Subgroups — C₂×C₁₉×#2141 #18039337

Order 4864.

1150 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₉×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#427	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#2141	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₉×#358	C₂	1	4	yes	maximal, complemented
6	C₂³×Q₈×C₁₉	—	2	1	no	—
7	C₂²×Q₁₆×C₁₉	—	2	1	no	—
8	C₂²×Q₁₆×C₁₉	—	2	2	no	—
9	C₂²×Q₁₆×C₁₉	—	2	4	no	—
10	C₂²×Q₁₆×C₁₉	C₄	1	8	yes	complemented
11	C₂²×Q₁₆×C₁₉	C₂²	1	2	yes	—
12	C₂²×Q₁₆×C₁₉	C₂²	1	4	yes	—
13	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
14	C₂×C₁₉×#117	—	2	1	no	—
15	C₂×C₁₉×M₃₂	—	2	4	no	—
16	C₂×C₁₉×M₃₂	C₂²	1	4	yes	—
17	C₂²×C₁₆×C₁₉	C₂²	1	2	yes	—
18	C₁₉×#427	—	2	2	no	—
19	C₁₉×#427	C₂²	1	4	yes	—
20	C₂×C₁₉×#107	C₂²	1	2	yes	—
21	C₂×C₁₉×#106	—	2	1	no	—
22	C₁₉×#358	C₂²	1	16	yes	complemented
23	C₂²×Q₈×C₁₉	—	2	8	no	—
24	C₂²×Q₈×C₁₉	—	2	4	no	—
25	C₂²×Q₈×C₁₉	—	2	2	no	—
26	C₂²×Q₈×C₁₉	—	4	2	no	—
27	C₂²×Q₈×C₁₉	—	4	1	no	—
28	C₂²×Q₈×C₁₉	—	4	4	no	—
29	C₂³×C₄×C₁₉	—	4	1	no	—
30	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
31	C₂×Q₁₆×C₁₉	—	2	16	no	—
32	C₂×Q₁₆×C₁₉	—	2	8	no	—
33	C₂×Q₁₆×C₁₉	—	2	4	no	—
34	C₂×Q₁₆×C₁₉	—	2	2	no	—
35	C₂×Q₁₆×C₁₉	—	2	4	no	—
36	C₂×Q₁₆×C₁₉	—	2	8	no	—
37	C₂×Q₁₆×C₁₉	C₂×C₄	1	8	yes	—
38	C₂×Q₁₆×C₁₉	C₂×C₄	1	16	yes	complemented
39	C₂×Q₁₆×C₁₉	C₂³	1	4	yes	—
40	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
41	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₁₉	D₄	1	2	yes	—
43	C₂²×C₈×C₁₉	D₄	1	4	yes	—
44	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
46	C₁₉×#117	—	2	4	no	—
47	C₁₉×#117	—	4	2	no	—
48	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₁₉	—	4	1	no	—
50	C₂×C₄²×C₁₉	—	4	1	no	—
51	C₁₉×M₃₂	—	2	8	no	—
52	C₂×C₁₆×C₁₉	—	2	4	no	—
53	C₂×C₁₆×C₁₉	C₂³	1	4	yes	—
54	C₁₉×#107	—	2	2	no	—
55	C₁₉×#107	C₂³	1	4	yes	—
56	C₁₉×#106	—	2	4	no	—
57	C₂⁴×C₁₉	D₈	1	1	yes	characteristic
58	C₂×Q₈×C₁₉	—	2	8	no	—
59	C₂×Q₈×C₁₉	—	2	16	no	—
60	C₂×Q₈×C₁₉	—	2	4	no	—
61	C₂×Q₈×C₁₉	—	4	16	no	—
62	C₂×Q₈×C₁₉	—	4	8	no	—
63	C₂×Q₈×C₁₉	—	4	4	no	—
64	C₂×Q₈×C₁₉	—	4	2	no	—
65	C₂×Q₈×C₁₉	—	4	8	no	—
66	C₂×Q₈×C₁₉	—	4	4	no	—
67	C₂²×C₄×C₁₉	—	4	8	no	—
68	C₂²×C₄×C₁₉	—	4	4	no	—
69	C₂²×C₄×C₁₉	—	4	2	no	—
70	C₂²×C₄×C₁₉	—	4	2	no	—
71	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
72	C₂²×C₄×C₁₉	D₈	1	2	yes	—
73	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	SD₁₆	1	4	yes	—
