Subgroups — C₁₉×#23824 #18029783

Order 4864.

882 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₉×#127	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₉×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₉×#2099	C₂	1	2	yes	maximal, complemented
4	C₁₉×#2073	C₂	1	2	yes	maximal, complemented
5	C₁₉×#2041	C₂	1	2	yes	maximal
6	C₁₉×#2025	C₂	1	2	yes	maximal, complemented
7	C₁₉×#1762	C₂	1	4	yes	maximal, complemented
8	C₁₉×#1745	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₉×#127	C₂²	1	1	yes	characteristic
10	C₂×C₁₉×#125	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₄×C₁₉	—	2	2	no	—
12	C₂×C₄×D₄×C₁₉	C₂²	1	2	yes	—
13	C₂×C₁₉×#102	C₂²	1	4	yes	—
14	C₂×C₁₉×#102	C₂²	1	4	yes	—
15	C₂×C₁₉×#97	C₂²	1	2	yes	—
16	C₁₉×#390	—	2	2	no	—
17	C₁₉×#390	C₂²	1	2	yes	—
18	C₁₉×#389	C₂²	1	2	yes	—
19	C₁₉×#387	—	2	2	no	—
20	C₁₉×#384	C₂²	1	1	yes	characteristic
21	C₁₉×#382	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₁₉	—	2	2	no	—
23	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
24	C₄³×C₁₉	C₂²	1	1	yes	characteristic
25	C₁₉×#340	C₂²	1	4	yes	—
26	C₁₉×#340	C₂²	1	4	yes	—
27	C₁₉×#336	C₂²	1	2	yes	—
28	C₁₉×#336	C₂²	1	2	yes	—
29	C₂²×D₄×C₁₉	—	2	2	no	—
30	C₂²×D₄×C₁₉	C₂³	1	2	yes	—
31	C₂³×C₄×C₁₉	—	2	2	no	—
32	C₂²×C₈×C₁₉	C₂³	1	4	yes	—
33	C₁₉×#127	—	2	4	no	—
34	C₁₉×#125	—	2	4	no	—
35	C₄×D₄×C₁₉	—	2	8	no	—
36	C₄×D₄×C₁₉	—	4	8	no	—
37	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
39	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
40	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
43	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
44	C₂×C₂²⋊C₄×C₁₉	—	2	4	no	—
45	C₂×C₄²×C₁₉	—	2	2	no	—
46	C₂×C₄²×C₁₉	D₄	1	2	yes	—
47	C₂×C₄²×C₁₉	D₄	1	2	yes	—
48	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
51	C₁₉×#102	—	2	8	no	—
52	C₁₉×#102	—	2	8	no	—
53	C₁₉×#97	—	2	4	no	—
54	C₁₉×#95	—	2	2	no	—
55	C₁₉×#95	—	2	2	no	—
56	C₁₉×#95	—	2	2	no	—
57	C₂⁴×C₁₉	—	2	2	no	—
58	C₂×D₄×C₁₉	—	2	8	no	—
59	C₂×D₄×C₁₉	—	2	4	no	—
60	C₂×D₄×C₁₉	—	2	4	no	—
61	C₂×D₄×C₁₉	—	2	2	no	—
62	C₂×D₄×C₁₉	—	4	4	no	—
63	C₂×D₄×C₁₉	—	4	4	no	—
64	C₂×D₄×C₁₉	#41	1	4	yes	—
65	C₂²×C₄×C₁₉	—	2	2	no	—
66	C₂²×C₄×C₁₉	—	2	2	no	—
67	C₂²×C₄×C₁₉	—	2	2	no	—
68	C₂²×C₄×C₁₉	—	4	8	no	—
69	C₂²×C₄×C₁₉	—	4	4	no	—
70	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
75	C₂×C₈×C₁₉	—	2	4	no	—
76	C₂×C₈×C₁₉	—	2	4	no	—
77	C₂×C₈×C₁₉	—	2	4	no	—
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	4	no	—
82	C₄⋊C₄×C₁₉	—	4	4	no	—
83	C₄⋊C₄×C₁₉	—	4	4	no	—
84	C₄⋊C₄×C₁₉	#41	1	4	yes	—
85	C₂²⋊C₄×C₁₉	—	4	8	no	—
86	C₂²⋊C₄×C₁₉	—	4	4	no	—
87	C₄²×C₁₉	—	2	4	no	—
88	C₄²×C₁₉	—	2	2	no	—
89	C₄²×C₁₉	—	2	2	no	—
90	C₄²×C₁₉	—	4	2	no	—
91	C₄²×C₁₉	—	4	4	no	—
92	C₄²×C₁₉	#41	1	4	yes	—
93	#23824	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₁₉	—	2	4	no	—
95	C₂³×C₁₉	—	4	2	no	—
96	C₂³×C₁₉	—	4	2	no	—
97	C₂³×C₁₉	—	4	4	no	—
98	C₂³×C₁₉	—	4	4	no	—
99	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
100	D₄×C₁₉	—	2	8	no	—
101	D₄×C₁₉	—	4	16	no	—
102	D₄×C₁₉	—	4	4	no	—
103	C₂×C₄×C₁₉	—	2	2	no	—
104	C₂×C₄×C₁₉	—	2	2	no	—
105	C₂×C₄×C₁₉	—	2	2	no	—
