Finite Group Databasebeta

← #27213406

Subgroups — #27213406

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#20270008C211yesmaximal, characteristic, complemented
2#20270003C211yesmaximal, characteristic
3#20269983C211yesmaximal, characteristic, complemented
4#20269913C211yesmaximal, characteristic, complemented
5#20269896C211yesmaximal, characteristic, complemented
6#20269894C211yesmaximal, characteristic, complemented
7C2×SD16×C211C211yesmaximal, characteristic, complemented
8#16809962C2211yescharacteristic
9#16809952—21no—
10#16809950—21no—
11#16809949—21no—
12#16809947—21no—
13#16809946C2211yescharacteristic
14#16809946C2211yescharacteristic
15#16809946C2211yescharacteristic
16#16809945—21no—
17C8×D211—21no—
18C2×Q8×C211C2211yescharacteristic
19C2×D4×C211C2211yescharacteristic
20SD16×C211—21no—
21SD16×C211—21no—
22C2×C8×C211C2211yescharacteristic
23Dic422D411yescharacteristic
24D844D411yescharacteristic
25C22⋊D211—21no—
26C4×D211—21no—
27C211⋊C8—21no—
28C211⋊C8—21no—
29C211⋊C8—21no—
30C23×C211—21no—
31Q8×C211—21no—
32Q8×C211—21no—
33D4×C211—21no—
34D4×C211—21no—
35C2×C4×C211—21no—
36C2×C4×C211C2311yescharacteristic
37C8×C211D411yescharacteristic
38C8×C211D411yescharacteristic
39C22×C211—41no—
40C22×C211—81no—
41C22×C211#4111yescharacteristic
42C2×D211—21no—
43C4×C211—41no—
44C4×C211C2×D411yescharacteristic
45C4×C211C2×D411yescharacteristic
46Dic211—21no—
47C2×C211—21no—
48C2×C211—81no—
49C2×C211#12111yescharacteristic
50D211—41no—
51C211#47111yescharacteristic, complemented, Sylow
52#471—2111nomaximal, Sylow
53#137—2111no—
54AGL(1,ℤ/8)—2111no—
55C2×SD16D21111yescharacteristic, complemented
56#131—2111no—
57#108—2111no—
58#101—2111no—
59#100—2111no—
60#41—2111no—
61C2×Q8C2×D21111yescharacteristic
62C2×D4C2×D21111yescharacteristic
63Q16—4221no—
64SD16—21no—
65SD16—21no—
66SD16—4221no—
67SD16—4221no—
68D8—4221no—
69M16—4221no—
70M16—2111no—
71M16—2111no—
72M16—2111no—
73C2×C8—4221no—
74C2×C8C2×D21111yescharacteristic
75C23—21no—
76Q8—21no—
77Q8—21no—
78Q8—2111no—
79D4—21no—
80D4—21no—
81D4—4221no—
82D4—2111no—
83C2×C4—21no—
84C2×C4—4221no—
85C2×C4C22×D21111yescharacteristic
86C8—4221no—
87C8—4221no—
88C8—4221no—
89C8C22⋊D21111yescharacteristic
90C8C22⋊D21111yescharacteristic
91C22—41no—
92C22—81no—
93C22—4221no—
94C22#1680996111yescharacteristic
95C4—41no—
96C4—4221no—
97C4D4×D21111yescharacteristic
98C4C2×C22⋊D21111yescharacteristic
99C2—21no—
100C2—81no—
101C2—8441no—
102C2#2026994811yescharacteristic
subgroupG/H
centerC2#20269948
commutator, cosocleC2×C4×C211C23
FrattiniC2×C4C22×D211
FittingC2×SD16×C211C2
socleC2×C211#121
prime radical#20270003C2
hypercenterC2×SD16D211
hypocenterC211#471
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. #20270008C2semidirect
3. #20269983C2semidirect
4. #20269913C2semidirect
5. #20269896C2semidirect
6. #20269894C2semidirect
7. C2×SD16×C211C2semidirect
51. C211#471semidirect
55. C2×SD16D211semidirect

Maximal subgroups

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

subgroupindex
#202700082
#202700032
#202699832
#202699132
#202698962
#202698942
C2×SD16×C2112
#471211