Finite Group Databasebeta

← C24×#11765864 #32911334

Subgroups — C24×#11765864 #32911334

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×#11765864C21120yesmaximal, direct product factor
2C24×Dic142C211yesmaximal, characteristic, complemented
3C24×C22⋊D71C212yesmaximal, complemented
4C25×Dic71C212yesmaximal, complemented
5C24×C4×D71C211yesmaximal, characteristic, complemented
6C24×D4×C71C211yesmaximal, characteristic, complemented
7C25×D71C2211yescharacteristic
8C22×#11765864C2212240yesdirect product factor
9C23×Dic142C22160yescomplemented
10C23×C22⋊D71C221120yescomplemented
11C24×Dic71C2211yescharacteristic
12C24×Dic71C2212yes—
13C24×Dic71C221120yescomplemented
14C23×C4×D71C22160yescomplemented
15C26×C71C2212yes—
16C23×D4×C71C22160yescomplemented
17C24×C4×C71C2211yescharacteristic
18C24×D71—216no—
19C24×D71C23130yes—
20C2×#11765864C2317680yesdirect product factor
21C22×Dic142C231560yescomplemented
22C22×C22⋊D71C2311120yescomplemented
23C23×Dic71C23130yes—
24C23×Dic71C23160yes—
25C23×Dic71C2311120yescomplemented
26C22×C4×D71C231560yescomplemented
27C25×C71—232no—
28C25×C71C23160yes—
29C25×C71C2311yescharacteristic
30C22×D4×C71C231560yescomplemented
31C23×C4×C71C23130yes—
32C23×D71—2240no—
33C23×D71C241140yes—
34#11765864C2414096yesdirect product factor
35C2×Dic142C241960yescomplemented
36C2×C22⋊D71C2411920yescomplemented
37C22×Dic71C241140yes—
38C22×Dic71C241280yes—
39C22×Dic71C2411920yescomplemented
40C2×C4×D71C241960yescomplemented
41C24×C71—2480no—
42C24×C71#41116yescomplemented
43C24×C71C241280yes—
44C24×C71C24115yes—
45C2×D4×C71C241960yescomplemented
46C22×C4×C71C241140yes—
47C23×C71—21120no—
48C23×C71C2×#411120yescomplemented
49C23×C71C251240yes—
50C23×C71C25135yes—
51C22×D71—2560no—
52C22×D71C251120yes—
53D4×C71C251256yescomplemented
54C2×C4×C71C251120yes—
55C22⋊D71C251512yescomplemented
56C2×Dic71C251120yes—
57C2×Dic71C251240yes—
58C2×Dic71C251512yescomplemented
59C4×D71C251256yescomplemented
60Dic142C251256yescomplemented
61C22×C71—2480no—
62C22×C71C22×#411140yescomplemented
63C22×C71C26132yes—
64C22×C71C26115yes—
65C2×D71—2240no—
66C2×D71C26116yes—
67C4×C71C26116yes—
68Dic71C26132yes—
69Dic71C26116yes—
70C24×#41—711nomaximal, Sylow
71C2×C71—232no—
72C2×C71C23×#41130yescomplemented
73C2×C71C2711yescharacteristic
74D71—216no—
75C23×#41—71120no—
76C24×Q8—711no—
77C24×D4—712no—
78C24×D4D7111yescharacteristic, complemented
79C25×C4—712no—
80C25×C4—711no—
81C71C24×#4111yescharacteristic, complemented, Sylow
82C26—711no—
83C26C2×D7112yes—
84C22×#41—712240no—
85C23×Q8—7160no—
86C23×D4—71120no—
87C23×D4C2×D71160yescomplemented
88C24×C4—71120no—
89C24×C4—711no—
90C24×C4—712no—
91C24×C4—7160no—
92C24×C4C2×D7111yescharacteristic
93C25—232no—
94C25—7130no—
95C25—14216no—
96C25C22×D71160yes—
97C25C22×D7111yescharacteristic
98C2×#41—717680no—
99C22×Q8—71560no—
100C22×D4—711120no—
101C22×D4C22×D711560yescomplemented
102C23×C4—7130no—
103C23×C4—7160no—
104C23×C4—711120no—
105C23×C4—71560no—
106C23×C4C22×D71130yes—
107C24—2480no—
108C24—71140no—
109C24—142240no—
110C24#11765864116yesdirect product factor
111C24C23×D711280yes—
112C24C23×D71115yes—
113#41—714096no—
114C2×Q8—71960no—
115C2×D4—711920no—
116C2×D4C23×D711960yescomplemented
117C22×C4—71140no—
118C22×C4—71280no—
119C22×C4—711920no—
120C22×C4—71960no—
121C22×C4C23×D711140yes—
122C23—21120no—
123C23—71120no—
124C23—142560no—
125C23C2×#117658641120yesdirect product factor
126C23C24×D711240yes—
127C23C24×D71135yes—
128Q8—71256no—
129D4—71512no—
130D4C24×D711256yescomplemented
131C2×C4—71120no—
132C2×C4—71240no—
133C2×C4—71512no—
134C2×C4—71256no—
135C2×C4C24×D711120yes—
136C22—2480no—
137C22—7116no—
138C22—142240no—
139C22C22×#117658641140yesdirect product factor
140C22C25×D71132yes—
141C22C25×D71115yes—
142C4—7116no—
143C4—7132no—
144C4C25×D71116yes—
145C2—232no—
146C2—14216no—
147C2C23×#11765864130yesdirect product factor
148C2C26×D7111yescharacteristic
subgroupG/H
centerC25C22×D71
commutator, cosocleC2×C71C27
FrattiniC2C26×D71
FittingC24×D4×C71C2
socleC25×C71C23
hypercenterC24×D4D71
hypocenterC71C24×#41
LCSG ▷ C2×C71 ▷ C71
UCSC1 ◁ C25 ◁ C24×D4

