Subgroups — C₂×C₄×C₁₉×2₋¹⁺⁴ #18068609

Order 4864.

11210 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₉×2₋¹⁺⁴	C₂	1	32	yes	maximal, direct product factor
3	C₂×C₁₉×#520	C₂	1	10	yes	maximal, complemented
4	C₂×C₁₉×#519	C₂	1	5	yes	maximal, complemented
5	C₂×C₄×#41×C₁₉	C₂	1	10	yes	maximal, complemented
6	C₂²×C₄×Q₈×C₁₉	C₂	1	5	yes	maximal, complemented
7	C₂×C₁₉×2₋¹⁺⁴	C₄	1	32	yes	direct product factor
8	C₂×C₁₉×2₋¹⁺⁴	C₂²	1	16	yes	—
9	C₂²×#41×C₁₉	C₂²	1	10	yes	—
10	C₂³×Q₈×C₁₉	C₂²	1	5	yes	—
11	C₁₉×#520	C₂²	1	160	yes	complemented
12	C₁₉×#519	C₂²	1	80	yes	complemented
13	C₄×#41×C₁₉	C₂²	1	160	yes	complemented
14	C₂×C₄×Q₈×C₁₉	C₂²	1	10	yes	—
15	C₂×C₄×Q₈×C₁₉	C₂²	1	20	yes	—
16	C₂×C₄×Q₈×C₁₉	C₂²	1	10	yes	—
17	C₂×C₄×Q₈×C₁₉	C₂²	1	80	yes	complemented
18	C₂×C₄×D₄×C₁₉	C₂²	1	10	yes	—
19	C₂×C₄×D₄×C₁₉	C₂²	1	30	yes	—
20	C₂×C₁₉×#117	C₂²	1	30	yes	—
21	C₂²×C₄⋊C₄×C₁₉	C₂²	1	15	yes	—
22	C₂²×C₄²×C₁₉	C₂²	1	15	yes	—
23	C₁₉×2₋¹⁺⁴	C₂×C₄	1	256	yes	direct product factor
24	C₂×#41×C₁₉	C₂×C₄	1	160	yes	complemented
25	C₂×#41×C₁₉	C₂³	1	80	yes	—
26	C₂²×Q₈×C₁₉	C₂×C₄	1	80	yes	complemented
27	C₂²×Q₈×C₁₉	C₂³	1	10	yes	—
28	C₂²×Q₈×C₁₉	C₂³	1	40	yes	—
29	C₂²×D₄×C₁₉	C₂³	1	10	yes	—
30	C₂³×C₄×C₁₉	C₂³	1	5	yes	—
31	C₂³×C₄×C₁₉	C₂³	1	15	yes	—
32	C₂³×C₄×C₁₉	C₂³	1	15	yes	—
33	C₄×Q₈×C₁₉	C₂³	1	80	yes	—
34	C₄×Q₈×C₁₉	C₂³	1	160	yes	—
35	C₄×Q₈×C₁₉	C₂³	1	80	yes	—
36	C₄×D₄×C₁₉	C₂³	1	80	yes	—
37	C₄×D₄×C₁₉	C₂³	1	240	yes	—
38	C₁₉×#117	C₂³	1	240	yes	—
39	C₂×C₄⋊C₄×C₁₉	C₂³	1	30	yes	—
40	C₂×C₄⋊C₄×C₁₉	C₂³	1	120	yes	—
41	C₂×C₄⋊C₄×C₁₉	C₂³	1	10	yes	—
42	C₂×C₂²⋊C₄×C₁₉	C₂³	1	20	yes	—
43	C₂×C₄²×C₁₉	C₂³	1	30	yes	—
44	C₂×C₄²×C₁₉	C₂³	1	120	yes	—
45	C₂×C₄²×C₁₉	C₂³	1	10	yes	—
46	C₂⁴×C₁₉	C₂⁴	1	5	yes	—
47	#41×C₁₉	C₂²×C₄	1	640	yes	complemented
48	C₂×Q₈×C₁₉	C₂²×C₄	1	80	yes	—
