Finite Group Databasebeta

← #36712581

Subgroups — #36712581

Order 22784.

462 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 127 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#26033402C211yesmaximal, characteristic, complemented
2#26033390C211yesmaximal, characteristic, complemented
3#26022571C212yesmaximal, complemented
4#26022569C211yesmaximal, characteristic, complemented
5#26022564C211yesmaximal, characteristic, complemented
6#26022563C212yesmaximal, complemented
7#26022560C212yesmaximal, complemented
8C2×#19151437C211yesmaximal, characteristic, complemented
9SD64×D89C212yesmaximal, complemented
10#26022545C211yesmaximal, characteristic, complemented
11C89×#2262C211yesmaximal, characteristic, complemented
12#19152412—21no—
13#19152400—21no—
14#19152038—21no—
15#19152037—22no—
16#19152035—22no—
17#19152033—21no—
18#19152031C2212yes—
19#19152027C2212yescomplemented
20C2×Dic712C2211yescharacteristic, complemented
21Q32×D89C2212yescomplemented
22C2×D1424C2211yescharacteristic
23D16×D89C2212yes—
24#19152014C2211yescharacteristic
25#19151438C2212yescomplemented
26#19151437C2212yescomplemented
27#19151437C2212yescomplemented
28#19151436C2212yescomplemented
29#19151435C2212yescomplemented
30#19151433C2212yescomplemented
31#19151432C2211yescharacteristic, complemented
32#19151431C2212yescomplemented
33C32×D89C2212yescomplemented
34#508×C89C2211yescharacteristic
35#508×C89C2211yescharacteristic, complemented
36Q64×C89C2211yescharacteristic, complemented
37SD64×C89C2211yescharacteristic, complemented
38SD64×C89C2211yescharacteristic, complemented
39D32×C89C2211yescharacteristic, complemented
40C2×C32×C89C2211yescharacteristic, complemented
41#15668059—41no—
42#15668058—41no—
43#15668042—41no—
44#15668041—42no—
45#15668039—42no—
46#15668037—41no—
47#15668035—22no—
48#15668031—22no—
49C2×Dic356—21no—
50Q16×D89—22no—
51C2×D712—21no—
52D8×D89—22no—
53#15668018D411yescharacteristic
54Dic712—21no—
55Dic712C2312yes—
56#15667958—22no—
57#15667956—22no—
58#15667955—22no—
59D1424—21no—
60D1424C2312yes—
61#15667953—22no—
62#15667952D411yescharacteristic
63#15667951D412yes—
64C16×D89C2312yes—
65#15667918C2312yes—
66#135×C89—21no—
67#135×C89—21no—
68Q32×C89C2311yescharacteristic
69Q32×C89C2311yescharacteristic
70SD32×C89—21no—
71SD32×C89—21no—
72D16×C89C2311yescharacteristic
73D16×C89C2311yescharacteristic
74C2×C16×C89C2311yescharacteristic
75C32×C89C2311yescharacteristic
76C32×C89C2311yescharacteristic
77#13083575—42no—
78#13083574—81no—
79#13083574—81no—
80#13083574D811yescharacteristic
81#13083573—42no—
82#13083573—82no—
83C2×Dic178—41no—
84C2×Dic178—81no—
85Q8×D89—42no—
86C2×D356—41no—
87C2×D356—81no—
88D4×D89—42no—
89D4×D89—82no—
90Dic356—22no—
91Dic356—41no—
92#13083564—42no—
93#13083562—42no—
94#13083561—42no—
95D712—22no—
96D712—41no—
97#13083559—42no—
98#13083558D811yescharacteristic
99#13083557D812yes—
100C8×D89C2×D412yes—
101#13083549C2×D412yes—
102#41×C89—41no—
103#41×C89—41no—
104Q16×C89—21no—
105Q16×C89—21no—
106SD16×C89—41no—
107SD16×C89—41no—
108D8×C89—21no—
109D8×C89—21no—
110C2×C8×C89C2×D411yescharacteristic
111C16×C89C2×D411yescharacteristic
112C16×C89C2411yescharacteristic
113C22×D89—81no—
114C22×D89—82no—
115Q8×C89—41no—
116Q8×C89—41no—
117D4×C89—41no—
118D4×C89—41no—
119D4×C89—81no—
120D4×C89—81no—
121C2×C4×C89—81no—
122C2×C4×C89—81no—
123C2×C4×C89C2×D811yescharacteristic
124C22⋊D89—82no—
125C22⋊D89—82no—
126C22⋊D89D1612yes—
127C2×Dic89—81no—
128C2×Dic89—82no—
129D356—42no—
130D356—82no—
131D356—81no—
132D356D1611yescharacteristic
133C4×D89—82no—
134C4×D89—82no—
135C4×D89C2×D812yes—
136Dic178—42no—
137Dic178—81no—
138Dic178—82no—
139Dic178D1611yescharacteristic
140C8×C89C2×D811yescharacteristic
141C8×C89C22×D411yescharacteristic
