Finite Group Databasebeta

← C24×#11765865 #32911335

Subgroups — C24×#11765865 #32911335

Order 18176.

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×#11765865C21120yesmaximal, direct product factor
2C24×Dic142C211yesmaximal, characteristic, complemented
3C24×D284C211yesmaximal, characteristic, complemented
4C24×C22⋊D71C212yesmaximal, complemented
5C24×C4×D71C212yesmaximal, complemented
6C25×C4×C71C211yesmaximal, characteristic, complemented
7C25×D71C2212yes—
8C22×#11765865C2212240yesdirect product factor
9C23×Dic142C22160yescomplemented
10C23×D284C22160yescomplemented
11C23×C22⋊D71C221120yescomplemented
12C24×Dic71C2212yes—
13C23×C4×D71C221120yescomplemented
14C26×C71C2211yescharacteristic
15C24×C4×C71C2211yescharacteristic
16C24×C4×C71C2211yescharacteristic
17C24×C4×C71C22160yescomplemented
18C24×D71—232no—
19C24×D71C23160yes—
20C2×#11765865C2317680yesdirect product factor
21C22×Dic142C231560yescomplemented
22C22×D284C231560yescomplemented
23C22×C22⋊D71C2311120yescomplemented
24C23×Dic71C23160yes—
25C22×C4×D71C2311120yescomplemented
26C25×C71—216no—
27C25×C71C23130yes—
28C25×C71C2311yescharacteristic
29C23×C4×C71C23130yes—
30C23×C4×C71C23130yes—
31C23×C4×C71C231560yescomplemented
32C23×D71—2480no—
33C23×D71C241280yes—
34#11765865C2414096yesdirect product factor
35C2×Dic142C241960yescomplemented
36C2×D284C241960yescomplemented
37C2×C22⋊D71C2411920yescomplemented
38C22×Dic71C241280yes—
39C2×C4×D71C2411920yescomplemented
40C24×C71—2240no—
41C24×C71#41116yescomplemented
42C24×C71C241140yes—
43C24×C71C24115yes—
44C22×C4×C71C241140yes—
45C22×C4×C71C241140yes—
46C22×C4×C71C241960yescomplemented
47C23×C71—2560no—
48C23×C71C2×#411120yescomplemented
49C23×C71C251120yes—
50C23×C71C25135yes—
51C22×D71—21120no—
52C22×D71C251240yes—
53C2×C4×C71C251120yes—
54C2×C4×C71C251120yes—
55C2×C4×C71C251256yescomplemented
56C22⋊D71C251512yescomplemented
57C2×Dic71C251240yes—
58D284C251256yescomplemented
59C4×D71C251512yescomplemented
60Dic142C251256yescomplemented
61C22×C71—2240no—
62C22×C71C22×#411140yescomplemented
63C22×C71C26116yes—
64C22×C71C26115yes—
65C2×D71—2480no—
66C2×D71C26132yes—
67C4×C71C26116yes—
68C4×C71C26116yes—
69Dic71C26132yes—
70C24×#41—711nomaximal, Sylow
71C2×C71—216no—
72C2×C71C23×#41130yescomplemented
73C2×C71C2711yescharacteristic
74D71—232no—
75C23×#41—71120no—
76C24×Q8—711no—
77C24×D4—712no—
78C24×D4—711no—
79C25×C4—712no—
80C25×C4D7111yescharacteristic, complemented
81C71C24×#4111yescharacteristic, complemented, Sylow
82C26—712no—
83C26C2×D7111yescharacteristic
84C22×#41—712240no—
85C23×Q8—7160no—
86C23×D4—71120no—
87C23×D4—7160no—
88C24×C4—712no—
89C24×C4—71120no—
90C24×C4C2×D7111yescharacteristic
91C24×C4C2×D7111yescharacteristic
92C24×C4C2×D71160yescomplemented
93C25—216no—
94C25—7160no—
95C25—14232no—
96C25C22×D71130yes—
97C25C22×D7111yescharacteristic
98C2×#41—717680no—
99C22×Q8—71560no—
100C22×D4—711120no—
101C22×D4—71560no—
102C23×C4—7160no—
103C23×C4—711120no—
104C23×C4C22×D71130yes—
105C23×C4C22×D71130yes—
106C23×C4C22×D711560yescomplemented
107C24—2240no—
108C24—71280no—
109C24—142480no—
110C24#11765865116yesdirect product factor
111C24C23×D711140yes—
112C24C23×D71115yes—
113#41—714096no—
114C2×Q8—71960no—
115C2×D4—711920no—
116C2×D4—71960no—
117C22×C4—71280no—
118C22×C4—711920no—
119C22×C4C23×D711140yes—
120C22×C4C23×D711140yes—
121C22×C4C23×D711960yescomplemented
122C23—2560no—
123C23—71240no—
124C23—1421120no—
125C23C2×#117658651120yesdirect product factor
126C23C24×D711120yes—
127C23C24×D71135yes—
128Q8—71256no—
129D4—71512no—
130D4—71256no—
131C2×C4—71240no—
132C2×C4—71512no—
133C2×C4C24×D711120yes—
134C2×C4C24×D711120yes—
135C2×C4C24×D711256yescomplemented
136C22—2240no—
137C22—7132no—
138C22—142480no—
139C22C22×#117658651140yesdirect product factor
140C22C25×D71116yes—
141C22C25×D71115yes—
142C4—7132no—
143C4C25×D71116yes—
144C4C25×D71116yes—
145C2—216no—
146C2—14232no—
147C2C23×#11765865130yesdirect product factor
148C2C26×D7111yescharacteristic
subgroupG/H
centerC24×C4C2×D71
commutator, cosocleC2×C71C27
FrattiniC2C26×D71
FittingC25×C4×C71C2
socleC25×C71C23
hypercenterC25×C4D71
hypocenterC71C24×#41
LCSG ▷ C2×C71 ▷ C71
UCSC1 ◁ C24×C4 ◁ C25×C4

