Finite Group Databasebeta

← #37544738

Subgroups — #37544738

Order 23168.

490 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 115 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#26041811C211yesmaximal, characteristic, complemented
3#26041667C212yesmaximal, complemented
4#26041642C212yesmaximal, complemented
5#26041480C211yesmaximal, characteristic, complemented
6#26041476C211yesmaximal, characteristic, complemented
7#26041474C211yesmaximal, characteristic, complemented
8#26041431C212yesmaximal, complemented
9#26041411C212yesmaximal, complemented
10#26041335C211yesmaximal, characteristic, complemented
11#459×C181C211yesmaximal, characteristic, complemented
12C2×D4×D181—21no—
13C2×D4×D181—21no—
14C2×D4×D181C2212yes—
15C2×#15668438C2212yescomplemented
16C2×#15668437C2212yescomplemented
17C2×#15668434C2212yescomplemented
18#19153813C2211yescharacteristic
19#19153812—21no—
20#19153812—21no—
21#19153812C2212yes—
22#19153809—21no—
23#19153807C2211yescharacteristic, complemented
24#19153803C2211yescharacteristic, complemented
25#19153798—21no—
26#19153796C2211yescharacteristic
27#19153787C2211yescharacteristic
28C4×Dic362C2211yescharacteristic, complemented
29C4×C22⋊D181—21no—
30#19153775—21no—
31C42×D181C2211yescharacteristic
32C4⋊#13089678C2211yescharacteristic, complemented
33#19153738C2212yescomplemented
34#19153737C2212yescomplemented
35#19153736C2212yescomplemented
36#19153728C2212yes—
37#19153726C2212yescomplemented
38C2×D8×C181C2212yescomplemented
39#127×C181C2211yescharacteristic
40C4×D4×C181C2211yescharacteristic, complemented
41C4⋊C8×C181C2211yescharacteristic, complemented
42#102×C181C2212yescomplemented
43C23×D181—22no—
44C2×Dic362C2311yescharacteristic
45C2×D724—21no—
46C2×D724—21no—
47C2×D724C2312yes—
48C2×C22⋊D181—21no—
49C2×C22⋊D181—22no—
50C2×C22⋊D181—22no—
51C2×C22⋊D181—22no—
52D4×D181—42no—
53D4×D181—42no—
54D4×D181—44no—
55D4×D181—44no—
56C2×C4×D181D411yescharacteristic
57C2×C4×D181D411yescharacteristic
58C2×C4×D181C2311yescharacteristic
59#15668438—42no—
60#15668437—42no—
61#15668434—42no—
62C2×#13089678C2312yes—
63C4⋊Dic181—21no—
64C4⋊Dic181—21no—
65C4⋊Dic181C2311yescharacteristic
66#15668428—21no—
67#15668428—21no—
68C22⋊Dic181—21no—
69C22⋊Dic181—21no—
70C22⋊Dic181—21no—
71#15668426—21no—
72C4×Dic181—21no—
73C4×Dic181D411yescharacteristic
74C4×Dic181D411yescharacteristic
75C4×Dic181C2311yescharacteristic
76C2×D4×C181—21no—
77C2×D4×C181—21no—
78C2×D4×C181C2312yes—
79C2×D4×C181C2311yescharacteristic
80C22×C4×C181—21no—
81D8×C181—42no—
82C2×C8×C181C2312yes—
83C4⋊C4×C181C2311yescharacteristic
84C22⋊C4×C181—21no—
85C42×C181C2311yescharacteristic
86C22×D181—22no—
87C22×D181—44no—
88C22×D181—44no—
89C22×D181—44no—
90C22×D181C2×D411yescharacteristic
91Dic362—21no—
92Dic362—21no—
93D724—22no—
94D724—22no—
