Finite Group Databasebeta

← #38742005

Subgroups — #38742005

Order 25216.

230 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 49 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#27161469C212yesmaximal, complemented
2#27161445C211yesmaximal, characteristic, complemented
3#27160821C211yesmaximal, characteristic, complemented
4#27160819C212yesmaximal, complemented
5#562×C197C211yesmaximal, characteristic, complemented
6C2×#15691345—22no—
7C2×#15691343—22no—
8#20263323C2211yescharacteristic
9#20263246—22no—
10#20263237C2211yescharacteristic
11#20263237C2212yes—
12#125×C197C2211yescharacteristic
13#124×C197C2212yes—
14#123×C197—22no—
15C22⋊Q8×C197—22no—
16#121×C197—22no—
17C2×D788—21no—
18C2×D788—22no—
19#15691345—44no—
20#15691343—44no—
21C2×#13096802—22no—
22C2×#13096802—21no—
23C2×Q8×C197D412yes—
24C2×D4×C197—22no—
25C2×D4×C197D412yes—
26C22×C4×C197—22no—
27C4⋊C4×C197—22no—
28C4⋊C4×C197—22no—
29C4⋊C4×C197D412yes—
30C22⋊C4×C197—22no—
31C22⋊C4×C197—22no—
32C22⋊C4×C197—22no—
33C42×C197C2311yescharacteristic
34C22×D197—41no—
35D788—42no—
36D788—42no—
37D788—42no—
38#13096802—24no—
39#13096802—41no—
40C23×C197—22no—
41Q8×C197—42no—
42D4×C197—42no—
43D4×C197—82no—
44C2×C4×C197—22no—
45C2×C4×C197—22no—
46C2×C4×C197—82no—
47C2×C4×C197C2×D411yescharacteristic
48C2×C4×C197C2×D412yes—
49C22×C197—42no—
50C22×C197—42no—
51C22×C197—42no—
52C22×C197#12011yescharacteristic
53C2×D197—82no—
54C2×D197—81no—
55C4×C197—22no—
56C4×C197—21no—
57C4×C197—42no—
58C4×C197—42no—
59C2×C197—82no—
60C2×C197#45312yes—
61C2×C197#45411yescharacteristic
62D197—161no—
63C197#165611yescharacteristic, complemented, Sylow
64#1656—1971nomaximal, Sylow
65#562D19711yescharacteristic, complemented
66#496—1972no—
67#490—1971no—
68#330—1971no—
69#329—1972no—
70C2×SD16—3942no—
71C2×D8—3942no—
72#127—1971no—
73#125C2×D19711yescharacteristic
74#124C2×D19712yes—
75#123—22no—
76C22⋊Q8—22no—
77#121—22no—
78#102—3942no—
79#97—1971no—
80#97—1972no—
81C2×Q8C22⋊D19712yes—
82C2×D4—22no—
83C2×D4—3941no—
84C2×D4—3942no—
85C2×D4C22⋊D19712yes—
86C22×C4—22no—
87SD16—7884no—
88D8—7884no—
89C2×C8—3941no—
90C2×C8—3942no—
91C4⋊C4—22no—
92C4⋊C4—22no—
93C4⋊C4C22⋊D19712yes—
94C22⋊C4—22no—
95C22⋊C4—22no—
96C22⋊C4—22no—
97C42C22×D19711yescharacteristic
98C23—22no—
99C23—7881no—
100Q8—42no—
101D4—42no—
102D4—82no—
103D4—7882no—
104D4—7882no—
105D4—7882no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—82no—
109C2×C4C2×C22⋊D19711yescharacteristic
110C2×C4C2×C22⋊D19712yes—
111C8—7881no—
112C8—3944no—
113C22—42no—
114C22—42no—
115C22—42no—
116C22—15762no—
117C22—15761no—
118C22#2026329511yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—42no—
123C2—82no—
124C2—31521no—
125C2#2716124712yes—
126C2#2716125211yescharacteristic
subgroupG/H
centerC22#20263295
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#562×C197C2
socleC22×C197#120
hypercenter#562D197
hypocenterC197#1656
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #562

Complements

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

NH ≅ G/Ntype
1. #27161469C2semidirect
2. #27161445C2semidirect
3. #27160821C2semidirect
4. #27160819C2semidirect
5. #562×C197C2semidirect
63. C197#1656semidirect
65. #562D197semidirect

Maximal subgroups

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

subgroupindex
#271614692
#271614452
#271608212
#271608192
#562×C1972
#1656197