Finite Group Databasebeta

← #37549144

Subgroups — #37549144

Order 23168.

208 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 56 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#26042302C211yesmaximal, characteristic
2#26042224C211yesmaximal, characteristic
3#26042132C214yesmaximal
4#26041834C211yesmaximal, characteristic
5#19153883—21no—
6#19153873C2212yes—
7#19153871—21no—
8(C2×#10624382)⋊C4—21no—
9(C2×#10624382)⋊C4—21no—
10(C2×#10624382)⋊C4C2212yes—
11#19153816C411yescharacteristic, complemented
12#19153807—21no—
13#19153796C2212yes—
14#19153792—21no—
15Q8×Dic181—21no—
16#19153781C412yescomplemented
17#19153774—21no—
18C42×D181C2211yescharacteristic
19C4⋊Q8×C181C411yescharacteristic, complemented
20C22×#10624382—22no—
21C2×#13089685—22no—
22C2×Dic362C2×C412yes—
23C2×C4×D181D411yescharacteristic
24C2×C4×D181D411yescharacteristic
25C2×C4×D181C2311yescharacteristic
26C4⋊Dic181—22no—
27C4⋊Dic181—21no—
28C4⋊Dic181—21no—
29#15668428—22no—
30#15668426—22no—
31C4×Dic181—22no—
32C4×Dic181C2×C411yescharacteristic
33C4×Dic181D411yescharacteristic
34C4×Dic181D411yescharacteristic
35C2×Q8×C181—21no—
36C4⋊C4×C181—21no—
37C4⋊C4×C181C2×C412yes—
38C42×C181C2×C411yescharacteristic
39C2×#10624382—42no—
40C2×#10624382—42no—
41C2×#10624382—42no—
42#13089685—42no—
43C22×D181C2×D411yescharacteristic
44Dic362—42no—
45C2×Dic181—22no—
46C2×Dic181C22⋊C411yescharacteristic
47C2×Dic181C22⋊C411yescharacteristic
48C2×Dic181C2×D411yescharacteristic
49C4×D181—22no—
50C4×D181—41no—
51C4×D181—41no—
52Q8×C181—42no—
53C2×C4×C181—22no—
54C2×C4×C181C22⋊C411yescharacteristic
55C2×C4×C181C22⋊C411yescharacteristic
56C2×C4×C181C22×C411yescharacteristic
57C22×C181C2×C22⋊C411yescharacteristic
58C2×D181—21no—
59C2×D181—21no—
60C2×D181#13712yes—
61#10624382—82no—
62C4×C181—21no—
63C4×C181—21no—
64C4×C181—21no—
65C4×C181—42no—
66Dic181—21no—
67Dic181—21no—
68Dic181—21no—
69Dic181—42no—
70C2×C181#41011yescharacteristic
71C2×C181#41911yescharacteristic
72C2×C181#42111yescharacteristic
73D181—22no—
74C181#151511yescharacteristic, complemented, Sylow
75#1515—1811nomaximal, Sylow
76#533—1811no—
77#432—1811no—
78#383—1811no—
79#328—1814no—
80C4⋊Q8—1811no—
81C4⋊Q8#1062438211yescharacteristic, complemented
82#125—3621no—
83C22⋊Q8—3621no—
84C22⋊Q8—1812no—
85C4×Q8—3621no—
86C4×Q8—1812no—
87#117—3621no—
88C2×C42—1811no—
89C4⋊C8—3621no—
90C22⋊C8—1812no—
91#97—3621no—
92#95—3621no—
93#95—3621no—
94#95—1812no—
95C2×Q8—21no—
96C2×Q8—1812no—
97C22×C4—3622no—
98C22×C4—1811no—
99C22×C4—1811no—
100C22×C4—1811no—
101C2×C8—3622no—
102C4⋊C4—21no—
103C4⋊C4—3622no—
104C4⋊C4—3622no—
105C4⋊C4—3621no—
106C4⋊C4—3621no—
107C4⋊C4C2×#1062438212yes—
108C22⋊C4—3622no—
109C42—3622no—
110C42—1811no—
111C42—1811no—
112C42—1811no—
113C42C2×#1062438211yescharacteristic
114C23—1811no—
115Q8—42no—
116Q8—7242no—
117C2×C4—22no—
118C2×C4—3622no—
119C2×C4—3622no—
120C2×C4—1811no—
121C2×C4—1811no—
122C2×C4—1811no—
123C2×C4—7241no—
124C2×C4—7241no—
125C2×C4—7242no—
126C2×C4—7242no—
127C2×C4—7242no—
128C2×C4C22⋊#1062438211yescharacteristic
129C2×C4C22⋊#1062438211yescharacteristic
130C2×C4C22×#1062438211yescharacteristic
131C8—7242no—
132C22—3621no—
133C22—3621no—
134C22—1812no—
135C22C2×C22⋊#1062438211yescharacteristic
136C4—21no—
137C4—21no—
138C4—21no—
139C4—42no—
140C4—3621no—
141C4—3621no—
142C4—3621no—
143C4—14482no—
144C4—7242no—
145C2—3622no—
146C2#2604226011yescharacteristic
147C2#2604227211yescharacteristic
148C2#2604227711yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10624382
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
FittingC4⋊Q8×C181C4
socleC22×C181C2×C22⋊C4
prime radicalC22×D181C2×D4
cosocleC2×C4×D181C23
hypercenterC4⋊Q8#10624382
hypocenterC181#1515
LCSG ▷ C2×C4×C181 ▷ C22×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
11. #19153816C4semidirect
16. #19153781C4semidirect
19. C4⋊Q8×C181C4semidirect
74. C181#1515semidirect
81. C4⋊Q8#10624382semidirect

Maximal subgroups

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

subgroupindex
#260423022
#260422242
#260421322
#260418342
#1515181