Finite Group Databasebeta

← #36712578

Subgroups — #36712578

Order 22784.

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

normal subgroups only

This group has 472 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#26033390C211yesmaximal, characteristic, complemented
2#26033389C211yesmaximal, characteristic, complemented
3#26022567C211yesmaximal, characteristic, complemented
4#26022566C212yesmaximal, complemented
5#26022563C212yesmaximal, complemented
6#26022560C212yesmaximal, complemented
7#26022559C211yesmaximal, characteristic, complemented
8C2×#19151433C211yesmaximal, characteristic, complemented
9D32×D89C212yesmaximal, complemented
10#26022547C211yesmaximal, characteristic, complemented
11C89×#2263C211yesmaximal, characteristic, complemented
12#19152400—21no—
13#19152399—21no—
14#19152035—22no—
15#19152034—21no—
16#19152033—21no—
17#19152032—22no—
18#19152031C2212yescomplemented
19#19152030C2211yescharacteristic, complemented
20#19152027C2212yescomplemented
21C2×D1424C2211yescharacteristic, complemented
22D16×D89C2212yescomplemented
23D16×D89C2212yescomplemented
24#19152014C2211yescharacteristic
25#19151438C2211yescharacteristic, complemented
26#19151437C2212yescomplemented
27#19151436C2211yescharacteristic, complemented
28#19151435C2211yescharacteristic, complemented
29D2848C2212yescomplemented
30#19151433C2211yescharacteristic, complemented
31#19151433C2212yescomplemented
32#19151433C2212yescomplemented
33#19151431C2212yescomplemented
34C2×#15667918C2211yescharacteristic, complemented
35C32×D89C2212yescomplemented
36#508×C89C2211yescharacteristic, complemented
37C2×D16×C89C2211yescharacteristic, complemented
38SD64×C89C2212yescomplemented
39D32×C89C2212yescomplemented
40M64×C89C2211yescharacteristic, complemented
41#15668058—41no—
42#15668057—41no—
43#15668039—42no—
44#15668038—41no—
45#15668037—41no—
46#15668036—42no—
47#15668035—22no—
48#15668034—21no—
49#15668031—22no—
50C2×D712—21no—
51D8×D89—22no—
52D8×D89—22no—
53#15668018D411yescharacteristic
54Dic712C2311yescharacteristic
55#15667957—21no—
56#15667956—22no—
57#15667955—22no—
58D1424—21no—
59D1424C2312yes—
60D1424C2311yescharacteristic
61#15667953—22no—
62#15667953—22no—
63#15667952D411yescharacteristic
64#15667951D412yes—
65C16×D89C2312yes—
66#15667918C2311yescharacteristic
67#15667918C2311yescharacteristic
68#135×C89—21no—
69C2×D8×C89—21no—
70Q32×C89C2311yescharacteristic
71SD32×C89—21no—
72D16×C89—21no—
73D16×C89C2312yes—
74D16×C89C2311yescharacteristic
75C2×C16×C89C2311yescharacteristic
76C32×C89C2312yes—
77#13083575—42no—
78#13083574—41no—
79#13083574—81no—
80#13083574D811yescharacteristic
81#13083573—42no—
82#13083573—81no—
83C2×D356—41no—
84C2×D356—81no—
85C2×C22⋊D89—82no—
86D4×D89—42no—
87D4×D89—42no—
88D4×D89—82no—
89D4×D89—81no—
90Dic356—21no—
91#13083563—41no—
92#13083562—42no—
93#13083561—42no—
94D712—21no—
95D712—22no—
96D712—41no—
97#13083559—42no—
98#13083559—42no—
99#13083558D811yescharacteristic
100#13083557D812yes—
101C8×D89C2×D412yes—
102#13083549C2×D412yes—
103#41×C89—41no—
104C2×D4×C89—41no—
105Q16×C89—21no—
106SD16×C89—41no—
107D8×C89—21no—
108D8×C89—22no—
109D8×C89—41no—
110C2×C8×C89C2×D411yescharacteristic
111C16×C89C2×D411yescharacteristic
112C16×C89C2411yescharacteristic
113C23×C89—81no—
114C22×D89—81no—
115C22×D89—82no—
116C22×D89—82no—
117Q8×C89—41no—
118D4×C89—42no—
119D4×C89—41no—
120D4×C89—81no—
121D4×C89—81no—
122C2×C4×C89—81no—
123C2×C4×C89C2×D811yescharacteristic
124C22⋊D89—81no—
125C22⋊D89—82no—
126C22⋊D89—82no—
127C22⋊D89—82no—
128C22⋊D89D1612yes—
129C2×Dic89—82no—
130D356—41no—
131D356—42no—
132D356—82no—
133D356—81no—
134D356D1611yescharacteristic
135C4×D89—81no—
136C4×D89—82no—
137C4×D89C2×D812yes—
138Dic178—41no—
139Dic178D1611yescharacteristic
