Finite Group Databasebeta

← #27157276

Subgroups — #27157276

Order 12416.

210 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 60 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#20261740C212yesmaximal
2#20261594C214yesmaximal, complemented
3#20261291C211yesmaximal, characteristic
4C2×#13095521—22no—
5#15690877—22no—
6#15690877C2214yes—
7#15690809C2212yes—
8#15690802C411yescharacteristic
9#15690788—22no—
10#15690775—22no—
11#15690768C412yes—
12C42×D97C2211yescharacteristic
13#125×C97C411yescharacteristic
14#13095521—44no—
15C2×#11715897—24no—
16C2×Dic194—22no—
17C2×C4×D97C2×C411yescharacteristic
18C2×C4×D97D412yes—
19C4⋊Dic97C2×C412yes—
20#13095490—22no—
21#13095490—22no—
22#13095488—22no—
23C4×Dic97—22no—
24C4×Dic97D412yes—
25C4×Dic97C2311yescharacteristic
26C4⋊C4×C97—22no—
27C4⋊C4×C97C2×C412yes—
28C42×C97C2×C411yescharacteristic
29C22×D97C22⋊C411yescharacteristic
30C2×C4×C97—22no—
31C2×C4×C97C22⋊C412yes—
32C2×C4×C97C22×C411yescharacteristic
33C2×Dic97—22no—
34C2×Dic97C22⋊C411yescharacteristic
35C2×Dic97C2×D412yes—
36C4×D97—22no—
37C4×D97—22no—
38C4×D97—22no—
39Dic194—42no—
40Dic194—42no—
41#11715897—44no—
42C22×C97C2×C22⋊C411yescharacteristic
43C2×D97—21no—
44C2×D97—22no—
45C4×C97—21no—
46C4×C97—22no—
47C4×C97—42no—
48Dic97—42no—
49Dic97#10112yes—
50Dic97AGL(1,ℤ/8)12yes—
51Dic97#13712yes—
52C2×C97C2×#10111yescharacteristic
53C2×C97#41712yes—
54D97—41no—
55#1561—971nomaximal, Sylow
56C97#156111yescharacteristic, complemented, Sylow
57#533—971no—
58#423—972no—
59#332—974no—
60C2×M16—1942no—
61C4⋊Q8—972no—
62#125—971no—
63#125#8760411yescharacteristic
64C22⋊Q8—1942no—
65C4×Q8—1942no—
66#117—972no—
67C2×C42—971no—
68C4⋊C8—974no—
69C4⋊C8—1942no—
70C2×Q8—1942no—
71C22×C4—972no—
72C22×C4—971no—
73M16—3884no—
74C2×C8—1944no—
75C4⋊C4—22no—
76C4⋊C4—972no—
77C4⋊C4—1942no—
78C4⋊C4—1942no—
79C4⋊C4C2×#8760412yes—
80C22⋊C4—1942no—
81C42—972no—
82C42—971no—
83C42—1942no—
84C42C2×#8760411yescharacteristic
85C23—971no—
86Q8—3882no—
87Q8—3882no—
88C2×C4—22no—
89C2×C4—971no—
90C2×C4—972no—
91C2×C4—1942no—
92C2×C4—1942no—
93C2×C4—1942no—
94C2×C4—1942no—
95C2×C4C22⋊#8760412yes—
96C2×C4C22×#8760411yescharacteristic
97C8—3884no—
98C22—1941no—
99C22—1942no—
100C22C2×C22⋊#8760411yescharacteristic
101C4—21no—
102C4—22no—
103C4—42no—
104C4—972no—
105C4—972no—
106C4—972no—
107C4—3882no—
108C2—3881no—
109C2D97⋊#10111yescharacteristic
110C2#2026172612yes—
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#125×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C22⋊C4
cosocleC4×Dic97C23
hypercenter#125#87604
hypocenterC97#1561
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
2. #20261594C2semidirect
56. C97#1561semidirect

Maximal subgroups

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

subgroupindex
#202617402
#202615942
#202612912
#156197