Finite Group Databasebeta

← #72259448

Subgroups — #72259448

Order 8112.

184 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 25 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
1#72259261C311yesmaximal, characteristic, complemented, Hall
2#72229258C211yesmaximal, characteristic, complemented
3#72229164C211yesmaximal, characteristic, complemented
4#72229020C211yesmaximal, characteristic, complemented
5C2×#11759817—21no—
6C2×#11759815—21no—
7C4×#92788C2211yescharacteristic
8C4⋊(C13⋊D13)C2×C311yescharacteristic, complemented
9C13×D52C2×C311yescharacteristic, complemented
10C13×#1055C2×C311yescharacteristic, complemented
11C2×#92788D411yescharacteristic
12#11759817—41no—
13#11759815—41no—
14C2×C13⋊D13—21no—
15C2×C13×D13—21no—
16C4×C132C22×C311yescharacteristic
17#6104⋊C3—131nomaximal
18D104⋊C3—131nomaximal
19#92788D811yescharacteristic, complemented, Hall
20C2×C132C3×D411yescharacteristic
21C13⋊D13—41no—
22C13×D13—41no—
23D4×C13⋊C3—131no—
24C4⋊(D13⋊C3)—131no—
25C4⋊(D13⋊C3)—131no—
26C4⋊(D13⋊C3)—266no—
27C4⋊(D13⋊C3)—261no—
28C8×C13⋊C3—131no—
29#1055⋊C3—261no—
30#6104—131no—
31D104—131no—
32C132C3×D811yescharacteristic, complemented, Sylow
33C22×C13⋊C3—261no—
34C2×D13⋊C3—261no—
35C2×D13⋊C3—261no—
36C2×D13⋊C3—261no—
37C2×D13⋊C3—2612no—
38C4×C13⋊C3—261no—
39C4×C13⋊C3—261no—
40C4×C13⋊C3—526no—
41D4×C13—131no—
42D52—131no—
43D52—131no—
44D52—266no—
45D52D13⋊C311yescharacteristic, complemented
46C8×C13—131no—
47#1055D13⋊C311yescharacteristic, complemented
48C2×C13⋊C3—261no—
49C2×C13⋊C3—261no—
50C2×C13⋊C3—526no—
51C2×C13⋊C3—521no—
52D13⋊C3—521no—
53D13⋊C3—521no—
54D13⋊C3—521no—
55D13⋊C3—5212no—
56C22×C13—261no—
57C2×D13—21no—
58C2×D13—2612no—
59C2×D13—261no—
60C2×D13—261no—
61C4×C13—26no—
62C4×C13C2×D13⋊C311yescharacteristic
63C4×C13C2×D13⋊C311yescharacteristic
64C3×D8—1691noHall
65C13⋊C3—261no—
66C13⋊C3—261no—
67C13⋊C3—526no—
68C2×C13—26no—
69C2×C13—521no—
70C2×C13C4⋊(D13⋊C3)11yescharacteristic
71C2×C13(C22⋊D13)⋊C311yescharacteristic
72D13—41no—
73D13—5212no—
74D13—521no—
75D13—521no—
76C3×D4—3381no—
77C3×D4—1691no—
78C3×C8—3381no—
79D8—1691noSylow
80C13—26no—
81C13D104⋊C311yescharacteristic, complemented
82C13#6104⋊C311yescharacteristic, complemented
83C22×C3—3381no—
84C22×C3—3381no—
85C3×C4—6761no—
86D4—131no—
87D4—1691no—
88C8—131no—
89C2×C3—6761no—
90C2×C3—6761no—
91C2×C3—6761no—
92C22—261no—
93C22—3381no—
94C4#11759815⋊C211yescharacteristic
95C3—6761noSylow
96C2—521no—
97C2—6761no—
98C2#7221409411yescharacteristic
subgroupG/H
centerC2#72214094
commutator, Fitting, cosocleC4×C132C22×C3
Frattini, hypercenterC4#11759815⋊C2
socleC2×C132C3×D4
hypocenterC132C3×D8
LCSG ▷ C4×C132 ▷ C2×C132 ▷ C132
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #72259261C3semidirect
2. #72229258C2semidirect
3. #72229164C2semidirect
4. #72229020C2semidirect
8. C4⋊(C13⋊D13)C2×C3semidirect
9. C13×D52C2×C3semidirect
10. C13×#1055C2×C3semidirect
19. #92788D8semidirect
32. C132C3×D8semidirect
45. D52D13⋊C3semidirect
47. #1055D13⋊C3semidirect
81. C13D104⋊C3semidirect
82. C13#6104⋊C3semidirect

Maximal subgroups

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

subgroupindex
#722592613
#722292582
#722291642
#722290202
#6104⋊C313
D104⋊C313