Subgroups — C₁₉×#3262 #14561119

Order 2432.

346 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#542	C₂	1	2	yes	maximal, complemented
2	C₁₉×#477	C₂	1	2	yes	maximal, complemented
3	C₁₉×#474	C₂	1	2	yes	maximal, complemented
4	C₁₉×#468	C₂	1	2	yes	maximal, complemented
5	C₁₉×#467	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#466	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#464	C₂	1	2	yes	maximal, complemented
8	C₁₉×#463	C₂	1	2	yes	maximal, complemented
9	C₁₉×#424	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×SD₁₆×C₁₉	C₂²	1	2	yes	—
12	C₂×D₈×C₁₉	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₁₉×C₄⋊Q₈	C₂²	1	2	yes	—
16	C₁₉×#126	—	2	2	no	—
17	C₁₉×#124	—	2	2	no	—
18	C₁₉×#124	C₂²	1	2	yes	—
19	C₁₉×#123	—	2	2	no	—
20	C₁₉×C₂²⋊Q₈	—	2	2	no	—
21	C₁₉×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₁₉×#121	C₂²	1	2	yes	—
23	C₄×Q₈×C₁₉	C₂²	1	2	yes	—
24	C₄×D₄×C₁₉	C₂²	1	2	yes	—
25	C₁₉×#117	C₂²	1	1	yes	characteristic
26	C₁₉×#107	C₂²	1	1	yes	characteristic
27	C₁₉×#107	C₂²	1	1	yes	characteristic
28	C₁₉×#107	C₂²	1	2	yes	—
29	C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
30	C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
31	C₁₉×#103	C₂²	1	2	yes	—
32	C₁₉×#103	C₂²	1	2	yes	—
33	C₁₉×#102	C₂²	1	2	yes	—
34	C₁₉×#102	C₂²	1	2	yes	—
35	C₂×Q₈×C₁₉	—	2	2	no	—
36	C₂×Q₈×C₁₉	C₂³	1	2	yes	—
37	C₂×D₄×C₁₉	—	2	2	no	—
38	C₂×D₄×C₁₉	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₉	—	2	2	no	—
40	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
41	Q₁₆×C₁₉	—	4	1	no	—
42	SD₁₆×C₁₉	—	4	2	no	—
43	D₈×C₁₉	—	4	1	no	—
44	M₁₆×C₁₉	—	4	1	no	—
45	C₂×C₈×C₁₉	—	4	1	no	—
46	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
48	C₂×C₈×C₁₉	C₂³	1	2	yes	—
49	C₄⋊C₄×C₁₉	—	2	2	no	—
50	C₄⋊C₄×C₁₉	—	2	2	no	—
51	C₄⋊C₄×C₁₉	—	2	2	no	—
52	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
53	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
54	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
55	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
56	C₂²⋊C₄×C₁₉	—	2	2	no	—
57	C₂²⋊C₄×C₁₉	—	2	2	no	—
58	C₂²⋊C₄×C₁₉	—	2	2	no	—
59	C₂²⋊C₄×C₁₉	—	2	2	no	—
60	C₂²⋊C₄×C₁₉	—	2	2	no	—
61	C₂²⋊C₄×C₁₉	D₄	1	2	yes	—
62	C₄²×C₁₉	—	2	2	no	—
63	C₄²×C₁₉	C₂³	1	2	yes	—
64	C₂³×C₁₉	—	2	2	no	—
65	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
66	Q₈×C₁₉	—	2	2	no	—
67	Q₈×C₁₉	—	2	2	no	—
68	Q₈×C₁₉	—	8	2	no	—
69	D₄×C₁₉	—	2	2	no	—
70	D₄×C₁₉	—	2	2	no	—
71	D₄×C₁₉	—	8	2	no	—
72	C₂×C₄×C₁₉	—	2	1	no	—
