Subgroups — C₁₉×#21712 #18027671

Order 4864.

822 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	C₂×C₁₉×#446	C₂	1	2	yes	maximal
2	C₄×C₁₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1984	C₂	1	2	yes	maximal
4	C₁₉×#1939	C₂	1	4	yes	maximal
5	C₁₉×#1842	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#1770	C₂	1	4	yes	maximal
7	C₄²×C₈×C₁₉	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₁₉	C₂²	1	2	yes	—
10	C₁₉×#446	—	2	8	no	—
11	C₂×C₁₉×C₄⋊C₈	C₂²	1	4	yes	—
12	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₁₉×C₄⋊C₈	C₂²	1	4	yes	—
14	C₂×C₁₉×#97	C₂²	1	4	yes	—
15	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
19	C₁₉×#391	C₄	1	2	yes	—
20	C₁₉×#389	C₄	1	4	yes	—
21	C₁₉×#384	C₄	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
23	C₄³×C₁₉	C₂²	1	1	yes	characteristic
24	C₁₉×#336	C₂²	1	8	yes	—
25	C₂²×Q₈×C₁₉	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₉	C₂³	1	4	yes	—
27	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
29	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
30	C₁₉×C₄⋊Q₈	—	2	4	no	—
31	C₄×Q₈×C₁₉	—	2	4	no	—
32	C₄×Q₈×C₁₉	—	2	4	no	—
33	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
34	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₉	C₂³	1	4	yes	—
41	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
43	C₁₉×C₄⋊C₈	—	2	8	no	—
44	C₁₉×C₄⋊C₈	—	2	4	no	—
45	C₁₉×C₄⋊C₈	—	2	8	no	—
46	C₁₉×#97	—	2	8	no	—
47	C₄×C₈×C₁₉	—	2	2	no	—
48	C₄×C₈×C₁₉	—	2	4	no	—
49	C₄×C₈×C₁₉	D₄	1	4	yes	—
50	C₄×C₈×C₁₉	Q₈	1	8	yes	complemented
51	C₁₉×#95	C₂×C₄	1	8	yes	—
52	C₂×Q₈×C₁₉	—	2	4	no	—
53	C₂×Q₈×C₁₉	—	2	4	no	—
54	C₂×Q₈×C₁₉	—	2	4	no	—
55	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
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	4	no	—
66	C₂×C₈×C₁₉	—	2	8	no	—
67	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₁₉	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₁₉	C₂×Q₈	1	4	yes	—
70	C₂×C₈×C₁₉	C₂×Q₈	1	4	yes	—
71	C₄⋊C₄×C₁₉	—	2	4	no	—
72	C₄⋊C₄×C₁₉	—	2	8	no	—
73	C₄⋊C₄×C₁₉	—	2	8	no	—
74	C₄⋊C₄×C₁₉	—	2	8	no	—
75	C₄²×C₁₉	—	2	2	no	—
76	C₄²×C₁₉	—	2	4	no	—
77	C₄²×C₁₉	—	2	4	no	—
78	C₄²×C₁₉	—	2	4	no	—
79	C₄²×C₁₉	M₁₆	1	4	yes	—
80	C₄²×C₁₉	C₂×D₄	1	2	yes	—
81	C₄²×C₁₉	C₂×Q₈	1	4	yes	—
82	C₄²×C₁₉	#41	1	2	yes	—
83	C₄²×C₁₉	#41	1	4	yes	—
84	#21712	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂³×C₁₉	C₂³×C₄	1	1	yes	characteristic
86	Q₈×C₁₉	—	4	8	no	—
87	C₂×C₄×C₁₉	—	2	2	no	—
88	C₂×C₄×C₁₉	—	2	2	no	—
89	C₂×C₄×C₁₉	—	2	2	no	—
90	C₂×C₄×C₁₉	—	2	4	no	—
91	C₂×C₄×C₁₉	—	2	4	no	—
92	C₂×C₄×C₁₉	—	2	4	no	—
93	C₂×C₄×C₁₉	—	2	8	no	—
94	C₂×C₄×C₁₉	—	2	4	no	—
95	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
97	C₂×C₄×C₁₉	C₄×Q₈	1	4	yes	—
98	C₂×C₄×C₁₉	C₄×Q₈	1	4	yes	—
99	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
100	C₂×C₄×C₁₉	C₂×M₁₆	1	4	yes	—
101	C₂×C₄×C₁₉	#131	1	2	yes	—
102	C₂×C₄×C₁₉	#131	1	4	yes	—
