Subgroups — C₁₉×#12018 #18017977

Order 4864.

236 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#2162	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#2150	C₂	1	1	yes	maximal, characteristic
3	C₈×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1370	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1367	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₉×M₃₂	—	2	1	no	—
7	C₁₉×#431	C₄	1	1	yes	characteristic
8	C₄×M₁₆×C₁₉	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
10	C₁₉×C₄⋊C₁₆	—	2	1	no	—
11	C₁₉×C₄⋊C₁₆	C₂²	1	2	yes	—
12	C₁₉×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₁₉×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₁₉×C₂²⋊C₁₆	—	2	1	no	—
15	C₄×C₁₆×C₁₉	—	2	1	no	—
16	C₁₉×#322	C₄	1	2	yes	—
17	C₈²×C₁₉	C₂²	1	2	yes	—
18	C₂×M₁₆×C₁₉	—	2	1	no	—
19	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₉	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₁₉×M₃₂	—	4	2	no	—
23	C₂×C₁₆×C₁₉	—	2	1	no	—
24	C₂×C₁₆×C₁₉	—	2	1	no	—
25	C₂×C₁₆×C₁₉	—	2	2	no	—
26	C₁₉×C₄⋊C₈	C₈	1	2	yes	—
27	C₁₉×C₂²⋊C₈	C₈	1	2	yes	—
28	C₁₉×#97	—	2	1	no	—
29	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
33	C₄×C₈×C₁₉	Q₈	1	1	yes	characteristic
34	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₉	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₁₉	M₁₆	1	1	yes	characteristic
38	M₁₆×C₁₉	—	2	4	no	—
39	C₂×C₈×C₁₉	—	2	2	no	—
40	C₂×C₈×C₁₉	—	2	1	no	—
41	C₂×C₈×C₁₉	—	2	1	no	—
42	C₂×C₈×C₁₉	—	4	1	no	—
43	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₉	C₂×C₈	1	4	yes	—
46	C₂×C₈×C₁₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₁₉	C₂×Q₈	1	1	yes	characteristic
48	C₄²×C₁₉	C₄⋊C₄	1	1	yes	characteristic
49	C₄²×C₁₉	C₂×C₈	1	1	yes	characteristic
50	C₄²×C₁₉	M₁₆	1	1	yes	characteristic
51	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
52	C₁₆×C₁₉	—	4	1	no	—
53	C₁₆×C₁₉	—	4	2	no	—
54	C₁₆×C₁₉	—	4	1	no	—
55	#12018	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂³×C₁₉	C₄⋊C₈	1	1	yes	characteristic
57	C₂×C₄×C₁₉	—	2	2	no	—
58	C₂×C₄×C₁₉	—	2	1	no	—
59	C₂×C₄×C₁₉	—	2	1	no	—
60	C₂×C₄×C₁₉	—	2	2	no	—
61	C₂×C₄×C₁₉	C₄⋊C₈	1	1	yes	characteristic
62	C₂×C₄×C₁₉	C₄⋊C₈	1	1	yes	characteristic
63	C₂×C₄×C₁₉	C₄⋊C₈	1	1	yes	characteristic
64	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
66	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
67	C₈×C₁₉	—	2	4	no	—
68	C₈×C₁₉	—	2	1	no	—
69	C₈×C₁₉	—	2	1	no	—
70	C₈×C₁₉	—	2	1	no	—
71	C₈×C₁₉	—	2	4	no	—
72	C₈×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₈×C₁₉	#137	1	1	yes	characteristic
74	#2162	C₂×C₁₉	1	1	yes	characteristic
75	#2150	C₂×C₁₉	1	1	yes	characteristic
76	C₈×M₁₆	C₂×C₁₉	1	1	yes	characteristic
77	#1370	C₂×C₁₉	1	2	yes	complemented
78	#1367	C₂×C₁₉	1	2	yes	complemented
79	C₂²×C₁₉	—	2	1	no	—
80	C₂²×C₁₉	—	2	1	no	—
81	C₂²×C₁₉	—	2	1	no	—
82	C₂²×C₁₉	C₂×C₄⋊C₈	1	1	yes	characteristic
83	C₄×C₁₉	—	2	1	no	—
84	C₄×C₁₉	—	2	1	no	—
85	C₄×C₁₉	—	2	1	no	—
86	C₄×C₁₉	#364	1	2	yes	—
87	C₄×C₁₉	#428	1	1	yes	characteristic
88	C₄×C₁₉	#430	1	1	yes	characteristic
89	C₄×C₁₉	#504	1	2	yes	—
90	C₂×M₃₂	—	2	1	no	—
91	#431	C₄×C₁₉	1	1	yes	characteristic
92	C₄×M₁₆	C₄×C₁₉	1	1	yes	characteristic
93	C₂×C₄×C₈	C₂²×C₁₉	1	1	yes	characteristic
94	C₄⋊C₁₆	—	2	1	no	—
