Finite Group Databasebeta

← #38742004

Subgroups — #38742004

Order 25216.

230 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 45 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#27161827C211yesmaximal, characteristic, complemented
2#27161466C212yesmaximal, complemented
3#27160821C211yesmaximal, characteristic, complemented
4#27160818C212yesmaximal, complemented
5#490×C197C211yesmaximal, characteristic, complemented
6C2×#15691346—22no—
7C2×#15691343—22no—
8#20263314C2212yes—
9#20263311—22no—
10#20263304—22no—
11#20263301—22no—
12#20263237C2212yes—
13#127×C197C2211yescharacteristic
14#125×C197C2211yescharacteristic
15#102×C197—22no—
16#97×C197C2211yescharacteristic
17C2×Dic394D412yes—
18C2×D788—22no—
19C2×D788D412yes—
20C2×C4×D197—22no—
21#15691346—44no—
22#15691343—44no—
23C2×#13096802—22no—
24#15691337—22no—
25#15691335—22no—
26#15691335—22no—
27#15691335—22no—
28C2×D4×C197—21no—
29C2×D4×C197—22no—
30C2×C8×C197—21no—
31C4⋊C4×C197—22no—
32C4⋊C4×C197D412yes—
33C42×C197C2311yescharacteristic
34C22×D197—22no—
35Dic394—42no—
36D788—42no—
37D788—82no—
38C2×Dic197—22no—
39C4×D197—82no—
40#13096802—24no—
41C23×C197—41no—
42D4×C197—42no—
43D4×C197—42no—
44D4×C197—42no—
45C2×C4×C197—22no—
46C2×C4×C197C2×D412yes—
47C2×C4×C197C2×D411yescharacteristic
48C8×C197—41no—
49C22×C197—82no—
50C22×C197—81no—
51C22×C197#12011yescharacteristic
52C2×D197—42no—
53C2×D197—42no—
54C2×D197—42no—
55C4×C197—22no—
56C4×C197—21no—
57C4×C197—42no—
58Dic197—42no—
59C2×C197—161no—
60C2×C197#45312yes—
61C2×C197#45411yescharacteristic
62D197—82no—
63C197#165611yescharacteristic, complemented, Sylow
64#1656—1971nomaximal, Sylow
65#562—1971no—
66#496—1972no—
67#490D19711yescharacteristic, complemented
68#330—1971no—
69#329—1972no—
70C2×SD16—3942no—
71C2×D8—3942no—
72#127C2×D19711yescharacteristic
73#125C2×D19711yescharacteristic
74#124—1972no—
75#123—3942no—
76C22⋊Q8—3942no—
77#121—3942no—
78#102—22no—
79#97—1972no—
80#97C2×D19711yescharacteristic
81C2×Q8—1972no—
82C2×D4—22no—
83C2×D4—21no—
84C2×D4—3942no—
85C2×D4—1972no—
86C22×C4—3942no—
87SD16—7884no—
88D8—7884no—
89C2×C8—21no—
90C2×C8—3942no—
91C4⋊C4—22no—
92C4⋊C4—3942no—
93C4⋊C4C22⋊D19712yes—
94C22⋊C4—3942no—
95C22⋊C4—3942no—
96C22⋊C4—3942no—
97C42C22×D19711yescharacteristic
98C23—41no—
99C23—3942no—
100Q8—7882no—
101D4—42no—
102D4—42no—
103D4—42no—
104D4—7882no—
105D4—15762no—
106C2×C4—22no—
107C2×C4—15762no—
108C2×C4—3942no—
109C2×C4D4×D19712yes—
110C2×C4C2×C22⋊D19711yescharacteristic
111C8—41no—
112C8—3944no—
113C22—82no—
114C22—81no—
115C22—7882no—
116C22—7882no—
117C22—7882no—
118C22#2026329411yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—7882no—
123C2—161no—
124C2—15762no—
125C2#2716124512yes—
126C2#2716125111yescharacteristic
subgroupG/H
centerC22#20263294
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#490×C197C2
socleC22×C197#120
hypercenter#490D197
hypocenterC197#1656
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #27161827C2semidirect
2. #27161466C2semidirect
3. #27160821C2semidirect
4. #27160818C2semidirect
5. #490×C197C2semidirect
63. C197#1656semidirect
67. #490D197semidirect

Maximal subgroups

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

subgroupindex
#271618272
#271614662
#271608212
#271608182
#490×C1972
#1656197