Finite Group Databasebeta

← #480×D181 #37543589

Subgroups — #480×D181 #37543589

Order 23168.

512 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 135 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#121×D181C211yesmaximal, characteristic
2C2×C4⋊C4×D181C211yesmaximal, characteristic, complemented
3#26041594C212yesmaximal, complemented
4#26041593C211yesmaximal, characteristic, complemented
5#26041592C211yesmaximal, characteristic, complemented
6#26041591C212yesmaximal, complemented
7#26041590C211yesmaximal, characteristic, complemented
8#107×D181C212yesmaximal, complemented
9#102×D181C212yesmaximal, complemented
10C22⋊C8×D181C211yesmaximal, characteristic, complemented
11#480×C181C211yesmaximal, characteristic, complemented
12C2×D4×D181—21no—
13C2×D4×D181C2211yescharacteristic
14C22×C4×D181—21no—
15C22×C4×D181C2211yescharacteristic
16C2×C8×D181C2212yes—
17#19153791C2211yescharacteristic
18#19153790C2211yescharacteristic
19#19153789—21no—
20#19153788—21no—
21#19153787C2211yescharacteristic
22C2×C4⋊Dic181C2211yescharacteristic, complemented
23C2×#15668428—21no—
24C4⋊C4×D181—21no—
25C4⋊C4×D181—21no—
26C4⋊C4×D181C2212yes—
27C4⋊C4×D181C2211yescharacteristic
28C4⋊C4×D181C2212yescomplemented
29C22⋊C4×D181—21no—
30#19153750C2212yescomplemented
31#19153749C2212yescomplemented
32#19153749C2212yescomplemented
33#19153738C2212yescomplemented
34#19153737C2212yescomplemented
35#19153736C2212yescomplemented
36C22⋊#13089678C2211yescharacteristic, complemented
37#19153728C2212yescomplemented
38#121×C181C2211yescharacteristic
39C2×C4⋊C4×C181C2211yescharacteristic, complemented
40#107×C181C2212yescomplemented
41#102×C181C2212yescomplemented
42C22⋊C8×C181C2211yescharacteristic, complemented
43C23×D181—21no—
44C23×D181D411yescharacteristic
45C2×D724—21no—
46C2×D724C2311yescharacteristic
47C2×C22⋊D181—21no—
48C2×C22⋊D181—21no—
49C2×C22⋊D181—21no—
50D4×D181—24no—
51D4×D181—44no—
52C22×Dic181—21no—
53C22×Dic181D411yescharacteristic
54C2×C4×D181—24no—
55C2×C4×D181—21no—
56C2×C4×D181—22no—
57C2×C4×D181—22no—
58C2×C4×D181—44no—
59C2×C4×D181D411yescharacteristic
60C2×C4×D181D411yescharacteristic
61C2×C4×D181C2311yescharacteristic
62C2×C4×D181C2311yescharacteristic
63C2×#13089678C2312yes—
64C8×D181—24no—
65C4⋊Dic181—21no—
66C4⋊Dic181C2312yes—
67C4⋊Dic181C2311yescharacteristic
68#15668428—22no—
69#15668428—22no—
70#15668428—21no—
71C22⋊Dic181—21no—
72#15668426—21no—
73#15668426—21no—
74C2×D4×C181—21no—
75C2×D4×C181C2311yescharacteristic
76C22×C4×C181—21no—
77C22×C4×C181C2311yescharacteristic
78C2×C8×C181C2312yes—
79C4⋊C4×C181—21no—
80C4⋊C4×C181C2312yes—
81C4⋊C4×C181C2311yescharacteristic
82C22⋊C4×C181—21no—
83C22×D181—22no—
84C22×D181—22no—
85C22×D181—21no—
86C22×D181—42no—
87C22×D181—42no—
88C22×D181—42no—
89C22×D181D814yes—
90C22×D181C2×D411yescharacteristic
