Finite Group Databasebeta

← #35202170

Subgroups — #35202170

Order 20224.

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#23778213C2128yesmaximal, complemented
2#23778210C2135yesmaximal, complemented
3C2×#19099398C2135yesmaximal, complemented
4#585×D79C2128yesmaximal, complemented
5C79×H(7,𝔽2)C211yesmaximal, characteristic, complemented
6#19099400C22156yescomplemented
7#19099399C221420yescomplemented
8#19099398C221280yescomplemented
9#19099398C221560yescomplemented
10C2×#14579208C221210yescomplemented
112−1+4×D79C221168yescomplemented
12C2×#14579206C221315yescomplemented
13H(5,𝔽2)×D79C221280yescomplemented
14C22×#11929623C221105yescomplemented
15C2×#41×D79C221210yescomplemented
16#585×C79C22128yescomplemented
17C2×H(5,𝔽2)×C79C22135yescomplemented
18#14579209C231280yescomplemented
19#14579208C2311680yescomplemented
20#14579207C231560yescomplemented
21#14579206C2312520yescomplemented
22C2×#11929625C23170yescomplemented
23C2×#11929624C231630yescomplemented
24C2×#11929623C2311260yescomplemented
25C2×#11929623C231630yescomplemented
26#41×D79C2311680yescomplemented
27#41×D79C231560yescomplemented
28C22×Dic158C231105yescomplemented
29C2×Q8×D79C231210yescomplemented
30C22×C22⋊D79C231210yescomplemented
31C2×D4×D79C231630yescomplemented
32C23×Dic79C23130yescomplemented
33C22×C4×D79C231105yescomplemented
342−1+4×C79C23156yescomplemented
35H(5,𝔽2)×C79C231280yescomplemented
36C2×#41×C79C231210yescomplemented
37C22×D4×C79C231105yescomplemented
38C23×D79C241105yes—
39#11929625C24156yes—
40#11929625C241280yes—
41#11929624C2411680yes—
42#11929623C241280yes—
43#11929623C2412520yes—
44C2×Dic158C241630yes—
45Q8×D79C241560yes—
46C2×D316C241210yes—
47C2×C22⋊D79C2411260yes—
48D4×D79C2411680yes—
49C22×Dic79C241210yes—
50C22×Dic79C241105yes—
51C2×C4×D79C241630yes—
52C2×C4×D79C241210yes—
53C24×C79C24130yes—
54#41×C79C241560yes—
55C2×Q8×C79C24170yes—
56C2×D4×C79C241630yes—
57C22×C4×C79C241105yes—
58C23×C79—830no—
59C23×C79C251105yes—
60C22×D79—8105no—
61C22×D79C251210yes—
62Q8×C79C25156yes—
63D4×C79C251280yes—
64C2×C4×C79C251210yes—
65C22⋊D79C251560yes—
66C2×Dic79C25135yes—
67C2×Dic79C251315yes—
68D316C251168yes—
69C4×D79C25128yes—
70C4×D79C251420yes—
71Dic158C251280yes—
72C22×C79—4105no—
73C22×C79C26135yes—
74C2×D79—4210no—
75C2×D79C26128yes—
76C4×C79C26128yes—
77Dic79C26135yes—
78Dic79C2611yescharacteristic
79#63103—791nomaximal, Sylow
80C2×C79—235no—
81C2×C79C2711yescharacteristic
82D79—228no—
832−1+6—7928no—
84H(7,𝔽2)—7935no—
85H(7,𝔽2)D7911yescharacteristic, complemented
86C2×#585—7928no—
87C2×#585—7935no—
88C79#6310311yescharacteristic, complemented, Sylow
89#585—7956no—
90#585—79280no—
91#585—79560no—
92#585—79420no—
93#585C2×D79128yescomplemented
94C2×2−1+4—79168no—
95C2×2−1+4—79210no—
96C2×H(5,𝔽2)—79280no—
97C2×H(5,𝔽2)—79315no—
98C2×H(5,𝔽2)C2×D79135yescomplemented
99C22×#41—79210no—
100C22×#41—79105no—
1012−1+4—791680no—
1022−1+4—79280no—
