Finite Group Databasebeta

← #27083041

Subgroups — #27083041

Order 12032.

4854 conjugacy classes of subgroups; only the 807 normal subgroups are shown.

normal subgroups only

This group has 4854 conjugacy classes of subgroups — too many to draw. Only its 807 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#20257028C213yesmaximal, complemented
2#20252794C214yesmaximal, complemented
3#20252785C218yesmaximal, complemented
4#20252780C214yesmaximal, complemented
5C2×#15689313C218yesmaximal, complemented
6#20251849C213yesmaximal, complemented
7C47×H(7,𝔽2)C211yesmaximal, characteristic, complemented
8C22×#10625352C2211yescharacteristic, complemented
9#15689772—212no—
10#15689767—212no—
11#15689694C2216yescomplemented
12#15689682C22112yescomplemented
13#15689630C2216yescomplemented
14C2×#13094005C22112yescomplemented
15#15689598C2213yescomplemented
16#15689313C22116yescomplemented
17#15689313C22132yescomplemented
18#15689310C22116yescomplemented
19#15689097C22112yescomplemented
20C2×#13093939C22124yescomplemented
21C47×#585—212no—
22C47×#585C2214yes—
23C2×H(5,𝔽2)×C47—212no—
24C2×H(5,𝔽2)×C47C2218yes—
25C2×H(5,𝔽2)×C47C2213yes—
26C2×#10625352—212no—
27C22×Dic94C2311yescharacteristic, complemented
28C22×C22⋊D47C2312yescomplemented
29C23×Dic47C2312yescomplemented
30C22×C4×D47C2311yescharacteristic, complemented
31#13094022—26no—
32#13094011—26no—
33#13094007—212no—
34#13094005—224no—
35#13094005C23112yescomplemented
36#13094005C23124yescomplemented
37#13094003—212no—
38#13094000—26no—
39#13093996—224no—
40#13093995—26no—
41#13093993—212no—
42#13093992C23112yescomplemented
43D4×Dic47—212no—
44C4×C22⋊D47—212no—
45#13093983C2316yescomplemented
46C2×Dic47⋊C4—212no—
47C2×C22⋊Dic47—212no—
48C2×C22⋊Dic47C23112yescomplemented
49#13093939C23148yescomplemented
502−1+4×C47—216no—
512−1+4×C47—212no—
52H(5,𝔽2)×C47—248no—
53H(5,𝔽2)×C47—212no—
54H(5,𝔽2)×C47—248no—
55H(5,𝔽2)×C47—224no—
56H(5,𝔽2)×C47C23116yes—
57C2×#41×C47—248no—
58C2×#41×C47—224no—
59C2×#41×C47—224no—
60C2×#41×C47D4112yescomplemented
61C2×#41×C47C2316yes—
62C22×D4×C47—216no—
63C22×D4×C47—224no—
64C22×D4×C47D4112yescomplemented
65C22×D4×C47C23112yes—
66C22×D4×C47C2311yescharacteristic
67C24×C47—28no—
68C24×C47C2×D4112yes—
69C24×C47C2412yes—
70C23×D47C2411yescharacteristic
71#41×C47—296no—
72#41×C47—248no—
73#41×C47—216no—
74#41×C47—248no—
75#41×C47—248no—
76#41×C47—224no—
77C2×Q8×C47—224no—
78C2×Q8×C47—28no—
79C2×Q8×C47C2×D416yes—
80C2×D4×C47—296no—
81C2×D4×C47—224no—
82C2×D4×C47—224no—
83C2×D4×C47—248no—
84C2×D4×C47—248no—
85C2×D4×C47—248no—
86C2×D4×C47C2×D416yes—
87C2×D4×C47C2×D4124yes—
88C2×D4×C47C2×D4112yes—
89C2×D4×C47C24112yes—
90C22×C4×C47—224no—
91C22×C4×C47—216no—
92C22×C4×C47C2×D4112yes—
93C22×C4×C47C2×D4112yes—
94C22×C4×C47C2411yescharacteristic
95C2×C22⋊D47—212no—
96C2×C22⋊D47—212no—
97C22×Dic47—212no—
