Subgroups — C₁₉×#9193 #18015152

Order 4864.

504 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2905	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2628	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1549	C₂	1	2	yes	maximal, complemented
4	C₁₉×#1458	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1286	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆×C₁₉	—	2	1	no	—
7	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
9	C₁₉×#431	C₂²	1	2	yes	—
10	C₁₉×#424	—	2	4	no	—
11	C₁₉×#423	—	2	1	no	—
12	C₁₉×#423	C₂²	1	2	yes	—
13	C₁₉×#400	C₄	1	2	yes	—
14	C₁₉×#399	C₄	1	2	yes	—
15	C₁₉×#395	—	2	1	no	—
16	C₁₉×#393	—	2	1	no	—
17	C₁₉×#389	—	2	1	no	—
18	C₁₉×#382	C₂²	1	1	yes	characteristic
19	C₁₉×#380	—	2	1	no	—
20	C₁₉×#332	—	2	4	no	—
21	C₁₉×#324	—	2	1	no	—
22	C₁₉×#323	—	2	1	no	—
23	C₂²×Q₈×C₁₉	—	2	1	no	—
24	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
25	C₂×M₁₆×C₁₉	—	2	2	no	—
26	C₂×M₁₆×C₁₉	—	2	2	no	—
27	C₂×M₁₆×C₁₉	—	4	2	no	—
28	C₂×M₁₆×C₁₉	—	4	2	no	—
29	C₂²×C₈×C₁₉	—	2	2	no	—
30	C₁₉×C₄⋊Q₈	—	2	2	no	—
31	C₁₉×C₄⋊Q₈	—	2	2	no	—
32	C₁₉×#117	—	2	2	no	—
33	C₁₉×#117	—	2	2	no	—
34	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
35	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
36	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
37	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
40	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
41	C₂×C₂²⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
45	C₁₉×C₄⋊C₈	—	2	4	no	—
46	C₁₉×C₄⋊C₈	—	2	2	no	—
47	C₁₉×C₄⋊C₈	—	2	2	no	—
48	C₁₉×C₄⋊C₈	D₄	1	4	yes	—
49	C₁₉×C₂²⋊C₈	—	2	2	no	—
50	C₄×C₈×C₁₉	—	2	2	no	—
51	C₁₉×#95	—	2	2	no	—
52	C₁₉×#95	—	2	2	no	—
53	C₁₉×#95	—	2	1	no	—
54	C₁₉×#95	C₂×C₄	1	2	yes	—
55	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Q₈×C₁₉	—	4	2	no	—
57	C₂×Q₈×C₁₉	—	8	2	no	—
58	C₂²×C₄×C₁₉	—	2	1	no	—
59	C₂²×C₄×C₁₉	—	2	1	no	—
60	C₂²×C₄×C₁₉	—	2	1	no	—
61	C₂²×C₄×C₁₉	—	2	1	no	—
62	C₂²×C₄×C₁₉	—	2	2	no	—
63	C₂²×C₄×C₁₉	—	2	2	no	—
64	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
68	C₂²×C₄×C₁₉	Q₁₆	1	2	yes	—
69	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
72	M₁₆×C₁₉	—	4	4	no	—
73	M₁₆×C₁₉	—	4	2	no	—
74	M₁₆×C₁₉	—	4	2	no	—
75	C₂×C₈×C₁₉	—	2	4	no	—
76	C₂×C₈×C₁₉	—	4	4	no	—
77	C₂×C₈×C₁₉	—	4	2	no	—
78	C₄⋊C₄×C₁₉	—	2	2	no	—
79	C₄⋊C₄×C₁₉	—	2	2	no	—
80	C₄⋊C₄×C₁₉	—	2	4	no	—
81	C₄⋊C₄×C₁₉	—	4	1	no	—
82	C₄⋊C₄×C₁₉	—	4	1	no	—
83	C₄⋊C₄×C₁₉	—	4	1	no	—
84	C₄⋊C₄×C₁₉	—	4	1	no	—
85	C₄⋊C₄×C₁₉	—	8	2	no	—
86	C₂²⋊C₄×C₁₉	—	2	2	no	—
87	C₂²⋊C₄×C₁₉	—	2	2	no	—
88	C₂²⋊C₄×C₁₉	—	8	2	no	—
89	C₄²×C₁₉	—	2	1	no	—
90	C₄²×C₁₉	—	2	1	no	—
91	C₄²×C₁₉	C₂×D₄	1	2	yes	—
92	C₄²×C₁₉	#41	1	2	yes	—
93	#9193	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₁₉	—	2	1	no	—
95	C₂³×C₁₉	—	2	1	no	—
96	C₂³×C₁₉	—	4	1	no	—
97	C₂³×C₁₉	—	4	1	no	—
98	C₂³×C₁₉	#103	1	2	yes	—
99	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	Q₈×C₁₉	—	8	4	no	—
