Finite Group Databasebeta

← #72241920

Subgroups — #72241920

Order 8112.

152 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 30 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
1C2×#11759818⋊C2C211yesmaximal, characteristic
2C2×#14523147C211yesmaximal, characteristic, complemented
3C2×#14523145C211yesmaximal, characteristic, complemented
4#72214177C311yesmaximal, characteristic, complemented, Hall
5C2×#11759818C412yescomplemented
6C2×#11759816C2211yescharacteristic
7#11759818⋊C2—21no—
8#11759818⋊C2—21no—
9#14523147—21no—
10#14523145—21no—
11C2×D132C2×C311yescharacteristic
12C2×#10621729C2×C311yescharacteristic, complemented
13C2×#10621728C2×C311yescharacteristic, complemented
14C2×#92789C2×C411yescharacteristic
15#11759818—22no—
16#11759816D411yescharacteristic
17#11759816D411yescharacteristic
18C2×C13⋊D13C22×C311yescharacteristic
19C2×C13×D13C3×C412yescomplemented
20D132—21no—
21D132—21no—
22#10621729—21no—
23#10621728—21no—
24(C22⋊#261)⋊C3—132nomaximal
25#92789C22⋊C411yescharacteristic, complemented, Hall
26C2×C132C2×C3×C411yescharacteristic
27C13⋊D13C3×D411yescharacteristic
28C13⋊D13C3×D411yescharacteristic
29C13×D13—22no—
30C22×D13⋊C3—132no—
31C2×AGL(1,𝔽13)—132no—
32C2×AGL(1,𝔽13)—132no—
33C22⋊#261—132no—
34C132C3×C22⋊C411yescharacteristic, complemented, Sylow
35C22×C13⋊C3—132no—
36C2×D13⋊C3—132no—
37C2×D13⋊C3—262no—
38C2×D13⋊C3—262no—
39C2×D13⋊C3—522no—
40AGL(1,𝔽13)—262no—
41AGL(1,𝔽13)—262no—
42C22×D13—132no—
43C2×#261—132no—
44C2×#261—132no—
45C2×#261—782no—
46C2×C13⋊C3—262no—
47C2×C13⋊C3—522no—
48D13⋊C3—132no—
49D13⋊C3—132no—
50D13⋊C3—522no—
51C22×C13—132no—
52C2×D13—132no—
53C2×D13—262no—
54C2×D13—262no—
55C2×D13—782no—
56C2×D13AGL(1,𝔽13)12yescomplemented
57#261—262no—
58#261—262no—
59#261—782no—
60#261—782no—
61C3×C22⋊C4—1691noHall
62C13⋊C3—522no—
63C2×C13—62no—
64C2×C13—262no—
65C2×C13C2×AGL(1,𝔽13)12yes—
66D13—22no—
67D13—132no—
68D13—132no—
69D13—782no—
70D13—782no—
71C23×C3—1691no—
72C2×C3×C4—1691no—
73C2×C3×C4—1691no—
74C22⋊C4—1691noSylow
75C13—62no—
76C13(C22⋊#261)⋊C312yescomplemented
77C22×C3—3381no—
78C22×C3—3381no—
79C22×C3—6762no—
80C22×C3—1691no—
81C3×C4—3381no—
82C3×C4—3381no—
83C23—1691no—
84C2×C4—1691no—
85C2×C4—1691no—
86C2×C3—6762no—
87C2×C3—6761no—
88C2×C3—1691no—
89C2×C3—1691no—
90C22—132no—
91C22—3381no—
92C22—3381no—
93C22—1691no—
94C4—3381no—
95C4—3381no—
96C3—6761noSylow
97C2—262no—
98C2—1691no—
99C2—1691no—
100C2#7222062211yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2#72220622
commutator, Fitting, socleC2×C132C2×C3×C4
prime radicalC2×#11759818⋊C2C2
cosocleC2×C13⋊D13C22×C3
hypocenterC132C3×C22⋊C4
LCSG ▷ C2×C132 ▷ C132
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. C2×#14523147C2semidirect
3. C2×#14523145C2semidirect
4. #72214177C3semidirect
5. C2×#11759818C4semidirect
12. C2×#10621729C2×C3semidirect
13. C2×#10621728C2×C3semidirect
19. C2×C13×D13C3×C4semidirect
25. #92789C22⋊C4semidirect
34. C132C3×C22⋊C4semidirect
56. C2×D13AGL(1,𝔽13)semidirect
76. C13(C22⋊#261)⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#11759818⋊C22
C2×#145231472
C2×#145231452
#722141773
(C22⋊#261)⋊C313