Subgroups — C₂×C₁₉×#1375 #18018505

Order 4864.

630 conjugacy classes of subgroups.

The subgroup lattice has 258 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	2	yes	maximal
2	C₂²×C₁₉×#97	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1375	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₈×C₁₉	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
6	C₂²×C₄²×C₁₉	C₄	1	1	yes	characteristic
7	C₂×C₁₉×M₃₂	—	2	4	no	—
8	C₂×C₁₉×M₃₂	—	2	4	no	—
9	C₂×C₁₉×M₃₂	C₄	1	4	yes	complemented
10	C₂×C₁₉×M₃₂	C₂²	1	4	yes	—
11	C₂²×C₁₆×C₁₉	C₄	1	4	yes	complemented
12	C₂×C₁₉×#97	—	2	4	no	—
13	C₂×C₁₉×#97	C₄	1	2	yes	—
14	C₂×C₁₉×#97	C₂²	1	2	yes	—
15	C₂³×C₄×C₁₉	—	2	1	no	—
16	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₁₉	—	2	2	no	—
18	C₂²×C₈×C₁₉	—	2	1	no	—
19	C₂²×C₈×C₁₉	—	2	1	no	—
20	C₂²×C₈×C₁₉	C₈	1	4	yes	—
21	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₉	D₄	1	2	yes	—
26	C₂²×C₈×C₁₉	D₄	1	4	yes	—
27	C₂²×C₈×C₁₉	Q₈	1	2	yes	—
28	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₁₉	—	2	2	no	—
30	C₂×C₄²×C₁₉	—	2	1	no	—
31	C₂×C₄²×C₁₉	C₈	1	4	yes	—
32	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
33	C₁₉×M₃₂	—	2	8	no	—
34	C₁₉×M₃₂	—	2	4	no	—
35	C₁₉×M₃₂	—	2	4	no	—
36	C₂×C₁₆×C₁₉	—	2	8	no	—
37	C₂×C₁₆×C₁₉	C₂×C₄	1	8	yes	complemented
38	C₁₉×#97	—	2	4	no	—
39	C₁₉×#97	—	2	4	no	—
40	C₂⁴×C₁₉	M₁₆	1	1	yes	characteristic
41	C₂²×C₄×C₁₉	—	2	4	no	—
42	C₂²×C₄×C₁₉	—	2	2	no	—
43	C₂²×C₄×C₁₉	—	4	4	no	—
44	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
45	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
46	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₁₉	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₁₉	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₁₉	M₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	M₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	M₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₁₉	—	2	4	no	—
55	C₂×C₈×C₁₉	—	2	4	no	—
56	C₂×C₈×C₁₉	—	2	2	no	—
57	C₂×C₈×C₁₉	—	2	2	no	—
58	C₂×C₈×C₁₉	C₄²	1	2	yes	—
59	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
61	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
62	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
63	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
64	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
65	C₂×C₈×C₁₉	C₂×C₈	1	4	yes	—
66	C₂×C₈×C₁₉	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
70	C₄²×C₁₉	—	2	4	no	—
71	C₄²×C₁₉	—	2	2	no	—
72	C₄²×C₁₉	C₂×C₈	1	4	yes	—
73	C₁₆×C₁₉	—	2	8	no	—
74	C₂×#1375	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₁₉	—	2	4	no	—
76	C₂³×C₁₉	#97	1	1	yes	characteristic
77	C₂³×C₁₉	#97	1	1	yes	characteristic
78	C₂³×C₁₉	C₂²⋊C₈	1	2	yes	—
79	C₂³×C₁₉	C₄⋊C₈	1	2	yes	—
80	C₂³×C₁₉	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₁₉	—	2	4	no	—
82	C₂×C₄×C₁₉	—	4	4	no	—
83	C₂×C₄×C₁₉	—	4	8	no	—
84	C₂×C₄×C₁₉	#95	1	2	yes	—
85	C₂×C₄×C₁₉	#95	1	2	yes	—
86	C₂×C₄×C₁₉	C₄×C₈	1	2	yes	—
87	C₂×C₄×C₁₉	C₄×C₈	1	2	yes	—
88	C₂×C₄×C₁₉	#97	1	1	yes	characteristic
89	C₂×C₄×C₁₉	#97	1	1	yes	characteristic
90	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
