Finite Group Databasebeta

← #29507906

Subgroups — #29507906

Order 15424.

160 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 39 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
1#21440687C211yesmaximal, characteristic
2#21440625C212yesmaximal, complemented
3#21440557C211yesmaximal, characteristic, complemented
4#21440556C212yesmaximal, complemented
5#123×C241C211yesmaximal, characteristic, complemented
6#16838770—21no—
7#16838769—22no—
8C2×C22⋊D241—21no—
9C2×C22⋊D241C2212yes—
10D4×D241—22no—
11#16838749—22no—
12C22⋊Dic241C2212yescomplemented
13#16838747—22no—
14C2×D4×C241C2211yescharacteristic
15C22×C4×C241C2211yescharacteristic, complemented
16C4⋊C4×C241—21no—
17C22⋊C4×C241—21no—
18C22⋊C4×C241C2211yescharacteristic, complemented
19C22×D241—21no—
20C22×D241D412yes—
21Dic482—21no—
22D964—21no—
23C22⋊D241—22no—
24C22⋊D241—22no—
25C22⋊D241—22no—
26C2×Dic241—21no—
27C2×Dic241—22no—
28C2×Dic241D412yes—
29C4×D241—22no—
30C23×C241D411yescharacteristic
31C23×C241C2311yescharacteristic
32D4×C241—22no—
33C2×C4×C241—21no—
34C2×C4×C241—21no—
35C2×C4×C241—22no—
36C2×C4×C241D411yescharacteristic
37C22×C241—21no—
38C22×C241—41no—
39C22×C241—41no—
40C22×C241C2×D412yes—
41C22×C241C2×D411yescharacteristic
42C2×D241—22no—
43C2×D241—42no—
44C2×D241—81no—
45C4×C241—21no—
46C4×C241—41no—
47C4×C241—41no—
48Dic241—22no—
49Dic241—42no—
50C2×C241—22no—
51C2×C241—21no—
52C2×C241—41no—
53C2×C241#12011yescharacteristic
54D241—42no—
55C241#45811yescharacteristic, complemented, Sylow
56#458—2411nomaximal, Sylow
57H(5,𝔽2)—2411no—
58#123—2412no—
59#123D24111yescharacteristic, complemented
60#99—2411no—
61#99—2412no—
62#41—4822no—
63#41—4821no—
64C2×D4—4822no—
65C2×D4—4821no—
66C2×D4—2412no—
67C2×D4C2×D24111yescharacteristic
68C22×C4C2×D24111yescharacteristic, complemented
69C4⋊C4—21no—
70C4⋊C4—4822no—
71C22⋊C4—21no—
72C22⋊C4—4822no—
73C22⋊C4—2412no—
74C22⋊C4C2×D24111yescharacteristic, complemented
75C23—4821no—
76C23—2412no—
77C23C22⋊D24111yescharacteristic
78C23C22×D24111yescharacteristic
79Q8—4821no—
80D4—22no—
81D4—4821no—
82D4—4822no—
83D4—4822no—
84D4—4822no—
85C2×C4—22no—
86C2×C4—21no—
87C2×C4—21no—
88C2×C4—4822no—
89C2×C4—4822no—
90C2×C4—4821no—
91C2×C4—2412no—
92C2×C4C22⋊D24111yescharacteristic
93C22—21no—
94C22—41no—
95C22—41no—
96C22—19281no—
97C22—9642no—
98C22—4822no—
99C22D4×D24112yes—
100C22C2×C22⋊D24111yescharacteristic
101C4—21no—
102C4—41no—
103C4—41no—
104C4—9642no—
105C4—4822no—
106C2—21no—
107C2—22no—
108C2—41no—
109C2—9642no—
110C2#2144061011yescharacteristic
subgroupG/H
centerC2#21440610
commutator, cosocleC23×C241C23
FrattiniC23C22×D241
Fitting#123×C241C2
socleC2×C241#120
prime radical#21440687C2
hypercenter#123D241
hypocenterC241#458
LCSG ▷ C23×C241 ▷ C2×C241 ▷ C241
UCSC1 ◁ C2 ◁ C23 ◁ #123

Complements

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

NH ≅ G/Ntype
2. #21440625C2semidirect
3. #21440557C2semidirect
4. #21440556C2semidirect
5. #123×C241C2semidirect
12. C22⋊Dic241C22semidirect
15. C22×C4×C241C22semidirect
18. C22⋊C4×C241C22semidirect
55. C241#458semidirect
59. #123D241semidirect
68. C22×C4C2×D241semidirect
74. C22⋊C4C2×D241semidirect

Maximal subgroups

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

subgroupindex
#214406872
#214406252
#214405572
#214405562
#123×C2412
#458241