Finite Group Databasebeta

← #27213415

Subgroups — #27213415

Order 13504.

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

normal subgroups only

This group has 108 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#20270005C211yesmaximal, characteristic, complemented
2#20270003C211yesmaximal, characteristic, complemented
3C2×#16809947C211yesmaximal, characteristic, complemented
4#20269985C211yesmaximal, characteristic, complemented
5#20269912C211yesmaximal, characteristic, complemented
6#20269893C211yesmaximal, characteristic, complemented
7#100×C211C211yesmaximal, characteristic, complemented
8#16809962C2211yescharacteristic
9C2×D844C2211yescharacteristic
10#16809951—21no—
11#16809950—21no—
12D1688—21no—
13#16809947—21no—
14#16809947—21no—
15#16809947—21no—
16#16809946C2211yescharacteristic
17#16809945—21no—
18C2×C211⋊C8C2211yescharacteristic
19C8×D211—21no—
20C2×D4×C211C2211yescharacteristic
21M16×C211C2211yescharacteristic
22M16×C211C2211yescharacteristic
23C22×D211—21no—
24Dic422D411yescharacteristic
25D844—21no—
26D844—21no—
27D844D411yescharacteristic
28C22⋊D211—21no—
29C4×D211—21no—
30C211⋊C8—21no—
31C211⋊C8D411yescharacteristic
32C211⋊C8D411yescharacteristic
33C23×C211—21no—
34D4×C211—21no—
35D4×C211—21no—
36C2×C4×C211C2311yescharacteristic
37C8×C211—21no—
38C8×C211—21no—
39C22×C211—41no—
40C22×C211—81no—
41C22×C211#4111yescharacteristic
42C2×D211—21no—
43C2×D211—41no—
44C2×D211—81no—
45C4×C211C2×D411yescharacteristic
46C4×C211C2×D411yescharacteristic
47Dic211—21no—
48C2×C211—21no—
49C2×C211—81no—
50C2×C211#12111yescharacteristic
51D211—41no—
52D211—81no—
53C211#47211yescharacteristic, complemented, Sylow
54#472—2111nomaximal, Sylow
55AGL(1,ℤ/8)—2111no—
56AGL(1,ℤ/8)—2111no—
57C2×D8—2111no—
58#131—2111no—
59#108—2111no—
60#100—2111no—
61#100D21111yescharacteristic, complemented
62#41—2111no—
63C2×D4—2111no—
64C2×D4C2×D21111yescharacteristic
65SD16—4221no—
66SD16—4221no—
67D8—4221no—
68D8—4221no—
69D8—4221no—
70D8—4221no—
71M16—4221no—
72M16—2111no—
73M16C2×D21111yescharacteristic
74M16C2×D21111yescharacteristic
75C2×C8—4221no—
76C2×C8—2111no—
77C23—21no—
78C23—4221no—
79Q8—2111no—
80D4—21no—
81D4—21no—
82D4—4221no—
83D4—4221no—
84D4—4221no—
85D4—2111no—
86C2×C4—4221no—
87C2×C4C22×D21111yescharacteristic
88C8—21no—
89C8—21no—
90C8—4221no—
91C8—2111no—
92C8—2111no—
93C22—41no—
94C22—81no—
95C22—8441no—
96C22—16881no—
97C22—4221no—
98C22#1680996211yescharacteristic
99C4—4221no—
100C4D4×D21111yescharacteristic
101C4D4×D21111yescharacteristic
102C2—21no—
103C2—81no—
104C2—8441no—
105C2—16881no—
106C2#2026995011yescharacteristic
subgroupG/H
centerC2#20269950
commutator, cosocleC2×C4×C211C23
FrattiniC2×C4C22×D211
Fitting#100×C211C2
socleC2×C211#121
hypercenter#100D211
hypocenterC211#472
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C2×C4 ◁ #100

Complements

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

NH ≅ G/Ntype
1. #20270005C2semidirect
2. #20270003C2semidirect
3. C2×#16809947C2semidirect
4. #20269985C2semidirect
5. #20269912C2semidirect
6. #20269893C2semidirect
7. #100×C211C2semidirect
53. C211#472semidirect
61. #100D211semidirect

Maximal subgroups

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

subgroupindex
#202700052
#202700032
C2×#168099472
#202699852
#202699122
#202698932
#100×C2112
#472211