Finite Group Databasebeta

← #27213320

Subgroups — #27213320

Order 13504.

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

normal subgroups only

This group has 146 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, complemented
2#20270005C211yesmaximal, characteristic, complemented
3#20270002C211yesmaximal, characteristic, complemented
4#20269965C211yesmaximal, characteristic, complemented
5#20269905C211yesmaximal, characteristic, complemented
6#20269895C211yesmaximal, characteristic, complemented
7#104×C211C211yesmaximal, characteristic, complemented
8#16809962—21no—
9#16809962C2211yescharacteristic
10#16809961—22no—
11#16809961—21no—
12C2×Dic422—21no—
13C2×Dic422C2211yescharacteristic
14Q8×D211—21no—
15C2×D844C2211yescharacteristic
16#16809951—21no—
17#16809949—21no—
18D1688—21no—
19#16809947—21no—
20#16809946C2211yescharacteristic
21(C2×D211)⋊C4—22no—
22#41×C211C2211yescharacteristic
23M16×C211C2211yescharacteristic
24C42×C211C2211yescharacteristic
25C22×D211—21no—
26Dic422—22no—
27Dic422—21no—
28Dic422—21no—
29Dic422D411yescharacteristic
30D844—21no—
31D844—21no—
32D844D411yescharacteristic
33C22⋊D211—22no—
34C22⋊D211—21no—
35C2×Dic211—22no—
36C2×Dic211D412yes—
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—21no—
50C2×D211—42no—
51C2×D211—41no—
52C4×C211—21no—
53C4×C211—41no—
54C4×C211C2×D411yescharacteristic
55C4×C211C2×D411yescharacteristic
56Dic211—21no—
57Dic211—22no—
58C2×C211—21no—
59C2×C211—41no—
60C2×C211#12011yescharacteristic
61D211—41no—
62D211—81no—
63C211#45411yescharacteristic, complemented, Sylow
64#454—2111nomaximal, Sylow
652−1+4—2111no—
66AGL(1,ℤ/8)—2111no—
67AGL(1,ℤ/8)—2111no—
68#124—2111no—
69#104—2111no—
70#104D21111yescharacteristic, complemented
71#101—2111no—
72#41—4221no—
73#41—4221no—
74#41—4222no—
75#41—2111no—
76#41C2×D21111yescharacteristic
77C2×Q8—4221no—
78C2×Q8—4221no—
79C2×Q8—2111no—
80C2×D4—2111no—
81SD16—4221no—
82SD16—4221no—
83D8—4221no—
84D8—4221no—
85M16—2111no—
86M16C2×D21111yescharacteristic
87C22⋊C4—4222no—
88C42C2×D21111yescharacteristic
89C23—4221no—
90Q8—4222no—
91Q8—4221no—
92Q8—4221no—
93Q8—2111no—
94Q8D84411yescharacteristic
95D4—21no—
96D4—4222no—
97D4—4221no—
98D4—4221no—
99D4—4221no—
100D4—2111no—
101D4D84411yescharacteristic
102C2×C4—21no—
103C2×C4—21no—
104C2×C4—4222no—
105C2×C4—4222no—
106C2×C4—4221no—
107C2×C4—2112no—
108C2×C4C22×D21111yescharacteristic
109C8—21no—
110C8—4221no—
111C22—21no—
112C22—8442no—
113C22—8441no—
114C22—4221no—
115C22D4×D21111yescharacteristic
116C4—21no—
117C4—41no—
118C4—4221no—
119C4—4222no—
120C4D4×D21111yescharacteristic
121C4C2×D84411yescharacteristic
122C2—21no—
123C2—41no—
124C2—8441no—
125C2—16881no—
126C2#2026994711yescharacteristic
subgroupG/H
centerC2#20269947
commutator, cosocleC2×C4×C211C23
FrattiniC2×C4C22×D211
Fitting#104×C211C2
socleC2×C211#120
hypercenter#104D211
hypocenterC211#454
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
1. #20270024C2semidirect
2. #20270005C2semidirect
3. #20270002C2semidirect
4. #20269965C2semidirect
5. #20269905C2semidirect
6. #20269895C2semidirect
7. #104×C211C2semidirect
63. C211#454semidirect
70. #104D211semidirect

Maximal subgroups

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

subgroupindex
#202700242
#202700052
#202700022
#202699652
#202699052
#202698952
#104×C2112
#454211