Finite Group Databasebeta

← C22×#22548761 #44565167

Subgroups — C22×#22548761 #44565167

Order 32128.

6560 conjugacy classes of subgroups; only the 3221 normal subgroups are shown.

normal subgroups only

This group has 6560 conjugacy classes of subgroups — too many to draw. Only its 3221 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
1C2×#22548761C2148yesmaximal, direct product factor
2C22×#16860811C212yesmaximal, complemented
3C22×#16860810C213yesmaximal, complemented
4C23×D1004C213yesmaximal, complemented
5C22×D4×D251C216yesmaximal, complemented
6C22×#41×C251C211yesmaximal, characteristic, complemented
7C24×D251C2216yescomplemented
8#22548761C221256yesdirect product factor
9C2×#16860811C22148yescomplemented
10C2×#16860810C22172yescomplemented
11C22×Dic502C2211yescharacteristic, complemented
12C22×D1004C2216yescomplemented
13C22×D1004C2213yescomplemented
14C22×D1004C22172yescomplemented
15C22×C22⋊D251C2216yescomplemented
16C2×D4×D251C221144yescomplemented
17C22×C4×D251C2216yescomplemented
18C2×#41×C251C22124yescomplemented
19C22×Q8×C251C2211yescharacteristic, complemented
20C22×D4×C251C2213yescomplemented
21C23×C4×C251C2213yescomplemented
22C23×D251—424no—
23C23×D251C2316yes—
24C23×D251C23172yescomplemented
25#16860811C231128yescomplemented
26#16860810C231192yescomplemented
27C2×Dic502C23112yescomplemented
28C2×D1004C23172yescomplemented
29C2×D1004C23136yescomplemented
30C2×D1004C231192yescomplemented
31C2×C22⋊D251C23172yescomplemented
32D4×D251C231384yescomplemented
33C22×Dic251C2312yes—
34C2×C4×D251C23172yescomplemented
35C24×C251C2313yes—
36#41×C251C23164yescomplemented
37C2×Q8×C251C23112yescomplemented
38C2×D4×C251C23136yescomplemented
39C22×C4×C251C2313yes—
40C22×C4×C251C2311yescharacteristic
41C22×C4×C251C23136yescomplemented
42C22×D251—224no—
43C22×D251—4144no—
44C22×D251C24136yes—
45C22×D251C24196yescomplemented
46Dic502C24116yescomplemented
47D1004C24196yescomplemented
48D1004C24148yescomplemented
49C22⋊D251C24196yescomplemented
50C2×Dic251C24112yes—
51C4×D251C24196yescomplemented
52C23×C251—212no—
53C23×C251C24118yes—
54C23×C251C2411yescharacteristic
55Q8×C251C24116yescomplemented
56D4×C251C24148yescomplemented
57C2×C4×C251C2416yes—
58C2×C4×C251C24118yes—
59C2×C4×C251C24148yescomplemented
60C22×C251—236no—
61C22×C251H(5,𝔽2)14yescomplemented
62C22×C251C25112yes—
63C22×C251C2513yes—
64C2×D251—272no—
65C2×D251—496no—
66C2×D251C25124yes—
67C4×C251C2514yes—
68C4×C251C25112yes—
69Dic251C2518yes—
70C2×C251—212no—
71C2×C251C2×H(5,𝔽2)16yescomplemented
72C2×C251C2611yescharacteristic
73D251—224no—
74C251C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
75C22×H(5,𝔽2)—2511nomaximal, Sylow
76C2×H(5,𝔽2)—25148no—
77C22×#41—2512no—
78C22×#41—2513no—
79C22×#41D25111yescharacteristic, complemented
80C23×D4—2516no—
81C23×D4—2513no—
82C25—2516no—
83H(5,𝔽2)—251256no—
84C2×#41—25148no—
85C2×#41—25172no—
86C2×#41C2×D251124yescomplemented
87C22×Q8—2511no—
88C22×Q8C2×D25111yescharacteristic, complemented
89C22×D4—2516no—
90C22×D4—251144no—
91C22×D4—2516no—
92C22×D4—2513no—
93C22×D4—25172no—
94C22×D4C2×D25113yescomplemented
95C23×C4—2516no—
96C23×C4C2×D25113yescomplemented
97C24—100424no—
98C24—25172no—
