Finite Group Databasebeta

← #27156859

Subgroups — #27156859

Order 12416.

112 conjugacy classes of subgroups; only the 20 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 20 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#20261641C211yesmaximal, characteristic, complemented
2#20261591C211yesmaximal, characteristic, complemented
3#20260908C211yesmaximal, characteristic
4#15690869—21no—
5#15690865—21no—
6#15690835—21no—
7#15690799C2211yescharacteristic
8#15690737—21no—
9#15690729C411yescharacteristic, complemented
102−1+4×C97C411yescharacteristic, complemented
11C2×#11715897—41no—
12C22⋊#87604—41no—
13C2×D388D411yescharacteristic
14#13095498—41no—
15#13095496—41no—
16#13095495—21no—
17#13095488—41no—
18C4×Dic97D411yescharacteristic
19#41×C97—21no—
20#41×C97—41no—
21#41×C97—41no—
22C2×Q8×C97—41no—
23C2×Q8×C97C2×C411yescharacteristic
24C22×D97—21no—
25C2×#87604—81no—
26Q8×C97—21no—
27Q8×C97—41no—
28Q8×C97—41no—
29D4×C97—21no—
30D4×C97—41no—
31D4×C97—41no—
32C2×C4×C97—21no—
33C2×C4×C97—41no—
34C2×C4×C97—42no—
35C2×C4×C97C22⋊C411yescharacteristic
36C2×Dic97—21no—
37D388—41no—
38#11715897—42no—
39#11715895—41no—
40C22×C97—41no—
41C22×C97#9911yescharacteristic
42C2×D97—41no—
43C2×D97—81no—
44#87604—161no—
45C4×C97—21no—
46C4×C97—41no—
47C4×C97—41no—
48Dic97—41no—
49C2×C97—21no—
50C2×C97—81no—
51C2×C97AΣL(1,𝔽16)11yescharacteristic
52D97—81no—
53#1404—971nomaximal, Sylow
54C97#140411yescharacteristic, complemented, Sylow
55#454—971no—
56#354—971no—
57#329—971no—
582−1+4#8760411yescharacteristic, complemented
59AGL(1,ℤ/8)—1941no—
60#124—971no—
61#104—1941no—
62#101—971no—
63#99—1941no—
64#97—1941no—
65#41—21no—
66#41—41no—
67#41—41no—
68C2×Q8—41no—
69C2×Q8C2×#8760411yescharacteristic
70C2×D4—971no—
71SD16—3881no—
72D8—3881no—
73M16—1941no—
74C2×C8—3881no—
75C22⋊C4—3881no—
76C22⋊C4—3881no—
77C42—971no—
78C23—1941no—
79Q8—21no—
80Q8—41no—
81Q8—41no—
82D4—21no—
83D4—41no—
84D4—41no—
85D4—3881no—
86C2×C4—21no—
87C2×C4—41no—
88C2×C4—42no—
89C2×C4—7761no—
90C2×C4—1941no—
91C2×C4C22⋊#8760411yescharacteristic
92C8—3881no—
93C8—3882no—
94C22—41no—
95C22—7761no—
96C22—3881no—
97C22#1569086811yescharacteristic
98C4—21no—
99C4—41no—
100C4—41no—
101C4—15521no—
102C4—3881no—
103C2—21no—
104C2—81no—
105C2—7761no—
106C2#2026163511yescharacteristic
subgroupG/H
centerC2#20261635
commutatorC2×Q8×C97C2×C4
FrattiniC2×Q8C2×#87604
Fitting2−1+4×C97C4
socleC2×C97AΣL(1,𝔽16)
prime radical#20260908C2
cosocle#15690799C22
hypercenter2−1+4#87604
hypocenterC97#1404
LCSG ▷ C2×Q8×C97 ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C2×Q8 ◁ 2−1+4

Complements

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

NH ≅ G/Ntype
1. #20261641C2semidirect
2. #20261591C2semidirect
9. #15690729C4semidirect
10. 2−1+4×C97C4semidirect
54. C97#1404semidirect
58. 2−1+4#87604semidirect

Maximal subgroups

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

subgroupindex
#202616412
#202615912
#202609082
#140497