Finite Group Databasebeta

← #38755850

Subgroups — #38755850

Order 25216.

370 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 113 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#27161881C212yesmaximal, complemented
2#27161516C214yesmaximal, complemented
3#27161514C212yesmaximal, complemented
4#27161512C212yesmaximal, complemented
5C2×D3152C211yesmaximal, characteristic, complemented
6D16×D197C212yesmaximal, complemented
7#27161495C211yesmaximal, characteristic, complemented
8#508×C197C211yesmaximal, characteristic, complemented
9#20263372—22no—
10#20263353—24no—
11#20263351—22no—
12#20263349C2214yescomplemented
13C2×D1576C2212yescomplemented
14D8×D197C2214yescomplemented
15#20263332C2211yescharacteristic
16#20263269C2214yescomplemented
17D3152C2212yescomplemented
18D3152C2212yescomplemented
19#20263267C2214yescomplemented
20#20263266C2211yescharacteristic, complemented
21#20263265C2212yescomplemented
22C16×D197C2212yescomplemented
23#135×C197C2212yescomplemented
24Q32×C197C2211yescharacteristic, complemented
25SD32×C197C2212yescomplemented
26D16×C197C2211yescharacteristic, complemented
27C2×C16×C197C2211yescharacteristic, complemented
28#15691359—24no—
29#15691358—42no—
30#15691358D411yescharacteristic
31C2×D788—22no—
32C2×D788—42no—
33D4×D197—24no—
34D4×D197—44no—
35#15691345—24no—
36D1576—22no—
37D1576C2314yes—
38#15691343—24no—
39#15691342D411yescharacteristic
40#15691341D412yes—
41C8×D197C2312yes—
42#15691333C2312yes—
43#41×C197—22no—
44Q16×C197C2312yes—
45SD16×C197—22no—
46D8×C197C2312yes—
47C2×C8×C197C2311yescharacteristic
48C16×C197C2311yescharacteristic
49C16×C197C2311yescharacteristic
50C22×D197—42no—
51C22×D197—44no—
52Dic394D811yescharacteristic
53D788—24no—
54D788—44no—
55D788—42no—
56D788D811yescharacteristic
57C22⋊D197—44no—
58C22⋊D197D812yes—
59C4×D197—44no—
60C4×D197C2×D412yes—
61#13096802C2×D412yes—
62Q8×C197—22no—
63D4×C197—22no—
64D4×C197—42no—
65C2×C4×C197—42no—
66C2×C4×C197C2×D411yescharacteristic
67C8×C197C2×D411yescharacteristic
68C8×C197C2411yescharacteristic
69C22×C197—42no—
70C22×C197C2×D811yescharacteristic
71C2×D197—44no—
72C2×D197—164no—
73C2×D197—162no—
74C2×D197C2×D812yes—
75C4×C197—42no—
76C4×C197C2×D811yescharacteristic
77C4×C197C22×D411yescharacteristic
78Dic197C2×D812yes—
79C2×C197—21no—
80C2×C197—82no—
81C2×C197C22×D811yescharacteristic
82D197—22no—
83D197—84no—
84C197#341511yescharacteristic, complemented, Sylow
85#3415—1971nomaximal, Sylow
86#576—1972no—
87#509—1972no—
88#509—1974no—
89#508—1972no—
90#508D19711yescharacteristic, complemented
91C2×D16—1972no—
92C2×D16—1971no—
93#504—1971no—
94H(5,𝔽2)—3942no—
95AGL(1,ℤ/8)—3944no—
96AGL(1,ℤ/8)—3942no—
97#135—1974no—
98#135C2×D19712yescomplemented
99C2×D8—1974no—
100C2×D8—1972no—
101#131—1971no—
102Q32C2×D19711yescharacteristic, complemented
103SD32—1974no—
104SD32C2×D19712yescomplemented
105D16—1974no—
106D16—1972no—
107D16—1972no—
108D16C2×D19711yescharacteristic, complemented
109M32—1971no—
110M32—1972no—
111C2×C16—1972no—
112C2×C16C2×D19711yescharacteristic, complemented
113#41—22no—
114#41—7882no—
115#41—3944no—
116#41—1971no—
117C2×D4—7882no—
118C2×D4—7884no—
119C2×D4—3944no—
120C2×D4—3942no—
121Q16C22×D19712yes—
122SD16—22no—
123SD16—3944no—
124D8—3944no—
125D8—3942no—
126D8—1974no—
127D8C22×D19712yes—
128M16—1971no—
129M16—1972no—
130C2×C8—1972no—
131C2×C8C22×D19711yescharacteristic
132C16—1972no—
133C16C22×D19711yescharacteristic
134C16C22×D19711yescharacteristic
135C23—7884no—
136C23—7882no—
137Q8—22no—
138Q8—1971no—
139D4—22no—
140D4—42no—
141D4—7884no—
142D4—7884no—
143D4—7882no—
144D4—3944no—
145D4—1972no—
146D4—1971no—
147C2×C4—42no—
148C2×C4—7884no—
149C2×C4—1972no—
150C2×C4D4×D19711yescharacteristic
151C8—1972no—
152C8D4×D19711yescharacteristic
153C8C23×D19711yescharacteristic
154C22—42no—
155C22—7884no—
156C22—31522no—
157C22—31524no—
158C22—1972no—
159C22D8×D19711yescharacteristic
160C4—42no—
161C4—1972no—
162C4D8×D19711yescharacteristic
163C4C2×D4×D19711yescharacteristic
164C2—21no—
165C2—82no—
166C2—15764no—
167C2—3942no—
168C2C2×D8×D19711yescharacteristic
subgroupG/H
centerC2C2×D8×D197
commutator, cosocleC8×C197C24
FrattiniC8C23×D197
Fitting#508×C197C2
socleC2×C197C22×D8
hypercenter#508D197
hypocenterC197#3415
LCSG ▷ C8×C197 ▷ C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #27161881C2semidirect
2. #27161516C2semidirect
3. #27161514C2semidirect
4. #27161512C2semidirect
5. C2×D3152C2semidirect
6. D16×D197C2semidirect
7. #27161495C2semidirect
8. #508×C197C2semidirect
12. #20263349C22semidirect
13. C2×D1576C22semidirect
14. D8×D197C22semidirect
16. #20263269C22semidirect
17. D3152C22semidirect
18. D3152C22semidirect
19. #20263267C22semidirect
20. #20263266C22semidirect
21. #20263265C22semidirect
22. C16×D197C22semidirect
23. #135×C197C22semidirect
24. Q32×C197C22semidirect
25. SD32×C197C22semidirect
26. D16×C197C22semidirect
27. C2×C16×C197C22semidirect
84. C197#3415semidirect
90. #508D197semidirect
98. #135C2×D197semidirect
102. Q32C2×D197semidirect
104. SD32C2×D197semidirect
108. D16C2×D197semidirect
112. C2×C16C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271618812
#271615162
#271615142
#271615122
C2×D31522
D16×D1972
#271614952
#508×C1972
#3415197