Finite Group Databasebeta

← #38740862

Subgroups — #38740862

Order 25216.

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

normal subgroups only

This group has 112 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#27161251C211yesmaximal, characteristic
2#27160819C211yesmaximal, characteristic, complemented
3#354×C197C211yesmaximal, characteristic, complemented
4#20263374C411yescharacteristic, complemented
5#20263352—21no—
6#20263254—21no—
7#20263245C411yescharacteristic, complemented
8#20263237—21no—
9#124×C197C2211yescharacteristic
10#99×C197—21no—
11#15691359—41no—
12#15691358—21no—
13#15691358—41no—
14Q8×D197—41no—
15#15691346—41no—
16#15691343—41no—
17#15691342—21no—
18C2×#13096802—41no—
19C2×Q8×C197C2×C411yescharacteristic
20C2×D4×C197D411yescharacteristic
21C22⋊C4×C197—41no—
22C22⋊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—42no—
33C23×C197—21no—
34Q8×C197—41no—
35D4×C197—41no—
36C2×C4×C197—21no—
37C2×C4×C197—21no—
38C2×C4×C197—81no—
39C2×C4×C197C22⋊C411yescharacteristic
40C22×C197—41no—
41C22×C197—81no—
42C22×C197#9911yescharacteristic
43C2×D197—41no—
44C4×C197—21no—
45C4×C197—41no—
46C4×C197—41no—
47C4×C197—161no—
48Dic197—41no—
49C2×C197—21no—
50C2×C197—81no—
51C2×C197AΣL(1,𝔽16)11yescharacteristic
52D197—81no—
53C197#140411yescharacteristic, complemented, Sylow
54#1404—1971nomaximal, Sylow
55#454—1971no—
56#354D19711yescharacteristic, complemented
57#329—1971no—
582−1+4—1971no—
59AGL(1,ℤ/8)—3941no—
60#124C2×D19711yescharacteristic
61#104—3941no—
62#101—1971no—
63#99—21no—
64#97—3941no—
65#41—7881no—
66#41—7881no—
67#41—3941no—
68C2×Q8—7881no—
69C2×Q8C4×D19711yescharacteristic
70C2×D4C22⋊D19711yescharacteristic
71SD16—7881no—
72D8—7881no—
73M16—3941no—
74C2×C8—7881no—
75C22⋊C4—41no—
76C22⋊C4—41no—
77C42D78811yescharacteristic
78C23—21no—
79Q8—41no—
80Q8—7881no—
81Q8—3941no—
82D4—41no—
83D4—7881no—
84D4—7881no—
85D4—3941no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—81no—
89C2×C4—7881no—
90C2×C4—7882no—
91C2×C4#1569133511yescharacteristic
92C8—7881no—
93C8—7882no—
94C22—41no—
95C22—81no—
96C22—7881no—
97C22#2026324011yescharacteristic
98C4—21no—
99C4—41no—
100C4—41no—
101C4—161no—
102C4—7881no—
103C2—21no—
104C2—81no—
105C2—15761no—
106C2#2716086511yescharacteristic
subgroupG/H
centerC2#27160865
commutatorC2×Q8×C197C2×C4
Frattini#124C2×D197
Fitting#354×C197C2
socleC2×C197AΣL(1,𝔽16)
prime radical#27161251C2
cosocle#124×C197C22
hypercenter#354D197
hypocenterC197#1404
LCSG ▷ C2×Q8×C197 ▷ C2×C4×C197 ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #124 ◁ #354

Complements

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

NH ≅ G/Ntype
2. #27160819C2semidirect
3. #354×C197C2semidirect
4. #20263374C4semidirect
7. #20263245C4semidirect
53. C197#1404semidirect
56. #354D197semidirect

Maximal subgroups

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

subgroupindex
#271612512
#271608192
#354×C1972
#1404197