Finite Group Databasebeta

← C2×#30632492 #44565118

Subgroups — C2×#30632492 #44565118

Order 32128.

1512 conjugacy classes of subgroups; only the 549 normal subgroups are shown.

normal subgroups only

This group has 1512 conjugacy classes of subgroups — too many to draw. Only its 549 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×#22548761C212yesmaximal, complemented
2#30632492C2116yesmaximal, direct product factor
3C2×#22548743C216yesmaximal, complemented
4C2×#22548737C213yesmaximal, complemented
5C22×D2008C213yesmaximal, complemented
6C2×#131×C251C211yesmaximal, characteristic, complemented
7#22548761—28no—
8#22548761C22116yescomplemented
9C2×#16860811—22no—
10C2×#16860810C2216yescomplemented
11C22×D1004C2216yescomplemented
12C2×D4×D251—26no—
13#22548743C22148yescomplemented
14#22548737C22124yescomplemented
15C2×Dic1004C2211yescharacteristic, complemented
16C2×#16860799C2216yescomplemented
17C2×D2008C2213yescomplemented
18C2×D2008C2216yescomplemented
19C2×D2008C22124yescomplemented
20C2×#41×C251C2211yescharacteristic
21#131×C251C2218yescomplemented
22C2×M16×C251C2213yescomplemented
23C22×C8×C251C2213yescomplemented
24C23×D251—26no—
25#16860811—216no—
26#16860810—212no—
27#16860810C23124yescomplemented
28C2×Dic502C2312yes—
29C2×D1004—26no—
30C2×D1004—212no—
31C2×D1004C2316yes—
32C2×D1004C23124yescomplemented
33C2×C22⋊D251—26no—
34D4×D251—248no—
35C2×C4×D251—26no—
36Dic1004C2314yescomplemented
37#16860799C23124yescomplemented
38D2008C23112yescomplemented
39D2008C23124yescomplemented
40#41×C251D414yes—
41#41×C251C2314yes—
42C2×Q8×C251D411yescharacteristic
43C2×D4×C251D413yes—
44C22×C4×C251C2313yes—
45M16×C251C23112yescomplemented
46C2×C8×C251C2311yescharacteristic
47C2×C8×C251C2313yes—
48C2×C8×C251C23112yescomplemented
49C22×D251—26no—
50C22×D251—224no—
51C22×D251—812no—
52Dic502—22no—
53Dic502C2414yes—
54D1004—224no—
55D1004—26no—
56D1004C24112yes—
57C22⋊D251—224no—
58C2×Dic251—22no—
59C4×D251—224no—
60C23×C251C2×D413yes—
61Q8×C251C2×D414yes—
62D4×C251C2×D4112yes—
63C2×C4×C251C2×D413yes—
64C2×C4×C251C2×D416yes—
65C2×C4×C251C2411yescharacteristic
66C2×C4×C251C2416yes—
67C8×C251C2412yes—
68C8×C251C2416yes—
69C22×C251—26no—
70C22×C251C22×D416yes—
71C22×C251C22×D411yescharacteristic
72C2×D251—212no—
73C2×D251—412no—
74C2×D251—824no—
75C4×C251C22×D411yescharacteristic
76C4×C251C22×D416yes—
77C4×C251C2511yescharacteristic
78Dic251—24no—
79C2×C251—26no—
80C2×C251#57612yescomplemented
81C2×C251C23×D411yescharacteristic
82D251—412no—
83C251C2×#57611yescharacteristic, complemented, Sylow
84C2×#576—2511nomaximal, Sylow
85C2×H(5,𝔽2)—2512no—
86#576—25116no—
87C2×AGL(1,ℤ/8)—2516no—
88C2×#135—2513no—
89C22×D8—2513no—
90C2×#131D25111yescharacteristic, complemented
91H(5,𝔽2)—5028no—
92H(5,𝔽2)—25116no—
93C2×#41—5022no—
94C2×#41—2516no—
95C2×#41C2×D25111yescharacteristic
96C22×D4—5026no—
97C22×D4—2516no—
98AGL(1,ℤ/8)—25148no—
99#135—25124no—
100C2×Q16—2511no—
101C2×SD16—2516no—
102C2×D8—2513no—
103C2×D8—2516no—
104C2×D8—25124no—
105#131C2×D25118yescomplemented
106C2×M16C2×D25113yescomplemented
107C22×C8C2×D25113yescomplemented
108C24—5026no—
109#41—50216no—
110#41—50212no—
111#41—25124no—
112#41D100414yes—
113#41C22×D25114yes—
114C2×Q8—2512no—
115C2×Q8D100411yescharacteristic
116C2×D4—5026no—
117C2×D4—50248no—
118C2×D4—5026no—
119C2×D4—50212no—
120C2×D4—2516no—
121C2×D4—25124no—
122C2×D4D100413yes—
123C22×C4—5026no—
124C22×C4C22×D25113yes—
125Q16—2514no—
126SD16—25124no—
127D8—25112no—
128D8—25124no—
129M16C22×D251112yescomplemented
130C2×C8C22×D25111yescharacteristic
131C2×C8C22×D25113yes—
132C2×C8C22×D251112yescomplemented
133C23—200812no—
134C23—50224no—
135C23—5026no—
136C23C2×D100413yes—
137Q8—5022no—
138Q8—2514no—
139Q8C2×D100414yes—
140D4—50224no—
141D4—50224no—
142D4—5026no—
143D4—25112no—
144D4C2×D1004112yes—
145C2×C4—5022no—
146C2×C4—50224no—
147C2×C4C2×D100416yes—
148C2×C4C2×D100413yes—
149C2×C4C23×D25116yes—
150C2×C4C23×D25111yescharacteristic
151C8C23×D25112yes—
152C8C23×D25116yes—
153C22—26no—
154C22—200824no—
155C22—100412no—
156C22—50212no—
157C22C22×D100416yes—
158C22C22×D100411yescharacteristic
159C4—5024no—
160C4C22×D100416yes—
161C4C22×D100411yescharacteristic
162C4C24×D25111yescharacteristic
163C2—26no—
164C2—100412no—
165C2#3063249212yesdirect product factor
166C2C23×D100411yescharacteristic
subgroupG/H
centerC22C22×D1004
commutator, cosocleC4×C251C25
FrattiniC4C24×D251
FittingC2×#131×C251C2
socleC22×C251C22×D4
hypercenterC2×#131D251
hypocenterC251C2×#576
LCSG ▷ C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131

