Finite Group Databasebeta

← #37533028

Subgroups — #37533028

Order 23168.

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

normal subgroups only

This group has 228 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#26042021C211yesmaximal, characteristic
2#26041674C211yesmaximal, characteristic, complemented
3#26041639C212yesmaximal, complemented
4#26041018C212yesmaximal, complemented
5#329×C181C211yesmaximal, characteristic, complemented
6C2×#15668439—21no—
7C2×#15668437—21no—
8#19153816C2211yescharacteristic, complemented
9#19153805—21no—
10#19153804C2212yes—
11#19153802—22no—
12#19153786—22no—
13#19153784—21no—
14#19153741—22no—
15#19153738—22no—
16#19153727C2211yescharacteristic, complemented
17#124×C181C2211yescharacteristic
18#97×C181C2212yescomplemented
19C23×D181—21no—
20C2×Dic362—21no—
21C2×Dic362D412yes—
22C2×D724—21no—
23C2×D724D412yes—
24C2×C22⋊D181—22no—
25#15668439—42no—
26#15668437—42no—
27C2×#13089678—21no—
28C4⋊Dic181—22no—
29C22⋊Dic181—21no—
30#15668426—22no—
31#15668426—22no—
32#15668426—22no—
33C4×Dic181—21no—
34C2×Q8×C181D411yescharacteristic
35C2×D4×C181D411yescharacteristic
36C2×C8×C181—22no—
37C22⋊C4×C181—22no—
38C42×C181C2311yescharacteristic
39C22×D181—22no—
40C22×D181—81no—
41C22×D181—82no—
42Dic362—42no—
43Dic362—41no—
44Dic362—41no—
45D724—42no—
46D724—81no—
47C22⋊D181—82no—
48C2×Dic181—22no—
49C2×Dic181—41no—
50#13089678—22no—
51C23×C181—21no—
52Q8×C181—41no—
53D4×C181—41no—
54C2×C4×C181—21no—
55C2×C4×C181C2×D412yes—
56C2×C4×C181C2×D411yescharacteristic
57C8×C181—42no—
58C22×C181—41no—
59C22×C181—42no—
60C22×C181#12011yescharacteristic
61C2×D181—42no—
62C2×D181—42no—
63C2×D181—42no—
64C2×D181—84no—
65C4×C181—21no—
66C4×C181—22no—
67C4×C181—41no—
68Dic181—42no—
69Dic181—81no—
70C2×C181—81no—
71C2×C181#45311yescharacteristic
72C2×C181#45512yes—
73D181—82no—
74C181#165911yescharacteristic, complemented, Sylow
75#1659—1811nomaximal, Sylow
76H(3,𝔽4)—1811no—
77#497—1811no—
78#489—1812no—
79#329—1812no—
80#329D18111yescharacteristic, complemented
81C2×Q16—3621no—
82C2×SD16—3621no—
83C4⋊Q8—1811no—
84#124—3622no—
85#124—3621no—
86#124—1812no—
87#124C2×D18111yescharacteristic
88#120—3622no—
89#120—3621no—
90#103—3622no—
91#102—3622no—
92#97—1811no—
93#97C2×D18112yescomplemented
94C24—3621no—
95C2×Q8—3621no—
96C2×Q8—1812no—
97C2×Q8D72411yescharacteristic
98C2×D4—3621no—
99C2×D4—3622no—
100C2×D4—1812no—
101C2×D4D72411yescharacteristic
102Q16—7242no—
103SD16—7242no—
104C2×C8—22no—
105C2×C8—3621no—
106C4⋊C4—3622no—
107C22⋊C4—22no—
108C22⋊C4—3622no—
109C22⋊C4—3621no—
110C22⋊C4—3622no—
111C22⋊C4—3622no—
112C42—3621no—
113C42C22×D18111yescharacteristic
114C23—21no—
115C23—3622no—
116C23—14481no—
117C23—14482no—
118Q8—41no—
119Q8—7241no—
120Q8—7241no—
121Q8—7242no—
122D4—41no—
123D4—14481no—
124D4—14482no—
125D4—7242no—
126C2×C4—21no—
127C2×C4—3622no—
128C2×C4—7241no—
129C2×C4D4×D18112yes—
130C2×C4C2×D72411yescharacteristic
131C8—42no—
132C8—3622no—
133C22—41no—
134C22—42no—
135C22—14484no—
136C22—7242no—
137C22—7242no—
138C22—7242no—
139C22#1915378611yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—14481no—
144C4—7242no—
145C2—81no—
146C2—14482no—
147C2#2604144911yescharacteristic
148C2#2604145812yes—
subgroupG/H
centerC22#19153786
commutator, cosocleC42×C181C23
FrattiniC42C22×D181
Fitting#329×C181C2
socleC22×C181#120
prime radical#26042021C2
hypercenter#329D181
hypocenterC181#1659
LCSG ▷ C42×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #26041674C2semidirect
3. #26041639C2semidirect
4. #26041018C2semidirect
5. #329×C181C2semidirect
8. #19153816C22semidirect
16. #19153727C22semidirect
18. #97×C181C22semidirect
74. C181#1659semidirect
80. #329D181semidirect
93. #97C2×D181semidirect

Maximal subgroups

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

subgroupindex
#260420212
#260416742
#260416392
#260410182
#329×C1812
#1659181