Finite Group Databasebeta

← #37535066

Subgroups — #37535066

Order 23168.

246 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 73 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#26041910C211yesmaximal, characteristic
2#26041319C212yesmaximal
3#26041048C211yesmaximal, characteristic
4#26041039C212yesmaximal
5#384×C181C211yesmaximal, characteristic
6C2×#15668433C2212yes—
7#19153815—22no—
8#19153807—22no—
9#19153795—22no—
10#19153775C2212yes—
11#19153749—22no—
12#19153747—22no—
13C2×C4⋊C4×C181—22no—
14C2×C4⋊C4×C181C2211yescharacteristic
15C2×C4⋊C4×C181C2211yescharacteristic
16C2×C42×C181C2211yescharacteristic
17C2×Dic362—22no—
18#15668433—22no—
19#15668433Q814yescomplemented
20C2×#13089678—22no—
21C4⋊Dic181—22no—
22C4⋊Dic181D412yes—
23C4⋊Dic181D412yes—
24#15668428—24no—
25C22⋊Dic181—22no—
26C4×Dic181—22no—
27C22×C4×C181—21no—
28C22×C4×C181—21no—
29C22×C4×C181D412yes—
30C22×C4×C181C2311yescharacteristic
31C4⋊C4×C181—22no—
32C4⋊C4×C181—22no—
33C4⋊C4×C181—44no—
34C42×C181—21no—
35C42×C181Q812yescomplemented
36Dic362—82no—
37C2×Dic181—22no—
38C2×Dic181—22no—
39#13089678—24no—
40C23×C181C2×D411yescharacteristic
41C2×C4×C181—22no—
42C2×C4×C181—22no—
43C2×C4×C181—22no—
44C2×C4×C181—22no—
45C2×C4×C181—42no—
46C2×C4×C181—42no—
47C2×C4×C181—41no—
48C2×C4×C181C2×D412yes—
49C2×C4×C181C2×Q812yes—
50C2×C4×C181C2×Q811yescharacteristic
51C2×C4×C181#4111yescharacteristic
52C22×C181—21no—
53C22×C181—21no—
54C22×C181#12011yescharacteristic
55C22×C181C22⋊Q811yescharacteristic
56C22×C181C4⋊Q811yescharacteristic
57C4×C181—24no—
58C4×C181—42no—
59C4×C181—81no—
60C4×C181C22⋊Q812yes—
61C4×C181C4⋊Q812yes—
62Dic181—42no—
63Dic181—42no—
64C2×C181—21no—
65C2×C181—21no—
66C2×C181#39511yescharacteristic
67C2×C181#45311yescharacteristic
68C2×C181#45611yescharacteristic
69C181#206011yescharacteristic, complemented, Sylow
70#2060—1811nomaximal, Sylow
71#544—1811no—
72#428—1812no—
73#384D18111yescharacteristic
74#343—1811no—
75#339—1812no—
76C2×M16—1812no—
77C4⋊Q8—3622no—
78#125—3622no—
79C22⋊Q8—3622no—
80#117—1812no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4C2×D18111yescharacteristic
83C2×C4⋊C4C2×D18111yescharacteristic
84C2×C42C2×D18111yescharacteristic
85#107—3622no—
86#106—3622no—
87C2×Q8—3622no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4C22⋊D18112yes—
91C22×C4C22×D18111yescharacteristic
92M16—3622no—
93M16—1814no—
94C2×C8—3622no—
95C4⋊C4—22no—
96C4⋊C4—22no—
97C4⋊C4—44no—
98C4⋊C4—3622no—
99C4⋊C4—3624no—
100C4⋊C4—1812no—
101C4⋊C4—1812no—
102C22⋊C4—3622no—
103C42—21no—
104C42—3622no—
105C42Dic36212yescomplemented
106C23C2×C22⋊D18111yescharacteristic
107Q8—14482no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—22no—
111C2×C4—22no—
112C2×C4—42no—
113C2×C4—42no—
114C2×C4—41no—
115C2×C4—3622no—
116C2×C4—3622no—
117C2×C4D4×D18112yes—
118C2×C4Q8×D18112yes—
119C2×C4C2×Dic36211yescharacteristic
120C2×C4#1566844911yescharacteristic
121C8—3624no—
122C22—21no—
123C22—21no—
124C22#1915378411yescharacteristic
125C22#1915379511yescharacteristic
126C22#1915381411yescharacteristic
127C4—24no—
128C4—42no—
129C4—81no—
130C4—7242no—
131C4—7242no—
132C4#1915379412yes—
133C4#1915381512yes—
134C2—21no—
135C2—21no—
136C2#2604119911yescharacteristic
137C2#2604144611yescharacteristic
138C2#2604146111yescharacteristic
subgroupG/H
centerC22#19153784
commutator, cosocleC22×C4×C181C23
FrattiniC22×C4C22×D181
Fitting#384×C181C2
socleC22×C181#120
prime radicalC23×C181C2×D4
hypercenter#384D181
hypocenterC181#2060
LCSG ▷ C22×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #384

Complements

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

NH ≅ G/Ntype
19. #15668433Q8semidirect
35. C42×C181Q8semidirect
69. C181#2060semidirect
105. C42Dic362semidirect

Maximal subgroups

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

subgroupindex
#260419102
#260413192
#260410482
#260410392
#384×C1812
#2060181