1032−1+4C22×D79156yescomplemented
104H(5,𝔽2)—79560no—
105H(5,𝔽2)—792520no—
106H(5,𝔽2)C22×D791280yescomplemented
107C2×#41—79560no—
108C2×#41—791680no—
109C2×#41—7970no—
110C2×#41—79630no—
111C2×#41—791260no—
112C2×#41—79630no—
113C2×#41C22×D791210yescomplemented
114C22×Q8—79210no—
115C22×Q8—79105no—
116C22×D4—79210no—
117C22×D4—79630no—
118C22×D4C22×D791105yescomplemented
119C23×C4—7930no—
120C23×C4—79105no—
121C24—79105no—
122C24C23×D79130yes—
123#41—7956no—
124#41—79280no—
125#41—79280no—
126#41—792520no—
127#41—791680no—
128#41C23×D791560yes—
129C2×Q8—79560no—
130C2×Q8—79630no—
131C2×Q8C23×D79170yes—
132C2×D4—791260no—
133C2×D4—791680no—
134C2×D4—79210no—
135C2×D4C23×D791630yes—
136C22×C4—79105no—
137C22×C4—79210no—
138C22×C4—79210no—
139C22×C4—79630no—
140C22×C4C23×D791105yes—
141C23—830no—
142C23—79210no—
143C23—632105no—
144C23C24×D791105yes—
145Q8—79280no—
146Q8C24×D79156yes—
147D4—79560no—
148D4—79168no—
149D4C24×D791280yes—
150C2×C4—7935no—
151C2×C4—79315no—
152C2×C4—7928no—
153C2×C4—79420no—
154C2×C4C24×D791210yes—
155C22—4105no—
156C22—316210no—
157C22—7928no—
158C22C25×D79135yes—
159C4—791no—
160C4—7935no—
161C4C25×D79128yes—
162C2—235no—
163C2—15828no—
164C2C26×D7911yescharacteristic
subgroupG/H
center, FrattiniC2C26×D79
commutator, socle, cosocleC2×C79C27
FittingC79×H(7,𝔽2)C2
hypercenterH(7,𝔽2)D79
hypocenterC79#63103
LCSG ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ H(7,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #23778213C2semidirect
2. #23778210C2semidirect
3. C2×#19099398C2semidirect
4. #585×D79C2semidirect
5. C79×H(7,𝔽2)C2semidirect
6. #19099400C22semidirect
7. #19099399C22semidirect
8. #19099398C22semidirect
9. #19099398C22semidirect
10. C2×#14579208C22semidirect
11. 2−1+4×D79C22semidirect
12. C2×#14579206C22semidirect
13. H(5,𝔽2)×D79C22semidirect
14. C22×#11929623C22semidirect
15. C2×#41×D79C22semidirect
16. #585×C79C22semidirect
17. C2×H(5,𝔽2)×C79C22semidirect
18. #14579209C23semidirect
19. #14579208C23semidirect
20. #14579207C23semidirect
21. #14579206C23semidirect
22. C2×#11929625C23semidirect
23. C2×#11929624C23semidirect
24. C2×#11929623C23semidirect
25. C2×#11929623C23semidirect
26. #41×D79C23semidirect
27. #41×D79C23semidirect
28. C22×Dic158C23semidirect
29. C2×Q8×D79C23semidirect
30. C22×C22⋊D79C23semidirect
31. C2×D4×D79C23semidirect
32. C23×Dic79C23semidirect
33. C22×C4×D79C23semidirect
34. 2−1+4×C79C23semidirect
35. H(5,𝔽2)×C79C23semidirect
36. C2×#41×C79C23semidirect
37. C22×D4×C79C23semidirect
85. H(7,𝔽2)D79semidirect
88. C79#63103semidirect
93. #585C2×D79semidirect
98. C2×H(5,𝔽2)C2×D79semidirect
103. 2−1+4C22×D79semidirect
106. H(5,𝔽2)C22×D79semidirect
113. C2×#41C22×D79semidirect
118. C22×D4C22×D79semidirect

Maximal subgroups

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

subgroupindex
#237782132
#237782102
C2×#190993982
#585×D792
C79×H(7,𝔽2)2
#6310379