Finite Group Databasebeta

← #24920601

Subgroups — #24920601

Order 10624.

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#19128503C212yesmaximal, complemented
2#19128138C214yesmaximal, complemented
3#19128136C212yesmaximal, complemented
4#19128134C212yesmaximal, complemented
5C2×D1328C211yesmaximal, characteristic, complemented
6D16×D83C212yesmaximal, complemented
7#19128117C211yesmaximal, characteristic, complemented
8#508×C83C211yesmaximal, characteristic, complemented
9#14581838—22no—
10#14581819—24no—
11#14581817—22no—
12#14581815C2214yescomplemented
13C2×D664C2212yescomplemented
14D8×D83C2214yescomplemented
15#14581798C2211yescharacteristic
16#14581735C2214yescomplemented
17D1328C2212yescomplemented
18D1328C2212yescomplemented
19#14581733C2214yescomplemented
20#14581732C2211yescharacteristic, complemented
21#14581731C2212yescomplemented
22C16×D83C2212yescomplemented
23#135×C83C2212yescomplemented
24Q32×C83C2211yescharacteristic, complemented
25SD32×C83C2212yescomplemented
26D16×C83C2211yescharacteristic, complemented
27C2×C16×C83C2211yescharacteristic, complemented
28#13050591—24no—
29#13050590—42no—
30#13050590D411yescharacteristic
31C2×D332—22no—
32C2×D332—42no—
33D4×D83—24no—
34D4×D83—44no—
35#13050577—24no—
36D664—22no—
37D664C2314yes—
38#13050575—24no—
39#13050574D411yescharacteristic
40#13050573D412yes—
41C8×D83C2312yes—
42C83⋊C16C2312yes—
43#41×C83—22no—
44Q16×C83C2312yes—
45SD16×C83—22no—
46D8×C83C2312yes—
47C2×C8×C83C2311yescharacteristic
48C16×C83C2311yescharacteristic
49C16×C83C2311yescharacteristic
50C22×D83—42no—
51C22×D83—44no—
52Q8×C83—22no—
53D4×C83—22no—
54D4×C83—42no—
55C2×C4×C83—42no—
56C2×C4×C83C2×D411yescharacteristic
57C22⋊D83—44no—
58C22⋊D83D812yes—
59D332—24no—
60D332—44no—
61D332—42no—
62D332D811yescharacteristic
63C4×D83—44no—
64C4×D83C2×D412yes—
65Dic166D811yescharacteristic
66C8×C83C2×D411yescharacteristic
67C8×C83C2411yescharacteristic
68C83⋊C8C2×D412yes—
69C22×C83—42no—
70C22×C83C2×D811yescharacteristic
71C2×D83—44no—
72C2×D83—164no—
73C2×D83—162no—
74C2×D83C2×D812yes—
75C4×C83—42no—
76C4×C83C2×D811yescharacteristic
77C4×C83C22×D411yescharacteristic
78Dic83C2×D812yes—
79C2×C83—21no—
80C2×C83—82no—
81C2×C83C22×D811yescharacteristic
82D83—22no—
83D83—84no—
84#3415—831nomaximal, Sylow
85C83#341511yescharacteristic, complemented, Sylow
86#576—832no—
87#509—832no—
88#509—834no—
89#508—832no—
90#508D8311yescharacteristic, complemented
91C2×D16—832no—
92C2×D16—831no—
93#504—831no—
94H(5,𝔽2)—1662no—
95AGL(1,ℤ/8)—1664no—
96AGL(1,ℤ/8)—1662no—
97#135—834no—
98#135C2×D8312yescomplemented
99C2×D8—834no—
100C2×D8—832no—
101#131—831no—
102Q32C2×D8311yescharacteristic, complemented
103SD32—834no—
104SD32C2×D8312yescomplemented
105D16—834no—
106D16—832no—
107D16—832no—
108D16C2×D8311yescharacteristic, complemented
109M32—831no—
110M32—832no—
111C2×C16—832no—
112C2×C16C2×D8311yescharacteristic, complemented
113#41—22no—
114#41—3322no—
115#41—831no—
116#41—1664no—
117C2×D4—3322no—
118C2×D4—3324no—
119C2×D4—1664no—
120C2×D4—1662no—
121Q16C22×D8312yes—
122SD16—22no—
123SD16—1664no—
124D8—834no—
125D8—1664no—
126D8—1662no—
127D8C22×D8312yes—
128M16—831no—
129M16—832no—
130C2×C8—832no—
131C2×C8C22×D8311yescharacteristic
132C16—832no—
133C16C22×D8311yescharacteristic
134C16C22×D8311yescharacteristic
135C23—3324no—
136C23—3322no—
137Q8—22no—
138Q8—831no—
139D4—22no—
140D4—42no—
141D4—3324no—
142D4—3324no—
143D4—3322no—
144D4—832no—
145D4—831no—
146D4—1664no—
147C2×C4—42no—
148C2×C4—3324no—
149C2×C4—832no—
150C2×C4D4×D8311yescharacteristic
151C8—832no—
152C8D4×D8311yescharacteristic
153C8C23×D8311yescharacteristic
154C22—42no—
155C22—13282no—
156C22—13284no—
157C22—3324no—
158C22—832no—
159C22D8×D8311yescharacteristic
160C4—42no—
161C4—832no—
162C4D8×D8311yescharacteristic
163C4C2×D4×D8311yescharacteristic
164C2—21no—
165C2—82no—
166C2—6644no—
167C2—1662no—
168C2C2×D8×D8311yescharacteristic
subgroupG/H
centerC2C2×D8×D83
commutator, cosocleC8×C83C24
FrattiniC8C23×D83
Fitting#508×C83C2
socleC2×C83C22×D8
hypercenter#508D83
hypocenterC83#3415
LCSG ▷ C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #19128503C2semidirect
2. #19128138C2semidirect
3. #19128136C2semidirect
4. #19128134C2semidirect
5. C2×D1328C2semidirect
6. D16×D83C2semidirect
7. #19128117C2semidirect
8. #508×C83C2semidirect
12. #14581815C22semidirect
13. C2×D664C22semidirect
14. D8×D83C22semidirect
16. #14581735C22semidirect
17. D1328C22semidirect
18. D1328C22semidirect
19. #14581733C22semidirect
20. #14581732C22semidirect
21. #14581731C22semidirect
22. C16×D83C22semidirect
23. #135×C83C22semidirect
24. Q32×C83C22semidirect
25. SD32×C83C22semidirect
26. D16×C83C22semidirect
27. C2×C16×C83C22semidirect
85. C83#3415semidirect
90. #508D83semidirect
98. #135C2×D83semidirect
102. Q32C2×D83semidirect
104. SD32C2×D83semidirect
108. D16C2×D83semidirect
112. C2×C16C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191285032
#191281382
#191281362
#191281342
C2×D13282
D16×D832
#191281172
#508×C832
#341583