49	C₂×Q₈×C₁₉	C₂²×C₄	1	320	yes	complemented
50	C₂×Q₈×C₁₉	C₂⁴	1	40	yes	—
51	C₂×D₄×C₁₉	C₂²×C₄	1	80	yes	—
52	C₂×D₄×C₁₉	C₂⁴	1	40	yes	—
53	C₂²×C₄×C₁₉	—	2	20	no	—
54	C₂²×C₄×C₁₉	—	4	30	no	—
55	C₂²×C₄×C₁₉	C₂²×C₄	1	120	yes	—
56	C₂²×C₄×C₁₉	C₂⁴	1	5	yes	—
57	C₂²×C₄×C₁₉	C₂⁴	1	20	yes	—
58	C₂²×C₄×C₁₉	C₂⁴	1	10	yes	—
59	C₂²×C₄×C₁₉	C₂⁴	1	60	yes	—
60	C₂²×C₄×C₁₉	C₂⁴	1	10	yes	—
61	C₂²×C₄×C₁₉	C₂⁴	1	60	yes	—
62	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
63	C₄⋊C₄×C₁₉	C₂⁴	1	120	yes	—
64	C₄⋊C₄×C₁₉	C₂⁴	1	40	yes	—
65	C₂²⋊C₄×C₁₉	C₂⁴	1	80	yes	—
66	C₄²×C₁₉	C₂⁴	1	40	yes	—
67	C₄²×C₁₉	C₂⁴	1	120	yes	—
68	C₂×C₄×2₋¹⁺⁴	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂³×C₁₉	—	2	10	no	—
70	C₂³×C₁₉	—	2	10	no	—
71	C₂³×C₁₉	C₂³×C₄	1	20	yes	—
72	C₂³×C₁₉	C₂⁵	1	10	yes	—
73	C₂³×C₁₉	C₂⁵	1	1	yes	characteristic
74	Q₈×C₁₉	C₂³×C₄	1	160	yes	—
75	D₄×C₁₉	C₂³×C₄	1	160	yes	—
76	C₂×C₄×C₁₉	—	2	10	no	—
77	C₂×C₄×C₁₉	—	2	20	no	—
78	C₂×C₄×C₁₉	—	2	40	no	—
79	C₂×C₄×C₁₉	—	4	60	no	—
80	C₂×C₄×C₁₉	C₂³×C₄	1	40	yes	—
81	C₂×C₄×C₁₉	C₂³×C₄	1	240	yes	—
82	C₂×C₄×C₁₉	2₋¹⁺⁴	1	4	yes	direct product factor
83	C₂×C₄×C₁₉	C₂⁵	1	10	yes	—
84	C₂×C₄×C₁₉	C₂⁵	1	20	yes	—
85	C₂×C₄×C₁₉	C₂⁵	1	20	yes	—
86	C₂×C₄×C₁₉	C₂⁵	1	2	yes	—
87	C₂²×2₋¹⁺⁴	C₂×C₁₉	1	1	yes	characteristic
88	C₄×2₋¹⁺⁴	C₂×C₁₉	1	32	yes	direct product factor
89	C₂×#520	C₂×C₁₉	1	10	yes	complemented
90	C₂×#519	C₂×C₁₉	1	5	yes	complemented
91	C₂×C₄×#41	C₂×C₁₉	1	10	yes	complemented
92	C₂²×C₄×Q₈	C₂×C₁₉	1	5	yes	complemented
93	C₂²×C₁₉	—	2	10	no	—
94	C₂²×C₁₉	—	2	10	no	—
95	C₂²×C₁₉	—	2	40	no	—
96	C₂²×C₁₉	C₂⁴×C₄	1	20	yes	—
97	C₂²×C₁₉	C₂⁴×C₄	1	2	yes	—
98	C₂²×C₁₉	C₂×2₋¹⁺⁴	1	2	yes	—
99	C₂²×C₁₉	#585	1	2	yes	—
100	C₂²×C₁₉	C₂⁶	1	1	yes	characteristic
101	C₄×C₁₉	—	2	10	no	—
102	C₄×C₁₉	—	2	20	no	—
