Finite Group Databasebeta

← #27213420

Subgroups — #27213420

Order 13504.

100 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 100 conjugacy classes of subgroups — too many to draw. Only its 35 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#20270010C211yesmaximal, characteristic
2#20270008C211yesmaximal, characteristic
3C2×#16809952C211yesmaximal, characteristic, complemented
4#20269985C211yesmaximal, characteristic
5#20269912C211yesmaximal, characteristic, complemented
6#20269895C211yesmaximal, characteristic, complemented
7#101×C211C211yesmaximal, characteristic, complemented
8#16809962C2211yescharacteristic
9C2×Dic422C2211yescharacteristic
10Dic844—21no—
11#16809952—21no—
12#16809952—21no—
13#16809952—21no—
14#16809951—21no—
15#16809949—21no—
16#16809946C2211yescharacteristic
17#16809945—21no—
18C2×C211⋊C8C2211yescharacteristic
19C8×D211—21no—
20C2×Q8×C211C2211yescharacteristic
21M16×C211C2211yescharacteristic
22M16×C211C2211yescharacteristic
23Dic422—21no—
24Dic422—21no—
25Dic422D411yescharacteristic
26D844D411yescharacteristic
27C22⋊D211—21no—
28C2×Dic211—21no—
29C4×D211—21no—
30C211⋊C8—21no—
31C211⋊C8D411yescharacteristic
32C211⋊C8D411yescharacteristic
33Q8×C211—21no—
34Q8×C211—21no—
35C2×C4×C211—21no—
36C2×C4×C211C2311yescharacteristic
37C8×C211—21no—
38C8×C211—21no—
39C22×C211#4111yescharacteristic
40C2×D211—21no—
41C4×C211—41no—
42C4×C211C2×D411yescharacteristic
43C4×C211C2×D411yescharacteristic
44Dic211—21no—
45Dic211—41no—
46C2×C211—21no—
47C2×C211#12111yescharacteristic
48D211—41no—
49C211#47311yescharacteristic, complemented, Sylow
50#473—2111nomaximal, Sylow
51#137—2111no—
52#137—2111no—
53C2×Q16—2111no—
54#131—2111no—
55#108—2111no—
56#101—2111no—
57#101D21111yescharacteristic, complemented
58#41—2111no—
59C2×Q8—2111no—
60C2×Q8C2×D21111yescharacteristic
61Q16—4221no—
62Q16—4221no—
63Q16—4221no—
64Q16—4221no—
65SD16—4221no—
66SD16—4221no—
67M16—4221no—
68M16—2111no—
69M16C2×D21111yescharacteristic
70M16C2×D21111yescharacteristic
71C2×C8—4221no—
72C2×C8—2111no—
73Q8—21no—
74Q8—21no—
75Q8—4221no—
76Q8—4221no—
77Q8—2111no—
78D4—4221no—
79D4—2111no—
80C2×C4—21no—
81C2×C4—4221no—
82C2×C4—4221no—
83C2×C4C22×D21111yescharacteristic
84C8—21no—
85C8—21no—
86C8—4221no—
87C8—2111no—
88C8—2111no—
89C22—4221no—
90C22#1680996211yescharacteristic
91C4—41no—
92C4—8441no—
93C4—4221no—
94C4D4×D21111yescharacteristic
95C4D4×D21111yescharacteristic
96C2—21no—
97C2—8441no—
98C2#2026995011yescharacteristic
subgroupG/H
centerC2#20269950
commutator, cosocleC2×C4×C211C23
FrattiniC2×C4C22×D211
Fitting#101×C211C2
socleC2×C211#121
prime radical#16809962C22
hypercenter#101D211
hypocenterC211#473
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C2×C4 ◁ #101

Complements

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

NH ≅ G/Ntype
3. C2×#16809952C2semidirect
5. #20269912C2semidirect
6. #20269895C2semidirect
7. #101×C211C2semidirect
49. C211#473semidirect
57. #101D211semidirect

Maximal subgroups

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

subgroupindex
#202700102
#202700082
C2×#168099522
#202699852
#202699122
#202698952
#101×C2112
#473211