73	C₂×C₄×C₁₉	—	2	1	no	—
74	C₂×C₄×C₁₉	—	2	2	no	—
75	C₂×C₄×C₁₉	—	2	2	no	—
76	C₂×C₄×C₁₉	—	2	2	no	—
77	C₂×C₄×C₁₉	—	8	2	no	—
78	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
82	C₈×C₁₉	—	2	1	no	—
83	C₈×C₁₉	—	2	1	no	—
84	C₈×C₁₉	—	2	2	no	—
85	#3262	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂²×C₁₉	—	2	1	no	—
87	C₂²×C₁₉	—	2	1	no	—
88	C₂²×C₁₉	—	2	1	no	—
89	C₂²×C₁₉	—	4	2	no	—
90	C₂²×C₁₉	—	4	2	no	—
91	C₂²×C₁₉	—	4	2	no	—
92	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
93	C₄×C₁₉	—	2	1	no	—
94	C₄×C₁₉	—	2	2	no	—
95	C₄×C₁₉	—	2	2	no	—
96	C₄×C₁₉	—	4	2	no	—
97	C₄×C₁₉	—	4	2	no	—
98	C₄×C₁₉	—	4	2	no	—
99	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
100	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
101	#542	C₂×C₁₉	1	2	yes	complemented
102	#477	C₂×C₁₉	1	2	yes	complemented
103	#474	C₂×C₁₉	1	2	yes	complemented
104	#468	C₂×C₁₉	1	2	yes	complemented
105	#467	C₂×C₁₉	1	1	yes	characteristic, complemented
106	#466	C₂×C₁₉	1	1	yes	characteristic
107	#464	C₂×C₁₉	1	2	yes	complemented
108	#463	C₂×C₁₉	1	2	yes	complemented
109	#424	C₂×C₁₉	1	1	yes	characteristic, complemented
110	C₂×C₁₉	—	4	1	no	—
111	C₂×C₁₉	—	8	2	no	—
112	C₂×C₁₉	#537	1	1	yes	characteristic
113	C₂×C₁₉	#576	1	1	yes	characteristic
114	C₂×C₁₉	#578	1	1	yes	characteristic
115	C₂×Q₁₆	C₂²×C₁₉	1	1	yes	characteristic
116	C₂×SD₁₆	C₂²×C₁₉	1	2	yes	—
117	C₂×D₈	C₂²×C₁₉	1	1	yes	characteristic
118	C₂×M₁₆	C₂²×C₁₉	1	1	yes	characteristic
119	C₂²×C₈	C₂²×C₁₉	1	1	yes	characteristic
120	C₄⋊Q₈	C₂²×C₁₉	1	2	yes	—
121	#126	—	2	2	no	—
122	#124	—	2	2	no	—
123	#124	C₂²×C₁₉	1	2	yes	—
124	#123	—	2	2	no	—
125	C₂²⋊Q₈	—	2	2	no	—
126	C₂²⋊Q₈	C₂²×C₁₉	1	2	yes	—
127	#121	C₂²×C₁₉	1	2	yes	—
128	C₄×Q₈	C₂²×C₁₉	1	2	yes	—
129	C₄×D₄	C₂²×C₁₉	1	2	yes	—
130	#117	C₂²×C₁₉	1	1	yes	characteristic
131	#107	C₂²×C₁₉	1	1	yes	characteristic
132	#107	C₂²×C₁₉	1	1	yes	characteristic
133	#107	C₂²×C₁₉	1	2	yes	—
134	C₄⋊C₈	C₂²×C₁₉	1	2	yes	—
135	C₄⋊C₈	C₂²×C₁₉	1	2	yes	—
136	#103	C₂²×C₁₉	1	2	yes	—
137	#103	C₂²×C₁₉	1	2	yes	—
138	#102	C₂²×C₁₉	1	2	yes	—
139	#102	C₂²×C₁₉	1	2	yes	—
140	C₁₉	#3262	1	1	yes	characteristic, direct product factor, Sylow
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	C₂³×C₁₉	1	2	yes	—
143	C₂×D₄	—	2	2	no	—
144	C₂×D₄	C₂³×C₁₉	1	2	yes	—
145	C₂²×C₄	—	2	2	no	—
146	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
147	Q₁₆	—	4	1	no	—
148	SD₁₆	—	4	2	no	—
149	D₈	—	4	1	no	—
150	M₁₆	—	4	1	no	—
151	C₂×C₈	—	4	1	no	—
152	C₂×C₈	C₂³×C₁₉	1	1	yes	characteristic
