Finite Group Databasebeta

← C23×#14524012 #31762462

Subgroups — C23×#14524012 #31762462

Order 17152.

42862 conjugacy classes of subgroups; only the 5451 normal subgroups are shown.

normal subgroups only

This group has 42862 conjugacy classes of subgroups — too many to draw. Only its 5451 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C22⋊D67C213yesmaximal, complemented
2C22×#14524012C2156yesmaximal, direct product factor
3C23×C22⋊Dic67C213yesmaximal, complemented
4C27×C67C211yesmaximal, characteristic, complemented
5C25×D67C2211yescharacteristic, complemented
6C23×C22⋊D67—248no—
7C23×C22⋊D67C22184yescomplemented
8C24×Dic67C2213yescomplemented
9C2×#14524012C221448yesdirect product factor
10C22×C22⋊Dic67C22184yescomplemented
11C26×C67—248no—
12C26×C67C2213yes—
13C26×C67C22128yescomplemented
14C24×D67—224no—
15C24×D67C23114yescomplemented
16C22×C22⋊D67—2672no—
17C22×C22⋊D67C231336yescomplemented
18C23×Dic67—224no—
19C23×Dic67C23142yescomplemented
20#14524012C231512yesdirect product factor
21C2×C22⋊Dic67C231336yescomplemented
22C25×C67—2512no—
23C25×C67—248no—
24C25×C67—2672no—
25C25×C67D4148yescomplemented
26C25×C67C23142yes—
27C25×C67C2311yescharacteristic
28C25×C67C231112yescomplemented
29C23×D67—2168no—
30C23×D67—464no—
31C23×D67C24128yescomplemented
32C2×C22⋊D67—21344no—
33C2×C22⋊D67C241192yescomplemented
34C22×Dic67—2168no—
35C22×Dic67C24184yescomplemented
36C22⋊Dic67C241192yescomplemented
37C24×C67—23584no—
38C24×C67—2384no—
39C24×C67—2336no—
40C24×C67—21344no—
41C24×C67C2×D4124yes—
42C24×C67C2×D41336yescomplemented
43C24×C67C24184yes—
44C24×C67C2417yes—
45C24×C67C24164yescomplemented
46C23×C67—23584no—
47C23×C67—21344no—
48C23×C67—2336no—
49C23×C67—2384no—
50C23×C67#120164yescomplemented
51C23×C67C22×D4184yes—
52C23×C67C22×D41336yescomplemented
53C23×C67C25124yes—
54C23×C67C2517yes—
55C22×D67—2168no—
56C22×D67—4224no—
57C22×D67C2518yescomplemented
58C22⋊D67—2384no—
59C2×Dic67—2168no—
60C2×Dic67C25124yescomplemented
61C22×C67—2512no—
62C22×C67—248no—
63C22×C67—2672no—
64C22×C67C2×#1201112yescomplemented
65C22×C67C23×D4142yes—
66C22×C67C23×D4148yescomplemented
67C22×C67C2611yescharacteristic
68C2×D67—224no—
69C2×D67—4112no—
70Dic67—224no—
71C23×#120—671nomaximal, Sylow
72C2×C67—248no—
73C2×C67C22×#120128yescomplemented
74C2×C67C24×D413yes—
75D67—48no—
76C27D6711yescharacteristic, complemented
77C24×D4—673no—
78C22×#120—6756no—
79C23×C22⋊C4—673no—
80C67C23×#12011yescharacteristic, complemented, Sylow
81C26—248no—
82C26—671no—
83C26C2×D6713yes—
84C26C2×D67128yescomplemented
85C23×D4—6784no—
86C23×D4—13448no—
87C24×C4—673no—
88C2×#120—67448no—
89C22×C22⋊C4—6784no—
90C25—2512no—
91C25—248no—
92C25—2672no—
93C25—6714no—
94C25—13424no—
95C25C22⋊D67148yescomplemented
96C25C22×D67142yes—
97C25C22×D6711yescharacteristic
98C25C22×D671112yescomplemented
99C22×D4—67336no—
100C22×D4—134672no—
101C23×C4—6742no—
102C23×C4—13424no—
103#120—67512no—
104C2×C22⋊C4—67336no—
105C24—23584no—
106C24—2384no—
107C24—2336no—
108C24—21344no—
109C24—26864no—
110C24—6728no—
111C24—134168no—
112C24C2×C22⋊D67124yes—
113C24C2×C22⋊D671336yescomplemented
114C24C23×D67184yes—
115C24C23×D6717yes—
116C24C23×D67164yescomplemented
117C2×D4—67192no—
118C2×D4—1341344no—
119C22×C4—6784no—
120C22×C4—134168no—
121C22⋊C4—67192no—
122C23—23584no—
123C23—21344no—
124C23—2336no—
125C23—2384no—
126C23—268224no—
127C23—678no—
128C23—134168no—
129C23#14524012164yesdirect product factor
130C23C22×C22⋊D67184yes—
131C23C22×C22⋊D671336yescomplemented
132C23C24×D67124yes—
133C23C24×D6717yes—
134D4—134384no—
135C2×C4—6724no—
136C2×C4—134168no—
137C22—2512no—
138C22—2672no—
139C22—248no—
140C22—268112no—
141C22—13424no—
142C22C2×#145240121112yesdirect product factor
143C22C23×C22⋊D67142yes—
144C22C23×C22⋊D67148yescomplemented
145C22C25×D6711yescharacteristic
146C4—13424no—
147C2—248no—
148C2—2688no—
149C2C22×#14524012128yesdirect product factor
150C2C24×C22⋊D6713yes—
subgroupG/H
centerC25C22×D67
commutator, cosocleC22×C67C26
FrattiniC22C25×D67
FittingC27×C67C2
socleC25×C67C23
hypercenterC27D67
hypocenterC67C23×#120
LCSG ▷ C22×C67 ▷ C67
UCSC1 ◁ C25 ◁ C27

