Finite Group Databasebeta

← #27080845

Subgroups — #27080845

Order 12032.

1878 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20256778C211yesmaximal, characteristic
2#20252388C218yesmaximal, complemented
3#20252378C218yesmaximal, complemented
4#20252328C211yesmaximal, characteristic, complemented
5#20252322C211yesmaximal, characteristic, complemented
6#20252107C211yesmaximal, characteristic, complemented
7#20252101C211yesmaximal, characteristic, complemented
8#20252091C212yesmaximal, complemented
9#20252082C212yesmaximal, complemented
10#20252030C212yesmaximal, complemented
11#20252026C211yesmaximal, characteristic, complemented
12#20251813C212yesmaximal, complemented
13C2×C47×#405C211yesmaximal, characteristic, complemented
14C2×#13094030C2212yes—
15C2×#13094030C2212yes—
16#15689744C414yes—
17#15689735C412yes—
18#15689728C412yes—
19#15689720C414yes—
20#15689711C414yes—
21#15689680C2211yescharacteristic
22#15689673C2211yescharacteristic
23#15689651C2218yes—
24#15689634C2212yes—
25C4×#10625350C2212yes—
26#15689253C2214yescomplemented
27#15689251C2218yescomplemented
28C8×Dic94C2214yescomplemented
29#15689202C2214yescomplemented
30#15689199C2218yes—
31#15689196C2218yescomplemented
32#15689194C2218yes—
33#15689192C2218yes—
34C8×D188C2214yescomplemented
35C8×C22⋊D47C2218yes—
36#15689184C2218yes—
37#15689182C2218yescomplemented
38#15689174C2218yescomplemented
39#15689149C2212yes—
40#15689149C2212yes—
41#15689147C2211yescharacteristic
42#15689147C2211yescharacteristic
43#15689089C2212yes—
44#15689089C2212yes—
45#15689087C2212yes—
46#15689080C2214yescomplemented
47#15689080C2214yescomplemented
48#15689078C2212yes—
49#15689078C2212yes—
50C4×#10625329C2212yes—
51#15689075C2214yescomplemented
52#97×D47C2214yescomplemented
53C22×M16×C47C2211yescharacteristic
54C23×C8×C47C2211yescharacteristic
55C2×#117×C47C2211yescharacteristic
56C47×#405C2214yes—
57C47×#405C2214yes—
58C2×#97×C47C2211yescharacteristic
59C2×#97×C47C2211yescharacteristic
60C2×C4×C8×C47C2211yescharacteristic
61C2×C4×C8×C47C2211yescharacteristic
62C2×#10625350C2312yes—
63#13094030—28no—
64#13094030—28no—
65C2×#10625329C2312yes—
66C2×#10625329C2312yes—
67C2×#10625324C2314yes—
68C2×#10625324C2314yes—
69C2×C8×D47C2314yes—
70C2×C8×D47C2314yes—
71#13094024C2×C411yescharacteristic
72#13094024C2×C412yes—
73#13094021C2×C411yescharacteristic
74#13094019C2×C418yes—
75#13094017C2×C418yes—
76#13094016C2×C412yes—
77#13094014C2×C418yes—
78#13094012C2×C412yes—
79#13094010C2×C418yes—
80#13094009C2×C418yes—
81#13094007C2×C414yes—
82#13094006C2×C418yes—
83#13094003C2×C414yes—
84#13094003C2×C414yes—
85#13094002C2×C418yes—
86#13093998C2×C414yes—
87#13093998C2×C414yes—
88#13093997C2×C418yes—
89#13093993C2×C414yes—
90C4×Dic94C2×C414yes—
91C4×Dic94C2314yes—
92C4×D188C2×C414yes—
93C4×D188C2314yes—
94C4×C22⋊D47C2×C418yes—
95C4×C22⋊D47C2318yes—
96#13093983C2312yes—
97#13093981C2314yes—
98C42×D47C2314yes—
99C4⋊(C47⋊C8)C2314yes—
100C4⋊(C47⋊C8)C2314yes—
101Dic47⋊C8C2318yes—
102Dic47⋊C8C2318yes—
103C22⋊(C47⋊C8)C2318yes—
104(C2×D47)⋊C8C2318yes—
105(C2×D47)⋊C8C2318yes—
106#13093935C2314yes—
107#13093934C2314yes—
108#13093934C2314yes—
109C4×C47⋊C8C2314yes—
