Subgroups — C₁₉×#9626 #18015585

Order 4864.

290 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#1950	C₂	1	2	yes	maximal
2	C₁₉×#1567	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉×#334	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1451	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1301	C₂	1	2	yes	maximal
6	C₁₉×#431	C₂²	1	1	yes	characteristic
7	C₂×C₁₉×#108	—	2	2	no	—
8	C₁₉×#423	C₂²	1	2	yes	—
9	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
10	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
12	C₁₉×#341	—	2	2	no	—
13	C₁₉×#335	—	2	1	no	—
14	C₁₉×#335	—	2	1	no	—
15	C₁₉×#334	C₄	1	2	yes	—
16	C₁₉×#334	C₄	1	2	yes	—
17	C₁₉×#322	—	2	1	no	—
18	C₁₉×#322	—	2	1	no	—
19	C₂×M₁₆×C₁₉	—	2	2	no	—
20	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
21	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₉	—	2	2	no	—
23	C₂²×C₈×C₁₉	Q₈	1	1	yes	characteristic
24	C₂²×C₈×C₁₉	Q₈	1	1	yes	characteristic
25	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
26	C₁₉×#108	—	4	4	no	—
27	C₁₉×C₄⋊C₈	—	2	2	no	—
28	C₁₉×C₄⋊C₈	—	2	2	no	—
29	C₁₉×C₄⋊C₈	—	2	1	no	—
30	C₁₉×C₄⋊C₈	—	2	2	no	—
31	C₁₉×C₄⋊C₈	C₂×C₄	1	4	yes	—
32	C₁₉×C₂²⋊C₈	Q₈	1	2	yes	—
33	C₁₉×#97	—	2	1	no	—
34	C₄×C₈×C₁₉	—	2	1	no	—
35	C₄×C₈×C₁₉	—	2	1	no	—
36	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
39	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
43	M₁₆×C₁₉	—	2	2	no	—
44	M₁₆×C₁₉	—	2	2	no	—
45	M₁₆×C₁₉	—	4	2	no	—
46	M₁₆×C₁₉	—	4	2	no	—
47	C₂×C₈×C₁₉	—	2	2	no	—
48	C₂×C₈×C₁₉	—	2	4	no	—
49	C₂×C₈×C₁₉	—	2	1	no	—
50	C₂×C₈×C₁₉	—	2	1	no	—
51	C₂×C₈×C₁₉	—	2	2	no	—
52	C₂×C₈×C₁₉	—	4	1	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	1	yes	characteristic
57	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
60	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
61	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₁₉	#41	1	1	yes	characteristic
63	C₄²×C₁₉	#41	1	1	yes	characteristic
64	#9626	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂³×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₁₉	—	2	1	no	—
67	C₂×C₄×C₁₉	—	2	1	no	—
68	C₂×C₄×C₁₉	—	4	1	no	—
69	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
73	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₁₉	#125	1	1	yes	characteristic
75	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
76	C₂×C₄×C₁₉	#137	1	2	yes	—
77	C₈×C₁₉	—	2	1	no	—
78	C₈×C₁₉	—	2	1	no	—
79	C₈×C₁₉	—	2	1	no	—
80	C₈×C₁₉	—	4	2	no	—
81	C₈×C₁₉	—	4	4	no	—
82	C₈×C₁₉	—	4	1	no	—
83	C₈×C₁₉	—	4	1	no	—
84	C₈×C₁₉	—	8	2	no	—
85	C₈×C₁₉	#108	1	2	yes	complemented
86	#1950	C₂×C₁₉	1	2	yes	—
87	#1567	C₂×C₁₉	1	1	yes	characteristic
88	C₂×#334	C₂×C₁₉	1	1	yes	characteristic
89	#1451	C₂×C₁₉	1	1	yes	characteristic
90	#1301	C₂×C₁₉	1	2	yes	—
91	C₂²×C₁₉	—	2	1	no	—
92	C₂²×C₁₉	—	2	1	no	—
93	C₂²×C₁₉	#389	1	1	yes	characteristic
94	C₂²×C₁₉	#501	1	2	yes	—
95	C₄×C₁₉	—	2	1	no	—
96	C₄×C₁₉	—	2	1	no	—
97	C₄×C₁₉	—	2	1	no	—
98	C₄×C₁₉	C₂×#108	1	1	yes	characteristic
99	C₄×C₁₉	#430	1	1	yes	characteristic
100	C₄×C₁₉	#440	1	2	yes	—
101	C₄×C₁₉	#469	1	1	yes	characteristic
102	C₄×C₁₉	#470	1	1	yes	characteristic
103	#431	C₂²×C₁₉	1	1	yes	characteristic
104	C₂×#108	—	2	2	no	—
105	#423	C₂²×C₁₉	1	2	yes	—
106	C₂×C₄⋊C₈	C₂²×C₁₉	1	2	yes	—
107	C₄×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
109	#341	—	2	2	no	—
110	#335	—	2	1	no	—
111	#335	—	2	1	no	—
112	#334	C₄×C₁₉	1	2	yes	—
113	#334	C₄×C₁₉	1	2	yes	—
114	#322	—	2	1	no	—
115	#322	—	2	1	no	—
