Finite Group Databasebeta

← #27213327

Subgroups — #27213327

Order 13504.

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

normal subgroups only

This group has 142 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#20270024C211yesmaximal, characteristic
2#20270008C211yesmaximal, characteristic, complemented
3#20269975C211yesmaximal, characteristic, complemented
4#20269904C211yesmaximal, characteristic, complemented
5#20269903C211yesmaximal, characteristic, complemented
6#20269895C211yesmaximal, characteristic, complemented
7#137×C211C211yesmaximal, characteristic, complemented
8#16809962—21no—
9#16809962C2211yescharacteristic
10#16809961—22no—
11#16809961—21no—
12C2×Dic422—21no—
13C2×Dic422C2211yescharacteristic
14Q8×D211—21no—
15#16809952—21no—
16#16809949—21no—
17#16809946C2211yescharacteristic
18Dic211⋊C4—22no—
19C4×Dic211C2211yescharacteristic
20#41×C211C2211yescharacteristic
21C2×Q8×C211C2211yescharacteristic
22Q16×C211—21no—
23SD16×C211—21no—
24M16×C211C2211yescharacteristic
25Dic422—22no—
26Dic422—21no—
27Dic422—21no—
28Dic422D411yescharacteristic
29D844D411yescharacteristic
30C22⋊D211—22no—
31C22⋊D211—21no—
32C2×Dic211—22no—
33C2×Dic211—21no—
34C2×Dic211D412yes—
35C4×D211—22no—
36C4×D211—21no—
37C211⋊C8—21no—
38Q8×C211—21no—
39Q8×C211—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—21no—
50C4×C211—21no—
51C4×C211—41no—
52C4×C211C2×D411yescharacteristic
53C4×C211C2×D411yescharacteristic
54Dic211—22no—
55Dic211—21no—
56Dic211—22no—
57C2×C211—21no—
58C2×C211—41no—
59C2×C211#12011yescharacteristic
60D211—41no—
61C211#45611yescharacteristic, complemented, Sylow
62#456—2111nomaximal, Sylow
632−1+4—2111no—
64#137—2111no—
65#137D21111yescharacteristic, complemented
66C4⋊Q8—2111no—
67#104—2111no—
68#104—2111no—
69#101—2111no—
70#41—4221no—
71#41—4221no—
72#41—4222no—
73#41—2111no—
74#41C2×D21111yescharacteristic
75C2×Q8—4221no—
76C2×Q8—4221no—
77C2×Q8—2111no—
78C2×Q8C2×D21111yescharacteristic
79Q16—21no—
80Q16—4221no—
81SD16—21no—
82SD16—4221no—
83M16—2111no—
84M16C2×D21111yescharacteristic
85C4⋊C4—4222no—
86C42—2111no—
87Q8—21no—
88Q8—21no—
89Q8—4222no—
90Q8—4221no—
91Q8—4221no—
92Q8—2111no—
93Q8C22⋊D21111yescharacteristic
94D4—21no—
95D4—4222no—
96D4—4221no—
97D4—2111no—
98D4C22⋊D21111yescharacteristic
99C2×C4—21no—
100C2×C4—21no—
101C2×C4—4222no—
102C2×C4—4222no—
103C2×C4—4221no—
104C2×C4—4221no—
105C2×C4—2112no—
106C2×C4C22×D21111yescharacteristic
107C8—21no—
108C8—4221no—
109C22—21no—
110C22—4221no—
111C22D4×D21111yescharacteristic
112C4—21no—
113C4—41no—
114C4—4222no—
115C4—4221no—
116C4—4222no—
117C4D4×D21111yescharacteristic
118C4C2×C22⋊D21111yescharacteristic
119C2—21no—
120C2—41no—
121C2—8441no—
122C2#2026994511yescharacteristic
subgroupG/H
centerC2#20269945
commutator, cosocleC2×C4×C211C23
FrattiniC2×C4C22×D211
Fitting#137×C211C2
socleC2×C211#120
prime radical#20270024C2
hypercenter#137D211
hypocenterC211#456
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
2. #20270008C2semidirect
3. #20269975C2semidirect
4. #20269904C2semidirect
5. #20269903C2semidirect
6. #20269895C2semidirect
7. #137×C211C2semidirect
61. C211#456semidirect
65. #137D211semidirect

Maximal subgroups

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

subgroupindex
#202700242
#202700082
#202699752
#202699042
#202699032
#202698952
#137×C2112
#456211