101	C₂×C₄×C₁₉	—	2	1	no	—
102	C₂×C₄×C₁₉	—	2	1	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	1	no	—
108	C₂×C₄×C₁₉	—	2	1	no	—
109	C₂×C₄×C₁₉	—	4	1	no	—
110	C₂×C₄×C₁₉	—	4	1	no	—
111	C₂×C₄×C₁₉	—	4	1	no	—
112	C₂×C₄×C₁₉	—	4	1	no	—
113	C₂×C₄×C₁₉	—	4	2	no	—
114	C₂×C₄×C₁₉	—	4	2	no	—
115	C₂×C₄×C₁₉	—	4	2	no	—
116	C₂×C₄×C₁₉	—	4	2	no	—
117	C₂×C₄×C₁₉	—	4	2	no	—
118	C₂×C₄×C₁₉	—	4	2	no	—
119	C₂×C₄×C₁₉	#103	1	2	yes	—
120	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
122	C₂×C₄×C₁₉	#121	1	2	yes	—
123	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
126	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
128	C₈×C₁₉	—	4	4	no	—
129	C₈×C₁₉	—	8	2	no	—
130	#2905	C₂×C₁₉	1	1	yes	characteristic
131	#2628	C₂×C₁₉	1	1	yes	characteristic
132	#1549	C₂×C₁₉	1	2	yes	complemented
133	#1458	C₂×C₁₉	1	1	yes	characteristic
134	#1286	C₂×C₁₉	1	2	yes	complemented
135	C₂²×C₁₉	—	2	1	no	—
136	C₂²×C₁₉	—	2	1	no	—
137	C₂²×C₁₉	—	2	2	no	—
138	C₂²×C₁₉	—	4	2	no	—
139	C₂²×C₁₉	—	4	1	no	—
140	C₂²×C₁₉	—	4	1	no	—
141	C₂²×C₁₉	—	8	1	no	—
142	C₂²×C₁₉	#386	1	1	yes	characteristic
143	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
144	C₂²×C₁₉	#417	1	1	yes	characteristic
145	C₄×C₁₉	—	2	2	no	—
146	C₄×C₁₉	—	2	2	no	—
147	C₄×C₁₉	—	2	2	no	—
148	C₄×C₁₉	—	4	1	no	—
149	C₄×C₁₉	—	4	1	no	—
150	C₄×C₁₉	—	8	2	no	—
151	C₄×C₁₉	—	8	2	no	—
152	C₄×C₁₉	#450	1	1	yes	characteristic
153	C₄×C₁₉	#451	1	1	yes	characteristic
154	C₄×C₁₉	#483	1	1	yes	characteristic
155	C₄×C₁₉	#484	1	1	yes	characteristic
156	C₂²×M₁₆	—	2	1	no	—
157	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
158	C₂×#117	C₂²×C₁₉	1	1	yes	characteristic
159	#431	C₂²×C₁₉	1	2	yes	—
160	#424	—	2	4	no	—
161	#423	—	2	1	no	—
162	#423	C₂²×C₁₉	1	2	yes	—
163	#400	C₄×C₁₉	1	2	yes	—
164	#399	C₄×C₁₉	1	2	yes	—
165	#395	—	2	1	no	—
166	#393	—	2	1	no	—
167	#389	—	2	1	no	—
168	#382	C₂²×C₁₉	1	1	yes	characteristic
169	#380	—	2	1	no	—
170	#332	—	2	4	no	—
171	#324	—	2	1	no	—
172	#323	—	2	1	no	—
173	C₂×C₁₉	—	2	1	no	—
174	C₂×C₁₉	—	2	1	no	—
175	C₂×C₁₉	—	4	2	no	—
176	C₂×C₁₉	#1872	1	1	yes	characteristic
177	C₂×C₁₉	#1910	1	1	yes	characteristic
178	C₂×C₁₉	#1915	1	1	yes	characteristic
179	C₂²×Q₈	—	2	1	no	—
180	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	4	2	no	—
184	C₂×M₁₆	—	4	2	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₄⋊Q₈	—	2	2	no	—
187	C₄⋊Q₈	—	2	2	no	—
188	#117	—	2	2	no	—
189	#117	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
194	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
195	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
196	C₂×C₂²⋊C₄	—	2	2	no	—
197	C₂×C₂²⋊C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
198	C₂×C₂²⋊C₄	D₄×C₁₉	1	1	yes	characteristic
199	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
200	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
201	C₄⋊C₈	—	2	4	no	—
202	C₄⋊C₈	—	2	2	no	—
