Finite Group Databasebeta

← #37548845

Subgroups — #37548845

Order 23168.

112 conjugacy classes of subgroups; only the 20 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 20 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#26042185C211yesmaximal, characteristic, complemented
2#26042135C211yesmaximal, characteristic, complemented
3#26041454C211yesmaximal, characteristic
4#19153875—21no—
5#19153871—21no—
6#19153864C411yescharacteristic, complemented
7#19153844—21no—
8#19153805C2211yescharacteristic
9#19153742—21no—
10#101×C181C411yescharacteristic, complemented
11C2×#13089685—41no—
12C22⋊#10624382—41no—
13#15668450—41no—
14#15668449—21no—
15#15668449—41no—
16Q8×D181—41no—
17C2×D724D411yescharacteristic
18#15668438—41no—
19D1448—41no—
20#15668426—41no—
21C4×Dic181D411yescharacteristic
22C2×Q8×C181C2×C411yescharacteristic
23M16×C181—21no—
24C2×#10624382—81no—
25#13089685—42no—
26C22×D181—21no—
27Dic362—21no—
28Dic362—41no—
29D724—21no—
30D724—41no—
31D724—41no—
32C22⋊D181—41no—
33C2×Dic181—21no—
34C4×D181—41no—
35C4×D181—42no—
36Q8×C181—41no—
37C2×C4×C181—21no—
38C2×C4×C181C22⋊C411yescharacteristic
39C8×C181—41no—
40C22×C181#9911yescharacteristic
41C2×D181—41no—
42C2×D181—41no—
43C2×D181—81no—
44#10624382—161no—
45C4×C181—21no—
46C4×C181—41no—
47Dic181—41no—
48Dic181—41no—
49C2×C181—21no—
50C2×C181AΣL(1,𝔽16)11yescharacteristic
51D181—81no—
52D181—81no—
53C181#140411yescharacteristic, complemented, Sylow
54#1404—1811nomaximal, Sylow
55#454—1811no—
56#354—1811no—
57#329—1811no—
582−1+4—1811no—
59AGL(1,ℤ/8)—3621no—
60#124—1811no—
61#104—3621no—
62#101#1062438211yescharacteristic, complemented
63#99—3621no—
64#97—3621no—
65#41—3621no—
66#41—7241no—
67#41—7241no—
68C2×Q8—7241no—
69C2×Q8C2×#1062438211yescharacteristic
70C2×D4—1811no—
71SD16—7241no—
72D8—7241no—
73M16—21no—
74C2×C8—7241no—
75C22⋊C4—7241no—
76C22⋊C4—7241no—
77C42—1811no—
78C23—3621no—
79Q8—41no—
80Q8—3621no—
81Q8—7241no—
82D4—3621no—
83D4—7241no—
84D4—7241no—
85D4—7241no—
86C2×C4—21no—
87C2×C4—3621no—
88C2×C4—14481no—
89C2×C4—7241no—
90C2×C4—7242no—
91C2×C4C22⋊#1062438211yescharacteristic
92C8—41no—
93C8—7242no—
94C22—14481no—
95C22—7241no—
96C22—7241no—
97C22#1915387511yescharacteristic
98C4—21no—
99C4—41no—
100C4—28961no—
101C4—7241no—
102C4—7241no—
103C2—21no—
104C2—14481no—
105C2—14481no—
106C2#2604218011yescharacteristic
subgroupG/H
centerC2#26042180
commutatorC2×Q8×C181C2×C4
FrattiniC2×Q8C2×#10624382
Fitting#101×C181C4
socleC2×C181AΣL(1,𝔽16)
prime radical#26041454C2
cosocle#19153805C22
hypercenter#101#10624382
hypocenterC181#1404
LCSG ▷ C2×Q8×C181 ▷ C2×C4×C181 ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C2×Q8 ◁ #101

Complements

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

NH ≅ G/Ntype
1. #26042185C2semidirect
2. #26042135C2semidirect
6. #19153864C4semidirect
10. #101×C181C4semidirect
53. C181#1404semidirect
62. #101#10624382semidirect

Maximal subgroups

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

subgroupindex
#260421852
#260421352
#260414542
#1404181