103	C₄×C₁₉	C₂⁴×C₄	1	40	yes	—
104	C₄×C₁₉	C₂×2₋¹⁺⁴	1	4	yes	direct product factor
105	C₂×2₋¹⁺⁴	C₄×C₁₉	1	32	yes	direct product factor
106	C₂×2₋¹⁺⁴	C₂²×C₁₉	1	16	yes	—
107	C₂²×#41	C₂²×C₁₉	1	10	yes	—
108	C₂³×Q₈	C₂²×C₁₉	1	5	yes	—
109	#520	C₂²×C₁₉	1	160	yes	complemented
110	#519	C₂²×C₁₉	1	80	yes	complemented
111	C₄×#41	C₂²×C₁₉	1	160	yes	complemented
112	C₂×C₄×Q₈	C₂²×C₁₉	1	10	yes	—
113	C₂×C₄×Q₈	C₂²×C₁₉	1	20	yes	—
114	C₂×C₄×Q₈	C₂²×C₁₉	1	10	yes	—
115	C₂×C₄×Q₈	C₂²×C₁₉	1	80	yes	complemented
116	C₂×C₄×D₄	C₂²×C₁₉	1	10	yes	—
117	C₂×C₄×D₄	C₂²×C₁₉	1	30	yes	—
118	C₂×#117	C₂²×C₁₉	1	30	yes	—
119	C₂²×C₄⋊C₄	C₂²×C₁₉	1	15	yes	—
120	C₂²×C₄²	C₂²×C₁₉	1	15	yes	—
121	C₂×C₁₉	—	2	20	no	—
122	C₂×C₁₉	C₄×2₋¹⁺⁴	1	4	yes	direct product factor
123	C₂×C₁₉	C₂⁵×C₄	1	1	yes	characteristic
124	C₂×C₁₉	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
125	C₂×C₁₉	C₂×#585	1	1	yes	characteristic
126	2₋¹⁺⁴	C₂×C₄×C₁₉	1	256	yes	direct product factor
127	C₂×#41	C₂×C₄×C₁₉	1	160	yes	complemented
128	C₂×#41	C₂³×C₁₉	1	80	yes	—
129	C₂²×Q₈	C₂×C₄×C₁₉	1	80	yes	complemented
130	C₂²×Q₈	C₂³×C₁₉	1	10	yes	—
131	C₂²×Q₈	C₂³×C₁₉	1	40	yes	—
132	C₂²×D₄	C₂³×C₁₉	1	10	yes	—
133	C₂³×C₄	C₂³×C₁₉	1	5	yes	—
134	C₂³×C₄	C₂³×C₁₉	1	15	yes	—
135	C₂³×C₄	C₂³×C₁₉	1	15	yes	—
136	C₄×Q₈	C₂³×C₁₉	1	80	yes	—
137	C₄×Q₈	C₂³×C₁₉	1	160	yes	—
138	C₄×Q₈	C₂³×C₁₉	1	80	yes	—
139	C₄×D₄	C₂³×C₁₉	1	80	yes	—
140	C₄×D₄	C₂³×C₁₉	1	240	yes	—
141	#117	C₂³×C₁₉	1	240	yes	—
142	C₂×C₄⋊C₄	C₂³×C₁₉	1	30	yes	—
143	C₂×C₄⋊C₄	C₂³×C₁₉	1	120	yes	—
144	C₂×C₄⋊C₄	C₂³×C₁₉	1	10	yes	—
145	C₂×C₂²⋊C₄	C₂³×C₁₉	1	20	yes	—
146	C₂×C₄²	C₂³×C₁₉	1	30	yes	—
147	C₂×C₄²	C₂³×C₁₉	1	10	yes	—
148	C₂×C₄²	C₂³×C₁₉	1	120	yes	—
149	C₁₉	C₂×C₄×2₋¹⁺⁴	1	1	yes	characteristic, direct product factor, Sylow
150	C₂⁴	C₂⁴×C₁₉	1	5	yes	—