91	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
92	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
93	C₂×C₄×C₁₉	C₄⋊C₈	1	2	yes	—
94	C₂×C₄×C₁₉	C₄⋊C₈	1	2	yes	—
95	C₂×C₄×C₁₉	C₄⋊C₈	1	2	yes	—
96	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
97	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
102	C₈×C₁₉	—	2	4	no	—
103	C₈×C₁₉	#95	1	4	yes	—
104	C₈×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
105	C₈×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
106	C₂²×M₃₂	C₂×C₁₉	1	2	yes	—
107	C₂²×#97	C₂×C₁₉	1	1	yes	characteristic
108	#1375	C₂×C₁₉	1	4	yes	direct product factor
109	C₂²×C₁₉	—	2	2	no	—
110	C₂²×C₁₉	—	2	2	no	—
111	C₂²×C₁₉	—	2	4	no	—
112	C₂²×C₁₉	—	2	4	no	—
113	C₂²×C₁₉	#336	1	2	yes	—
114	C₂²×C₁₉	#336	1	2	yes	—
115	C₂²×C₁₉	#347	1	4	yes	—
116	C₂²×C₁₉	C₂×#97	1	1	yes	characteristic
117	C₂²×C₁₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
118	C₂²×C₁₉	C₂×C₄⋊C₈	1	1	yes	characteristic
119	C₄×C₁₉	—	4	4	no	—
120	C₄×C₁₉	#336	1	2	yes	—
121	C₄×C₁₉	#336	1	2	yes	—
122	C₄×C₁₉	C₂×#95	1	1	yes	characteristic
123	C₄×C₁₉	C₂×C₄×C₈	1	1	yes	characteristic
124	C₄×C₁₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
125	C₄×C₁₉	C₂×C₄⋊C₈	1	1	yes	characteristic
126	C₂³×C₈	C₄×C₁₉	1	1	yes	characteristic
127	C₂³×C₈	C₂²×C₁₉	1	1	yes	characteristic
128	C₂²×C₄²	C₄×C₁₉	1	1	yes	characteristic
129	C₂×M₃₂	—	2	4	no	—
130	C₂×M₃₂	—	2	4	no	—
131	C₂×M₃₂	C₄×C₁₉	1	4	yes	complemented
132	C₂×M₃₂	C₂²×C₁₉	1	4	yes	—
133	C₂²×C₁₆	C₄×C₁₉	1	4	yes	complemented
134	C₂×#97	—	2	4	no	—
135	C₂×#97	C₄×C₁₉	1	2	yes	—
136	C₂×#97	C₂²×C₁₉	1	2	yes	—
137	C₂×C₁₉	—	2	1	no	—
138	C₂×C₁₉	—	2	2	no	—
139	C₂×C₁₉	—	2	1	no	—
140	C₂×C₁₉	#1375	1	4	yes	direct product factor
141	C₂×C₁₉	C₂×#336	1	1	yes	characteristic
142	C₂×C₁₉	C₂×#347	1	2	yes	—
143	C₂³×C₄	—	2	1	no	—
144	C₂³×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
145	C₂²×C₈	—	2	2	no	—
146	C₂²×C₈	—	2	1	no	—
147	C₂²×C₈	—	2	1	no	—
148	C₂²×C₈	C₈×C₁₉	1	4	yes	—
149	C₂²×C₈	C₂×C₄×C₁₉	1	2	yes	—
150	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
151	C₂²×C₈	C₂×C₄×C₁₉	1	2	yes	—
152	C₂²×C₈	C₂×C₄×C₁₉	1	2	yes	—
153	C₂²×C₈	D₄×C₁₉	1	2	yes	—
154	C₂²×C₈	D₄×C₁₉	1	4	yes	—
155	C₂²×C₈	Q₈×C₁₉	1	2	yes	—
156	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
157	C₂×C₄²	—	2	2	no	—
158	C₂×C₄²	—	2	1	no	—
159	C₂×C₄²	C₈×C₁₉	1	4	yes	—
160	C₂×C₄²	C₂×C₄×C₁₉	1	2	yes	—
161	M₃₂	—	2	8	no	—
162	M₃₂	—	2	4	no	—
163	M₃₂	—	2	4	no	—
164	C₂×C₁₆	—	2	8	no	—
165	C₂×C₁₆	C₂×C₄×C₁₉	1	8	yes	complemented
166	#97	—	2	4	no	—
167	#97	—	2	4	no	—
168	C₁₉	C₂×#1375	1	1	yes	characteristic, direct product factor, Sylow
169	C₂⁴	M₁₆×C₁₉	1	1	yes	characteristic
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	—	4	4	no	—
173	C₂²×C₄	C₄²×C₁₉	1	1	yes	characteristic
174	C₂²×C₄	C₄²×C₁₉	1	1	yes	characteristic
175	C₂²×C₄	C₂²⋊C₄×C₁₉	1	2	yes	—
176	C₂²×C₄	C₄⋊C₄×C₁₉	1	2	yes	—
177	C₂²×C₄	C₂×C₈×C₁₉	1	2	yes	—
178	C₂²×C₄	C₂×C₈×C₁₉	1	2	yes	—
179	C₂²×C₄	M₁₆×C₁₉	1	1	yes	characteristic
180	C₂²×C₄	M₁₆×C₁₉	1	1	yes	characteristic
181	C₂²×C₄	M₁₆×C₁₉	1	1	yes	characteristic
182	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	C₄²×C₁₉	1	2	yes	—
188	C₂×C₈	C₂²⋊C₄×C₁₉	1	4	yes	—
