Finite Group Databasebeta

← #37544740

Subgroups — #37544740

Order 23168.

490 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 113 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#127×D181C211yesmaximal, characteristic, complemented
2#26041814C211yesmaximal, characteristic, complemented
3#26041668C211yesmaximal, characteristic, complemented
4#26041666C211yesmaximal, characteristic, complemented
5#26041645C211yesmaximal, characteristic, complemented
6#26041478C212yesmaximal, complemented
7#26041473C212yesmaximal, complemented
8#26041431C212yesmaximal, complemented
9#26041414C212yesmaximal, complemented
10#26041334C211yesmaximal, characteristic, complemented
11#490×C181C211yesmaximal, characteristic, complemented
12C2×D4×D181—21no—
13C2×D4×D181—21no—
14C2×D4×D181C2212yes—
15C2×#15668438C2212yescomplemented
16C2×#15668437C2212yescomplemented
17C2×D1448C2212yescomplemented
18C2×#15668434C2212yescomplemented
19#19153813C2211yescharacteristic
20#19153812—21no—
21#19153812—21no—
22#19153812C2212yes—
23#19153811—21no—
24#19153804C2211yescharacteristic, complemented
25#19153801—21no—
26#19153796C2211yescharacteristic
27#19153791C2211yescharacteristic
28#19153781C2211yescharacteristic, complemented
29#19153778—21no—
30#19153778C2211yescharacteristic, complemented
31#19153774—21no—
32C42×D181C2211yescharacteristic
33#19153745C2212yescomplemented
34#19153736C2212yescomplemented
35#19153736C2212yescomplemented
36#19153728C2212yes—
37#19153727C2211yescharacteristic, complemented
38#127×C181C2211yescharacteristic
39#125×C181C2211yescharacteristic, complemented
40#102×C181C2212yescomplemented
41#102×C181C2212yescomplemented
42#97×C181C2211yescharacteristic, complemented
43C23×D181—22no—
44C2×Dic362C2311yescharacteristic
45C2×D724—21no—
46C2×D724—21no—
47C2×D724—21no—
48C2×D724C2312yes—
49C2×D724C2311yescharacteristic
50C2×C22⋊D181—22no—
51C2×C22⋊D181—22no—
52C2×C22⋊D181—22no—
53D4×D181—42no—
54D4×D181—42no—
55D4×D181—44no—
56D4×D181—44no—
57C2×C4×D181—21no—
58C2×C4×D181D411yescharacteristic
59C2×C4×D181D411yescharacteristic
60C2×C4×D181C2311yescharacteristic
61#15668438—42no—
62#15668437—42no—
63D1448—42no—
64#15668434—42no—
65C2×#13089678C2312yes—
66C4⋊Dic181—21no—
67#15668428—21no—
68#15668426—21no—
69#15668426—21no—
70#15668426—21no—
71#15668426—21no—
72#15668426—21no—
73C4×Dic181—21no—
74C4×Dic181D411yescharacteristic
75C4×Dic181D411yescharacteristic
76C4×Dic181C2311yescharacteristic
77C2×D4×C181—21no—
78C2×D4×C181—21no—
79C2×D4×C181C2312yes—
80C2×C8×C181C2312yes—
81C4⋊C4×C181—21no—
82C4⋊C4×C181—21no—
83C4⋊C4×C181C2311yescharacteristic
84C4⋊C4×C181C2311yescharacteristic
85C42×C181C2311yescharacteristic
86C22×D181—21no—
87C22×D181—22no—
88C22×D181—44no—
89C22×D181—44no—
90C22×D181—44no—
91C22×D181C2×D411yescharacteristic
92Dic362—22no—
93D724—22no—
94D724—22no—
95D724—22no—
96D724—42no—
97D724—42no—
98D724—42no—
99C22⋊D181—44no—
100C22⋊D181—44no—
101C22⋊D181—44no—
102C2×Dic181—21no—
103C2×Dic181C2×D411yescharacteristic
104C2×Dic181C2×D411yescharacteristic
105C2×Dic181C2×D411yescharacteristic
106C4×D181—22no—
107C4×D181—22no—
108C4×D181—22no—
109C4×D181—42no—
110#13089678—22no—
111C23×C181—22no—
112D4×C181—22no—
113D4×C181—22no—
114D4×C181—42no—
115D4×C181—42no—
116C2×C4×C181—21no—
117C2×C4×C181—21no—
118C2×C4×C181C2×D411yescharacteristic
119C2×C4×C181C2×D411yescharacteristic
120C2×C4×C181C2411yescharacteristic
121C8×C181—22no—
122C22×C181—42no—
123C22×C181—42no—
124C22×C181—42no—
125C22×C181C22×D411yescharacteristic
126C2×D181—21no—
127C2×D181—21no—
128C2×D181—42no—
129C2×D181—41no—
130C2×D181—41no—
131C2×D181—42no—
132C2×D181—42no—
133C2×D181—41no—
134C2×D181—88no—
135C2×D181AGL(1,ℤ/8)12yes—
136C4×C181—21no—
137C4×C181—21no—
138C4×C181—41no—
139C4×C181—41no—
140C4×C181H(5,𝔽2)12yes—
141Dic181—21no—
142Dic181—21no—
143Dic181—41no—
144Dic181AGL(1,ℤ/8)12yes—
145C2×C181—82no—
146C2×C181#53511yescharacteristic
147C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
148C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
149D181—22no—
150D181—81no—
151D181—82no—