75	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
77	Q₁₆×C₁₉	—	2	8	no	—
78	Q₁₆×C₁₉	—	2	16	no	—
79	Q₁₆×C₁₉	—	2	4	no	—
80	Q₁₆×C₁₉	C₂²×C₄	1	8	yes	—
81	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
82	C₂×C₈×C₁₉	C₂²⋊C₄	1	8	yes	—
83	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₁₉	C₂×D₄	1	4	yes	—
89	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
90	C₄⋊C₄×C₁₉	—	2	4	no	—
91	C₄⋊C₄×C₁₉	—	4	2	no	—
92	C₂²⋊C₄×C₁₉	—	4	4	no	—
93	C₄²×C₁₉	—	4	4	no	—
94	C₁₆×C₁₉	—	2	4	no	—
95	C₂×#2141	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂³×C₁₉	—	2	4	no	—
97	C₂³×C₁₉	#102	1	4	yes	—
98	C₂³×C₁₉	C₂×D₈	1	2	yes	—
99	C₂³×C₁₉	C₂×D₈	1	1	yes	characteristic
100	Q₈×C₁₉	—	2	8	no	—
101	Q₈×C₁₉	—	4	4	no	—
102	Q₈×C₁₉	—	4	8	no	—
103	Q₈×C₁₉	—	4	16	no	—
104	C₂×C₄×C₁₉	—	4	4	no	—
105	C₂×C₄×C₁₉	—	4	16	no	—
106	C₂×C₄×C₁₉	—	4	8	no	—
107	C₂×C₄×C₁₉	—	4	4	no	—
108	C₂×C₄×C₁₉	—	8	4	no	—
109	C₂×C₄×C₁₉	#102	1	8	yes	—
110	C₂×C₄×C₁₉	#102	1	4	yes	—
111	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	4	yes	—
112	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
113	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
114	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₂×SD₁₆	1	4	yes	—
116	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
117	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
118	C₈×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
119	C₈×C₁₉	C₂×C₂²⋊C₄	1	4	yes	—
120	C₈×C₁₉	C₂³×C₄	1	1	yes	characteristic
121	C₈×C₁₉	C₂²×D₄	1	1	yes	characteristic
122	C₂³×Q₁₆	C₂×C₁₉	1	1	yes	characteristic
123	C₂²×M₃₂	C₂×C₁₉	1	1	yes	characteristic
124	C₂×#427	C₂×C₁₉	1	1	yes	characteristic
125	#2141	C₂×C₁₉	1	8	yes	direct product factor
126	C₂×#358	C₂×C₁₉	1	4	yes	complemented
127	C₂²×C₁₉	—	2	4	no	—
128	C₂²×C₁₉	—	2	8	no	—
129	C₂²×C₁₉	C₂×#102	1	4	yes	—
130	C₂²×C₁₉	C₂×#102	1	2	yes	—
131	C₂²×C₁₉	#510	1	4	yes	—
132	C₂²×C₁₉	C₂²×D₈	1	1	yes	characteristic
133	C₄×C₁₉	—	4	8	no	—
134	C₄×C₁₉	—	8	4	no	—
135	C₄×C₁₉	C₂×#102	1	4	yes	—
136	C₄×C₁₉	C₂×#102	1	2	yes	—
137	C₄×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
138	C₄×C₁₉	C₂²×SD₁₆	1	1	yes	characteristic
139	C₂³×Q₈	—	2	1	no	—
140	C₂²×Q₁₆	—	2	1	no	—
141	C₂²×Q₁₆	—	2	2	no	—
142	C₂²×Q₁₆	—	2	4	no	—
143	C₂²×Q₁₆	C₄×C₁₉	1	8	yes	complemented
144	C₂²×Q₁₆	C₂²×C₁₉	1	2	yes	—
145	C₂²×Q₁₆	C₂²×C₁₉	1	4	yes	—
146	C₂³×C₈	C₂²×C₁₉	1	1	yes	characteristic
