Finite Group Databasebeta

← #38743235

Subgroups — #38743235

Order 25216.

312 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 93 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#27161846C211yesmaximal, characteristic
2#27161088C212yesmaximal, complemented
3#27161053C211yesmaximal, characteristic
4#27160835C212yesmaximal, complemented
5#536×C197C211yesmaximal, characteristic
6#20263333—24no—
7C2×#15691342—22no—
8C2×#15691342C2212yes—
9#20263285C2212yes—
10#20263250—24no—
11#20263241—22no—
12C2×#41×C197C2211yescharacteristic
13C22×Q8×C197C2211yescharacteristic
14C4⋊Q8×C197C412yescomplemented
15#124×C197C412yescomplemented
16#123×C197—22no—
17C22⋊Q8×C197—22no—
18#117×C197C2211yescharacteristic
19#15691342—24no—
20#15691342—22no—
21#15691342—22no—
22C2×#13096802—24no—
23C2×#13096802D414yes—
24C4⋊Dic197—22no—
25C22⋊Dic197—22no—
26C4×Dic197—22no—
27#41×C197—22no—
28#41×C197—22no—
29C2×Q8×C197—21no—
30C2×Q8×C197—22no—
31C2×Q8×C197—22no—
32C2×Q8×C197C2×C412yes—
33C2×Q8×C197C2×C411yescharacteristic
34C2×D4×C197C2×C411yescharacteristic
35C2×D4×C197D412yes—
36C22×C4×C197—21no—
37C22×C4×C197D412yes—
38C22×C4×C197C2311yescharacteristic
39C4⋊C4×C197—24no—
40C4⋊C4×C197—21no—
41C22⋊C4×C197—24no—
42C22⋊C4×C197—21no—
43C42×C197C2×C412yes—
44C2×Dic197—42no—
45#13096802—24no—
46C23×C197C22⋊C412yes—
47C23×C197C2×D411yescharacteristic
48Q8×C197—22no—
49Q8×C197—24no—
50Q8×C197—41no—
51Q8×C197—41no—
52D4×C197—24no—
53D4×C197—22no—
54C2×C4×C197—22no—
55C2×C4×C197—22no—
56C2×C4×C197—22no—
57C2×C4×C197—22no—
58C2×C4×C197—42no—
59C2×C4×C197C22⋊C412yes—
60C2×C4×C197C22×C411yescharacteristic
61C2×C4×C197C2×D412yes—
62C2×C4×C197C2×D411yescharacteristic
63C2×C4×C197#4112yes—
64C22×C197—22no—
65C22×C197—41no—
66C22×C197—42no—
67C22×C197C2×C22⋊C411yescharacteristic
68C22×C197#12112yes—
69C4×C197—22no—
70C4×C197—42no—
71C4×C197—42no—
72C4×C197C4×D412yes—
73C4×C197#12011yescharacteristic
74C4×C197#12311yescharacteristic
75Dic197—82no—
76C2×C197—22no—
77C2×C197—21no—
78C2×C197—42no—
79C2×C197#38611yescharacteristic
80C197#190911yescharacteristic, complemented, Sylow
81#1909—1971nomaximal, Sylow
82#568—1971no—
83#536D19711yescharacteristic
84#419—1972no—
85#410—1971no—
86H(3,ℤ/4)—1972no—
87C2×#41C2×D19711yescharacteristic
88C22×Q8C2×D19711yescharacteristic
89#131—3944no—
90C2×M16—3942no—
91C2×M16—1972no—
92C4⋊Q8Dic19712yescomplemented
93#124Dic19712yescomplemented
94#123—22no—
95C22⋊Q8—22no—
96#117—1972no—
97#117C2×D19711yescharacteristic
98#103—3944no—
99#99—3942no—
100#41—22no—
101#41—22no—
102C2×Q8—21no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8C2×Dic19712yes—
106C2×Q8C2×Dic19711yescharacteristic
107C2×D4C2×Dic19711yescharacteristic
108C2×D4C22⋊D19712yes—
109C22×C4—21no—
110C22×C4C22⋊D19712yes—
111C22×C4C22×D19711yescharacteristic
112M16—3944no—
113M16—3942no—
114M16—3942no—
115C2×C8—3944no—
116C2×C8—1974no—
117C4⋊C4—24no—
118C4⋊C4—21no—
119C4⋊C4—3942no—
120C22⋊C4—24no—
121C22⋊C4—21no—
122C22⋊C4—3942no—
123C42—3942no—
124C42C2×Dic19712yes—
125C23C22⋊Dic19712yes—
126C23C2×C22⋊D19711yescharacteristic
127Q8—22no—
128Q8—24no—
129Q8—41no—
130Q8—41no—
131D4—24no—
132D4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—42no—
138C2×C4—7882no—
139C2×C4C22⋊Dic19712yes—
140C2×C4C22×Dic19711yescharacteristic
141C2×C4D4×D19712yes—
142C2×C4C2×C22⋊D19711yescharacteristic
143C2×C4#1569135712yes—
144C8—3944no—
145C22—22no—
146C22—41no—
147C22—42no—
148C22C2×C22⋊Dic19711yescharacteristic
149C22#2026329812yes—
150C4—22no—
151C4—42no—
152C4—42no—
153C4—15762no—
154C4D4×Dic19712yes—
155C4#2026329411yescharacteristic
156C4#2026330711yescharacteristic
157C2—21no—
158C2—22no—
159C2—42no—
160C2#2716095911yescharacteristic
subgroupG/H
centerC2#27160959
commutatorC2×C4×C197C22×C4
FrattiniC22×C4C22×D197
Fitting#536×C197C2
socleC2×C197#386
prime radicalC2×#41×C197C22
cosocleC22×C4×C197C23
hypercenter#536D197
hypocenterC197#1909
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #27161088C2semidirect
4. #27160835C2semidirect
14. C4⋊Q8×C197C4semidirect
15. #124×C197C4semidirect
80. C197#1909semidirect
92. C4⋊Q8Dic197semidirect
93. #124Dic197semidirect

Maximal subgroups

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

subgroupindex
#271618462
#271610882
#271610532
#271608352
#536×C1972
#1909197