Subgroups — C₁₉×#13180 #18019139

Order 4864.

294 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2159	C₂	1	2	yes	maximal
2	C₂×C₁₉×#364	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#1846	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1772	C₂	1	1	yes	maximal, characteristic
5	C₁₉×#1387	C₂	1	2	yes	maximal
6	C₂×C₁₉×M₃₂	C₂²	1	2	yes	—
7	C₂×C₁₉×#108	C₂²	1	1	yes	characteristic
8	C₁₉×#427	C₂²	1	1	yes	characteristic
9	C₂×C₁₉×#107	C₂²	1	2	yes	—
10	C₂×C₁₉×#106	—	2	1	no	—
11	C₁₉×#405	—	2	1	no	—
12	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
13	C₁₉×#382	—	2	1	no	—
14	C₁₉×#367	—	2	2	no	—
15	C₁₉×#366	—	2	2	no	—
16	C₁₉×#364	C₄	1	4	yes	complemented
17	C₁₉×#339	—	2	1	no	—
18	C₂×M₁₆×C₁₉	—	2	1	no	—
19	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
21	C₁₉×#117	—	2	1	no	—
22	C₂×C₄⋊C₄×C₁₉	—	2	2	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	1	no	—
28	C₁₉×#108	Q₈	1	2	yes	—
29	C₁₉×#107	—	2	2	no	—
30	C₁₉×#107	—	2	2	no	—
31	C₁₉×#107	D₄	1	2	yes	—
32	C₁₉×#106	—	2	1	no	—
33	C₁₉×#106	—	4	1	no	—
34	C₁₉×#106	—	4	1	no	—
35	C₁₉×#97	—	4	1	no	—
36	C₄×C₈×C₁₉	—	4	1	no	—
37	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₁₉	Q₈	1	2	yes	—
39	C₁₉×#95	—	4	2	no	—
40	C₂²×C₄×C₁₉	—	2	1	no	—
41	C₂²×C₄×C₁₉	—	4	2	no	—
42	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
43	M₁₆×C₁₉	—	2	2	no	—
44	M₁₆×C₁₉	—	4	1	no	—
45	C₂×C₈×C₁₉	—	4	1	no	—
46	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₁₉	D₈	1	1	yes	characteristic
48	C₂×C₈×C₁₉	D₈	1	1	yes	characteristic
49	C₂×C₈×C₁₉	Q₁₆	1	2	yes	—
50	C₂×C₈×C₁₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
55	C₂×C₈×C₁₉	#41	1	1	yes	characteristic
56	C₄⋊C₄×C₁₉	—	2	2	no	—
57	C₄⋊C₄×C₁₉	—	4	2	no	—
58	C₄⋊C₄×C₁₉	—	4	2	no	—
59	C₂²⋊C₄×C₁₉	—	4	1	no	—
60	C₄²×C₁₉	—	2	1	no	—
61	C₄²×C₁₉	—	4	1	no	—
62	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
63	C₁₆×C₁₉	—	2	4	no	—
64	#13180	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂³×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂×C₄×C₁₉	—	2	2	no	—
67	C₂×C₄×C₁₉	—	4	2	no	—
68	C₂×C₄×C₁₉	—	4	2	no	—
69	C₂×C₄×C₁₉	—	8	4	no	—
70	C₂×C₄×C₁₉	—	8	2	no	—
71	C₂×C₄×C₁₉	C₂×C₄⋊C₄	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	C₂×D₈	1	1	yes	characteristic
75	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
77	C₈×C₁₉	—	4	2	no	—
78	C₈×C₁₉	#107	1	2	yes	—
79	C₈×C₁₉	#107	1	2	yes	—
80	C₈×C₁₉	C₄×D₄	1	1	yes	characteristic
81	C₈×C₁₉	C₄×Q₈	1	1	yes	characteristic
82	C₈×C₁₉	#125	1	1	yes	characteristic
83	C₈×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
84	#2159	C₂×C₁₉	1	2	yes	—
85	C₂×#364	C₂×C₁₉	1	1	yes	characteristic
86	#1846	C₂×C₁₉	1	1	yes	characteristic
87	#1772	C₂×C₁₉	1	1	yes	characteristic
88	#1387	C₂×C₁₉	1	2	yes	—
89	C₂²×C₁₉	—	2	2	no	—
90	C₂²×C₁₉	#428	1	1	yes	characteristic
91	C₂²×C₁₉	#459	1	1	yes	characteristic
92	C₂²×C₁₉	#477	1	1	yes	characteristic
93	C₄×C₁₉	—	4	2	no	—
94	C₄×C₁₉	—	8	2	no	—
95	C₄×C₁₉	—	16	2	no	—
96	C₄×C₁₉	#389	1	1	yes	characteristic
97	C₄×C₁₉	C₂×#107	1	1	yes	characteristic
98	C₄×C₁₉	#462	1	1	yes	characteristic
99	C₄×C₁₉	#474	1	1	yes	characteristic
100	C₂×M₃₂	C₂²×C₁₉	1	2	yes	—
101	C₂×#108	C₂²×C₁₉	1	1	yes	characteristic
102	#427	C₂²×C₁₉	1	1	yes	characteristic
103	C₂×#107	C₂²×C₁₉	1	2	yes	—
104	C₂×#106	—	2	1	no	—
105	#405	—	2	1	no	—
106	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
107	#382	—	2	1	no	—
108	#367	—	2	2	no	—