95D724—42no—
96D724—42no—
97C22⋊D181—44no—
98C22⋊D181—44no—
99C22⋊D181—44no—
100C22⋊D181—42no—
101C2×Dic181—21no—
102C2×Dic181—21no—
103C2×Dic181C2×D411yescharacteristic
104C2×Dic181C2×D411yescharacteristic
105C2×Dic181C2×D411yescharacteristic
106C4×D181—22no—
107C4×D181—22no—
108C4×D181—21no—
109C4×D181—21no—
110#13089678—22no—
111C23×C181—21no—
112C23×C181—22no—
113D4×C181—21no—
114D4×C181—22no—
115D4×C181—21no—
116D4×C181—22no—
117D4×C181—42no—
118D4×C181—42no—
119C2×C4×C181—21no—
120C2×C4×C181—42no—
121C2×C4×C181C2×D411yescharacteristic
122C2×C4×C181C2×D411yescharacteristic
123C2×C4×C181C2411yescharacteristic
124C8×C181—22no—
125C22×C181—41no—
126C22×C181—42no—
127C22×C181—41no—
128C22×C181—42no—
129C22×C181—42no—
130C22×C181—41no—
131C22×C181C22×D411yescharacteristic
132C2×D181—21no—
133C2×D181—21no—
134C2×D181—42no—
135C2×D181—42no—
136C2×D181—42no—
137C2×D181—88no—
138C2×D181AGL(1,ℤ/8)12yes—
139C4×C181—21no—
140C4×C181—41no—
141C4×C181AGL(1,ℤ/8)12yes—
142C4×C181H(5,𝔽2)11yescharacteristic
143C4×C181H(5,𝔽2)11yescharacteristic
144Dic181—21no—
145Dic181—21no—
146Dic181—21no—
147Dic181—41no—
148Dic181—41no—
149C2×C181—82no—
150C2×C181—81no—
151C2×C181#53511yescharacteristic
152C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
153C2×C181C2×AGL(1,ℤ/8)11yescharacteristic
154D181—22no—
155D181—82no—
156C181#321711yescharacteristic, complemented, Sylow
157#3217—1811nomaximal, Sylow
158C2×#127—1811no—
159#529—1811no—
160#496—1812no—
161#490—1812no—
162#460—1811no—
163#460—1811no—
164#459—1811no—
165#459D18111yescharacteristic, complemented
166#450—1812no—
167#447—1812no—
168#432—1811no—
169C22×D4—3621no—
170C22×D4—3621no—
171C22×D4—1812no—
172C2×SD16—1812no—
173C2×SD16—1812no—
174C2×D8—1812no—
175C2×D8C2×D18112yescomplemented
176#127—3621no—
177#127—3621no—
178#127—1812no—
179#127—1811no—
180#127C2×D18111yescharacteristic
181#126—3621no—
182#125—1811no—
183#124—1811no—
184#123—3621no—
185C22⋊Q8—1811no—
186#121—1811no—
187C4×Q8—1811no—
188C4×D4—3621no—
189C4×D4C2×D18111yescharacteristic, complemented
190#117—3621no—
191C2×C42—1811no—
192C4⋊C8—1811no—
193C4⋊C8C2×D18111yescharacteristic, complemented
194#102—1812no—
195#102—1812no—
196#102—1812no—
197#102C2×D18112yescomplemented
198C22⋊C8—1812no—
199#97—1812no—
200C24—3622no—
201C2×Q8—1811no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—3621no—
205C2×D4—3622no—
206C2×D4—3622no—
207C2×D4—3622no—
208C2×D4—3621no—
209C2×D4—3621no—
210C2×D4—1812no—
211C2×D4—7242no—
212C2×D4—7242no—
213C2×D4—7244no—
214C2×D4—7244no—
215C2×D4C22×D18112yes—
216C2×D4C22×D18111yescharacteristic
217C22×C4—21no—
218C22×C4—1811no—
219C22×C4—1811no—
220C22×C4—1811no—
221SD16—7242no—
222SD16—7242no—
223D8—42no—
224D8—7242no—
225C2×C8—1812no—
226C2×C8C22×D18112yes—
227C4⋊C4—3621no—
228C4⋊C4—3621no—
