Subgroups — C₁₉×#20508 #18026467

Order 4864.

1206 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	C₂×C₄×D₈×C₁₉	C₂	1	3	yes	maximal
2	C₁₉×#2275	C₂	1	2	yes	maximal, complemented
3	C₄×C₁₉×#107	C₂	1	3	yes	maximal, complemented
4	C₁₉×#1762	C₂	1	6	yes	maximal, complemented
5	C₁₉×#1726	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₈×C₁₉	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₄×C₁₉	C₂²	1	6	yes	—
8	C₄×D₈×C₁₉	C₄	1	24	yes	complemented
9	C₂×C₁₉×#107	C₂²	1	3	yes	—
10	C₂×C₁₉×#102	C₂²	1	6	yes	—
11	C₂×C₁₉×#97	C₂²	1	3	yes	—
12	C₂×C₄×C₈×C₁₉	C₂²	1	3	yes	—
13	C₄×C₄⋊C₄×C₁₉	C₂²	1	6	yes	—
14	C₄×C₂²⋊C₄×C₁₉	—	2	6	no	—
15	C₁₉×#376	C₂²	1	1	yes	characteristic
16	C₂×C₁₉×#95	—	2	2	no	—
17	C₁₉×#340	C₂²	1	6	yes	—
18	C₂²×D₄×C₁₉	C₂³	1	2	yes	—
19	C₂³×C₄×C₁₉	—	2	6	no	—
20	C₂×D₈×C₁₉	C₂×C₄	1	12	yes	—
21	C₂²×C₈×C₁₉	C₂³	1	3	yes	—
22	C₂²×C₈×C₁₉	C₂³	1	1	yes	characteristic
23	C₄×D₄×C₁₉	—	2	12	no	—
24	C₄×D₄×C₁₉	C₂×C₄	1	24	yes	complemented
25	C₂×C₄⋊C₄×C₁₉	C₂³	1	6	yes	—
26	C₂×C₂²⋊C₄×C₁₉	—	2	6	no	—
27	C₂×C₄²×C₁₉	—	2	6	no	—
28	C₂×C₄²×C₁₉	C₂³	1	3	yes	—
29	C₁₉×#107	C₂×C₄	1	12	yes	complemented
30	C₁₉×#102	C₂×C₄	1	24	yes	complemented
31	C₁₉×#97	—	2	6	no	—
32	C₄×C₈×C₁₉	C₂×C₄	1	12	yes	complemented
33	C₁₉×#95	—	2	6	no	—
34	C₁₉×#95	D₄	1	1	yes	characteristic
35	C₁₉×#95	D₄	1	3	yes	—
36	C₂⁴×C₁₉	—	2	2	no	—
37	C₂×D₄×C₁₉	—	2	6	no	—
38	C₂×D₄×C₁₉	C₂²×C₄	1	12	yes	—
39	C₂²×C₄×C₁₉	—	2	6	no	—
40	C₂²×C₄×C₁₉	—	2	24	no	—
41	C₂²×C₄×C₁₉	—	8	12	no	—
42	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
43	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
44	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
45	D₈×C₁₉	C₄²	1	16	yes	complemented
46	C₂×C₈×C₁₉	—	2	6	no	—
47	C₂×C₈×C₁₉	C₂²×C₄	1	6	yes	—
48	C₂×C₈×C₁₉	C₂²×C₄	1	6	yes	—
49	C₄⋊C₄×C₁₉	—	2	6	no	—
50	C₄⋊C₄×C₁₉	C₂²×C₄	1	12	yes	—
51	C₂²⋊C₄×C₁₉	—	2	24	no	—
52	C₄²×C₁₉	—	8	6	no	—
53	C₄²×C₁₉	C₂²×C₄	1	6	yes	—
54	C₄²×C₁₉	#41	1	6	yes	—
55	#20508	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂³×C₁₉	—	2	12	no	—
57	C₂³×C₁₉	—	4	6	no	—
58	C₂³×C₁₉	—	4	2	no	—
59	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
60	D₄×C₁₉	—	2	12	no	—
61	D₄×C₁₉	C₂×C₄²	1	8	yes	—
62	C₂×C₄×C₁₉	—	2	3	no	—
63	C₂×C₄×C₁₉	—	2	6	no	—
64	C₂×C₄×C₁₉	—	2	3	no	—