95	C₄⋊C₁₆	C₂²×C₁₉	1	1	yes	characteristic
96	C₄⋊C₁₆	C₂²×C₁₉	1	2	yes	—
97	C₄⋊C₁₆	C₂²×C₁₉	1	1	yes	characteristic
98	C₂²⋊C₁₆	—	2	1	no	—
99	C₄×C₁₆	—	2	1	no	—
100	#322	C₄×C₁₉	1	2	yes	—
101	C₈²	C₂²×C₁₉	1	2	yes	—
102	C₂×C₁₉	—	4	1	no	—
103	C₂×C₁₉	#1565	1	1	yes	characteristic
104	C₂×C₁₉	#2151	1	1	yes	characteristic
105	C₂×C₁₉	C₂×#364	1	1	yes	characteristic
106	C₂×M₁₆	—	2	1	no	—
107	C₂²×C₈	D₄×C₁₉	1	1	yes	characteristic
108	C₂²×C₈	Q₈×C₁₉	1	1	yes	characteristic
109	C₂×C₄²	C₂×C₄×C₁₉	1	1	yes	characteristic
110	M₃₂	—	4	2	no	—
111	C₂×C₁₆	—	2	1	no	—
112	C₂×C₁₆	—	2	1	no	—
113	C₂×C₁₆	—	2	2	no	—
114	C₄⋊C₈	C₈×C₁₉	1	2	yes	—
115	C₂²⋊C₈	C₈×C₁₉	1	2	yes	—
116	#97	—	2	1	no	—
117	C₄×C₈	C₂×C₄×C₁₉	1	2	yes	—
118	C₄×C₈	C₂×C₄×C₁₉	1	1	yes	characteristic
119	C₄×C₈	C₂×C₄×C₁₉	1	2	yes	—
120	C₄×C₈	D₄×C₁₉	1	1	yes	characteristic
121	C₄×C₈	Q₈×C₁₉	1	1	yes	characteristic
122	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
123	C₁₉	#12018	1	1	yes	characteristic, direct product factor, Sylow
124	C₂²×C₄	C₄⋊C₄×C₁₉	1	1	yes	characteristic
125	C₂²×C₄	C₂×C₈×C₁₉	1	1	yes	characteristic
126	C₂²×C₄	M₁₆×C₁₉	1	1	yes	characteristic
127	M₁₆	—	2	4	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	—	4	1	no	—
132	C₂×C₈	C₄⋊C₄×C₁₉	1	1	yes	characteristic
133	C₂×C₈	C₄⋊C₄×C₁₉	1	1	yes	characteristic
134	C₂×C₈	C₂×C₈×C₁₉	1	4	yes	—
135	C₂×C₈	C₂×D₄×C₁₉	1	1	yes	characteristic
136	C₂×C₈	C₂×Q₈×C₁₉	1	1	yes	characteristic
137	C₄²	C₄⋊C₄×C₁₉	1	1	yes	characteristic
138	C₄²	C₂×C₈×C₁₉	1	1	yes	characteristic
139	C₄²	M₁₆×C₁₉	1	1	yes	characteristic
140	C₄²	C₂²×C₄×C₁₉	1	1	yes	characteristic
141	C₁₆	—	4	1	no	—
142	C₁₆	—	4	2	no	—
143	C₁₆	—	4	1	no	—
144	C₂³	C₁₉×C₄⋊C₈	1	1	yes	characteristic
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	2	2	no	—
149	C₂×C₄	C₁₉×C₄⋊C₈	1	1	yes	characteristic
150	C₂×C₄	C₁₉×C₄⋊C₈	1	1	yes	characteristic
151	C₂×C₄	C₁₉×C₄⋊C₈	1	1	yes	characteristic
152	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	1	yes	characteristic
153	C₂×C₄	C₂²×C₈×C₁₉	1	1	yes	characteristic
154	C₂×C₄	C₂×M₁₆×C₁₉	1	1	yes	characteristic
155	C₈	—	2	4	no	—
156	C₈	—	2	1	no	—
157	C₈	—	2	1	no	—
158	C₈	—	2	1	no	—
159	C₈	—	2	4	no	—
160	C₈	C₁₉×AGL(1,ℤ/8)	1	1	yes	characteristic
161	C₈	C₁₉×#137	1	1	yes	characteristic
162	C₂²	—	2	1	no	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	C₂×C₁₉×C₄⋊C₈	1	1	yes	characteristic
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	C₁₉×#364	1	2	yes	—
170	C₄	C₁₉×#428	1	1	yes	characteristic
171	C₄	C₁₉×#430	1	1	yes	characteristic
172	C₄	C₁₉×#504	1	2	yes	—
173	C₂	—	4	1	no	—
174	C₂	C₁₉×#1565	1	1	yes	characteristic
175	C₂	C₁₉×#2151	1	1	yes	characteristic
176	C₂	C₂×C₁₉×#364	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₉
Frattini, cosocle	C₄×C₈	C₂³×C₁₉
socle	C₂²×C₁₉	C₂×C₄⋊C₈
prime radical	C₂³×C₁₉	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₉×#1370	C₂	semidirect
5. C₁₉×#1367	C₂	semidirect
55. #12018	C₁₉	direct (N × H)
77. #1370	C₂×C₁₉	semidirect
78. #1367	C₂×C₁₉	semidirect
123. C₁₉	#12018	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×#2162	2
C₁₉×#2150	2
C₈×M₁₆×C₁₉	2
C₁₉×#1370	2
C₁₉×#1367	2
#12018	19

Subgroups — C19×#12018 #18017977

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#12018 #18017977