Complements

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

NH ≅ G/Ntype
1. C2×#22548761C2semidirect
2. #30632492C2direct (N × H)
3. C2×#22548743C2semidirect
4. C2×#22548737C2semidirect
5. C22×D2008C2semidirect
6. C2×#131×C251C2semidirect
8. #22548761C22semidirect
10. C2×#16860810C22semidirect
11. C22×D1004C22semidirect
13. #22548743C22semidirect
14. #22548737C22semidirect
15. C2×Dic1004C22semidirect
16. C2×#16860799C22semidirect
17. C2×D2008C22semidirect
18. C2×D2008C22semidirect
19. C2×D2008C22semidirect
21. #131×C251C22semidirect
22. C2×M16×C251C22semidirect
23. C22×C8×C251C22semidirect
27. #16860810C23semidirect
32. C2×D1004C23semidirect
36. Dic1004C23semidirect
37. #16860799C23semidirect
38. D2008C23semidirect
39. D2008C23semidirect
45. M16×C251C23semidirect
48. C2×C8×C251C23semidirect
80. C2×C251#576semidirect
83. C251C2×#576semidirect
90. C2×#131D251semidirect
105. #131C2×D251semidirect
106. C2×M16C2×D251semidirect
107. C22×C8C2×D251semidirect
129. M16C22×D251semidirect
132. C2×C8C22×D251semidirect
165. C2#30632492direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#225487612
#306324922
C2×#225487432
C2×#225487372
C22×D20082
C2×#131×C2512
C2×#576251