99C24—2516no—
100C24C22×D25113yes—
101#41—251128no—
102#41—251192no—
103#41C22×D251164yescomplemented
104C2×Q8—25112no—
105C2×Q8C22×D251112yescomplemented
106C2×D4—25172no—
107C2×D4—251384no—
108C2×D4—25172no—
109C2×D4—25136no—
110C2×D4—251192no—
111C2×D4C22×D251136yescomplemented
112C22×C4—2512no—
113C22×C4—25172no—
114C22×C4C22×D25113yes—
115C22×C4C22×D25111yescharacteristic
116C22×C4C22×D251136yescomplemented
117C23—212no—
118C23—1004144no—
119C23—50224no—
120C23—25196no—
121C23—25136no—
122C23C23×D251118yes—
123C23C23×D25111yescharacteristic
124Q8—25116no—
125Q8C23×D251116yescomplemented
126D4—25196no—
127D4—25196no—
128D4—25148no—
129D4C23×D251148yescomplemented
130C2×C4—25112no—
131C2×C4—25196no—
132C2×C4C23×D251118yes—
133C2×C4C23×D25116yes—
134C2×C4C23×D251148yescomplemented
135C22—236no—
136C22—100496no—
137C22—50272no—
138C22—25124no—
139C22#2254876114yesdirect product factor
140C22C24×D251112yes—
141C22C24×D25113yes—
142C4—2518no—
143C4C24×D251112yes—
144C4C24×D25114yes—
145C2—212no—
146C2—50224no—
147C2C2×#2254876116yesdirect product factor
148C2C25×D25111yescharacteristic
subgroupG/H
centerC23C23×D251
commutator, cosocleC2×C251C26
FrattiniC2C25×D251
FittingC22×#41×C251C2
socleC23×C251C24
hypercenterC22×#41D251
hypocenterC251C22×H(5,𝔽2)
LCSG ▷ C2×C251 ▷ C251
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#22548761C2direct (N × H)
2. C22×#16860811C2semidirect
3. C22×#16860810C2semidirect
4. C23×D1004C2semidirect
5. C22×D4×D251C2semidirect
6. C22×#41×C251C2semidirect
7. C24×D251C22semidirect
8. #22548761C22direct (N × H)
9. C2×#16860811C22semidirect
10. C2×#16860810C22semidirect
11. C22×Dic502C22semidirect
12. C22×D1004C22semidirect
13. C22×D1004C22semidirect
14. C22×D1004C22semidirect
15. C22×C22⋊D251C22semidirect
16. C2×D4×D251C22semidirect
17. C22×C4×D251C22semidirect
18. C2×#41×C251C22semidirect
19. C22×Q8×C251C22semidirect
20. C22×D4×C251C22semidirect
21. C23×C4×C251C22semidirect
24. C23×D251C23semidirect
25. #16860811C23semidirect
26. #16860810C23semidirect
27. C2×Dic502C23semidirect
28. C2×D1004C23semidirect
29. C2×D1004C23semidirect
30. C2×D1004C23semidirect
31. C2×C22⋊D251C23semidirect
32. D4×D251C23semidirect
34. C2×C4×D251C23semidirect
36. #41×C251C23semidirect
37. C2×Q8×C251C23semidirect
38. C2×D4×C251C23semidirect
41. C22×C4×C251C23semidirect
45. C22×D251C24semidirect
46. Dic502C24semidirect
47. D1004C24semidirect
48. D1004C24semidirect
49. C22⋊D251C24semidirect
51. C4×D251C24semidirect
55. Q8×C251C24semidirect
56. D4×C251C24semidirect
59. C2×C4×C251C24semidirect
61. C22×C251H(5,𝔽2)semidirect
71. C2×C251C2×H(5,𝔽2)semidirect
74. C251C22×H(5,𝔽2)semidirect
79. C22×#41D251semidirect
86. C2×#41C2×D251semidirect
88. C22×Q8C2×D251semidirect
94. C22×D4C2×D251semidirect
96. C23×C4C2×D251semidirect
103. #41C22×D251semidirect
105. C2×Q8C22×D251semidirect
111. C2×D4C22×D251semidirect
116. C22×C4C22×D251semidirect
125. Q8C23×D251semidirect
129. D4C23×D251semidirect
134. C2×C4C23×D251semidirect
139. C22#22548761direct (N × H)
147. C2C2×#22548761direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#225487612
C22×#168608112
C22×#168608102
C23×D10042
C22×D4×D2512
C22×#41×C2512
C22×H(5,𝔽2)251