Finite Group Databasebeta

← #27213414

Subgroups — #27213414

Order 13504.

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

normal subgroups only

This group has 108 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#20270002C211yesmaximal, characteristic, complemented
2C2×#16809947C211yesmaximal, characteristic, complemented
3#20269984C211yesmaximal, characteristic, complemented
4#20269912C211yesmaximal, characteristic, complemented
5#20269894C211yesmaximal, characteristic, complemented
6#20269893C211yesmaximal, characteristic, complemented
7AGL(1,ℤ/8)×C211C211yesmaximal, characteristic, complemented
8C2×D844C2211yescharacteristic
9#16809949—21no—
10#16809947—21no—
11#16809947—21no—
12#16809947—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
23C22×D211—21no—
24D844—21no—
25D844—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
43C2×D211—41no—
44C2×D211—81no—
45C4×C211—21no—
46C4×C211C2×D411yescharacteristic
47C4×C211C2×D411yescharacteristic
48C2×C211—21no—
49C2×C211—41no—
50C2×C211—81no—
51C2×C211#12111yescharacteristic
52D211—81no—
53C211#47211yescharacteristic, complemented, Sylow
54#472—2111nomaximal, Sylow
55AGL(1,ℤ/8)—2111no—
56AGL(1,ℤ/8)D21111yescharacteristic, complemented
57C2×D8—2111no—
58#131—2111no—
59#108—2111no—
60#100—2111no—
61#100—2111no—
62#41C2×D21111yescharacteristic
63C2×D4—2111no—
64C2×D4C2×D21111yescharacteristic
65SD16—21no—
66SD16—4221no—
67D8—21no—
68D8—4221no—
69D8—4221no—
70D8—4221no—
71M16—4221no—
72M16—2111no—
73M16—2111no—
74M16C2×D21111yescharacteristic
75C2×C8—4221no—
76C2×C8—2111no—
77C23—21no—
78C23—4221no—
79Q8C22⋊D21111yescharacteristic
80D4—21no—
81D4—21no—
82D4—21no—
83D4—4221no—
84D4—4221no—
85D4C22⋊D21111yescharacteristic
86C2×C4—21no—
87C2×C4C22×D21111yescharacteristic
88C8—21no—
89C8—4221no—
90C8—4221no—
91C8—2111no—
92C8—2111no—
93C22—21no—
94C22—41no—
95C22—81no—
96C22—8441no—
97C22—16881no—
98C22#1680996111yescharacteristic
99C4—21no—
100C4D4×D21111yescharacteristic
101C4C2×C22⋊D21111yescharacteristic
102C2—21no—
103C2—41no—
104C2—81no—
105C2—16881no—
106C2#2026994911yescharacteristic
subgroupG/H
centerC2#20269949
commutator, cosocleC2×C4×C211C23
FrattiniC2×C4C22×D211
FittingAGL(1,ℤ/8)×C211C2
socleC2×C211#121
hypercenterAGL(1,ℤ/8)D211
hypocenterC211#472
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. #20270002C2semidirect
2. C2×#16809947C2semidirect
3. #20269984C2semidirect
4. #20269912C2semidirect
5. #20269894C2semidirect
6. #20269893C2semidirect
7. AGL(1,ℤ/8)×C211C2semidirect
53. C211#472semidirect
56. AGL(1,ℤ/8)D211semidirect

Maximal subgroups

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

subgroupindex
#202700022
C2×#168099472
#202699842
#202699122
#202698942
#202698932
AGL(1,ℤ/8)×C2112
#472211