Finite Group Databasebeta

← #26374428

Subgroups — #26374428

Order 12032.

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#20256526C211yesmaximal, characteristic, complemented
2#20256525C211yesmaximal, characteristic, complemented
3#20245703C211yesmaximal, characteristic, complemented
4#20245702C212yesmaximal, complemented
5#20245699C212yesmaximal, complemented
6#20245696C212yesmaximal, complemented
7#20245695C211yesmaximal, characteristic, complemented
8C2×#15688958C211yesmaximal, characteristic, complemented
9D32×D47C212yesmaximal, complemented
10#20245683C211yesmaximal, characteristic, complemented
11C47×#2263C211yesmaximal, characteristic, complemented
12#15689925—21no—
13#15689924—21no—
14#15689560—22no—
15#15689559—21no—
16#15689558—21no—
17#15689557—22no—
18#15689556C2212yescomplemented
19#15689555C2211yescharacteristic, complemented
20#15689552C2212yescomplemented
21C2×D752C2211yescharacteristic, complemented
22D16×D47C2212yescomplemented
23D16×D47C2212yescomplemented
24#15689539C2211yescharacteristic
25#15688963C2211yescharacteristic, complemented
26#15688962C2212yescomplemented
27#15688961C2211yescharacteristic, complemented
28#15688960C2211yescharacteristic, complemented
29D1504C2212yescomplemented
30#15688958C2211yescharacteristic, complemented
31#15688958C2212yescomplemented
32#15688958C2212yescomplemented
33#15688956C2212yescomplemented
34C2×C47⋊C32C2211yescharacteristic, complemented
35C32×D47C2212yescomplemented
36C47×#508C2211yescharacteristic, complemented
37C2×D16×C47C2211yescharacteristic, complemented
38C47×SD64C2212yescomplemented
39C47×D32C2212yescomplemented
40C47×M64C2211yescharacteristic, complemented
41#13094070—41no—
42#13094069—41no—
43#13094051—42no—
44#13094050—41no—
45#13094049—41no—
46#13094048—42no—
47#13094047—22no—
48#13094046—21no—
49#13094043—22no—
50C2×D376—21no—
51D8×D47—22no—
52D8×D47—22no—
53#13094030D411yescharacteristic
54Dic376C2311yescharacteristic
55#13093969—21no—
56#13093968—22no—
57#13093967—22no—
58D752—21no—
59D752C2311yescharacteristic
60D752C2312yes—
61#13093965—22no—
62#13093965—22no—
63#13093964D411yescharacteristic
64#13093963D412yes—
65C16×D47C2312yes—
66C47⋊C32C2311yescharacteristic
67C47⋊C32C2311yescharacteristic
68#135×C47—21no—
69C2×D8×C47—21no—
70Q32×C47C2311yescharacteristic
71SD32×C47—21no—
72D16×C47—21no—
73D16×C47C2311yescharacteristic
74D16×C47C2312yes—
75C2×C16×C47C2311yescharacteristic
76C32×C47C2312yes—
77#41×C47—41no—
78C2×D4×C47—41no—
79C2×C22⋊D47—82no—
80#10625354—42no—
81#10625352—42no—
82#10625352—81no—
83D4×D47—42no—
84D4×D47—42no—
85D4×D47—82no—
86D4×D47—81no—
87#10625350—41no—
88#10625350—81no—
89#10625350D811yescharacteristic
90C2×D188—41no—
91C2×D188—81no—
92Q16×C47—21no—
93SD16×C47—41no—
94D8×C47—22no—
95D8×C47—21no—
96D8×C47—41no—
97C2×C8×C47C2×D411yescharacteristic
98#10625336—42no—
99#10625335—42no—
100#10625334—42no—
101#10625334—42no—
102#10625329D811yescharacteristic
103Dic188—21no—
104D376—21no—
105D376—22no—
106D376—41no—
107#10625325—41no—
108#10625324D812yes—
109C8×D47C2×D412yes—
110C16×C47C2×D411yescharacteristic
111C16×C47C2411yescharacteristic
112C47⋊C16C2×D412yes—
113C23×C47—81no—
114C22×D47—81no—
115C22×D47—82no—
116C22×D47—82no—
117Q8×C47—41no—
118D4×C47—41no—
119D4×C47—42no—
120D4×C47—81no—
121D4×C47—81no—
122C2×C4×C47—81no—
123C2×C4×C47C2×D811yescharacteristic
124C22⋊D47—81no—
125C22⋊D47—82no—
126C22⋊D47—82no—
127C22⋊D47—82no—
128C22⋊D47D1612yes—
129C2×Dic47—82no—
130D188—41no—
131D188—42no—
132D188—82no—
133D188—81no—
134D188D1611yescharacteristic
135C4×D47—81no—
136C4×D47—82no—
137C4×D47C2×D812yes—
138Dic94—41no—
139Dic94D1611yescharacteristic
140C8×C47C2×D811yescharacteristic
