Subgroups — C₁₉×#13188 #18019147

Order 4864.

280 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×#2155	C₂	1	2	yes	maximal
2	C₂×C₁₉×#364	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1848	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1772	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1388	C₂	1	2	yes	maximal
6	C₂×C₁₉×M₃₂	C₂²	1	2	yes	—
7	C₂×C₁₉×#108	—	2	1	no	—
8	C₂×C₁₉×#108	C₂²	1	2	yes	—
9	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
10	C₁₉×#427	C₂²	1	1	yes	characteristic
11	C₁₉×#423	—	2	1	no	—
12	C₁₉×#405	—	2	1	no	—
13	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
14	C₁₉×#365	—	2	4	no	—
15	C₁₉×#364	C₄	1	2	yes	complemented
16	C₁₉×#364	C₄	1	2	yes	complemented
17	C₁₉×#339	—	2	1	no	—
18	C₂×M₁₆×C₁₉	—	2	1	no	—
19	C₂×M₁₆×C₁₉	—	2	2	no	—
20	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
22	C₁₉×#117	—	2	1	no	—
23	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
24	C₁₉×M₃₂	—	2	2	no	—
25	C₁₉×M₃₂	C₂×C₄	1	4	yes	—
26	C₂×C₁₆×C₁₉	—	2	2	no	—
27	C₁₉×#108	—	2	2	no	—
28	C₁₉×#108	—	2	1	no	—
29	C₁₉×#108	—	2	2	no	—
30	C₁₉×#108	—	4	1	no	—
31	C₁₉×#108	—	4	1	no	—
32	C₁₉×#108	Q₈	1	2	yes	—
33	C₁₉×#107	—	2	1	no	—
34	C₁₉×#106	D₄	1	2	yes	—
35	C₁₉×C₄⋊C₈	—	4	2	no	—
36	C₁₉×#97	—	4	1	no	—
37	C₄×C₈×C₁₉	—	4	1	no	—
38	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₁₉	Q₈	1	1	yes	characteristic
41	C₄×C₈×C₁₉	Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	—	2	1	no	—
43	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
44	M₁₆×C₁₉	—	2	2	no	—
45	M₁₆×C₁₉	—	4	1	no	—
46	M₁₆×C₁₉	—	4	2	no	—
47	M₁₆×C₁₉	—	4	2	no	—
48	C₂×C₈×C₁₉	—	4	1	no	—
49	C₂×C₈×C₁₉	—	4	2	no	—
50	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₉	SD₁₆	1	1	yes	characteristic
53	C₂×C₈×C₁₉	SD₁₆	1	1	yes	characteristic
54	C₂×C₈×C₁₉	SD₁₆	1	1	yes	characteristic
55	C₂×C₈×C₁₉	SD₁₆	1	1	yes	characteristic
56	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
61	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
62	C₄⋊C₄×C₁₉	—	2	2	no	—
63	C₂²⋊C₄×C₁₉	—	4	1	no	—
64	C₄²×C₁₉	—	2	1	no	—
65	C₄²×C₁₉	—	4	1	no	—
66	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
67	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
68	C₁₆×C₁₉	—	2	4	no	—
69	#13188	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂³×C₁₉	#137	1	1	yes	characteristic
71	C₂×C₄×C₁₉	—	2	1	no	—
72	C₂×C₄×C₁₉	—	2	1	no	—
73	C₂×C₄×C₁₉	—	4	2	no	—
74	C₂×C₄×C₁₉	—	4	2	no	—
75	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
77	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
80	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₈×C₁₉	—	4	2	no	—
82	C₈×C₁₉	—	8	2	no	—
83	C₈×C₁₉	#106	1	1	yes	characteristic
84	C₈×C₁₉	#106	1	1	yes	characteristic
85	C₈×C₁₉	#106	1	1	yes	characteristic
86	C₈×C₁₉	#106	1	1	yes	characteristic
87	C₈×C₁₉	C₄×D₄	1	1	yes	characteristic
88	C₈×C₁₉	C₄×Q₈	1	1	yes	characteristic
89	C₈×C₁₉	#125	1	1	yes	characteristic
90	C₈×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
91	#2155	C₂×C₁₉	1	2	yes	—
92	C₂×#364	C₂×C₁₉	1	1	yes	characteristic
93	#1848	C₂×C₁₉	1	1	yes	characteristic
94	#1772	C₂×C₁₉	1	1	yes	characteristic
95	#1388	C₂×C₁₉	1	2	yes	—
96	C₂²×C₁₉	—	4	1	no	—
97	C₂²×C₁₉	#428	1	1	yes	characteristic
98	C₂²×C₁₉	#461	1	1	yes	characteristic
99	C₂²×C₁₉	#475	1	1	yes	characteristic
100	C₄×C₁₉	—	2	2	no	—
101	C₄×C₁₉	—	4	1	no	—
102	C₄×C₁₉	—	8	2	no	—
103	C₄×C₁₉	#389	1	1	yes	characteristic
104	C₄×C₁₉	C₂×#106	1	1	yes	characteristic
105	C₄×C₁₉	#460	1	1	yes	characteristic
106	C₄×C₁₉	#476	1	1	yes	characteristic
107	C₂×M₃₂	C₂²×C₁₉	1	2	yes	—
108	C₂×#108	—	2	1	no	—
109	C₂×#108	C₂²×C₁₉	1	2	yes	—
110	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
111	#427	C₂²×C₁₉	1	1	yes	characteristic
112	#423	—	2	1	no	—
113	#405	—	2	1	no	—
114	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