103	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₉	C₂²×Q₈	1	2	yes	—
105	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
106	C₂×C₄×C₁₉	C₂×#41	1	1	yes	characteristic
107	C₈×C₁₉	—	2	2	no	—
108	C₈×C₁₉	—	2	4	no	—
109	C₈×C₁₉	—	4	4	no	—
110	C₈×C₁₉	C₄⋊Q₈	1	4	yes	complemented
111	C₂×#446	C₂×C₁₉	1	2	yes	—
112	C₄×C₄⋊Q₈	C₂×C₁₉	1	1	yes	characteristic
113	#1984	C₂×C₁₉	1	2	yes	—
114	#1939	C₂×C₁₉	1	4	yes	—
115	#1842	C₂×C₁₉	1	1	yes	characteristic
116	#1770	C₂×C₁₉	1	4	yes	—
117	C₄²×C₈	C₂×C₁₉	1	1	yes	characteristic
118	C₂²×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
119	C₂²×C₁₉	C₂×C₄×Q₈	1	2	yes	—
120	C₂²×C₁₉	C₂²×M₁₆	1	1	yes	characteristic
121	C₂²×C₁₉	C₂×#131	1	1	yes	characteristic
122	C₂²×C₁₉	C₂×#131	1	2	yes	—
123	C₄×C₁₉	—	2	2	no	—
124	C₄×C₁₉	—	2	4	no	—
125	C₄×C₁₉	—	4	4	no	—
126	C₄×C₁₉	—	4	4	no	—
127	C₄×C₁₉	#436	1	4	yes	—
128	C₄×C₁₉	#446	1	8	yes	complemented
129	C₄×C₁₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
130	C₄×C₁₉	#532	1	1	yes	characteristic
131	C₄×C₁₉	#533	1	2	yes	—
132	C₂×C₄⋊Q₈	C₄×C₁₉	1	1	yes	characteristic
133	C₂×C₄×Q₈	C₂²×C₁₉	1	2	yes	—
134	#446	—	2	8	no	—
135	C₂×C₄⋊C₈	C₂²×C₁₉	1	2	yes	—
136	C₂×C₄⋊C₈	C₂²×C₁₉	1	4	yes	—
137	C₂×C₄⋊C₈	C₂²×C₁₉	1	4	yes	—
138	C₂×#97	C₂²×C₁₉	1	4	yes	—
139	C₂×C₄×C₈	C₂²×C₁₉	1	2	yes	—
140	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
141	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
142	C₂×C₄×C₈	C₂²×C₁₉	1	2	yes	—
143	#391	C₄×C₁₉	1	2	yes	—
144	#389	C₄×C₁₉	1	4	yes	—
145	#384	C₄×C₁₉	1	1	yes	characteristic
146	C₄×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
147	C₄³	C₂²×C₁₉	1	1	yes	characteristic
148	#336	C₂²×C₁₉	1	8	yes	—
149	C₂×C₁₉	C₄×C₄⋊Q₈	1	1	yes	characteristic
150	C₂×C₁₉	C₂×#436	1	1	yes	characteristic
151	C₂×C₁₉	#2931	1	1	yes	characteristic
152	C₂×C₁₉	C₂×#446	1	2	yes	—
153	C₂×C₁₉	#2960	1	2	yes	—
154	C₂²×Q₈	C₂×C₄×C₁₉	1	2	yes	—
155	C₂²×C₈	C₂³×C₁₉	1	4	yes	—
156	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
157	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
158	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
159	C₄⋊Q₈	—	2	4	no	—
160	C₄×Q₈	—	2	4	no	—
161	C₄×Q₈	—	2	4	no	—
162	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
163	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	4	yes	—
164	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	4	yes	—
165	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	4	yes	—
166	C₂×C₄²	C₂×C₄×C₁₉	1	2	yes	—
167	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
168	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
169	C₂×C₄²	C₂³×C₁₉	1	4	yes	—
170	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
171	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
172	C₄⋊C₈	—	2	8	no	—
173	C₄⋊C₈	—	2	4	no	—
174	C₄⋊C₈	—	2	8	no	—
175	#97	—	2	8	no	—
176	C₄×C₈	—	2	2	no	—
177	C₄×C₈	—	2	4	no	—
178	C₄×C₈	D₄×C₁₉	1	4	yes	—
179	C₄×C₈	Q₈×C₁₉	1	8	yes	complemented
