Finite Group Databasebeta

← D53×H(7,𝔽2) #28297987

Subgroups — D53×H(7,𝔽2) #28297987

Order 13568.

60570 conjugacy classes of subgroups; only the 32041 normal subgroups are shown.

normal subgroups only

This group has 60570 conjugacy classes of subgroups — too many to draw. Only its 32041 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#20289526C2128yesmaximal, complemented
2#20289525C2135yesmaximal, complemented
3#585×D53C2128yesmaximal, complemented
4C2×H(5,𝔽2)×D53C2135yesmaximal, complemented
5C53×H(7,𝔽2)C211yesmaximal, characteristic, complemented
6#16811460C221168yescomplemented
7#16811459C221420yescomplemented
8#16811458C221280yescomplemented
92−1+4×D53C22156yescomplemented
10C2×#13133263C221210yescomplemented
11C2×#13133262C221315yescomplemented
12H(5,𝔽2)×D53C221280yescomplemented
13H(5,𝔽2)×D53C221560yescomplemented
14C2×#41×D53C221210yescomplemented
15C22×D4×D53C221105yescomplemented
16C53×#585C22128yescomplemented
17C2×H(5,𝔽2)×C53C22135yescomplemented
18C24×D53C23130yescomplemented
19#13133265C231280yescomplemented
20#13133264C231560yescomplemented
21#13133263C2311680yescomplemented
22#13133262C2312520yescomplemented
23C2×#11719949C231210yescomplemented
24C2×#11719945C231630yescomplemented
25C2×#11719947C231630yescomplemented
26#41×D53C231560yescomplemented
27#41×D53C2311680yescomplemented
28C2×Q8×D53C23170yescomplemented
29C22×D212C231105yescomplemented
30C22×C22⋊D53C231210yescomplemented
31C2×D4×D53C231630yescomplemented
32C2×D4×D53C2311260yescomplemented
33C22×C4×D53C231105yescomplemented
342−1+4×C53C23156yescomplemented
35H(5,𝔽2)×C53C231280yescomplemented
36C2×#41×C53C231210yescomplemented
37C22×D4×C53C231105yescomplemented
38C24×C53C24130yescomplemented
39C23×D53—860no—
40C23×D53C241105yes—
41C23×D53C241210yescomplemented
42#41×C53C241560yescomplemented
43C2×Q8×C53C24170yescomplemented
44C2×D4×C53C241630yescomplemented
45C22×C4×C53C241105yescomplemented
46C2×C22⋊D53C2411260yescomplemented
47C22×Dic53C241105yescomplemented
48#11719949C241560yescomplemented
49Q8×D53C24156yes—
50Q8×D53C241280yescomplemented
51#11719947C2411680yescomplemented
52D4×D53C241280yes—
53D4×D53C2412520yescomplemented
54#11719945C2411680yescomplemented
55C2×D212C241630yescomplemented
56C2×C4×D53C241210yes—
57C2×C4×D53C241630yescomplemented
58C2×Dic106C241210yescomplemented
59C23×C53—830no—
60C23×C53C251105yes—
61C22×D53—4210no—
62C22×D53—8210no—
63C22×D53C25135yes—
64C22×D53C251315yes—
65Q8×C53C25156yes—
66D4×C53C251280yes—
67C2×C4×C53C251210yes—
68C22⋊D53C251560yes—
69C2×Dic53C251210yes—
70D212C251280yes—
71C4×D53C25128yes—
72C4×D53C251420yes—
73Dic106C251168yes—
74C2×H(7,𝔽2)—531nomaximal, Sylow
75C22×C53—4105no—
76C22×C53C26135yes—
77C2×D53—270no—
78C2×D53—4315no—
79C2×D53C26135yes—
80C2×D53C2611yescharacteristic
81C4×C53C26128yes—
82Dic53C26128yes—
83H(7,𝔽2)—5328no—
84H(7,𝔽2)—5335no—
85H(7,𝔽2)D5311yescharacteristic, direct product factor
86C2×#585—5328no—
87C22×H(5,𝔽2)—5335no—
88C2×C53—235no—
89C2×C53C2711yescharacteristic
90D53—235no—
91D53H(7,𝔽2)12yesdirect product factor
92#585—53168no—
93#585—53280no—
94#585—53420no—
95#585C2×D53128yescomplemented
96C2×2−1+4—5356no—
97C2×H(5,𝔽2)—53280no—
98C2×H(5,𝔽2)—53560no—
99C2×H(5,𝔽2)—53210no—
100C2×H(5,𝔽2)—53315no—
101C2×H(5,𝔽2)C2×D53135yescomplemented
102C22×#41—53210no—
103C23×D4—53105no—
104C53C2×H(7,𝔽2)11yescharacteristic, complemented, Sylow
105C25—5330no—
1062−1+4—53560no—
1072−1+4—53280no—
1082−1+4C22×D53156yescomplemented
109H(5,𝔽2)—531680no—
110H(5,𝔽2)—532520no—
