Finite Group Databasebeta

← #37549757

Subgroups — #37549757

Order 23168.

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

normal subgroups only

This group has 310 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#26042347C211yesmaximal, characteristic, complemented
2#26042277C211yesmaximal, characteristic
3#26042272C211yesmaximal, characteristic, complemented
4#26042259C211yesmaximal, characteristic
5#26042161C211yesmaximal, characteristic, complemented
6#26042147C211yesmaximal, characteristic, complemented
7C2×#19153845C211yesmaximal, characteristic
8D181⋊M16C2211yescharacteristic
9Q8×#10624382—22no—
10Q8×#10624382—22no—
11#19153895—22no—
12#19153895C2211yescharacteristic
13#19153895C2211yescharacteristic
14#19153895C2211yescharacteristic
15#19153880—22no—
16#19153879—22no—
17#19153874—22no—
18C2×#15668448C2211yescharacteristic
19C2×Q8×D181C2211yescharacteristic
20#19153845—22no—
21#19153845—22no—
22C2×Dic724C411yescharacteristic, complemented
23C2×#15668439C411yescharacteristic, complemented
24C2×#15668437C411yescharacteristic, complemented
25C2×#15668432C2211yescharacteristic
26C2×SD16×C181C411yescharacteristic, complemented
27#15668459—21no—
28#15668458—21no—
29D181⋊C8—21no—
30C4⋊#10624382—24no—
31C4⋊#10624382—21no—
32C4⋊#10624382—21no—
33C4⋊#10624382—21no—
34C22⋊#10624382—21no—
35C22⋊#10624382—21no—
36C22⋊#10624382D412yes—
37C4×#10624382—24no—
38C4×#10624382—21no—
39C4×#10624382—21no—
40C4×#10624382D412yes—
41#15668448—21no—
42#15668448—21no—
43#15668448—21no—
44#15668448—21no—
45C2×Dic362—21no—
46C2×Dic362C2×C411yescharacteristic
47C2×Dic362C2×C411yescharacteristic
48Q8×D181—21no—
49Q8×D181—21no—
50Q8×D181D412yes—
51Q8×D181D412yes—
52C2×C22⋊D181—21no—
53C22×Dic181—21no—
54C2×C4×D181—21no—
55C2×C4×D181C2311yescharacteristic
56Dic724—22no—
57#15668439—22no—
58#15668437—22no—
59#15668432—21no—
60#15668432—21no—
61C2×#13089678C2×C411yescharacteristic
62C2×Q8×C181C2×C411yescharacteristic
63C2×D4×C181C2×C411yescharacteristic
64SD16×C181—22no—
65C2×C8×C181C2×C411yescharacteristic
66C2×#10624382—22no—
67C2×#10624382—22no—
68C2×#10624382—41no—
69#13089685—41no—
70C22×D181C2×D411yescharacteristic
71Dic362—21no—
72Dic362—22no—
73Dic362—22no—
74Dic362—21no—
75Dic362—21no—
76Dic362C22⋊C412yes—
77C22⋊D181—41no—
78C22⋊D181—41no—
79C2×Dic181—21no—
80C2×Dic181—21no—
81C2×Dic181—41no—
82C2×Dic181—41no—
83C2×Dic181#4111yescharacteristic
84C4×D181—22no—
85C4×D181—22no—
86C4×D181C2×D411yescharacteristic
87C4×D181C2×D411yescharacteristic
88C4×D181C2×D411yescharacteristic
89C4×D181#4111yescharacteristic
90#13089678—21no—
91C23×C181—21no—
92Q8×C181—21no—
93Q8×C181C22⋊C412yes—
94D4×C181—21no—
95D4×C181—21no—
96C2×C4×C181—21no—
97C2×C4×C181C22×C411yescharacteristic
98C8×C181—21no—
99C22×C181—41no—
100C22×C181—81no—
101C22×C181C4×D411yescharacteristic
102C2×D181—41no—
103C2×D181#12011yescharacteristic
104C2×D181#12111yescharacteristic
105#10624382—42no—
106#10624382—42no—
107#10624382—81no—
108C4×C181—22no—
109C4×C181C2×C22⋊C411yescharacteristic
110C4×C181C4×D411yescharacteristic
111Dic181—22no—
112Dic181—41no—