Complements

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

NH ≅ G/Ntype
1. C24×C22⋊D67C2semidirect
2. C22×#14524012C2direct (N × H)
3. C23×C22⋊Dic67C2semidirect
4. C27×C67C2semidirect
5. C25×D67C22semidirect
7. C23×C22⋊D67C22semidirect
8. C24×Dic67C22semidirect
9. C2×#14524012C22direct (N × H)
10. C22×C22⋊Dic67C22semidirect
13. C26×C67C22semidirect
15. C24×D67C23semidirect
17. C22×C22⋊D67C23semidirect
19. C23×Dic67C23semidirect
20. #14524012C23direct (N × H)
21. C2×C22⋊Dic67C23semidirect
25. C25×C67D4semidirect
28. C25×C67C23semidirect
31. C23×D67C24semidirect
33. C2×C22⋊D67C24semidirect
35. C22×Dic67C24semidirect
36. C22⋊Dic67C24semidirect
42. C24×C67C2×D4semidirect
45. C24×C67C24semidirect
50. C23×C67#120semidirect
52. C23×C67C22×D4semidirect
57. C22×D67C25semidirect
60. C2×Dic67C25semidirect
64. C22×C67C2×#120semidirect
66. C22×C67C23×D4semidirect
73. C2×C67C22×#120semidirect
76. C27D67semidirect
80. C67C23×#120semidirect
84. C26C2×D67semidirect
95. C25C22⋊D67semidirect
98. C25C22×D67semidirect
113. C24C2×C22⋊D67semidirect
116. C24C23×D67semidirect
129. C23#14524012direct (N × H)
131. C23C22×C22⋊D67semidirect
142. C22C2×#14524012direct (N × H)
144. C22C23×C22⋊D67semidirect
149. C2C22×#14524012direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C22⋊D672
C22×#145240122
C23×C22⋊Dic672
C27×C672
C23×#12067