Subgroups — C₁₉×#9596 #18015555

Order 4864.

382 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#1939	C₂	1	2	yes	maximal
2	C₄×C₁₉×#106	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#335	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₉×#322	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1279	C₂	1	2	yes	maximal
6	C₂×C₁₉×#106	C₄	1	1	yes	characteristic
7	C₂×C₁₉×#106	C₄	1	1	yes	characteristic
8	C₂×C₁₉×C₄⋊C₈	—	2	2	no	—
9	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
14	C₁₉×#340	C₄	1	2	yes	—
15	C₁₉×#336	—	2	2	no	—
16	C₁₉×#335	—	2	2	no	—
17	C₁₉×#322	—	2	2	no	—
18	C₂²×C₈×C₁₉	—	2	1	no	—
19	C₂²×C₈×C₁₉	—	2	2	no	—
20	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
25	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₁₉	—	2	2	no	—
27	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
28	C₁₉×#106	—	2	2	no	—
29	C₁₉×#106	C₈	1	4	yes	complemented
30	C₁₉×C₄⋊C₈	—	2	2	no	—
31	C₁₉×C₄⋊C₈	—	2	2	no	—
32	C₁₉×C₄⋊C₈	—	4	4	no	—
33	C₄×C₈×C₁₉	—	2	1	no	—
34	C₄×C₈×C₁₉	—	2	1	no	—
35	C₄×C₈×C₁₉	—	2	1	no	—
36	C₄×C₈×C₁₉	D₄	1	2	yes	—
37	C₄×C₈×C₁₉	Q₈	1	2	yes	—
38	C₄×C₈×C₁₉	Q₈	1	2	yes	—
39	C₁₉×#95	—	2	2	no	—
40	C₂²×C₄×C₁₉	—	2	2	no	—
41	C₂²×C₄×C₁₉	—	2	2	no	—
42	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
45	C₂×C₈×C₁₉	—	2	1	no	—
46	C₂×C₈×C₁₉	—	2	1	no	—
47	C₂×C₈×C₁₉	—	2	1	no	—
48	C₂×C₈×C₁₉	—	2	1	no	—
49	C₂×C₈×C₁₉	—	2	2	no	—
50	C₂×C₈×C₁₉	—	2	2	no	—
51	C₂×C₈×C₁₉	—	2	2	no	—
52	C₂×C₈×C₁₉	—	4	2	no	—
53	C₂×C₈×C₁₉	—	4	2	no	—
54	C₂×C₈×C₁₉	—	4	2	no	—
55	C₂×C₈×C₁₉	—	4	2	no	—
56	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
58	C₂×C₈×C₁₉	C₂×C₈	1	2	yes	—
59	C₂×C₈×C₁₉	M₁₆	1	2	yes	—
60	C₄⋊C₄×C₁₉	—	2	2	no	—
61	C₄⋊C₄×C₁₉	—	2	2	no	—
62	C₄⋊C₄×C₁₉	—	2	2	no	—
63	C₄⋊C₄×C₁₉	C₂×C₈	1	4	yes	—
64	C₄²×C₁₉	—	4	2	no	—
65	C₄²×C₁₉	—	4	2	no	—
66	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
67	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
68	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
69	C₄²×C₁₉	#41	1	1	yes	characteristic
70	#9596	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₁₉	—	2	2	no	—
73	C₂×C₄×C₁₉	—	2	4	no	—
74	C₂×C₄×C₁₉	—	4	2	no	—
75	C₂×C₄×C₁₉	—	4	2	no	—
76	C₂×C₄×C₁₉	—	4	2	no	—
77	C₂×C₄×C₁₉	—	4	2	no	—
78	C₂×C₄×C₁₉	—	4	2	no	—
79	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
81	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
82	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₁₉	#125	1	1	yes	characteristic
84	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
87	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
88	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
89	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
90	C₂×C₄×C₁₉	#137	1	2	yes	—
91	C₈×C₁₉	—	2	2	no	—
92	C₈×C₁₉	—	4	2	no	—
93	C₈×C₁₉	—	4	1	no	—
94	C₈×C₁₉	—	4	1	no	—
95	C₈×C₁₉	—	8	2	no	—
96	C₈×C₁₉	C₄⋊C₈	1	4	yes	complemented
97	#1939	C₂×C₁₉	1	2	yes	—
98	C₄×#106	C₂×C₁₉	1	1	yes	characteristic
99	C₂×#335	C₂×C₁₉	1	1	yes	characteristic
100	C₂×#322	C₂×C₁₉	1	1	yes	characteristic
101	#1279	C₂×C₁₉	1	2	yes	—
102	C₂²×C₁₉	#389	1	1	yes	characteristic
103	C₂²×C₁₉	C₈×D₄	1	1	yes	characteristic
104	C₂²×C₁₉	#436	1	1	yes	characteristic
105	C₂²×C₁₉	#440	1	2	yes	—
106	C₂²×C₁₉	#444	1	2	yes	—
107	C₄×C₁₉	—	2	1	no	—
108	C₄×C₁₉	—	2	1	no	—
109	C₄×C₁₉	—	4	4	no	—
110	C₄×C₁₉	—	8	2	no	—
111	C₄×C₁₉	C₂×C₄⋊C₈	1	1	yes	characteristic
112	C₄×C₁₉	#424	1	1	yes	characteristic
113	C₄×C₁₉	C₈×Q₈	1	1	yes	characteristic
114	C₄×C₁₉	#446	1	1	yes	characteristic
115	C₄×C₁₉	#469	1	1	yes	characteristic
116	C₄×C₁₉	#470	1	1	yes	characteristic
117	C₄×C₁₉	#501	1	2	yes	—
118	C₂×#106	C₄×C₁₉	1	1	yes	characteristic
119	C₂×#106	C₄×C₁₉	1	1	yes	characteristic
120	C₂×C₄⋊C₈	—	2	2	no	—
121	C₂×C₄⋊C₈	C₂²×C₁₉	1	2	yes	—
122	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
123	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
124	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