110C8×Dic47C2314yes—
111C8×Dic47C2314yes—
112C23×C4×C47C2311yescharacteristic
113C2×M16×C47—24no—
114C2×M16×C47C2312yes—
115C2×M16×C47C2312yes—
116C22×C8×C47—24no—
117C22×C8×C47C2311yescharacteristic
118C22×C8×C47C2314yes—
119C22×C8×C47C2311yescharacteristic
120C22×C8×C47C2311yescharacteristic
121C22×C8×C47C2311yescharacteristic
122#117×C47C2×C412yes—
123#117×C47C2×C412yes—
124#117×C47C2314yes—
125C2×C4⋊C4×C47C2×C412yes—
126C2×C22⋊C4×C47C2×C412yes—
127C2×C42×C47C2311yescharacteristic
128C2×C42×C47C2311yescharacteristic
129#97×C47C2314yes—
130#97×C47C2312yes—
131#97×C47C2312yes—
132C4×C8×C47C2312yes—
133C4×C8×C47C2312yes—
134C4×C8×C47C2314yes—
135C24×C47C22×C411yescharacteristic
136C22×C4×C47—22no—
137C22×C4×C47—22no—
138C22×C4×C47C22×C411yescharacteristic
139C22×C4×C47C22×C412yes—
140C22×C4×C47C22×C412yes—
141C22×C4×C47C22×C412yes—
142C22×C4×C47C2412yes—
143C22×C4×C47C2411yescharacteristic
144C2×C22⋊D47C22×C414yes—
145C2×C22⋊D47C22×C418yes—
146#10625350—24no—
147#10625350—24no—
148C2×D188C22×C412yes—
149C2×D188C22×C414yes—
150C2×C4×D47C22×C418yes—
151C2×C4×D47C2414yes—
152C2×Dic94C22×C412yes—
153C2×Dic94C22×C414yes—
154M16×C47—24no—
155M16×C47—24no—
156C2×C8×C47—22no—
157C2×C8×C47—22no—
158C2×C8×C47—28no—
159C2×C8×C47#4114yes—
160C2×C8×C47#4114yes—
161C2×C8×C47C2412yes—
162C2×C8×C47C2412yes—
163C2×C8×C47C2412yes—
164C2×C8×C47C2411yescharacteristic
165C2×C8×C47C2411yescharacteristic
166C4⋊C4×C47C22×C414yes—
167C4⋊C4×C47C22×C414yes—
168C22⋊C4×C47C22×C414yes—
169C22⋊C4×C47C22×C414yes—
170C42×C47C22×C412yes—
171C42×C47C22×C411yescharacteristic
172C42×C47C22×C411yescharacteristic
173C42×C47C2412yes—
174C42×C47C2412yes—
175C22⋊Dic47C22×C414yes—
176C22⋊Dic47C22×C418yes—
177(C2×D47)⋊C4C22×C418yes—
178(C2×D47)⋊C4C22×C418yes—
179(C2×D47)⋊C4C22×C418yes—
180C4⋊Dic47C22×C414yes—
181C4⋊Dic47C22×C414yes—
182C4⋊Dic47C22×C414yes—
183Dic47⋊C4C22×C418yes—
184Dic47⋊C4C22×C418yes—
185Dic47⋊C4C22×C418yes—
186C4×Dic47C22×C418yes—
187C4×Dic47C2414yes—
188#10625329—24no—
189#10625329—24no—
190C2×C47⋊C8C2414yes—
191C2×C47⋊C8C2414yes—
192#10625324—28no—
193#10625324—28no—
194C8×D47—28no—
195C8×D47—28no—
196C23×C47—22no—
197C23×C47—42no—
198C23×C47C23×C412yes—
199C23×C47C23×C411yescharacteristic
200C22×D47C23×C414yes—
201C2×C4×C47—22no—
202C2×C4×C47—24no—
203C2×C4×C47—22no—
204C2×C4×C47—22no—
205C2×C4×C47—42no—
206C2×C4×C47—42no—
207C2×C4×C47C23×C414yes—
208C2×C4×C47C23×C412yes—
209C2×C4×C47C23×C412yes—
210C2×C4×C47C23×C412yes—
211C2×C4×C47C23×C411yescharacteristic
212C2×C4×C47C2×#4112yes—
213C2×C4×C47C2×#4112yes—
214C2×C4×C47C2511yescharacteristic
215C22⋊D47—28no—
216C22⋊D47—48no—
217C2×Dic47C23×C414yes—
218C2×Dic47C23×C418yes—
219D188—24no—
220D188—44no—
221C4×D47—24no—
222C4×D47—24no—
223C4×D47—48no—
224Dic94—24no—
225Dic94—44no—
226C8×C47—22no—
227C8×C47—22no—
228C8×C47C2×#4112yes—
229C8×C47C2×#4114yes—
230C8×C47C2×#4112yes—
231C47⋊C8—24no—
232C47⋊C8—24no—
233#60209—471nomaximal, Sylow
234C22×C47—22no—
235C22×C47—22no—
236C22×C47—24no—
237C22×C47—44no—
238C22×C47C4×#4112yes—
239C22×C47C24×C411yescharacteristic
240C2×D47—24no—
241C2×D47—28no—
242C4×C47—22no—
243C4×C47—22no—
244C4×C47—24no—
