Finite Group Databasebeta

← #37549764

Subgroups — #37549764

Order 23168.

344 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 66 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#26042357C211yesmaximal, characteristic, complemented
2#26042271C212yesmaximal, complemented
3#26042259C211yesmaximal, characteristic, complemented
4#26042146C212yesmaximal, complemented
5#26041842C211yesmaximal, characteristic
6#19153903—24no—
7D181⋊M16C2212yes—
8C2×#15668458—22no—
9#19153895C2212yes—
10#19153878—24no—
11#19153874—22no—
12C2×#15668448C2211yescharacteristic
13C2×D4×D181C2211yescharacteristic
14#19153812C411yescharacteristic, complemented
15#19153804C411yescharacteristic, complemented
16#19153803C411yescharacteristic, complemented
17#19153788—22no—
18#19153784—22no—
19#19153773C2211yescharacteristic
20#127×C181C411yescharacteristic, complemented
21#15668459—22no—
22#15668458—24no—
23#15668458—22no—
24D181⋊C8D414yescomplemented
25C2×#13089685—24no—
26C4⋊#10624382—22no—
27C22⋊#10624382—22no—
28C4×#10624382—22no—
29C23×D181—21no—
30#15668448—22no—
31#15668448—22no—
32C2×Dic362C2×C411yescharacteristic
33C2×D724C2×C411yescharacteristic
34C2×C22⋊D181—21no—
35C2×C22⋊D181—22no—
36C2×C22⋊D181D412yes—
37D4×D181—22no—
38D4×D181—22no—
39C22×Dic181D412yes—
40C2×C4×D181C2311yescharacteristic
41#15668428—21no—
42C22⋊Dic181—22no—
43#15668426—21no—
44#15668426—22no—
45C4×Dic181C2×C411yescharacteristic
46C2×D4×C181—22no—
47C2×D4×C181C2×C411yescharacteristic
48C2×D4×C181C2×C411yescharacteristic
49C42×C181C2×C411yescharacteristic
50C2×#10624382—42no—
51#13089685—24no—
52C22×D181—21no—
53C22×D181—42no—
54C22×D181—82no—
55C22×D181C2×D411yescharacteristic
56Dic362—22no—
57D724—22no—
58C22⋊D181—24no—
59C22⋊D181—41no—
60C22⋊D181—41no—
61C22⋊D181—82no—
62C2×Dic181—22no—
63C2×Dic181—22no—
64C2×Dic181—21no—
65C2×Dic181C22⋊C412yes—
66C2×Dic181C2×D411yescharacteristic
67C4×D181C2×D412yes—
68C4×D181#4112yes—
69C23×C181—21no—
70C23×C181C22⋊C412yes—
71D4×C181—22no—
72D4×C181—22no—
73D4×C181—82no—
74C2×C4×C181—21no—
75C2×C4×C181C22×C411yescharacteristic
76C22×C181—22no—
77C22×C181—42no—
78C22×C181—42no—
79C22×C181—41no—
80C22×C181C2×C22⋊C411yescharacteristic
81C2×D181—41no—
82C2×D181—42no—
83C2×D181—41no—
84C2×D181—84no—
85C2×D181#12112yes—
86#10624382—82no—
87C4×C181—41no—
88C4×C181C4×D412yes—
89Dic181—22no—
90Dic181—41no—
91Dic181#12011yescharacteristic
92Dic181#12311yescharacteristic
93C2×C181—21no—
94C2×C181—42no—
95C2×C181—81no—
96C2×C181#38611yescharacteristic
97D181—22no—
98D181—81no—
99C181#191011yescharacteristic, complemented, Sylow
100#1910—1811nomaximal, Sylow
101#568—1811no—
102#535—1811no—
103#418—1812no—
104#410—1811no—
105H(3,ℤ/4)—1812no—
106C2×#41—1811no—
107C22×D4—1811no—
108#131—3624no—
109C2×M16—3622no—
110C2×M16—1812no—
111#127—1811no—
112#127#1062438211yescharacteristic, complemented
113#124—1811no—
114#124—1811no—
115#121—3622no—
116#120—3622no—
117#117—1812no—
118#117—1811no—
119#102—3624no—
120#99—3622no—
121C24—3621no—
122#41—3622no—
123#41—3622no—
124C2×Q8—1811no—
125C2×D4—22no—
126C2×D4—3622no—
127C2×D4—3622no—
128C2×D4—3622no—
129C2×D4—3621no—
130C2×D4—1812no—
131C2×D4—1811no—
132C2×D4C2×#1062438211yescharacteristic
133C2×D4C2×#1062438211yescharacteristic
134C22×C4—1812no—
135C22×C4—1811no—
136M16—3624no—
137M16—3622no—
138M16—3622no—
139C2×C8—3624no—
140C2×C8—1814no—
141C4⋊C4—3621no—
142C4⋊C4—3622no—
143C22⋊C4—3621no—
144C22⋊C4—3622no—
145C22⋊C4—3622no—
146C22⋊C4—3622no—
147C42—3622no—
148C42—1811no—
149C42C2×#1062438211yescharacteristic
150C23—21no—
151C23—3621no—
152C23—14482no—
153C23—1811no—
154C23—7242no—
155C23C22⋊#1062438212yes—
156Q8—3622no—
157D4—22no—
158D4—22no—
159D4—82no—
160D4—3624no—
161D4—3622no—
162D4—14482no—
163D4—7241no—
164D4—7241no—
165C2×C4—21no—
166C2×C4—3622no—
167C2×C4—3622no—
168C2×C4—3621no—
169C2×C4—1812no—
170C2×C4—1811no—
171C2×C4—1812no—
172C2×C4—1812no—
173C2×C4—7242no—
174C2×C4C22×#1062438211yescharacteristic
175C8—3624no—
176C22—22no—
177C22—42no—
178C22—42no—
179C22—41no—
180C22—14484no—
181C22—1812no—
182C22—7241no—
183C22—7242no—
184C22—7241no—
185C22C2×C22⋊#1062438211yescharacteristic
186C4—41no—
187C4—3622no—
188C4—14482no—
189C4—1811no—
190C4—1811no—
191C4—7241no—
192C4D4×#1062438212yes—
193C2—21no—
194C2—42no—
195C2—81no—
196C2—3622no—
197C2—14481no—
198C2#2604222711yescharacteristic
subgroupG/H
centerC2#26042227
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
Fitting#127×C181C4
socleC2×C181#386
prime radical#26041842C2
cosocleC2×C4×D181C23
hypercenter#127#10624382
hypocenterC181#1910
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #26042357C2semidirect
2. #26042271C2semidirect
3. #26042259C2semidirect
4. #26042146C2semidirect
14. #19153812C4semidirect
15. #19153804C4semidirect
16. #19153803C4semidirect
20. #127×C181C4semidirect
24. D181⋊C8D4semidirect
99. C181#1910semidirect
112. #127#10624382semidirect

Maximal subgroups

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

subgroupindex
#260423572
#260422712
#260422592
#260421462
#260418422
#1910181