Finite Group Databasebeta

← #37533042

Subgroups — #37533042

Order 23168.

196 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 45 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#26042027C211yesmaximal, characteristic
2#26041641C212yesmaximal, complemented
3#26041021C211yesmaximal, characteristic, complemented
4#26041018C212yesmaximal, complemented
5#491×C181C211yesmaximal, characteristic, complemented
6#19153804C2212yes—
7#19153801—22no—
8#19153794—22no—
9#19153791—22no—
10#19153739—22no—
11#19153736—22no—
12#19153727C2212yes—
13C4⋊Q8×C181C2211yescharacteristic
14#125×C181C2211yescharacteristic
15#103×C181—22no—
16#97×C181C2211yescharacteristic
17C2×Dic362D412yes—
18C2×D724—22no—
19C2×D724D412yes—
20C2×C4×D181—22no—
21C2×#13089678—22no—
22#15668428—22no—
23#15668426—22no—
24#15668426—22no—
25#15668426—22no—
26C2×Q8×C181—21no—
27C2×C8×C181—21no—
28C4⋊C4×C181—22no—
29C4⋊C4×C181—22no—
30C4⋊C4×C181D412yes—
31C42×C181C2311yescharacteristic
32C22×D181—22no—
33Dic362—42no—
34D724—42no—
35D724—82no—
36C2×Dic181—22no—
37C4×D181—82no—
38#13089678—42no—
39Q8×C181—42no—
40C2×C4×C181—22no—
41C2×C4×C181—41no—
42C2×C4×C181C2×D412yes—
43C2×C4×C181C2×D411yescharacteristic
44C8×C181—41no—
45C22×C181#12011yescharacteristic
46C2×D181—42no—
47C2×D181—42no—
48C2×D181—42no—
49C4×C181—21no—
50C4×C181—22no—
51C4×C181—42no—
52C4×C181—81no—
53Dic181—42no—
54C2×C181#45411yescharacteristic
55C2×C181#45512yes—
56D181—82no—
57C181#166211yescharacteristic, complemented, Sylow
58#1662—1811nomaximal, Sylow
59#562—1811no—
60#491D18111yescharacteristic, complemented
61#489—1812no—
62#330—1811no—
63#329—1812no—
64C4⋊Q8C2×D18111yescharacteristic
65#125C2×D18111yescharacteristic
66#124—1812no—
67#123—3622no—
68C22⋊Q8—3622no—
69#121—3622no—
70#103—22no—
71#103—3622no—
72#102—3622no—
73#97—1812no—
74#97C2×D18111yescharacteristic
75C2×Q8—21no—
76C2×Q8—1812no—
77C2×D4—3622no—
78C2×D4—1812no—
79C22×C4—3622no—
80C2×C8—21no—
81C2×C8—3622no—
82C4⋊C4—22no—
83C4⋊C4—22no—
84C4⋊C4—3622no—
85C4⋊C4C22⋊D18112yes—
86C22⋊C4—3622no—
87C22⋊C4—3622no—
88C22⋊C4—3622no—
89C42C22×D18111yescharacteristic
90C23—3622no—
91Q8—42no—
92Q8—7242no—
93D4—14482no—
94D4—7242no—
95C2×C4—22no—
96C2×C4—41no—
97C2×C4—3622no—
98C2×C4—14482no—
99C2×C4D4×D18112yes—
100C2×C4C2×C22⋊D18111yescharacteristic
101C8—41no—
102C8—7242no—
103C22—7242no—
104C22—7242no—
105C22—7242no—
106C22#1915378411yescharacteristic
107C4—21no—
108C4—22no—
109C4—42no—
110C4—81no—
111C4—7242no—
112C2—14482no—
113C2#2604145111yescharacteristic
114C2#2604145512yes—
subgroupG/H
centerC22#19153784
commutator, cosocleC42×C181C23
FrattiniC42C22×D181
Fitting#491×C181C2
socleC22×C181#120
prime radical#26042027C2
hypercenter#491D181
hypocenterC181#1662
LCSG ▷ C42×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
2. #26041641C2semidirect
3. #26041021C2semidirect
4. #26041018C2semidirect
5. #491×C181C2semidirect
57. C181#1662semidirect
60. #491D181semidirect

Maximal subgroups

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

subgroupindex
#260420272
#260416412
#260410212
#260410182
#491×C1812
#1662181