106	C₂×C₄×C₁₉	—	2	2	no	—
107	C₂×C₄×C₁₉	—	2	2	no	—
108	C₂×C₄×C₁₉	—	2	4	no	—
109	C₂×C₄×C₁₉	—	2	2	no	—
110	C₂×C₄×C₁₉	—	2	2	no	—
111	C₂×C₄×C₁₉	—	4	2	no	—
112	C₂×C₄×C₁₉	—	4	2	no	—
113	C₂×C₄×C₁₉	—	4	2	no	—
114	C₂×C₄×C₁₉	—	4	2	no	—
115	C₂×C₄×C₁₉	—	4	4	no	—
116	C₂×C₄×C₁₉	—	4	2	no	—
117	C₂×C₄×C₁₉	—	4	2	no	—
118	C₂×C₄×C₁₉	—	8	8	no	—
119	C₂×C₄×C₁₉	#124	1	2	yes	—
120	C₂×C₄×C₁₉	#124	1	2	yes	—
121	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	4	yes	—
122	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
123	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
124	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
125	C₈×C₁₉	—	4	4	no	—
126	C₄×#127	C₂×C₁₉	1	1	yes	characteristic
127	C₄×#125	C₂×C₁₉	1	1	yes	characteristic, complemented
128	#2099	C₂×C₁₉	1	2	yes	complemented
129	#2073	C₂×C₁₉	1	2	yes	complemented
130	#2041	C₂×C₁₉	1	2	yes	—
131	#2025	C₂×C₁₉	1	2	yes	complemented
132	#1762	C₂×C₁₉	1	4	yes	complemented
133	#1745	C₂×C₁₉	1	1	yes	characteristic, complemented
134	C₂²×C₁₉	—	4	8	no	—
135	C₂²×C₁₉	—	4	4	no	—
136	C₂²×C₁₉	—	8	4	no	—
137	C₂²×C₁₉	—	8	2	no	—
138	C₂²×C₁₉	—	8	2	no	—
139	C₂²×C₁₉	#525	1	1	yes	characteristic
140	C₂²×C₁₉	#525	1	1	yes	characteristic
141	C₂²×C₁₉	C₂×#124	1	1	yes	characteristic
142	C₂²×C₁₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
143	C₂²×C₁₉	#575	1	2	yes	—
144	C₄×C₁₉	—	2	4	no	—
145	C₄×C₁₉	—	2	2	no	—
146	C₄×C₁₉	—	2	2	no	—
147	C₄×C₁₉	—	4	2	no	—
148	C₄×C₁₉	—	4	4	no	—
149	C₄×C₁₉	—	8	2	no	—
150	C₄×C₁₉	—	8	2	no	—
151	C₄×C₁₉	#529	1	4	yes	—
152	C₂×#127	C₂²×C₁₉	1	1	yes	characteristic
153	C₂×#125	C₂²×C₁₉	1	1	yes	characteristic
154	C₂×C₄×D₄	—	2	2	no	—
155	C₂×C₄×D₄	C₂²×C₁₉	1	2	yes	—
156	C₂×#102	C₂²×C₁₉	1	4	yes	—
157	C₂×#102	C₂²×C₁₉	1	4	yes	—
158	C₂×#97	C₂²×C₁₉	1	2	yes	—
159	#390	—	2	2	no	—
160	#390	C₂²×C₁₉	1	2	yes	—
161	#389	C₂²×C₁₉	1	2	yes	—
162	#387	—	2	2	no	—
163	#384	C₂²×C₁₉	1	1	yes	characteristic
164	#382	C₂²×C₁₉	1	1	yes	characteristic
165	C₄×C₄⋊C₄	—	2	2	no	—
166	C₄×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
167	C₄³	C₂²×C₁₉	1	1	yes	characteristic
168	#340	C₂²×C₁₉	1	4	yes	—
169	#340	C₂²×C₁₉	1	4	yes	—
170	#336	C₂²×C₁₉	1	2	yes	—
171	#336	C₂²×C₁₉	1	2	yes	—
172	C₂×C₁₉	—	8	4	no	—
173	C₂×C₁₉	#2398	1	1	yes	characteristic
174	C₂×C₁₉	#3114	1	2	yes	—
175	C₂×C₁₉	#3115	1	2	yes	—
176	C₂×C₁₉	#3138	1	1	yes	characteristic
177	C₂×C₁₉	#3142	1	1	yes	characteristic
178	C₂²×D₄	—	2	2	no	—
179	C₂²×D₄	C₂³×C₁₉	1	2	yes	—
180	C₂³×C₄	—	2	2	no	—
181	C₂²×C₈	C₂³×C₁₉	1	4	yes	—
182	#127	—	2	4	no	—
183	#125	—	2	4	no	—
184	C₄×D₄	—	2	8	no	—
185	C₄×D₄	—	4	8	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
190	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
191	C₂×C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
192	C₂×C₂²⋊C₄	—	2	4	no	—
193	C₂×C₂²⋊C₄	—	2	2	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	D₄×C₁₉	1	2	yes	—
196	C₂×C₄²	D₄×C₁₉	1	2	yes	—
197	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
198	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
199	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
200	#102	—	2	8	no	—