98C22×Dic47—212no—
99C22×Dic47C2411yescharacteristic
100C22×Dic47C2412yes—
101#10625352—416no—
102C2×C4×D47—26no—
103C2×Dic94—26no—
104C22⋊Dic47—212no—
105C22⋊Dic47—224no—
106C22⋊Dic47—212no—
107C22⋊Dic47C24112yes—
108(C2×D47)⋊C4—212no—
109C4⋊Dic47—26no—
110Dic47⋊C4—212no—
111Dic47⋊C4—224no—
112C4×Dic47—26no—
113C23×C47—216no—
114C23×C47—224no—
115C23×C47—82no—
116C23×C47—812no—
117C23×C47—168no—
118C23×C47C22×D4112yes—
119C23×C47C22×D4112yes—
120C23×C47C2511yescharacteristic
121C22×D47—23no—
122C22×D47—81no—
123Q8×C47—216no—
124Q8×C47—212no—
125D4×C47—248no—
126D4×C47—248no—
127D4×C47—224no—
128D4×C47—212no—
129D4×C47#120116yes—
130C2×C4×C47—248no—
131C2×C4×C47—224no—
132C2×C4×C47—224no—
133C2×C4×C47C22×D4112yes—
134C2×C4×C47C22×D416yes—
135C22⋊D47—48no—
136C22⋊D47—812no—
137C2×Dic47—212no—
138C2×Dic47—26no—
139C2×Dic47—23no—
140C2×Dic47—48no—
141C2×Dic47—812no—
142C4×D47—44no—
143Dic94—44no—
144#60401—471nomaximal, Sylow
145C22×C47—212no—
146C22×C47—412no—
147C22×C47—41no—
148C22×C47—412no—
149C22×C47—824no—
150C22×C47—816no—
151C22×C47C2×#12018yes—
152C22×C47C23×D413yes—
153C2×D47—46no—
154C2×D47—41no—
155C4×C47—212no—
156C4×C47C2×#12014yes—
157Dic47—412no—
158Dic47—42no—
159Dic47—41no—
160H(7,𝔽2)D4711yescharacteristic, complemented
161#3485—473no—
162#3028—474no—
163#3026—478no—
164#3025—474no—
165C2×#458—478no—
166#2883—473no—
167C2×C47—23no—
168C2×C47—28no—
169C2×C47—412no—
170C2×C47C22×#12011yescharacteristic
171D47—81no—
172#585—212no—
173#585C2×D4714yes—
174C2×H(5,𝔽2)—212no—
175C2×H(5,𝔽2)C2×D4718yes—
176C2×H(5,𝔽2)C2×D4713yes—
177C22×#41—471no—
178#547—9412no—
179#546—9412no—
180#536—476no—
181#534—4712no—
182#525—476no—
183C2×#123—4712no—
184#518—473no—
185#458—4716no—
186#458—4732no—
187B(4,𝔽2)—4716no—
188#410—4712no—
189C2×#99—4724no—
190C47#6040111yescharacteristic, complemented, Sylow
1912−1+4—216no—
1922−1+4—212no—
193H(5,𝔽2)—248no—
194H(5,𝔽2)—212no—
195H(5,𝔽2)—248no—
196H(5,𝔽2)—224no—
197H(5,𝔽2)C22×D47116yes—
198C2×#41—248no—
199C2×#41—224no—
200C2×#41—224no—
201C2×#41—9412no—
202C2×#41C22⋊D47112yescomplemented
203C2×#41C22×D4716yes—
204C22×Q8—471no—
205C22×D4—216no—
206C22×D4—224no—
207C22×D4—472no—
208C22×D4C22⋊D47112yescomplemented
209C22×D4C22×D47112yes—
210C22×D4C22×D4711yescharacteristic
211C23×C4—472no—
212C23×C4—471no—
213C4⋊Q8—946no—
214#124—946no—
215#123—4712no—
216#123—4724no—
217#123—9424no—
218#123—9412no—
219C22⋊Q8—946no—
220C22⋊Q8—9412no—
221#121—946no—
222#121—9424no—
223#120—4712no—
224#120—9412no—
225C4×D4—9412no—
226C4×D4—9412no—
227#117—476no—
228C2×C4⋊C4—9412no—
229C2×C22⋊C4—4712no—
230C2×C22⋊C4—9412no—
231#99—4748no—
232C24—28no—
233C24—471no—
234C24C2×C22⋊D47112yes—
235C24C23×D4712yes—
236#41—296no—
237#41—248no—
238#41—216no—
239#41—248no—
240#41—248no—
241#41—224no—
242#41—18816no—
243C2×Q8—224no—
244C2×Q8—28no—
245C2×Q8—946no—
