Finite Group Databasebeta

← #37533014

Subgroups — #37533014

Order 23168.

230 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 49 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#26041669C212yesmaximal, complemented
2#26041645C211yesmaximal, characteristic, complemented
3#26041021C211yesmaximal, characteristic, complemented
4#26041019C212yesmaximal, complemented
5#562×C181C211yesmaximal, characteristic, complemented
6C2×#15668436—22no—
7C2×#15668434—22no—
8#19153813C2211yescharacteristic
9#19153736—22no—
10#19153727C2211yescharacteristic
11#19153727C2212yes—
12#125×C181C2211yescharacteristic
13#124×C181C2212yes—
14#123×C181—22no—
15C22⋊Q8×C181—22no—
16#121×C181—22no—
17C2×D724—21no—
18C2×D724—22no—
19#15668436—44no—
20#15668434—44no—
21C2×#13089678—22no—
22C2×#13089678—21no—
23C2×Q8×C181D412yes—
24C2×D4×C181—22no—
25C2×D4×C181D412yes—
26C22×C4×C181—22no—
27C4⋊C4×C181—22no—
28C4⋊C4×C181—22no—
29C4⋊C4×C181D412yes—
30C22⋊C4×C181—22no—
31C22⋊C4×C181—22no—
32C22⋊C4×C181—22no—
33C42×C181C2311yescharacteristic
34C22×D181—41no—
35D724—42no—
36D724—42no—
37D724—42no—
38#13089678—24no—
39#13089678—41no—
40C23×C181—22no—
41Q8×C181—42no—
42D4×C181—42no—
43D4×C181—82no—
44C2×C4×C181—22no—
45C2×C4×C181—22no—
46C2×C4×C181—82no—
47C2×C4×C181C2×D411yescharacteristic
48C2×C4×C181C2×D412yes—
49C22×C181—42no—
50C22×C181—42no—
51C22×C181—42no—
52C22×C181#12011yescharacteristic
53C2×D181—82no—
54C2×D181—81no—
55C4×C181—22no—
56C4×C181—21no—
57C4×C181—42no—
58C4×C181—42no—
59C2×C181—82no—
60C2×C181#45312yes—
61C2×C181#45411yescharacteristic
62D181—161no—
63C181#165611yescharacteristic, complemented, Sylow
64#1656—1811nomaximal, Sylow
65#562D18111yescharacteristic, complemented
66#496—1812no—
67#490—1811no—
68#330—1811no—
69#329—1812no—
70C2×SD16—3622no—
71C2×D8—3622no—
72#127—1811no—
73#125C2×D18111yescharacteristic
74#124C2×D18112yes—
75#123—22no—
76C22⋊Q8—22no—
77#121—22no—
78#102—3622no—
79#97—1811no—
80#97—1812no—
81C2×Q8C22⋊D18112yes—
82C2×D4—22no—
83C2×D4—3621no—
84C2×D4—3622no—
85C2×D4C22⋊D18112yes—
86C22×C4—22no—
87SD16—7244no—
88D8—7244no—
89C2×C8—3622no—
90C2×C8—3621no—
91C4⋊C4—22no—
92C4⋊C4—22no—
93C4⋊C4C22⋊D18112yes—
94C22⋊C4—22no—
95C22⋊C4—22no—
96C22⋊C4—22no—
97C42C22×D18111yescharacteristic
98C23—22no—
99C23—7241no—
100Q8—42no—
101D4—42no—
102D4—82no—
103D4—7242no—
104D4—7242no—
105D4—7242no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—82no—
109C2×C4C2×C22⋊D18111yescharacteristic
110C2×C4C2×C22⋊D18112yes—
111C8—3624no—
112C8—7241no—
113C22—42no—
114C22—42no—
115C22—42no—
116C22—14482no—
117C22—14481no—
118C22#1915378511yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—42no—
123C2—82no—
124C2—28961no—
125C2#2604144712yes—
126C2#2604145211yescharacteristic
subgroupG/H
centerC22#19153785
commutator, cosocleC42×C181C23
FrattiniC42C22×D181
Fitting#562×C181C2
socleC22×C181#120
hypercenter#562D181
hypocenterC181#1656
LCSG ▷ C42×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #562

Complements

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

NH ≅ G/Ntype
1. #26041669C2semidirect
2. #26041645C2semidirect
3. #26041021C2semidirect
4. #26041019C2semidirect
5. #562×C181C2semidirect
63. C181#1656semidirect
65. #562D181semidirect

Maximal subgroups

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

subgroupindex
#260416692
#260416452
#260410212
#260410192
#562×C1812
#1656181