Finite Group Databasebeta

← #28297993

Subgroups — #28297993

Order 13568.

59452 conjugacy classes of subgroups; only the 32039 normal subgroups are shown.

normal subgroups only

This group has 59452 conjugacy classes of subgroups — too many to draw. Only its 32039 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#20289528C2128yesmaximal, complemented
2#20289525C2135yesmaximal, complemented
3C2×#16811458C2135yesmaximal, complemented
4#585×D53C2128yesmaximal, complemented
5C53×H(7,𝔽2)C211yesmaximal, characteristic, complemented
6#16811460C22156yescomplemented
7#16811459C221420yescomplemented
8#16811458C221280yescomplemented
9#16811458C221560yescomplemented
10C2×#13133264C221210yescomplemented
112−1+4×D53C221168yescomplemented
12C2×#13133262C221315yescomplemented
13H(5,𝔽2)×D53C221280yescomplemented
14C22×#11719947C221105yescomplemented
15C2×#41×D53C221210yescomplemented
16C53×#585C22128yescomplemented
17C2×H(5,𝔽2)×C53C22135yescomplemented
18#13133265C231280yescomplemented
19#13133264C2311680yescomplemented
20#13133263C231560yescomplemented
21#13133262C2312520yescomplemented
22C2×#11719949C23170yescomplemented
23C2×#11719945C231630yescomplemented
24C2×#11719947C231630yescomplemented
25C2×#11719947C2311260yescomplemented
26#41×D53C231560yescomplemented
27#41×D53C2311680yescomplemented
28C22×Dic106C231105yescomplemented
29C2×Q8×D53C231210yescomplemented
30C22×C22⋊D53C231210yescomplemented
31C2×D4×D53C231630yescomplemented
32C23×Dic53C23130yescomplemented
33C22×C4×D53C231105yescomplemented
342−1+4×C53C23156yescomplemented
35H(5,𝔽2)×C53C231280yescomplemented
36C2×#41×C53C231210yescomplemented
37C22×D4×C53C231105yescomplemented
38C24×C53C24130yes—
39C23×D53C241105yes—
40#41×C53C241560yes—
41C2×Q8×C53C24170yes—
42C2×D4×C53C241630yes—
43C22×C4×C53C241105yes—
44C2×C22⋊D53C2411260yes—
45C22×Dic53C241105yes—
46C22×Dic53C241210yes—
47#11719949C24156yes—
48#11719949C241280yes—
49Q8×D53C241560yes—
50#11719947C241280yes—
51#11719947C2412520yes—
52D4×D53C2411680yes—
53#11719945C2411680yes—
54C2×D212C241210yes—
55C2×C4×D53C241210yes—
56C2×C4×D53C241630yes—
57C2×Dic106C241630yes—
58C23×C53—830no—
59C23×C53C251105yes—
60C22×D53—8105no—
61C22×D53C251210yes—
62Q8×C53C25156yes—
63D4×C53C251280yes—
64C2×C4×C53C251210yes—
65C22⋊D53C251560yes—
66C2×Dic53C251315yes—
67C2×Dic53C25135yes—
68D212C251168yes—
69C4×D53C25128yes—
70C4×D53C251420yes—
71Dic106C251280yes—
72#63103—531nomaximal, Sylow
73C22×C53—4105no—
74C22×C53C26135yes—
75C2×D53—4210no—
76C2×D53C26128yes—
77C4×C53C26128yes—
78Dic53C26135yes—
79Dic53C2611yescharacteristic
802−1+6—5328no—
81H(7,𝔽2)—5335no—
82H(7,𝔽2)D5311yescharacteristic, complemented
83C2×#585—5328no—
84C2×#585—5335no—
85C2×C53—235no—
86C2×C53C2711yescharacteristic
87D53—228no—
88#585—5356no—
89#585—53280no—
90#585—53560no—
91#585—53420no—
92#585C2×D53128yescomplemented
93C2×2−1+4—53168no—
94C2×2−1+4—53210no—
95C2×H(5,𝔽2)—53280no—
96C2×H(5,𝔽2)—53315no—
97C2×H(5,𝔽2)C2×D53135yescomplemented
98C22×#41—53210no—
