Subgroups — C₁₉×#62654 #18068613

Order 4864.

11222 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#585	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₉×2₋¹⁺⁴	C₂	1	12	yes	maximal, complemented
3	C₄×C₁₉×H(5,𝔽₂)	C₂	1	20	yes	maximal, complemented
4	C₁₉×#3428	C₂	1	15	yes	maximal, complemented
5	C₂×C₁₉×#520	C₂	1	15	yes	maximal, complemented
6	C₁₉×#585	C₄	1	32	yes	complemented
7	C₂×C₁₉×2₋¹⁺⁴	C₂²	1	6	yes	—
8	C₂×C₁₉×H(5,𝔽₂)	C₂²	1	10	yes	—
9	C₂²×#41×C₁₉	C₂²	1	15	yes	—
10	C₁₉×#520	C₂²	1	60	yes	complemented
11	C₁₉×#520	C₂²	1	120	yes	complemented
12	C₁₉×#520	C₂²	1	20	yes	complemented
13	C₁₉×#519	C₂²	1	30	yes	complemented
14	C₁₉×#518	C₂²	1	90	yes	complemented
15	C₄×#41×C₁₉	C₂²	1	120	yes	complemented
16	C₂×C₄×Q₈×C₁₉	C₂²	1	30	yes	complemented
17	C₂×C₄×D₄×C₁₉	C₂²	1	90	yes	complemented
18	C₂×C₁₉×#117	C₂²	1	45	yes	complemented
19	C₂²×C₄⋊C₄×C₁₉	C₂²	1	15	yes	complemented
20	C₁₉×2₋¹⁺⁴	C₂×C₄	1	96	yes	complemented
21	C₁₉×H(5,𝔽₂)	C₂×C₄	1	160	yes	complemented
22	C₂×#41×C₁₉	C₂×C₄	1	240	yes	complemented
23	C₂×#41×C₁₉	C₂³	1	60	yes	—
24	C₂×#41×C₁₉	C₂³	1	20	yes	—
25	C₂²×Q₈×C₁₉	C₂³	1	15	yes	—
26	C₂²×D₄×C₁₉	C₂³	1	45	yes	—
27	C₂³×C₄×C₁₉	C₂³	1	30	yes	—
28	C₂³×C₄×C₁₉	C₂³	1	15	yes	—
29	C₄×Q₈×C₁₉	C₂³	1	120	yes	—
30	C₄×Q₈×C₁₉	C₂³	1	40	yes	—
31	C₄×D₄×C₁₉	C₂³	1	360	yes	—
32	C₄×D₄×C₁₉	C₂³	1	120	yes	—
33	C₁₉×#117	C₂³	1	60	yes	—
34	C₁₉×#117	C₂³	1	180	yes	—
35	C₂×C₄⋊C₄×C₁₉	C₂³	1	45	yes	—
36	C₂×C₄⋊C₄×C₁₉	C₂³	1	90	yes	—
37	C₂×C₄⋊C₄×C₁₉	C₂³	1	15	yes	—
38	C₂×C₂²⋊C₄×C₁₉	C₂³	1	90	yes	—
39	C₂×C₄²×C₁₉	C₂³	1	90	yes	—
40	C₂⁴×C₁₉	C₂⁴	1	15	yes	—
41	#41×C₁₉	C₂²×C₄	1	160	yes	complemented
42	#41×C₁₉	C₂²×C₄	1	480	yes	complemented
43	C₂×Q₈×C₁₉	C₂²×C₄	1	120	yes	complemented
44	C₂×Q₈×C₁₉	C₂⁴	1	20	yes	—
45	C₂×D₄×C₁₉	C₂²×C₄	1	360	yes	complemented
46	C₂×D₄×C₁₉	C₂⁴	1	60	yes	—
47	C₂²×C₄×C₁₉	—	8	30	no	—
48	C₂²×C₄×C₁₉	C₂²×C₄	1	120	yes	complemented
49	C₂²×C₄×C₁₉	C₂⁴	1	30	yes	—
50	C₂²×C₄×C₁₉	C₂⁴	1	90	yes	—
51	C₂²×C₄×C₁₉	C₂⁴	1	45	yes	—
