Subgroups — C₁₉×#10684 #18016643

Order 4864.

426 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#1990	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#1990	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1956	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1845	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1844	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1834	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1273	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₁₉×#107	—	2	2	no	—
10	C₂×C₁₉×#106	—	2	2	no	—
11	C₂×C₁₉×#103	—	2	2	no	—
12	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×#384	C₂²	1	1	yes	characteristic
16	C₁₉×#384	C₂²	1	1	yes	characteristic
17	C₁₉×#384	C₂²	1	1	yes	characteristic
18	C₁₉×#340	—	2	2	no	—
19	C₁₉×#340	—	2	2	no	—
20	C₁₉×#340	—	2	2	no	—
21	C₂²×Q₈×C₁₉	—	2	1	no	—
22	C₂²×C₈×C₁₉	D₄	1	2	yes	—
23	C₂²×C₈×C₁₉	D₄	1	2	yes	—
24	C₂²×C₈×C₁₉	D₄	1	2	yes	—
25	C₁₉×C₄⋊Q₈	—	2	1	no	—
26	C₁₉×C₄⋊Q₈	—	2	1	no	—
27	C₁₉×C₄⋊Q₈	—	2	1	no	—
28	C₁₉×C₄⋊Q₈	Q₈	1	2	yes	—
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	—	2	1	no	—
35	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
36	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
37	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
38	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
39	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
40	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
41	C₁₉×#107	—	4	4	no	—
42	C₁₉×#106	—	4	4	no	—
43	C₁₉×#103	—	4	4	no	—
44	C₄×C₈×C₁₉	—	2	1	no	—
45	C₄×C₈×C₁₉	—	2	1	no	—
46	C₄×C₈×C₁₉	—	2	1	no	—
47	C₄×C₈×C₁₉	—	2	1	no	—
48	C₄×C₈×C₁₉	—	2	1	no	—
49	C₄×C₈×C₁₉	—	2	1	no	—
50	C₂×Q₈×C₁₉	—	2	2	no	—
51	C₂×Q₈×C₁₉	—	4	1	no	—
52	C₂×Q₈×C₁₉	—	4	2	no	—
53	C₂×Q₈×C₁₉	—	4	2	no	—
54	C₂²×C₄×C₁₉	—	4	1	no	—
55	C₂²×C₄×C₁₉	—	4	1	no	—
56	C₂²×C₄×C₁₉	—	4	1	no	—
57	C₂²×C₄×C₁₉	—	4	1	no	—
58	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₁₉	—	2	2	no	—
62	C₂×C₈×C₁₉	—	2	2	no	—
63	C₂×C₈×C₁₉	—	2	2	no	—
64	C₂×C₈×C₁₉	—	2	2	no	—
65	C₂×C₈×C₁₉	—	2	2	no	—
66	C₂×C₈×C₁₉	—	2	2	no	—
67	C₂×C₈×C₁₉	—	2	2	no	—
68	C₂×C₈×C₁₉	SD₁₆	1	4	yes	—
69	C₂×C₈×C₁₉	Q₁₆	1	4	yes	—
70	C₄⋊C₄×C₁₉	—	2	2	no	—
71	C₄⋊C₄×C₁₉	—	2	2	no	—
72	C₄⋊C₄×C₁₉	—	4	1	no	—
73	C₄⋊C₄×C₁₉	—	4	1	no	—
74	C₄⋊C₄×C₁₉	—	4	2	no	—
75	C₄⋊C₄×C₁₉	—	4	2	no	—
76	C₄⋊C₄×C₁₉	—	4	2	no	—
77	C₄⋊C₄×C₁₉	—	4	2	no	—
78	C₄⋊C₄×C₁₉	—	4	2	no	—
79	C₄⋊C₄×C₁₉	—	4	2	no	—
80	C₄⋊C₄×C₁₉	—	4	2	no	—
81	C₄⋊C₄×C₁₉	—	4	2	no	—
82	C₄⋊C₄×C₁₉	—	4	2	no	—
83	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
84	C₄²×C₁₉	#41	1	1	yes	characteristic
85	C₄²×C₁₉	#41	1	1	yes	characteristic
86	C₄²×C₁₉	#41	1	1	yes	characteristic
87	#10684	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₁₉	#127	1	1	yes	characteristic
89	Q₈×C₁₉	—	4	4	no	—
90	Q₈×C₁₉	—	8	2	no	—
91	C₂×C₄×C₁₉	—	4	2	no	—
92	C₂×C₄×C₁₉	—	8	1	no	—
93	C₂×C₄×C₁₉	—	8	1	no	—
94	C₂×C₄×C₁₉	—	8	1	no	—