115	#365	—	2	4	no	—
116	#364	C₄×C₁₉	1	2	yes	complemented
117	#364	C₄×C₁₉	1	2	yes	complemented
118	#339	—	2	1	no	—
119	C₂×C₁₉	—	2	1	no	—
120	C₂×C₁₉	—	2	1	no	—
121	C₂×C₁₉	—	2	1	no	—
122	C₂×C₁₉	#1948	1	1	yes	characteristic
123	C₂×M₁₆	—	2	1	no	—
124	C₂×M₁₆	—	2	2	no	—
125	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
126	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
127	#117	—	2	1	no	—
128	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
129	M₃₂	—	2	2	no	—
130	M₃₂	C₂×C₄×C₁₉	1	4	yes	—
131	C₂×C₁₆	—	2	2	no	—
132	#108	—	2	2	no	—
133	#108	—	2	1	no	—
134	#108	—	2	2	no	—
135	#108	—	4	1	no	—
136	#108	—	4	1	no	—
137	#108	Q₈×C₁₉	1	2	yes	—
138	#107	—	2	1	no	—
139	#106	D₄×C₁₉	1	2	yes	—
140	C₄⋊C₈	—	4	2	no	—
141	#97	—	4	1	no	—
142	C₄×C₈	—	4	1	no	—
143	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
144	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
145	C₄×C₈	Q₈×C₁₉	1	1	yes	characteristic
146	C₄×C₈	Q₈×C₁₉	1	1	yes	characteristic
147	C₁₉	#13188	1	1	yes	characteristic, direct product factor, Sylow
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
150	M₁₆	—	2	2	no	—
151	M₁₆	—	4	1	no	—
152	M₁₆	—	4	2	no	—
153	M₁₆	—	4	2	no	—
154	C₂×C₈	—	4	1	no	—
155	C₂×C₈	—	4	2	no	—
156	C₂×C₈	C₄⋊C₄×C₁₉	1	1	yes	characteristic
157	C₂×C₈	C₄⋊C₄×C₁₉	1	1	yes	characteristic
158	C₂×C₈	SD₁₆×C₁₉	1	1	yes	characteristic
159	C₂×C₈	SD₁₆×C₁₉	1	1	yes	characteristic
160	C₂×C₈	SD₁₆×C₁₉	1	1	yes	characteristic
161	C₂×C₈	SD₁₆×C₁₉	1	1	yes	characteristic
162	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
163	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
164	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
165	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
166	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
167	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
168	C₄⋊C₄	—	2	2	no	—
169	C₂²⋊C₄	—	4	1	no	—
170	C₄²	—	2	1	no	—
171	C₄²	—	4	1	no	—
172	C₄²	C₄⋊C₄×C₁₉	1	1	yes	characteristic
173	C₄²	C₄⋊C₄×C₁₉	1	1	yes	characteristic
174	C₁₆	—	2	4	no	—
175	C₂³	C₁₉×#137	1	1	yes	characteristic
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
181	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
182	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
183	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
184	C₂×C₄	C₂×SD₁₆×C₁₉	1	1	yes	characteristic
185	C₂×C₄	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
186	C₈	—	4	2	no	—
187	C₈	—	8	2	no	—
188	C₈	C₁₉×#106	1	1	yes	characteristic
189	C₈	C₁₉×#106	1	1	yes	characteristic
190	C₈	C₁₉×#106	1	1	yes	characteristic
191	C₈	C₁₉×#106	1	1	yes	characteristic
192	C₈	C₄×D₄×C₁₉	1	1	yes	characteristic
193	C₈	C₄×Q₈×C₁₉	1	1	yes	characteristic
194	C₈	C₁₉×#125	1	1	yes	characteristic
195	C₈	C₁₉×C₄⋊Q₈	1	1	yes	characteristic
196	C₂²	—	4	1	no	—
197	C₂²	C₁₉×#428	1	1	yes	characteristic
198	C₂²	C₁₉×#461	1	1	yes	characteristic
199	C₂²	C₁₉×#475	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	4	1	no	—
202	C₄	—	8	2	no	—
203	C₄	C₁₉×#389	1	1	yes	characteristic
204	C₄	C₂×C₁₉×#106	1	1	yes	characteristic
205	C₄	C₁₉×#460	1	1	yes	characteristic
206	C₄	C₁₉×#476	1	1	yes	characteristic
207	C₂	—	2	1	no	—
208	C₂	—	2	1	no	—
209	C₂	—	2	1	no	—
210	C₂	C₁₉×#1948	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₉	#476
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
socle	C₂×C₁₉	#1948
prime radical	C₂³×C₁₉	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₁₉×#364	C₄	semidirect
16. C₁₉×#364	C₄	semidirect
69. #13188	C₁₉	direct (N × H)
116. #364	C₄×C₁₉	semidirect
117. #364	C₄×C₁₉	semidirect
147. C₁₉	#13188	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×#2155	2
C₂×C₁₉×#364	2
C₁₉×#1848	2
C₁₉×#1772	2
C₁₉×#1388	2
#13188	19

Subgroups — C19×#13188 #18019147

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13188 #18019147