Finite Group Databasebeta

← #38742007

Subgroups — #38742007

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#27161468C212yesmaximal, complemented
3#27161445C211yesmaximal, characteristic, complemented
4#27160818C212yesmaximal, complemented
5#330×C197C211yesmaximal, characteristic, complemented
6C2×#15691347—22no—
7C2×D1576—22no—
8#20263323C2211yescharacteristic
9#20263314C2212yes—
10#20263311—22no—
11#20263304—22no—
12#20263301—22no—
13#20263246—22no—
14#20263237C2211yescharacteristic
15#125×C197C2211yescharacteristic
16#97×C197C2212yes—
17C2×Dic394D412yes—
18C2×D788—22no—
19C2×D788—21no—
20C2×D788—22no—
21C2×D788D412yes—
22C2×C4×D197—22no—
23#15691347—44no—
24D1576—44no—
25C2×#13096802—21no—
26#15691337—22no—
27#15691335—22no—
28#15691335—22no—
29#15691335—22no—
30C2×C8×C197—22no—
31C4⋊C4×C197—22no—
32C4⋊C4×C197D412yes—
33C42×C197C2311yescharacteristic
34C22×D197—22no—
35C22×D197—41no—
36Dic394—42no—
37D788—42no—
38D788—42no—
39D788—42no—
40D788—42no—
41D788—82no—
42C2×Dic197—22no—
43C4×D197—82no—
44#13096802—41no—
45C2×C4×C197—22no—
46C2×C4×C197C2×D412yes—
47C2×C4×C197C2×D411yescharacteristic
48C8×C197—24no—
49C22×C197#12011yescharacteristic
50C2×D197—42no—
51C2×D197—42no—
52C2×D197—42no—
53C2×D197—82no—
54C2×D197—81no—
55C4×C197—22no—
56C4×C197—21no—
57C4×C197—42no—
58Dic197—42no—
59C2×C197#45312yes—
60C2×C197#45411yescharacteristic
61D197—82no—
62D197—161no—
63C197#165611yescharacteristic, complemented, Sylow
64#1656—1971nomaximal, Sylow
65#562—1971no—
66#496—1972no—
67#490—1971no—
68#330D19711yescharacteristic, complemented
69#329—1972no—
70C2×SD16—3942no—
71C2×D8—3942no—
72#127—1971no—
73#125C2×D19711yescharacteristic
74#124—1972no—
75#123—3942no—
76C22⋊Q8—3942no—
77#121—3942no—
78#102—3942no—
79#97—1971no—
80#97C2×D19712yes—
81C2×Q8—1972no—
82C2×D4—3942no—
83C2×D4—3942no—
84C2×D4—3941no—
85C2×D4—1972no—
86C22×C4—3942no—
87SD16—7884no—
88D8—7884no—
89C2×C8—22no—
90C2×C8—3941no—
91C4⋊C4—22no—
92C4⋊C4—3942no—
93C4⋊C4D78812yes—
94C22⋊C4—3942no—
95C22⋊C4—3942no—
96C22⋊C4—3942no—
97C42C22×D19711yescharacteristic
98C23—7881no—
99C23—3942no—
100Q8—7882no—
101D4—7882no—
102D4—7882no—
103D4—7882no—
104D4—7882no—
105D4—15762no—
106C2×C4—22no—
107C2×C4—15762no—
108C2×C4—3942no—
109C2×C4D4×D19712yes—
110C2×C4C2×D78811yescharacteristic
111C8—24no—
112C8—7881no—
113C22—7882no—
114C22—7882no—
115C22—7882no—
116C22—15762no—
117C22—15761no—
118C22#2026329611yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—7882no—
123C2—15762no—
124C2—31521no—
125C2#2716124812yes—
126C2#2716125411yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#330×C197C2
socleC22×C197#120
hypercenter#330D197
hypocenterC197#1656
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
1. #27161827C2semidirect
2. #27161468C2semidirect
3. #27161445C2semidirect
4. #27160818C2semidirect
5. #330×C197C2semidirect
63. C197#1656semidirect
68. #330D197semidirect

Maximal subgroups

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

subgroupindex
#271618272
#271614682
#271614452
#271608182
#330×C1972
#1656197