151	#41	C₂²×C₄×C₁₉	1	640	yes	complemented
152	C₂×Q₈	C₂²×C₄×C₁₉	1	80	yes	—
153	C₂×Q₈	C₂²×C₄×C₁₉	1	320	yes	complemented
154	C₂×Q₈	C₂⁴×C₁₉	1	40	yes	—
155	C₂×D₄	C₂²×C₄×C₁₉	1	80	yes	—
156	C₂×D₄	C₂⁴×C₁₉	1	40	yes	—
157	C₂²×C₄	—	2	20	no	—
158	C₂²×C₄	—	4	30	no	—
159	C₂²×C₄	C₂²×C₄×C₁₉	1	120	yes	—
160	C₂²×C₄	C₂⁴×C₁₉	1	5	yes	—
161	C₂²×C₄	C₂⁴×C₁₉	1	20	yes	—
162	C₂²×C₄	C₂⁴×C₁₉	1	10	yes	—
163	C₂²×C₄	C₂⁴×C₁₉	1	60	yes	—
164	C₂²×C₄	C₂⁴×C₁₉	1	10	yes	—
165	C₂²×C₄	C₂⁴×C₁₉	1	60	yes	—
166	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
167	C₄⋊C₄	C₂⁴×C₁₉	1	120	yes	—
168	C₄⋊C₄	C₂⁴×C₁₉	1	40	yes	—
169	C₂²⋊C₄	C₂⁴×C₁₉	1	80	yes	—
170	C₄²	C₂⁴×C₁₉	1	40	yes	—
171	C₄²	C₂⁴×C₁₉	1	120	yes	—
172	C₂³	—	2	10	no	—
173	C₂³	—	2	10	no	—
174	C₂³	C₂³×C₄×C₁₉	1	20	yes	—
175	C₂³	C₂⁵×C₁₉	1	10	yes	—
176	C₂³	C₂⁵×C₁₉	1	1	yes	characteristic
177	Q₈	C₂³×C₄×C₁₉	1	160	yes	—
178	D₄	C₂³×C₄×C₁₉	1	160	yes	—
179	C₂×C₄	—	2	10	no	—
180	C₂×C₄	—	2	20	no	—
181	C₂×C₄	—	2	40	no	—
182	C₂×C₄	—	4	60	no	—
183	C₂×C₄	C₂³×C₄×C₁₉	1	40	yes	—
184	C₂×C₄	C₂³×C₄×C₁₉	1	240	yes	—
185	C₂×C₄	C₁₉×2₋¹⁺⁴	1	4	yes	direct product factor
186	C₂×C₄	C₂⁵×C₁₉	1	10	yes	—
187	C₂×C₄	C₂⁵×C₁₉	1	20	yes	—
188	C₂×C₄	C₂⁵×C₁₉	1	20	yes	—
189	C₂×C₄	C₂⁵×C₁₉	1	2	yes	—
190	C₂²	—	2	40	no	—
191	C₂²	—	2	10	no	—
192	C₂²	—	2	10	no	—
193	C₂²	C₂⁴×C₄×C₁₉	1	20	yes	—
194	C₂²	C₂⁴×C₄×C₁₉	1	2	yes	—
195	C₂²	C₂×C₁₉×2₋¹⁺⁴	1	2	yes	—
196	C₂²	C₁₉×#585	1	2	yes	—
197	C₂²	C₂⁶×C₁₉	1	1	yes	characteristic
198	C₄	—	2	10	no	—
199	C₄	—	2	20	no	—
200	C₄	C₂⁴×C₄×C₁₉	1	40	yes	—
201	C₄	C₂×C₁₉×2₋¹⁺⁴	1	4	yes	direct product factor
202	C₂	—	2	20	no	—
203	C₂	C₄×C₁₉×2₋¹⁺⁴	1	4	yes	direct product factor
204	C₂	C₂⁵×C₄×C₁₉	1	1	yes	characteristic
205	C₂	C₂²×C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