245C4×C47—22no—
246C4×C47C4×#4112yes—
247C4×C47C22×#4111yescharacteristic
248C4×C47C22×#4111yescharacteristic
249Dic47—24no—
250Dic47—28no—
251#3455—471no—
252#2969—478no—
253#2968—478no—
254#2961—471no—
255#2960—471no—
256#2932—471no—
257#2931—471no—
258#2930—472no—
259#2929—472no—
260#2922—472no—
261#2921—471no—
262#2875—472no—
263C2×#405D4711yescharacteristic, complemented
264C2×C47—22no—
265C2×C47—42no—
266C2×C47C2×C4×#4111yescharacteristic
267C2×C47#357312yes—
268D47—44no—
269C2×#131—472no—
270C2×#131—472no—
271C22×M16C2×D4711yescharacteristic
272C23×C8C2×D4711yescharacteristic
273#542—474no—
274#541—472no—
275#540—472no—
276#539—474no—
277#538—474no—
278#533—471no—
279#532—471no—
280#529—478no—
281#525—472no—
282C4×#41—472no—
283C2×#117C2×D4711yescharacteristic
284#446—478no—
285#446—474no—
286C8×Q8—474no—
287#436—478no—
288#436—474no—
289#435—478no—
290#435—478no—
291#435—478no—
292C8×D4—478no—
293C8×D4—474no—
294#433—478no—
295#433—478no—
296#431—478no—
297#424—472no—
298#424—472no—
299#423—471no—
300#423—471no—
301#408—472no—
302#408—472no—
303#407—472no—
304#405—472no—
305#405—472no—
306#405—474no—
307#405—474no—
308#405C2×D4714yes—
309#405C2×D4714yes—
310C4×M16—472no—
311C4×M16—474no—
312C2×#97—474no—
313C2×#97C2×D4711yescharacteristic
314C2×#97C2×D4711yescharacteristic
315C2×C4×C8C2×D4711yescharacteristic
316C2×C4×C8C2×D4711yescharacteristic
317C47#6020911yescharacteristic, complemented, Sylow
318C2×#41—472no—
319C23×C4C22×D4711yescharacteristic
320#131—948no—
321#131—948no—
322C2×M16—24no—
323C2×M16—472no—
324C2×M16—472no—
325C2×M16—474no—
326C2×M16—474no—
327C2×M16C22×D4712yes—
328C2×M16C22×D4712yes—
329C22×C8—24no—
330C22×C8—474no—
331C22×C8—474no—
332C22×C8C22×D4711yescharacteristic
333C22×C8C22×D4714yes—
334C22×C8C22×D4711yescharacteristic
335C22×C8C22×D4711yescharacteristic
336C22×C8C22×D4711yescharacteristic
337C4⋊Q8—471no—
338C4⋊Q8—472no—
339#127—471no—
340#126—478no—
341#126—478no—
342#125—478no—
343#125—472no—
344#124—478no—
345#124—472no—
346#123—474no—
347#123—478no—
348#123—478no—
349C22⋊Q8—474no—
350C22⋊Q8—474no—
351C22⋊Q8—478no—
352#121—474no—
353#121—474no—
354#121—478no—
355#120—474no—
356C4×Q8—474no—
357C4×Q8—474no—
358C4×D4—478no—
359C4×D4—478no—
360C4×D4—474no—
361C4×D4—474no—
362#117—472no—
363#117—474no—
364#117C4×D4712yes—
365#117C4×D4712yes—
366#117C22×D4714yes—
367C2×C4⋊C4C4×D4712yes—
368C2×C22⋊C4C4×D4712yes—
369C2×C42—474no—
370C2×C42C22×D4711yescharacteristic
371C2×C42C22×D4711yescharacteristic
372C4⋊C8—478no—
373C4⋊C8—478no—
374C4⋊C8—474no—
375C4⋊C8—474no—
376C22⋊C8—478no—
377C22⋊C8—478no—
378C22⋊C8—478no—
379#97—474no—
380#97—474no—
381#97—474no—
382#97C22×D4712yes—
383#97C22×D4712yes—
384#97C22×D4714yes—
385C4×C8—474no—
386C4×C8—474no—
387C4×C8—474no—
388C4×C8C22×D4714yes—
389C4×C8C22×D4712yes—
390C4×C8C22×D4712yes—
391C24C2×C4×D4711yescharacteristic
392#41—944no—
393#41—944no—
394C2×Q8—472no—
395C2×Q8—474no—
396C2×D4—474no—
397C2×D4—478no—
398C2×D4—472no—
399C2×D4—474no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—474no—
403C22×C4—478no—
404C22×C4C2×C4×D4711yescharacteristic
405C22×C4C2×C4×D4712yes—
406C22×C4C2×C4×D4712yes—
