Finite Group Databasebeta

← #38742002

Subgroups — #38742002

Order 25216.

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

normal subgroups only

This group has 262 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, complemented
2#27161469C211yesmaximal, characteristic, complemented
3#27161468C212yesmaximal, complemented
4#27160818C212yesmaximal, complemented
5#329×C197C211yesmaximal, characteristic, complemented
6C2×#15691347—22no—
7C2×#15691345—21no—
8C2×D1576—22no—
9C2×#15691343—21no—
10#20263323C2211yescharacteristic, complemented
11#20263315—21no—
12#20263314C2212yes—
13#20263312—22no—
14#20263296—22no—
15#20263294—21no—
16#20263237C2211yescharacteristic, complemented
17#124×C197C2211yescharacteristic
18#97×C197C2212yescomplemented
19C23×D197—21no—
20C2×Dic394D412yes—
21C2×D788—21no—
22C2×D788—21no—
23C2×D788—22no—
24C2×D788D412yes—
25C2×C22⋊D197—22no—
26#15691347—44no—
27#15691345—42no—
28D1576—44no—
29#15691343—42no—
30C2×#13096802—21no—
31C22⋊Dic197—21no—
32#15691335—22no—
33#15691335—22no—
34#15691335—22no—
35C4×Dic197—21no—
36C2×Q8×C197D411yescharacteristic
37C2×D4×C197D411yescharacteristic
38C2×C8×C197—22no—
39C22⋊C4×C197—22no—
40C42×C197C2311yescharacteristic
41C22×D197—22no—
42C22×D197—41no—
43C22×D197—82no—
44C22×D197—81no—
45Dic394—42no—
46D788—42no—
47D788—41no—
48D788—42no—
49D788—41no—
50D788—42no—
51D788—81no—
52C22⋊D197—82no—
53C2×Dic197—22no—
54#13096802—22no—
55C23×C197—21no—
56Q8×C197—41no—
57D4×C197—41no—
58C2×C4×C197—21no—
59C2×C4×C197C2×D412yes—
60C2×C4×C197C2×D411yescharacteristic
61C8×C197—24no—
62C22×C197—41no—
63C22×C197—42no—
64C22×C197#12011yescharacteristic
65C2×D197—42no—
66C2×D197—42no—
67C2×D197—42no—
68C2×D197—84no—
69C2×D197—82no—
70C2×D197—81no—
71C4×C197—22no—
72C4×C197—21no—
73C4×C197—41no—
74Dic197—42no—
75C2×C197—81no—
76C2×C197#45312yes—
77C2×C197#45311yescharacteristic
78D197—82no—
79D197—161no—
80C197#165511yescharacteristic, complemented, Sylow
81#1655—1971nomaximal, Sylow
82H(3,𝔽4)—1971no—
83#496—1971no—
84#496—1972no—
85#329—1972no—
86#329D19711yescharacteristic, complemented
87C2×SD16—3942no—
88C2×SD16—3941no—
89C2×D8—3942no—
90C2×D8—3941no—
91#127—1971no—
92#124—3942no—
93#124—3941no—
94#124—1972no—
95#124C2×D19711yescharacteristic
96#120—3942no—
97#120—3941no—
98#97—1971no—
99#97C2×D19712yescomplemented
100C24—3941no—
101C2×Q8—1972no—
102C2×Q8D78811yescharacteristic
103C2×D4—3941no—
104C2×D4—3942no—
105C2×D4—3942no—
106C2×D4—3941no—
107C2×D4—1972no—
108C2×D4D78811yescharacteristic
109SD16—7884no—
110SD16—7882no—
111D8—7884no—
112D8—7882no—
113C2×C8—22no—
114C2×C8—3941no—
115C22⋊C4—22no—
116C22⋊C4—3942no—
117C22⋊C4—3941no—
118C22⋊C4—3942no—
119C22⋊C4—3942no—
120C42—3941no—
121C42C22×D19711yescharacteristic
122C23—21no—
123C23—7881no—
124C23—15762no—
125C23—15761no—
126C23—3942no—
127Q8—41no—
128Q8—7882no—
129D4—41no—
130D4—7882no—
131D4—7882no—
132D4—7881no—
133D4—7882no—
134D4—7881no—
135D4—15761no—
136D4—15762no—
137C2×C4—21no—
138C2×C4—3942no—
139C2×C4D4×D19712yes—
140C2×C4C2×D78811yescharacteristic
141C8—24no—
142C8—3942no—
143C22—42no—
144C22—41no—
145C22—7882no—
146C22—7882no—
147C22—7882no—
148C22—15764no—
149C22—15762no—
150C22—15761no—
151C22#2026329611yescharacteristic
152C4—22no—
153C4—21no—
154C4—41no—
155C4—7882no—
156C2—81no—
157C2—15762no—
158C2—31521no—
159C2#2716124812yes—
160C2#2716124911yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#329×C197C2
socleC22×C197#120
hypercenter#329D197
hypocenterC197#1655
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
1. #27161821C2semidirect
2. #27161469C2semidirect
3. #27161468C2semidirect
4. #27160818C2semidirect
5. #329×C197C2semidirect
10. #20263323C22semidirect
16. #20263237C22semidirect
18. #97×C197C22semidirect
80. C197#1655semidirect
86. #329D197semidirect
99. #97C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271618212
#271614692
#271614682
#271608182
#329×C1972
#1655197