Finite Group Databasebeta

← #37533019

Subgroups — #37533019

Order 23168.

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

normal subgroups only

This group has 206 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#26041669C211yesmaximal, characteristic
2#26041645C212yesmaximal, complemented
3#26041021C212yesmaximal, complemented
4#26041019C211yesmaximal, characteristic, complemented
5#563×C181C211yesmaximal, characteristic, complemented
6C2×#15668436—21no—
7C2×#15668434—21no—
8#19153813C2211yescharacteristic
9#19153736—22no—
10#19153736—22no—
11#19153727C2212yes—
12#19153727C2211yescharacteristic
13C4⋊Q8×C181—21no—
14#126×C181—22no—
15#125×C181C2212yes—
16#124×C181C2211yescharacteristic
17#123×C181—21no—
18C22⋊Q8×C181—22no—
19C2×D724—22no—
20C2×D724—21no—
21#15668436—42no—
22#15668434—42no—
23C2×#13089678—21no—
24C2×#13089678—22no—
25C2×Q8×C181—21no—
26C2×Q8×C181D411yescharacteristic
27C2×D4×C181D411yescharacteristic
28C22×C4×C181—21no—
29C4⋊C4×C181—22no—
30C4⋊C4×C181—22no—
31C4⋊C4×C181—22no—
32C4⋊C4×C181D412yes—
33C4⋊C4×C181D412yes—
34C22⋊C4×C181—22no—
35C22⋊C4×C181—21no—
36C22⋊C4×C181—22no—
37C42×C181—21no—
38C42×C181C2311yescharacteristic
39C22×D181—41no—
40D724—41no—
41D724—41no—
42D724—44no—
43#13089678—22no—
44#13089678—42no—
45C23×C181—21no—
46Q8×C181—41no—
47Q8×C181—81no—
48D4×C181—41no—
49C2×C4×C181—22no—
50C2×C4×C181—21no—
51C2×C4×C181—22no—
52C2×C4×C181—81no—
53C2×C4×C181C2×D412yes—
54C2×C4×C181C2×D411yescharacteristic
55C22×C181—41no—
56C22×C181—42no—
57C22×C181#12011yescharacteristic
58C2×D181—81no—
59C2×D181—82no—
60C4×C181—21no—
61C4×C181—22no—
62C4×C181—42no—
63C4×C181—41no—
64C4×C181—42no—
65C2×C181—81no—
66C2×C181#45311yescharacteristic
67C2×C181#45412yes—
68D181—161no—
69C181#165711yescharacteristic, complemented, Sylow
70#1657—1811nomaximal, Sylow
71#563D18111yescharacteristic, complemented
72#496—1811no—
73#490—1812no—
74#330—1812no—
75#329—1811no—
76C2×SD16—3621no—
77C2×D8—3621no—
78C4⋊Q8—21no—
79#127—1811no—
80#126—22no—
81#125C2×D18112yes—
82#124C2×D18111yescharacteristic
83#123—21no—
84C22⋊Q8—22no—
85#102—3622no—
86#102—3622no—
87#97—1812no—
88#97—1811no—
89C2×Q8—21no—
90C2×Q8C22⋊D18111yescharacteristic
91C2×D4—3622no—
92C2×D4—3621no—
93C2×D4C22⋊D18111yescharacteristic
94C22×C4—21no—
95SD16—7242no—
96D8—7242no—
97C2×C8—3621no—
98C2×C8—3622no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4C22⋊D18112yes—
103C4⋊C4C22⋊D18112yes—
104C22⋊C4—22no—
105C22⋊C4—21no—
106C22⋊C4—22no—
107C42—21no—
108C42C22×D18111yescharacteristic
109C23—21no—
110C23—7241no—
111Q8—41no—
112Q8—81no—
113D4—41no—
114D4—7241no—
115D4—7241no—
116D4—7244no—
117C2×C4—22no—
118C2×C4—21no—
119C2×C4—22no—
120C2×C4—81no—
121C2×C4C2×C22⋊D18112yes—
122C2×C4C2×C22⋊D18111yescharacteristic
123C8—3622no—
124C8—7242no—
125C22—41no—
126C22—42no—
127C22—14481no—
128C22—14482no—
129C22#1915378511yescharacteristic
130C4—21no—
131C4—22no—
132C4—42no—
133C4—41no—
134C4—42no—
135C2—81no—
136C2—28961no—
137C2#2604144711yescharacteristic
138C2#2604145212yes—
subgroupG/H
centerC22#19153785
commutator, cosocleC42×C181C23
FrattiniC42C22×D181
Fitting#563×C181C2
socleC22×C181#120
prime radical#26041669C2
hypercenter#563D181
hypocenterC181#1657
LCSG ▷ C42×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
2. #26041645C2semidirect
3. #26041021C2semidirect
4. #26041019C2semidirect
5. #563×C181C2semidirect
69. C181#1657semidirect
71. #563D181semidirect

Maximal subgroups

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

subgroupindex
#260416692
#260416452
#260410212
#260410192
#563×C1812
#1657181