Finite Group Databasebeta

← #38743237

Subgroups — #38743237

Order 25216.

344 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 85 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, complemented
2#27161643C211yesmaximal, characteristic
3#27161083C212yesmaximal, complemented
4#27160835C212yesmaximal, complemented
5#410×C197C211yesmaximal, characteristic, complemented
6C22×D788C2211yescharacteristic
7#20263333—24no—
8C2×#15691342—22no—
9C2×#15691342C2212yes—
10#20263322C412yescomplemented
11#20263315C412yescomplemented
12#20263300—22no—
13#20263294—22no—
14#20263285C2211yescharacteristic
15#20263246—24no—
16C2×#41×C197C2211yescharacteristic
17#117×C197C2212yes—
18#99×C197—22no—
19C23×D197—21no—
20C2×D788—22no—
21C2×D788—21no—
22C2×D788—22no—
23C2×D788C2×C412yes—
24C2×C22⋊D197—24no—
25#15691342—24no—
26#15691342—22no—
27#15691342—22no—
28C2×#13096802—24no—
29C2×#13096802D414yescomplemented
30C4⋊Dic197—21no—
31C22⋊Dic197—21no—
32#15691335—24no—
33C4×Dic197C2×C412yes—
34#41×C197—22no—
35#41×C197—22no—
36C2×Q8×C197C2×C411yescharacteristic
37C2×D4×C197C2×C411yescharacteristic
38C2×D4×C197D412yes—
39C22×C4×C197D412yes—
40C22×C4×C197C2311yescharacteristic
41C4⋊C4×C197—22no—
42C22⋊C4×C197—22no—
43C42×C197—22no—
44C22×D197—22no—
45C22×D197—42no—
46C22×D197—82no—
47D788—24no—
48D788—41no—
49D788—41no—
50C22⋊D197—84no—
51C2×Dic197—22no—
52#13096802—24no—
53C23×C197C22⋊C412yes—
54C23×C197C2×D411yescharacteristic
55Q8×C197—22no—
56D4×C197—24no—
57D4×C197—22no—
58C2×C4×C197—22no—
59C2×C4×C197—22no—
60C2×C4×C197—42no—
61C2×C4×C197C22⋊C412yes—
62C2×C4×C197C22×C411yescharacteristic
63C2×C4×C197C2×D412yes—
64C2×C4×C197C2×D411yescharacteristic
65C2×C4×C197#4112yes—
66C22×C197—22no—
67C22×C197—41no—
68C22×C197—42no—
69C22×C197C2×C22⋊C411yescharacteristic
70C22×C197#12112yes—
71C2×D197—44no—
72C2×D197—42no—
73C2×D197—84no—
74C4×C197—22no—
75C4×C197—82no—
76C4×C197C4×D412yes—
77C4×C197#12011yescharacteristic
78C4×C197#12311yescharacteristic
79Dic197—42no—
80C2×C197—21no—
81C2×C197—22no—
82C2×C197—42no—
83C2×C197#38611yescharacteristic
84D197—82no—
85C197#191011yescharacteristic, complemented, Sylow
86#1910—1971nomaximal, Sylow
87#568—1971no—
88#535—1971no—
89#418—1972no—
90#410D19711yescharacteristic, complemented
91H(3,ℤ/4)—1972no—
92C2×#41C2×D19711yescharacteristic
93C22×D4—1971no—
94#131—3944no—
95C2×M16—3942no—
96C2×M16—1972no—
97#127—1972no—
98#124—1972no—
99#121—3942no—
100#120—3942no—
101#117—1971no—
102#117C2×D19712yes—
103#102—3944no—
104#99—22no—
105C24—3941no—
106#41—22no—
107#41—22no—
108C2×Q8C4×D19711yescharacteristic
109C2×D4—3944no—
110C2×D4—3942no—
111C2×D4—3941no—
112C2×D4—3942no—
113C2×D4—1972no—
114C2×D4C4×D19711yescharacteristic
115C2×D4C22⋊D19712yes—
116C22×C4D78812yes—
117C22×C4C22×D19711yescharacteristic
118M16—3944no—
119M16—3942no—
120M16—3942no—
121C2×C8—3944no—
122C2×C8—1974no—
123C4⋊C4—22no—
124C4⋊C4—3941no—
125C22⋊C4—22no—
126C22⋊C4—3941no—
127C22⋊C4—3944no—
128C42—22no—
129C42—1972no—
130C23—7882no—
131C23—15762no—
132C23—3942no—
133C23#1569133512yes—
134C23C2×C22⋊D19711yescharacteristic
135Q8—22no—
136D4—24no—
137D4—22no—
138D4—7881no—
139D4—7881no—
140D4—15764no—
141D4—3944no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—42no—
145C2×C4—3942no—
146C2×C4#1569133512yes—
147C2×C4C2×C4×D19711yescharacteristic
148C2×C4D4×D19712yes—
149C2×C4C2×D78811yescharacteristic
150C2×C4#1569135712yes—
151C8—3944no—
152C22—22no—
153C22—41no—
154C22—42no—
155C22—7884no—
156C22—7882no—
157C22—15764no—
158C22C2×#1569133511yescharacteristic
159C22#2026329812yes—
160C4—22no—
161C4—82no—
162C4—7882no—
163C4#2026328612yes—
164C4#2026329611yescharacteristic
165C4#2026331011yescharacteristic
166C2—22no—
167C2—21no—
168C2—42no—
169C2—15762no—
170C2#2716095611yescharacteristic
subgroupG/H
centerC2#27160956
commutatorC2×C4×C197C22×C4
FrattiniC22×C4C22×D197
Fitting#410×C197C2
socleC2×C197#386
prime radical#27161643C2
cosocleC22×C4×C197C23
hypercenter#410D197
hypocenterC197#1910
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #27161846C2semidirect
3. #27161083C2semidirect
4. #27160835C2semidirect
5. #410×C197C2semidirect
10. #20263322C4semidirect
11. #20263315C4semidirect
29. C2×#13096802D4semidirect
85. C197#1910semidirect
90. #410D197semidirect

Maximal subgroups

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

subgroupindex
#271618462
#271616432
#271610832
#271608352
#410×C1972
#1910197