Subgroups — C₁₉×#3208 #14561065

Order 2432.

488 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₉×#127	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#492	C₂	1	2	yes	maximal, complemented
4	C₁₉×#486	C₂	1	2	yes	maximal, complemented
5	C₁₉×#460	C₂	1	4	yes	maximal, complemented
6	C₁₉×#450	C₂	1	4	yes	maximal, complemented
7	C₁₉×#431	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄×C₁₉	—	2	1	no	—
9	C₂²×D₄×C₁₉	—	2	1	no	—
10	C₂²×D₄×C₁₉	C₂²	1	2	yes	—
11	C₂×SD₁₆×C₁₉	C₂²	1	8	yes	complemented
12	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
13	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
14	C₁₉×#127	—	2	1	no	—
15	C₁₉×#127	—	2	1	no	—
16	C₁₉×#127	C₂²	1	2	yes	—
17	C₁₉×#127	C₂²	1	2	yes	—
18	C₁₉×#125	—	2	1	no	—
19	C₁₉×C₂²⋊Q₈	—	2	1	no	—
20	C₁₉×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₄×Q₈×C₁₉	—	2	1	no	—
22	C₄×Q₈×C₁₉	C₂²	1	2	yes	complemented
23	C₁₉×#117	—	2	1	no	—
24	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
25	C₁₉×C₄⋊C₈	C₂²	1	2	yes	complemented
26	C₁₉×#102	C₂²	1	8	yes	complemented
27	C₁₉×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₄×C₈×C₁₉	C₂²	1	2	yes	complemented
29	C₂⁴×C₁₉	—	2	2	no	—
30	C₂×Q₈×C₁₉	—	2	1	no	—
31	C₂×Q₈×C₁₉	C₂³	1	2	yes	—
32	C₂×D₄×C₁₉	—	2	2	no	—
33	C₂×D₄×C₁₉	—	2	2	no	—
34	C₂×D₄×C₁₉	—	2	2	no	—
35	C₂×D₄×C₁₉	—	2	2	no	—
36	C₂×D₄×C₁₉	—	2	2	no	—
37	C₂×D₄×C₁₉	—	4	4	no	—
38	C₂×D₄×C₁₉	—	4	4	no	—
39	C₂×D₄×C₁₉	—	4	4	no	—
40	C₂×D₄×C₁₉	C₂³	1	4	yes	—
41	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
44	SD₁₆×C₁₉	—	4	8	no	—
45	C₂×C₈×C₁₉	C₂³	1	4	yes	—
46	C₄⋊C₄×C₁₉	—	2	1	no	—
47	C₄⋊C₄×C₁₉	—	2	1	no	—
48	C₄⋊C₄×C₁₉	—	2	1	no	—
49	C₄⋊C₄×C₁₉	—	2	2	no	—
50	C₄⋊C₄×C₁₉	—	2	2	no	—
51	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
52	C₂²⋊C₄×C₁₉	—	2	2	no	—
53	C₄²×C₁₉	—	2	2	no	—
54	C₄²×C₁₉	D₄	1	1	yes	characteristic
55	C₄²×C₁₉	D₄	1	1	yes	characteristic
56	C₄²×C₁₉	C₂³	1	1	yes	characteristic
57	C₄²×C₁₉	C₂³	1	1	yes	characteristic
58	C₂³×C₁₉	—	2	4	no	—
59	C₂³×C₁₉	—	4	4	no	—
60	C₂³×C₁₉	—	4	4	no	—
61	C₂³×C₁₉	—	4	4	no	—
62	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
63	Q₈×C₁₉	—	2	4	no	—
64	Q₈×C₁₉	—	4	2	no	—
65	D₄×C₁₉	—	2	8	no	—
66	D₄×C₁₉	—	4	4	no	—
67	D₄×C₁₉	—	4	4	no	—
68	D₄×C₁₉	—	4	4	no	—
69	D₄×C₁₉	—	4	4	no	—
70	D₄×C₁₉	—	4	4	no	—
71	C₂×C₄×C₁₉	—	2	2	no	—
72	C₂×C₄×C₁₉	—	2	2	no	—
73	C₂×C₄×C₁₉	—	2	2	no	—
74	C₂×C₄×C₁₉	—	2	2	no	—
75	C₂×C₄×C₁₉	SD₁₆	1	4	yes	complemented
76	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
82	C₈×C₁₉	—	2	4	no	—
