Finite Group Databasebeta

← C24×#11929624 #35202152

Subgroups — C24×#11929624 #35202152

Order 20224.

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×#11929624C21120yesmaximal, direct product factor
2C24×Dic158C211yesmaximal, characteristic, complemented
3C24×D316C211yesmaximal, characteristic, complemented
4C24×C22⋊D79C212yesmaximal, complemented
5C24×C4×D79C212yesmaximal, complemented
6C25×C4×C79C211yesmaximal, characteristic, complemented
7C25×D79C2212yes—
8C22×#11929624C2212240yesdirect product factor
9C23×Dic158C22160yescomplemented
10C23×D316C22160yescomplemented
11C23×C22⋊D79C221120yescomplemented
12C24×Dic79C2212yes—
13C23×C4×D79C221120yescomplemented
14C26×C79C2211yescharacteristic
15C24×C4×C79C2211yescharacteristic
16C24×C4×C79C2211yescharacteristic
17C24×C4×C79C22160yescomplemented
18C24×D79—232no—
19C24×D79C23160yes—
20C2×#11929624C2317680yesdirect product factor
21C22×Dic158C231560yescomplemented
22C22×D316C231560yescomplemented
23C22×C22⋊D79C2311120yescomplemented
24C23×Dic79C23160yes—
25C22×C4×D79C2311120yescomplemented
26C25×C79—216no—
27C25×C79C23130yes—
28C25×C79C2311yescharacteristic
29C23×C4×C79C23130yes—
30C23×C4×C79C23130yes—
31C23×C4×C79C231560yescomplemented
32C23×D79—2480no—
33C23×D79C241280yes—
34#11929624C2414096yesdirect product factor
35C2×Dic158C241960yescomplemented
36C2×D316C241960yescomplemented
37C2×C22⋊D79C2411920yescomplemented
38C22×Dic79C241280yes—
39C2×C4×D79C2411920yescomplemented
40C24×C79—2240no—
41C24×C79#41116yescomplemented
42C24×C79C241140yes—
43C24×C79C24115yes—
44C22×C4×C79C241140yes—
45C22×C4×C79C241140yes—
46C22×C4×C79C241960yescomplemented
47C23×C79—2560no—
48C23×C79C2×#411120yescomplemented
49C23×C79C251120yes—
50C23×C79C25135yes—
51C22×D79—21120no—
52C22×D79C251240yes—
53C2×C4×C79C251120yes—
54C2×C4×C79C251120yes—
55C2×C4×C79C251256yescomplemented
56C22⋊D79C251512yescomplemented
57C2×Dic79C251240yes—
58D316C251256yescomplemented
59C4×D79C251512yescomplemented
60Dic158C251256yescomplemented
61C22×C79—2240no—
62C22×C79C22×#411140yescomplemented
63C22×C79C26116yes—
64C22×C79C26115yes—
65C2×D79—2480no—
66C2×D79C26132yes—
67C4×C79C26116yes—
68C4×C79C26116yes—
69Dic79C26132yes—
70C24×#41—791nomaximal, Sylow
71C2×C79—216no—
72C2×C79C23×#41130yescomplemented
73C2×C79C2711yescharacteristic
74D79—232no—
75C23×#41—79120no—
76C24×Q8—791no—
77C24×D4—792no—
78C24×D4—791no—
79C25×C4—792no—
80C25×C4D7911yescharacteristic, complemented
81C79C24×#4111yescharacteristic, complemented, Sylow
82C26—792no—
83C26C2×D7911yescharacteristic
84C22×#41—792240no—
85C23×Q8—7960no—
86C23×D4—79120no—
87C23×D4—7960no—
88C24×C4—792no—
89C24×C4—79120no—
90C24×C4C2×D7911yescharacteristic
91C24×C4C2×D7911yescharacteristic
92C24×C4C2×D79160yescomplemented
93C25—216no—
94C25—7960no—
95C25—15832no—
96C25C22×D79130yes—
97C25C22×D7911yescharacteristic
98C2×#41—797680no—
99C22×Q8—79560no—
100C22×D4—791120no—
101C22×D4—79560no—
102C23×C4—7960no—
103C23×C4—791120no—
104C23×C4C22×D79130yes—
105C23×C4C22×D79130yes—
106C23×C4C22×D791560yescomplemented
107C24—2240no—
108C24—79280no—
109C24—158480no—
110C24#11929624116yesdirect product factor
111C24C23×D791140yes—
112C24C23×D79115yes—
113#41—794096no—
114C2×Q8—79960no—
115C2×D4—791920no—
116C2×D4—79960no—
117C22×C4—79280no—
118C22×C4—791920no—
119C22×C4C23×D791140yes—
120C22×C4C23×D791140yes—
121C22×C4C23×D791960yescomplemented
122C23—2560no—
123C23—79240no—
124C23—1581120no—
125C23C2×#119296241120yesdirect product factor
126C23C24×D791120yes—
127C23C24×D79135yes—
128Q8—79256no—
129D4—79512no—
130D4—79256no—
131C2×C4—79240no—
132C2×C4—79512no—
133C2×C4C24×D791120yes—
134C2×C4C24×D791120yes—
135C2×C4C24×D791256yescomplemented
136C22—2240no—
137C22—7932no—
138C22—158480no—
139C22C22×#119296241140yesdirect product factor
140C22C25×D79116yes—
141C22C25×D79115yes—
142C4—7932no—
143C4C25×D79116yes—
144C4C25×D79116yes—
145C2—216no—
146C2—15832no—
147C2C23×#11929624130yesdirect product factor
148C2C26×D7911yescharacteristic
subgroupG/H
centerC24×C4C2×D79
commutator, cosocleC2×C79C27
FrattiniC2C26×D79
FittingC25×C4×C79C2
socleC25×C79C23
hypercenterC25×C4D79
hypocenterC79C24×#41
LCSG ▷ C2×C79 ▷ C79
UCSC1 ◁ C24×C4 ◁ C25×C4

