Finite Group Databasebeta

← #27156860

Subgroups — #27156860

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#20261641C211yesmaximal, characteristic, complemented
2#20261591C211yesmaximal, characteristic, complemented
3#20260910C211yesmaximal, characteristic
4#15690869—21no—
5#15690865—21no—
6#15690858C411yescharacteristic, complemented
7#15690838—21no—
8#15690799C2211yescharacteristic
9#15690736—21no—
10#101×C97C411yescharacteristic, complemented
11C2×#11715897—41no—
12C22⋊#87604—41no—
13#13095512—41no—
14#13095511—21no—
15#13095511—41no—
16Q8×D97—41no—
17C2×D388D411yescharacteristic
18#13095500—41no—
19D776—41no—
20#13095488—41no—
21C4×Dic97D411yescharacteristic
22C2×Q8×C97C2×C411yescharacteristic
23M16×C97—21no—
24C22×D97—21no—
25C2×#87604—81no—
26Q8×C97—41no—
27C2×C4×C97—21no—
28C2×C4×C97C22⋊C411yescharacteristic
29C22⋊D97—41no—
30C2×Dic97—21no—
31D388—21no—
32D388—41no—
33D388—41no—
34C4×D97—41no—
35C4×D97—42no—
36Dic194—21no—
37Dic194—41no—
38#11715897—42no—
39C8×C97—41no—
40C22×C97#9911yescharacteristic
41C2×D97—41no—
42C2×D97—41no—
43C2×D97—81no—
44#87604—161no—
45C4×C97—21no—
46C4×C97—41no—
47Dic97—41no—
48Dic97—41no—
49C2×C97—21no—
50C2×C97AΣL(1,𝔽16)11yescharacteristic
51D97—81no—
52D97—81no—
53#1404—971nomaximal, Sylow
54C97#140411yescharacteristic, complemented, Sylow
55#454—971no—
56#354—971no—
57#329—971no—
582−1+4—971no—
59AGL(1,ℤ/8)—1941no—
60#124—971no—
61#104—1941no—
62#101#8760411yescharacteristic, complemented
63#99—1941no—
64#97—1941no—
65#41—3881no—
66#41—3881no—
67#41—1941no—
68C2×Q8—3881no—
69C2×Q8C2×#8760411yescharacteristic
70C2×D4—971no—
71SD16—3881no—
72D8—3881no—
73M16—21no—
74C2×C8—3881no—
75C22⋊C4—3881no—
76C22⋊C4—3881no—
77C42—971no—
78C23—1941no—
79Q8—41no—
80Q8—3881no—
81Q8—1941no—
82D4—3881no—
83D4—3881no—
84D4—3881no—
85D4—1941no—
86C2×C4—21no—
87C2×C4—7761no—
88C2×C4—3881no—
89C2×C4—3882no—
90C2×C4—1941no—
91C2×C4C22⋊#8760411yescharacteristic
92C8—41no—
93C8—3882no—
94C22—7761no—
95C22—3881no—
96C22—3881no—
97C22#1569086911yescharacteristic
98C4—21no—
99C4—41no—
100C4—15521no—
101C4—3881no—
102C4—3881no—
103C2—21no—
104C2—7761no—
105C2—7761no—
106C2#2026163611yescharacteristic
subgroupG/H
centerC2#20261636
commutatorC2×Q8×C97C2×C4
FrattiniC2×Q8C2×#87604
Fitting#101×C97C4
socleC2×C97AΣL(1,𝔽16)
prime radical#20260910C2
cosocle#15690799C22
hypercenter#101#87604
hypocenterC97#1404
LCSG ▷ C2×Q8×C97 ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C2×Q8 ◁ #101

Complements

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

NH ≅ G/Ntype
1. #20261641C2semidirect
2. #20261591C2semidirect
6. #15690858C4semidirect
10. #101×C97C4semidirect
54. C97#1404semidirect
62. #101#87604semidirect

Maximal subgroups

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

subgroupindex
#202616412
#202615912
#202609102
#140497