Subgroups — C₁₉×#11983 #18017942

Order 4864.

314 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3170	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1567	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1365	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1362	C₂	1	2	yes	maximal, complemented
6	C₁₉×#533	—	2	1	no	—
7	C₂²×C₁₆×C₁₉	—	2	1	no	—
8	C₁₉×#500	C₂²	1	1	yes	characteristic
9	C₁₉×#494	C₂²	1	1	yes	characteristic
10	C₁₉×#487	—	2	1	no	—
11	C₁₉×#467	—	2	1	no	—
12	C₁₉×#467	C₄	1	2	yes	—
13	C₄×Q₁₆×C₁₉	—	2	1	no	—
14	C₄×Q₁₆×C₁₉	C₄	1	2	yes	—
15	C₁₉×#423	—	2	1	no	—
16	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×C₄⋊C₁₆	—	2	1	no	—
18	C₁₉×C₄⋊C₁₆	C₂²	1	2	yes	—
19	C₁₉×#335	—	2	1	no	—
20	C₁₉×#335	C₂²	1	2	yes	—
21	C₂×Q₁₆×C₁₉	—	2	1	no	—
22	C₂×Q₁₆×C₁₉	C₂×C₄	1	2	yes	—
23	C₂×M₁₆×C₁₉	—	4	1	no	—
24	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
26	C₁₉×C₄⋊Q₈	—	2	1	no	—
27	C₁₉×C₄⋊Q₈	—	2	1	no	—
28	C₁₉×#125	—	4	1	no	—
29	C₁₉×C₂²⋊Q₈	—	2	2	no	—
30	C₁₉×C₂²⋊Q₈	—	4	1	no	—
31	C₄×Q₈×C₁₉	—	2	2	no	—
32	C₄×Q₈×C₁₉	—	4	1	no	—
33	C₁₉×#117	—	4	1	no	—
34	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
35	C₂×C₁₆×C₁₉	—	2	2	no	—
36	C₂×C₁₆×C₁₉	—	4	2	no	—
37	C₁₉×#107	—	2	1	no	—
38	C₁₉×#107	C₂×C₄	1	2	yes	—
39	C₁₉×C₄⋊C₈	—	2	2	no	—
40	C₁₉×C₄⋊C₈	—	4	1	no	—
41	C₁₉×#103	—	2	2	no	—
42	C₁₉×#103	—	2	2	no	—
43	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
45	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
46	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
47	C₂×Q₈×C₁₉	—	2	2	no	—
48	C₂×Q₈×C₁₉	—	4	1	no	—
49	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	D₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	SD₁₆	1	1	yes	characteristic
52	Q₁₆×C₁₉	—	4	2	no	—
53	Q₁₆×C₁₉	—	4	1	no	—
54	Q₁₆×C₁₉	—	4	1	no	—
55	M₁₆×C₁₉	—	8	2	no	—
56	C₂×C₈×C₁₉	—	2	1	no	—
57	C₂×C₈×C₁₉	—	2	1	no	—
58	C₂×C₈×C₁₉	—	4	1	no	—
59	C₂×C₈×C₁₉	—	4	2	no	—
60	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
64	C₄⋊C₄×C₁₉	—	2	2	no	—
65	C₄⋊C₄×C₁₉	—	4	2	no	—
66	C₄⋊C₄×C₁₉	—	4	2	no	—
67	C₄⋊C₄×C₁₉	—	4	1	no	—
68	C₄⋊C₄×C₁₉	—	4	1	no	—
69	C₄⋊C₄×C₁₉	—	4	1	no	—
70	C₂²⋊C₄×C₁₉	—	4	2	no	—
71	C₄²×C₁₉	—	4	2	no	—
72	C₄²×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₁₉	D₈	1	1	yes	characteristic
74	C₄²×C₁₉	SD₁₆	1	1	yes	characteristic
75	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
76	C₁₆×C₁₉	—	4	2	no	—
77	#11983	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₁₉	#102	1	1	yes	characteristic
79	Q₈×C₁₉	—	4	2	no	—
80	Q₈×C₁₉	—	4	2	no	—
81	Q₈×C₁₉	—	8	2	no	—
82	C₂×C₄×C₁₉	—	2	1	no	—
83	C₂×C₄×C₁₉	—	2	1	no	—
84	C₂×C₄×C₁₉	—	4	1	no	—
85	C₂×C₄×C₁₉	—	4	2	no	—
86	C₂×C₄×C₁₉	—	4	2	no	—
