Finite Group Databasebeta

← #37550004

Subgroups — #37550004

Order 23168.

238 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 52 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#26042260C212yesmaximal
2#26042186C212yesmaximal, complemented
3#26042185C212yesmaximal, complemented
4#26041854C211yesmaximal, characteristic
5#19153895—22no—
6#19153875—22no—
7#19153875C2214yes—
8C2×#15668450C2211yescharacteristic
9C2×Q8×D181C411yescharacteristic, complemented
10#19153814C411yescharacteristic, complemented
11#19153805C2211yescharacteristic
12#19153804C2211yescharacteristic
13#19153801—22no—
14#19153794—22no—
15#19153773C411yescharacteristic, complemented
16C4⋊Q8×C181C411yescharacteristic, complemented
17C4⋊#10624382—42no—
18C22⋊#10624382—24no—
19C4×#10624382—42no—
20#15668450—42no—
21C2×Dic362—21no—
22C2×Dic362C2×C411yescharacteristic
23Q8×D181—42no—
24C2×D724D412yes—
25C2×D724C2311yescharacteristic
26C2×C4×D181—21no—
27C2×C4×D181C2×C411yescharacteristic
28C2×C4×D181D412yes—
29#15668428—21no—
30#15668428—22no—
31#15668426—21no—
32#15668426—22no—
33#15668426—22no—
34C4×Dic181C2×C411yescharacteristic
35C2×Q8×C181C2×C411yescharacteristic
36C2×Q8×C181C2×C411yescharacteristic
37C4⋊C4×C181—22no—
38C42×C181C2×C411yescharacteristic
39C2×#10624382—44no—
40C22×D181C22⋊C411yescharacteristic
41C22×D181C2×D412yes—
42Dic362—42no—
43Dic362—41no—
44D724—42no—
45D724—41no—
46C2×Dic181—21no—
47C2×Dic181—21no—
48C2×Dic181C22⋊C411yescharacteristic
49C4×D181—22no—
50C4×D181—22no—
51C4×D181—22no—
52C4×D181—42no—
53Q8×C181—41no—
54Q8×C181—41no—
55C2×C4×C181—21no—
56C2×C4×C181—21no—
57C2×C4×C181C22⋊C412yes—
58C2×C4×C181C22×C411yescharacteristic
59C22×C181C2×C22⋊C411yescharacteristic
60C2×D181—21no—
61C2×D181—22no—
62C2×D181—41no—
63C2×D181—42no—
64#10624382—84no—
65C4×C181—22no—
66C4×C181—21no—
67C4×C181—41no—
68C4×C181—41no—
69Dic181—41no—
70Dic181—41no—
71Dic181#9912yes—
72C2×C181—21no—
73C2×C181C2×#9911yescharacteristic
74D181—42no—
75D181—41no—
76C181#212911yescharacteristic, complemented, Sylow
77#2129—1811nomaximal, Sylow
78#536—1811no—
79#410—1812no—
80#354—1812no—
81#354—1812no—
82C2×#41—1811no—
83C22×Q8—1811no—
84C4⋊Q8—1811no—
85C4⋊Q8#1062438211yescharacteristic, complemented
86#124—1811no—
87#124—1811no—
88#123—3622no—
89C22⋊Q8—3622no—
90#117—3622no—
91#117—1811no—
92#99—3622no—
93#99—1814no—
94#41—7242no—
95C2×Q8—3621no—
96C2×Q8—1811no—
97C2×Q8—7242no—
98C2×Q8C2×#1062438211yescharacteristic
99C2×Q8C2×#1062438211yescharacteristic
100C2×D4—1812no—
101C2×D4—1811no—
102C22×C4—3621no—
103C22×C4—1812no—
104C22×C4—1811no—
105C4⋊C4—22no—
106C4⋊C4—3621no—
107C4⋊C4—3622no—
108C4⋊C4—7242no—
109C22⋊C4—3621no—
110C22⋊C4—3622no—
111C22⋊C4—3622no—
112C22⋊C4—3624no—
113C42—1811no—
114C42—7242no—
115C42C2×#1062438211yescharacteristic
116C23—1812no—
117C23—1811no—
118Q8—41no—
119Q8—41no—
120Q8—7241no—
121Q8—7242no—
122D4—7242no—
123D4—7241no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—3622no—
127C2×C4—3622no—
128C2×C4—3622no—
129C2×C4—3621no—
130C2×C4—3621no—
131C2×C4—1811no—
132C2×C4—7242no—
133C2×C4—7244no—
134C2×C4C22⋊#1062438212yes—
135C2×C4C22×#1062438211yescharacteristic
136C22—3622no—
137C22—3621no—
138C22—7242no—
139C22—7241no—
140C22C2×C22⋊#1062438211yescharacteristic
141C4—22no—
142C4—21no—
143C4—41no—
144C4—41no—
145C4—14484no—
146C4—1812no—
147C4—7241no—
148C4—7241no—
149C2—21no—
150C2—7242no—
151C2—7241no—
152C2D181⋊#9911yescharacteristic
subgroupG/H
centerC2D181⋊#99
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
FittingC4⋊Q8×C181C4
socleC2×C181C2×#99
prime radicalC2×#15668450C22
cosocleC2×D724C23
hypercenterC4⋊Q8#10624382
hypocenterC181#2129
LCSG ▷ C2×C4×C181 ▷ C22×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #26042186C2semidirect
3. #26042185C2semidirect
9. C2×Q8×D181C4semidirect
10. #19153814C4semidirect
15. #19153773C4semidirect
16. C4⋊Q8×C181C4semidirect
76. C181#2129semidirect
85. C4⋊Q8#10624382semidirect

Maximal subgroups

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

subgroupindex
#260422602
#260421862
#260421852
#260418542
#2129181