Subgroups — C₁₉×#10654 #18016613

Order 4864.

350 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×#1948	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1948	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1939	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1844	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1296	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1296	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1277	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₉×#106	—	2	1	no	—
9	C₂×C₁₉×#106	—	2	1	no	—
10	C₂×C₁₉×#106	—	2	1	no	—
11	C₂×C₁₉×#106	—	2	1	no	—
12	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
13	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#384	C₂²	1	1	yes	characteristic
18	C₁₉×#384	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
20	C₁₉×#340	—	2	1	no	—
21	C₁₉×#340	—	2	1	no	—
22	C₁₉×#336	—	2	1	no	—
23	C₂²×C₈×C₁₉	—	2	1	no	—
24	C₂²×C₈×C₁₉	—	2	1	no	—
25	C₂²×C₈×C₁₉	—	2	1	no	—
26	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
32	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
33	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
34	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
35	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₉	—	2	1	no	—
37	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
38	C₁₉×#106	—	4	2	no	—
39	C₁₉×#106	—	4	2	no	—
40	C₁₉×#106	—	4	2	no	—
41	C₁₉×#106	—	4	2	no	—
42	C₁₉×C₄⋊C₈	—	2	1	no	—
43	C₁₉×C₄⋊C₈	—	2	1	no	—
44	C₁₉×C₄⋊C₈	—	2	1	no	—
45	C₁₉×C₄⋊C₈	—	2	1	no	—
46	C₁₉×C₄⋊C₈	—	4	2	no	—
47	C₁₉×C₄⋊C₈	Q₈	1	2	yes	—
48	C₁₉×C₄⋊C₈	Q₈	1	2	yes	—
49	C₄×C₈×C₁₉	—	2	1	no	—
50	C₄×C₈×C₁₉	—	2	1	no	—
51	C₁₉×#95	—	2	1	no	—
52	C₂²×C₄×C₁₉	—	2	1	no	—
53	C₂²×C₄×C₁₉	—	2	1	no	—
54	C₂²×C₄×C₁₉	—	4	1	no	—
55	C₂²×C₄×C₁₉	—	4	1	no	—
56	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
59	C₂×C₈×C₁₉	—	2	1	no	—
60	C₂×C₈×C₁₉	—	2	1	no	—
61	C₂×C₈×C₁₉	—	2	1	no	—
62	C₂×C₈×C₁₉	—	2	1	no	—
63	C₂×C₈×C₁₉	—	2	2	no	—
64	C₂×C₈×C₁₉	—	2	2	no	—
65	C₂×C₈×C₁₉	—	4	1	no	—
66	C₂×C₈×C₁₉	—	4	1	no	—
67	C₂×C₈×C₁₉	—	4	1	no	—
68	C₂×C₈×C₁₉	—	4	1	no	—
69	C₂×C₈×C₁₉	—	4	1	no	—
70	C₂×C₈×C₁₉	—	4	1	no	—
71	C₂×C₈×C₁₉	—	4	1	no	—
72	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
73	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
74	C₄⋊C₄×C₁₉	—	2	1	no	—
75	C₄⋊C₄×C₁₉	—	2	1	no	—
76	C₄⋊C₄×C₁₉	—	2	1	no	—
77	C₄⋊C₄×C₁₉	—	4	1	no	—
78	C₄⋊C₄×C₁₉	—	4	1	no	—
79	C₄⋊C₄×C₁₉	—	4	1	no	—
80	C₄⋊C₄×C₁₉	—	4	1	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	1	no	—
86	C₄⋊C₄×C₁₉	—	8	1	no	—
87	C₄⋊C₄×C₁₉	SD₁₆	1	2	yes	—
88	C₄²×C₁₉	—	4	1	no	—
89	C₄²×C₁₉	—	4	1	no	—
90	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
91	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
92	C₄²×C₁₉	#41	1	1	yes	characteristic
93	C₄²×C₁₉	#41	1	1	yes	characteristic
94	#10654	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂³×C₁₉	#121	1	1	yes	characteristic
96	C₂×C₄×C₁₉	—	2	1	no	—
97	C₂×C₄×C₁₉	—	2	1	no	—
98	C₂×C₄×C₁₉	—	2	2	no	—
99	C₂×C₄×C₁₉	—	4	1	no	—
100	C₂×C₄×C₁₉	—	4	1	no	—
101	C₂×C₄×C₁₉	—	4	1	no	—
102	C₂×C₄×C₁₉	—	4	1	no	—