91C22×D181C2×D411yescharacteristic
92D724—22no—
93D724—42no—
94C22⋊D181—42no—
95C22⋊D181—42no—
96C22⋊D181—42no—
97C2×Dic181—22no—
98C2×Dic181—21no—
99C2×Dic181—22no—
100C2×Dic181—42no—
101C2×Dic181C2×D411yescharacteristic
102C2×Dic181C2×D411yescharacteristic
103C4×D181—22no—
104C4×D181—22no—
105C4×D181—22no—
106C4×D181—44no—
107C4×D181—44no—
108C4×D181—42no—
109C4×D181#4114yes—
110#13089678—22no—
111C23×C181—21no—
112C23×C181C2×D411yescharacteristic
113D4×C181—22no—
114D4×C181—42no—
115C2×C4×C181—22no—
116C2×C4×C181—21no—
117C2×C4×C181—22no—
118C2×C4×C181—42no—
119C2×C4×C181C2×D411yescharacteristic
120C2×C4×C181C2411yescharacteristic
121C8×C181—22no—
122C22×C181—21no—
123C22×C181—21no—
124C22×C181—41no—
125C22×C181—41no—
126C22×C181—41no—
127C22×C181C2×D812yes—
128C22×C181C22×D411yescharacteristic
129C2×D181—21no—
130C2×D181—21no—
131C2×D181—28no—
132C2×D181—41no—
133C2×D181—41no—
134C2×D181—41no—
135C2×D181—84no—
136C2×D181#12312yes—
137C2×D181C2×D812yes—
138C2×D181C2×D812yes—
139C2×D181#13712yes—
140C4×C181—21no—
141C4×C181—42no—
142C4×C181—41no—
143C4×C181C2×#4112yes—
144Dic181—21no—
145Dic181—42no—
146Dic181—41no—
147Dic181#12312yes—
148C2×C181—22no—
149C2×C181—81no—
150C2×C181C2×#12311yescharacteristic
151C2×C181C22×D811yescharacteristic
152C2×C181C2×#13711yescharacteristic
153D181—22no—
154D181—81no—
155D181#48014yesdirect product factor
156C181C2×#48011yescharacteristic, complemented, Sylow
157C2×#480—1811nomaximal, Sylow
158C2×#121—1811no—
159C22×C4⋊C4—1811no—
160#480—1811no—
161#480—1811no—
162#480—1812no—
163#480—1812no—
164#480—1811no—
165#480D18111yescharacteristic, direct product factor
166C2×#107—1812no—
167C2×#102—1812no—
168C2×C22⋊C8—1811no—
169C22×D4—3621no—
170C22×D4—1811no—
171C23×C4—3621no—
172C23×C4—1811no—
173C22×C8—1812no—
174#121—3621no—
175#121—3621no—
176#121—1811no—
177#121—1811no—
178#121—1811no—
179#121C2×D18111yescharacteristic
180C2×C4⋊C4—3621no—
181C2×C4⋊C4—3621no—
182C2×C4⋊C4—3621no—
183C2×C4⋊C4—1811no—
184C2×C4⋊C4—1812no—
185C2×C4⋊C4—1812no—
186C2×C4⋊C4—1811no—
187C2×C4⋊C4C2×D18111yescharacteristic, complemented
188C2×C22⋊C4—3621no—
189#107—1812no—
190#107—1812no—
191#107—1812no—
192#107C2×D18112yescomplemented
193#102—1812no—
194#102—1812no—
195#102—1812no—
196#102C2×D18112yescomplemented
197C22⋊C8—1811no—
198C22⋊C8—1812no—
199C22⋊C8C2×D18111yescharacteristic, complemented
200C24—3621no—
201C24—1811no—
202C2×D4—21no—
203C2×D4—3621no—
204C2×D4—3621no—
205C2×D4—3621no—
206C2×D4—3624no—
207C2×D4—3621no—
208C2×D4—1811no—
209C2×D4—7244no—
210C2×D4C22×D18111yescharacteristic
211C22×C4—21no—
212C22×C4—3624no—
213C22×C4—3621no—
214C22×C4—3622no—
215C22×C4—3622no—
216C22×C4—3621no—
217C22×C4—1811no—
218C22×C4—1811no—
219C22×C4—1811no—
220C22×C4—1811no—
221C22×C4—1811no—
222C22×C4—7244no—
223C22×C4C22×D18111yescharacteristic