142#10623494C2×D812yes—
143C22×C89—81no—
144C22×C89—81no—
145C22×C89C2×D1611yescharacteristic
146C2×D89—82no—
147C2×D89—322no—
148C2×D89—321no—
149C2×D89C2×D1612yes—
150C4×C89—81no—
151C4×C89—81no—
152C4×C89C2×D1611yescharacteristic
153C4×C89C22×D811yescharacteristic
154Dic89—82no—
155Dic89C2×D1612yes—
156#33983—891nomaximal, Sylow
157C2×C89—21no—
158C2×C89—161no—
159C2×C89—161no—
160C2×C89C22×D1611yescharacteristic
161D89—22no—
162D89—162no—
163#3417—891no—
164#3415—891no—
165#2264—891no—
166#2264—892no—
167#2263—891no—
168#2263—892no—
169#2262—892no—
170#2262D8911yescharacteristic, complemented
171C2×SD64—892no—
172C2×SD64—891no—
173#2258—891no—
174C89#3398311yescharacteristic, complemented, Sylow
175#578—1781no—
176#576—1781no—
177#510—1782no—
178#510—1781no—
179#509—1782no—
180#509—1781no—
181#508—892no—
182#508—892no—
183#508C2×D8911yescharacteristic
184#508C2×D8911yescharacteristic, complemented
185C2×Q32—892no—
186C2×Q32—891no—
187C2×D16—892no—
188C2×D16—891no—
189#504—891no—
190Q64—892no—
191Q64C2×D8911yescharacteristic, complemented
192SD64—892no—
193SD64—892no—
194SD64—892no—
195SD64—892no—
196SD64C2×D8911yescharacteristic, complemented
197SD64C2×D8911yescharacteristic, complemented
198D32—892no—
199D32C2×D8911yescharacteristic, complemented
200M64—891no—
201M64—892no—
202C2×C32—892no—
203C2×C32C2×D8911yescharacteristic, complemented
2042−1+4—3561no—
205H(5,𝔽2)—3561no—
206#137—3562no—
207#137—3561no—
208AGL(1,ℤ/8)—3562no—
209AGL(1,ℤ/8)—3561no—
210#135—21no—
211#135—21no—
212#135—1782no—
213#135—1782no—
214C2×Q16—1782no—
215C2×Q16—1781no—
216C2×D8—1782no—
217C2×D8—1781no—
218#131—891no—
219Q32—892no—
220Q32—1782no—
221Q32—1781no—
222Q32C22×D8911yescharacteristic
223Q32C22×D8911yescharacteristic
224SD32—21no—
225SD32—21no—
226SD32—1782no—
227SD32—1782no—
228D16—892no—
229D16—1782no—
230D16—1781no—
231D16C22×D8911yescharacteristic
232D16C22×D8911yescharacteristic
233M32—891no—
234M32—892no—
235C2×C16—892no—
236C2×C16C22×D8911yescharacteristic
237C32—892no—
238C32C22×D8911yescharacteristic
239C32C22×D8911yescharacteristic
240#41—41no—
241#41—41no—
242#41—891no—
243#41—3562no—
244#41—3562no—
245#41—7121no—
246#41—7121no—
247#41—7122no—
248C2×Q8—3562no—
249C2×Q8—3561no—
250C2×Q8—7121no—
251C2×D4—3562no—
252C2×D4—3561no—
253C2×D4—7121no—
254C2×D4—7122no—
255Q16—21no—
256Q16—21no—
257Q16—3562no—
258Q16—3561no—
259Q16—1782no—
260SD16—41no—
261SD16—41no—
262SD16—3562no—
263SD16—3562no—
264D8—21no—
265D8—21no—
266D8—3562no—
267D8—3561no—
268D8—1782no—
269M16—891no—
270M16—892no—
271C2×C8—892no—
272C2×C8D4×D8911yescharacteristic
273C16—892no—
274C16D4×D8911yescharacteristic
275C16C23×D8911yescharacteristic
276C23—7122no—
277C23—7121no—
278Q8—41no—
279Q8—41no—
280Q8—891no—
281Q8—3562no—
282Q8—7122no—
283Q8—7121no—
284D4—41no—
285D4—41no—
286D4—81no—
287D4—81no—
288D4—892no—
289D4—891no—
290D4—3562no—
291D4—7122no—
292D4—7122no—
293D4—7122no—
294D4—7121no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—892no—
298C2×C4—7122no—
299C2×C4—7122no—
300C2×C4—7122no—
301C2×C4—7121no—
302C2×C4D8×D8911yescharacteristic
303C8—892no—
304C8D8×D8911yescharacteristic
305C8C2×D4×D8911yescharacteristic
306C22—81no—
307C22—81no—
308C22—28481no—
309C22—28482no—
310C22—892no—
311C22—7122no—
312C22D16×D8911yescharacteristic
313C4—81no—
314C4—81no—
315C4—892no—
316C4—7122no—
317C4D16×D8911yescharacteristic
318C4C2×D8×D8911yescharacteristic
319C2—21no—
320C2—161no—
321C2—161no—
322C2—14242no—
323C2—1782no—
324C2C2×D16×D8911yescharacteristic
subgroupG/H
centerC2C2×D16×D89
commutator, cosocleC16×C89C24
FrattiniC16C23×D89
FittingC89×#2262C2
socleC2×C89C22×D16
hypercenter#2262D89
hypocenterC89#33983
LCSG ▷ C16×C89 ▷ C8×C89 ▷ C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #2262

