Finite Group Databasebeta

← #37559990

Subgroups — #37559990

Order 23872.

334 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 103 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
1C2×#20238611C211yesmaximal, characteristic, complemented
2C2×D4×D373C211yesmaximal, characteristic, complemented
3#26045789C211yesmaximal, characteristic, complemented
4#26045788C211yesmaximal, characteristic, complemented
5#26045781C211yesmaximal, characteristic, complemented
6#26045765C214yesmaximal, complemented
7#26045762C214yesmaximal, complemented
8#26045749C211yesmaximal, characteristic, complemented
9#124×C373C211yesmaximal, characteristic, complemented
10C23×D373C2212yes—
11#20238611—24no—
12C2×D1492C2212yes—
13C2×D1492C2211yescharacteristic
14C2×D1492C2211yescharacteristic, complemented
15C2×D1492C2214yescomplemented
16C2×C22⋊D373C2212yes—
17C2×C22⋊D373C2214yescomplemented
18D4×D373—24no—
19C2×C4×D373C2211yescharacteristic
20C2×C4×D373C2212yescomplemented
21#20238589C2212yescomplemented
22#20238587C2212yes—
23#20238587C2214yescomplemented
24C4×Dic373C2211yescharacteristic, complemented
25C2×Q8×C373C2211yescharacteristic, complemented
26C2×D4×C373C2211yescharacteristic
27C22⋊C4×C373C2214yescomplemented
28C42×C373C2211yescharacteristic, complemented
29C22×D373—24no—
30C22×D373—44no—
31C22×D373C2312yes—
32C22×D373C2312yes—
33C22×D373C2311yescharacteristic
34D1492—24no—
35D1492—22no—
36D1492—22no—
37D1492—44no—
38C22⋊D373—24no—
39C22⋊D373—44no—
40C2×Dic373C2312yes—
41C2×Dic373C2311yescharacteristic
42C4×D373—24no—
43C4×D373D414yescomplemented
44C23×C373C2312yes—
45Q8×C373—22no—
46D4×C373—22no—
47C2×C4×C373C2312yes—
48C2×C4×C373C2312yes—
49C2×C4×C373C2311yescharacteristic
50C22×C373—22no—
51C22×C373—24no—
52C22×C373C2411yescharacteristic
53C2×D373—22no—
54C2×D373—22no—
55C2×D373—22no—
56C2×D373—24no—
57C2×D373—24no—
58C2×D373—48no—
59C2×D373C2×D412yes—
60C4×C373—22no—
61C4×C373—22no—
62C4×C373C2×D412yes—
63Dic373—22no—
64Dic373C2×D412yes—
65C2×C373—42no—
66C2×C373C22×D411yescharacteristic
67C2×C373H(5,𝔽2)12yes—
68D373—22no—
69D373—42no—
70D373—42no—
71C373#53511yescharacteristic, complemented, Sylow
72#535—3731nomaximal, Sylow
73C2×#41—3731no—
74C22×D4—3731no—
75#127—3731no—
76#127—3731no—
77#124—3731no—
78#124D37311yescharacteristic, complemented
79#121—3734no—
80#120—3734no—
81#117—3731no—
82C24—3732no—
83#41—7464no—
84C2×Q8C2×D37311yescharacteristic, complemented
85C2×D4—3734no—
86C2×D4—3732no—
87C2×D4—3734no—
88C2×D4—3732no—
89C2×D4—3731no—
90C2×D4—3731no—
91C2×D4—7464no—
92C2×D4C2×D37311yescharacteristic
93C22×C4—3731no—
94C22×C4—3732no—
95C4⋊C4—3732no—
96C22⋊C4—3732no—
97C22⋊C4—3734no—
98C22⋊C4C2×D37314yescomplemented
99C42—3731no—
100C42C2×D37311yescharacteristic, complemented
101C23—3731no—
102C23—3732no—
103C23—3732no—
104C23—14924no—
105C23—7464no—
106C23C22×D37312yes—
107Q8—22no—
108D4—22no—
109D4—14924no—
110D4—14924no—
111D4—7464no—
112D4—7464no—
113D4—7462no—
114D4—7462no—
115C2×C4—3731no—
116C2×C4—3734no—
117C2×C4—3732no—
118C2×C4—7464no—
119C2×C4C22×D37312yes—
120C2×C4C22×D37312yes—
121C2×C4C22×D37311yescharacteristic
122C22—22no—
123C22—24no—
124C22—3732no—
125C22—14928no—
126C22—7462no—
127C22—7462no—
128C22—7464no—
129C22—7464no—
130C22—7462no—
131C22C23×D37311yescharacteristic
132C4—22no—
133C4—22no—
134C4—3732no—
135C4—7462no—
136C4D4×D37312yes—
137C2—42no—
138C2—14922no—
139C2—14922no—
140C2—7462no—
141C2C2×D4×D37311yescharacteristic
142C2#2604583812yes—
subgroupG/H
center, FrattiniC22C23×D373
commutator, socle, cosocleC22×C373C24
Fitting#124×C373C2
hypercenter#124D373
hypocenterC373#535
LCSG ▷ C22×C373 ▷ C373
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C2×#20238611C2semidirect
2. C2×D4×D373C2semidirect
3. #26045789C2semidirect
4. #26045788C2semidirect
5. #26045781C2semidirect
6. #26045765C2semidirect
7. #26045762C2semidirect
8. #26045749C2semidirect
9. #124×C373C2semidirect
14. C2×D1492C22semidirect
15. C2×D1492C22semidirect
17. C2×C22⋊D373C22semidirect
20. C2×C4×D373C22semidirect
21. #20238589C22semidirect
23. #20238587C22semidirect
24. C4×Dic373C22semidirect
25. C2×Q8×C373C22semidirect
27. C22⋊C4×C373C22semidirect
28. C42×C373C22semidirect
43. C4×D373D4semidirect
71. C373#535semidirect
78. #124D373semidirect
84. C2×Q8C2×D373semidirect
98. C22⋊C4C2×D373semidirect
100. C42C2×D373semidirect

Maximal subgroups

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

subgroupindex
C2×#202386112
C2×D4×D3732
#260457892
#260457882
#260457812
#260457652
#260457622
#260457492
#124×C3732
#535373