83	#3208	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂²×C₁₉	—	2	1	no	—
85	C₂²×C₁₉	—	2	1	no	—
86	C₂²×C₁₉	—	4	4	no	—
87	C₂²×C₁₉	—	4	4	no	—
88	C₂²×C₁₉	—	4	4	no	—
89	C₂²×C₁₉	—	8	8	no	—
90	C₂²×C₁₉	C₂×SD₁₆	1	2	yes	—
91	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
92	C₄×C₁₉	—	2	1	no	—
93	C₄×C₁₉	—	2	1	no	—
94	C₄×C₁₉	—	4	2	no	—
95	C₄×C₁₉	—	4	2	no	—
96	C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
97	C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
98	C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
99	C₄×C₁₉	H(5,𝔽₂)	1	2	yes	—
100	#533	C₂×C₁₉	1	1	yes	characteristic, complemented
101	C₂×#127	C₂×C₁₉	1	1	yes	characteristic
102	#492	C₂×C₁₉	1	2	yes	complemented
103	#486	C₂×C₁₉	1	2	yes	complemented
104	#460	C₂×C₁₉	1	4	yes	complemented
105	#450	C₂×C₁₉	1	4	yes	complemented
106	#431	C₂×C₁₉	1	1	yes	characteristic, complemented
107	C₂×C₁₉	—	2	2	no	—
108	C₂×C₁₉	—	8	4	no	—
109	C₂×C₁₉	#535	1	1	yes	characteristic
110	C₂×C₁₉	C₂²×SD₁₆	1	1	yes	characteristic
111	C₂×C₁₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₂²×D₄	—	2	1	no	—
113	C₂²×D₄	—	2	1	no	—
114	C₂²×D₄	C₂²×C₁₉	1	2	yes	—
115	C₂×SD₁₆	C₂²×C₁₉	1	8	yes	complemented
116	C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic, complemented
117	C₄⋊Q₈	C₂²×C₁₉	1	1	yes	characteristic, complemented
118	#127	—	2	1	no	—
119	#127	—	2	1	no	—
120	#127	C₂²×C₁₉	1	2	yes	—
121	#127	C₂²×C₁₉	1	2	yes	—
122	#125	—	2	1	no	—
123	C₂²⋊Q₈	—	2	1	no	—
124	C₂²⋊Q₈	C₂²×C₁₉	1	2	yes	—
125	C₄×Q₈	—	2	1	no	—
126	C₄×Q₈	C₂²×C₁₉	1	2	yes	complemented
127	#117	—	2	1	no	—
128	C₂×C₄²	C₂²×C₁₉	1	1	yes	characteristic
129	C₄⋊C₈	C₂²×C₁₉	1	2	yes	complemented
130	#102	C₂²×C₁₉	1	8	yes	complemented
131	C₂²⋊C₈	C₂²×C₁₉	1	2	yes	—
132	C₄×C₈	C₂²×C₁₉	1	2	yes	complemented
133	C₁₉	#3208	1	1	yes	characteristic, direct product factor, Sylow
134	C₂⁴	—	2	2	no	—
135	C₂×Q₈	—	2	1	no	—
136	C₂×Q₈	C₂³×C₁₉	1	2	yes	—
137	C₂×D₄	—	2	2	no	—
138	C₂×D₄	—	2	2	no	—
139	C₂×D₄	—	2	2	no	—
140	C₂×D₄	—	2	2	no	—
141	C₂×D₄	—	2	2	no	—
142	C₂×D₄	—	4	4	no	—
143	C₂×D₄	—	4	4	no	—
144	C₂×D₄	—	4	4	no	—
145	C₂×D₄	C₂³×C₁₉	1	4	yes	—
146	C₂²×C₄	D₄×C₁₉	1	1	yes	characteristic
147	C₂²×C₄	D₄×C₁₉	1	1	yes	characteristic
148	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
149	SD₁₆	—	4	8	no	—
150	C₂×C₈	C₂³×C₁₉	1	4	yes	—
151	C₄⋊C₄	—	2	1	no	—
152	C₄⋊C₄	—	2	1	no	—
153	C₄⋊C₄	—	2	1	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
157	C₂²⋊C₄	—	2	2	no	—
158	C₄²	—	2	2	no	—
159	C₄²	D₄×C₁₉	1	1	yes	characteristic
160	C₄²	D₄×C₁₉	1	1	yes	characteristic
161	C₄²	C₂³×C₁₉	1	1	yes	characteristic