141C8×C47C22×D411yescharacteristic
142C47⋊C8C2×D812yes—
143#33982—471nomaximal, Sylow
144C22×C47—82no—
145C22×C47—81no—
146C22×C47—321no—
147C22×C47C2×D1611yescharacteristic
148C2×D47—82no—
149C2×D47—81no—
150C2×D47—322no—
151C2×D47—322no—
152C2×D47—321no—
153C2×D47C2×D1612yes—
154C4×C47—81no—
155C4×C47C2×D1611yescharacteristic
156C4×C47C22×D811yescharacteristic
157Dic47—81no—
158Dic47C2×D1612yes—
159#3415—471no—
160#3415—471no—
161#2263—472no—
162#2263—472no—
163#2263—471no—
164#2263D4711yescharacteristic, complemented
165#2262—471no—
166#2262—472no—
167C2×D32—471no—
168C2×D32—472no—
169#2258—471no—
170C2×C47—21no—
171C2×C47—161no—
172C2×C47—162no—
173C2×C47C22×D1611yescharacteristic
174D47—22no—
175D47—162no—
176D47—161no—
177#576—941no—
178#576—941no—
179#509—942no—
180#509—942no—
181#509—941no—
182#509—941no—
183#508—472no—
184#508—472no—
185#508—471no—
186#508C2×D4711yescharacteristic, complemented
187C2×D16—472no—
188C2×D16—472no—
189C2×D16—471no—
190C2×D16C2×D4711yescharacteristic, complemented
191#504—471no—
192Q64—471no—
193SD64—471no—
194SD64—471no—
195SD64—472no—
196SD64C2×D4712yescomplemented
197D32—472no—
198D32—471no—
199D32—472no—
200D32—472no—
201D32C2×D4712yescomplemented
202M64—472no—
203M64C2×D4711yescharacteristic, complemented
204C2×C32—471no—
205C2×C32—472no—
206C47#3398211yescharacteristic, complemented, Sylow
207H(5,𝔽2)—1881no—
208H(5,𝔽2)—1881no—
209AGL(1,ℤ/8)—1882no—
210AGL(1,ℤ/8)—1882no—
211AGL(1,ℤ/8)—1881no—
212AGL(1,ℤ/8)—1881no—
213#135—21no—
214#135—942no—
215#135—942no—
216#135—941no—
217C2×D8—21no—
218C2×D8—942no—
219C2×D8—942no—
220C2×D8—941no—
221#131—471no—
222Q32—471no—
223Q32C22×D4711yescharacteristic
224SD32—21no—
225SD32—942no—
226SD32—942no—
227SD32—941no—
228D16—21no—
229D16—471no—
230D16—472no—
231D16—942no—
232D16—942no—
233D16—941no—
234D16C22×D4712yes—
235D16C22×D4711yescharacteristic
236M32—471no—
237M32—472no—
238C2×C16—472no—
239C2×C16C22×D4711yescharacteristic
240C32—471no—
241C32—471no—
242C32C22×D4712yes—
243#41—41no—
244#41—471no—
245#41—3761no—
246#41—3761no—
247#41—1882no—
248#41—1882no—
249#41—1881no—
250C2×D4—41no—
251C2×D4—3761no—
252C2×D4—3762no—
253C2×D4—3761no—
254C2×D4—3762no—
255C2×D4—1882no—
256C2×D4—1882no—
257C2×D4—1881no—
258Q16—21no—
259Q16—941no—
260SD16—41no—
261SD16—1882no—
262SD16—1882no—
263SD16—1881no—
264D8—22no—
265D8—21no—
266D8—41no—
267D8—941no—
268D8—942no—
269D8—1882no—
270D8—1882no—
271D8—1881no—
272M16—471no—
273M16—472no—
274C2×C8—472no—
275C2×C8D4×D4711yescharacteristic
276C16—472no—
277C16D4×D4711yescharacteristic
278C16C23×D4711yescharacteristic
279C23—81no—
280C23—3762no—
281C23—3762no—
282C23—3761no—
283Q8—41no—
284Q8—471no—
285Q8—1881no—
286D4—42no—
287D4—41no—
288D4—81no—
289D4—81no—
290D4—472no—
291D4—471no—
292D4—3762no—
293D4—3762no—
294D4—3762no—
295D4—3762no—
296D4—3761no—
297D4—3761no—
298D4—1881no—
299D4—1882no—
300C2×C4—81no—
301C2×C4—472no—
302C2×C4—3762no—
303C2×C4—3762no—
304C2×C4—3761no—
305C2×C4D8×D4711yescharacteristic
306C8—472no—
307C8D8×D4711yescharacteristic
308C8C2×D4×D4711yescharacteristic
309C22—81no—
310C22—82no—
311C22—321no—
312C22—472no—
313C22—3762no—
314C22—3761no—
315C22—15041no—
316C22—15042no—
317C22—15042no—
318C22D16×D4711yescharacteristic
319C4—81no—
320C4—472no—
321C4—3761no—
322C4D16×D4711yescharacteristic
323C4C2×D8×D4711yescharacteristic
324C2—21no—
325C2—162no—
326C2—161no—
327C2—942no—
328C2—7522no—
329C2—7521no—
330C2C2×D16×D4711yescharacteristic
subgroupG/H
centerC2C2×D16×D47
commutator, cosocleC16×C47C24
FrattiniC16C23×D47
FittingC47×#2263C2
socleC2×C47C22×D16
hypercenter#2263D47
hypocenterC47#33982
LCSG ▷ C16×C47 ▷ C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #2263