111H(5,𝔽2)C22×D531280yescomplemented
112C2×#41—53560no—
113C2×#41—531680no—
114C2×#41—53210no—
115C2×#41—53630no—
116C2×#41—53630no—
117C2×#41C22×D531210yescomplemented
118C22×Q8—5370no—
119C22×D4—53210no—
120C22×D4—53630no—
121C22×D4—531260no—
122C22×D4—53105no—
123C22×D4C22×D531105yescomplemented
124C23×C4—53105no—
125C24—53210no—
126C24—53105no—
127C24—42460no—
128C24C23×D53130yescomplemented
129#41—53560no—
130#41—531680no—
131#41—531680no—
132#41C23×D531560yescomplemented
133C2×Q8—5356no—
134C2×Q8—53280no—
135C2×Q8—53210no—
136C2×Q8C23×D53170yescomplemented
137C2×D4—531260no—
138C2×D4—53280no—
139C2×D4—532520no—
140C2×D4—53630no—
141C2×D4C23×D531630yescomplemented
142C22×C4—53105no—
143C22×C4—53210no—
144C22×C4—53630no—
145C22×C4C23×D531105yescomplemented
146C23—830no—
147C23—5335no—
148C23—53315no—
149C23—424210no—
150C23—212210no—
151C23C24×D531105yes—
152Q8—53168no—
153Q8C24×D53156yes—
154D4—53560no—
155D4—53280no—
156D4C24×D531280yes—
157C2×C4—53210no—
158C2×C4—5328no—
159C2×C4—53420no—
160C2×C4C24×D531210yes—
161C22—4105no—
162C22—531no—
163C22—5335no—
164C22—10670no—
165C22—212315no—
166C22C25×D53135yes—
167C4—5328no—
168C4C25×D53128yes—
169C2—235no—
170C2—532no—
171C2—10635no—
172C2C26×D5311yescharacteristic
subgroupG/H
center, FrattiniC2C26×D53
commutator, socle, cosocleC2×C53C27
FittingC53×H(7,𝔽2)C2
hypercenterH(7,𝔽2)D53
hypocenterC53C2×H(7,𝔽2)
LCSG ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ H(7,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #20289526C2semidirect
2. #20289525C2semidirect
3. #585×D53C2semidirect
4. C2×H(5,𝔽2)×D53C2semidirect
5. C53×H(7,𝔽2)C2semidirect
6. #16811460C22semidirect
7. #16811459C22semidirect
8. #16811458C22semidirect
9. 2−1+4×D53C22semidirect
10. C2×#13133263C22semidirect
11. C2×#13133262C22semidirect
12. H(5,𝔽2)×D53C22semidirect
13. H(5,𝔽2)×D53C22semidirect
14. C2×#41×D53C22semidirect
15. C22×D4×D53C22semidirect
16. C53×#585C22semidirect
17. C2×H(5,𝔽2)×C53C22semidirect
18. C24×D53C23semidirect
19. #13133265C23semidirect
20. #13133264C23semidirect
21. #13133263C23semidirect
22. #13133262C23semidirect
23. C2×#11719949C23semidirect
24. C2×#11719945C23semidirect
25. C2×#11719947C23semidirect
26. #41×D53C23semidirect
27. #41×D53C23semidirect
28. C2×Q8×D53C23semidirect
29. C22×D212C23semidirect
30. C22×C22⋊D53C23semidirect
31. C2×D4×D53C23semidirect
32. C2×D4×D53C23semidirect
33. C22×C4×D53C23semidirect
34. 2−1+4×C53C23semidirect
35. H(5,𝔽2)×C53C23semidirect
36. C2×#41×C53C23semidirect
37. C22×D4×C53C23semidirect
38. C24×C53C24semidirect
41. C23×D53C24semidirect
42. #41×C53C24semidirect
43. C2×Q8×C53C24semidirect
44. C2×D4×C53C24semidirect
45. C22×C4×C53C24semidirect
46. C2×C22⋊D53C24semidirect
47. C22×Dic53C24semidirect
48. #11719949C24semidirect
50. Q8×D53C24semidirect
51. #11719947C24semidirect
53. D4×D53C24semidirect
54. #11719945C24semidirect
55. C2×D212C24semidirect
57. C2×C4×D53C24semidirect
58. C2×Dic106C24semidirect
85. H(7,𝔽2)D53direct (N × H)
91. D53H(7,𝔽2)direct (N × H)
95. #585C2×D53semidirect
101. C2×H(5,𝔽2)C2×D53semidirect
104. C53C2×H(7,𝔽2)semidirect
108. 2−1+4C22×D53semidirect
111. H(5,𝔽2)C22×D53semidirect
117. C2×#41C22×D53semidirect
123. C22×D4C22×D53semidirect
128. C24C23×D53semidirect
132. #41C23×D53semidirect
136. C2×Q8C23×D53semidirect
141. C2×D4C23×D53semidirect
145. C22×C4C23×D53semidirect

Maximal subgroups

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

subgroupindex
#202895262
#202895252
#585×D532
C2×H(5,𝔽2)×D532
C53×H(7,𝔽2)2
C2×H(7,𝔽2)53