Finite Group Databasebeta

← C24×#11761628 #31769714

Subgroups — C24×#11761628 #31769714

Order 17152.

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×#11761628C21120yesmaximal, direct product factor
2C24×Dic134C211yesmaximal, characteristic, complemented
3C24×D268C211yesmaximal, characteristic, complemented
4C24×C22⋊D67C212yesmaximal, complemented
5C24×C4×D67C212yesmaximal, complemented
6C25×C4×C67C211yesmaximal, characteristic, complemented
7C25×D67C2212yes—
8C22×#11761628C2212240yesdirect product factor
9C23×Dic134C22160yescomplemented
10C23×D268C22160yescomplemented
11C23×C22⋊D67C221120yescomplemented
12C24×Dic67C2212yes—
13C23×C4×D67C221120yescomplemented
14C26×C67C2211yescharacteristic
15C24×C4×C67C2211yescharacteristic
16C24×C4×C67C2211yescharacteristic
17C24×C4×C67C22160yescomplemented
18C24×D67—232no—
19C24×D67C23160yes—
20C2×#11761628C2317680yesdirect product factor
21C22×Dic134C231560yescomplemented
22C22×D268C231560yescomplemented
23C22×C22⋊D67C2311120yescomplemented
24C23×Dic67C23160yes—
25C22×C4×D67C2311120yescomplemented
26C25×C67—216no—
27C25×C67C23130yes—
28C25×C67C2311yescharacteristic
29C23×C4×C67C23130yes—
30C23×C4×C67C23130yes—
31C23×C4×C67C231560yescomplemented
32C23×D67—2480no—
33C23×D67C241280yes—
34#11761628C2414096yesdirect product factor
35C2×Dic134C241960yescomplemented
36C2×D268C241960yescomplemented
37C2×C22⋊D67C2411920yescomplemented
38C22×Dic67C241280yes—
39C2×C4×D67C2411920yescomplemented
40C24×C67—2240no—
41C24×C67#41116yescomplemented
42C24×C67C241140yes—
43C24×C67C24115yes—
44C22×C4×C67C241140yes—
45C22×C4×C67C241140yes—
46C22×C4×C67C241960yescomplemented
47C23×C67—2560no—
48C23×C67C2×#411120yescomplemented
49C23×C67C251120yes—
50C23×C67C25135yes—
51C22×D67—21120no—
52C22×D67C251240yes—
53C2×C4×C67C251120yes—
54C2×C4×C67C251120yes—
55C2×C4×C67C251256yescomplemented
56C22⋊D67C251512yescomplemented
57C2×Dic67C251240yes—
58D268C251256yescomplemented
59C4×D67C251512yescomplemented
60Dic134C251256yescomplemented
61C22×C67—2240no—
62C22×C67C22×#411140yescomplemented
63C22×C67C26116yes—
64C22×C67C26115yes—
65C2×D67—2480no—
66C2×D67C26132yes—
67C4×C67C26116yes—
68C4×C67C26116yes—
69Dic67C26132yes—
70C24×#41—671nomaximal, Sylow
71C2×C67—216no—
72C2×C67C23×#41130yescomplemented
73C2×C67C2711yescharacteristic
74D67—232no—
75C23×#41—67120no—
76C24×Q8—671no—
77C24×D4—672no—
78C24×D4—671no—
79C25×C4—672no—
80C25×C4D6711yescharacteristic, complemented
81C67C24×#4111yescharacteristic, complemented, Sylow
82C26—672no—
83C26C2×D6711yescharacteristic
84C22×#41—672240no—
85C23×Q8—6760no—
86C23×D4—67120no—
87C23×D4—6760no—
88C24×C4—672no—
89C24×C4—67120no—
90C24×C4C2×D6711yescharacteristic
91C24×C4C2×D6711yescharacteristic
92C24×C4C2×D67160yescomplemented
93C25—216no—
94C25—6760no—
95C25—13432no—
96C25C22×D67130yes—
97C25C22×D6711yescharacteristic
98C2×#41—677680no—
99C22×Q8—67560no—
100C22×D4—671120no—
101C22×D4—67560no—
102C23×C4—6760no—
103C23×C4—671120no—
104C23×C4C22×D67130yes—
105C23×C4C22×D67130yes—
106C23×C4C22×D671560yescomplemented
107C24—2240no—
108C24—67280no—
109C24—134480no—
110C24#11761628116yesdirect product factor
111C24C23×D671140yes—
112C24C23×D67115yes—
113#41—674096no—
114C2×Q8—67960no—
115C2×D4—671920no—
116C2×D4—67960no—
117C22×C4—67280no—
118C22×C4—671920no—
119C22×C4C23×D671140yes—
120C22×C4C23×D671140yes—
121C22×C4C23×D671960yescomplemented
122C23—2560no—
123C23—67240no—
124C23—1341120no—
125C23C2×#117616281120yesdirect product factor
126C23C24×D671120yes—
127C23C24×D67135yes—
128Q8—67256no—
129D4—67512no—
130D4—67256no—
131C2×C4—67240no—
132C2×C4—67512no—
133C2×C4C24×D671120yes—
134C2×C4C24×D671120yes—
135C2×C4C24×D671256yescomplemented
136C22—2240no—
137C22—6732no—
138C22—134480no—
139C22C22×#117616281140yesdirect product factor
140C22C25×D67116yes—
141C22C25×D67115yes—
142C4—6732no—
143C4C25×D67116yes—
144C4C25×D67116yes—
145C2—216no—
146C2—13432no—
147C2C23×#11761628130yesdirect product factor
148C2C26×D6711yescharacteristic
subgroupG/H
centerC24×C4C2×D67
commutator, cosocleC2×C67C27
FrattiniC2C26×D67
FittingC25×C4×C67C2
socleC25×C67C23
hypercenterC25×C4D67
hypocenterC67C24×#41
LCSG ▷ C2×C67 ▷ C67
UCSC1 ◁ C24×C4 ◁ C25×C4

