Finite Group Databasebeta

← #27156710

Subgroups — #27156710

Order 12416.

110 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 110 conjugacy classes of subgroups — too many to draw. Only its 26 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#20261632C211yesmaximal, characteristic, complemented
2#20261604C211yesmaximal, characteristic, complemented
3#20260985C211yesmaximal, characteristic
4D97⋊M16—21no—
5#15690884—21no—
6C2×#13095494C2211yescharacteristic
7#15690781—21no—
8#15690742C411yescharacteristic
9#15690731—21no—
10C2×D8×C97C411yescharacteristic
11#13095521—41no—
12#13095520—41no—
13#13095519—22no—
14#13095514—41no—
15C2×C22⋊D97—41no—
16C2×C4×D97D411yescharacteristic
17#13095494—41no—
18C2×#11715895D411yescharacteristic
19C4⋊Dic97—21no—
20C22⋊Dic97—41no—
21C2×D4×C97—21no—
22D8×C97—41no—
23C2×C8×C97C2×C411yescharacteristic
24C23×C97—41no—
25C22×D97SD1611yescharacteristic
26D4×C97—41no—
27D4×C97—41no—
28C2×C4×C97C22⋊C411yescharacteristic
29C22⋊D97—82no—
30C2×Dic97—41no—
31C2×Dic97D811yescharacteristic
32C4×D97—21no—
33C4×D97—21no—
34#11715897—42no—
35C8×C97—21no—
36#11715895—21no—
37C22×C97—81no—
38C22×C97—81no—
39C22×C97—81no—
40C22×C97#10211yescharacteristic
41C2×D97—21no—
42C2×D97—41no—
43C4×C97#9911yescharacteristic
44C4×C97#10411yescharacteristic
45Dic97—21no—
46Dic97—81no—
47C2×C97—21no—
48C2×C97—161no—
49C2×C97#32711yescharacteristic
50D97—41no—
51#1341—971nomaximal, Sylow
52C97#134111yescharacteristic, complemented, Sylow
53#469—971no—
54#349—971no—
55#338—971no—
56C2×D8#8760411yescharacteristic
57C2×M16—971no—
58C2×M16—1941no—
59#121—1941no—
60M32—1941no—
61#106—971no—
62#102—1941no—
63C2×D4—21no—
64C2×D4—3881no—
65C22×C4—971no—
66D8—41no—
67M16—3881no—
68M16—3881no—
69M16—3881no—
70C2×C8—971no—
71C2×C8—1942no—
72C2×C8C2×#8760411yescharacteristic
73C4⋊C4—1941no—
74C22⋊C4—3881no—
75C16—3881no—
76C23—41no—
77C23—971no—
78D4—41no—
79D4—41no—
80D4—7762no—
81C2×C4—971no—
82C2×C4—3881no—
83C2×C4—1941no—
84C2×C4—1941no—
85C2×C4C22⋊#8760411yescharacteristic
86C8—21no—
87C8—3882no—
88C8—1941no—
89C22—81no—
90C22—81no—
91C22—81no—
92C22—3881no—
93C22—1941no—
94C22#1569087211yescharacteristic
95C4—7761no—
96C4—1941no—
97C4#1569086811yescharacteristic
98C4#1569087411yescharacteristic
99C2—21no—
100C2—161no—
101C2—3881no—
102C2#2026158611yescharacteristic
subgroupG/H
centerC2#20261586
commutatorC2×C8×C97C2×C4
FrattiniC2×C8C2×#87604
FittingC2×D8×C97C4
socleC2×C97#327
prime radical#20260985C2
cosocleC2×#13095494C22
hypercenterC2×D8#87604
hypocenterC97#1341
LCSG ▷ C2×C8×C97 ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×D8

Complements

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

NH ≅ G/Ntype
1. #20261632C2semidirect
2. #20261604C2semidirect
52. C97#1341semidirect

Maximal subgroups

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

subgroupindex
#202616322
#202616042
#202609852
#134197