147	C₂×#117	—	2	1	no	—
148	C₂×M₃₂	—	2	4	no	—
149	C₂×M₃₂	C₂²×C₁₉	1	4	yes	—
150	C₂²×C₁₆	C₂²×C₁₉	1	2	yes	—
151	#427	—	2	2	no	—
152	#427	C₂²×C₁₉	1	4	yes	—
153	C₂×#107	C₂²×C₁₉	1	2	yes	—
154	C₂×#106	—	2	1	no	—
155	#358	C₂²×C₁₉	1	16	yes	complemented
156	C₂×C₁₉	—	2	4	no	—
157	C₂×C₁₉	#2141	1	4	yes	direct product factor
158	C₂×C₁₉	C₂²×#102	1	1	yes	characteristic
159	C₂×C₁₉	C₂×#510	1	2	yes	—
160	C₂²×Q₈	—	2	8	no	—
161	C₂²×Q₈	—	2	2	no	—
162	C₂²×Q₈	—	2	4	no	—
163	C₂²×Q₈	—	4	1	no	—
164	C₂²×Q₈	—	4	2	no	—
165	C₂²×Q₈	—	4	4	no	—
166	C₂³×C₄	—	4	1	no	—
167	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
168	C₂×Q₁₆	—	2	16	no	—
169	C₂×Q₁₆	—	2	8	no	—
170	C₂×Q₁₆	—	2	4	no	—
171	C₂×Q₁₆	—	2	2	no	—
172	C₂×Q₁₆	—	2	4	no	—
173	C₂×Q₁₆	—	2	8	no	—
174	C₂×Q₁₆	C₂×C₄×C₁₉	1	8	yes	—
175	C₂×Q₁₆	C₂×C₄×C₁₉	1	16	yes	complemented
176	C₂×Q₁₆	C₂³×C₁₉	1	4	yes	—
177	C₂²×C₈	C₂×C₄×C₁₉	1	4	yes	—
178	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
179	C₂²×C₈	D₄×C₁₉	1	2	yes	—
180	C₂²×C₈	D₄×C₁₉	1	4	yes	—
181	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
182	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
183	#117	—	2	4	no	—
184	#117	—	4	2	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₂²⋊C₄	—	4	1	no	—
187	C₂×C₄²	—	4	1	no	—
188	M₃₂	—	2	8	no	—
189	C₂×C₁₆	—	2	4	no	—
190	C₂×C₁₆	C₂³×C₁₉	1	4	yes	—
191	#107	—	2	2	no	—
192	#107	C₂³×C₁₉	1	4	yes	—
193	#106	—	2	4	no	—
194	C₁₉	C₂×#2141	1	1	yes	characteristic, direct product factor, Sylow
195	C₂⁴	D₈×C₁₉	1	1	yes	characteristic
196	C₂×Q₈	—	2	8	no	—
197	C₂×Q₈	—	2	16	no	—
198	C₂×Q₈	—	2	4	no	—
199	C₂×Q₈	—	4	4	no	—
200	C₂×Q₈	—	4	2	no	—
201	C₂×Q₈	—	4	8	no	—
202	C₂×Q₈	—	4	4	no	—
203	C₂×Q₈	—	4	16	no	—
204	C₂×Q₈	—	4	8	no	—
205	C₂²×C₄	—	4	4	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	—	4	8	no	—
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	C₂²⋊C₄×C₁₉	1	4	yes	—
210	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
211	C₂²×C₄	D₈×C₁₉	1	2	yes	—
212	C₂²×C₄	SD₁₆×C₁₉	1	4	yes	—
213	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
215	Q₁₆	—	2	8	no	—
216	Q₁₆	—	2	16	no	—
217	Q₁₆	—	2	4	no	—
218	Q₁₆	C₂²×C₄×C₁₉	1	8	yes	—
219	C₂×C₈	C₂²⋊C₄×C₁₉	1	4	yes	—
220	C₂×C₈	C₂²⋊C₄×C₁₉	1	8	yes	—
221	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
222	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
223	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