Complements

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

NH ≅ G/Ntype
1. #26033402C2semidirect
2. #26033390C2semidirect
3. #26022571C2semidirect
4. #26022569C2semidirect
5. #26022564C2semidirect
6. #26022563C2semidirect
7. #26022560C2semidirect
8. C2×#19151437C2semidirect
9. SD64×D89C2semidirect
10. #26022545C2semidirect
11. C89×#2262C2semidirect
19. #19152027C22semidirect
20. C2×Dic712C22semidirect
21. Q32×D89C22semidirect
25. #19151438C22semidirect
26. #19151437C22semidirect
27. #19151437C22semidirect
28. #19151436C22semidirect
29. #19151435C22semidirect
30. #19151433C22semidirect
31. #19151432C22semidirect
32. #19151431C22semidirect
33. C32×D89C22semidirect
35. #508×C89C22semidirect
36. Q64×C89C22semidirect
37. SD64×C89C22semidirect
38. SD64×C89C22semidirect
39. D32×C89C22semidirect
40. C2×C32×C89C22semidirect
170. #2262D89semidirect
174. C89#33983semidirect
184. #508C2×D89semidirect
191. Q64C2×D89semidirect
196. SD64C2×D89semidirect
197. SD64C2×D89semidirect
199. D32C2×D89semidirect
203. C2×C32C2×D89semidirect

Maximal subgroups

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

subgroupindex
#260334022
#260333902
#260225712
#260225692
#260225642
#260225632
#260225602
C2×#191514372
SD64×D892
#260225452
C89×#22622
#3398389