153	C₂×C₈	C₂³×C₁₉	1	2	yes	—
154	C₂×C₈	C₂³×C₁₉	1	1	yes	characteristic
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	—	2	2	no	—
158	C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
159	C₄⋊C₄	D₄×C₁₉	1	1	yes	characteristic
160	C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
161	C₄⋊C₄	C₂³×C₁₉	1	2	yes	—
162	C₂²⋊C₄	—	2	2	no	—
163	C₂²⋊C₄	—	2	2	no	—
164	C₂²⋊C₄	—	2	2	no	—
165	C₂²⋊C₄	—	2	2	no	—
166	C₂²⋊C₄	—	2	2	no	—
167	C₂²⋊C₄	D₄×C₁₉	1	2	yes	—
168	C₄²	—	2	2	no	—
169	C₄²	C₂³×C₁₉	1	2	yes	—
170	C₂³	—	2	2	no	—
171	C₂³	C₂×D₄×C₁₉	1	1	yes	characteristic
172	Q₈	—	2	2	no	—
173	Q₈	—	2	2	no	—
174	Q₈	—	8	2	no	—
175	D₄	—	2	2	no	—
176	D₄	—	2	2	no	—
177	D₄	—	8	2	no	—
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	8	2	no	—
184	C₂×C₄	C₂×D₄×C₁₉	1	2	yes	—
185	C₂×C₄	C₂×D₄×C₁₉	1	2	yes	—
186	C₂×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
187	C₂×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
188	C₈	—	2	1	no	—
189	C₈	—	2	1	no	—
190	C₈	—	2	2	no	—
191	C₂²	—	2	1	no	—
192	C₂²	—	2	1	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	4	2	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	4	2	no	—
197	C₂²	C₂²×D₄×C₁₉	1	1	yes	characteristic
198	C₄	—	2	1	no	—
199	C₄	—	2	2	no	—
200	C₄	—	2	2	no	—
201	C₄	—	4	2	no	—
202	C₄	—	4	2	no	—
203	C₄	—	4	2	no	—
204	C₄	C₁₉×H(5,𝔽₂)	1	1	yes	characteristic
205	C₄	C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
206	C₂	—	4	1	no	—
207	C₂	—	8	2	no	—
208	C₂	C₁₉×#537	1	1	yes	characteristic
209	C₂	C₁₉×#576	1	1	yes	characteristic
210	C₂	C₁₉×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₉
prime radical	C₁₉×#466	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₁₉×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₉×#542	C₂	semidirect
2. C₁₉×#477	C₂	semidirect
3. C₁₉×#474	C₂	semidirect
4. C₁₉×#468	C₂	semidirect
5. C₁₉×#467	C₂	semidirect
7. C₁₉×#464	C₂	semidirect
8. C₁₉×#463	C₂	semidirect
9. C₁₉×#424	C₂	semidirect
85. #3262	C₁₉	direct (N × H)
101. #542	C₂×C₁₉	semidirect
102. #477	C₂×C₁₉	semidirect
103. #474	C₂×C₁₉	semidirect
104. #468	C₂×C₁₉	semidirect
105. #467	C₂×C₁₉	semidirect
107. #464	C₂×C₁₉	semidirect
108. #463	C₂×C₁₉	semidirect
109. #424	C₂×C₁₉	semidirect
140. C₁₉	#3262	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×#542	2
C₁₉×#477	2
C₁₉×#474	2
C₁₉×#468	2
C₁₉×#467	2
C₁₉×#466	2
C₁₉×#464	2
C₁₉×#463	2
C₁₉×#424	2
#3262	19

Subgroups — C19×#3262 #14561119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#3262 #14561119