116	C₂×C₁₉	—	2	2	no	—
117	C₂×C₁₉	#1944	1	1	yes	characteristic
118	C₂×C₁₉	#1950	1	1	yes	characteristic
119	C₂×C₁₉	#1953	1	1	yes	characteristic
120	C₂×M₁₆	—	2	2	no	—
121	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
122	C₂×M₁₆	D₄×C₁₉	1	1	yes	characteristic
123	C₂²×C₈	—	2	2	no	—
124	C₂²×C₈	Q₈×C₁₉	1	1	yes	characteristic
125	C₂²×C₈	Q₈×C₁₉	1	1	yes	characteristic
126	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
127	#108	—	4	4	no	—
128	C₄⋊C₈	—	2	2	no	—
129	C₄⋊C₈	—	2	2	no	—
130	C₄⋊C₈	—	2	1	no	—
131	C₄⋊C₈	—	2	2	no	—
132	C₄⋊C₈	C₂×C₄×C₁₉	1	4	yes	—
133	C₂²⋊C₈	Q₈×C₁₉	1	2	yes	—
134	#97	—	2	1	no	—
135	C₄×C₈	—	2	1	no	—
136	C₄×C₈	—	2	1	no	—
137	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
138	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
139	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
140	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
141	C₁₉	#9626	1	1	yes	characteristic, direct product factor, Sylow
142	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
143	C₂²×C₄	C₂×Q₈×C₁₉	1	1	yes	characteristic
144	C₂²×C₄	C₂×Q₈×C₁₉	1	1	yes	characteristic
145	M₁₆	—	2	2	no	—
146	M₁₆	—	2	2	no	—
147	M₁₆	—	4	2	no	—
148	M₁₆	—	4	2	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	4	1	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	—	4	2	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	C₄⋊C₄×C₁₉	1	1	yes	characteristic
159	C₂×C₈	C₄⋊C₄×C₁₉	1	1	yes	characteristic
160	C₂×C₈	C₄⋊C₄×C₁₉	1	1	yes	characteristic
161	C₂×C₈	C₄⋊C₄×C₁₉	1	1	yes	characteristic
162	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
163	C₄²	C₂×D₄×C₁₉	1	1	yes	characteristic
164	C₄²	#41×C₁₉	1	1	yes	characteristic
165	C₄²	#41×C₁₉	1	1	yes	characteristic
166	C₂³	C₁₉×C₄⋊Q₈	1	1	yes	characteristic
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	1	no	—
169	C₂×C₄	—	4	1	no	—
170	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
171	C₂×C₄	C₄×D₄×C₁₉	1	1	yes	characteristic
172	C₂×C₄	C₄×Q₈×C₁₉	1	1	yes	characteristic
173	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
174	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
175	C₂×C₄	C₁₉×#125	1	1	yes	characteristic
176	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	2	yes	—
177	C₂×C₄	C₁₉×#137	1	2	yes	—
178	C₈	—	2	1	no	—
179	C₈	—	2	1	no	—
180	C₈	—	2	1	no	—
181	C₈	—	4	2	no	—
182	C₈	—	4	4	no	—
183	C₈	—	4	1	no	—
184	C₈	—	4	1	no	—
185	C₈	—	8	2	no	—
186	C₈	C₁₉×#108	1	2	yes	complemented
187	C₂²	—	2	1	no	—
188	C₂²	—	2	1	no	—
189	C₂²	C₁₉×#389	1	1	yes	characteristic
190	C₂²	C₁₉×#501	1	2	yes	—
191	C₄	—	2	1	no	—
192	C₄	—	2	1	no	—
193	C₄	—	2	1	no	—
194	C₄	C₂×C₁₉×#108	1	1	yes	characteristic
195	C₄	C₁₉×#430	1	1	yes	characteristic
196	C₄	C₁₉×#440	1	2	yes	—
197	C₄	C₁₉×#469	1	1	yes	characteristic
198	C₄	C₁₉×#470	1	1	yes	characteristic
199	C₂	—	2	2	no	—
200	C₂	C₁₉×#1944	1	1	yes	characteristic
201	C₂	C₁₉×#1950	1	1	yes	characteristic
202	C₂	C₁₉×#1953	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₉	#121
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
socle	C₂²×C₁₉	#389
prime radical	C₂³×C₁₉	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
64. #9626	C₁₉	direct (N × H)
85. C₈×C₁₉	#108	semidirect
141. C₁₉	#9626	direct (N × H)
186. C₈	C₁₉×#108	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×#1950	2
C₁₉×#1567	2
C₂×C₁₉×#334	2
C₁₉×#1451	2
C₁₉×#1301	2
#9626	19

Subgroups — C19×#9626 #18015585

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#9626 #18015585