Subgroups — C₁₉×#20338 #18026297

Order 4864.

2036 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#568	C₂	1	1	yes	maximal, characteristic
2	C₁₉×#3428	C₂	1	2	yes	maximal
3	C₂×C₁₉×H(3,ℤ/4)	C₂	1	12	yes	maximal, complemented
4	C₂²×#41×C₁₉	C₂²	1	1	yes	characteristic
5	C₁₉×#568	—	2	8	no	—
6	C₂×C₁₉×#131	C₄	1	8	yes	complemented
7	C₂²×M₁₆×C₁₉	C₂²	1	6	yes	—
8	C₁₉×#520	—	2	8	no	—
9	C₁₉×#519	—	2	2	no	—
10	C₁₉×#518	—	2	6	no	—
11	C₄×#41×C₁₉	C₄	1	16	yes	complemented
12	C₂×C₁₉×#117	C₂²	1	12	yes	—
13	C₁₉×H(3,ℤ/4)	C₂²	1	16	yes	—
14	C₂×#41×C₁₉	C₂×C₄	1	8	yes	—
15	C₂×#41×C₁₉	C₂×C₄	1	4	yes	—
16	C₂×#41×C₁₉	D₄	1	8	yes	—
17	C₂×#41×C₁₉	Q₈	1	4	yes	—
18	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
19	C₂²×D₄×C₁₉	D₄	1	3	yes	—
20	C₂³×C₄×C₁₉	C₂³	1	3	yes	—
21	C₁₉×#131	—	2	32	no	—
22	C₂×M₁₆×C₁₉	—	2	24	no	—
23	C₂×M₁₆×C₁₉	C₂×C₄	1	12	yes	complemented
24	C₂×M₁₆×C₁₉	C₂³	1	4	yes	—
25	C₂²×C₈×C₁₉	C₂×C₄	1	12	yes	complemented
26	C₄×Q₈×C₁₉	—	2	16	no	—
27	C₄×D₄×C₁₉	—	2	48	no	—
28	C₁₉×#117	—	2	12	no	—
29	C₁₉×#117	—	2	12	no	—
30	C₁₉×#117	C₂×C₄	1	24	yes	complemented
31	C₁₉×#117	C₂³	1	8	yes	—
32	C₂×C₄⋊C₄×C₁₉	—	2	12	no	—
33	C₂×C₂²⋊C₄×C₁₉	—	2	12	no	—
34	C₂×C₄²×C₁₉	C₂×C₄	1	24	yes	complemented
35	C₂⁴×C₁₉	C₂×D₄	1	3	yes	—
36	#41×C₁₉	C₄²	1	16	yes	—
37	#41×C₁₉	C₂²⋊C₄	1	16	yes	—
38	#41×C₁₉	C₄⋊C₄	1	32	yes	—
39	C₂×Q₈×C₁₉	C₂²⋊C₄	1	8	yes	—
40	C₂×Q₈×C₁₉	C₄⋊C₄	1	4	yes	—
41	C₂×D₄×C₁₉	C₂²⋊C₄	1	24	yes	—
42	C₂×D₄×C₁₉	C₄⋊C₄	1	12	yes	—
43	C₂²×C₄×C₁₉	—	2	24	no	—
44	C₂²×C₄×C₁₉	C₂²×C₄	1	6	yes	—
45	C₂²×C₄×C₁₉	C₂²×C₄	1	12	yes	—
46	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
47	C₂²×C₄×C₁₉	C₂×D₄	1	12	yes	—
48	C₂²×C₄×C₁₉	C₂×Q₈	1	6	yes	—
49	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
50	M₁₆×C₁₉	—	2	24	no	—
51	M₁₆×C₁₉	—	2	8	no	—
52	C₂×C₈×C₁₉	—	2	24	no	—
53	C₂×C₈×C₁₉	C₂²×C₄	1	8	yes	—
54	C₄⋊C₄×C₁₉	—	2	24	no	—
55	C₄⋊C₄×C₁₉	—	2	8	no	—
56	C₂²⋊C₄×C₁₉	—	2	24	no	—
57	C₂²⋊C₄×C₁₉	—	2	8	no	—
58	C₄²×C₁₉	—	2	24	no	—
59	C₄²×C₁₉	C₂²×C₄	1	16	yes	—
60	#20338	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₁₉	—	8	3	no	—
62	C₂³×C₁₉	C₂×C₂²⋊C₄	1	12	yes	—
63	C₂³×C₁₉	C₂×C₄⋊C₄	1	6	yes	—
64	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
65	Q₈×C₁₉	#95	1	16	yes	—
66	D₄×C₁₉	#95	1	48	yes	—
67	C₂×C₄×C₁₉	—	2	16	no	—
68	C₂×C₄×C₁₉	—	8	24	no	—
69	C₂×C₄×C₁₉	C₂×C₄²	1	12	yes	—
70	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	12	yes	—
71	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	12	yes	—
