Finite Group Databasebeta

← #27213325

Subgroups — #27213325

Order 13504.

152 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 39 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#20270023C211yesmaximal, characteristic
2#20270010C211yesmaximal, characteristic, complemented
3#20270007C211yesmaximal, characteristic, complemented
4#20269965C211yesmaximal, characteristic, complemented
5#20269905C211yesmaximal, characteristic, complemented
6#20269893C211yesmaximal, characteristic, complemented
7#104×C211C211yesmaximal, characteristic, complemented
8#16809963—21no—
9#16809962—21no—
10#16809962C2211yescharacteristic
11C2×Dic422C2211yescharacteristic, complemented
12C2×D844—21no—
13C2×D844C2211yescharacteristic
14D4×D211—22no—
15D4×D211—21no—
16Dic844—21no—
17#16809952—21no—
18#16809951—21no—
19#16809950—21no—
20#16809946C2211yescharacteristic, complemented
21(C2×D211)⋊C4—22no—
22#41×C211C2211yescharacteristic
23M16×C211C2211yescharacteristic, complemented
24C42×C211C2211yescharacteristic, complemented
25C22×D211—22no—
26C22×D211D412yes—
27Dic422—21no—
28Dic422—21no—
29Dic422D411yescharacteristic
30D844—22no—
31D844—21no—
32D844—21no—
33D844D411yescharacteristic
34C22⋊D211—22no—
35C22⋊D211—21no—
36C2×Dic211—21no—
37C4×D211—22no—
38C4×D211—21no—
39C211⋊C8—21no—
40Q8×C211D411yescharacteristic
41D4×C211—21no—
42D4×C211D411yescharacteristic
43C2×C4×C211—21no—
44C2×C4×C211—21no—
45C2×C4×C211C2311yescharacteristic
46C8×C211—21no—
47C22×C211—21no—
48C22×C211C2×D411yescharacteristic
49C2×D211—22no—
50C2×D211—21no—
51C2×D211—42no—
52C2×D211—82no—
53C4×C211—21no—
54C4×C211—41no—
55C4×C211C2×D411yescharacteristic
56C4×C211C2×D411yescharacteristic
57Dic211—21no—
58Dic211—41no—
59C2×C211—21no—
60C2×C211—41no—
61C2×C211#12011yescharacteristic
62D211—42no—
63D211—41no—
64C211#45511yescharacteristic, complemented, Sylow
65#455—2111nomaximal, Sylow
66H(5,𝔽2)—2111no—
67#137—2111no—
68#137—2111no—
69#124—2111no—
70#104—2111no—
71#104D21111yescharacteristic, complemented
72#100—2111no—
73#41—4221no—
74#41—4221no—
75#41—2111no—
76#41C2×D21111yescharacteristic
77C2×Q8—2111no—
78C2×D4—4221no—
79C2×D4—4222no—
80C2×D4—4221no—
81C2×D4—2111no—
82Q16—4221no—
83Q16—4221no—
84SD16—4221no—
85SD16—4221no—
86M16—2111no—
87M16C2×D21111yescharacteristic, complemented
88C22⋊C4—4222no—
89C42C2×D21111yescharacteristic, complemented
90C23—4222no—
91C23—2112no—
92Q8—4221no—
93Q8—4221no—
94Q8—2111no—
95Q8D84411yescharacteristic
96D4—21no—
97D4—4222no—
98D4—4222no—
99D4—4221no—
100D4—4221no—
101D4—4221no—
102D4—2111no—
103D4D84411yescharacteristic
104C2×C4—21no—
105C2×C4—21no—
106C2×C4—4222no—
107C2×C4—4221no—
108C2×C4—4221no—
109C2×C4C22×D21111yescharacteristic
110C8—21no—
111C8—4221no—
112C22—21no—
113C22—8442no—
114C22—16882no—
115C22—4222no—
116C22—4221no—
117C22D4×D21111yescharacteristic
118C4—21no—
119C4—41no—
120C4—8441no—
121C4—4221no—
122C4D4×D21111yescharacteristic
123C4C2×D84411yescharacteristic
124C2—21no—
125C2—41no—
126C2—8442no—
127C2—8441no—
128C2#2026994711yescharacteristic
subgroupG/H
centerC2#20269947
commutator, cosocleC2×C4×C211C23
FrattiniC2×C4C22×D211
Fitting#104×C211C2
socleC2×C211#120
prime radical#20270023C2
hypercenter#104D211
hypocenterC211#455
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C2×C4 ◁ #104

Complements

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

NH ≅ G/Ntype
2. #20270010C2semidirect
3. #20270007C2semidirect
4. #20269965C2semidirect
5. #20269905C2semidirect
6. #20269893C2semidirect
7. #104×C211C2semidirect
11. C2×Dic422C22semidirect
20. #16809946C22semidirect
23. M16×C211C22semidirect
24. C42×C211C22semidirect
64. C211#455semidirect
71. #104D211semidirect
87. M16C2×D211semidirect
89. C42C2×D211semidirect

Maximal subgroups

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

subgroupindex
#202700232
#202700102
#202700072
#202699652
#202699052
#202698932
#104×C2112
#455211