407C22×C4C2×C4×D4712yes—
408C22×C4C23×D4712yes—
409C22×C4C23×D4711yescharacteristic
410M16—24no—
411M16—24no—
412M16—944no—
413M16—944no—
414M16—948no—
415M16—948no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—28no—
419C2×C8—474no—
420C2×C8—474no—
421C2×C8—948no—
422C2×C8—948no—
423C2×C8#1062535014yes—
424C2×C8#1062535014yes—
425C2×C8C23×D4712yes—
426C2×C8C23×D4712yes—
427C2×C8C23×D4711yescharacteristic
428C2×C8C23×D4712yes—
429C2×C8C23×D4711yescharacteristic
430C4⋊C4—478no—
431C4⋊C4—478no—
432C4⋊C4—478no—
433C4⋊C4—474no—
434C4⋊C4—474no—
435C4⋊C4—474no—
436C4⋊C4C2×C4×D4714yes—
437C4⋊C4C2×C4×D4714yes—
438C22⋊C4—474no—
439C22⋊C4—478no—
440C22⋊C4—478no—
441C22⋊C4—478no—
442C22⋊C4—478no—
443C22⋊C4C2×C4×D4714yes—
444C22⋊C4C2×C4×D4714yes—
445C42—478no—
446C42—474no—
447C42C2×C4×D4712yes—
448C42C2×C4×D4711yescharacteristic
449C42C2×C4×D4711yescharacteristic
450C42C23×D4712yes—
451C42C23×D4712yes—
452C23—22no—
453C23—42no—
454C23—474no—
455C23C22×C4×D4712yes—
456C23C22×C4×D4711yescharacteristic
457Q8—944no—
458Q8—1884no—
459D4—948no—
460D4—944no—
461D4—1884no—
462D4—1888no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—478no—
470C2×C4—474no—
471C2×C4—944no—
472C2×C4—944no—
473C2×C4—1888no—
474C2×C4C22×C4×D4714yes—
475C2×C4C22×C4×D4712yes—
476C2×C4C22×C4×D4712yes—
477C2×C4C22×C4×D4712yes—
478C2×C4C22×C4×D4711yescharacteristic
479C2×C4C2×#1062535012yes—
480C2×C4C2×#1062535012yes—
481C2×C4C24×D4711yescharacteristic
482C8—22no—
483C8—22no—
484C8—944no—
485C8—944no—
486C8C2×#1062535014yes—
487C8C2×#1062535012yes—
488C8C2×#1062535012yes—
489C22—22no—
490C22—24no—
491C22—22no—
492C22—44no—
493C22—948no—
494C22—944no—
495C22C4×#1062535012yes—
496C22C23×C4×D4711yescharacteristic
497C4—24no—
498C4—22no—
499C4—22no—
500C4—22no—
501C4—948no—
502C4—944no—
503C4C4×#1062535012yes—
504C4C22×#1062535011yescharacteristic
505C4C22×#1062535011yescharacteristic
506C2—22no—
507C2—42no—
508C2—1884no—
509C2C2×C4×#1062535011yescharacteristic
510C2#2025809912yes—
subgroupG/H
center, FrattiniC2×C4C24×D47
commutator, socleC22×C47C24×C4
FittingC2×C47×#405C2
prime radical#20256778C2
cosocleC2×C4×C47C25
hypercenterC2×#405D47
hypocenterC47#60209
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
2. #20252388C2semidirect
3. #20252378C2semidirect
4. #20252328C2semidirect
5. #20252322C2semidirect
6. #20252107C2semidirect
7. #20252101C2semidirect
8. #20252091C2semidirect
9. #20252082C2semidirect
10. #20252030C2semidirect
11. #20252026C2semidirect
12. #20251813C2semidirect
13. C2×C47×#405C2semidirect
26. #15689253C22semidirect
27. #15689251C22semidirect
28. C8×Dic94C22semidirect
29. #15689202C22semidirect
31. #15689196C22semidirect
34. C8×D188C22semidirect
37. #15689182C22semidirect
38. #15689174C22semidirect
46. #15689080C22semidirect
47. #15689080C22semidirect
51. #15689075C22semidirect
52. #97×D47C22semidirect
263. C2×#405D47semidirect
317. C47#60209semidirect

Maximal subgroups

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

subgroupindex
#202567782
#202523882
#202523782
#202523282
#202523222
#202521072
#202521012
#202520912
#202520822
#202520302
#202520262
#202518132
C2×C47×#4052
#6020947