Complements

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

NH ≅ G/Ntype
1. C23×#11765865C2direct (N × H)
2. C24×Dic142C2semidirect
3. C24×D284C2semidirect
4. C24×C22⋊D71C2semidirect
5. C24×C4×D71C2semidirect
6. C25×C4×C71C2semidirect
8. C22×#11765865C22direct (N × H)
9. C23×Dic142C22semidirect
10. C23×D284C22semidirect
11. C23×C22⋊D71C22semidirect
13. C23×C4×D71C22semidirect
17. C24×C4×C71C22semidirect
20. C2×#11765865C23direct (N × H)
21. C22×Dic142C23semidirect
22. C22×D284C23semidirect
23. C22×C22⋊D71C23semidirect
25. C22×C4×D71C23semidirect
31. C23×C4×C71C23semidirect
34. #11765865C24direct (N × H)
35. C2×Dic142C24semidirect
36. C2×D284C24semidirect
37. C2×C22⋊D71C24semidirect
39. C2×C4×D71C24semidirect
41. C24×C71#41semidirect
46. C22×C4×C71C24semidirect
48. C23×C71C2×#41semidirect
55. C2×C4×C71C25semidirect
56. C22⋊D71C25semidirect
58. D284C25semidirect
59. C4×D71C25semidirect
60. Dic142C25semidirect
62. C22×C71C22×#41semidirect
72. C2×C71C23×#41semidirect
80. C25×C4D71semidirect
81. C71C24×#41semidirect
92. C24×C4C2×D71semidirect
106. C23×C4C22×D71semidirect
110. C24#11765865direct (N × H)
121. C22×C4C23×D71semidirect
125. C23C2×#11765865direct (N × H)
135. C2×C4C24×D71semidirect
139. C22C22×#11765865direct (N × H)
147. C2C23×#11765865direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117658652
C24×Dic1422
C24×D2842
C24×C22⋊D712
C24×C4×D712
C25×C4×C712
C24×#4171