Finite Group Databasebeta

← #27156857

Subgroups — #27156857

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#20260913C211yesmaximal, characteristic
4#15690871—21no—
5#15690865—21no—
6#15690840—21no—
7#15690797C2211yescharacteristic
8#15690737—21no—
9#15690727C411yescharacteristic
10H(5,𝔽2)×C97C411yescharacteristic
11#13095520—41no—
12C2×#11715897—41no—
13C2×Dic194D411yescharacteristic
14#13095501—41no—
15#13095499—41no—
16#13095495—21no—
17C22⋊Dic97—41no—
18C4×Dic97D411yescharacteristic
19#41×C97—21no—
20#41×C97—41no—
21C2×D4×C97—41no—
22C2×D4×C97—41no—
23C2×D4×C97C2×C411yescharacteristic
24C23×C97—21no—
25C23×C97—41no—
26Q8×C97—21no—
27D4×C97—21no—
28D4×C97—41no—
29D4×C97—42no—
30D4×C97—41no—
31C2×C4×C97—41no—
32C2×C4×C97—41no—
33C2×C4×C97C22⋊C411yescharacteristic
34C2×Dic97—21no—
35C2×Dic97—21no—
36Dic194—41no—
37#11715897—42no—
38#11715897—81no—
39#11715895—41no—
40C22×C97—41no—
41C22×C97—41no—
42C22×C97—81no—
43C22×C97—161no—
44C22×C97#9911yescharacteristic
45C4×C97—21no—
46C4×C97—41no—
47Dic97—41no—
48Dic97—41no—
49C2×C97—21no—
50C2×C97—81no—
51C2×C97—81no—
52C2×C97AΣL(1,𝔽16)11yescharacteristic
53#1403—971nomaximal, Sylow
54C97#140311yescharacteristic, complemented, Sylow
55#455—971no—
56#355—971no—
57#329—971no—
58H(5,𝔽2)#8760411yescharacteristic
59#137—1941no—
60#124—971no—
61#104—1941no—
62#101—1941no—
63#100—971no—
64#97—1941no—
65#41—21no—
66#41—41no—
67C2×Q8—971no—
68C2×D4—41no—
69C2×D4—41no—
70C2×D4C2×#8760411yescharacteristic
71Q16—3881no—
72SD16—3881no—
73M16—3881no—
74M16—1941no—
75C2×C8—3881no—
76C22⋊C4—3881no—
77C42—971no—
78C23—21no—
79C23—41no—
80Q8—21no—
81Q8—3881no—
82D4—21no—
83D4—41no—
84D4—42no—
85D4—41no—
86C2×C4—41no—
87C2×C4—41no—
88C2×C4—1941no—
89C2×C4—1941no—
90C2×C4C22⋊#8760411yescharacteristic
91C8—7761no—
92C8—3881no—
93C8—3882no—
94C22—41no—
95C22—41no—
96C22—81no—
97C22—161no—
98C22#1569086811yescharacteristic
99C4—21no—
100C4—41no—
101C4—3881no—
102C4—3881no—
103C2—21no—
104C2—81no—
105C2—81no—
106C2#2026163511yescharacteristic
subgroupG/H
centerC2#20261635
commutatorC2×D4×C97C2×C4
FrattiniC2×D4C2×#87604
Fitting, prime radicalH(5,𝔽2)×C97C4
socleC2×C97AΣL(1,𝔽16)
cosocle#15690797C22
hypercenterH(5,𝔽2)#87604
hypocenterC97#1403
LCSG ▷ C2×D4×C97 ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C2×D4 ◁ H(5,𝔽2)

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
#202609132
#140397