140C8×C89C2×D811yescharacteristic
141C8×C89C22×D411yescharacteristic
142#10623494C2×D812yes—
143C22×C89—82no—
144C22×C89—81no—
145C22×C89—321no—
146C22×C89C2×D1611yescharacteristic
147C2×D89—82no—
148C2×D89—81no—
149C2×D89—322no—
150C2×D89—322no—
151C2×D89—321no—
152C2×D89C2×D1612yes—
153C4×C89—81no—
154C4×C89C2×D1611yescharacteristic
155C4×C89C22×D811yescharacteristic
156Dic89—81no—
157Dic89C2×D1612yes—
158#33982—891nomaximal, Sylow
159C2×C89—21no—
160C2×C89—161no—
161C2×C89—162no—
162C2×C89C22×D1611yescharacteristic
163D89—22no—
164D89—162no—
165D89—161no—
166#3415—891no—
167#3415—891no—
168#2263—892no—
169#2263—892no—
170#2263—891no—
171#2263D8911yescharacteristic, complemented
172#2262—891no—
173#2262—892no—
174C2×D32—891no—
175C2×D32—892no—
176#2258—891no—
177C89#3398211yescharacteristic, complemented, Sylow
178#576—1781no—
179#576—1781no—
180#509—1782no—
181#509—1782no—
182#509—1781no—
183#509—1781no—
184#508—892no—
185#508—892no—
186#508—891no—
187#508C2×D8911yescharacteristic, complemented
188C2×D16—892no—
189C2×D16—892no—
190C2×D16—891no—
191C2×D16C2×D8911yescharacteristic, complemented
192#504—891no—
193Q64—891no—
194SD64—891no—
195SD64—891no—
196SD64—892no—
197SD64C2×D8912yescomplemented
198D32—892no—
199D32—891no—
200D32—892no—
201D32—892no—
202D32C2×D8912yescomplemented
203M64—892no—
204M64C2×D8911yescharacteristic, complemented
205C2×C32—891no—
206C2×C32—892no—
207H(5,𝔽2)—3561no—
208H(5,𝔽2)—3561no—
209AGL(1,ℤ/8)—3562no—
210AGL(1,ℤ/8)—3562no—
211AGL(1,ℤ/8)—3561no—
212AGL(1,ℤ/8)—3561no—
213#135—21no—
214#135—1782no—
215#135—1782no—
216#135—1781no—
217C2×D8—21no—
218C2×D8—1782no—
219C2×D8—1782no—
220C2×D8—1781no—
221#131—891no—
222Q32—891no—
223Q32C22×D8911yescharacteristic
224SD32—21no—
225SD32—1782no—
226SD32—1782no—
227SD32—1781no—
228D16—21no—
229D16—892no—
230D16—891no—
231D16—1782no—
232D16—1782no—
233D16—1781no—
234D16C22×D8912yes—
235D16C22×D8911yescharacteristic
236M32—891no—
237M32—892no—
238C2×C16—892no—
239C2×C16C22×D8911yescharacteristic
240C32—891no—
241C32—891no—
242C32C22×D8912yes—
243#41—41no—
244#41—891no—
245#41—3562no—
246#41—3562no—
247#41—3561no—
248#41—7121no—
249#41—7121no—
250C2×D4—41no—
251C2×D4—3562no—
252C2×D4—3562no—
253C2×D4—3561no—
254C2×D4—7121no—
255C2×D4—7122no—
256C2×D4—7121no—
257C2×D4—7122no—
258Q16—21no—
259Q16—1781no—
260SD16—41no—
261SD16—3562no—
262SD16—3562no—
263SD16—3561no—
264D8—22no—
265D8—21no—
266D8—41no—
267D8—3562no—
268D8—3562no—
269D8—3561no—
270D8—1781no—
271D8—1782no—
272M16—891no—
273M16—892no—
274C2×C8—892no—
275C2×C8D4×D8911yescharacteristic
276C16—892no—
277C16D4×D8911yescharacteristic
278C16C23×D8911yescharacteristic
279C23—81no—
280C23—7122no—
281C23—7122no—
282C23—7121no—
283Q8—41no—
284Q8—891no—
285Q8—3561no—
286D4—42no—
287D4—41no—
288D4—81no—
289D4—81no—
290D4—892no—
291D4—891no—
292D4—3561no—
293D4—3562no—
294D4—7122no—
295D4—7122no—
296D4—7122no—
297D4—7122no—
298D4—7121no—
299D4—7121no—
300C2×C4—81no—
301C2×C4—892no—
302C2×C4—7122no—
303C2×C4—7122no—
304C2×C4—7121no—
305C2×C4D8×D8911yescharacteristic
306C8—892no—
307C8D8×D8911yescharacteristic
308C8C2×D4×D8911yescharacteristic
309C22—81no—
310C22—82no—
311C22—321no—
312C22—28481no—
313C22—28482no—
314C22—28482no—
315C22—892no—
316C22—7122no—
317C22—7121no—
318C22D16×D8911yescharacteristic
319C4—81no—
320C4—892no—
321C4—7121no—
322C4D16×D8911yescharacteristic
323C4C2×D8×D8911yescharacteristic
324C2—21no—
325C2—162no—
326C2—161no—
327C2—14242no—
328C2—14241no—
329C2—1782no—
330C2C2×D16×D8911yescharacteristic
subgroupG/H
centerC2C2×D16×D89
commutator, cosocleC16×C89C24
FrattiniC16C23×D89
FittingC89×#2263C2
socleC2×C89C22×D16
hypercenter#2263D89
hypocenterC89#33982
LCSG ▷ C16×C89 ▷ C8×C89 ▷ C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #2263

