Finite Group Databasebeta

← #38742016

Subgroups — #38742016

Order 25216.

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

normal subgroups only

This group has 228 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#27161821C211yesmaximal, characteristic
2#27161441C212yesmaximal, complemented
3#27160819C211yesmaximal, characteristic, complemented
4#27160818C212yesmaximal, complemented
5#497×C197C211yesmaximal, characteristic, complemented
6#20263315—21no—
7#20263314C2212yes—
8#20263312—22no—
9#20263296—22no—
10#20263294—21no—
11#20263249—22no—
12#20263246—22no—
13#20263237C2212yescomplemented
14C2×Q16×C197—21no—
15C2×SD16×C197—21no—
16C4⋊Q8×C197C2211yescharacteristic, complemented
17#124×C197C2211yescharacteristic
18#97×C197C2211yescharacteristic, complemented
19C23×D197—21no—
20C2×Dic394D412yes—
21C2×D788—21no—
22C2×D788D412yes—
23C2×C22⋊D197—22no—
24C2×#13096802—22no—
25C22⋊Dic197—21no—
26#15691335—22no—
27#15691335—22no—
28#15691335—22no—
29C4×Dic197—21no—
30C2×Q8×C197—21no—
31C2×Q8×C197D411yescharacteristic
32C2×D4×C197D411yescharacteristic
33Q16×C197—42no—
34SD16×C197—42no—
35C2×C8×C197—21no—
36C4⋊C4×C197—22no—
37C22⋊C4×C197—22no—
38C42×C197C2311yescharacteristic
39C22×D197—22no—
40C22×D197—81no—
41C22×D197—82no—
42Dic394—42no—
43D788—42no—
44D788—81no—
45C22⋊D197—82no—
46C2×Dic197—22no—
47#13096802—42no—
48C23×C197—21no—
49Q8×C197—41no—
50Q8×C197—41no—
51Q8×C197—41no—
52D4×C197—41no—
53C2×C4×C197—21no—
54C2×C4×C197—41no—
55C2×C4×C197C2×D412yes—
56C2×C4×C197C2×D411yescharacteristic
57C8×C197—22no—
58C22×C197—41no—
59C22×C197—42no—
60C22×C197#12011yescharacteristic
61C2×D197—42no—
62C2×D197—42no—
63C2×D197—42no—
64C2×D197—84no—
65C4×C197—21no—
66C4×C197—22no—
67C4×C197—41no—
68C4×C197—81no—
69Dic197—42no—
70C2×C197—81no—
71C2×C197#45311yescharacteristic
72C2×C197#45512yes—
73D197—82no—
74C197#165911yescharacteristic, complemented, Sylow
75#1659—1971nomaximal, Sylow
76H(3,𝔽4)—1971no—
77#497D19711yescharacteristic, complemented
78#489—1972no—
79#329—1971no—
80#329—1972no—
81C2×Q16—21no—
82C2×SD16—21no—
83C4⋊Q8C2×D19711yescharacteristic, complemented
84#124—3942no—
85#124—3941no—
86#124—1972no—
87#124C2×D19711yescharacteristic
88#120—3942no—
89#120—3941no—
90#103—3942no—
91#102—3942no—
92#97—1972no—
93#97C2×D19711yescharacteristic, complemented
94C24—3941no—
95C2×Q8—21no—
96C2×Q8—1972no—
97C2×Q8C22⋊D19711yescharacteristic
98C2×D4—3941no—
99C2×D4—3942no—
100C2×D4—1972no—
101C2×D4C22⋊D19711yescharacteristic
102Q16—42no—
103SD16—42no—
104C2×C8—21no—
105C2×C8—3942no—
106C4⋊C4—22no—
107C22⋊C4—22no—
108C22⋊C4—3942no—
109C22⋊C4—3941no—
110C22⋊C4—3942no—
111C22⋊C4—3942no—
112C42—3941no—
113C42C22×D19711yescharacteristic
114C23—21no—
115C23—15761no—
116C23—15762no—
117C23—3942no—
118Q8—41no—
119Q8—41no—
120Q8—41no—
121Q8—7882no—
122D4—41no—
123D4—7882no—
124D4—15761no—
125D4—15762no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—3942no—
129C2×C4D4×D19712yes—
130C2×C4C2×C22⋊D19711yescharacteristic
131C8—22no—
132C8—7882no—
133C22—41no—
134C22—42no—
135C22—7882no—
136C22—7882no—
137C22—7882no—
138C22—15764no—
139C22#2026329411yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—81no—
144C4—7882no—
145C2—81no—
146C2—15762no—
147C2#2716124611yescharacteristic
148C2#2716125512yes—
subgroupG/H
centerC22#20263294
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#497×C197C2
socleC22×C197#120
prime radical#27161821C2
hypercenter#497D197
hypocenterC197#1659
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
2. #27161441C2semidirect
3. #27160819C2semidirect
4. #27160818C2semidirect
5. #497×C197C2semidirect
13. #20263237C22semidirect
16. C4⋊Q8×C197C22semidirect
18. #97×C197C22semidirect
74. C197#1659semidirect
77. #497D197semidirect
83. C4⋊Q8C2×D197semidirect
93. #97C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271618212
#271614412
#271608192
#271608182
#497×C1972
#1659197