Complements

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

NH ≅ G/Ntype
1. C23×#11765864C2direct (N × H)
2. C24×Dic142C2semidirect
3. C24×C22⋊D71C2semidirect
4. C25×Dic71C2semidirect
5. C24×C4×D71C2semidirect
6. C24×D4×C71C2semidirect
8. C22×#11765864C22direct (N × H)
9. C23×Dic142C22semidirect
10. C23×C22⋊D71C22semidirect
13. C24×Dic71C22semidirect
14. C23×C4×D71C22semidirect
16. C23×D4×C71C22semidirect
20. C2×#11765864C23direct (N × H)
21. C22×Dic142C23semidirect
22. C22×C22⋊D71C23semidirect
25. C23×Dic71C23semidirect
26. C22×C4×D71C23semidirect
30. C22×D4×C71C23semidirect
34. #11765864C24direct (N × H)
35. C2×Dic142C24semidirect
36. C2×C22⋊D71C24semidirect
39. C22×Dic71C24semidirect
40. C2×C4×D71C24semidirect
42. C24×C71#41semidirect
45. C2×D4×C71C24semidirect
48. C23×C71C2×#41semidirect
53. D4×C71C25semidirect
55. C22⋊D71C25semidirect
58. C2×Dic71C25semidirect
59. C4×D71C25semidirect
60. Dic142C25semidirect
62. C22×C71C22×#41semidirect
72. C2×C71C23×#41semidirect
78. C24×D4D71semidirect
81. C71C24×#41semidirect
87. C23×D4C2×D71semidirect
101. C22×D4C22×D71semidirect
110. C24#11765864direct (N × H)
116. C2×D4C23×D71semidirect
125. C23C2×#11765864direct (N × H)
130. D4C24×D71semidirect
139. C22C22×#11765864direct (N × H)
147. C2C23×#11765864direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117658642
C24×Dic1422
C24×C22⋊D712
C25×Dic712
C24×C4×D712
C24×D4×C712
C24×#4171