Subgroups — C₂×C₁₉×#1992 #18027278

Order 4864.

1294 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉×#125	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₉×#97	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#384	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1992	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₉×#340	C₂	1	4	yes	maximal
6	C₂³×C₈×C₁₉	C₂²	1	2	yes	—
7	C₂×C₁₉×#125	—	2	2	no	—
8	C₂×C₁₉×#125	—	2	4	no	—
9	C₂×C₁₉×#125	C₄	1	8	yes	complemented
10	C₂×C₁₉×#125	C₂²	1	4	yes	—
11	C₂²×C₄⋊C₄×C₁₉	—	2	2	no	—
12	C₂²×C₄⋊C₄×C₁₉	—	2	2	no	—
13	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
14	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
15	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
16	C₂×C₁₉×#97	—	2	4	no	—
17	C₂×C₁₉×#97	C₂²	1	4	yes	—
18	C₁₉×#384	—	2	2	no	—
19	C₁₉×#384	C₂²	1	4	yes	—
20	C₁₉×#340	C₂²	1	16	yes	—
21	C₂³×C₄×C₁₉	—	2	2	no	—
22	C₂³×C₄×C₁₉	—	2	2	no	—
23	C₂³×C₄×C₁₉	D₄	1	2	yes	—
24	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₁₉	—	2	4	no	—
26	C₂²×C₈×C₁₉	—	2	4	no	—
27	C₂²×C₈×C₁₉	Q₈	1	8	yes	—
28	C₂²×C₈×C₁₉	C₂³	1	4	yes	—
29	C₁₉×#125	—	2	16	no	—
30	C₁₉×#125	—	2	8	no	—
31	C₁₉×#125	C₂×C₄	1	16	yes	complemented
32	C₂×C₄⋊C₄×C₁₉	—	2	16	no	—
33	C₂×C₄⋊C₄×C₁₉	—	2	8	no	—
34	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
35	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
36	C₂×C₄⋊C₄×C₁₉	—	2	8	no	—
37	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₉	—	2	8	no	—
39	C₂×C₄⋊C₄×C₁₉	—	4	8	no	—
40	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
42	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
43	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₁₉	D₄	1	2	yes	—
45	C₂×C₄²×C₁₉	D₄	1	4	yes	—
46	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
47	C₁₉×#97	—	2	8	no	—
48	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	—	2	4	no	—
50	C₂²×C₄×C₁₉	—	2	4	no	—
51	C₂²×C₄×C₁₉	—	2	8	no	—
52	C₂²×C₄×C₁₉	—	2	4	no	—
53	C₂²×C₄×C₁₉	—	2	4	no	—
54	C₂²×C₄×C₁₉	—	4	4	no	—
55	C₂²×C₄×C₁₉	—	4	4	no	—
56	C₂²×C₄×C₁₉	—	4	4	no	—
57	C₂²×C₄×C₁₉	—	4	8	no	—
58	C₂²×C₄×C₁₉	C₂²⋊C₄	1	8	yes	—
59	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
61	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₉	C₂×Q₈	1	4	yes	—
64	C₂²×C₄×C₁₉	#41	1	4	yes	—
65	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
66	C₂×C₈×C₁₉	—	2	16	no	—
67	C₂×C₈×C₁₉	—	2	4	no	—
68	C₂×C₈×C₁₉	—	2	4	no	—
69	C₂×C₈×C₁₉	C₂×Q₈	1	8	yes	—
70	C₄⋊C₄×C₁₉	—	2	32	no	—
71	C₄⋊C₄×C₁₉	—	2	8	no	—
72	C₄⋊C₄×C₁₉	—	2	4	no	—
73	C₄⋊C₄×C₁₉	—	2	8	no	—
74	C₄⋊C₄×C₁₉	—	4	16	no	—
75	C₄⋊C₄×C₁₉	—	4	4	no	—
76	C₄⋊C₄×C₁₉	C₂²×C₄	1	8	yes	—
77	C₄²×C₁₉	C₂²⋊C₄	1	8	yes	—
78	C₄²×C₁₉	C₂²×C₄	1	4	yes	—
79	C₄²×C₁₉	C₂×D₄	1	4	yes	—
80	C₂×#1992	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂³×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
82	C₂³×C₁₉	#124	1	2	yes	—
83	C₂³×C₁₉	C₄⋊Q₈	1	2	yes	—
84	C₂³×C₁₉	AGL(1,ℤ/8)	1	4	yes	—
85	C₂³×C₁₉	#137	1	4	yes	—
86	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₉	—	2	4	no	—
88	C₂×C₄×C₁₉	—	2	4	no	—
89	C₂×C₄×C₁₉	—	2	16	no	—
90	C₂×C₄×C₁₉	—	2	8	no	—
91	C₂×C₄×C₁₉	—	4	4	no	—
92	C₂×C₄×C₁₉	—	4	4	no	—
93	C₂×C₄×C₁₉	—	4	16	no	—
