Finite Group Databasebeta

← #37533044

Subgroups — #37533044

Order 23168.

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

normal subgroups only

This group has 196 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#26041649C211yesmaximal, characteristic, complemented
2#26041638C212yesmaximal, complemented
3#26041021C211yesmaximal, characteristic, complemented
4#26041019C212yesmaximal, complemented
5#562×C181C211yesmaximal, characteristic
6#19153816C2211yescharacteristic
7#19153740—22no—
8#19153739—22no—
9#19153737—22no—
10#19153727C2212yes—
11#19153727C2211yescharacteristic
12#125×C181C2211yescharacteristic
13#124×C181C2212yes—
14#123×C181—22no—
15C22⋊Q8×C181—22no—
16#121×C181—22no—
17C2×Dic362—21no—
18C2×#13089678—22no—
19C2×#13089678—21no—
20C4⋊Dic181—22no—
21C2×Q8×C181D412yes—
22C2×D4×C181—22no—
23C2×D4×C181D412yes—
24C22×C4×C181—22no—
25C4⋊C4×C181—22no—
26C4⋊C4×C181—22no—
27C4⋊C4×C181D412yes—
28C22⋊C4×C181—22no—
29C22⋊C4×C181—22no—
30C22⋊C4×C181—22no—
31C42×C181C2311yescharacteristic
32Dic362—42no—
33C2×Dic181—41no—
34#13089678—41no—
35#13089678—42no—
36C23×C181—22no—
37Q8×C181—42no—
38D4×C181—42no—
39D4×C181—82no—
40C2×C4×C181—22no—
41C2×C4×C181—22no—
42C2×C4×C181—82no—
43C2×C4×C181C2×D411yescharacteristic
44C2×C4×C181C2×D412yes—
45C22×C181—42no—
46C22×C181—42no—
47C22×C181—42no—
48C22×C181#12011yescharacteristic
49C4×C181—21no—
50C4×C181—22no—
51C4×C181—42no—
52C4×C181—42no—
53Dic181—81no—
54C2×C181—82no—
55C2×C181#45411yescharacteristic
56C2×C181#45512yes—
57C181#166211yescharacteristic, complemented, Sylow
58#1662—1811nomaximal, Sylow
59#562D18111yescharacteristic
60#491—1811no—
61#489—1812no—
62#330—1811no—
63#329—1812no—
64C4⋊Q8—1811no—
65#125C2×D18111yescharacteristic
66#124C2×D18112yes—
67#123—22no—
68C22⋊Q8—22no—
69#121—22no—
70#103—3622no—
71#103—3622no—
72#102—3622no—
73#97—1811no—
74#97—1812no—
75C2×Q8—3621no—
76C2×Q8C22⋊D18112yes—
77C2×D4—22no—
78C2×D4C22⋊D18112yes—
79C22×C4—22no—
80C2×C8—3622no—
81C2×C8—3621no—
82C4⋊C4—22no—
83C4⋊C4—22no—
84C4⋊C4—3622no—
85C4⋊C4C22⋊D18112yes—
86C22⋊C4—22no—
87C22⋊C4—22no—
88C22⋊C4—22no—
89C42C22×D18111yescharacteristic
90C23—22no—
91Q8—42no—
92Q8—7242no—
93D4—42no—
94D4—82no—
95C2×C4—22no—
96C2×C4—22no—
97C2×C4—82no—
98C2×C4—7241no—
99C2×C4C2×C22⋊D18111yescharacteristic
100C2×C4C2×C22⋊D18112yes—
101C8—7241no—
102C8—7242no—
103C22—42no—
104C22—42no—
105C22—42no—
106C22#1915378511yescharacteristic
107C4—21no—
108C4—22no—
109C4—42no—
110C4—42no—
111C4—14481no—
112C2—82no—
113C2#2604145211yescharacteristic
114C2#2604145712yes—
subgroupG/H
centerC22#19153785
commutator, cosocleC42×C181C23
FrattiniC42C22×D181
Fitting, prime radical#562×C181C2
socleC22×C181#120
hypercenter#562D181
hypocenterC181#1662
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. #26041649C2semidirect
2. #26041638C2semidirect
3. #26041021C2semidirect
4. #26041019C2semidirect
57. C181#1662semidirect

Maximal subgroups

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

subgroupindex
#260416492
#260416382
#260410212
#260410192
#562×C1812
#1662181