103	C₂×C₄×C₁₉	—	4	1	no	—
104	C₂×C₄×C₁₉	—	8	1	no	—
105	C₂×C₄×C₁₉	—	8	1	no	—
106	C₂×C₄×C₁₉	—	8	1	no	—
107	C₂×C₄×C₁₉	—	8	1	no	—
108	C₂×C₄×C₁₉	—	8	1	no	—
109	C₂×C₄×C₁₉	—	8	1	no	—
110	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
113	C₂×C₄×C₁₉	#125	1	1	yes	characteristic
114	C₂×C₄×C₁₉	#125	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
120	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
123	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
124	C₈×C₁₉	—	2	2	no	—
125	C₈×C₁₉	—	2	2	no	—
126	C₈×C₁₉	—	4	2	no	—
127	C₈×C₁₉	—	4	2	no	—
128	C₈×C₁₉	—	8	1	no	—
129	#1948	C₂×C₁₉	1	1	yes	characteristic
130	#1948	C₂×C₁₉	1	1	yes	characteristic
131	#1939	C₂×C₁₉	1	1	yes	characteristic
132	#1844	C₂×C₁₉	1	1	yes	characteristic
133	#1296	C₂×C₁₉	1	1	yes	characteristic
134	#1296	C₂×C₁₉	1	1	yes	characteristic
135	#1277	C₂×C₁₉	1	1	yes	characteristic
136	C₂²×C₁₉	#398	1	1	yes	characteristic
137	C₂²×C₁₉	#460	1	1	yes	characteristic
138	C₂²×C₁₉	#461	1	1	yes	characteristic
139	C₂²×C₁₉	#465	1	1	yes	characteristic
140	C₂²×C₁₉	#468	1	1	yes	characteristic
141	C₂²×C₁₉	#471	1	1	yes	characteristic
142	C₂²×C₁₉	#471	1	1	yes	characteristic
143	C₄×C₁₉	—	2	1	no	—
144	C₄×C₁₉	—	2	1	no	—
145	C₄×C₁₉	—	4	2	no	—
146	C₄×C₁₉	—	8	1	no	—
147	C₄×C₁₉	—	16	1	no	—
148	C₄×C₁₉	—	16	1	no	—
149	C₄×C₁₉	#475	1	1	yes	characteristic
150	C₄×C₁₉	#476	1	1	yes	characteristic
151	C₄×C₁₉	#478	1	1	yes	characteristic
152	C₄×C₁₉	#479	1	1	yes	characteristic
153	C₄×C₁₉	#481	1	1	yes	characteristic
154	C₄×C₁₉	#483	1	1	yes	characteristic
155	C₄×C₁₉	#498	1	1	yes	characteristic
156	C₄×C₁₉	#501	1	1	yes	characteristic
157	C₂×#106	—	2	1	no	—
158	C₂×#106	—	2	1	no	—
159	C₂×#106	—	2	1	no	—
160	C₂×#106	—	2	1	no	—
161	C₂×C₄⋊C₈	—	2	1	no	—
162	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
163	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
164	C₂×C₄⋊C₈	C₂²×C₁₉	1	1	yes	characteristic
165	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
166	#384	C₂²×C₁₉	1	1	yes	characteristic
167	#384	C₂²×C₁₉	1	1	yes	characteristic
168	C₄×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
169	#340	—	2	1	no	—
170	#340	—	2	1	no	—
171	#336	—	2	1	no	—
172	C₂×C₁₉	#1666	1	1	yes	characteristic
173	C₂×C₁₉	#1671	1	1	yes	characteristic
174	C₂×C₁₉	#1686	1	1	yes	characteristic
175	C₂×C₁₉	#1691	1	1	yes	characteristic
176	C₂×C₁₉	#2084	1	1	yes	characteristic
177	C₂×C₁₉	#2088	1	1	yes	characteristic
178	C₂×C₁₉	#2090	1	1	yes	characteristic
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
183	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
184	C₂×C₄⋊C₄	—	2	1	no	—
185	C₂×C₄⋊C₄	—	2	1	no	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
191	C₂×C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
194	#106	—	4	2	no	—
195	#106	—	4	2	no	—
196	#106	—	4	2	no	—
197	#106	—	4	2	no	—
198	C₄⋊C₈	—	2	1	no	—
199	C₄⋊C₈	—	2	1	no	—
200	C₄⋊C₈	—	2	1	no	—
201	C₄⋊C₈	—	2	1	no	—
202	C₄⋊C₈	—	4	2	no	—
203	C₄⋊C₈	Q₈×C₁₉	1	2	yes	—
204	C₄⋊C₈	Q₈×C₁₉	1	2	yes	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	2	1	no	—
207	#95	—	2	1	no	—
208	C₁₉	#10654	1	1	yes	characteristic, direct product factor, Sylow
