Subgroups — C₂×C₁₉×#2159 #18039367

Order 4864.

718 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×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×#427	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₉×#107	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#2159	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₉×#367	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₉×#366	C₂	1	2	yes	maximal, complemented
7	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
8	C₂×C₁₉×#117	—	2	1	no	—
9	C₂²×C₄⋊C₄×C₁₉	—	2	1	no	—
10	C₂×C₁₉×M₃₂	C₄	1	8	yes	complemented
11	C₂×C₁₉×M₃₂	C₂²	1	4	yes	—
12	C₂²×C₁₆×C₁₉	C₂²	1	2	yes	—
13	C₁₉×#427	—	2	2	no	—
14	C₁₉×#427	C₂²	1	4	yes	—
15	C₂×C₁₉×#107	—	2	1	no	—
16	C₂×C₁₉×#107	—	2	2	no	—
17	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
18	C₂×C₁₉×#107	C₂²	1	2	yes	—
19	C₂×C₁₉×#107	C₂²	1	1	yes	characteristic
20	C₂×C₁₉×#107	C₂²	1	4	yes	—
21	C₂×C₁₉×#106	—	2	1	no	—
22	C₁₉×#367	C₂²	1	8	yes	complemented
23	C₁₉×#366	C₂²	1	8	yes	complemented
24	C₂³×C₄×C₁₉	—	4	1	no	—
25	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₉	D₄	1	2	yes	—
29	C₂²×C₈×C₁₉	Q₈	1	4	yes	—
30	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
32	C₁₉×#117	—	2	4	no	—
33	C₁₉×#117	—	4	2	no	—
34	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
35	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
36	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
37	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₉	—	4	2	no	—
39	C₂×C₄⋊C₄×C₁₉	—	4	1	no	—
40	C₂×C₂²⋊C₄×C₁₉	—	4	1	no	—
41	C₂×C₄²×C₁₉	—	4	1	no	—
42	C₁₉×M₃₂	C₂×C₄	1	16	yes	complemented
43	C₂×C₁₆×C₁₉	C₂×C₄	1	8	yes	complemented
44	C₂×C₁₆×C₁₉	C₂³	1	4	yes	—
45	C₁₉×#107	—	2	4	no	—
46	C₁₉×#107	—	2	2	no	—
47	C₁₉×#107	—	2	1	no	—
48	C₁₉×#107	—	2	1	no	—
49	C₁₉×#107	C₂³	1	2	yes	—
50	C₁₉×#107	C₂³	1	4	yes	—
51	C₁₉×#107	C₂³	1	2	yes	—
52	C₁₉×#106	—	2	4	no	—
53	C₂⁴×C₁₉	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	—	4	4	no	—
55	C₂²×C₄×C₁₉	—	4	1	no	—
56	C₂²×C₄×C₁₉	—	4	1	no	—
57	C₂²×C₄×C₁₉	—	4	2	no	—
58	C₂²×C₄×C₁₉	—	8	4	no	—
59	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
60	C₂²×C₄×C₁₉	D₈	1	2	yes	—
61	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	Q₁₆	1	4	yes	—
63	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
66	C₂×C₈×C₁₉	C₄⋊C₄	1	8	yes	—
67	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
69	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
70	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₁₉	C₂×Q₈	1	2	yes	—
72	C₂×C₈×C₁₉	C₂×Q₈	1	4	yes	—
73	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
74	C₄⋊C₄×C₁₉	—	2	4	no	—
75	C₄⋊C₄×C₁₉	—	2	2	no	—
76	C₄⋊C₄×C₁₉	—	2	2	no	—
77	C₄⋊C₄×C₁₉	—	4	4	no	—
78	C₄⋊C₄×C₁₉	—	4	2	no	—
79	C₄⋊C₄×C₁₉	—	4	1	no	—
80	C₄⋊C₄×C₁₉	—	4	1	no	—
81	C₂²⋊C₄×C₁₉	—	4	4	no	—
82	C₄²×C₁₉	—	4	4	no	—
83	C₁₆×C₁₉	C₂²×C₄	1	8	yes	complemented
84	C₂×#2159	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂³×C₁₉	—	2	2	no	—
86	C₂³×C₁₉	—	2	2	no	—
87	C₂³×C₁₉	#107	1	4	yes	—
88	C₂³×C₁₉	C₂×D₈	1	2	yes	—
89	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₁₉	—	4	2	no	—
91	C₂×C₄×C₁₉	—	4	2	no	—
92	C₂×C₄×C₁₉	—	4	4	no	—
93	C₂×C₄×C₁₉	—	8	8	no	—
94	C₂×C₄×C₁₉	—	8	2	no	—
95	C₂×C₄×C₁₉	—	8	4	no	—
96	C₂×C₄×C₁₉	—	8	2	no	—
97	C₂×C₄×C₁₉	#107	1	8	yes	—
98	C₂×C₄×C₁₉	#107	1	4	yes	—