52	C₂²×C₄×C₁₉	C₂⁴	1	15	yes	—
53	C₄⋊C₄×C₁₉	C₂⁴	1	10	yes	—
54	C₄⋊C₄×C₁₉	C₂⁴	1	6	yes	—
55	C₄⋊C₄×C₁₉	C₂⁴	1	90	yes	—
56	C₄⋊C₄×C₁₉	C₂⁴	1	30	yes	—
57	C₂²⋊C₄×C₁₉	C₂⁴	1	120	yes	—
58	C₄²×C₁₉	C₂⁴	1	120	yes	—
59	#62654	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂³×C₁₉	—	4	15	no	—
61	C₂³×C₁₉	—	4	15	no	—
62	C₂³×C₁₉	C₂³×C₄	1	60	yes	—
63	C₂³×C₁₉	C₂⁵	1	15	yes	—
64	Q₈×C₁₉	C₂³×C₄	1	80	yes	—
65	D₄×C₁₉	C₂³×C₄	1	240	yes	—
66	C₂×C₄×C₁₉	—	4	90	no	—
67	C₂×C₄×C₁₉	—	4	30	no	—
68	C₂×C₄×C₁₉	C₂³×C₄	1	60	yes	—
69	C₂×C₄×C₁₉	C₂³×C₄	1	180	yes	—
70	C₂×C₄×C₁₉	C₂⁵	1	12	yes	—
71	C₂×C₄×C₁₉	C₂⁵	1	20	yes	—
72	C₂×C₄×C₁₉	C₂⁵	1	15	yes	—
73	C₂×C₄×C₁₉	C₂⁵	1	1	yes	characteristic
74	C₂×#585	C₂×C₁₉	1	1	yes	characteristic
75	C₄×2₋¹⁺⁴	C₂×C₁₉	1	12	yes	complemented
76	C₄×H(5,𝔽₂)	C₂×C₁₉	1	20	yes	complemented
77	#3428	C₂×C₁₉	1	15	yes	complemented
78	C₂×#520	C₂×C₁₉	1	15	yes	complemented
79	C₂²×C₁₉	—	2	15	no	—
80	C₂²×C₁₉	—	2	15	no	—
81	C₂²×C₁₉	—	4	60	no	—
82	C₂²×C₁₉	C₂⁴×C₄	1	30	yes	—
83	C₂²×C₁₉	C₂⁶	1	1	yes	characteristic
84	C₄×C₁₉	—	2	20	no	—
85	C₄×C₁₉	—	2	12	no	—
86	C₄×C₁₉	C₂⁴×C₄	1	2	yes	—
87	C₄×C₁₉	C₂⁴×C₄	1	30	yes	—
88	#585	C₄×C₁₉	1	32	yes	complemented
89	C₂×2₋¹⁺⁴	C₂²×C₁₉	1	6	yes	—
90	C₂×H(5,𝔽₂)	C₂²×C₁₉	1	10	yes	—
91	C₂²×#41	C₂²×C₁₉	1	15	yes	—
92	#520	C₂²×C₁₉	1	60	yes	complemented
93	#520	C₂²×C₁₉	1	120	yes	complemented
94	#520	C₂²×C₁₉	1	20	yes	complemented
95	#519	C₂²×C₁₉	1	30	yes	complemented
96	#518	C₂²×C₁₉	1	90	yes	complemented
97	C₄×#41	C₂²×C₁₉	1	120	yes	complemented
98	C₂×C₄×Q₈	C₂²×C₁₉	1	30	yes	complemented
99	C₂×C₄×D₄	C₂²×C₁₉	1	90	yes	complemented
100	C₂×#117	C₂²×C₁₉	1	45	yes	complemented
101	C₂²×C₄⋊C₄	C₂²×C₁₉	1	15	yes	complemented
102	C₂×C₁₉	—	2	30	no	—
103	C₂×C₁₉	C₂⁵×C₄	1	1	yes	characteristic
104	C₂×C₁₉	H(7,𝔽₂)	1	1	yes	characteristic
105	C₂×C₁₉	2₋¹⁺⁶	1	1	yes	characteristic
106	2₋¹⁺⁴	C₂×C₄×C₁₉	1	96	yes	complemented
107	H(5,𝔽₂)	C₂×C₄×C₁₉	1	160	yes	complemented
108	C₂×#41	C₂×C₄×C₁₉	1	240	yes	complemented
109	C₂×#41	C₂³×C₁₉	1	60	yes	—
