Finite Group Databasebeta

← #37559985

Subgroups — #37559985

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×#20238609C211yesmaximal, characteristic, complemented
2C2×D4×D373C211yesmaximal, characteristic, complemented
3#26045788C211yesmaximal, characteristic, complemented
4#26045780C211yesmaximal, characteristic, complemented
5#26045779C211yesmaximal, characteristic, complemented
6#26045764C214yesmaximal, complemented
7#26045760C214yesmaximal, complemented
8#26045749C211yesmaximal, characteristic, complemented
9#127×C373C211yesmaximal, characteristic, complemented
10C23×D373C2212yes—
11#20238609—24no—
12C2×Dic746C2211yescharacteristic, complemented
13C2×D1492C2211yescharacteristic
14C2×C22⋊D373C2212yes—
15C2×C22⋊D373C2212yes—
16C2×C22⋊D373C2214yescomplemented
17D4×D373—24no—
18C22×Dic373C2212yescomplemented
19C2×C4×D373C2211yescharacteristic
20#20238589C2212yescomplemented
21C22⋊Dic373C2214yescomplemented
22#20238587C2212yes—
23#20238587C2214yescomplemented
24C4×Dic373C2211yescharacteristic, complemented
25C2×D4×C373C2211yescharacteristic
26C2×D4×C373C2211yescharacteristic, complemented
27C2×D4×C373C2214yescomplemented
28C42×C373C2211yescharacteristic, complemented
29C22×D373—24no—
30C22×D373—44no—
31C22×D373C2312yes—
32C22×D373C2311yescharacteristic
33Dic746—22no—
34D1492—22no—
35C22⋊D373—24no—
36C22⋊D373—24no—
37C22⋊D373—44no—
38C2×Dic373—24no—
39C2×Dic373C2312yes—
40C2×Dic373C2312yes—
41C2×Dic373C2311yescharacteristic
42C4×D373D414yescomplemented
43C23×C373C2312yes—
44C23×C373C2312yes—
45D4×C373—22no—
46D4×C373—22no—
47D4×C373—44no—
48C2×C4×C373C2312yes—
49C2×C4×C373C2311yescharacteristic
50C22×C373—22no—
51C22×C373—22no—
52C22×C373—24no—
53C22×C373—24no—
54C22×C373C2411yescharacteristic
55C2×D373—22no—
56C2×D373—22no—
57C2×D373—24no—
58C2×D373—48no—
59C2×D373C2×D412yes—
60C4×C373—22no—
61C4×C373C2×D412yes—
62Dic373—22no—
63Dic373—22no—
64Dic373C2×D412yes—
65C2×C373—42no—
66C2×C373—42no—
67C2×C373C22×D411yescharacteristic
68C2×C373H(5,𝔽2)12yes—
69D373—22no—
70D373—42no—
71C373#53511yescharacteristic, complemented, Sylow
72#535—3731nomaximal, Sylow
73C2×#41—3731no—
74C22×D4—3731no—
75#127—3731no—
76#127D37311yescharacteristic, complemented
77#124—3731no—
78#124—3731no—
79#121—3734no—
80#120—3734no—
81#117—3731no—
82C24—3732no—
83#41—7464no—
84C2×Q8—3731no—
85C2×D4—3734no—
86C2×D4—3732no—
87C2×D4—3732no—
88C2×D4—3731no—
89C2×D4—7464no—
90C2×D4C2×D37311yescharacteristic
91C2×D4C2×D37311yescharacteristic, complemented
92C2×D4C2×D37314yescomplemented
93C22×C4—3731no—
94C22×C4—3732no—
95C4⋊C4—3732no—
96C22⋊C4—3734no—
97C22⋊C4—3732no—
98C22⋊C4—3734no—
99C42—3731no—
100C42C2×D37311yescharacteristic, complemented
101C23—3731no—
102C23—3732no—
103C23—14924no—
104C23—7464no—
105C23C22×D37312yes—
106C23C22×D37312yes—
107Q8—7462no—
108D4—22no—
109D4—22no—
110D4—44no—
111D4—14924no—
112D4—7464no—
113D4—7464no—
114D4—7462no—
115C2×C4—3731no—
116C2×C4—3734no—
117C2×C4—3732no—
118C2×C4—3732no—
119C2×C4—7464no—
120C2×C4C22×D37312yes—
121C2×C4C22×D37311yescharacteristic
122C22—22no—
123C22—22no—
124C22—24no—
125C22—24no—
126C22—3732no—
127C22—14928no—
128C22—7462no—
129C22—7464no—
130C22—7462no—
131C22C23×D37311yescharacteristic
132C4—22no—
133C4—3732no—
134C4—7462no—
135C4—7462no—
136C4D4×D37312yes—
137C2—42no—
138C2—42no—
139C2—14922no—
140C2—7462no—
141C2C2×D4×D37311yescharacteristic
142C2#2604583712yes—
subgroupG/H
center, FrattiniC22C23×D373
commutator, socle, cosocleC22×C373C24
Fitting#127×C373C2
hypercenter#127D373
hypocenterC373#535
LCSG ▷ C22×C373 ▷ C373
UCSC1 ◁ C22 ◁ #127

Complements

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

NH ≅ G/Ntype
1. C2×#20238609C2semidirect
2. C2×D4×D373C2semidirect
3. #26045788C2semidirect
4. #26045780C2semidirect
5. #26045779C2semidirect
6. #26045764C2semidirect
7. #26045760C2semidirect
8. #26045749C2semidirect
9. #127×C373C2semidirect
12. C2×Dic746C22semidirect
16. C2×C22⋊D373C22semidirect
18. C22×Dic373C22semidirect
20. #20238589C22semidirect
21. C22⋊Dic373C22semidirect
23. #20238587C22semidirect
24. C4×Dic373C22semidirect
26. C2×D4×C373C22semidirect
27. C2×D4×C373C22semidirect
28. C42×C373C22semidirect
42. C4×D373D4semidirect
71. C373#535semidirect
76. #127D373semidirect
91. C2×D4C2×D373semidirect
92. C2×D4C2×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×#202386092
C2×D4×D3732
#260457882
#260457802
#260457792
#260457642
#260457602
#260457492
#127×C3732
#535373