Finite Group Databasebeta

← #27156858

Subgroups — #27156858

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#20261643C211yesmaximal, characteristic, complemented
2#20261590C211yesmaximal, characteristic, complemented
3#20260915C211yesmaximal, characteristic
4#15690871—21no—
5#15690865—21no—
6#15690855C411yescharacteristic
7#15690843—21no—
8#15690797C2211yescharacteristic
9#15690736—21no—
10#100×C97C411yescharacteristic
11#13095520—41no—
12C2×#11715897—41no—
13#13095511—21no—
14#13095510—41no—
15C2×Dic194D411yescharacteristic
16C2×C22⋊D97—41no—
17D4×D97—41no—
18Dic388—41no—
19#13095500—41no—
20C22⋊Dic97—41no—
21C4×Dic97D411yescharacteristic
22C2×D4×C97C2×C411yescharacteristic
23M16×C97—21no—
24C23×C97—21no—
25C22×D97—41no—
26D4×C97—41no—
27C2×C4×C97C22⋊C411yescharacteristic
28C22⋊D97—42no—
29C22⋊D97—41no—
30C2×Dic97—21no—
31C2×Dic97—21no—
32C2×Dic97—41no—
33D388—21no—
34C4×D97—41no—
35Dic194—21no—
36Dic194—41no—
37#11715897—42no—
38#11715897—81no—
39C8×C97—41no—
40C22×C97—41no—
41C22×C97—81no—
42C22×C97#9911yescharacteristic
43C2×D97—41no—
44C2×D97—161no—
45C4×C97—21no—
46Dic97—41no—
47Dic97—41no—
48Dic97—41no—
49C2×C97—21no—
50C2×C97—81no—
51C2×C97AΣL(1,𝔽16)11yescharacteristic
52D97—81no—
53#1403—971nomaximal, Sylow
54C97#140311yescharacteristic, complemented, Sylow
55#455—971no—
56#355—971no—
57#329—971no—
58H(5,𝔽2)—971no—
59#137—1941no—
60#124—971no—
61#104—1941no—
62#101—1941no—
63#100#8760411yescharacteristic
64#97—1941no—
65#41—3881no—
66#41—1941no—
67C2×Q8—971no—
68C2×D4—3881no—
69C2×D4—3881no—
70C2×D4C2×#8760411yescharacteristic
71Q16—3881no—
72SD16—3881no—
73M16—21no—
74M16—3881no—
75C2×C8—3881no—
76C22⋊C4—3881no—
77C42—971no—
78C23—21no—
79C23—3881no—
80Q8—3881no—
81Q8—1941no—
82D4—41no—
83D4—3882no—
84D4—3881no—
85D4—1941no—
86C2×C4—3881no—
87C2×C4—3881no—
88C2×C4—1941no—
89C2×C4—1941no—
90C2×C4C22⋊#8760411yescharacteristic
91C8—41no—
92C8—7761no—
93C8—3882no—
94C22—41no—
95C22—81no—
96C22—15521no—
97C22—3881no—
98C22#1569086911yescharacteristic
99C4—21no—
100C4—3881no—
101C4—3881no—
102C4—3881no—
103C2—21no—
104C2—81no—
105C2—7761no—
106C2#2026163611yescharacteristic
subgroupG/H
centerC2#20261636
commutatorC2×D4×C97C2×C4
FrattiniC2×D4C2×#87604
Fitting#100×C97C4
socleC2×C97AΣL(1,𝔽16)
prime radical#15690855C4
cosocle#15690797C22
hypercenter#100#87604
hypocenterC97#1403
LCSG ▷ C2×D4×C97 ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C2×D4 ◁ #100

Complements

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

NH ≅ G/Ntype
1. #20261643C2semidirect
2. #20261590C2semidirect
54. C97#1403semidirect

Maximal subgroups

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

subgroupindex
#202616432
#202615902
#202609152
#140397