Finite Group Databasebeta

← #37544735

Subgroups — #37544735

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#26041808C211yesmaximal, characteristic, complemented
3#26041669C211yesmaximal, characteristic, complemented
4#26041644C211yesmaximal, characteristic, complemented
5#26041643C211yesmaximal, characteristic, complemented
6#26041477C212yesmaximal, complemented
7#26041472C212yesmaximal, complemented
8#26041428C212yesmaximal, complemented
9#26041411C212yesmaximal, complemented
10#26041334C211yesmaximal, characteristic, complemented
11#496×C181C211yesmaximal, characteristic, complemented
12C2×D4×D181—21no—
13C2×D4×D181—21no—
14C2×D4×D181C2212yes—
15C2×#15668436C2212yescomplemented
16C2×#15668434C2212yescomplemented
17#19153813C2211yescharacteristic
18#19153812—21no—
19#19153812—21no—
20#19153812C2212yes—
21#19153809—21no—
22#19153808C2211yescharacteristic, complemented
23#19153798—21no—
24#19153792C2211yescharacteristic
25#19153787C2211yescharacteristic
26Q8×Dic181C2211yescharacteristic, complemented
27D4×Dic181C2211yescharacteristic, complemented
28#19153776—21no—
29#19153772—21no—
30C42×D181C2211yescharacteristic
31#19153745C2212yescomplemented
32#19153738C2212yescomplemented
33#19153738C2212yescomplemented
34#19153737C2212yescomplemented
35#19153737C2212yescomplemented
36#19153728C2212yes—
37#19153727C2211yescharacteristic, complemented
38C2×SD16×C181C2212yescomplemented
39C2×D8×C181C2212yescomplemented
40#127×C181C2211yescharacteristic
41#124×C181C2211yescharacteristic, complemented
42#97×C181C2211yescharacteristic, complemented
43C23×D181—22no—
44C2×D724—21no—
45C2×D724—21no—
46C2×D724C2312yes—
47C2×C22⋊D181—21no—
48C2×C22⋊D181—22no—
49C2×C22⋊D181—22no—
50C2×C22⋊D181—22no—
51D4×D181—42no—
52D4×D181—42no—
53D4×D181—44no—
54D4×D181—44no—
55C22×Dic181—21no—
56C2×C4×D181D411yescharacteristic
57C2×C4×D181D411yescharacteristic
58C2×C4×D181C2311yescharacteristic
59#15668436—42no—
60#15668434—42no—
61C2×#13089678C2312yes—
62C4⋊Dic181—21no—
63C4⋊Dic181C2311yescharacteristic
64C4⋊Dic181C2311yescharacteristic
65#15668428—21no—
66#15668428—21no—
67#15668428—21no—
68C22⋊Dic181—21no—
69C22⋊Dic181—21no—
70#15668426—21no—
71C4×Dic181—21no—
72C4×Dic181D411yescharacteristic
73C4×Dic181D411yescharacteristic
74C4×Dic181C2311yescharacteristic
75C2×Q8×C181C2311yescharacteristic
76C2×D4×C181—21no—
77C2×D4×C181—21no—
78C2×D4×C181C2311yescharacteristic
79C2×D4×C181C2312yes—
80SD16×C181—42no—
81D8×C181—42no—
82C2×C8×C181C2312yes—
83C22⋊C4×C181—21no—
84C22⋊C4×C181—21no—
85C42×C181C2311yescharacteristic
86C22×D181—22no—
87C22×D181—44no—
88C22×D181—44no—
89C22×D181—44no—
90C22×D181C2×D411yescharacteristic
91D724—22no—
92D724—22no—
93D724—42no—
94D724—42no—
95C22⋊D181—44no—
96C22⋊D181—44no—
97C22⋊D181—42no—
98C22⋊D181—44no—
99C2×Dic181—21no—
100C2×Dic181—21no—
101C2×Dic181—42no—
102C2×Dic181C2×D411yescharacteristic
103C2×Dic181C2×D411yescharacteristic
104C2×Dic181C2×D411yescharacteristic
105C4×D181—22no—
106C4×D181—22no—
107C4×D181—22no—
108#13089678—22no—
109C23×C181—21no—
110C23×C181—22no—
111Q8×C181—22no—
112D4×C181—22no—
113D4×C181—22no—
114D4×C181—22no—
115D4×C181—42no—
116D4×C181—42no—
117C2×C4×C181—21no—
118C2×C4×C181C2×D411yescharacteristic
119C2×C4×C181C2×D411yescharacteristic
120C2×C4×C181C2411yescharacteristic
121C8×C181—22no—
122C22×C181—41no—
123C22×C181—42no—
124C22×C181—42no—
125C22×C181—42no—
126C22×C181—41no—
127C22×C181—41no—
128C22×C181C22×D411yescharacteristic
129C2×D181—21no—
130C2×D181—21no—
131C2×D181—42no—
132C2×D181—42no—
133C2×D181—42no—
134C2×D181—88no—
135C2×D181AGL(1,ℤ/8)12yes—
136C4×C181—21no—
137C4×C181—21no—
138C4×C181—41no—
139C4×C181H(5,𝔽2)12yes—
140Dic181—21no—
141Dic181—21no—
142Dic181—41no—
