Subgroups — C₂×C₁₉×#544 #14561314

Order 2432.

1080 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#544	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₁₉×C₄⋊Q₈	C₂	1	4	yes	maximal, complemented
3	C₂×C₁₉×#125	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₉×C₂²⋊Q₈	C₂	1	4	yes	maximal
5	C₂×C₁₉×#117	C₂	1	2	yes	maximal
6	C₂²×C₄⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₁₉	C₂²	1	4	yes	—
8	C₂³×C₄×C₁₉	C₂²	1	2	yes	—
9	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
10	C₁₉×C₄⋊Q₈	C₂²	1	32	yes	complemented
11	C₁₉×#125	C₂²	1	32	yes	complemented
12	C₁₉×C₂²⋊Q₈	C₂²	1	32	yes	—
13	C₁₉×#117	C₂²	1	16	yes	—
14	C₂×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
15	C₂×C₄⋊C₄×C₁₉	C₂²	1	8	yes	—
16	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
19	C₂×C₄⋊C₄×C₁₉	C₂²	1	8	yes	—
20	C₂×C₂²⋊C₄×C₁₉	C₂²	1	4	yes	—
21	C₂×C₄²×C₁₉	C₂²	1	4	yes	—
22	C₂⁴×C₁₉	C₂³	1	1	yes	characteristic
23	C₂×Q₈×C₁₉	—	4	8	no	—
24	C₂×Q₈×C₁₉	C₂³	1	16	yes	—
25	C₂²×C₄×C₁₉	—	4	4	no	—
26	C₂²×C₄×C₁₉	Q₈	1	8	yes	complemented
27	C₂²×C₄×C₁₉	C₂³	1	4	yes	—
28	C₂²×C₄×C₁₉	C₂³	1	4	yes	—
29	C₂²×C₄×C₁₉	C₂³	1	4	yes	—
30	C₂²×C₄×C₁₉	C₂³	1	8	yes	—
31	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
32	C₂²×C₄×C₁₉	C₂³	1	4	yes	—
33	C₄⋊C₄×C₁₉	C₂³	1	16	yes	—
34	C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
35	C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
36	C₄⋊C₄×C₁₉	C₂³	1	16	yes	—
37	C₄⋊C₄×C₁₉	C₂³	1	32	yes	—
38	C₂²⋊C₄×C₁₉	C₂³	1	16	yes	—
39	C₄²×C₁₉	C₂³	1	16	yes	—
40	C₂³×C₁₉	—	2	2	no	—
41	C₂³×C₁₉	—	2	2	no	—
42	C₂³×C₁₉	C₂×Q₈	1	4	yes	—
43	C₂³×C₁₉	C₂⁴	1	2	yes	—
44	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
45	Q₈×C₁₉	—	4	16	no	—
46	C₂×C₄×C₁₉	—	2	8	no	—
47	C₂×C₄×C₁₉	—	2	8	no	—
48	C₂×C₄×C₁₉	—	2	8	no	—
49	C₂×C₄×C₁₉	—	4	8	no	—
50	C₂×C₄×C₁₉	C₂×Q₈	1	8	yes	—
51	C₂×C₄×C₁₉	C₂×Q₈	1	16	yes	complemented
52	C₂×C₄×C₁₉	C₂⁴	1	8	yes	—
53	C₂×C₄×C₁₉	C₂⁴	1	8	yes	—
54	C₂×C₄×C₁₉	C₂⁴	1	8	yes	—
55	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
56	C₂×#544	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂²×C₁₉	—	2	8	no	—
58	C₂²×C₁₉	—	2	2	no	—
59	C₂²×C₁₉	—	2	2	no	—
60	C₂²×C₁₉	C₂²×Q₈	1	4	yes	—
61	C₂²×C₁₉	C₂²×Q₈	1	2	yes	—
62	C₂²×C₁₉	H(5,𝔽₂)	1	2	yes	—
63	C₂²×C₁₉	2₋¹⁺⁴	1	2	yes	—
64	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
65	C₄×C₁₉	—	2	8	no	—
66	C₄×C₁₉	—	2	8	no	—
67	C₄×C₁₉	—	2	8	no	—
68	C₄×C₁₉	C₂²×Q₈	1	8	yes	—
69	#544	C₂×C₁₉	1	16	yes	direct product factor
70	C₂×C₄⋊Q₈	C₂×C₁₉	1	4	yes	complemented
71	C₂×#125	C₂×C₁₉	1	4	yes	complemented
72	C₂×C₂²⋊Q₈	C₂×C₁₉	1	4	yes	—
73	C₂×#117	C₂×C₁₉	1	2	yes	—
74	C₂²×C₄⋊C₄	C₂×C₁₉	1	1	yes	characteristic
75	C₂×C₁₉	—	2	4	no	—
76	C₂×C₁₉	#544	1	4	yes	direct product factor
77	C₂×C₁₉	C₂³×Q₈	1	1	yes	characteristic
78	C₂×C₁₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
79	C₂×C₁₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
80	C₂²×Q₈	C₂²×C₁₉	1	4	yes	—
81	C₂³×C₄	C₂²×C₁₉	1	2	yes	—
82	C₂³×C₄	C₂²×C₁₉	1	1	yes	characteristic
