Finite Group Databasebeta

← #38740865

Subgroups — #38740865

Order 25216.

106 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 106 conjugacy classes of subgroups — too many to draw. Only its 23 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#27161261C211yesmaximal, characteristic
2#27160828C211yesmaximal, characteristic, complemented
3#356×C197C211yesmaximal, characteristic, complemented
4#20263374C411yescharacteristic
5#20263357—21no—
6C4⋊#13096802—21no—
7#20263254—21no—
8#20263245C411yescharacteristic
9C4⋊Q8×C197C2211yescharacteristic
10#101×C197—21no—
11#15691359—41no—
12#15691358—21no—
13#15691358—41no—
14Q8×D197—41no—
15#15691348—41no—
16#15691345—41no—
17#15691342—21no—
18C2×#13096802—41no—
19C2×Q8×C197C2×C411yescharacteristic
20C2×Q8×C197D411yescharacteristic
21M16×C197—41no—
22C4⋊C4×C197—41no—
23C42×C197D411yescharacteristic
24Dic394—21no—
25Dic394—41no—
26D788—21no—
27D788—41no—
28C22⋊D197—41no—
29C4×D197—41no—
30C4×D197—42no—
31#13096802—41no—
32#13096802—81no—
33Q8×C197—41no—
34Q8×C197—41no—
35C2×C4×C197—21no—
36C2×C4×C197—21no—
37C2×C4×C197—21no—
38C2×C4×C197C22⋊C411yescharacteristic
39C8×C197—81no—
40C22×C197#9911yescharacteristic
41C2×D197—41no—
42C4×C197—21no—
43C4×C197—21no—
44C4×C197—21no—
45C4×C197—41no—
46C4×C197—41no—
47Dic197—41no—
48C2×C197—21no—
49C2×C197AΣL(1,𝔽16)11yescharacteristic
50D197—81no—
51C197#140511yescharacteristic, complemented, Sylow
52#1405—1971nomaximal, Sylow
53#456—1971no—
54#356D19711yescharacteristic, complemented
55#333—1971no—
562−1+4—1971no—
57#137—3941no—
58C4⋊Q8C2×D19711yescharacteristic
59C4⋊C8—3941no—
60#104—3941no—
61#101—21no—
62#101—1971no—
63#41—7881no—
64#41—7881no—
65#41—3941no—
66C2×Q8—7881no—
67C2×Q8C4×D19711yescharacteristic
68C2×Q8C22⋊D19711yescharacteristic
69Q16—7881no—
70SD16—7881no—
71M16—41no—
72M16—3941no—
73C2×C8—7881no—
74C4⋊C4—41no—
75C42D78811yescharacteristic
76Q8—41no—
77Q8—41no—
78Q8—7881no—
79Q8—3941no—
80D4—7881no—
81D4—7881no—
82D4—3941no—
83C2×C4—21no—
84C2×C4—21no—
85C2×C4—21no—
86C2×C4—7881no—
87C2×C4—7882no—
88C2×C4#1569133511yescharacteristic
89C8—81no—
90C8—7881no—
91C8—15761no—
92C22—7881no—
93C22#2026324011yescharacteristic
94C4—21no—
95C4—21no—
96C4—21no—
97C4—41no—
98C4—41no—
99C4—7881no—
100C2—21no—
101C2—15761no—
102C2#2716086511yescharacteristic
subgroupG/H
centerC2#27160865
commutatorC2×Q8×C197C2×C4
FrattiniC4⋊Q8C2×D197
Fitting#356×C197C2
socleC2×C197AΣL(1,𝔽16)
prime radical#20263374C4
cosocleC4⋊Q8×C197C22
hypercenter#356D197
hypocenterC197#1405
LCSG ▷ C2×Q8×C197 ▷ C2×C4×C197 ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
2. #27160828C2semidirect
3. #356×C197C2semidirect
51. C197#1405semidirect
54. #356D197semidirect

Maximal subgroups

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

subgroupindex
#271612612
#271608282
#356×C1972
#1405197