143Dic181—41no—
144Dic181AGL(1,ℤ/8)12yes—
145C2×C181—82no—
146C2×C181—81no—
147C2×C181#53511yescharacteristic
148C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
149C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
150D181—22no—
151D181—82no—
152C181#321711yescharacteristic, complemented, Sylow
153#3217—1811nomaximal, Sylow
154C2×#127—1811no—
155#529—1811no—
156#496—1811no—
157#496D18111yescharacteristic, complemented
158#490—1811no—
159#490—1811no—
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×SD16C2×D18112yescomplemented
170C2×D8—1812no—
171C2×D8C2×D18112yescomplemented
172#127—3621no—
173#127—3621no—
174#127—1812no—
175#127—1811no—
176#127C2×D18111yescharacteristic
177#126—3621no—
178#125—1811no—
179#124C2×D18111yescharacteristic, complemented
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#102—1812no—
192#102—1812no—
193C22⋊C8—1812no—
194#97—1811no—
195#97C2×D18111yescharacteristic, complemented
196C24—3622no—
197C2×Q8C22×D18111yescharacteristic
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—7242no—
208C2×D4—7242no—
209C2×D4—7244no—
210C2×D4—7244no—
211C2×D4C22×D18112yes—
212C2×D4C22×D18111yescharacteristic
213C22×C4—3621no—
214C22×C4—1811no—
215C22×C4—1811no—
216C22×C4—1811no—
217SD16—42no—
218SD16—7242no—
219D8—42no—
220D8—7242no—
221C2×C8—1812no—
222C2×C8C22×D18112yes—
223C4⋊C4—3621no—
224C4⋊C4—3621no—
225C4⋊C4—3621no—
226C4⋊C4—3621no—
227C4⋊C4—1811no—
228C4⋊C4—1811no—
229C22⋊C4—21no—
230C22⋊C4—21no—
231C22⋊C4—3621no—
232C22⋊C4—3621no—
233C22⋊C4—3621no—
234C42—3621no—
235C42—1811no—
236C42—1811no—
237C42—1811no—
238C42C22×D18111yescharacteristic
239C23—21no—
240C23—22no—
241C23—3622no—
242C23—1811no—
243C23—7244no—
244C23—7244no—
245C23—7244no—
246Q8—22no—
247D4—22no—
248D4—22no—
249D4—22no—
250D4—42no—
251D4—42no—
252D4—3622no—
253D4—3622no—
254D4—7242no—
255D4—7242no—
256D4—7244no—
257D4—7244no—
258D4—7244no—
259D4—7242no—
260C2×C4—21no—
261C2×C4—3622no—
262C2×C4—3622no—
263C2×C4—3622no—
264C2×C4—3621no—
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—41no—
276C22—42no—
277C22—42no—
278C22—41no—
279C22—42no—
280C22—41no—
281C22—3621no—
282C22—3621no—
283C22—14488no—
284C22—1812no—
285C22—7242no—
286C22—7242no—
287C22—7242no—
288C22C2×D4×D18111yescharacteristic
289C4—21no—
290C4—21no—
291C4—41no—
292C4—3621no—
293C4—3621no—
294C4—1812no—
295C4—7241no—
296C4—7241no—
297C4#1915386112yes—
298C2—82no—
299C2—81no—
300C2—3622no—
301C2—14482no—
302C2#2604184211yescharacteristic
303C2AGL(1,ℤ/8)×D18111yescharacteristic
304C2AGL(1,ℤ/8)×D18111yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#496×C181C2
socleC22×C181C22×D4
hypercenter#496D181
hypocenterC181#3217
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #127×D181C2semidirect
2. #26041808C2semidirect
3. #26041669C2semidirect
4. #26041644C2semidirect
5. #26041643C2semidirect
6. #26041477C2semidirect
7. #26041472C2semidirect
8. #26041428C2semidirect
9. #26041411C2semidirect
10. #26041334C2semidirect
11. #496×C181C2semidirect
15. C2×#15668436C22semidirect
16. C2×#15668434C22semidirect
22. #19153808C22semidirect
26. Q8×Dic181C22semidirect
27. D4×Dic181C22semidirect
31. #19153745C22semidirect
32. #19153738C22semidirect
33. #19153738C22semidirect
34. #19153737C22semidirect
35. #19153737C22semidirect
37. #19153727C22semidirect
38. C2×SD16×C181C22semidirect
39. C2×D8×C181C22semidirect
41. #124×C181C22semidirect
42. #97×C181C22semidirect
152. C181#3217semidirect
157. #496D181semidirect
169. C2×SD16C2×D181semidirect
171. C2×D8C2×D181semidirect
179. #124C2×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
#260418082
#260416692
#260416442
#260416432
#260414772
#260414722
#260414282
#260414112
#260413342
#496×C1812
#3217181