	characteristic
180	C₂²×D₄	C₂×C₄×C₁₉	1	4	yes	—
181	C₂²×D₄	C₂×C₄×C₁₉	1	2	yes	—
182	C₂²×D₄	C₂×C₄×C₁₉	1	1	yes	characteristic
183	C₂³×C₄	—	4	4	no	—
184	C₂³×C₄	C₂×C₄×C₁₉	1	2	yes	—
185	C₂³×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
186	C₂³×C₄	D₄×C₁₉	1	8	yes	—
187	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
188	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
189	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
190	C₂²×C₈	—	2	8	no	—
191	C₂²×C₈	—	4	8	no	—
192	C₂²×C₈	C₂³×C₁₉	1	8	yes	—
193	#123	—	2	8	no	—
194	C₂²⋊Q₈	—	2	8	no	—
195	#121	—	2	8	no	—
196	#120	—	2	8	no	—
197	C₄×D₄	—	2	8	no	—
198	C₄×D₄	—	2	8	no	—
199	#117	—	2	4	no	—
200	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
201	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	4	yes	—
202	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
203	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	4	yes	—
204	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	4	yes	—
205	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
206	C₂²⋊C₈	—	2	8	no	—
207	C₂²⋊C₈	—	2	8	no	—
208	C₂²⋊C₈	—	4	8	no	—
209	C₁₉	#21739	1	1	yes	characteristic, direct product factor, Sylow
210	C₂⁴	—	2	8	no	—
211	C₂⁴	—	4	2	no	—
212	C₂⁴	C₂²⋊C₄×C₁₉	1	4	yes	—
213	C₂⁴	C₂²×C₄×C₁₉	1	2	yes	—
214	C₂⁴	C₂²×C₄×C₁₉	1	1	yes	characteristic
215	C₂⁴	C₂²×C₄×C₁₉	1	2	yes	—
216	#41	—	4	16	no	—
217	C₂×Q₈	—	2	2	no	—
218	C₂×Q₈	—	4	2	no	—
219	C₂×D₄	—	2	8	no	—
220	C₂×D₄	—	2	4	no	—
221	C₂×D₄	—	2	16	no	—
222	C₂×D₄	—	2	2	no	—
223	C₂×D₄	—	4	2	no	—
224	C₂×D₄	—	4	4	no	—
225	C₂²×C₄	—	2	8	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	2	8	no	—
229	C₂²×C₄	—	4	16	no	—
230	C₂²×C₄	—	4	4	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	C₂²⋊C₄×C₁₉	1	8	yes	—
233	C₂²×C₄	C₂²⋊C₄×C₁₉	1	4	yes	—
234	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
235	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
236	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
237	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
239	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
240	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
241	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
242	C₂×C₈	—	2	8	no	—
243	C₂×C₈	—	2	16	no	—
244	C₂×C₈	—	4	16	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	8	no	—
247	C₂²⋊C₄	—	2	4	no	—
248	C₂²⋊C₄	—	2	8	no	—
249	C₂²⋊C₄	—	2	8	no	—
250	C₄²	—	2	4	no	—
251	C₂³	—	2	4	no	—
252	C₂³	—	2	4	no	—
253	C₂³	—	2	16	no	—
254	C₂³	—	2	2	no	—
255	C₂³	—	2	4	no	—
256	C₂³	—	2	2	no	—
257	C₂³	—	2	4	no	—
258	C₂³	—	2	8	no	—
259	C₂³	—	4	4	no	—
260	C₂³	—	4	16	no	—
261	C₂³	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
262	C₂³	C₂×C₂²⋊C₄×C₁₉	1	4	yes	—
263	C₂³	C₁₉×#131	1	8	yes	—
264	C₂³	C₂³×C₄×C₁₉	1	1	yes	characteristic
265	Q₈	—	4	4	no	—
266	D₄	—	4	4	no	—
267	D₄	—	4	8	no	—
268	D₄	—	4	16	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	2	8	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	4	8	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	8	no	—
281	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	4	yes	—
282	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
283	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
284	C₂×C₄	C₁₉×H(5,𝔽₂)	1	4	yes	—
285	C₈	—	4	8	no	—
286	C₂²	—	2	16	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	4	16	no	—
292	C₂²	—	4	4	no	—
293	C₂²	—	4	8	no	—
294	C₂²	—	4	2	no	—
295	C₂²	—	4	4	no	—
296	C₂²	C₁₉×#408	1	8	yes	—
297	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
298	C₂²	C₁₉×#518	1	2	yes	—
299	C₂²	C₂×C₁₉×#131	1	4	yes	—
300	C₄	—	2	2	no	—
301	C₄	—	2	4	no	—
302	C₄	—	4	2	no	—
303	C₄	—	4	4	no	—
304	C₄	C₁₉×#534	1	4	yes	—
305	C₂	—	2	8	no	—
306	C₂	—	4	2	no	—
307	C₂	—	8	2	no	—
308	C₂	C₁₉×#2308	1	1	yes	characteristic
309	C₂	C₂×C₁₉×#408	1	2	yes	—
310	C₂	C₁₉×#2977	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₉	C₂⁴
commutator	C₂³	C₂³×C₄×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
socle	C₂³×C₁₉	C₂³×C₄
prime radical	C₂×C₁₉×#525	C₂
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. C₂²×C₁₉×C₂²⋊C₈	C₂	semidirect
3. C₁₉×#1869	C₂	semidirect
4. C₁₉×#1818	C₂	semidirect
5. C₁₉×#1780	C₂	semidirect
19. C₂×C₁₉×C₂²⋊C₈	C₂²	semidirect
20. C₁₉×#336	C₂²	semidirect
95. #21739	C₁₉	direct (N × H)
132. C₂²×C₂²⋊C₈	C₂×C₁₉	semidirect
133. #1869	C₂×C₁₉	semidirect
134. #1818	C₂×C₁₉	semidirect
135. #1780	C₂×C₁₉	semidirect
168. C₂×C₂²⋊C₈	C₂²×C₁₉	semidirect
169. #336	C₂²×C₁₉	semidirect
209. C₁₉	#21739	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×#525	2
C₂²×C₁₉×C₂²⋊C₈	2
C₁₉×#1869	2
C₁₉×#1818	2
C₁₉×#1780	2
#21739	19

Subgroups — C19×#21739 #18027698

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#21739 #18027698