152C181#321711yescharacteristic, complemented, Sylow
153#3217—1811nomaximal, Sylow
154C2×#127—1811no—
155#529—1811no—
156#496—1811no—
157#496—1811no—
158#490—1811no—
159#490D18111yescharacteristic, complemented
160#460—1812no—
161#459—1812no—
162#450—1812no—
163#447—1812no—
164#432—1811no—
165C22×D4—3621no—
166C22×D4—3621no—
167C22×D4—1812no—
168C2×SD16—1812no—
169C2×SD16—1812no—
170C2×D8—1812no—
171C2×D8—1812no—
172#127—3621no—
173#127—3621no—
174#127—1812no—
175#127—1811no—
176#127C2×D18111yescharacteristic
177#126—3621no—
178#125C2×D18111yescharacteristic, complemented
179#124—1811no—
180#123—3621no—
181C22⋊Q8—1811no—
182#121—1811no—
183C4×Q8—1811no—
184C4×D4—3621no—
185C4×D4—1811no—
186#117—3621no—
187C2×C42—1811no—
188C4⋊C8—1812no—
189#102—1812no—
190#102—1812no—
191#102C2×D18112yescomplemented
192#102C2×D18112yescomplemented
193C22⋊C8—1812no—
194#97—1811no—
195#97C2×D18111yescharacteristic, complemented
196C24—3622no—
197C2×Q8—1811no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—3621no—
201C2×D4—3622no—
202C2×D4—3622no—
203C2×D4—3622no—
204C2×D4—3621no—
205C2×D4—3621no—
206C2×D4—1812no—
207C2×D4—1811no—
208C2×D4—7242no—
209C2×D4—7242no—
210C2×D4—7244no—
211C2×D4—7244no—
212C2×D4C22×D18112yes—
213C22×C4—3621no—
214C22×C4—1811no—
215C22×C4—1811no—
216C22×C4—1811no—
217SD16—7242no—
218SD16—7242no—
219D8—7242no—
220D8—7242no—
221C2×C8—1812no—
222C2×C8C22×D18112yes—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—3621no—
226C4⋊C4—3621no—
227C4⋊C4C22×D18111yescharacteristic
228C4⋊C4C22×D18111yescharacteristic
229C22⋊C4—3621no—
230C22⋊C4—3621no—
231C22⋊C4—3621no—
232C22⋊C4—3621no—
233C22⋊C4—3621no—
234C42—3621no—
235C42—1811no—
236C42—1811no—
237C42—1811no—
238C42C22×D18111yescharacteristic
239C23—22no—
240C23—3621no—
241C23—3622no—
242C23—1811no—
243C23—7244no—
244C23—7244no—
245C23—7244no—
246Q8—3622no—
247D4—22no—
248D4—22no—
249D4—42no—
250D4—42no—
251D4—3622no—
252D4—3622no—
253D4—3622no—
254D4—7242no—
255D4—7242no—
256D4—7242no—
257D4—7244no—
258D4—7244no—
259D4—7244no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—3622no—
263C2×C4—3622no—
264C2×C4—3622no—
265C2×C4—3621no—
266C2×C4—1811no—
267C2×C4—1811no—
268C2×C4—1811no—
269C2×C4—7242no—
270C2×C4D4×D18111yescharacteristic
271C2×C4D4×D18111yescharacteristic
272C2×C4C23×D18111yescharacteristic
273C8—22no—
274C8—3622no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—3621no—
279C22—3621no—
280C22—14488no—
281C22—1812no—
282C22—7241no—
283C22—7242no—
284C22—7242no—
285C22—7241no—
286C22—7242no—
287C22—7241no—
288C22C2×D4×D18111yescharacteristic
289C4—21no—
290C4—21no—
291C4—41no—
292C4—41no—
293C4—3621no—
294C4—3621no—
295C4—1812no—
296C4—7241no—
297C4#1915386212yes—
298C2—82no—
299C2—3622no—
300C2—14482no—
301C2—14481no—
302C2#2604184711yescharacteristic
303C2AGL(1,ℤ/8)×D18111yescharacteristic
304C2AGL(1,ℤ/8)×D18111yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#490×C181C2
socleC22×C181C22×D4
hypercenter#490D181
hypocenterC181#3217
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #127×D181C2semidirect
2. #26041814C2semidirect
3. #26041668C2semidirect
4. #26041666C2semidirect
5. #26041645C2semidirect
6. #26041478C2semidirect
7. #26041473C2semidirect
8. #26041431C2semidirect
9. #26041414C2semidirect
10. #26041334C2semidirect
11. #490×C181C2semidirect
15. C2×#15668438C22semidirect
16. C2×#15668437C22semidirect
17. C2×D1448C22semidirect
18. C2×#15668434C22semidirect
24. #19153804C22semidirect
28. #19153781C22semidirect
30. #19153778C22semidirect
33. #19153745C22semidirect
34. #19153736C22semidirect
35. #19153736C22semidirect
37. #19153727C22semidirect
39. #125×C181C22semidirect
40. #102×C181C22semidirect
41. #102×C181C22semidirect
42. #97×C181C22semidirect
152. C181#3217semidirect
159. #490D181semidirect
178. #125C2×D181semidirect
191. #102C2×D181semidirect
192. #102C2×D181semidirect
195. #97C2×D181semidirect

Maximal subgroups

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

subgroupindex
#127×D1812
#260418142
#260416682
#260416662
#260416452
#260414782
#260414732
#260414312
#260414142
#260413342
#490×C1812
#3217181