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	—	4	1	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
215	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	SD₁₆×C₁₉	1	2	yes	—
230	C₂×C₈	SD₁₆×C₁₉	1	2	yes	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	4	1	no	—
235	C₄⋊C₄	—	4	1	no	—
236	C₄⋊C₄	—	4	1	no	—
237	C₄⋊C₄	—	4	1	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	1	no	—
243	C₄⋊C₄	—	8	1	no	—
244	C₄⋊C₄	SD₁₆×C₁₉	1	2	yes	—
245	C₄²	—	4	1	no	—
246	C₄²	—	4	1	no	—
247	C₄²	C₂×Q₈×C₁₉	1	1	yes	characteristic
248	C₄²	C₂×Q₈×C₁₉	1	1	yes	characteristic
249	C₄²	#41×C₁₉	1	1	yes	characteristic
250	C₄²	#41×C₁₉	1	1	yes	characteristic
251	C₂³	C₁₉×#121	1	1	yes	characteristic
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	8	1	no	—
261	C₂×C₄	—	8	1	no	—
262	C₂×C₄	—	8	1	no	—
263	C₂×C₄	—	8	1	no	—
264	C₂×C₄	—	8	1	no	—
265	C₂×C₄	—	8	1	no	—
266	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
267	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
268	C₂×C₄	C₁₉×#123	1	1	yes	characteristic
269	C₂×C₄	C₁₉×#125	1	1	yes	characteristic
270	C₂×C₄	C₁₉×#125	1	1	yes	characteristic
271	C₂×C₄	C₁₉×C₄⋊Q₈	1	1	yes	characteristic
272	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
273	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
274	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
275	C₂×C₄	C₁₉×#135	1	1	yes	characteristic
276	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
277	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
278	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
279	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
280	C₈	—	2	2	no	—
281	C₈	—	2	2	no	—
282	C₈	—	4	2	no	—
283	C₈	—	4	2	no	—
284	C₈	—	8	1	no	—
285	C₂²	C₁₉×#398	1	1	yes	characteristic
286	C₂²	C₁₉×#460	1	1	yes	characteristic
287	C₂²	C₁₉×#461	1	1	yes	characteristic
288	C₂²	C₁₉×#465	1	1	yes	characteristic
289	C₂²	C₁₉×#468	1	1	yes	characteristic
290	C₂²	C₁₉×#471	1	1	yes	characteristic
291	C₂²	C₁₉×#471	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	4	2	no	—
295	C₄	—	8	1	no	—
296	C₄	—	16	1	no	—
297	C₄	—	16	1	no	—
298	C₄	C₁₉×#475	1	1	yes	characteristic
299	C₄	C₁₉×#476	1	1	yes	characteristic
300	C₄	C₁₉×#478	1	1	yes	characteristic
301	C₄	C₁₉×#479	1	1	yes	characteristic
302	C₄	C₁₉×#481	1	1	yes	characteristic
303	C₄	C₁₉×#483	1	1	yes	characteristic
304	C₄	C₁₉×#498	1	1	yes	characteristic
305	C₄	C₁₉×#501	1	1	yes	characteristic
306	C₂	C₁₉×#1666	1	1	yes	characteristic
307	C₂	C₁₉×#1671	1	1	yes	characteristic
308	C₂	C₁₉×#1686	1	1	yes	characteristic
309	C₂	C₁₉×#1691	1	1	yes	characteristic
310	C₂	C₁₉×#2084	1	1	yes	characteristic
311	C₂	C₁₉×#2088	1	1	yes	characteristic
312	C₂	C₁₉×#2090	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₉	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
94. #10654	C₁₉	direct (N × H)
208. C₁₉	#10654	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×#1948	2
C₁₉×#1939	2
C₁₉×#1844	2
C₁₉×#1296	2
C₁₉×#1277	2
#10654	19

Subgroups — C19×#10654 #18016613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#10654 #18016613