99	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	4	yes	—
100	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
102	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₁₉	C₂×Q₁₆	1	4	yes	—
104	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
105	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
106	C₈×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
107	C₈×C₁₉	C₂×C₄⋊C₄	1	4	yes	—
108	C₈×C₁₉	C₂³×C₄	1	1	yes	characteristic
109	C₈×C₁₉	C₂²×Q₈	1	1	yes	characteristic
110	C₂²×M₃₂	C₂×C₁₉	1	1	yes	characteristic
111	C₂×#427	C₂×C₁₉	1	1	yes	characteristic
112	C₂²×#107	C₂×C₁₉	1	1	yes	characteristic
113	#2159	C₂×C₁₉	1	8	yes	direct product factor
114	C₂×#367	C₂×C₁₉	1	2	yes	complemented
115	C₂×#366	C₂×C₁₉	1	2	yes	complemented
116	C₂²×C₁₉	—	2	2	no	—
117	C₂²×C₁₉	—	2	2	no	—
118	C₂²×C₁₉	—	2	8	no	—
119	C₂²×C₁₉	C₂×#107	1	4	yes	—
120	C₂²×C₁₉	C₂×#107	1	2	yes	—
121	C₂²×C₁₉	#509	1	2	yes	—
122	C₂²×C₁₉	#510	1	2	yes	—
123	C₂²×C₁₉	C₂²×D₈	1	1	yes	characteristic
124	C₄×C₁₉	—	8	2	no	—
125	C₄×C₁₉	—	8	4	no	—
126	C₄×C₁₉	—	8	2	no	—
127	C₄×C₁₉	C₂×#107	1	4	yes	—
128	C₄×C₁₉	C₂×#107	1	2	yes	—
129	C₄×C₁₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
130	C₄×C₁₉	C₂²×Q₁₆	1	1	yes	characteristic
131	C₂³×C₈	C₂²×C₁₉	1	1	yes	characteristic
132	C₂×#117	—	2	1	no	—
133	C₂²×C₄⋊C₄	—	2	1	no	—
134	C₂×M₃₂	C₄×C₁₉	1	8	yes	complemented
135	C₂×M₃₂	C₂²×C₁₉	1	4	yes	—
136	C₂²×C₁₆	C₂²×C₁₉	1	2	yes	—
137	#427	—	2	2	no	—
138	#427	C₂²×C₁₉	1	4	yes	—
139	C₂×#107	—	2	1	no	—
140	C₂×#107	—	2	2	no	—
141	C₂×#107	C₂²×C₁₉	1	2	yes	—
142	C₂×#107	C₂²×C₁₉	1	4	yes	—
143	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
144	C₂×#107	C₂²×C₁₉	1	1	yes	characteristic
145	C₂×#106	—	2	1	no	—
146	#367	C₂²×C₁₉	1	8	yes	complemented
147	#366	C₂²×C₁₉	1	8	yes	complemented
148	C₂×C₁₉	—	2	4	no	—
149	C₂×C₁₉	#2159	1	4	yes	direct product factor
150	C₂×C₁₉	C₂²×#107	1	1	yes	characteristic
151	C₂×C₁₉	C₂×#509	1	1	yes	characteristic
152	C₂×C₁₉	C₂×#510	1	1	yes	characteristic
153	C₂³×C₄	—	4	1	no	—
154	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
155	C₂²×C₈	C₂×C₄×C₁₉	1	4	yes	—
156	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
157	C₂²×C₈	D₄×C₁₉	1	2	yes	—
158	C₂²×C₈	Q₈×C₁₉	1	4	yes	—
159	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
160	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
161	#117	—	2	4	no	—
162	#117	—	4	2	no	—
163	C₂×C₄⋊C₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	4	no	—
165	C₂×C₄⋊C₄	—	2	1	no	—
166	C₂×C₄⋊C₄	—	2	1	no	—
167	C₂×C₄⋊C₄	—	4	1	no	—
168	C₂×C₄⋊C₄	—	4	2	no	—
169	C₂×C₂²⋊C₄	—	4	1	no	—
170	C₂×C₄²	—	4	1	no	—
171	M₃₂	C₂×C₄×C₁₉	1	16	yes	complemented
172	C₂×C₁₆	C₂×C₄×C₁₉	1	8	yes	complemented
173	C₂×C₁₆	C₂³×C₁₉	1	4	yes	—
174	#107	—	2	4	no	—
175	#107	—	2	2	no	—
176	#107	—	2	1	no	—
177	#107	—	2	1	no	—
178	#107	C₂³×C₁₉	1	2	yes	—
179	#107	C₂³×C₁₉	1	4	yes	—
180	#107	C₂³×C₁₉	1	2	yes	—
181	#106	—	2	4	no	—
182	C₁₉	C₂×#2159	1	1	yes	characteristic, direct product factor, Sylow
183	C₂⁴	D₈×C₁₉	1	1	yes	characteristic
184	C₂²×C₄	—	4	1	no	—
185	C₂²×C₄	—	4	1	no	—
186	C₂²×C₄	—	4	4	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	—	8	4	no	—
189	C₂²×C₄	C₄⋊C₄×C₁₉	1	4	yes	—
190	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
191	C₂²×C₄	D₈×C₁₉	1	2	yes	—
192	C₂²×C₄	Q₁₆×C₁₉	1	4	yes	—
