Finite Group Databasebeta

← #37542966

Subgroups — #37542966

Order 23168.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#15668449C211yesmaximal, characteristic, complemented
2C2×#19153849C211yesmaximal, characteristic, complemented
3C2×#19153844C211yesmaximal, characteristic, complemented
4#26041790C211yesmaximal, characteristic, complemented
5#26041443C214yesmaximal, complemented
6#26041426C214yesmaximal, complemented
7#26041289C211yesmaximal, characteristic, complemented
8#26041285C211yesmaximal, characteristic, complemented
9#407×C181C211yesmaximal, characteristic, complemented
10C2×#15668449—24no—
11C2×#15668449—24no—
12C2×#15668449—22no—
13C2×#15668449C2214yes—
14C2×#15668449C2211yescharacteristic
15C22×Dic362C2211yescharacteristic
16C22×D724C2211yescharacteristic
17C22×C22⋊D181—21no—
18C22×C4×D181—21no—
19#19153849—24no—
20#19153844—24no—
21C2×Dic724C2212yescomplemented
22C2×#15668438C2212yescomplemented
23C2×#15668438C2212yescomplemented
24C2×D1448C2212yescomplemented
25#19153799—21no—
26#19153795C2212yescomplemented
27#19153790C2212yescomplemented
28#19153785—21no—
29C4×C22⋊D181—22no—
30#19153775C2211yescharacteristic
31#19153741C2214yescomplemented
32#19153738C2214yescomplemented
33C23×C4×C181C2211yescharacteristic
34C2×M16×C181C2211yescharacteristic
35C2×M16×C181C2211yescharacteristic
36C22⋊C8×C181C2214yescomplemented
37C23×D181—21no—
38#15668449—216no—
39#15668449—28no—
40#15668449—28no—
41#15668449—22no—
42#15668449—22no—
43C2×Dic362—22no—
44C2×Dic362—21no—
45C2×Dic362D414yescomplemented
46C2×Dic362C2311yescharacteristic
47C2×Dic362C2312yes—
48C2×D724—22no—
49C2×D724—21no—
50C2×D724D414yescomplemented
51C2×D724C2311yescharacteristic
52C2×D724C2312yes—
53C2×C22⋊D181—28no—
54C2×C22⋊D181—24no—
55C2×C22⋊D181—22no—
56C2×C22⋊D181—21no—
57C22×Dic181—21no—
58C2×C4×D181—28no—
59C2×C4×D181—24no—
60C2×C4×D181—21no—
61Dic724—24no—
62#15668438—24no—
63#15668438—24no—
64D1448—24no—
65C4⋊Dic181C2312yes—
66#15668428—22no—
67C22⋊Dic181—21no—
68C22⋊Dic181—22no—
69#15668426—22no—
70C4×Dic181—21no—
71C24×C181D411yescharacteristic
72C22×C4×C181—22no—
73C22×C4×C181—22no—
74C22×C4×C181D412yes—
75C22×C4×C181D411yescharacteristic
76C22×C4×C181C2312yes—
77C22×C4×C181C2311yescharacteristic
78M16×C181—22no—
79M16×C181—22no—
80C2×C8×C181C2312yes—
81C2×C8×C181C2312yes—
82C22×D181—24no—
83C22×D181—22no—
84C22×D181—21no—
85C22×D181—44no—
86Dic362—24no—
87Dic362—22no—
88Dic362—21no—
89Dic362—21no—
90Dic362C2×D414yes—
91D724—24no—
92D724—22no—
93D724—21no—
94D724—21no—
95D724C2×D414yes—
96C22⋊D181—216no—
97C22⋊D181—44no—
98C22⋊D181—42no—
99C2×Dic181—24no—
100C2×Dic181—22no—
101C2×Dic181—22no—
102C2×Dic181—21no—
103C4×D181—216no—
104C4×D181—42no—
105C23×C181—22no—
106C23×C181—42no—
107C23×C181C2×D412yes—
108C23×C181C2×D411yescharacteristic
109C2×C4×C181—22no—
110C2×C4×C181—24no—
111C2×C4×C181—22no—
112C2×C4×C181—22no—
113C2×C4×C181C2×D412yes—
114C2×C4×C181C2×D412yes—
115C2×C4×C181C2×D412yes—
116C2×C4×C181C2×D411yescharacteristic
117C2×C4×C181C2411yescharacteristic
118C8×C181—22no—
119C8×C181—22no—
120C22×C181—22no—
121C22×C181—22no—
122C22×C181—22no—
123C22×C181—22no—
124C22×C181—44no—
125C22×C181#12012yes—
126C22×C181C22×D411yescharacteristic
127C2×D181—24no—
128C2×D181—48no—
129C2×D181—44no—
130C2×D181—41no—
131C2×D181—42no—
132C4×C181—22no—
133C4×C181#12012yes—
134C4×C181C22×D411yescharacteristic
135C4×C181C22×D411yescharacteristic
136Dic181—24no—
137Dic181—41no—
138Dic181—42no—
139C2×C181—22no—
140C2×C181—42no—
141C2×C181C2×#12011yescharacteristic
142C2×C181#57512yes—
143D181—44no—
144D181—81no—
145C181#300511yescharacteristic, complemented, Sylow
146#3005—1811nomaximal, Sylow
147C22×#41—1811no—
