Subgroups — C₁₉×#2566 #14560423

Order 2432.

548 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₄×C₁₉	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#396	C₂	1	4	yes	maximal, complemented
4	C₁₉×#390	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#390	C₂	1	1	yes	maximal, characteristic
6	C₁₉×#385	C₂	1	4	yes	maximal, complemented
7	C₁₉×#384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₉×#382	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₄⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₄×C₁₉	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₉	C₂²	1	2	yes	—
12	C₁₉×#125	—	2	4	no	—
13	C₄×D₄×C₁₉	—	2	4	no	—
14	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
21	C₂×C₂²⋊C₄×C₁₉	C₂²	1	4	yes	—
22	C₂×C₂²⋊C₄×C₁₉	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₁₉	C₂²	1	4	yes	—
24	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
27	C₁₉×#95	C₂²	1	2	yes	—
28	C₁₉×#95	C₂²	1	4	yes	—
29	C₂⁴×C₁₉	C₂³	1	2	yes	—
30	C₂×D₄×C₁₉	—	2	2	no	—
31	C₂×D₄×C₁₉	—	2	2	no	—
32	C₂×D₄×C₁₉	Q₈	1	4	yes	—
33	C₂²×C₄×C₁₉	—	2	4	no	—
34	C₂²×C₄×C₁₉	—	4	4	no	—
35	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
36	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
43	C₄⋊C₄×C₁₉	—	2	2	no	—
44	C₄⋊C₄×C₁₉	—	2	4	no	—
45	C₄⋊C₄×C₁₉	—	2	4	no	—
46	C₄⋊C₄×C₁₉	—	2	2	no	—
47	C₄⋊C₄×C₁₉	—	2	2	no	—
48	C₄⋊C₄×C₁₉	—	2	4	no	—
49	C₄⋊C₄×C₁₉	—	4	4	no	—
50	C₂²⋊C₄×C₁₉	—	2	4	no	—
51	C₂²⋊C₄×C₁₉	—	2	8	no	—
52	C₂²⋊C₄×C₁₉	—	4	4	no	—
53	C₄²×C₁₉	—	2	2	no	—
54	C₄²×C₁₉	—	2	2	no	—
55	C₄²×C₁₉	D₄	1	4	yes	—
56	C₂³×C₁₉	—	2	2	no	—
57	C₂³×C₁₉	—	2	4	no	—
58	C₂³×C₁₉	—	2	2	no	—
59	C₂³×C₁₉	—	2	4	no	—
60	C₂³×C₁₉	C₂×Q₈	1	4	yes	—
61	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
62	D₄×C₁₉	—	2	8	no	—
63	C₂×C₄×C₁₉	—	2	2	no	—
64	C₂×C₄×C₁₉	—	2	2	no	—
65	C₂×C₄×C₁₉	—	2	2	no	—
66	C₂×C₄×C₁₉	—	2	2	no	—
67	C₂×C₄×C₁₉	—	2	2	no	—
68	C₂×C₄×C₁₉	—	2	2	no	—
69	C₂×C₄×C₁₉	—	2	2	no	—
70	C₂×C₄×C₁₉	—	2	2	no	—
71	C₂×C₄×C₁₉	—	2	4	no	—
72	C₂×C₄×C₁₉	—	2	2	no	—
73	C₂×C₄×C₁₉	—	2	2	no	—
74	C₂×C₄×C₁₉	—	2	4	no	—
75	C₂×C₄×C₁₉	—	2	4	no	—
76	C₂×C₄×C₁₉	—	4	8	no	—
77	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₁₉	C₂×Q₈	1	2	yes	—
81	C₂×C₄×C₁₉	#41	1	4	yes	—
82	C₂×C₄×C₁₉	#41	1	2	yes	—
83	#2566	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂²×C₁₉	—	2	4	no	—
85	C₂²×C₁₉	—	2	8	no	—
86	C₂²×C₁₉	—	4	2	no	—
87	C₂²×C₁₉	—	4	2	no	—
88	C₂²×C₁₉	—	4	4	no	—
89	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
90	C₂²×C₁₉	C₂²×Q₈	1	1	yes	characteristic
91	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
92	C₂²×C₁₉	C₂×#41	1	2	yes	—
93	C₂²×C₁₉	H(5,𝔽₂)	1	2	yes	—
94	C₄×C₁₉	—	2	2	no	—
95	C₄×C₁₉	—	2	2	no	—
96	C₄×C₁₉	—	2	4	no	—
97	C₄×C₁₉	—	4	2	no	—
98	C₄×C₁₉	—	4	2	no	—
99	C₄×C₁₉	—	4	2	no	—
100	C₄×C₁₉	—	4	2	no	—
101	C₄×C₁₉	C₂²⋊Q₈	1	4	yes	—
102	C₂×#125	C₂×C₁₉	1	1	yes	characteristic, complemented
103	C₂×C₄×D₄	C₂×C₁₉	1	1	yes	characteristic
104	#396	C₂×C₁₉	1	4	yes	complemented
105	#390	C₂×C₁₉	1	1	yes	characteristic
106	#390	C₂×C₁₉	1	1	yes	characteristic
107	#385	C₂×C₁₉	1	4	yes	complemented
108	#384	C₂×C₁₉	1	1	yes	characteristic, complemented
109	#382	C₂×C₁₉	1	1	yes	characteristic, complemented
110	C₄×C₄⋊C₄	C₂×C₁₉	1	1	yes	characteristic, complemented
111	C₂×C₁₉	—	4	4	no	—
112	C₂×C₁₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₁₉	#532	1	1	yes	characteristic
114	C₂×C₁₉	#535	1	1	yes	characteristic
115	C₂×C₁₉	#553	1	2	yes	—
116	C₂×C₁₉	#554	1	2	yes	—