224	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
225	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
226	C₂×C₈	C₂×D₄×C₁₉	1	4	yes	—
227	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₂²⋊C₄	—	4	4	no	—
231	C₄²	—	4	4	no	—
232	C₁₆	—	2	4	no	—
233	C₂³	—	2	4	no	—
234	C₂³	C₁₉×#102	1	4	yes	—
235	C₂³	C₂×D₈×C₁₉	1	2	yes	—
236	C₂³	C₂×D₈×C₁₉	1	1	yes	characteristic
237	Q₈	—	2	8	no	—
238	Q₈	—	4	8	no	—
239	Q₈	—	4	16	no	—
240	Q₈	—	4	4	no	—
241	C₂×C₄	—	4	16	no	—
242	C₂×C₄	—	4	8	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	8	4	no	—
246	C₂×C₄	C₁₉×#102	1	8	yes	—
247	C₂×C₄	C₁₉×#102	1	4	yes	—
248	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	4	yes	—
249	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
250	C₂×C₄	C₂×D₈×C₁₉	1	2	yes	—
251	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
252	C₂×C₄	C₂×SD₁₆×C₁₉	1	4	yes	—
253	C₂×C₄	C₂×SD₁₆×C₁₉	1	2	yes	—
254	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
255	C₈	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
256	C₈	C₂×C₂²⋊C₄×C₁₉	1	4	yes	—
257	C₈	C₂³×C₄×C₁₉	1	1	yes	characteristic
258	C₈	C₂²×D₄×C₁₉	1	1	yes	characteristic
259	C₂²	—	2	4	no	—
260	C₂²	—	2	8	no	—
261	C₂²	C₂×C₁₉×#102	1	4	yes	—
262	C₂²	C₂×C₁₉×#102	1	2	yes	—
263	C₂²	C₁₉×#510	1	4	yes	—
264	C₂²	C₂²×D₈×C₁₉	1	1	yes	characteristic
265	C₄	—	4	8	no	—
266	C₄	—	8	4	no	—
267	C₄	C₂×C₁₉×#102	1	4	yes	—
268	C₄	C₂×C₁₉×#102	1	2	yes	—
269	C₄	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
270	C₄	C₂²×SD₁₆×C₁₉	1	1	yes	characteristic
271	C₂	—	2	4	no	—
272	C₂	C₁₉×#2141	1	4	yes	direct product factor
273	C₂	C₂²×C₁₉×#102	1	1	yes	characteristic
274	C₂	C₂×C₁₉×#510	1	2	yes	—

	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×#2141	C₂	direct (N × H)
5. C₂×C₁₉×#358	C₂	semidirect
10. C₂²×Q₁₆×C₁₉	C₄	semidirect
22. C₁₉×#358	C₂²	semidirect
38. C₂×Q₁₆×C₁₉	C₂×C₄	semidirect
95. C₂×#2141	C₁₉	direct (N × H)
125. #2141	C₂×C₁₉	direct (N × H)
126. C₂×#358	C₂×C₁₉	semidirect
143. C₂²×Q₁₆	C₄×C₁₉	semidirect
155. #358	C₂²×C₁₉	semidirect
157. C₂×C₁₉	#2141	direct (N × H)
175. C₂×Q₁₆	C₂×C₄×C₁₉	semidirect
194. C₁₉	C₂×#2141	direct (N × H)
272. C₂	C₁₉×#2141	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Q₁₆×C₁₉	2
C₂²×C₁₉×M₃₂	2
C₂×C₁₉×#427	2
C₁₉×#2141	2
C₂×C₁₉×#358	2
C₂×#2141	19

Subgroups — C2×C19×#2141 #18039337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#2141 #18039337