Finite Group Databasebeta

← #38740856

Subgroups — #38740856

Order 25216.

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

normal subgroups only

This group has 122 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#27161246C211yesmaximal, characteristic
2#27160823C211yesmaximal, characteristic, complemented
3#353×C197C211yesmaximal, characteristic, complemented
4#20263371C411yescharacteristic, complemented
5#20263352—21no—
6C4⋊#13096802—21no—
7#20263254—21no—
8#20263243C411yescharacteristic, complemented
9#127×C197C2211yescharacteristic
10#99×C197—21no—
11#15691358—21no—
12#15691357—41no—
13C2×C22⋊D197—41no—
14D4×D197—41no—
15#15691346—41no—
16#15691343—41no—
17#15691342—21no—
18C2×#13096802—41no—
19C2×D4×C197—41no—
20C2×D4×C197C2×C411yescharacteristic
21C2×D4×C197D411yescharacteristic
22C22⋊C4×C197—41no—
23C42×C197D411yescharacteristic
24C22×D197—41no—
25Dic394—21no—
26D788—21no—
27C22⋊D197—42no—
28C22⋊D197—41no—
29C2×Dic197—41no—
30C4×D197—41no—
31#13096802—41no—
32#13096802—81no—
33C23×C197—21no—
34C23×C197—21no—
35D4×C197—41no—
36D4×C197—41no—
37D4×C197—81no—
38D4×C197—81no—
39C2×C4×C197—21no—
40C2×C4×C197—81no—
41C2×C4×C197C22⋊C411yescharacteristic
42C22×C197—41no—
43C22×C197—41no—
44C22×C197—81no—
45C22×C197—81no—
46C22×C197#9911yescharacteristic
47C2×D197—41no—
48C2×D197—161no—
49C4×C197—21no—
50C4×C197—21no—
51C4×C197—21no—
52C4×C197—161no—
53Dic197—41no—
54C2×C197—21no—
55C2×C197—81no—
56C2×C197—81no—
57C2×C197AΣL(1,𝔽16)11yescharacteristic
58D197—81no—
59C197#140211yescharacteristic, complemented, Sylow
60#1402—1971nomaximal, Sylow
61#453—1971no—
62#353D19711yescharacteristic, complemented
63#331—1971no—
64H(5,𝔽2)—1971no—
65AGL(1,ℤ/8)—3941no—
66#127C2×D19711yescharacteristic
67C4⋊C8—3941no—
68#104—3941no—
69#100—1971no—
70#99—21no—
71#41—7881no—
72#41—3941no—
73C2×D4—41no—
74C2×D4—7881no—
75C2×D4—7881no—
76C2×D4C4×D19711yescharacteristic
77C2×D4C22⋊D19711yescharacteristic
78SD16—7881no—
79D8—7881no—
80M16—3941no—
81C2×C8—7881no—
82C22⋊C4—41no—
83C42D78811yescharacteristic
84C23—21no—
85C23—21no—
86C23—7881no—
87Q8—3941no—
88D4—41no—
89D4—41no—
90D4—81no—
91D4—81no—
92D4—7882no—
93D4—7881no—
94D4—3941no—
95C2×C4—21no—
96C2×C4—81no—
97C2×C4—7881no—
98C2×C4—7881no—
99C2×C4#1569133511yescharacteristic
100C8—7881no—
101C8—15761no—
102C22—41no—
103C22—41no—
104C22—81no—
105C22—81no—
106C22—7881no—
107C22—31521no—
108C22#2026324011yescharacteristic
109C4—21no—
110C4—21no—
111C4—21no—
112C4—161no—
113C4—7881no—
114C2—21no—
115C2—81no—
116C2—81no—
117C2—15761no—
118C2#2716086511yescharacteristic
subgroupG/H
centerC2#27160865
commutatorC2×D4×C197C2×C4
Frattini#127C2×D197
Fitting#353×C197C2
socleC2×C197AΣL(1,𝔽16)
prime radical#27161246C2
cosocle#127×C197C22
hypercenter#353D197
hypocenterC197#1402
LCSG ▷ C2×D4×C197 ▷ C2×C4×C197 ▷ C22×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #127 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #27160823C2semidirect
3. #353×C197C2semidirect
4. #20263371C4semidirect
8. #20263243C4semidirect
59. C197#1402semidirect
62. #353D197semidirect

Maximal subgroups

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

subgroupindex
#271612462
#271608232
#353×C1972
#1402197