148C2×#137—1811no—
149C2×AGL(1,ℤ/8)—1811no—
150#525—1811no—
151#452—1814no—
152#449—1814no—
153#419—1811no—
154#418—1811no—
155#407D18111yescharacteristic, complemented
156C2×#41—3624no—
157C2×#41—3624no—
158C2×#41—3622no—
159C2×#41—1811no—
160C2×#41—1814no—
161C22×Q8—1811no—
162C22×D4—3621no—
163C22×D4—1811no—
164C23×C4—3621no—
165C23×C4C2×D18111yescharacteristic
166#137—3624no—
167AGL(1,ℤ/8)—3624no—
168C2×Q16—1812no—
169C2×SD16—1812no—
170C2×SD16—1812no—
171C2×D8—1812no—
172C2×M16C2×D18111yescharacteristic
173C2×M16C2×D18111yescharacteristic
174#123—3621no—
175C22⋊Q8—1812no—
176#121—1812no—
177#120—3621no—
178C4×D4—3622no—
179#117—1811no—
180#103—1814no—
181#102—1814no—
182C22⋊C8C2×D18114yescomplemented
183C24—3621no—
184C24D72411yescharacteristic
185#41—36216no—
186#41—3628no—
187#41—3628no—
188#41—3622no—
189#41—3622no—
190C2×Q8—3622no—
191C2×Q8—3621no—
192C2×Q8—1814no—
193C2×Q8—1811no—
194C2×Q8—1812no—
195C2×D4—3628no—
196C2×D4—3624no—
197C2×D4—3622no—
198C2×D4—3621no—
199C2×D4—3622no—
200C2×D4—3621no—
201C2×D4—1814no—
202C2×D4—1811no—
203C2×D4—1812no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—3628no—
207C22×C4—3621no—
208C22×C4—3624no—
209C22×C4—3621no—
210C22×C4D72412yes—
211C22×C4D72411yescharacteristic
212C22×C4C22×D18112yes—
213C22×C4C22×D18111yescharacteristic
214Q16—3624no—
215SD16—3624no—
216SD16—3624no—
217D8—3624no—
218M16—22no—
219M16—22no—
220C2×C8C22×D18112yes—
221C2×C8C22×D18112yes—
222C4⋊C4—3622no—
223C4⋊C4—1812no—
224C22⋊C4—3621no—
225C22⋊C4—3622no—
226C22⋊C4—3622no—
227C42—3621no—
228C23—22no—
229C23—42no—
230C23—3624no—
231C23—3622no—
232C23—3621no—
233C23—7244no—
234C23C2×D72412yes—
235C23C2×D72411yescharacteristic
236Q8—3624no—
237Q8—3622no—
238Q8—3621no—
239Q8—3621no—
240Q8—1814no—
241D4—36216no—
242D4—3624no—
243D4—3622no—
244D4—3621no—
245D4—3621no—
246D4—1814no—
247D4—7244no—
248D4—7242no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—36216no—
254C2×C4—3624no—
255C2×C4—3622no—
256C2×C4—3622no—
257C2×C4—3621no—
258C2×C4—7242no—
259C2×C4D4×D18112yes—
260C2×C4D4×D18112yes—
261C2×C4C2×D72412yes—
262C2×C4C2×D72411yescharacteristic
263C2×C4C23×D18111yescharacteristic
264C8—22no—
265C8—22no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—44no—
271C22—3624no—
272C22—7248no—
273C22—7241no—
274C22—7242no—
275C22—7244no—
276C22#1915378612yes—
277C22C22×D72411yescharacteristic
278C4—22no—
279C4—3624no—
280C4—7241no—
281C4—7242no—
282C4#1915378612yes—
283C4C2×D4×D18111yescharacteristic
284C4C2×D4×D18111yescharacteristic
285C2—22no—
286C2—42no—
287C2—14481no—
288C2—7244no—
289C2C2×#1915378611yescharacteristic
290C2#2604207912yes—
subgroupG/H
centerC22C22×D724
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#407×C181C2
socleC22×C181C22×D4
hypercenter#407D181
hypocenterC181#3005
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C22×#15668449C2semidirect
2. C2×#19153849C2semidirect
3. C2×#19153844C2semidirect
4. #26041790C2semidirect
5. #26041443C2semidirect
6. #26041426C2semidirect
7. #26041289C2semidirect
8. #26041285C2semidirect
9. #407×C181C2semidirect
21. C2×Dic724C22semidirect
22. C2×#15668438C22semidirect
23. C2×#15668438C22semidirect
24. C2×D1448C22semidirect
26. #19153795C22semidirect
27. #19153790C22semidirect
31. #19153741C22semidirect
32. #19153738C22semidirect
36. C22⋊C8×C181C22semidirect
45. C2×Dic362D4semidirect
50. C2×D724D4semidirect
145. C181#3005semidirect
155. #407D181semidirect
182. C22⋊C8C2×D181semidirect

Maximal subgroups

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

subgroupindex
C22×#156684492
C2×#191538492
C2×#191538442
#260417902
#260414432
#260414262
#260412892
#260412852
#407×C1812
#3005181