117	C₂²×D₄	C₂²×C₁₉	1	1	yes	characteristic
118	C₂³×C₄	C₂²×C₁₉	1	2	yes	—
119	#125	—	2	4	no	—
120	C₄×D₄	—	2	4	no	—
121	C₂×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
122	C₂×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
123	C₂×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
124	C₂×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
125	C₂×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
126	C₂×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
127	C₂×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
128	C₂×C₂²⋊C₄	C₂²×C₁₉	1	2	yes	—
129	C₂×C₂²⋊C₄	C₂²×C₁₉	1	4	yes	—
130	C₂×C₂²⋊C₄	C₂²×C₁₉	1	4	yes	—
131	C₂×C₄²	C₂²×C₁₉	1	1	yes	characteristic
132	C₂×C₄²	C₂²×C₁₉	1	1	yes	characteristic
133	C₂×C₄²	C₂²×C₁₉	1	1	yes	characteristic
134	#95	C₂²×C₁₉	1	2	yes	—
135	#95	C₂²×C₁₉	1	4	yes	—
136	C₁₉	#2566	1	1	yes	characteristic, direct product factor, Sylow
137	C₂⁴	C₂³×C₁₉	1	2	yes	—
138	C₂×D₄	—	2	2	no	—
139	C₂×D₄	—	2	2	no	—
140	C₂×D₄	Q₈×C₁₉	1	4	yes	—
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	4	4	no	—
143	C₂²×C₄	C₂³×C₁₉	1	2	yes	—
144	C₂²×C₄	C₂³×C₁₉	1	2	yes	—
145	C₂²×C₄	C₂³×C₁₉	1	2	yes	—
146	C₂²×C₄	C₂³×C₁₉	1	2	yes	—
147	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
148	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
149	C₂²×C₄	C₂³×C₁₉	1	2	yes	—
150	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	4	no	—
157	C₄⋊C₄	—	4	4	no	—
158	C₂²⋊C₄	—	2	4	no	—
159	C₂²⋊C₄	—	2	8	no	—
160	C₂²⋊C₄	—	4	4	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	2	2	no	—
163	C₄²	D₄×C₁₉	1	4	yes	—
164	C₂³	—	2	2	no	—
165	C₂³	—	2	4	no	—
166	C₂³	—	2	2	no	—
167	C₂³	—	2	4	no	—
168	C₂³	C₂×Q₈×C₁₉	1	4	yes	—
169	C₂³	C₂⁴×C₁₉	1	1	yes	characteristic
170	D₄	—	2	8	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	4	8	no	—
185	C₂×C₄	C₂×D₄×C₁₉	1	2	yes	—
186	C₂×C₄	C₂×D₄×C₁₉	1	2	yes	—
187	C₂×C₄	C₂×D₄×C₁₉	1	2	yes	—
188	C₂×C₄	C₂×Q₈×C₁₉	1	2	yes	—
189	C₂×C₄	#41×C₁₉	1	4	yes	—
190	C₂×C₄	#41×C₁₉	1	2	yes	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	8	no	—
193	C₂²	—	4	2	no	—
194	C₂²	—	4	2	no	—
195	C₂²	—	4	4	no	—
196	C₂²	C₂²×D₄×C₁₉	1	1	yes	characteristic
197	C₂²	C₂²×Q₈×C₁₉	1	1	yes	characteristic
198	C₂²	C₂×#41×C₁₉	1	1	yes	characteristic
199	C₂²	C₂×#41×C₁₉	1	2	yes	—
200	C₂²	C₁₉×H(5,𝔽₂)	1	2	yes	—
201	C₄	—	2	2	no	—
202	C₄	—	2	2	no	—
203	C₄	—	2	4	no	—
204	C₄	—	4	2	no	—
205	C₄	—	4	2	no	—
206	C₄	—	4	2	no	—
207	C₄	—	4	2	no	—
208	C₄	C₁₉×C₂²⋊Q₈	1	4	yes	—
209	C₂	—	4	4	no	—
210	C₂	C₂×C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
211	C₂	C₁₉×#532	1	1	yes	characteristic
212	C₂	C₁₉×#535	1	1	yes	characteristic
213	C₂	C₁₉×#553	1	2	yes	—
214	C₂	C₁₉×#554	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₉
prime radical	C₂²×D₄×C₁₉	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₉×#125	C₂	semidirect
3. C₁₉×#396	C₂	semidirect
6. C₁₉×#385	C₂	semidirect
7. C₁₉×#384	C₂	semidirect
8. C₁₉×#382	C₂	semidirect
9. C₄×C₄⋊C₄×C₁₉	C₂	semidirect
83. #2566	C₁₉	direct (N × H)
102. C₂×#125	C₂×C₁₉	semidirect
104. #396	C₂×C₁₉	semidirect
107. #385	C₂×C₁₉	semidirect
108. #384	C₂×C₁₉	semidirect
109. #382	C₂×C₁₉	semidirect
110. C₄×C₄⋊C₄	C₂×C₁₉	semidirect
136. C₁₉	#2566	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#125	2
C₂×C₄×D₄×C₁₉	2
C₁₉×#396	2
C₁₉×#390	2
C₁₉×#385	2
C₁₉×#384	2
C₁₉×#382	2
C₄×C₄⋊C₄×C₁₉	2
#2566	19

Subgroups — C19×#2566 #14560423

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#2566 #14560423