180	#95	C₂×C₄×C₁₉	1	8	yes	—
181	C₁₉	#21712	1	1	yes	characteristic, direct product factor, Sylow
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	2	4	no	—
184	C₂×Q₈	—	2	4	no	—
185	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
186	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
187	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
188	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
189	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
190	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
191	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	2	8	no	—
197	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
198	C₂×C₈	C₂×D₄×C₁₉	1	2	yes	—
199	C₂×C₈	C₂×Q₈×C₁₉	1	4	yes	—
200	C₂×C₈	C₂×Q₈×C₁₉	1	4	yes	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	—	2	8	no	—
203	C₄⋊C₄	—	2	8	no	—
204	C₄⋊C₄	—	2	8	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	4	no	—
207	C₄²	—	2	4	no	—
208	C₄²	—	2	4	no	—
209	C₄²	M₁₆×C₁₉	1	4	yes	—
210	C₄²	C₂×D₄×C₁₉	1	2	yes	—
211	C₄²	C₂×Q₈×C₁₉	1	4	yes	—
212	C₄²	#41×C₁₉	1	2	yes	—
213	C₄²	#41×C₁₉	1	4	yes	—
214	C₂³	C₂³×C₄×C₁₉	1	1	yes	characteristic
215	Q₈	—	4	8	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
225	C₂×C₄	C₄×D₄×C₁₉	1	2	yes	—
226	C₂×C₄	C₄×Q₈×C₁₉	1	4	yes	—
227	C₂×C₄	C₄×Q₈×C₁₉	1	4	yes	—
228	C₂×C₄	C₂×M₁₆×C₁₉	1	2	yes	—
229	C₂×C₄	C₂×M₁₆×C₁₉	1	4	yes	—
230	C₂×C₄	C₁₉×#131	1	2	yes	—
231	C₂×C₄	C₁₉×#131	1	4	yes	—
232	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
233	C₂×C₄	C₂²×Q₈×C₁₉	1	2	yes	—
234	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
235	C₂×C₄	C₂×#41×C₁₉	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	—	2	4	no	—
238	C₈	—	4	4	no	—
239	C₈	C₁₉×C₄⋊Q₈	1	4	yes	complemented
240	C₂²	C₂×C₄×D₄×C₁₉	1	1	yes	characteristic
241	C₂²	C₂×C₄×Q₈×C₁₉	1	2	yes	—
242	C₂²	C₂²×M₁₆×C₁₉	1	1	yes	characteristic
243	C₂²	C₂×C₁₉×#131	1	1	yes	characteristic
244	C₂²	C₂×C₁₉×#131	1	2	yes	—
245	C₄	—	2	2	no	—
246	C₄	—	2	4	no	—
247	C₄	—	4	4	no	—
248	C₄	—	4	4	no	—
249	C₄	C₁₉×#436	1	4	yes	—
250	C₄	C₁₉×#446	1	8	yes	complemented
251	C₄	C₂×C₁₉×C₄⋊Q₈	1	1	yes	characteristic
252	C₄	C₁₉×#532	1	1	yes	characteristic
253	C₄	C₁₉×#533	1	2	yes	—
254	C₂	C₄×C₁₉×C₄⋊Q₈	1	1	yes	characteristic
255	C₂	C₂×C₁₉×#436	1	1	yes	characteristic
256	C₂	C₁₉×#2931	1	1	yes	characteristic
257	C₂	C₂×C₁₉×#446	1	2	yes	—
258	C₂	C₁₉×#2960	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
50. C₄×C₈×C₁₉	Q₈	semidirect
84. #21712	C₁₉	direct (N × H)
110. C₈×C₁₉	C₄⋊Q₈	semidirect
128. C₄×C₁₉	#446	semidirect
179. C₄×C₈	Q₈×C₁₉	semidirect
181. C₁₉	#21712	direct (N × H)
239. C₈	C₁₉×C₄⋊Q₈	semidirect
250. C₄	C₁₉×#446	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#446	2
C₄×C₁₉×C₄⋊Q₈	2
C₁₉×#1984	2
C₁₉×#1939	2
C₁₉×#1842	2
C₁₉×#1770	2
C₄²×C₈×C₁₉	2
#21712	19

Subgroups — C19×#21712 #18027671

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#21712 #18027671