162	C₄²	C₂³×C₁₉	1	1	yes	characteristic
163	C₂³	—	2	4	no	—
164	C₂³	—	4	4	no	—
165	C₂³	—	4	4	no	—
166	C₂³	—	4	4	no	—
167	C₂³	C₂×D₄×C₁₉	1	1	yes	characteristic
168	Q₈	—	2	4	no	—
169	Q₈	—	4	2	no	—
170	D₄	—	2	8	no	—
171	D₄	—	4	4	no	—
172	D₄	—	4	4	no	—
173	D₄	—	4	4	no	—
174	D₄	—	4	4	no	—
175	D₄	—	4	4	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	SD₁₆×C₁₉	1	4	yes	complemented
181	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
182	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
183	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
184	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
185	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
186	C₂×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
187	C₈	—	2	4	no	—
188	C₂²	—	2	1	no	—
189	C₂²	—	2	1	no	—
190	C₂²	—	4	4	no	—
191	C₂²	—	4	4	no	—
192	C₂²	—	4	4	no	—
193	C₂²	—	8	8	no	—
194	C₂²	C₂×SD₁₆×C₁₉	1	2	yes	—
195	C₂²	C₂²×D₄×C₁₉	1	1	yes	characteristic
196	C₄	—	2	1	no	—
197	C₄	—	2	1	no	—
198	C₄	—	4	2	no	—
199	C₄	—	4	2	no	—
200	C₄	C₂×SD₁₆×C₁₉	1	2	yes	—
201	C₄	C₂×SD₁₆×C₁₉	1	2	yes	—
202	C₄	C₁₉×AGL(1,ℤ/8)	1	2	yes	—
203	C₄	C₁₉×H(5,𝔽₂)	1	2	yes	—
204	C₂	—	2	2	no	—
205	C₂	—	8	4	no	—
206	C₂	C₁₉×#535	1	1	yes	characteristic
207	C₂	C₂²×SD₁₆×C₁₉	1	1	yes	characteristic
208	C₂	C₂×C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₉
prime radical	C₂×C₁₉×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₉×#533	C₂	semidirect
3. C₁₉×#492	C₂	semidirect
4. C₁₉×#486	C₂	semidirect
5. C₁₉×#460	C₂	semidirect
6. C₁₉×#450	C₂	semidirect
7. C₁₉×#431	C₂	semidirect
11. C₂×SD₁₆×C₁₉	C₂²	semidirect
12. C₁₉×C₄⋊Q₈	C₂²	semidirect
13. C₁₉×C₄⋊Q₈	C₂²	semidirect
22. C₄×Q₈×C₁₉	C₂²	semidirect
25. C₁₉×C₄⋊C₈	C₂²	semidirect
26. C₁₉×#102	C₂²	semidirect
28. C₄×C₈×C₁₉	C₂²	semidirect
75. C₂×C₄×C₁₉	SD₁₆	semidirect
83. #3208	C₁₉	direct (N × H)
100. #533	C₂×C₁₉	semidirect
102. #492	C₂×C₁₉	semidirect
103. #486	C₂×C₁₉	semidirect
104. #460	C₂×C₁₉	semidirect
105. #450	C₂×C₁₉	semidirect
106. #431	C₂×C₁₉	semidirect
115. C₂×SD₁₆	C₂²×C₁₉	semidirect
116. C₄⋊Q₈	C₂²×C₁₉	semidirect
117. C₄⋊Q₈	C₂²×C₁₉	semidirect
126. C₄×Q₈	C₂²×C₁₉	semidirect
129. C₄⋊C₈	C₂²×C₁₉	semidirect
130. #102	C₂²×C₁₉	semidirect
132. C₄×C₈	C₂²×C₁₉	semidirect
133. C₁₉	#3208	direct (N × H)
180. C₂×C₄	SD₁₆×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×#533	2
C₂×C₁₉×#127	2
C₁₉×#492	2
C₁₉×#486	2
C₁₉×#460	2
C₁₉×#450	2
C₁₉×#431	2
#3208	19

Subgroups — C19×#3208 #14561065

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#3208 #14561065