99C22×#41—53105no—
100C53#6310311yescharacteristic, complemented, Sylow
1012−1+4—531680no—
1022−1+4—53280no—
1032−1+4C22×D53156yescomplemented
104H(5,𝔽2)—53560no—
105H(5,𝔽2)—532520no—
106H(5,𝔽2)C22×D531280yescomplemented
107C2×#41—53560no—
108C2×#41—531680no—
109C2×#41—5370no—
110C2×#41—53630no—
111C2×#41—531260no—
112C2×#41—53630no—
113C2×#41C22×D531210yescomplemented
114C22×Q8—53210no—
115C22×Q8—53105no—
116C22×D4—53210no—
117C22×D4—53630no—
118C22×D4C22×D531105yescomplemented
119C23×C4—5330no—
120C23×C4—53105no—
121C24—53105no—
122C24C23×D53130yes—
123#41—5356no—
124#41—53280no—
125#41—53280no—
126#41—532520no—
127#41—531680no—
128#41C23×D531560yes—
129C2×Q8—53560no—
130C2×Q8—53630no—
131C2×Q8C23×D53170yes—
132C2×D4—531260no—
133C2×D4—531680no—
134C2×D4—53210no—
135C2×D4C23×D531630yes—
136C22×C4—53105no—
137C22×C4—53210no—
138C22×C4—53210no—
139C22×C4—53630no—
140C22×C4C23×D531105yes—
141C23—830no—
142C23—53210no—
143C23—424105no—
144C23C24×D531105yes—
145Q8—53280no—
146Q8C24×D53156yes—
147D4—53560no—
148D4—53168no—
149D4C24×D531280yes—
150C2×C4—5335no—
151C2×C4—53315no—
152C2×C4—5328no—
153C2×C4—53420no—
154C2×C4C24×D531210yes—
155C22—4105no—
156C22—5328no—
157C22—212210no—
158C22C25×D53135yes—
159C4—531no—
160C4—5335no—
161C4C25×D53128yes—
162C2—235no—
163C2—10628no—
164C2C26×D5311yescharacteristic
subgroupG/H
center, FrattiniC2C26×D53
commutator, socle, cosocleC2×C53C27
FittingC53×H(7,𝔽2)C2
hypercenterH(7,𝔽2)D53
hypocenterC53#63103
LCSG ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ H(7,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #20289528C2semidirect
2. #20289525C2semidirect
3. C2×#16811458C2semidirect
4. #585×D53C2semidirect
5. C53×H(7,𝔽2)C2semidirect
6. #16811460C22semidirect
7. #16811459C22semidirect
8. #16811458C22semidirect
9. #16811458C22semidirect
10. C2×#13133264C22semidirect
11. 2−1+4×D53C22semidirect
12. C2×#13133262C22semidirect
13. H(5,𝔽2)×D53C22semidirect
14. C22×#11719947C22semidirect
15. C2×#41×D53C22semidirect
16. C53×#585C22semidirect
17. C2×H(5,𝔽2)×C53C22semidirect
18. #13133265C23semidirect
19. #13133264C23semidirect
20. #13133263C23semidirect
21. #13133262C23semidirect
22. C2×#11719949C23semidirect
23. C2×#11719945C23semidirect
24. C2×#11719947C23semidirect
25. C2×#11719947C23semidirect
26. #41×D53C23semidirect
27. #41×D53C23semidirect
28. C22×Dic106C23semidirect
29. C2×Q8×D53C23semidirect
30. C22×C22⋊D53C23semidirect
31. C2×D4×D53C23semidirect
32. C23×Dic53C23semidirect
33. C22×C4×D53C23semidirect
34. 2−1+4×C53C23semidirect
35. H(5,𝔽2)×C53C23semidirect
36. C2×#41×C53C23semidirect
37. C22×D4×C53C23semidirect
82. H(7,𝔽2)D53semidirect
92. #585C2×D53semidirect
97. C2×H(5,𝔽2)C2×D53semidirect
100. C53#63103semidirect
103. 2−1+4C22×D53semidirect
106. H(5,𝔽2)C22×D53semidirect
113. C2×#41C22×D53semidirect
118. C22×D4C22×D53semidirect

Maximal subgroups

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

subgroupindex
#202895282
#202895252
C2×#168114582
#585×D532
C53×H(7,𝔽2)2
#6310353