201	#102	—	2	8	no	—
202	#97	—	2	4	no	—
203	#95	—	2	2	no	—
204	#95	—	2	2	no	—
205	#95	—	2	2	no	—
206	C₁₉	#23824	1	1	yes	characteristic, direct product factor, Sylow
207	C₂⁴	—	2	2	no	—
208	C₂×D₄	—	2	8	no	—
209	C₂×D₄	—	2	4	no	—
210	C₂×D₄	—	2	4	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	—	4	4	no	—
213	C₂×D₄	—	4	4	no	—
214	C₂×D₄	#41×C₁₉	1	4	yes	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	4	8	no	—
219	C₂²×C₄	—	4	4	no	—
220	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
221	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
224	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	4	no	—
228	C₄⋊C₄	—	2	8	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	4	4	no	—
233	C₄⋊C₄	—	4	4	no	—
234	C₄⋊C₄	#41×C₁₉	1	4	yes	—
235	C₂²⋊C₄	—	4	8	no	—
236	C₂²⋊C₄	—	4	4	no	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	4	2	no	—
241	C₄²	—	4	4	no	—
242	C₄²	#41×C₁₉	1	4	yes	—
243	C₂³	—	2	4	no	—
244	C₂³	—	4	2	no	—
245	C₂³	—	4	2	no	—
246	C₂³	—	4	4	no	—
247	C₂³	—	4	4	no	—
248	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
249	D₄	—	2	8	no	—
250	D₄	—	4	16	no	—
251	D₄	—	4	4	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	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	—	4	4	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	8	8	no	—
268	C₂×C₄	C₁₉×#124	1	2	yes	—
269	C₂×C₄	C₁₉×#124	1	2	yes	—
270	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	4	yes	—
271	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
272	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
273	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
274	C₈	—	4	4	no	—
275	C₂²	—	4	8	no	—
276	C₂²	—	4	4	no	—
277	C₂²	—	8	4	no	—
278	C₂²	—	8	2	no	—
279	C₂²	—	8	2	no	—
280	C₂²	C₁₉×#525	1	1	yes	characteristic
281	C₂²	C₁₉×#525	1	1	yes	characteristic
282	C₂²	C₂×C₁₉×#124	1	1	yes	characteristic
283	C₂²	C₂×C₁₉×AGL(1,ℤ/8)	1	2	yes	—
284	C₂²	C₁₉×#575	1	2	yes	—
285	C₄	—	2	4	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	4	no	—
290	C₄	—	8	2	no	—
291	C₄	—	8	2	no	—
292	C₄	C₁₉×#529	1	4	yes	—
293	C₂	—	8	4	no	—
294	C₂	C₁₉×#2398	1	1	yes	characteristic
295	C₂	C₁₉×#3114	1	2	yes	—
296	C₂	C₁₉×#3115	1	2	yes	—
297	C₂	C₁₉×#3138	1	1	yes	characteristic
298	C₂	C₁₉×#3142	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
prime radical	C₂×C₁₉×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₄³×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₁₉×#125	C₂	semidirect
3. C₁₉×#2099	C₂	semidirect
4. C₁₉×#2073	C₂	semidirect
6. C₁₉×#2025	C₂	semidirect
7. C₁₉×#1762	C₂	semidirect
8. C₁₉×#1745	C₂	semidirect
93. #23824	C₁₉	direct (N × H)
127. C₄×#125	C₂×C₁₉	semidirect
128. #2099	C₂×C₁₉	semidirect
129. #2073	C₂×C₁₉	semidirect
131. #2025	C₂×C₁₉	semidirect
132. #1762	C₂×C₁₉	semidirect
133. #1745	C₂×C₁₉	semidirect
206. C₁₉	#23824	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₁₉×#127	2
C₄×C₁₉×#125	2
C₁₉×#2099	2
C₁₉×#2073	2
C₁₉×#2041	2
C₁₉×#2025	2
C₁₉×#1762	2
C₁₉×#1745	2
#23824	19

Subgroups — C19×#23824 #18029783

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#23824 #18029783