Complements

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

NH ≅ G/Ntype
1. C23×#11929624C2direct (N × H)
2. C24×Dic158C2semidirect
3. C24×D316C2semidirect
4. C24×C22⋊D79C2semidirect
5. C24×C4×D79C2semidirect
6. C25×C4×C79C2semidirect
8. C22×#11929624C22direct (N × H)
9. C23×Dic158C22semidirect
10. C23×D316C22semidirect
11. C23×C22⋊D79C22semidirect
13. C23×C4×D79C22semidirect
17. C24×C4×C79C22semidirect
20. C2×#11929624C23direct (N × H)
21. C22×Dic158C23semidirect
22. C22×D316C23semidirect
23. C22×C22⋊D79C23semidirect
25. C22×C4×D79C23semidirect
31. C23×C4×C79C23semidirect
34. #11929624C24direct (N × H)
35. C2×Dic158C24semidirect
36. C2×D316C24semidirect
37. C2×C22⋊D79C24semidirect
39. C2×C4×D79C24semidirect
41. C24×C79#41semidirect
46. C22×C4×C79C24semidirect
48. C23×C79C2×#41semidirect
55. C2×C4×C79C25semidirect
56. C22⋊D79C25semidirect
58. D316C25semidirect
59. C4×D79C25semidirect
60. Dic158C25semidirect
62. C22×C79C22×#41semidirect
72. C2×C79C23×#41semidirect
80. C25×C4D79semidirect
81. C79C24×#41semidirect
92. C24×C4C2×D79semidirect
106. C23×C4C22×D79semidirect
110. C24#11929624direct (N × H)
121. C22×C4C23×D79semidirect
125. C23C2×#11929624direct (N × H)
135. C2×C4C24×D79semidirect
139. C22C22×#11929624direct (N × H)
147. C2C23×#11929624direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#119296242
C24×Dic1582
C24×D3162
C24×C22⋊D792
C24×C4×D792
C25×C4×C792
C24×#4179