Finite Group Databasebeta

← #37533016

Subgroups — #37533016

Order 23168.

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

normal subgroups only

This group has 230 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, complemented
2#26041668C212yesmaximal, complemented
3#26041645C211yesmaximal, characteristic, complemented
4#26041018C212yesmaximal, complemented
5#330×C181C211yesmaximal, characteristic, complemented
6C2×#15668438—22no—
7C2×D1448—22no—
8#19153813C2211yescharacteristic
9#19153804C2212yes—
10#19153801—22no—
11#19153794—22no—
12#19153791—22no—
13#19153736—22no—
14#19153727C2211yescharacteristic
15#125×C181C2211yescharacteristic
16#97×C181C2212yes—
17C2×Dic362D412yes—
18C2×D724—22no—
19C2×D724—21no—
20C2×D724—22no—
21C2×D724D412yes—
22C2×C4×D181—22no—
23#15668438—44no—
24D1448—44no—
25C2×#13089678—21no—
26#15668428—22no—
27#15668426—22no—
28#15668426—22no—
29#15668426—22no—
30C2×C8×C181—22no—
31C4⋊C4×C181—22no—
32C4⋊C4×C181D412yes—
33C42×C181C2311yescharacteristic
34C22×D181—22no—
35C22×D181—41no—
36Dic362—42no—
37D724—42no—
38D724—42no—
39D724—42no—
40D724—42no—
41D724—82no—
42C2×Dic181—22no—
43C4×D181—82no—
44#13089678—41no—
45C2×C4×C181—22no—
46C2×C4×C181C2×D412yes—
47C2×C4×C181C2×D411yescharacteristic
48C8×C181—24no—
49C22×C181#12011yescharacteristic
50C2×D181—42no—
51C2×D181—42no—
52C2×D181—42no—
53C2×D181—82no—
54C2×D181—81no—
55C4×C181—22no—
56C4×C181—21no—
57C4×C181—42no—
58Dic181—42no—
59C2×C181#45312yes—
60C2×C181#45411yescharacteristic
61D181—82no—
62D181—161no—
63C181#165611yescharacteristic, complemented, Sylow
64#1656—1811nomaximal, Sylow
65#562—1811no—
66#496—1812no—
67#490—1811no—
68#330D18111yescharacteristic, complemented
69#329—1812no—
70C2×SD16—3622no—
71C2×D8—3622no—
72#127—1811no—
73#125C2×D18111yescharacteristic
74#124—1812no—
75#123—3622no—
76C22⋊Q8—3622no—
77#121—3622no—
78#102—3622no—
79#97—1811no—
80#97C2×D18112yes—
81C2×Q8—1812no—
82C2×D4—3622no—
83C2×D4—3622no—
84C2×D4—3621no—
85C2×D4—1812no—
86C22×C4—3622no—
87SD16—7244no—
88D8—7244no—
89C2×C8—22no—
90C2×C8—3621no—
91C4⋊C4—22no—
92C4⋊C4—3622no—
93C4⋊C4D72412yes—
94C22⋊C4—3622no—
95C22⋊C4—3622no—
96C22⋊C4—3622no—
97C42C22×D18111yescharacteristic
98C23—3622no—
99C23—7241no—
100Q8—7242no—
101D4—14482no—
102D4—7242no—
103D4—7242no—
104D4—7242no—
105D4—7242no—
106C2×C4—22no—
107C2×C4—3622no—
108C2×C4—14482no—
109C2×C4D4×D18112yes—
110C2×C4C2×D72411yescharacteristic
111C8—24no—
112C8—7241no—
113C22—14482no—
114C22—14481no—
115C22—7242no—
116C22—7242no—
117C22—7242no—
118C22#1915378611yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—7242no—
123C2—28961no—
124C2—14482no—
125C2#2604144812yes—
126C2#2604145411yescharacteristic
subgroupG/H
centerC22#19153786
commutator, cosocleC42×C181C23
FrattiniC42C22×D181
Fitting#330×C181C2
socleC22×C181#120
hypercenter#330D181
hypocenterC181#1656
LCSG ▷ C42×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
1. #26042027C2semidirect
2. #26041668C2semidirect
3. #26041645C2semidirect
4. #26041018C2semidirect
5. #330×C181C2semidirect
63. C181#1656semidirect
68. #330D181semidirect

Maximal subgroups

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

subgroupindex
#260420272
#260416682
#260416452
#260410182
#330×C1812
#1656181