113Dic181#12111yescharacteristic
114Dic181#12311yescharacteristic
115C2×C181—21no—
116C2×C181—81no—
117C2×C181#38611yescharacteristic
118D181—21no—
119D181—21no—
120C181#190511yescharacteristic, complemented, Sylow
121#1905—1811nomaximal, Sylow
122C2×#137—1811no—
123#519—1811no—
124#421—1811no—
125#419—1811no—
126#410—1811no—
127#344—1811no—
128H(3,ℤ/4)—1811no—
129C2×#41—1811no—
130C22×Q8—1811no—
131#137—3622no—
132#137—3622no—
133C2×Q16—1811no—
134C2×Q16—1811no—
135C2×SD16—1811no—
136C2×SD16#1062438211yescharacteristic, complemented
137C2×M16—1811no—
138C2×M16—1811no—
139C4×Q8—3622no—
140C4×Q8—3622no—
141#117—3622no—
142#117—1811no—
143#117—1811no—
144#117—1811no—
145#104—3622no—
146#103—3622no—
147#99—3622no—
148#41—3621no—
149#41—3621no—
150#41—3621no—
151#41—3621no—
152C2×Q8—3621no—
153C2×Q8—3621no—
154C2×Q8—3621no—
155C2×Q8—1812no—
156C2×Q8—1812no—
157C2×Q8—1811no—
158C2×Q8—1811no—
159C2×Q8C2×#1062438211yescharacteristic
160C2×D4—3621no—
161C2×D4C2×#1062438211yescharacteristic
162C22×C4—3621no—
163C22×C4—3621no—
164C22×C4—1811no—
165Q16—3622no—
166Q16—3622no—
167SD16—22no—
168SD16—3622no—
169M16—3621no—
170M16—3621no—
171M16—3621no—
172M16—3621no—
173C2×C8—3621no—
174C2×C8—1811no—
175C2×C8C2×#1062438211yescharacteristic
176C4⋊C4—3624no—
177C4⋊C4—3621no—
178C4⋊C4—3621no—
179C4⋊C4—3621no—
180C22⋊C4—3621no—
181C22⋊C4—3621no—
182C22⋊C4—1812no—
183C42—3624no—
184C42—3621no—
185C42—3621no—
186C42—1812no—
187C23—21no—
188C23—1811no—
189Q8—21no—
190Q8—3622no—
191Q8—3622no—
192Q8—3621no—
193Q8—3621no—
194Q8—3621no—
195Q8—1812no—
196Q8C22⋊#1062438212yes—
197D4—21no—
198D4—21no—
199D4—7241no—
200D4—7241no—
201C2×C4—21no—
202C2×C4—3622no—
203C2×C4—3622no—
204C2×C4—3621no—
205C2×C4—3622no—
206C2×C4—3622no—
207C2×C4—3621no—
208C2×C4—1811no—
209C2×C4—1811no—
210C2×C4—1811no—
211C2×C4—1811no—
212C2×C4—1811no—
213C2×C4—7241no—
214C2×C4—7241no—
215C2×C4—7241no—
216C2×C4C22×#1062438211yescharacteristic
217C8—21no—
218C8—3621no—
219C8—7241no—
220C22—41no—
221C22—81no—
222C22—1811no—
223C22—1811no—
224C22—7241no—
225C22D4×#1062438211yescharacteristic
226C4—22no—
227C4—3622no—
228C4—14481no—
229C4—1811no—
230C4—1811no—
231C4—7242no—
232C4—7242no—
233C4—7241no—
234C4C2×C22⋊#1062438211yescharacteristic
235C4D4×#1062438211yescharacteristic
236C2—21no—
237C2—81no—
238C2—3621no—
239C2—3621no—
240C2#2604222711yescharacteristic
subgroupG/H
centerC2#26042227
commutatorC2×C4×C181C22×C4
FrattiniC2×C4C22×#10624382
FittingC2×SD16×C181C4
socleC2×C181#386
prime radicalC2×#15668448C22
cosocleC2×C4×D181C23
hypercenterC2×SD16#10624382
hypocenterC181#1905
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. #26042347C2semidirect
3. #26042272C2semidirect
5. #26042161C2semidirect
6. #26042147C2semidirect
22. C2×Dic724C4semidirect
23. C2×#15668439C4semidirect
24. C2×#15668437C4semidirect
26. C2×SD16×C181C4semidirect
120. C181#1905semidirect
136. C2×SD16#10624382semidirect

Maximal subgroups

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

subgroupindex
#260423472
#260422772
#260422722
#260422592
#260421612
#260421472
C2×#191538452
#1905181