224C2×C8—3624no—
225C2×C8—1812no—
226C2×C8C22×D18112yes—
227C4⋊C4—21no—
228C4⋊C4—3622no—
229C4⋊C4—3622no—
230C4⋊C4—3621no—
231C4⋊C4—3621no—
232C4⋊C4—1812no—
233C4⋊C4—1811no—
234C4⋊C4C22×D18112yes—
235C4⋊C4C22×D18111yescharacteristic
236C22⋊C4—21no—
237C22⋊C4—3621no—
238C22⋊C4—3621no—
239C22⋊C4—3621no—
240C23—21no—
241C23—3622no—
242C23—3622no—
243C23—3621no—
244C23—1814no—
245C23—1811no—
246C23—1811no—
247C23—7242no—
248C23—7242no—
249C23—7242no—
250C23D4×D18111yescharacteristic
251D4—22no—
252D4—42no—
253D4—3622no—
254D4—7242no—
255D4—7242no—
256D4—7242no—
257D4—7242no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—42no—
262C2×C4—3622no—
263C2×C4—3622no—
264C2×C4—3622no—
265C2×C4—3622no—
266C2×C4—3621no—
267C2×C4—3622no—
268C2×C4—1811no—
269C2×C4—1811no—
270C2×C4—1814no—
271C2×C4—7244no—
272C2×C4—7244no—
273C2×C4—7242no—
274C2×C4—7242no—
275C2×C4D4×D18111yescharacteristic
276C2×C4C23×D18111yescharacteristic
277C8—22no—
278C8—3622no—
279C22—21no—
280C22—21no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—3628no—
285C22—3621no—
286C22—3621no—
287C22—14484no—
288C22—1812no—
289C22—1812no—
290C22—1812no—
291C22—1812no—
292C22—7241no—
293C22—7241no—
294C22—7241no—
295C22D8×D18112yes—
296C22C2×D4×D18111yescharacteristic
297C4—21no—
298C4—42no—
299C4—41no—
300C4—3621no—
301C4—1812no—
302C4—7242no—
303C4—7241no—
304C4#41×D18112yes—
305C2—22no—
306C2—81no—
307C2—3622no—
308C2—14481no—
309C2—1814no—
310C2#123×D18111yescharacteristic
311C2C2×D8×D18111yescharacteristic
312C2#137×D18111yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#480×C181C2
socleC22×C181C22×D4
prime radical#121×D181C2
hypercenter#480D181
hypocenterC181C2×#480
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D181C2semidirect
3. #26041594C2semidirect
4. #26041593C2semidirect
5. #26041592C2semidirect
6. #26041591C2semidirect
7. #26041590C2semidirect
8. #107×D181C2semidirect
9. #102×D181C2semidirect
10. C22⋊C8×D181C2semidirect
11. #480×C181C2semidirect
22. C2×C4⋊Dic181C22semidirect
28. C4⋊C4×D181C22semidirect
30. #19153750C22semidirect
31. #19153749C22semidirect
32. #19153749C22semidirect
33. #19153738C22semidirect
34. #19153737C22semidirect
35. #19153736C22semidirect
36. C22⋊#13089678C22semidirect
37. #19153728C22semidirect
39. C2×C4⋊C4×C181C22semidirect
40. #107×C181C22semidirect
41. #102×C181C22semidirect
42. C22⋊C8×C181C22semidirect
155. D181#480direct (N × H)
156. C181C2×#480semidirect
165. #480D181direct (N × H)
187. C2×C4⋊C4C2×D181semidirect
192. #107C2×D181semidirect
196. #102C2×D181semidirect
199. C22⋊C8C2×D181semidirect

Maximal subgroups

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

subgroupindex
#121×D1812
C2×C4⋊C4×D1812
#260415942
#260415932
#260415922
#260415912
#260415902
#107×D1812
#102×D1812
C22⋊C8×D1812
#480×C1812
C2×#480181