206	C₂	C₂×C₁₉×#585	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₉	C₂⁴
commutator	C₂	C₂⁵×C₄×C₁₉
Frattini, cosocle	C₂²	C₂⁶×C₁₉
socle	C₂³×C₁₉	C₂⁵
prime radical	C₂²×C₁₉×2₋¹⁺⁴	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₁₉×2₋¹⁺⁴	C₂	direct (N × H)
3. C₂×C₁₉×#520	C₂	semidirect
4. C₂×C₁₉×#519	C₂	semidirect
5. C₂×C₄×#41×C₁₉	C₂	semidirect
6. C₂²×C₄×Q₈×C₁₉	C₂	semidirect
7. C₂×C₁₉×2₋¹⁺⁴	C₄	direct (N × H)
11. C₁₉×#520	C₂²	semidirect
12. C₁₉×#519	C₂²	semidirect
13. C₄×#41×C₁₉	C₂²	semidirect
17. C₂×C₄×Q₈×C₁₉	C₂²	semidirect
23. C₁₉×2₋¹⁺⁴	C₂×C₄	direct (N × H)
24. C₂×#41×C₁₉	C₂×C₄	semidirect
26. C₂²×Q₈×C₁₉	C₂×C₄	semidirect
47. #41×C₁₉	C₂²×C₄	semidirect
49. C₂×Q₈×C₁₉	C₂²×C₄	semidirect
68. C₂×C₄×2₋¹⁺⁴	C₁₉	direct (N × H)
82. C₂×C₄×C₁₉	2₋¹⁺⁴	direct (N × H)
88. C₄×2₋¹⁺⁴	C₂×C₁₉	direct (N × H)
89. C₂×#520	C₂×C₁₉	semidirect
90. C₂×#519	C₂×C₁₉	semidirect
91. C₂×C₄×#41	C₂×C₁₉	semidirect
92. C₂²×C₄×Q₈	C₂×C₁₉	semidirect
104. C₄×C₁₉	C₂×2₋¹⁺⁴	direct (N × H)
105. C₂×2₋¹⁺⁴	C₄×C₁₉	direct (N × H)
109. #520	C₂²×C₁₉	semidirect
110. #519	C₂²×C₁₉	semidirect
111. C₄×#41	C₂²×C₁₉	semidirect
115. C₂×C₄×Q₈	C₂²×C₁₉	semidirect
122. C₂×C₁₉	C₄×2₋¹⁺⁴	direct (N × H)
126. 2₋¹⁺⁴	C₂×C₄×C₁₉	direct (N × H)
127. C₂×#41	C₂×C₄×C₁₉	semidirect
129. C₂²×Q₈	C₂×C₄×C₁₉	semidirect
149. C₁₉	C₂×C₄×2₋¹⁺⁴	direct (N × H)
151. #41	C₂²×C₄×C₁₉	semidirect
153. C₂×Q₈	C₂²×C₄×C₁₉	semidirect
185. C₂×C₄	C₁₉×2₋¹⁺⁴	direct (N × H)
201. C₄	C₂×C₁₉×2₋¹⁺⁴	direct (N × H)
203. C₂	C₄×C₁₉×2₋¹⁺⁴	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₉×2₋¹⁺⁴	2
C₄×C₁₉×2₋¹⁺⁴	2
C₂×C₁₉×#520	2
C₂×C₁₉×#519	2
C₂×C₄×#41×C₁₉	2
C₂²×C₄×Q₈×C₁₉	2
C₂×C₄×2₋¹⁺⁴	19

Subgroups — C2×C4×C19×2−1+4 #18068609

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₁₉×2₋¹⁺⁴ #18068609