Finite Group Databasebeta

← #38742011

Subgroups — #38742011

Order 25216.

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

normal subgroups only

This group has 206 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#27161829C211yesmaximal, characteristic, complemented
2#27161469C211yesmaximal, characteristic
3#27161444C212yesmaximal, complemented
4#27160820C212yesmaximal, complemented
5#329×C197C211yesmaximal, characteristic, complemented
6C2×#15691345—21no—
7C2×#15691343—21no—
8#20263324—21no—
9#20263323C2211yescharacteristic
10#20263319—22no—
11#20263318C2212yes—
12#20263307—21no—
13#20263303—22no—
14#20263248—22no—
15#20263248—22no—
16#20263237C2211yescharacteristic
17#124×C197C2211yescharacteristic
18#97×C197C2212yes—
19C2×Dic394—21no—
20C2×D788—22no—
21C2×D788—21no—
22C22×Dic197—21no—
23#15691345—42no—
24#15691343—42no—
25C2×#13096802—21no—
26C4⋊Dic197D412yes—
27C4⋊Dic197D412yes—
28#15691337—22no—
29#15691337—22no—
30#15691337—22no—
31C22⋊Dic197—22no—
32C22⋊Dic197—21no—
33C4×Dic197—21no—
34C2×Q8×C197D411yescharacteristic
35C2×D4×C197D411yescharacteristic
36C2×C8×C197—22no—
37C22⋊C4×C197—22no—
38C42×C197C2311yescharacteristic
39C22×D197—41no—
40Dic394—81no—
41D788—41no—
42D788—41no—
43D788—44no—
44C2×Dic197—22no—
45C2×Dic197—22no—
46C2×Dic197—81no—
47#13096802—22no—
48C23×C197—21no—
49Q8×C197—41no—
50D4×C197—41no—
51C2×C4×C197—21no—
52C2×C4×C197C2×D412yes—
53C2×C4×C197C2×D411yescharacteristic
54C8×C197—42no—
55C22×C197—41no—
56C22×C197—42no—
57C22×C197#12011yescharacteristic
58C2×D197—81no—
59C2×D197—82no—
60C4×C197—21no—
61C4×C197—22no—
62C4×C197—41no—
63Dic197—42no—
64Dic197—42no—
65C2×C197—81no—
66C2×C197#45311yescharacteristic
67C2×C197#45412yes—
68D197—161no—
69C197#165711yescharacteristic, complemented, Sylow
70#1657—1971nomaximal, Sylow
71#563—1971no—
72#496—1971no—
73#490—1972no—
74#330—1972no—
75#329D19711yescharacteristic, complemented
76C2×SD16—3941no—
77C2×D8—3941no—
78C4⋊Q8—3941no—
79#127—1971no—
80#126—3942no—
81#125—1972no—
82#124C2×D19711yescharacteristic
83#123—3941no—
84C22⋊Q8—3942no—
85#102—3942no—
86#102—3942no—
87#97—1971no—
88#97C2×D19712yes—
89C2×Q8—3941no—
90C2×Q8D78811yescharacteristic
91C2×D4—3942no—
92C2×D4—3941no—
93C2×D4D78811yescharacteristic
94C22×C4—3941no—
95SD16—7882no—
96D8—7882no—
97C2×C8—22no—
98C2×C8—3941no—
99C4⋊C4—3942no—
100C4⋊C4—3942no—
101C4⋊C4—3942no—
102C4⋊C4—1972no—
103C4⋊C4—1972no—
104C22⋊C4—22no—
105C22⋊C4—3942no—
106C22⋊C4—3941no—
107C42—3941no—
108C42C22×D19711yescharacteristic
109C23—21no—
110C23—7881no—
111Q8—41no—
112Q8—15761no—
113D4—41no—
114D4—7881no—
115D4—7881no—
116D4—7884no—
117C2×C4—21no—
118C2×C4—15761no—
119C2×C4—3942no—
120C2×C4—3942no—
121C2×C4D4×D19712yes—
122C2×C4C2×D78811yescharacteristic
123C8—42no—
124C8—3942no—
125C22—41no—
126C22—42no—
127C22—15761no—
128C22—15762no—
129C22#2026329611yescharacteristic
130C4—21no—
131C4—22no—
132C4—41no—
133C4—7882no—
134C4—7882no—
135C2—81no—
136C2—31521no—
137C2#2716124911yescharacteristic
138C2#2716125312yes—
subgroupG/H
centerC22#20263296
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#329×C197C2
socleC22×C197#120
prime radical#27161469C2
hypercenter#329D197
hypocenterC197#1657
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. #27161829C2semidirect
3. #27161444C2semidirect
4. #27160820C2semidirect
5. #329×C197C2semidirect
69. C197#1657semidirect
75. #329D197semidirect

Maximal subgroups

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

subgroupindex
#271618292
#271614692
#271614442
#271608202
#329×C1972
#1657197