87	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
88	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
89	C₂×C₄×C₁₉	#102	1	1	yes	characteristic
90	C₂×C₄×C₁₉	SD₃₂	1	2	yes	—
91	C₂×C₄×C₁₉	Q₃₂	1	2	yes	—
92	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
95	C₈×C₁₉	—	2	1	no	—
96	C₈×C₁₉	—	2	1	no	—
97	C₈×C₁₉	—	2	1	no	—
98	C₈×C₁₉	—	8	2	no	—
99	C₈×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
100	C₈×C₁₉	#137	1	1	yes	characteristic
101	#3170	C₂×C₁₉	1	1	yes	characteristic
102	C₂×C₄⋊C₁₆	C₂×C₁₉	1	1	yes	characteristic
103	#1567	C₂×C₁₉	1	1	yes	characteristic
104	#1365	C₂×C₁₉	1	2	yes	complemented
105	#1362	C₂×C₁₉	1	2	yes	complemented
106	C₂²×C₁₉	—	2	1	no	—
107	C₂²×C₁₉	—	2	1	no	—
108	C₂²×C₁₉	#358	1	2	yes	—
109	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
110	C₄×C₁₉	—	2	1	no	—
111	C₄×C₁₉	—	2	1	no	—
112	C₄×C₁₉	—	2	1	no	—
113	C₄×C₁₉	—	8	2	no	—
114	C₄×C₁₉	—	8	2	no	—
115	C₄×C₁₉	#358	1	2	yes	—
116	C₄×C₁₉	#413	1	1	yes	characteristic
117	C₄×C₁₉	#417	1	1	yes	characteristic
118	C₄×C₁₉	C₂×SD₃₂	1	1	yes	characteristic
119	C₄×C₁₉	C₂×Q₃₂	1	1	yes	characteristic
120	#533	—	2	1	no	—
121	C₂²×C₁₆	—	2	1	no	—
122	#500	C₂²×C₁₉	1	1	yes	characteristic
123	#494	C₂²×C₁₉	1	1	yes	characteristic
124	#487	—	2	1	no	—
125	#467	—	2	1	no	—
126	#467	C₄×C₁₉	1	2	yes	—
127	C₄×Q₁₆	—	2	1	no	—
128	C₄×Q₁₆	C₄×C₁₉	1	2	yes	—
129	#423	—	2	1	no	—
130	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
131	C₄⋊C₁₆	—	2	1	no	—
132	C₄⋊C₁₆	C₂²×C₁₉	1	2	yes	—
133	#335	—	2	1	no	—
134	#335	C₂²×C₁₉	1	2	yes	—
135	C₂×C₁₉	—	2	2	no	—
136	C₂×C₁₉	#1547	1	1	yes	characteristic
137	C₂×C₁₉	C₂×#358	1	1	yes	characteristic
138	C₂×C₁₉	#2148	1	1	yes	characteristic
139	C₂×Q₁₆	—	2	1	no	—
140	C₂×Q₁₆	C₂×C₄×C₁₉	1	2	yes	—
141	C₂×M₁₆	—	4	1	no	—
142	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
143	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
144	C₄⋊Q₈	—	2	1	no	—
145	C₄⋊Q₈	—	2	1	no	—
146	#125	—	4	1	no	—
147	C₂²⋊Q₈	—	2	2	no	—
148	C₂²⋊Q₈	—	4	1	no	—
149	C₄×Q₈	—	2	2	no	—
150	C₄×Q₈	—	4	1	no	—
151	#117	—	4	1	no	—
152	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
153	C₂×C₁₆	—	2	2	no	—
154	C₂×C₁₆	—	4	2	no	—
155	#107	—	2	1	no	—
156	#107	C₂×C₄×C₁₉	1	2	yes	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	4	1	no	—
159	#103	—	2	2	no	—
160	#103	—	2	2	no	—
161	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
162	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
163	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
164	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
165	C₁₉	#11983	1	1	yes	characteristic, direct product factor, Sylow
166	C₂×Q₈	—	2	2	no	—
