Finite Group Databasebeta

← C24×#11719945 #28297975

Subgroups — C24×#11719945 #28297975

Order 13568.

65470 conjugacy classes of subgroups; only the 32651 normal subgroups are shown.

normal subgroups only

This group has 65470 conjugacy classes of subgroups — too many to draw. Only its 32651 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
1C23×#11719945C21120yesmaximal, direct product factor
2C24×Dic106C211yesmaximal, characteristic, complemented
3C24×D212C211yesmaximal, characteristic, complemented
4C24×C22⋊D53C212yesmaximal, complemented
5C24×C4×D53C212yesmaximal, complemented
6C25×C4×C53C211yesmaximal, characteristic, complemented
7C25×D53C2212yes—
8C22×#11719945C2212240yesdirect product factor
9C23×Dic106C22160yescomplemented
10C23×D212C22160yescomplemented
11C23×C22⋊D53C221120yescomplemented
12C24×Dic53C2212yes—
13C23×C4×D53C221120yescomplemented
14C26×C53C2211yescharacteristic
15C24×C4×C53C2211yescharacteristic
16C24×C4×C53C2211yescharacteristic
17C24×C4×C53C22160yescomplemented
18C24×D53—232no—
19C24×D53C23160yes—
20C2×#11719945C2317680yesdirect product factor
21C22×Dic106C231560yescomplemented
22C22×D212C231560yescomplemented
23C22×C22⋊D53C2311120yescomplemented
24C23×Dic53C23160yes—
25C22×C4×D53C2311120yescomplemented
26C25×C53—216no—
27C25×C53C23130yes—
28C25×C53C2311yescharacteristic
29C23×C4×C53C23130yes—
30C23×C4×C53C23130yes—
31C23×C4×C53C231560yescomplemented
32C24×C53—2240no—
33C24×C53#41116yescomplemented
34C24×C53C241140yes—
35C24×C53C24115yes—
36C23×D53—2480no—
37C23×D53C241280yes—
38C22×C4×C53C241140yes—
39C22×C4×C53C241140yes—
40C22×C4×C53C241960yescomplemented
41C2×C22⋊D53C2411920yescomplemented
42C22×Dic53C241280yes—
43#11719945C2414096yesdirect product factor
44C2×D212C241960yescomplemented
45C2×C4×D53C2411920yescomplemented
46C2×Dic106C241960yescomplemented
47C23×C53—2560no—
48C23×C53C2×#411120yescomplemented
49C23×C53C251120yes—
50C23×C53C25135yes—
51C22×D53—21120no—
52C22×D53C251240yes—
53C2×C4×C53C251120yes—
54C2×C4×C53C251120yes—
55C2×C4×C53C251256yescomplemented
56C22⋊D53C251512yescomplemented
57C2×Dic53C251240yes—
58D212C251256yescomplemented
59C4×D53C251512yescomplemented
60Dic106C251256yescomplemented
61C24×#41—531nomaximal, Sylow
62C22×C53—2240no—
63C22×C53C22×#411140yescomplemented
64C22×C53C26116yes—
65C22×C53C26115yes—
66C2×D53—2480no—
67C2×D53C26132yes—
68C4×C53C26116yes—
69C4×C53C26116yes—
70Dic53C26132yes—
71C23×#41—53120no—
72C24×Q8—531no—
73C24×D4—532no—
74C24×D4—531no—
75C25×C4—532no—
76C25×C4D5311yescharacteristic, complemented
77C2×C53—216no—
78C2×C53C23×#41130yescomplemented
79C2×C53C2711yescharacteristic
80D53—232no—
81C26—532no—
82C26C2×D5311yescharacteristic
83C22×#41—532240no—
84C23×Q8—5360no—
85C23×D4—53120no—
86C23×D4—5360no—
87C24×C4—532no—
88C24×C4—53120no—
89C24×C4C2×D5311yescharacteristic
90C24×C4C2×D5311yescharacteristic
91C24×C4C2×D53160yescomplemented
92C53C24×#4111yescharacteristic, complemented, Sylow
93C25—216no—
94C25—5360no—
95C25—10632no—
96C25C22×D53130yes—
97C25C22×D5311yescharacteristic
98C2×#41—537680no—
99C22×Q8—53560no—
100C22×D4—531120no—
101C22×D4—53560no—
102C23×C4—5360no—
103C23×C4—531120no—
104C23×C4C22×D53130yes—
105C23×C4C22×D53130yes—
106C23×C4C22×D531560yescomplemented
107C24—2240no—
108C24—53280no—
109C24—106480no—
110C24#11719945116yesdirect product factor
111C24C23×D531140yes—
112C24C23×D53115yes—
113#41—534096no—
114C2×Q8—53960no—
115C2×D4—531920no—
116C2×D4—53960no—
117C22×C4—53280no—
118C22×C4—531920no—
119C22×C4C23×D531140yes—
120C22×C4C23×D531140yes—
121C22×C4C23×D531960yescomplemented
122C23—2560no—
123C23—53240no—
124C23—1061120no—
125C23C2×#117199451120yesdirect product factor
126C23C24×D531120yes—
127C23C24×D53135yes—
128Q8—53256no—
129D4—53512no—
130D4—53256no—
131C2×C4—53240no—
132C2×C4—53512no—
133C2×C4C24×D531120yes—
134C2×C4C24×D531120yes—
135C2×C4C24×D531256yescomplemented
136C22—2240no—
137C22—5332no—
138C22—106480no—
139C22C22×#117199451140yesdirect product factor
140C22C25×D53116yes—
141C22C25×D53115yes—
142C4—5332no—
143C4C25×D53116yes—
144C4C25×D53116yes—
145C2—216no—
146C2—10632no—
147C2C23×#11719945130yesdirect product factor
148C2C26×D5311yescharacteristic
subgroupG/H
centerC24×C4C2×D53
commutator, cosocleC2×C53C27
FrattiniC2C26×D53
FittingC25×C4×C53C2
socleC25×C53C23
hypercenterC25×C4D53
hypocenterC53C24×#41
LCSG ▷ C2×C53 ▷ C53
UCSC1 ◁ C24×C4 ◁ C25×C4