65	C₂×C₄×C₁₉	—	2	12	no	—
66	C₂×C₄×C₁₉	—	4	12	no	—
67	C₂×C₄×C₁₉	—	8	24	no	—
68	C₂×C₄×C₁₉	C₄×D₄	1	6	yes	—
69	C₂×C₄×C₁₉	C₄×D₄	1	6	yes	—
70	C₂×C₄×C₁₉	C₂³×C₄	1	3	yes	—
71	C₂×C₄×C₁₉	C₂×#41	1	3	yes	—
72	C₈×C₁₉	—	2	6	no	—
73	C₈×C₁₉	C₂×C₄²	1	4	yes	—
74	C₂×C₄×D₈	C₂×C₁₉	1	3	yes	—
75	#2275	C₂×C₁₉	1	2	yes	complemented
76	C₄×#107	C₂×C₁₉	1	3	yes	complemented
77	#1762	C₂×C₁₉	1	6	yes	complemented
78	#1726	C₂×C₁₉	1	1	yes	characteristic, complemented
79	C₂²×C₁₉	—	2	8	no	—
80	C₂²×C₁₉	—	4	12	no	—
81	C₂²×C₁₉	—	4	12	no	—
82	C₂²×C₁₉	C₂×C₄×D₄	1	3	yes	—
83	C₂²×C₁₉	#576	1	3	yes	—
84	C₂²×C₁₉	#578	1	1	yes	characteristic
85	C₄×C₁₉	—	2	6	no	—
86	C₄×C₁₉	—	2	6	no	—
87	C₄×C₁₉	—	4	12	no	—
88	C₄×C₁₉	C₂²×C₄²	1	1	yes	characteristic
89	C₄×C₁₉	C₄×#41	1	3	yes	—
90	C₂²×D₈	C₂²×C₁₉	1	1	yes	characteristic
91	C₂×C₄×D₄	C₂²×C₁₉	1	6	yes	—
92	C₄×D₈	C₄×C₁₉	1	24	yes	complemented
93	C₂×#107	C₂²×C₁₉	1	3	yes	—
94	C₂×#102	C₂²×C₁₉	1	6	yes	—
95	C₂×#97	C₂²×C₁₉	1	3	yes	—
96	C₂×C₄×C₈	C₂²×C₁₉	1	3	yes	—
97	C₄×C₄⋊C₄	C₂²×C₁₉	1	6	yes	—
98	C₄×C₂²⋊C₄	—	2	6	no	—
99	#376	C₂²×C₁₉	1	1	yes	characteristic
100	C₂×#95	—	2	2	no	—
101	#340	C₂²×C₁₉	1	6	yes	—
102	C₂×C₁₉	—	4	8	no	—
103	C₂×C₁₉	C₄²×D₄	1	1	yes	characteristic
104	C₂×C₁₉	#2948	1	3	yes	—
105	C₂×C₁₉	#2951	1	3	yes	—
106	C₂²×D₄	C₂³×C₁₉	1	2	yes	—
107	C₂³×C₄	—	2	6	no	—
108	C₂×D₈	C₂×C₄×C₁₉	1	12	yes	—
109	C₂²×C₈	C₂³×C₁₉	1	3	yes	—
110	C₂²×C₈	C₂³×C₁₉	1	1	yes	characteristic
111	C₄×D₄	—	2	12	no	—
112	C₄×D₄	C₂×C₄×C₁₉	1	24	yes	complemented
113	C₂×C₄⋊C₄	C₂³×C₁₉	1	6	yes	—
114	C₂×C₂²⋊C₄	—	2	6	no	—
115	C₂×C₄²	—	2	6	no	—
116	C₂×C₄²	C₂³×C₁₉	1	3	yes	—
117	#107	C₂×C₄×C₁₉	1	12	yes	complemented
118	#102	C₂×C₄×C₁₉	1	24	yes	complemented
119	#97	—	2	6	no	—
120	C₄×C₈	C₂×C₄×C₁₉	1	12	yes	complemented
121	#95	—	2	6	no	—
122	#95	D₄×C₁₉	1	1	yes	characteristic
123	#95	D₄×C₁₉	1	3	yes	—
124	C₁₉	#20508	1	1	yes	characteristic, direct product factor, Sylow
125	C₂⁴	—	2	2	no	—
126	C₂×D₄	—	2	6	no	—
127	C₂×D₄	C₂²×C₄×C₁₉	1	12	yes	—
128	C₂²×C₄	—	2	6	no	—
129	C₂²×C₄	—	2	24	no	—
130	C₂²×C₄	—	8	12	no	—
131	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
132	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
133	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
134	D₈	C₄²×C₁₉	1	16	yes	complemented
135	C₂×C₈	—	2	6	no	—