229C4⋊C4—3621no—
230C4⋊C4—3621no—
231C4⋊C4—1811no—
232C4⋊C4C22×D18111yescharacteristic
233C22⋊C4—21no—
234C22⋊C4—3621no—
235C22⋊C4—3621no—
236C22⋊C4—3621no—
237C22⋊C4—3621no—
238C42—3621no—
239C42—1811no—
240C42—1811no—
241C42—1811no—
242C42C22×D18111yescharacteristic
243C23—21no—
244C23—22no—
245C23—3622no—
246C23—1811no—
247C23—7244no—
248C23—7244no—
249C23—7244no—
250Q8—3621no—
251Q8—3621no—
252D4—21no—
253D4—22no—
254D4—21no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—3622no—
259D4—3622no—
260D4—7242no—
261D4—7244no—
262D4—7244no—
263D4—7244no—
264D4—7242no—
265D4—7242no—
266C2×C4—21no—
267C2×C4—42no—
268C2×C4—3622no—
269C2×C4—3622no—
270C2×C4—3621no—
271C2×C4—3621no—
272C2×C4—3621no—
273C2×C4—3621no—
274C2×C4—1811no—
275C2×C4—1811no—
276C2×C4—1811no—
277C2×C4D4×D18111yescharacteristic
278C2×C4D4×D18111yescharacteristic
279C2×C4C23×D18111yescharacteristic
280C8—22no—
281C8—3622no—
282C22—42no—
283C22—41no—
284C22—41no—
285C22—42no—
286C22—42no—
287C22—41no—
288C22—3621no—
289C22—3621no—
290C22—14488no—
291C22—1812no—
292C22—7242no—
293C22—7242no—
294C22—7242no—
295C22C2×D4×D18111yescharacteristic
296C4—21no—
297C4—41no—
298C4—3621no—
299C4—3621no—
300C4—3621no—
301C4—7241no—
302C4—7241no—
303C4#1915384012yes—
304C4#1915386111yescharacteristic
305C4#1915386211yescharacteristic
306C2—82no—
307C2—81no—
308C2—3622no—
309C2—14482no—
310C2#2604184511yescharacteristic
311C2AGL(1,ℤ/8)×D18111yescharacteristic
312C2C2×#1915384011yescharacteristic
subgroupG/H
centerC22C2×D4×D181
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
Fitting#459×C181C2
socleC22×C181C22×D4
hypercenter#459D181
hypocenterC181#3217
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #127×D181C2semidirect
2. #26041811C2semidirect
3. #26041667C2semidirect
4. #26041642C2semidirect
5. #26041480C2semidirect
6. #26041476C2semidirect
7. #26041474C2semidirect
8. #26041431C2semidirect
9. #26041411C2semidirect
10. #26041335C2semidirect
11. #459×C181C2semidirect
15. C2×#15668438C22semidirect
16. C2×#15668437C22semidirect
17. C2×#15668434C22semidirect
23. #19153807C22semidirect
24. #19153803C22semidirect
28. C4×Dic362C22semidirect
32. C4⋊#13089678C22semidirect
33. #19153738C22semidirect
34. #19153737C22semidirect
35. #19153736C22semidirect
37. #19153726C22semidirect
38. C2×D8×C181C22semidirect
40. C4×D4×C181C22semidirect
41. C4⋊C8×C181C22semidirect
42. #102×C181C22semidirect
156. C181#3217semidirect
165. #459D181semidirect
175. C2×D8C2×D181semidirect
189. C4×D4C2×D181semidirect
193. C4⋊C8C2×D181semidirect
197. #102C2×D181semidirect

Maximal subgroups

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

subgroupindex
#127×D1812
#260418112
#260416672
#260416422
#260414802
#260414762
#260414742
#260414312
#260414112
#260413352
#459×C1812
#3217181