94	C₂×C₄×C₁₉	—	4	4	no	—
95	C₂×C₄×C₁₉	—	4	8	no	—
96	C₂×C₄×C₁₉	—	8	8	no	—
97	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	8	yes	—
98	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
99	C₂×C₄×C₁₉	C₄×Q₈	1	8	yes	—
100	C₂×C₄×C₁₉	C₂²⋊Q₈	1	8	yes	—
101	C₂×C₄×C₁₉	#124	1	2	yes	—
102	C₂×C₄×C₁₉	C₄⋊Q₈	1	2	yes	—
103	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
104	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₉	C₂²×Q₈	1	2	yes	—
106	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
107	C₈×C₁₉	—	2	8	no	—
108	C₂²×#125	C₂×C₁₉	1	1	yes	characteristic
109	C₂²×#97	C₂×C₁₉	1	1	yes	characteristic
110	C₂×#384	C₂×C₁₉	1	1	yes	characteristic
111	#1992	C₂×C₁₉	1	8	yes	direct product factor
112	C₂×#340	C₂×C₁₉	1	4	yes	—
113	C₂²×C₁₉	#391	1	4	yes	—
114	C₂²×C₁₉	#417	1	8	yes	—
115	C₂²×C₁₉	#490	1	4	yes	—
116	C₂²×C₁₉	#491	1	4	yes	—
117	C₂²×C₁₉	#501	1	8	yes	—
118	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
119	C₂²×C₁₉	C₂×#124	1	1	yes	characteristic
120	C₂²×C₁₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
121	C₂²×C₁₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
122	C₂²×C₁₉	C₂×#137	1	2	yes	—
123	C₄×C₁₉	—	2	8	no	—
124	C₄×C₁₉	—	4	8	no	—
125	C₄×C₁₉	—	8	4	no	—
126	C₄×C₁₉	#391	1	4	yes	—
127	C₄×C₁₉	C₂×C₄×Q₈	1	2	yes	—
128	C₄×C₁₉	C₂×C₂²⋊Q₈	1	2	yes	—
129	C₂³×C₈	C₂²×C₁₉	1	2	yes	—
130	C₂×#125	—	2	2	no	—
131	C₂×#125	—	2	4	no	—
132	C₂×#125	C₄×C₁₉	1	8	yes	complemented
133	C₂×#125	C₂²×C₁₉	1	4	yes	—
134	C₂²×C₄⋊C₄	—	2	2	no	—
135	C₂²×C₄⋊C₄	—	2	2	no	—
136	C₂²×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
137	C₂²×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
138	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
139	C₂×#97	—	2	4	no	—
140	C₂×#97	C₂²×C₁₉	1	4	yes	—
141	#384	—	2	2	no	—
142	#384	C₂²×C₁₉	1	4	yes	—
143	#340	C₂²×C₁₉	1	16	yes	—
144	C₂×C₁₉	#1992	1	8	yes	direct product factor
145	C₂×C₁₉	C₂×#391	1	1	yes	characteristic
146	C₂×C₁₉	C₂×#417	1	2	yes	—
147	C₂×C₁₉	C₂×#490	1	1	yes	characteristic
148	C₂×C₁₉	C₂×#491	1	1	yes	characteristic
149	C₂×C₁₉	C₂×#501	1	2	yes	—
150	C₂³×C₄	—	2	2	no	—
151	C₂³×C₄	—	2	2	no	—
152	C₂³×C₄	D₄×C₁₉	1	2	yes	—
153	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
154	C₂²×C₈	—	2	4	no	—
155	C₂²×C₈	—	2	4	no	—
156	C₂²×C₈	Q₈×C₁₉	1	8	yes	—
157	C₂²×C₈	C₂³×C₁₉	1	4	yes	—
158	#125	—	2	16	no	—
159	#125	—	2	8	no	—
160	#125	C₂×C₄×C₁₉	1	16	yes	complemented
161	C₂×C₄⋊C₄	—	2	16	no	—
162	C₂×C₄⋊C₄	—	2	8	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	4	no	—
165	C₂×C₄⋊C₄	—	2	8	no	—
166	C₂×C₄⋊C₄	—	2	8	no	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	—	4	8	no	—
169	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	8	yes	—
170	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
171	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
172	C₂×C₄²	C₂×C₄×C₁₉	1	4	yes	—
173	C₂×C₄²	D₄×C₁₉	1	2	yes	—
174	C₂×C₄²	D₄×C₁₉	1	4	yes	—
175	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
176	#97	—	2	8	no	—
177	C₁₉	C₂×#1992	1	1	yes	characteristic, direct product factor, Sylow
178	C₂⁴	C₂×D₄×C₁₉	1	1	yes	characteristic
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	4	no	—
181	C₂²×C₄	—	2	8	no	—
