Finite Group Databasebeta

← #37534243

Subgroups — #37534243

Order 23168.

312 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 89 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#26041850C211yesmaximal, characteristic
2#26041289C212yesmaximal, complemented
3#26041253C211yesmaximal, characteristic
4#26041036C212yesmaximal, complemented
5#568×C181C211yesmaximal, characteristic
6C22×Dic362C2211yescharacteristic
7#19153814C412yescomplemented
8#19153803C412yescomplemented
9#19153797—22no—
10#19153795—22no—
11#19153775C2212yes—
12#19153775C2211yescharacteristic
13#19153741—24no—
14#19153731—22no—
15C2×#41×C181C2211yescharacteristic
16#131×C181—24no—
17C2×M16×C181—22no—
18C2×M16×C181C2212yes—
19C2×Dic362—21no—
20C2×Dic362—22no—
21C2×Dic362—22no—
22C2×Dic362C2×C412yes—
23C22×Dic181—21no—
24C4⋊Dic181—21no—
25C4⋊Dic181—22no—
26#15668428—24no—
27C22⋊Dic181—24no—
28C22⋊Dic181—21no—
29C22⋊Dic181—22no—
30C4×Dic181—22no—
31C4×Dic181C2×C412yes—
32#41×C181—22no—
33#41×C181—22no—
34C2×Q8×C181C2×C411yescharacteristic
35C2×D4×C181C2×C411yescharacteristic
36C2×D4×C181D412yes—
37C22×C4×C181D412yes—
38C22×C4×C181C2311yescharacteristic
39M16×C181—24no—
40M16×C181—22no—
41M16×C181—22no—
42C2×C8×C181—24no—
43C2×C8×C181D414yes—
44Dic362—24no—
45Dic362—41no—
46Dic362—41no—
47C2×Dic181—22no—
48C2×Dic181—22no—
49C2×Dic181—42no—
50C2×Dic181—42no—
51C23×C181C22⋊C412yes—
52C23×C181C2×D411yescharacteristic
53Q8×C181—22no—
54D4×C181—24no—
55D4×C181—22no—
56C2×C4×C181—22no—
57C2×C4×C181—22no—
58C2×C4×C181C22⋊C412yes—
59C2×C4×C181C22×C411yescharacteristic
60C2×C4×C181C2×D412yes—
61C2×C4×C181C2×D411yescharacteristic
62C2×C4×C181#4112yes—
63C8×C181—24no—
64C22×C181—22no—
65C22×C181—41no—
66C22×C181—42no—
67C22×C181C2×C22⋊C411yescharacteristic
68C22×C181#12112yes—
69C4×C181—22no—
70C4×C181C4×D412yes—
71C4×C181#12011yescharacteristic
72C4×C181#12311yescharacteristic
73Dic181—42no—
74Dic181—42no—
75Dic181—82no—
76C2×C181—21no—
77C2×C181—22no—
78C2×C181—42no—
79C2×C181#38611yescharacteristic
80C181#190911yescharacteristic, complemented, Sylow
81#1909—1811nomaximal, Sylow
82#568D18111yescharacteristic
83#536—1811no—
84#419—1812no—
85#410—1811no—
86H(3,ℤ/4)—1812no—
87C2×#41C2×D18111yescharacteristic
88C22×Q8—1811no—
89#131—24no—
90C2×M16—22no—
91C2×M16C2×D18112yes—
92C4⋊Q8—1812no—
93#124—1812no—
94#123—3622no—
95C22⋊Q8—3622no—
96#117—1812no—
97#117—1811no—
98#103—3624no—
99#99—3622no—
100#41—22no—
101#41—22no—
102C2×Q8—3621no—
103C2×Q8—3622no—
104C2×Q8—3622no—
105C2×Q8—1812no—
106C2×Q8C4×D18111yescharacteristic
107C2×D4C4×D18111yescharacteristic
108C2×D4D72412yes—
109C22×C4—3621no—
110C22×C4C22⋊D18112yes—
111C22×C4C22×D18111yescharacteristic
112M16—24no—
113M16—22no—
114M16—22no—
115C2×C8—24no—
116C2×C8C22⋊D18114yes—
117C4⋊C4—3624no—
118C4⋊C4—3621no—
119C4⋊C4—3622no—
120C22⋊C4—3624no—
121C22⋊C4—3621no—
122C22⋊C4—3622no—
123C42—3622no—
124C42—1812no—
125C23#1566842612yes—
126C23C2×D72411yescharacteristic
127Q8—22no—
128Q8—3624no—
129Q8—7241no—
130Q8—7241no—
131D4—24no—
132D4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—3622no—
136C2×C4—3622no—
137C2×C4—7242no—
138C2×C4—7242no—
139C2×C4#1566842612yes—
140C2×C4C2×C4×D18111yescharacteristic
141C2×C4C2×C22⋊D18112yes—
142C2×C4C2×C22⋊D18111yescharacteristic
143C2×C4#1566844912yes—
144C8—24no—
145C22—22no—
146C22—41no—
147C22—42no—
148C22C2×#1566842611yescharacteristic
149C22#1915379012yes—
150C4—22no—
151C4—14482no—
152C4—7242no—
153C4—7242no—
154C4C4×C22⋊D18112yes—
155C4#1915378511yescharacteristic
156C4#1915379911yescharacteristic
157C2—22no—
158C2—21no—
159C2—42no—
160C2#2604115811yescharacteristic
subgroupG/H
centerC2#26041158
commutatorC2×C4×C181C22×C4
FrattiniC22×C4C22×D181
Fitting#568×C181C2
socleC2×C181#386
prime radicalC2×#41×C181C22
cosocleC22×C4×C181C23
hypercenter#568D181
hypocenterC181#1909
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #26041289C2semidirect
4. #26041036C2semidirect
7. #19153814C4semidirect
8. #19153803C4semidirect
80. C181#1909semidirect

Maximal subgroups

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

subgroupindex
#260418502
#260412892
#260412532
#260410362
#568×C1812
#1909181