Finite Group Databasebeta

← #27213390

Subgroups — #27213390

Order 13504.

120 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 41 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
1C2×#16809950C211yesmaximal, characteristic, complemented
2C2×#16809946C211yesmaximal, characteristic, complemented
3#20269956C211yesmaximal, characteristic, complemented
4#20269910C211yesmaximal, characteristic, complemented
5#20269900C211yesmaximal, characteristic, complemented
6#20269898C211yesmaximal, characteristic, complemented
7#121×C211C211yesmaximal, characteristic
8C2×Dic422C2211yescharacteristic
9#16809950—22no—
10#16809946—22no—
11C2×C211⋊C8C2211yescharacteristic
12C2×C211⋊C8C2211yescharacteristic
13C4⋊Dic211C2211yescharacteristic
14Dic211⋊C4—21no—
15C22⋊Dic211—21no—
16C2×D4×C211—21no—
17C2×D4×C211C2211yescharacteristic
18C22×C4×C211C2211yescharacteristic
19C4⋊C4×C211C2211yescharacteristic
20C22⋊C4×C211—21no—
21Dic422—21no—
22Dic422—21no—
23C2×Dic211—21no—
24C2×Dic211—21no—
25C211⋊C8—21no—
26C211⋊C8D412yescomplemented
27C23×C211—21no—
28C23×C211D411yescharacteristic
29D4×C211—21no—
30D4×C211—21no—
31D4×C211—42no—
32C2×C4×C211—21no—
33C2×C4×C211—21no—
34C2×C4×C211—21no—
35C2×C4×C211D411yescharacteristic
36C2×C4×C211C2311yescharacteristic
37C22×C211—21no—
38C22×C211—21no—
39C22×C211—21no—
40C22×C211—41no—
41C22×C211—41no—
42C22×C211—41no—
43C22×C211C2×D411yescharacteristic
44C4×C211—21no—
45C4×C211—41no—
46C4×C211C2×D411yescharacteristic
47C4×C211#4111yescharacteristic
48Dic211—41no—
49Dic211—41no—
50C2×C211—41no—
51C2×C211—81no—
52C2×C211#12111yescharacteristic
53C2×C211AGL(1,ℤ/8)11yescharacteristic
54C2×C211#13711yescharacteristic
55C211#46811yescharacteristic, complemented, Sylow
56#468—2111nomaximal, Sylow
57C2×SD16—2111no—
58C2×M16—2111no—
59C22⋊Q8—2111no—
60#121D21111yescharacteristic
61#107—2111no—
62#103—2111no—
63#102—2111no—
64C2×Q8—2111no—
65C2×D4—21no—
66C2×D4C2×D21111yescharacteristic
67C22×C4C2×D21111yescharacteristic
68SD16—4222no—
69M16—4222no—
70C2×C8—2111no—
71C2×C8—2111no—
72C4⋊C4—4221no—
73C4⋊C4—2111no—
74C4⋊C4C2×D21111yescharacteristic
75C22⋊C4—21no—
76C22⋊C4—4221no—
77C23—21no—
78C23C22⋊D21111yescharacteristic
79Q8—4221no—
80Q8—4221no—
81D4—21no—
82D4—21no—
83D4—42no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—21no—
87C2×C4—4221no—
88C2×C4—4221no—
89C2×C4C22⋊D21111yescharacteristic
90C2×C4C22×D21111yescharacteristic
91C8—4221no—
92C8—2112no—
93C22—21no—
94C22—21no—
95C22—21no—
96C22—41no—
97C22—41no—
98C22—41no—
99C22C2×C22⋊D21111yescharacteristic
100C4—21no—
101C4—41no—
102C4—8441no—
103C4—8441no—
104C4D4×D21111yescharacteristic
105C4#1680996111yescharacteristic
106C2—41no—
107C2—81no—
108C2#2026994911yescharacteristic
109C2#2027000311yescharacteristic
110C2#2027000711yescharacteristic
subgroupG/H
centerC22C2×C22⋊D211
commutator, cosocleC2×C4×C211C23
FrattiniC2×C4C22×D211
Fitting, prime radical#121×C211C2
socleC22×C211C2×D4
hypercenter#121D211
hypocenterC211#468
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#16809950C2semidirect
2. C2×#16809946C2semidirect
3. #20269956C2semidirect
4. #20269910C2semidirect
5. #20269900C2semidirect
6. #20269898C2semidirect
26. C211⋊C8D4semidirect
55. C211#468semidirect

Maximal subgroups

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

subgroupindex
C2×#168099502
C2×#168099462
#202699562
#202699102
#202699002
#202698982
#121×C2112
#468211