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	—	4	4	no	—
185	C₂²×C₄	—	4	4	no	—
186	C₂²×C₄	—	4	4	no	—
187	C₂²×C₄	—	4	8	no	—
188	C₂²×C₄	C₂²⋊C₄×C₁₉	1	8	yes	—
189	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
190	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
191	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
193	C₂²×C₄	C₂×Q₈×C₁₉	1	4	yes	—
194	C₂²×C₄	#41×C₁₉	1	4	yes	—
195	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
196	C₂×C₈	—	2	16	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	C₂×Q₈×C₁₉	1	8	yes	—
200	C₄⋊C₄	—	2	32	no	—
201	C₄⋊C₄	—	2	8	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	8	no	—
204	C₄⋊C₄	—	4	16	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₄⋊C₄	C₂²×C₄×C₁₉	1	8	yes	—
207	C₄²	C₂²⋊C₄×C₁₉	1	8	yes	—
208	C₄²	C₂²×C₄×C₁₉	1	4	yes	—
209	C₄²	C₂×D₄×C₁₉	1	4	yes	—
210	C₂³	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
211	C₂³	C₁₉×#124	1	2	yes	—
212	C₂³	C₁₉×C₄⋊Q₈	1	2	yes	—
213	C₂³	C₁₉×AGL(1,ℤ/8)	1	4	yes	—
214	C₂³	C₁₉×#137	1	4	yes	—
215	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
216	C₂×C₄	—	2	16	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	2	8	no	—
220	C₂×C₄	—	4	16	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	4	8	no	—
225	C₂×C₄	—	8	8	no	—
226	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	8	yes	—
227	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
228	C₂×C₄	C₄×Q₈×C₁₉	1	8	yes	—
229	C₂×C₄	C₁₉×C₂²⋊Q₈	1	8	yes	—
230	C₂×C₄	C₁₉×#124	1	2	yes	—
231	C₂×C₄	C₁₉×C₄⋊Q₈	1	2	yes	—
232	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
233	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
234	C₂×C₄	C₂²×Q₈×C₁₉	1	2	yes	—
235	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
236	C₈	—	2	8	no	—
237	C₂²	C₁₉×#391	1	4	yes	—
238	C₂²	C₁₉×#417	1	8	yes	—
239	C₂²	C₁₉×#490	1	4	yes	—
240	C₂²	C₁₉×#491	1	4	yes	—
241	C₂²	C₁₉×#501	1	8	yes	—
242	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
243	C₂²	C₂×C₁₉×#124	1	1	yes	characteristic
244	C₂²	C₂×C₁₉×C₄⋊Q₈	1	1	yes	characteristic
245	C₂²	C₂×C₁₉×AGL(1,ℤ/8)	1	2	yes	—
246	C₂²	C₂×C₁₉×#137	1	2	yes	—
247	C₄	—	2	8	no	—
248	C₄	—	4	8	no	—
249	C₄	—	8	4	no	—
250	C₄	C₁₉×#391	1	4	yes	—
251	C₄	C₂×C₄×Q₈×C₁₉	1	2	yes	—
252	C₄	C₂×C₁₉×C₂²⋊Q₈	1	2	yes	—
253	C₂	C₁₉×#1992	1	8	yes	direct product factor
254	C₂	C₂×C₁₉×#391	1	1	yes	characteristic
255	C₂	C₂×C₁₉×#417	1	2	yes	—
256	C₂	C₂×C₁₉×#490	1	1	yes	characteristic
257	C₂	C₂×C₁₉×#491	1	1	yes	characteristic
258	C₂	C₂×C₁₉×#501	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₉	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₉ ◁ C₂²×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₉×#1992	C₂	direct (N × H)
9. C₂×C₁₉×#125	C₄	semidirect
31. C₁₉×#125	C₂×C₄	semidirect
80. C₂×#1992	C₁₉	direct (N × H)
111. #1992	C₂×C₁₉	direct (N × H)
132. C₂×#125	C₄×C₁₉	semidirect
144. C₂×C₁₉	#1992	direct (N × H)
160. #125	C₂×C₄×C₁₉	semidirect
177. C₁₉	C₂×#1992	direct (N × H)
253. C₂	C₁₉×#1992	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₉×#125	2
C₂²×C₁₉×#97	2
C₂×C₁₉×#384	2
C₁₉×#1992	2
C₂×C₁₉×#340	2
C₂×#1992	19

Subgroups — C2×C19×#1992 #18027278

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#1992 #18027278