95	C₂×C₄×C₁₉	—	8	1	no	—
96	C₂×C₄×C₁₉	—	8	1	no	—
97	C₂×C₄×C₁₉	—	8	1	no	—
98	C₂×C₄×C₁₉	—	8	1	no	—
99	C₂×C₄×C₁₉	—	8	1	no	—
100	C₂×C₄×C₁₉	—	8	1	no	—
101	C₂×C₄×C₁₉	—	8	1	no	—
102	C₂×C₄×C₁₉	—	8	1	no	—
103	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
107	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
108	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
109	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
110	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
111	C₂×C₄×C₁₉	#135	1	2	yes	—
112	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
113	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
114	C₂×C₄×C₁₉	#137	1	2	yes	—
115	C₈×C₁₉	—	2	4	no	—
116	C₈×C₁₉	—	2	4	no	—
117	C₈×C₁₉	—	4	2	no	—
118	#1990	C₂×C₁₉	1	1	yes	characteristic
119	#1990	C₂×C₁₉	1	1	yes	characteristic
120	#1956	C₂×C₁₉	1	1	yes	characteristic
121	#1845	C₂×C₁₉	1	1	yes	characteristic
122	#1844	C₂×C₁₉	1	1	yes	characteristic
123	#1834	C₂×C₁₉	1	1	yes	characteristic
124	#1273	C₂×C₁₉	1	1	yes	characteristic
125	C₂²×C₁₉	#399	1	1	yes	characteristic
126	C₂²×C₁₉	#492	1	1	yes	characteristic
127	C₂²×C₁₉	#494	1	1	yes	characteristic
128	C₂²×C₁₉	#495	1	1	yes	characteristic
129	C₂²×C₁₉	#496	1	1	yes	characteristic
130	C₂²×C₁₉	#496	1	1	yes	characteristic
131	C₂²×C₁₉	#497	1	1	yes	characteristic
132	C₄×C₁₉	—	8	2	no	—
133	C₄×C₁₉	—	16	1	no	—
134	C₄×C₁₉	—	16	1	no	—
135	C₄×C₁₉	—	16	1	no	—
136	C₄×C₁₉	#476	1	2	yes	—
137	C₄×C₁₉	#477	1	2	yes	—
138	C₄×C₁₉	#479	1	2	yes	—
139	C₄×C₁₉	#481	1	2	yes	—
140	C₄×C₁₉	#484	1	2	yes	—
141	C₄×C₁₉	#485	1	2	yes	—
142	C₂×C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic
143	C₂×#107	—	2	2	no	—
144	C₂×#106	—	2	2	no	—
145	C₂×#103	—	2	2	no	—
146	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
148	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
149	#384	C₂²×C₁₉	1	1	yes	characteristic
150	#384	C₂²×C₁₉	1	1	yes	characteristic
151	#384	C₂²×C₁₉	1	1	yes	characteristic
152	#340	—	2	2	no	—
153	#340	—	2	2	no	—
154	#340	—	2	2	no	—
155	C₂×C₁₉	#1707	1	1	yes	characteristic
156	C₂×C₁₉	#1714	1	1	yes	characteristic
157	C₂×C₁₉	#1717	1	1	yes	characteristic
158	C₂×C₁₉	#1723	1	1	yes	characteristic
159	C₂×C₁₉	#2094	1	1	yes	characteristic
160	C₂×C₁₉	#2095	1	1	yes	characteristic
161	C₂×C₁₉	#2097	1	1	yes	characteristic
162	C₂²×Q₈	—	2	1	no	—
163	C₂²×C₈	D₄×C₁₉	1	2	yes	—
164	C₂²×C₈	D₄×C₁₉	1	2	yes	—
165	C₂²×C₈	D₄×C₁₉	1	2	yes	—
166	C₄⋊Q₈	—	2	1	no	—
167	C₄⋊Q₈	—	2	1	no	—
168	C₄⋊Q₈	—	2	1	no	—
169	C₄⋊Q₈	Q₈×C₁₉	1	2	yes	—
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄⋊C₄	—	2	1	no	—
172	C₂×C₄⋊C₄	—	2	1	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	—	2	1	no	—
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	—	2	1	no	—
177	C₂×C₄⋊C₄	—	2	1	no	—
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
182	#107	—	4	4	no	—
183	#106	—	4	4	no	—
184	#103	—	4	4	no	—
185	C₄×C₈	—	2	1	no	—