203	C₄⋊C₈	—	2	2	no	—
204	C₄⋊C₈	D₄×C₁₉	1	4	yes	—
205	C₂²⋊C₈	—	2	2	no	—
206	C₄×C₈	—	2	2	no	—
207	#95	—	2	2	no	—
208	#95	—	2	2	no	—
209	#95	—	2	1	no	—
210	#95	C₂×C₄×C₁₉	1	2	yes	—
211	C₁₉	#9193	1	1	yes	characteristic, direct product factor, Sylow
212	C₂⁴	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
213	C₂×Q₈	—	4	2	no	—
214	C₂×Q₈	—	8	2	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
222	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
223	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
224	C₂²×C₄	SD₁₆×C₁₉	1	2	yes	—
225	C₂²×C₄	Q₁₆×C₁₉	1	2	yes	—
226	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
229	M₁₆	—	4	4	no	—
230	M₁₆	—	4	2	no	—
231	M₁₆	—	4	2	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	4	4	no	—
234	C₂×C₈	—	4	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	4	1	no	—
239	C₄⋊C₄	—	4	1	no	—
240	C₄⋊C₄	—	4	1	no	—
241	C₄⋊C₄	—	4	1	no	—
242	C₄⋊C₄	—	8	2	no	—
243	C₂²⋊C₄	—	2	2	no	—
244	C₂²⋊C₄	—	2	2	no	—
245	C₂²⋊C₄	—	8	2	no	—
246	C₄²	—	2	1	no	—
247	C₄²	—	2	1	no	—
248	C₄²	C₂×D₄×C₁₉	1	2	yes	—
249	C₄²	#41×C₁₉	1	2	yes	—
250	C₂³	—	2	1	no	—
251	C₂³	—	2	1	no	—
252	C₂³	—	4	1	no	—
253	C₂³	—	4	1	no	—
254	C₂³	C₁₉×#103	1	2	yes	—
255	C₂³	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
256	Q₈	—	8	4	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	C₁₉×#103	1	2	yes	—
276	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
277	C₂×C₄	C₁₉×#120	1	1	yes	characteristic
278	C₂×C₄	C₁₉×#121	1	2	yes	—
279	C₂×C₄	C₁₉×#123	1	1	yes	characteristic
280	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
281	C₂×C₄	C₂×Q₁₆×C₁₉	1	1	yes	characteristic
282	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
283	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
284	C₈	—	4	4	no	—
285	C₈	—	8	2	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	1	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	8	1	no	—
293	C₂²	C₁₉×#386	1	1	yes	characteristic
294	C₂²	C₂×C₁₉×#103	1	1	yes	characteristic
295	C₂²	C₁₉×#417	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	8	2	no	—
302	C₄	—	8	2	no	—
303	C₄	C₁₉×#450	1	1	yes	characteristic
304	C₄	C₁₉×#451	1	1	yes	characteristic
305	C₄	C₁₉×#483	1	1	yes	characteristic
306	C₄	C₁₉×#484	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	—	4	2	no	—
310	C₂	C₁₉×#1872	1	1	yes	characteristic
311	C₂	C₁₉×#1910	1	1	yes	characteristic
312	C₂	C₁₉×#1915	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	#386
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₉×#1549	C₂	semidirect
5. C₁₉×#1286	C₂	semidirect
93. #9193	C₁₉	direct (N × H)
132. #1549	C₂×C₁₉	semidirect
134. #1286	C₂×C₁₉	semidirect
211. C₁₉	#9193	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×#2905	2
C₁₉×#2628	2
C₁₉×#1549	2
C₁₉×#1458	2
C₁₉×#1286	2
#9193	19

Subgroups — C19×#9193 #18015152

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#9193 #18015152