125	C₄×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
126	#340	C₄×C₁₉	1	2	yes	—
127	#336	—	2	2	no	—
128	#335	—	2	2	no	—
129	#322	—	2	2	no	—
130	C₂×C₁₉	SD₁₆⋊C₈	1	2	yes	—
131	C₂×C₁₉	#1592	1	2	yes	—
132	C₂×C₁₉	#1939	1	1	yes	characteristic
133	C₂×C₁₉	#1944	1	1	yes	characteristic
134	C₂×C₁₉	#1946	1	1	yes	characteristic
135	C₂²×C₈	—	2	1	no	—
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
138	C₂²×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
139	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
140	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
141	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
142	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	2	yes	—
143	C₂×C₄²	—	2	2	no	—
144	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
145	#106	—	2	2	no	—
146	#106	C₈×C₁₉	1	4	yes	complemented
147	C₄⋊C₈	—	2	2	no	—
148	C₄⋊C₈	—	2	2	no	—
149	C₄⋊C₈	—	4	4	no	—
150	C₄×C₈	—	2	1	no	—
151	C₄×C₈	—	2	1	no	—
152	C₄×C₈	—	2	1	no	—
153	C₄×C₈	D₄×C₁₉	1	2	yes	—
154	C₄×C₈	Q₈×C₁₉	1	2	yes	—
155	C₄×C₈	Q₈×C₁₉	1	2	yes	—
156	#95	—	2	2	no	—
157	C₁₉	#9596	1	1	yes	characteristic, direct product factor, Sylow
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	C₂²×C₄×C₁₉	1	1	yes	characteristic
161	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
162	C₂²×C₄	#41×C₁₉	1	1	yes	characteristic
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	4	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	—	4	2	no	—
174	C₂×C₈	C₄⋊C₄×C₁₉	1	2	yes	—
175	C₂×C₈	C₄⋊C₄×C₁₉	1	2	yes	—
176	C₂×C₈	C₂×C₈×C₁₉	1	2	yes	—
177	C₂×C₈	M₁₆×C₁₉	1	2	yes	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	C₂×C₈×C₁₉	1	4	yes	—
182	C₄²	—	4	2	no	—
183	C₄²	—	4	2	no	—
184	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
185	C₄²	C₂×Q₈×C₁₉	1	1	yes	characteristic
186	C₄²	C₂×Q₈×C₁₉	1	1	yes	characteristic
187	C₄²	#41×C₁₉	1	1	yes	characteristic
188	C₂³	C₄×D₄×C₁₉	1	1	yes	characteristic
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
197	C₂×C₄	C₄×Q₈×C₁₉	1	1	yes	characteristic
198	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
199	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
200	C₂×C₄	C₁₉×#125	1	1	yes	characteristic
201	C₂×C₄	C₁₉×C₄⋊Q₈	1	1	yes	characteristic
202	C₂×C₄	C₂²×C₈×C₁₉	1	1	yes	characteristic
203	C₂×C₄	C₂×M₁₆×C₁₉	1	1	yes	characteristic
204	C₂×C₄	C₁₉×#131	1	1	yes	characteristic
205	C₂×C₄	C₁₉×#131	1	1	yes	characteristic
206	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	2	yes	—
207	C₂×C₄	C₁₉×#137	1	2	yes	—
208	C₈	—	2	2	no	—
209	C₈	—	4	2	no	—
210	C₈	—	4	1	no	—
211	C₈	—	4	1	no	—
212	C₈	—	8	2	no	—
213	C₈	C₁₉×C₄⋊C₈	1	4	yes	complemented
214	C₂²	C₁₉×#389	1	1	yes	characteristic
215	C₂²	C₈×D₄×C₁₉	1	1	yes	characteristic
216	C₂²	C₁₉×#436	1	1	yes	characteristic
217	C₂²	C₁₉×#440	1	2	yes	—
218	C₂²	C₁₉×#444	1	2	yes	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	4	4	no	—
222	C₄	—	8	2	no	—
223	C₄	C₂×C₁₉×C₄⋊C₈	1	1	yes	characteristic
224	C₄	C₁₉×#424	1	1	yes	characteristic
225	C₄	C₈×Q₈×C₁₉	1	1	yes	characteristic
226	C₄	C₁₉×#446	1	1	yes	characteristic
227	C₄	C₁₉×#469	1	1	yes	characteristic
228	C₄	C₁₉×#470	1	1	yes	characteristic
229	C₄	C₁₉×#501	1	2	yes	—
230	C₂	C₁₉×SD₁₆⋊C₈	1	2	yes	—
231	C₂	C₁₉×#1592	1	2	yes	—
232	C₂	C₁₉×#1939	1	1	yes	characteristic
233	C₂	C₁₉×#1944	1	1	yes	characteristic
234	C₂	C₁₉×#1946	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
socle, prime radical	C₂³×C₁₉	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
29. C₁₉×#106	C₈	semidirect
70. #9596	C₁₉	direct (N × H)
96. C₈×C₁₉	C₄⋊C₈	semidirect
146. #106	C₈×C₁₉	semidirect
157. C₁₉	#9596	direct (N × H)
213. C₈	C₁₉×C₄⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×#1939	2
C₄×C₁₉×#106	2
C₂×C₁₉×#335	2
C₂×C₁₉×#322	2
C₁₉×#1279	2
#9596	19

Subgroups — C19×#9596 #18015555

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#9596 #18015555