Complements

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

NH ≅ G/Ntype
1. C23×#11719945C2direct (N × H)
2. C24×Dic106C2semidirect
3. C24×D212C2semidirect
4. C24×C22⋊D53C2semidirect
5. C24×C4×D53C2semidirect
6. C25×C4×C53C2semidirect
8. C22×#11719945C22direct (N × H)
9. C23×Dic106C22semidirect
10. C23×D212C22semidirect
11. C23×C22⋊D53C22semidirect
13. C23×C4×D53C22semidirect
17. C24×C4×C53C22semidirect
20. C2×#11719945C23direct (N × H)
21. C22×Dic106C23semidirect
22. C22×D212C23semidirect
23. C22×C22⋊D53C23semidirect
25. C22×C4×D53C23semidirect
31. C23×C4×C53C23semidirect
33. C24×C53#41semidirect
40. C22×C4×C53C24semidirect
41. C2×C22⋊D53C24semidirect
43. #11719945C24direct (N × H)
44. C2×D212C24semidirect
45. C2×C4×D53C24semidirect
46. C2×Dic106C24semidirect
48. C23×C53C2×#41semidirect
55. C2×C4×C53C25semidirect
56. C22⋊D53C25semidirect
58. D212C25semidirect
59. C4×D53C25semidirect
60. Dic106C25semidirect
63. C22×C53C22×#41semidirect
76. C25×C4D53semidirect
78. C2×C53C23×#41semidirect
91. C24×C4C2×D53semidirect
92. C53C24×#41semidirect
106. C23×C4C22×D53semidirect
110. C24#11719945direct (N × H)
121. C22×C4C23×D53semidirect
125. C23C2×#11719945direct (N × H)
135. C2×C4C24×D53semidirect
139. C22C22×#11719945direct (N × H)
147. C2C23×#11719945direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117199452
C24×Dic1062
C24×D2122
C24×C22⋊D532
C24×C4×D532
C25×C4×C532
C24×#4153