109	#366	—	2	2	no	—
110	#364	C₄×C₁₉	1	4	yes	complemented
111	#339	—	2	1	no	—
112	C₂×C₁₉	—	2	1	no	—
113	C₂×C₁₉	—	2	1	no	—
114	C₂×C₁₉	#1947	1	1	yes	characteristic
115	C₂×C₁₉	#2212	1	2	yes	—
116	C₂×M₁₆	—	2	1	no	—
117	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
118	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
119	#117	—	2	1	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄²	D₄×C₁₉	1	1	yes	characteristic
122	M₃₂	—	2	2	no	—
123	M₃₂	C₂×C₄×C₁₉	1	4	yes	—
124	C₂×C₁₆	—	2	2	no	—
125	#108	—	2	1	no	—
126	#108	Q₈×C₁₉	1	2	yes	—
127	#107	—	2	2	no	—
128	#107	—	2	2	no	—
129	#107	D₄×C₁₉	1	2	yes	—
130	#106	—	2	1	no	—
131	#106	—	4	1	no	—
132	#106	—	4	1	no	—
133	#97	—	4	1	no	—
134	C₄×C₈	—	4	1	no	—
135	C₄×C₈	C₂×C₄×C₁₉	1	2	yes	—
136	C₄×C₈	Q₈×C₁₉	1	2	yes	—
137	#95	—	4	2	no	—
138	C₁₉	#13180	1	1	yes	characteristic, direct product factor, Sylow
139	C₂²×C₄	—	2	1	no	—
140	C₂²×C₄	—	4	2	no	—
141	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
142	M₁₆	—	2	2	no	—
143	M₁₆	—	4	1	no	—
144	C₂×C₈	—	4	1	no	—
145	C₂×C₈	C₄⋊C₄×C₁₉	1	2	yes	—
146	C₂×C₈	D₈×C₁₉	1	1	yes	characteristic
147	C₂×C₈	D₈×C₁₉	1	1	yes	characteristic
148	C₂×C₈	Q₁₆×C₁₉	1	2	yes	—
149	C₂×C₈	C₂²×C₄×C₁₉	1	1	yes	characteristic
150	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
151	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
152	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
153	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
154	C₂×C₈	#41×C₁₉	1	1	yes	characteristic
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	4	2	no	—
157	C₄⋊C₄	—	4	2	no	—
158	C₂²⋊C₄	—	4	1	no	—
159	C₄²	—	2	1	no	—
160	C₄²	—	4	1	no	—
161	C₄²	C₄⋊C₄×C₁₉	1	2	yes	—
162	C₁₆	—	2	4	no	—
163	C₂³	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	4	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	8	4	no	—
168	C₂×C₄	—	8	2	no	—
169	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
170	C₂×C₄	C₁₉×#121	1	1	yes	characteristic
171	C₂×C₄	C₁₉×C₂²⋊Q₈	1	1	yes	characteristic
172	C₂×C₄	C₂×D₈×C₁₉	1	1	yes	characteristic
173	C₂×C₄	C₂×Q₁₆×C₁₉	1	1	yes	characteristic
174	C₂×C₄	C₁₉×#137	1	1	yes	characteristic
175	C₈	—	4	2	no	—
176	C₈	C₁₉×#107	1	2	yes	—
177	C₈	C₁₉×#107	1	2	yes	—
178	C₈	C₄×D₄×C₁₉	1	1	yes	characteristic
179	C₈	C₄×Q₈×C₁₉	1	1	yes	characteristic
180	C₈	C₁₉×#125	1	1	yes	characteristic
181	C₈	C₁₉×C₄⋊Q₈	1	1	yes	characteristic
182	C₂²	—	2	2	no	—
183	C₂²	C₁₉×#428	1	1	yes	characteristic
184	C₂²	C₁₉×#459	1	1	yes	characteristic
185	C₂²	C₁₉×#477	1	1	yes	characteristic
186	C₄	—	4	2	no	—
187	C₄	—	8	2	no	—
188	C₄	—	16	2	no	—
189	C₄	C₁₉×#389	1	1	yes	characteristic
190	C₄	C₂×C₁₉×#107	1	1	yes	characteristic
191	C₄	C₁₉×#462	1	1	yes	characteristic
192	C₄	C₁₉×#474	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	1	no	—
195	C₂	C₁₉×#1947	1	1	yes	characteristic
196	C₂	C₁₉×#2212	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	#459
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂³×C₁₉	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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
16. C₁₉×#364	C₄	semidirect
64. #13180	C₁₉	direct (N × H)
110. #364	C₄×C₁₉	semidirect
138. C₁₉	#13180	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×#2159	2
C₂×C₁₉×#364	2
C₁₉×#1846	2
C₁₉×#1772	2
C₁₉×#1387	2
#13180	19

Subgroups — C19×#13180 #18019139

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#13180 #18019139