Finite Group Databasebeta

← #27156861

Subgroups — #27156861

Order 12416.

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

normal subgroups only

This group has 106 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#20261645C211yesmaximal, characteristic, complemented
2#20261595C211yesmaximal, characteristic, complemented
3#20260918C211yesmaximal, characteristic
4#15690877—21no—
5#15690871—21no—
6#15690840—21no—
7#15690808C2211yescharacteristic
8#15690737—21no—
9#15690729C411yescharacteristic
102−1+4×C97C411yescharacteristic
11#13095520—41no—
12C2×#11715897—41no—
13C2×Dic194D411yescharacteristic
14#13095501—41no—
15#13095499—41no—
16#13095495—21no—
17#13095490—41no—
18C4×Dic97D411yescharacteristic
19#41×C97—21no—
20#41×C97—41no—
21#41×C97—41no—
22C2×Q8×C97—41no—
23C2×Q8×C97C2×C411yescharacteristic
24Q8×C97—21no—
25Q8×C97—41no—
26Q8×C97—41no—
27D4×C97—21no—
28D4×C97—41no—
29D4×C97—41no—
30C2×C4×C97—21no—
31C2×C4×C97—41no—
32C2×C4×C97—42no—
33C2×C4×C97C22⋊C411yescharacteristic
34C2×Dic97—21no—
35C2×Dic97—21no—
36Dic194—41no—
37#11715897—81no—
38#11715897—81no—
39#11715895—41no—
40C22×C97—41no—
41C22×C97#9911yescharacteristic
42C4×C97—21no—
43C4×C97—41no—
44C4×C97—41no—
45Dic97—21no—
46Dic97—21no—
47Dic97—41no—
48C2×C97—21no—
49C2×C97—81no—
50C2×C97AΣL(1,𝔽16)11yescharacteristic
51#1405—971nomaximal, Sylow
52C97#140511yescharacteristic, complemented, Sylow
53#456—971no—
54#356—971no—
55#333—971no—
562−1+4#8760411yescharacteristic
57#137—1941no—
58C4⋊Q8—971no—
59C4⋊C8—1941no—
60#104—1941no—
61#101—971no—
62#101—1941no—
63#41—21no—
64#41—41no—
65#41—41no—
66C2×Q8—41no—
67C2×Q8—971no—
68C2×Q8C2×#8760411yescharacteristic
69Q16—3881no—
70SD16—3881no—
71M16—3881no—
72M16—1941no—
73C2×C8—3881no—
74C4⋊C4—3881no—
75C42—971no—
76Q8—21no—
77Q8—41no—
78Q8—41no—
79Q8—3881no—
80D4—21no—
81D4—41no—
82D4—41no—
83C2×C4—21no—
84C2×C4—41no—
85C2×C4—42no—
86C2×C4—1941no—
87C2×C4—1941no—
88C2×C4C22⋊#8760411yescharacteristic
89C8—7761no—
90C8—7761no—
91C8—3881no—
92C22—41no—
93C22#1569086811yescharacteristic
94C4—21no—
95C4—41no—
96C4—41no—
97C4—3881no—
98C4—1941no—
99C4—1941no—
100C2—21no—
101C2—81no—
102C2#2026163511yescharacteristic
subgroupG/H
centerC2#20261635
commutatorC2×Q8×C97C2×C4
FrattiniC2×Q8C2×#87604
Fitting, prime radical2−1+4×C97C4
socleC2×C97AΣL(1,𝔽16)
cosocle#15690808C22
hypercenter2−1+4#87604
hypocenterC97#1405
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. #20261645C2semidirect
2. #20261595C2semidirect
52. C97#1405semidirect

Maximal subgroups

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

subgroupindex
#202616452
#202615952
#202609182
#140597