Finite Group Databasebeta

← #37560009

Subgroups — #37560009

Order 23872.

250 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 95 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#26045787C212yesmaximal, complemented
2#26045778C212yesmaximal, complemented
3#26045776C212yesmaximal, complemented
4#26045771C211yesmaximal, characteristic, complemented
5#26045764C212yesmaximal, complemented
6#26045760C212yesmaximal, complemented
7D4×Dic373C212yesmaximal, complemented
8C2×#20238587C211yesmaximal, characteristic, complemented
9#121×C373C211yesmaximal, characteristic, complemented
10C23×D373C2211yescharacteristic
11C2×Dic746C2211yescharacteristic, complemented
12C2×C22⋊D373C2212yes—
13C2×C22⋊D373C2212yes—
14C22×Dic373C2211yescharacteristic
15C22×Dic373C2212yescomplemented
16C4⋊Dic373C2212yescomplemented
17C4⋊Dic373C2211yescharacteristic, complemented
18#20238589C2212yescomplemented
19C22⋊Dic373C2212yes—
20C22⋊Dic373C2212yescomplemented
21#20238587C2212yes—
22#20238587C2212yes—
23#20238587C2212yes—
24#20238587C2212yes—
25C4×Dic373C2212yescomplemented
26C2×D4×C373C2212yes—
27C2×D4×C373C2211yescharacteristic, complemented
28C22×C4×C373C2211yescharacteristic
29C4⋊C4×C373C2211yescharacteristic, complemented
30C22⋊C4×C373C2212yescomplemented
31C22×D373—22no—
32C22×D373—41no—
33C22×D373—41no—
34C22×D373C2312yes—
35Dic746—41no—
36C22⋊D373—42no—
37C22⋊D373—42no—
38C2×Dic373—41no—
39C2×Dic373—42no—
40C2×Dic373C2312yes—
41C2×Dic373C2312yes—
42C2×Dic373C2312yes—
43C23×C373C2312yes—
44C23×C373C2311yescharacteristic
45D4×C373—24no—
46D4×C373—41no—
47C2×C4×C373—41no—
48C2×C4×C373C2311yescharacteristic
49C2×C4×C373C2312yes—
50C2×C4×C373C2311yescharacteristic
51C22×C373—21no—
52C22×C373—21no—
53C22×C373—22no—
54C22×C373—22no—
55C22×C373—22no—
56C22×C373#4112yes—
57C22×C373C2411yescharacteristic
58C2×D373—22no—
59C2×D373—22no—
60C2×D373—22no—
61C2×D373—44no—
62C4×C373—22no—
63C4×C373—21no—
64C4×C373—21no—
65Dic373—22no—
66Dic373—22no—
67Dic373—22no—
68C2×C373—22no—
69C2×C373—42no—
70C2×C373C2×#4111yescharacteristic
71C2×C373H(5,𝔽2)11yescharacteristic
72C2×C373H(5,𝔽2)11yescharacteristic
73D373—42no—
74C373#53811yescharacteristic, complemented, Sylow
75#538—3731nomaximal, Sylow
76#126—3732no—
77#124—3732no—
78#123—3732no—
79C22⋊Q8—3731no—
80#121—3732no—
81#121D37311yescharacteristic, complemented
82#120—3732no—
83C4×D4—3732no—
84C2×C22⋊C4—3731no—
85C24—3731no—
86C2×Q8—3731no—
87C2×D4—3732no—
88C2×D4—3732no—
89C2×D4C2×D37312yes—
90C2×D4C2×D37311yescharacteristic, complemented
91C22×C4—3731no—
92C22×C4—3732no—
93C22×C4C2×D37311yescharacteristic
94C4⋊C4—3732no—
95C4⋊C4—3732no—
96C4⋊C4—3731no—
97C4⋊C4C2×D37311yescharacteristic, complemented
98C22⋊C4—3732no—
99C22⋊C4—3732no—
100C22⋊C4—3732no—
101C22⋊C4—3732no—
102C22⋊C4—3732no—
103C22⋊C4—3732no—
104C22⋊C4C2×D37312yescomplemented
105C42—3732no—
106C23—3732no—
107C23—14921no—
108C23—14921no—
109C23—7462no—
110C23C22×D37312yes—
111C23C22×D37311yescharacteristic
112Q8—14921no—
113D4—24no—
114D4—41no—
115D4—14922no—
116D4—14922no—
117C2×C4—41no—
118C2×C4—3732no—
119C2×C4—3732no—
120C2×C4—3732no—
121C2×C4—14921no—
122C2×C4—14922no—
123C2×C4C22×D37312yes—
124C2×C4C22×D37311yescharacteristic
125C2×C4C22×D37311yescharacteristic
126C22—21no—
127C22—21no—
128C22—22no—
129C22—22no—
130C22—22no—
131C22—14924no—
132C22—7462no—
133C22—7462no—
134C22—7462no—
135C22#2023860912yes—
136C22C23×D37311yescharacteristic
137C4—22no—
138C4—21no—
139C4—21no—
140C4—7462no—
141C4—7462no—
142C4—7462no—
143C2—22no—
144C2—42no—
145C2—14922no—
146C2C2×#2023860911yescharacteristic
147C2#2604583711yescharacteristic
148C2#2604583811yescharacteristic
subgroupG/H
center, FrattiniC22C23×D373
commutator, socle, cosocleC22×C373C24
Fitting#121×C373C2
hypercenter#121D373
hypocenterC373#538
LCSG ▷ C22×C373 ▷ C373
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #26045787C2semidirect
2. #26045778C2semidirect
3. #26045776C2semidirect
4. #26045771C2semidirect
5. #26045764C2semidirect
6. #26045760C2semidirect
7. D4×Dic373C2semidirect
8. C2×#20238587C2semidirect
9. #121×C373C2semidirect
11. C2×Dic746C22semidirect
15. C22×Dic373C22semidirect
16. C4⋊Dic373C22semidirect
17. C4⋊Dic373C22semidirect
18. #20238589C22semidirect
20. C22⋊Dic373C22semidirect
25. C4×Dic373C22semidirect
27. C2×D4×C373C22semidirect
29. C4⋊C4×C373C22semidirect
30. C22⋊C4×C373C22semidirect
74. C373#538semidirect
81. #121D373semidirect
90. C2×D4C2×D373semidirect
97. C4⋊C4C2×D373semidirect
104. C22⋊C4C2×D373semidirect

Maximal subgroups

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

subgroupindex
#260457872
#260457782
#260457762
#260457712
#260457642
#260457602
D4×Dic3732
C2×#202385872
#121×C3732
#538373