Complements

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

NH ≅ G/Ntype
1. #20256526C2semidirect
2. #20256525C2semidirect
3. #20245703C2semidirect
4. #20245702C2semidirect
5. #20245699C2semidirect
6. #20245696C2semidirect
7. #20245695C2semidirect
8. C2×#15688958C2semidirect
9. D32×D47C2semidirect
10. #20245683C2semidirect
11. C47×#2263C2semidirect
18. #15689556C22semidirect
19. #15689555C22semidirect
20. #15689552C22semidirect
21. C2×D752C22semidirect
22. D16×D47C22semidirect
23. D16×D47C22semidirect
25. #15688963C22semidirect
26. #15688962C22semidirect
27. #15688961C22semidirect
28. #15688960C22semidirect
29. D1504C22semidirect
30. #15688958C22semidirect
31. #15688958C22semidirect
32. #15688958C22semidirect
33. #15688956C22semidirect
34. C2×C47⋊C32C22semidirect
35. C32×D47C22semidirect
36. C47×#508C22semidirect
37. C2×D16×C47C22semidirect
38. C47×SD64C22semidirect
39. C47×D32C22semidirect
40. C47×M64C22semidirect
164. #2263D47semidirect
186. #508C2×D47semidirect
190. C2×D16C2×D47semidirect
196. SD64C2×D47semidirect
201. D32C2×D47semidirect
203. M64C2×D47semidirect
206. C47#33982semidirect

Maximal subgroups

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

subgroupindex
#202565262
#202565252
#202457032
#202457022
#202456992
#202456962
#202456952
C2×#156889582
D32×D472
#202456832
C47×#22632
#3398247