72	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	24	yes	—
73	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	6	yes	—
74	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₉	C₂³×C₄	1	2	yes	—
76	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
77	C₂×C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
78	C₈×C₁₉	—	2	8	no	—
79	C₂×#568	C₂×C₁₉	1	1	yes	characteristic
80	#3428	C₂×C₁₉	1	2	yes	—
81	C₂×H(3,ℤ/4)	C₂×C₁₉	1	12	yes	complemented
82	C₂²×C₁₉	—	4	12	no	—
83	C₂²×C₁₉	—	4	1	no	—
84	C₂²×C₁₉	—	4	6	no	—
85	C₂²×C₁₉	C₂×#95	1	12	yes	—
86	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	2	yes	—
87	C₂²×C₁₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
88	C₄×C₁₉	—	4	16	no	—
89	C₄×C₁₉	C₂×#95	1	12	yes	—
90	C₄×C₁₉	C₂²×C₄²	1	1	yes	characteristic
91	C₄×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
92	C₄×C₁₉	C₂²×C₄⋊C₄	1	2	yes	—
93	C₂²×#41	C₂²×C₁₉	1	1	yes	characteristic
94	#568	—	2	8	no	—
95	C₂×#131	C₄×C₁₉	1	8	yes	complemented
96	C₂²×M₁₆	C₂²×C₁₉	1	6	yes	—
97	#520	—	2	8	no	—
98	#519	—	2	2	no	—
99	#518	—	2	6	no	—
100	C₄×#41	C₄×C₁₉	1	16	yes	complemented
101	C₂×#117	C₂²×C₁₉	1	12	yes	—
102	H(3,ℤ/4)	C₂²×C₁₉	1	16	yes	—
103	C₂×C₁₉	—	2	2	no	—
104	C₂×C₁₉	—	2	1	no	—
105	C₂×C₁₉	—	2	12	no	—
106	C₂×C₁₉	C₂²×#95	1	1	yes	characteristic
107	C₂×#41	C₂×C₄×C₁₉	1	8	yes	—
108	C₂×#41	C₂×C₄×C₁₉	1	4	yes	—
109	C₂×#41	D₄×C₁₉	1	8	yes	—
110	C₂×#41	Q₈×C₁₉	1	4	yes	—
111	C₂²×Q₈	D₄×C₁₉	1	1	yes	characteristic
112	C₂²×D₄	D₄×C₁₉	1	3	yes	—
113	C₂³×C₄	C₂³×C₁₉	1	3	yes	—
114	#131	—	2	32	no	—
115	C₂×M₁₆	—	2	24	no	—
116	C₂×M₁₆	C₂×C₄×C₁₉	1	12	yes	complemented
117	C₂×M₁₆	C₂³×C₁₉	1	4	yes	—
118	C₂²×C₈	C₂×C₄×C₁₉	1	12	yes	complemented
119	C₄×Q₈	—	2	16	no	—
120	C₄×D₄	—	2	48	no	—
121	#117	—	2	12	no	—
122	#117	—	2	12	no	—
123	#117	C₂×C₄×C₁₉	1	24	yes	complemented
124	#117	C₂³×C₁₉	1	8	yes	—
125	C₂×C₄⋊C₄	—	2	12	no	—
126	C₂×C₂²⋊C₄	—	2	12	no	—
127	C₂×C₄²	C₂×C₄×C₁₉	1	24	yes	complemented
128	C₁₉	#20338	1	1	yes	characteristic, direct product factor, Sylow
129	C₂⁴	C₂×D₄×C₁₉	1	3	yes	—
130	#41	C₄²×C₁₉	1	16	yes	—
131	#41	C₂²⋊C₄×C₁₉	1	16	yes	—
132	#41	C₄⋊C₄×C₁₉	1	32	yes	—
133	C₂×Q₈	C₂²⋊C₄×C₁₉	1	8	yes	—
134	C₂×Q₈	C₄⋊C₄×C₁₉	1	4	yes	—
135	C₂×D₄	C₂²⋊C₄×C₁₉	1	24	yes	—
136	C₂×D₄	C₄⋊C₄×C₁₉	1	12	yes	—
137	C₂²×C₄	—	2	24	no	—
138	C₂²×C₄	C₂²×C₄×C₁₉	1	6	yes	—
139	C₂²×C₄	C₂²×C₄×C₁₉	1	12	yes	—
140	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
141	C₂²×C₄	C₂×D₄×C₁₉	1	12	yes	—
142	C₂²×C₄	C₂×Q₈×C₁₉	1	6	yes	—