193	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
195	C₂×C₈	C₄⋊C₄×C₁₉	1	4	yes	—
196	C₂×C₈	C₄⋊C₄×C₁₉	1	8	yes	—
197	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
198	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
199	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
200	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
201	C₂×C₈	C₂×Q₈×C₁₉	1	2	yes	—
202	C₂×C₈	C₂×Q₈×C₁₉	1	4	yes	—
203	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	4	4	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	—	4	1	no	—
210	C₄⋊C₄	—	4	1	no	—
211	C₂²⋊C₄	—	4	4	no	—
212	C₄²	—	4	4	no	—
213	C₁₆	C₂²×C₄×C₁₉	1	8	yes	complemented
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	C₁₉×#107	1	4	yes	—
217	C₂³	C₂×D₈×C₁₉	1	2	yes	—
218	C₂³	C₂×D₈×C₁₉	1	1	yes	characteristic
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	8	8	no	—
223	C₂×C₄	—	8	4	no	—
224	C₂×C₄	—	8	2	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	C₁₉×#107	1	8	yes	—
227	C₂×C₄	C₁₉×#107	1	4	yes	—
228	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	4	yes	—
229	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
230	C₂×C₄	C₂×D₈×C₁₉	1	2	yes	—
231	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
232	C₂×C₄	C₂×Q₁₆×C₁₉	1	4	yes	—
233	C₂×C₄	C₂×Q₁₆×C₁₉	1	2	yes	—
234	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
235	C₈	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
236	C₈	C₂×C₄⋊C₄×C₁₉	1	4	yes	—
237	C₈	C₂³×C₄×C₁₉	1	1	yes	characteristic
238	C₈	C₂²×Q₈×C₁₉	1	1	yes	characteristic
239	C₂²	—	2	8	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	C₂×C₁₉×#107	1	4	yes	—
243	C₂²	C₂×C₁₉×#107	1	2	yes	—
244	C₂²	C₁₉×#509	1	2	yes	—
245	C₂²	C₁₉×#510	1	2	yes	—
246	C₂²	C₂²×D₈×C₁₉	1	1	yes	characteristic
247	C₄	—	8	4	no	—
248	C₄	—	8	2	no	—
249	C₄	—	8	2	no	—
250	C₄	C₂×C₁₉×#107	1	4	yes	—
251	C₄	C₂×C₁₉×#107	1	2	yes	—
252	C₄	C₂²×C₄⋊C₄×C₁₉	1	1	yes	characteristic
253	C₄	C₂²×Q₁₆×C₁₉	1	1	yes	characteristic
254	C₂	—	2	4	no	—
255	C₂	C₁₉×#2159	1	4	yes	direct product factor
256	C₂	C₂²×C₁₉×#107	1	1	yes	characteristic
257	C₂	C₂×C₁₉×#509	1	1	yes	characteristic
258	C₂	C₂×C₁₉×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂×D₈
commutator	C₈	C₂³×C₄×C₁₉
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₉
prime radical	C₂⁴×C₁₉	D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂³×C₄×C₁₉ ◁ C₂³×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₉×#2159	C₂	direct (N × H)
5. C₂×C₁₉×#367	C₂	semidirect
6. C₂×C₁₉×#366	C₂	semidirect
10. C₂×C₁₉×M₃₂	C₄	semidirect
22. C₁₉×#367	C₂²	semidirect
23. C₁₉×#366	C₂²	semidirect
42. C₁₉×M₃₂	C₂×C₄	semidirect
43. C₂×C₁₆×C₁₉	C₂×C₄	semidirect
83. C₁₆×C₁₉	C₂²×C₄	semidirect
84. C₂×#2159	C₁₉	direct (N × H)
113. #2159	C₂×C₁₉	direct (N × H)
114. C₂×#367	C₂×C₁₉	semidirect
115. C₂×#366	C₂×C₁₉	semidirect
134. C₂×M₃₂	C₄×C₁₉	semidirect
146. #367	C₂²×C₁₉	semidirect
147. #366	C₂²×C₁₉	semidirect
149. C₂×C₁₉	#2159	direct (N × H)
171. M₃₂	C₂×C₄×C₁₉	semidirect
172. C₂×C₁₆	C₂×C₄×C₁₉	semidirect
182. C₁₉	C₂×#2159	direct (N × H)
213. C₁₆	C₂²×C₄×C₁₉	semidirect
255. C₂	C₁₉×#2159	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×M₃₂	2
C₂×C₁₉×#427	2
C₂²×C₁₉×#107	2
C₁₉×#2159	2
C₂×C₁₉×#367	2
C₂×C₁₉×#366	2
C₂×#2159	19

Subgroups — C2×C19×#2159 #18039367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#2159 #18039367