83	C₄⋊Q₈	C₂²×C₁₉	1	32	yes	complemented
84	#125	C₂²×C₁₉	1	32	yes	complemented
85	C₂²⋊Q₈	C₂²×C₁₉	1	32	yes	—
86	#117	C₂²×C₁₉	1	16	yes	—
87	C₂×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
88	C₂×C₄⋊C₄	C₂²×C₁₉	1	8	yes	—
89	C₂×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
90	C₂×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
91	C₂×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
92	C₂×C₄⋊C₄	C₂²×C₁₉	1	8	yes	—
93	C₂×C₂²⋊C₄	C₂²×C₁₉	1	4	yes	—
94	C₂×C₄²	C₂²×C₁₉	1	4	yes	—
95	C₁₉	C₂×#544	1	1	yes	characteristic, direct product factor, Sylow
96	C₂⁴	C₂³×C₁₉	1	1	yes	characteristic
97	C₂×Q₈	—	4	8	no	—
98	C₂×Q₈	C₂³×C₁₉	1	16	yes	—
99	C₂²×C₄	—	4	4	no	—
100	C₂²×C₄	Q₈×C₁₉	1	8	yes	complemented
101	C₂²×C₄	C₂³×C₁₉	1	4	yes	—
102	C₂²×C₄	C₂³×C₁₉	1	4	yes	—
103	C₂²×C₄	C₂³×C₁₉	1	4	yes	—
104	C₂²×C₄	C₂³×C₁₉	1	8	yes	—
105	C₂²×C₄	C₂³×C₁₉	1	2	yes	—
106	C₂²×C₄	C₂³×C₁₉	1	4	yes	—
107	C₄⋊C₄	C₂³×C₁₉	1	16	yes	—
108	C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
109	C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
110	C₄⋊C₄	C₂³×C₁₉	1	16	yes	—
111	C₄⋊C₄	C₂³×C₁₉	1	32	yes	—
112	C₂²⋊C₄	C₂³×C₁₉	1	16	yes	—
113	C₄²	C₂³×C₁₉	1	16	yes	—
114	C₂³	—	2	2	no	—
115	C₂³	—	2	2	no	—
116	C₂³	C₂×Q₈×C₁₉	1	4	yes	—
117	C₂³	C₂⁴×C₁₉	1	2	yes	—
118	C₂³	C₂⁴×C₁₉	1	1	yes	characteristic
119	Q₈	—	4	16	no	—
120	C₂×C₄	—	2	8	no	—
121	C₂×C₄	—	2	8	no	—
122	C₂×C₄	—	2	8	no	—
123	C₂×C₄	—	4	8	no	—
124	C₂×C₄	C₂×Q₈×C₁₉	1	8	yes	—
125	C₂×C₄	C₂×Q₈×C₁₉	1	16	yes	complemented
126	C₂×C₄	C₂⁴×C₁₉	1	8	yes	—
127	C₂×C₄	C₂⁴×C₁₉	1	8	yes	—
128	C₂×C₄	C₂⁴×C₁₉	1	8	yes	—
129	C₂×C₄	C₂⁴×C₁₉	1	4	yes	—
130	C₂²	—	2	8	no	—
131	C₂²	—	2	2	no	—
132	C₂²	—	2	2	no	—
133	C₂²	C₂²×Q₈×C₁₉	1	4	yes	—
134	C₂²	C₂²×Q₈×C₁₉	1	2	yes	—
135	C₂²	C₁₉×H(5,𝔽₂)	1	2	yes	—
136	C₂²	C₁₉×2₋¹⁺⁴	1	2	yes	—
137	C₂²	C₂⁵×C₁₉	1	1	yes	characteristic
138	C₄	—	2	8	no	—
139	C₄	—	2	8	no	—
140	C₄	—	2	8	no	—
141	C₄	C₂²×Q₈×C₁₉	1	8	yes	—
142	C₂	—	2	4	no	—
143	C₂	C₁₉×#544	1	4	yes	direct product factor
144	C₂	C₂³×Q₈×C₁₉	1	1	yes	characteristic
145	C₂	C₂×C₁₉×H(5,𝔽₂)	1	1	yes	characteristic
146	C₂	C₂×C₁₉×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₉
prime radical	C₂⁴×C₁₉	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₉×#544	C₂	direct (N × H)
2. C₂×C₁₉×C₄⋊Q₈	C₂	semidirect
3. C₂×C₁₉×#125	C₂	semidirect
10. C₁₉×C₄⋊Q₈	C₂²	semidirect
11. C₁₉×#125	C₂²	semidirect
26. C₂²×C₄×C₁₉	Q₈	semidirect
51. C₂×C₄×C₁₉	C₂×Q₈	semidirect
56. C₂×#544	C₁₉	direct (N × H)
69. #544	C₂×C₁₉	direct (N × H)
70. C₂×C₄⋊Q₈	C₂×C₁₉	semidirect
71. C₂×#125	C₂×C₁₉	semidirect
76. C₂×C₁₉	#544	direct (N × H)
83. C₄⋊Q₈	C₂²×C₁₉	semidirect
84. #125	C₂²×C₁₉	semidirect
95. C₁₉	C₂×#544	direct (N × H)
100. C₂²×C₄	Q₈×C₁₉	semidirect
125. C₂×C₄	C₂×Q₈×C₁₉	semidirect
143. C₂	C₁₉×#544	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×#544	2
C₂×C₁₉×C₄⋊Q₈	2
C₂×C₁₉×#125	2
C₂×C₁₉×C₂²⋊Q₈	2
C₂×C₁₉×#117	2
C₂²×C₄⋊C₄×C₁₉	2
C₂×#544	19

Subgroups — C2×C19×#544 #14561314

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#544 #14561314