Complements

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

NH ≅ G/Ntype
1. C23×#11761628C2direct (N × H)
2. C24×Dic134C2semidirect
3. C24×D268C2semidirect
4. C24×C22⋊D67C2semidirect
5. C24×C4×D67C2semidirect
6. C25×C4×C67C2semidirect
8. C22×#11761628C22direct (N × H)
9. C23×Dic134C22semidirect
10. C23×D268C22semidirect
11. C23×C22⋊D67C22semidirect
13. C23×C4×D67C22semidirect
17. C24×C4×C67C22semidirect
20. C2×#11761628C23direct (N × H)
21. C22×Dic134C23semidirect
22. C22×D268C23semidirect
23. C22×C22⋊D67C23semidirect
25. C22×C4×D67C23semidirect
31. C23×C4×C67C23semidirect
34. #11761628C24direct (N × H)
35. C2×Dic134C24semidirect
36. C2×D268C24semidirect
37. C2×C22⋊D67C24semidirect
39. C2×C4×D67C24semidirect
41. C24×C67#41semidirect
46. C22×C4×C67C24semidirect
48. C23×C67C2×#41semidirect
55. C2×C4×C67C25semidirect
56. C22⋊D67C25semidirect
58. D268C25semidirect
59. C4×D67C25semidirect
60. Dic134C25semidirect
62. C22×C67C22×#41semidirect
72. C2×C67C23×#41semidirect
80. C25×C4D67semidirect
81. C67C24×#41semidirect
92. C24×C4C2×D67semidirect
106. C23×C4C22×D67semidirect
110. C24#11761628direct (N × H)
121. C22×C4C23×D67semidirect
125. C23C2×#11761628direct (N × H)
135. C2×C4C24×D67semidirect
139. C22C22×#11761628direct (N × H)
147. C2C23×#11761628direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117616282
C24×Dic1342
C24×D2682
C24×C22⋊D672
C24×C4×D672
C25×C4×C672
C24×#4167