189	C₂×C₈	C₂²⋊C₄×C₁₉	1	4	yes	—
190	C₂×C₈	C₂²⋊C₄×C₁₉	1	2	yes	—
191	C₂×C₈	C₄⋊C₄×C₁₉	1	4	yes	—
192	C₂×C₈	C₄⋊C₄×C₁₉	1	4	yes	—
193	C₂×C₈	C₄⋊C₄×C₁₉	1	2	yes	—
194	C₂×C₈	C₂×C₈×C₁₉	1	4	yes	—
195	C₂×C₈	C₂²×C₄×C₁₉	1	2	yes	—
196	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
197	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
198	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
199	C₄²	—	2	2	no	—
200	C₄²	—	2	4	no	—
201	C₄²	C₂×C₈×C₁₉	1	4	yes	—
202	C₁₆	—	2	8	no	—
203	C₂³	—	2	4	no	—
204	C₂³	C₁₉×#97	1	1	yes	characteristic
205	C₂³	C₁₉×#97	1	1	yes	characteristic
206	C₂³	C₁₉×C₂²⋊C₈	1	2	yes	—
207	C₂³	C₁₉×C₄⋊C₈	1	2	yes	—
208	C₂³	C₂×M₁₆×C₁₉	1	1	yes	characteristic
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	4	8	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	C₁₉×#95	1	2	yes	—
213	C₂×C₄	C₁₉×#95	1	2	yes	—
214	C₂×C₄	C₄×C₈×C₁₉	1	2	yes	—
215	C₂×C₄	C₄×C₈×C₁₉	1	2	yes	—
216	C₂×C₄	C₁₉×#97	1	1	yes	characteristic
217	C₂×C₄	C₁₉×#97	1	1	yes	characteristic
218	C₂×C₄	C₁₉×C₂²⋊C₈	1	2	yes	—
219	C₂×C₄	C₁₉×C₂²⋊C₈	1	2	yes	—
220	C₂×C₄	C₁₉×C₂²⋊C₈	1	2	yes	—
221	C₂×C₄	C₁₉×C₄⋊C₈	1	2	yes	—
222	C₂×C₄	C₁₉×C₄⋊C₈	1	2	yes	—
223	C₂×C₄	C₁₉×C₄⋊C₈	1	2	yes	—
224	C₂×C₄	C₂×C₄²×C₁₉	1	1	yes	characteristic
225	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
226	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
227	C₂×C₄	C₂²×C₈×C₁₉	1	1	yes	characteristic
228	C₂×C₄	C₂²×C₈×C₁₉	1	1	yes	characteristic
229	C₂×C₄	C₂×M₁₆×C₁₉	1	1	yes	characteristic
230	C₈	—	2	4	no	—
231	C₈	C₁₉×#95	1	4	yes	—
232	C₈	C₂×C₂²⋊C₄×C₁₉	1	2	yes	—
233	C₈	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
234	C₂²	—	2	4	no	—
235	C₂²	—	2	4	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	C₁₉×#336	1	2	yes	—
239	C₂²	C₁₉×#336	1	2	yes	—
240	C₂²	C₁₉×#347	1	4	yes	—
241	C₂²	C₂×C₁₉×#97	1	1	yes	characteristic
242	C₂²	C₂×C₁₉×C₂²⋊C₈	1	1	yes	characteristic
243	C₂²	C₂×C₁₉×C₄⋊C₈	1	1	yes	characteristic
244	C₄	—	4	4	no	—
245	C₄	C₁₉×#336	1	2	yes	—
246	C₄	C₁₉×#336	1	2	yes	—
247	C₄	C₂×C₁₉×#95	1	1	yes	characteristic
248	C₄	C₂×C₄×C₈×C₁₉	1	1	yes	characteristic
249	C₄	C₂×C₁₉×C₂²⋊C₈	1	1	yes	characteristic
250	C₄	C₂×C₁₉×C₄⋊C₈	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	1	no	—
253	C₂	—	2	2	no	—
254	C₂	C₁₉×#1375	1	4	yes	direct product factor
255	C₂	C₂×C₁₉×#336	1	1	yes	characteristic
256	C₂	C₂×C₁₉×#347	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₉	C₄²
commutator	C₄	C₂×C₄×C₈×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
socle	C₂³×C₁₉	#97
prime radical	C₂⁴×C₁₉	M₁₆
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₉×#1375	C₂	direct (N × H)
9. C₂×C₁₉×M₃₂	C₄	semidirect
11. C₂²×C₁₆×C₁₉	C₄	semidirect
37. C₂×C₁₆×C₁₉	C₂×C₄	semidirect
74. C₂×#1375	C₁₉	direct (N × H)
108. #1375	C₂×C₁₉	direct (N × H)
131. C₂×M₃₂	C₄×C₁₉	semidirect
133. C₂²×C₁₆	C₄×C₁₉	semidirect
140. C₂×C₁₉	#1375	direct (N × H)
165. C₂×C₁₆	C₂×C₄×C₁₉	semidirect
168. C₁₉	C₂×#1375	direct (N × H)
254. C₂	C₁₉×#1375	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
C₂²×C₁₉×#97	2
C₁₉×#1375	2
C₂×#1375	19

Subgroups — C2×C19×#1375 #18018505

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#1375 #18018505