186	C₄×C₈	—	2	1	no	—
187	C₄×C₈	—	2	1	no	—
188	C₄×C₈	—	2	1	no	—
189	C₄×C₈	—	2	1	no	—
190	C₄×C₈	—	2	1	no	—
191	C₁₉	#10684	1	1	yes	characteristic, direct product factor, Sylow
192	C₂×Q₈	—	2	2	no	—
193	C₂×Q₈	—	4	1	no	—
194	C₂×Q₈	—	4	2	no	—
195	C₂×Q₈	—	4	2	no	—
196	C₂²×C₄	—	4	1	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	—	4	1	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	SD₁₆×C₁₉	1	4	yes	—
211	C₂×C₈	Q₁₆×C₁₉	1	4	yes	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	4	1	no	—
215	C₄⋊C₄	—	4	1	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄²	C₂×Q₈×C₁₉	1	1	yes	characteristic
226	C₄²	#41×C₁₉	1	1	yes	characteristic
227	C₄²	#41×C₁₉	1	1	yes	characteristic
228	C₄²	#41×C₁₉	1	1	yes	characteristic
229	C₂³	C₁₉×#127	1	1	yes	characteristic
230	Q₈	—	4	4	no	—
231	Q₈	—	8	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	8	1	no	—
234	C₂×C₄	—	8	1	no	—
235	C₂×C₄	—	8	1	no	—
236	C₂×C₄	—	8	1	no	—
237	C₂×C₄	—	8	1	no	—
238	C₂×C₄	—	8	1	no	—
239	C₂×C₄	—	8	1	no	—
240	C₂×C₄	—	8	1	no	—
241	C₂×C₄	—	8	1	no	—
242	C₂×C₄	—	8	1	no	—
243	C₂×C₄	—	8	1	no	—
244	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
245	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
246	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
247	C₂×C₄	C₁₉×#123	1	1	yes	characteristic
248	C₂×C₄	C₁₉×#123	1	1	yes	characteristic
249	C₂×C₄	C₁₉×#123	1	1	yes	characteristic
250	C₂×C₄	C₂×SD₁₆×C₁₉	1	2	yes	—
251	C₂×C₄	C₂×Q₁₆×C₁₉	1	2	yes	—
252	C₂×C₄	C₁₉×#135	1	2	yes	—
253	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	2	yes	—
254	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	2	yes	—
255	C₂×C₄	C₁₉×#137	1	2	yes	—
256	C₈	—	2	4	no	—
257	C₈	—	2	4	no	—
258	C₈	—	4	2	no	—
259	C₂²	C₁₉×#399	1	1	yes	characteristic
260	C₂²	C₁₉×#492	1	1	yes	characteristic
261	C₂²	C₁₉×#494	1	1	yes	characteristic
262	C₂²	C₁₉×#495	1	1	yes	characteristic
263	C₂²	C₁₉×#496	1	1	yes	characteristic
264	C₂²	C₁₉×#496	1	1	yes	characteristic
265	C₂²	C₁₉×#497	1	1	yes	characteristic
266	C₄	—	8	2	no	—
267	C₄	—	16	1	no	—
268	C₄	—	16	1	no	—
269	C₄	—	16	1	no	—
270	C₄	C₁₉×#476	1	2	yes	—
271	C₄	C₁₉×#477	1	2	yes	—
272	C₄	C₁₉×#479	1	2	yes	—
273	C₄	C₁₉×#481	1	2	yes	—
274	C₄	C₁₉×#484	1	2	yes	—
275	C₄	C₁₉×#485	1	2	yes	—
276	C₂	C₁₉×#1707	1	1	yes	characteristic
277	C₂	C₁₉×#1714	1	1	yes	characteristic
278	C₂	C₁₉×#1717	1	1	yes	characteristic
279	C₂	C₁₉×#1723	1	1	yes	characteristic
280	C₂	C₁₉×#2094	1	1	yes	characteristic
281	C₂	C₁₉×#2095	1	1	yes	characteristic
282	C₂	C₁₉×#2097	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₉	#127
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
87. #10684	C₁₉	direct (N × H)
191. C₁₉	#10684	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×#1990	2
C₁₉×#1956	2
C₁₉×#1845	2
C₁₉×#1844	2
C₁₉×#1834	2
C₁₉×#1273	2
#10684	19

Subgroups — C19×#10684 #18016643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#10684 #18016643