Complements

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

NH ≅ G/Ntype
1. #26033390C2semidirect
2. #26033389C2semidirect
3. #26022567C2semidirect
4. #26022566C2semidirect
5. #26022563C2semidirect
6. #26022560C2semidirect
7. #26022559C2semidirect
8. C2×#19151433C2semidirect
9. D32×D89C2semidirect
10. #26022547C2semidirect
11. C89×#2263C2semidirect
18. #19152031C22semidirect
19. #19152030C22semidirect
20. #19152027C22semidirect
21. C2×D1424C22semidirect
22. D16×D89C22semidirect
23. D16×D89C22semidirect
25. #19151438C22semidirect
26. #19151437C22semidirect
27. #19151436C22semidirect
28. #19151435C22semidirect
29. D2848C22semidirect
30. #19151433C22semidirect
31. #19151433C22semidirect
32. #19151433C22semidirect
33. #19151431C22semidirect
34. C2×#15667918C22semidirect
35. C32×D89C22semidirect
36. #508×C89C22semidirect
37. C2×D16×C89C22semidirect
38. SD64×C89C22semidirect
39. D32×C89C22semidirect
40. M64×C89C22semidirect
171. #2263D89semidirect
177. C89#33982semidirect
187. #508C2×D89semidirect
191. C2×D16C2×D89semidirect
197. SD64C2×D89semidirect
202. D32C2×D89semidirect
204. M64C2×D89semidirect

Maximal subgroups

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

subgroupindex
#260333902
#260333892
#260225672
#260225662
#260225632
#260225602
#260225592
C2×#191514332
D32×D892
#260225472
C89×#22632
#3398289