110	C₂×#41	C₂³×C₁₉	1	20	yes	—
111	C₂²×Q₈	C₂³×C₁₉	1	15	yes	—
112	C₂²×D₄	C₂³×C₁₉	1	45	yes	—
113	C₂³×C₄	C₂³×C₁₉	1	30	yes	—
114	C₂³×C₄	C₂³×C₁₉	1	15	yes	—
115	C₄×Q₈	C₂³×C₁₉	1	120	yes	—
116	C₄×Q₈	C₂³×C₁₉	1	40	yes	—
117	C₄×D₄	C₂³×C₁₉	1	360	yes	—
118	C₄×D₄	C₂³×C₁₉	1	120	yes	—
119	#117	C₂³×C₁₉	1	60	yes	—
120	#117	C₂³×C₁₉	1	180	yes	—
121	C₂×C₄⋊C₄	C₂³×C₁₉	1	45	yes	—
122	C₂×C₄⋊C₄	C₂³×C₁₉	1	90	yes	—
123	C₂×C₄⋊C₄	C₂³×C₁₉	1	15	yes	—
124	C₂×C₂²⋊C₄	C₂³×C₁₉	1	90	yes	—
125	C₂×C₄²	C₂³×C₁₉	1	90	yes	—
126	C₁₉	#62654	1	1	yes	characteristic, direct product factor, Sylow
127	C₂⁴	C₂⁴×C₁₉	1	15	yes	—
128	#41	C₂²×C₄×C₁₉	1	160	yes	complemented
129	#41	C₂²×C₄×C₁₉	1	480	yes	complemented
130	C₂×Q₈	C₂²×C₄×C₁₉	1	120	yes	complemented
131	C₂×Q₈	C₂⁴×C₁₉	1	20	yes	—
132	C₂×D₄	C₂²×C₄×C₁₉	1	360	yes	complemented
133	C₂×D₄	C₂⁴×C₁₉	1	60	yes	—
134	C₂²×C₄	—	8	30	no	—
135	C₂²×C₄	C₂²×C₄×C₁₉	1	120	yes	complemented
136	C₂²×C₄	C₂⁴×C₁₉	1	30	yes	—
137	C₂²×C₄	C₂⁴×C₁₉	1	90	yes	—
138	C₂²×C₄	C₂⁴×C₁₉	1	45	yes	—
139	C₂²×C₄	C₂⁴×C₁₉	1	15	yes	—
140	C₄⋊C₄	C₂⁴×C₁₉	1	10	yes	—
141	C₄⋊C₄	C₂⁴×C₁₉	1	6	yes	—
142	C₄⋊C₄	C₂⁴×C₁₉	1	90	yes	—
143	C₄⋊C₄	C₂⁴×C₁₉	1	30	yes	—
144	C₂²⋊C₄	C₂⁴×C₁₉	1	120	yes	—
145	C₄²	C₂⁴×C₁₉	1	120	yes	—
146	C₂³	—	4	15	no	—
147	C₂³	—	4	15	no	—
148	C₂³	C₂³×C₄×C₁₉	1	60	yes	—
149	C₂³	C₂⁵×C₁₉	1	15	yes	—
150	Q₈	C₂³×C₄×C₁₉	1	80	yes	—
151	D₄	C₂³×C₄×C₁₉	1	240	yes	—
152	C₂×C₄	—	4	90	no	—
153	C₂×C₄	—	4	30	no	—
154	C₂×C₄	C₂³×C₄×C₁₉	1	60	yes	—
155	C₂×C₄	C₂³×C₄×C₁₉	1	180	yes	—
156	C₂×C₄	C₂⁵×C₁₉	1	12	yes	—
157	C₂×C₄	C₂⁵×C₁₉	1	20	yes	—
158	C₂×C₄	C₂⁵×C₁₉	1	15	yes	—
159	C₂×C₄	C₂⁵×C₁₉	1	1	yes	characteristic
160	C₂²	—	2	15	no	—
161	C₂²	—	2	15	no	—
162	C₂²	—	4	60	no	—
163	C₂²	C₂⁴×C₄×C₁₉	1	30	yes	—
164	C₂²	C₂⁶×C₁₉	1	1	yes	characteristic
165	C₄	—	2	20	no	—
166	C₄	—	2	12	no	—
167	C₄	C₂⁴×C₄×C₁₉	1	2	yes	—
168	C₄	C₂⁴×C₄×C₁₉	1	30	yes	—
169	C₂	—	2	30	no	—
170	C₂	C₂⁵×C₄×C₁₉	1	1	yes	characteristic
171	C₂	C₁₉×H(7,𝔽₂)	1	1	yes	characteristic