167	C₂×Q₈	—	4	1	no	—
168	C₂²×C₄	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
169	C₂²×C₄	D₈×C₁₉	1	1	yes	characteristic
170	C₂²×C₄	SD₁₆×C₁₉	1	1	yes	characteristic
171	Q₁₆	—	4	2	no	—
172	Q₁₆	—	4	1	no	—
173	Q₁₆	—	4	1	no	—
174	M₁₆	—	8	2	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	4	1	no	—
178	C₂×C₈	—	4	2	no	—
179	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
180	C₂×C₈	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
181	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
182	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	4	1	no	—
185	C₄⋊C₄	—	4	1	no	—
186	C₄⋊C₄	—	4	1	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₂²⋊C₄	—	4	2	no	—
190	C₄²	—	4	2	no	—
191	C₄²	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
192	C₄²	D₈×C₁₉	1	1	yes	characteristic
193	C₄²	SD₁₆×C₁₉	1	1	yes	characteristic
194	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
195	C₁₆	—	4	2	no	—
196	C₂³	C₁₉×#102	1	1	yes	characteristic
197	Q₈	—	4	2	no	—
198	Q₈	—	4	2	no	—
199	Q₈	—	8	2	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
206	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
207	C₂×C₄	C₁₉×#102	1	1	yes	characteristic
208	C₂×C₄	C₁₉×SD₃₂	1	2	yes	—
209	C₂×C₄	C₁₉×Q₃₂	1	2	yes	—
210	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
211	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
212	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
213	C₈	—	2	1	no	—
214	C₈	—	2	1	no	—
215	C₈	—	2	1	no	—
216	C₈	—	8	2	no	—
217	C₈	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
218	C₈	C₁₉×#137	1	1	yes	characteristic
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	C₁₉×#358	1	2	yes	—
222	C₂²	C₂×C₁₉×#102	1	1	yes	characteristic
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	—	8	2	no	—
227	C₄	—	8	2	no	—
228	C₄	C₁₉×#358	1	2	yes	—
229	C₄	C₁₉×#413	1	1	yes	characteristic
230	C₄	C₁₉×#417	1	1	yes	characteristic
231	C₄	C₂×C₁₉×SD₃₂	1	1	yes	characteristic
232	C₄	C₂×C₁₉×Q₃₂	1	1	yes	characteristic
233	C₂	—	2	2	no	—
234	C₂	C₁₉×#1547	1	1	yes	characteristic
235	C₂	C₂×C₁₉×#358	1	1	yes	characteristic
236	C₂	C₁₉×#2148	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₄×C₈	C₂³×C₁₉
prime radical	C₂³×C₁₉	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₉×#1365	C₂	semidirect
5. C₁₉×#1362	C₂	semidirect
77. #11983	C₁₉	direct (N × H)
104. #1365	C₂×C₁₉	semidirect
105. #1362	C₂×C₁₉	semidirect
165. C₁₉	#11983	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₉×#3170	2
C₂×C₁₉×C₄⋊C₁₆	2
C₁₉×#1567	2
C₁₉×#1365	2
C₁₉×#1362	2
#11983	19

Subgroups — C19×#11983 #18017942

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#11983 #18017942