136	C₂×C₈	C₂²×C₄×C₁₉	1	6	yes	—
137	C₂×C₈	C₂²×C₄×C₁₉	1	6	yes	—
138	C₄⋊C₄	—	2	6	no	—
139	C₄⋊C₄	C₂²×C₄×C₁₉	1	12	yes	—
140	C₂²⋊C₄	—	2	24	no	—
141	C₄²	—	8	6	no	—
142	C₄²	C₂²×C₄×C₁₉	1	6	yes	—
143	C₄²	#41×C₁₉	1	6	yes	—
144	C₂³	—	2	12	no	—
145	C₂³	—	4	6	no	—
146	C₂³	—	4	2	no	—
147	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
148	D₄	—	2	12	no	—
149	D₄	C₂×C₄²×C₁₉	1	8	yes	—
150	C₂×C₄	—	2	3	no	—
151	C₂×C₄	—	2	6	no	—
152	C₂×C₄	—	2	3	no	—
153	C₂×C₄	—	2	12	no	—
154	C₂×C₄	—	4	12	no	—
155	C₂×C₄	—	8	24	no	—
156	C₂×C₄	C₄×D₄×C₁₉	1	6	yes	—
157	C₂×C₄	C₄×D₄×C₁₉	1	6	yes	—
158	C₂×C₄	C₂³×C₄×C₁₉	1	3	yes	—
159	C₂×C₄	C₂×#41×C₁₉	1	3	yes	—
160	C₈	—	2	6	no	—
161	C₈	C₂×C₄²×C₁₉	1	4	yes	—
162	C₂²	—	2	8	no	—
163	C₂²	—	4	12	no	—
164	C₂²	—	4	12	no	—
165	C₂²	C₂×C₄×D₄×C₁₉	1	3	yes	—
166	C₂²	C₁₉×#576	1	3	yes	—
167	C₂²	C₁₉×#578	1	1	yes	characteristic
168	C₄	—	2	6	no	—
169	C₄	—	2	6	no	—
170	C₄	—	4	12	no	—
171	C₄	C₂²×C₄²×C₁₉	1	1	yes	characteristic
172	C₄	C₄×#41×C₁₉	1	3	yes	—
173	C₂	—	4	8	no	—
174	C₂	C₄²×D₄×C₁₉	1	1	yes	characteristic
175	C₂	C₁₉×#2948	1	3	yes	—
176	C₂	C₁₉×#2951	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂²×D₄
commutator	C₄	C₂²×C₄²×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
prime radical	C₂²×D₈×C₁₉	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₁₉×#376 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₉×#2275	C₂	semidirect
3. C₄×C₁₉×#107	C₂	semidirect
4. C₁₉×#1762	C₂	semidirect
5. C₁₉×#1726	C₂	semidirect
8. C₄×D₈×C₁₉	C₄	semidirect
24. C₄×D₄×C₁₉	C₂×C₄	semidirect
29. C₁₉×#107	C₂×C₄	semidirect
30. C₁₉×#102	C₂×C₄	semidirect
32. C₄×C₈×C₁₉	C₂×C₄	semidirect
45. D₈×C₁₉	C₄²	semidirect
55. #20508	C₁₉	direct (N × H)
75. #2275	C₂×C₁₉	semidirect
76. C₄×#107	C₂×C₁₉	semidirect
77. #1762	C₂×C₁₉	semidirect
78. #1726	C₂×C₁₉	semidirect
92. C₄×D₈	C₄×C₁₉	semidirect
112. C₄×D₄	C₂×C₄×C₁₉	semidirect
117. #107	C₂×C₄×C₁₉	semidirect
118. #102	C₂×C₄×C₁₉	semidirect
120. C₄×C₈	C₂×C₄×C₁₉	semidirect
124. C₁₉	#20508	direct (N × H)
134. D₈	C₄²×C₁₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×D₈×C₁₉	2
C₁₉×#2275	2
C₄×C₁₉×#107	2
C₁₉×#1762	2
C₁₉×#1726	2
#20508	19

Subgroups — C19×#20508 #18026467

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#20508 #18026467