172	C₂	C₁₉×2₋¹⁺⁶	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂⁶
commutator	C₂	C₂⁵×C₄×C₁₉
Frattini, cosocle	C₂²	C₂⁶×C₁₉
prime radical	C₂×C₁₉×#585	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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	semidirect
3. C₄×C₁₉×H(5,𝔽₂)	C₂	semidirect
4. C₁₉×#3428	C₂	semidirect
5. C₂×C₁₉×#520	C₂	semidirect
6. C₁₉×#585	C₄	semidirect
10. C₁₉×#520	C₂²	semidirect
11. C₁₉×#520	C₂²	semidirect
12. C₁₉×#520	C₂²	semidirect
13. C₁₉×#519	C₂²	semidirect
14. C₁₉×#518	C₂²	semidirect
15. C₄×#41×C₁₉	C₂²	semidirect
16. C₂×C₄×Q₈×C₁₉	C₂²	semidirect
17. C₂×C₄×D₄×C₁₉	C₂²	semidirect
18. C₂×C₁₉×#117	C₂²	semidirect
19. C₂²×C₄⋊C₄×C₁₉	C₂²	semidirect
20. C₁₉×2₋¹⁺⁴	C₂×C₄	semidirect
21. C₁₉×H(5,𝔽₂)	C₂×C₄	semidirect
22. C₂×#41×C₁₉	C₂×C₄	semidirect
41. #41×C₁₉	C₂²×C₄	semidirect
42. #41×C₁₉	C₂²×C₄	semidirect
43. C₂×Q₈×C₁₉	C₂²×C₄	semidirect
45. C₂×D₄×C₁₉	C₂²×C₄	semidirect
48. C₂²×C₄×C₁₉	C₂²×C₄	semidirect
59. #62654	C₁₉	direct (N × H)
75. C₄×2₋¹⁺⁴	C₂×C₁₉	semidirect
76. C₄×H(5,𝔽₂)	C₂×C₁₉	semidirect
77. #3428	C₂×C₁₉	semidirect
78. C₂×#520	C₂×C₁₉	semidirect
88. #585	C₄×C₁₉	semidirect
92. #520	C₂²×C₁₉	semidirect
93. #520	C₂²×C₁₉	semidirect
94. #520	C₂²×C₁₉	semidirect
95. #519	C₂²×C₁₉	semidirect
96. #518	C₂²×C₁₉	semidirect
97. C₄×#41	C₂²×C₁₉	semidirect
98. C₂×C₄×Q₈	C₂²×C₁₉	semidirect
99. C₂×C₄×D₄	C₂²×C₁₉	semidirect
100. C₂×#117	C₂²×C₁₉	semidirect
101. C₂²×C₄⋊C₄	C₂²×C₁₉	semidirect
106. 2₋¹⁺⁴	C₂×C₄×C₁₉	semidirect
107. H(5,𝔽₂)	C₂×C₄×C₁₉	semidirect
108. C₂×#41	C₂×C₄×C₁₉	semidirect
126. C₁₉	#62654	direct (N × H)
128. #41	C₂²×C₄×C₁₉	semidirect
129. #41	C₂²×C₄×C₁₉	semidirect
130. C₂×Q₈	C₂²×C₄×C₁₉	semidirect
132. C₂×D₄	C₂²×C₄×C₁₉	semidirect
135. C₂²×C₄	C₂²×C₄×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#585	2
C₄×C₁₉×2₋¹⁺⁴	2
C₄×C₁₉×H(5,𝔽₂)	2
C₁₉×#3428	2
C₂×C₁₉×#520	2
#62654	19

Subgroups — C19×#62654 #18068613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#62654 #18068613