246C2×Q8C2×C22⋊D4716yes—
247C2×D4—296no—
248C2×D4—224no—
249C2×D4—224no—
250C2×D4—248no—
251C2×D4—248no—
252C2×D4—248no—
253C2×D4—9412no—
254C2×D4—9412no—
255C2×D4C2×C22⋊D47124yes—
256C2×D4C2×C22⋊D4716yes—
257C2×D4C2×C22⋊D47112yes—
258C2×D4C23×D47112yes—
259C22×C4—224no—
260C22×C4—216no—
261C22×C4—471no—
262C22×C4—472no—
263C22×C4—9412no—
264C22×C4—9412no—
265C22×C4—946no—
266C22×C4C2×C22⋊D47112yes—
267C22×C4C2×C22⋊D47112yes—
268C22×C4C23×D4711yescharacteristic
269C4⋊C4—9412no—
270C4⋊C4—9424no—
271C4⋊C4—946no—
272C22⋊C4—4712no—
273C22⋊C4—9412no—
274C22⋊C4—9412no—
275C22⋊C4—9424no—
276C22⋊C4—9412no—
277C42—946no—
278C23—216no—
279C23—224no—
280C23—82no—
281C23—812no—
282C23—168no—
283C23—943no—
284C23—3761no—
285C23C22×C22⋊D47112yes—
286C23C22×C22⋊D47112yes—
287C23C24×D4711yescharacteristic
288Q8—216no—
289Q8—212no—
290Q8—1884no—
291D4—248no—
292D4—248no—
293D4—224no—
294D4—212no—
295D4—37612no—
296D4—1888no—
297D4#13093993116yes—
298C2×C4—248no—
299C2×C4—224no—
300C2×C4—224no—
301C2×C4—9412no—
302C2×C4—943no—
303C2×C4—946no—
304C2×C4—37612no—
305C2×C4—1888no—
306C2×C4—1884no—
307C2×C4C22×C22⋊D47112yes—
308C2×C4C22×C22⋊D4716yes—
309C22—212no—
310C22—412no—
311C22—41no—
312C22—412no—
313C22—824no—
314C22—816no—
315C22—1886no—
316C22—1881no—
317C22C2×#1309399318yes—
318C22C23×C22⋊D4713yes—
319C4—212no—
320C4—18812no—
321C4—1881no—
322C4—1882no—
323C4C2×#1309399314yes—
324C2—23no—
325C2—28no—
326C2—412no—
327C2—3761no—
328C2C22×#1309399311yescharacteristic
subgroupG/H
centerC2C22×#13093993
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC47×H(7,𝔽2)C2
socleC2×C47C22×#120
hypercenterH(7,𝔽2)D47
hypocenterC47#60401
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C22×D4 ◁ H(7,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #20257028C2semidirect
2. #20252794C2semidirect
3. #20252785C2semidirect
4. #20252780C2semidirect
5. C2×#15689313C2semidirect
6. #20251849C2semidirect
7. C47×H(7,𝔽2)C2semidirect
8. C22×#10625352C22semidirect
11. #15689694C22semidirect
12. #15689682C22semidirect
13. #15689630C22semidirect
14. C2×#13094005C22semidirect
15. #15689598C22semidirect
16. #15689313C22semidirect
17. #15689313C22semidirect
18. #15689310C22semidirect
19. #15689097C22semidirect
20. C2×#13093939C22semidirect
27. C22×Dic94C23semidirect
28. C22×C22⋊D47C23semidirect
29. C23×Dic47C23semidirect
30. C22×C4×D47C23semidirect
35. #13094005C23semidirect
36. #13094005C23semidirect
42. #13093992C23semidirect
45. #13093983C23semidirect
48. C2×C22⋊Dic47C23semidirect
49. #13093939C23semidirect
60. C2×#41×C47D4semidirect
64. C22×D4×C47D4semidirect
160. H(7,𝔽2)D47semidirect
190. C47#60401semidirect
202. C2×#41C22⋊D47semidirect
208. C22×D4C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
#202570282
#202527942
#202527852
#202527802
C2×#156893132
#202518492
C47×H(7,𝔽2)2
#6040147