143	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
144	M₁₆	—	2	24	no	—
145	M₁₆	—	2	8	no	—
146	C₂×C₈	—	2	24	no	—
147	C₂×C₈	C₂²×C₄×C₁₉	1	8	yes	—
148	C₄⋊C₄	—	2	24	no	—
149	C₄⋊C₄	—	2	8	no	—
150	C₂²⋊C₄	—	2	24	no	—
151	C₂²⋊C₄	—	2	8	no	—
152	C₄²	—	2	24	no	—
153	C₄²	C₂²×C₄×C₁₉	1	16	yes	—
154	C₂³	—	8	3	no	—
155	C₂³	C₂×C₂²⋊C₄×C₁₉	1	12	yes	—
156	C₂³	C₂×C₄⋊C₄×C₁₉	1	6	yes	—
157	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
158	Q₈	C₁₉×#95	1	16	yes	—
159	D₄	C₁₉×#95	1	48	yes	—
160	C₂×C₄	—	2	16	no	—
161	C₂×C₄	—	8	24	no	—
162	C₂×C₄	C₂×C₄²×C₁₉	1	12	yes	—
163	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	12	yes	—
164	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	12	yes	—
165	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	6	yes	—
166	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	24	yes	—
167	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
168	C₂×C₄	C₂³×C₄×C₁₉	1	2	yes	—
169	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
170	C₂×C₄	C₂²×Q₈×C₁₉	1	1	yes	characteristic
171	C₈	—	2	8	no	—
172	C₂²	—	4	12	no	—
173	C₂²	—	4	1	no	—
174	C₂²	—	4	6	no	—
175	C₂²	C₂×C₁₉×#95	1	12	yes	—
176	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	2	yes	—
177	C₂²	C₂²×C₄⋊C₄×C₁₉	1	1	yes	characteristic
178	C₄	—	4	16	no	—
179	C₄	C₂×C₁₉×#95	1	12	yes	—
180	C₄	C₂²×C₄²×C₁₉	1	1	yes	characteristic
181	C₄	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
182	C₄	C₂²×C₄⋊C₄×C₁₉	1	2	yes	—
183	C₂	—	2	2	no	—
184	C₂	—	2	1	no	—
185	C₂	—	2	12	no	—
186	C₂	C₂²×C₁₉×#95	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₉	C₂²×C₄²
commutator	C₄	C₂²×C₄²×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
socle	C₂×C₁₉	C₂²×#95
prime radical	C₂²×#41×C₁₉	C₂²
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₁₉×H(3,ℤ/4)	C₂	semidirect
6. C₂×C₁₉×#131	C₄	semidirect
11. C₄×#41×C₁₉	C₄	semidirect
23. C₂×M₁₆×C₁₉	C₂×C₄	semidirect
25. C₂²×C₈×C₁₉	C₂×C₄	semidirect
30. C₁₉×#117	C₂×C₄	semidirect
34. C₂×C₄²×C₁₉	C₂×C₄	semidirect
60. #20338	C₁₉	direct (N × H)
81. C₂×H(3,ℤ/4)	C₂×C₁₉	semidirect
95. C₂×#131	C₄×C₁₉	semidirect
100. C₄×#41	C₄×C₁₉	semidirect
116. C₂×M₁₆	C₂×C₄×C₁₉	semidirect
118. C₂²×C₈	C₂×C₄×C₁₉	semidirect
123. #117	C₂×C₄×C₁₉	semidirect
127. C₂×C₄²	C₂×C₄×C₁₉	semidirect
128. C₁₉	#20338	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×#568	2
C₁₉×#3428	2
C₂×C₁₉×H(3,ℤ/4)	2
#20338	19

Subgroups — C19×#20338 #18026297

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#20338 #18026297