Finite Group Databasebeta

← #27213407

Subgroups — #27213407

Order 13504.

104 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 37 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#20270007C211yesmaximal, characteristic, complemented
2C2×#16809950C211yesmaximal, characteristic, complemented
3#20269984C211yesmaximal, characteristic, complemented
4#20269912C211yesmaximal, characteristic, complemented
5#20269895C211yesmaximal, characteristic, complemented
6#20269894C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×C211C211yesmaximal, characteristic
8C2×Dic422C2211yescharacteristic
9#16809952—21no—
10#16809950—21no—
11#16809950—21no—
12#16809950—21no—
13#16809946—21no—
14#16809946C2211yescharacteristic
15#16809946C2211yescharacteristic
16C2×C211⋊C8—21no—
17C2×C211⋊C8C2211yescharacteristic
18#41×C211C2211yescharacteristic
19C2×D4×C211C2211yescharacteristic
20SD16×C211—21no—
21D8×C211—21no—
22M16×C211C2211yescharacteristic
23Dic422—21no—
24Dic422—21no—
25C2×Dic211—21no—
26C211⋊C8—21no—
27C211⋊C8—21no—
28C211⋊C8D411yescharacteristic
29C211⋊C8D411yescharacteristic
30C23×C211—21no—
31Q8×C211D411yescharacteristic
32D4×C211—21no—
33D4×C211—21no—
34D4×C211—21no—
35D4×C211D411yescharacteristic
36C2×C4×C211—21no—
37C2×C4×C211C2311yescharacteristic
38C8×C211—21no—
39C22×C211—21no—
40C22×C211—41no—
41C22×C211—81no—
42C22×C211#4111yescharacteristic
43C4×C211—21no—
44C4×C211C2×D411yescharacteristic
45C4×C211C2×D411yescharacteristic
46Dic211—41no—
47C2×C211—21no—
48C2×C211—41no—
49C2×C211—81no—
50C2×C211#12111yescharacteristic
51C211#47111yescharacteristic, complemented, Sylow
52#471—2111nomaximal, Sylow
53#137—2111no—
54AGL(1,ℤ/8)D21111yescharacteristic
55C2×SD16—2111no—
56#131—2111no—
57#108—2111no—
58#101—2111no—
59#100—2111no—
60#41C2×D21111yescharacteristic
61C2×Q8—2111no—
62C2×D4C2×D21111yescharacteristic
63Q16—4221no—
64SD16—21no—
65SD16—4221no—
66SD16—4221no—
67SD16—4221no—
68D8—21no—
69M16—4221no—
70M16—2111no—
71M16—2111no—
72M16C2×D21111yescharacteristic
73C2×C8—4221no—
74C2×C8—2111no—
75C23—21no—
76Q8—4221no—
77Q8—4221no—
78Q8C22⋊D21111yescharacteristic
79D4—21no—
80D4—21no—
81D4—21no—
82D4C22⋊D21111yescharacteristic
83C2×C4—21no—
84C2×C4—4221no—
85C2×C4C22×D21111yescharacteristic
86C8—21no—
87C8—4221no—
88C8—4221no—
89C8—2111no—
90C8—2111no—
91C22—21no—
92C22—41no—
93C22—81no—
94C22#1680996111yescharacteristic
95C4—21no—
96C4—8441no—
97C4D4×D21111yescharacteristic
98C4C2×C22⋊D21111yescharacteristic
99C2—21no—
100C2—41no—
101C2—81no—
102C2#2026994911yescharacteristic
subgroupG/H
centerC2#20269949
commutator, cosocleC2×C4×C211C23
FrattiniC2×C4C22×D211
Fitting, prime radicalAGL(1,ℤ/8)×C211C2
socleC2×C211#121
hypercenterAGL(1,ℤ/8)D211
hypocenterC211#471
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #20270007C2semidirect
2. C2×#16809950C2semidirect
3. #20269984C2semidirect
4. #20269912C2semidirect
5. #20269895C2semidirect
6. #20269894C2semidirect
51. C211#471semidirect

Maximal subgroups

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

subgroupindex
#202700072
C2×#168099502
#202699842
#202699122
#202698952
#202698942
AGL(1,ℤ/8)×C2112
#471211