Finite Group Databasebeta

← #26112213

Subgroups — #26112213

Order 12032.

620 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252852C211yesmaximal, characteristic
2#20241947C211yesmaximal, characteristic, complemented
3#20241882C211yesmaximal, characteristic, complemented
4#20241779C211yesmaximal, characteristic, complemented
5#20241291C211yesmaximal, characteristic, complemented
6#20241246C211yesmaximal, characteristic, complemented
7C47×#2349C211yesmaximal, characteristic, complemented
8C2×#13094050—21no—
9#15689673C2211yescharacteristic
10#15689338—22no—
11#15689328C412yescomplemented
12#15689318C412yescomplemented
13#15689176C2211yescharacteristic
14#15689176C2211yescharacteristic
15#15689147C2211yescharacteristic
16#15689122—21no—
17#15688856—22no—
18#15688856—21no—
19#15688856—21no—
20D4⋊(C47⋊C8)—22no—
21C2×C4×D4×C47—21no—
22C2×C4×D4×C47C2211yescharacteristic
23C47×#389—21no—
24C47×#385—22no—
25C47×#382C2211yescharacteristic
26C47×#380—21no—
27C4×C4⋊C4×C47C2211yescharacteristic
28C2×#95×C47—21no—
29C2×#10625350—21no—
30#13094050—42no—
31#13094050—42no—
32C2×#10625335—22no—
33C2×#10625334—22no—
34C2×#10625329—21no—
35#13094024C2×C411yescharacteristic, complemented
36#13094021C2×C411yescharacteristic, complemented
37#13094012—21no—
38#13093998—21no—
39#13093998—21no—
40C4×D188—21no—
41C4×D188—21no—
42C4⋊(C47⋊C8)—21no—
43C4⋊(C47⋊C8)—21no—
44C4⋊(C47⋊C8)—21no—
45C4⋊(C47⋊C8)C2×C412yescomplemented
46#13093947—22no—
47#13093944—22no—
48C22⋊(C47⋊C8)—21no—
49C22⋊(C47⋊C8)—21no—
50C4×C47⋊C8—21no—
51C4×C47⋊C8—21no—
52C22×D4×C47D411yescharacteristic
53C23×C4×C47—21no—
54C23×C4×C47—22no—
55C4×D4×C47—21no—
56C4×D4×C47—21no—
57C4×D4×C47—42no—
58C4×D4×C47—42no—
59C4×D4×C47C2×C412yes—
60C4×D4×C47D412yes—
61C2×C4⋊C4×C47—21no—
62C2×C4⋊C4×C47—21no—
63C2×C4⋊C4×C47—22no—
64C2×C4⋊C4×C47D411yescharacteristic
65C2×C22⋊C4×C47—21no—
66C2×C22⋊C4×C47—22no—
67C2×C42×C47—21no—
68C2×C42×C47C2311yescharacteristic
69#95×C47—22no—
70#95×C47—21no—
71#95×C47D411yescharacteristic
72#95×C47D411yescharacteristic
73#95×C47D412yes—
74C24×C47—21no—
75C2×D4×C47—21no—
76C2×D4×C47—21no—
77C2×D4×C47—21no—
78C2×D4×C47C22⋊C412yes—
79C2×D4×C47D812yes—
80C2×D4×C47SD1612yes—
81C22×C4×C47—21no—
82C22×C4×C47—21no—
83C22×C4×C47—22no—
84C22×C4×C47—22no—
85C22×C4×C47—21no—
86C22×C4×C47—22no—
87C22×C4×C47—44no—
88C22×C4×C47—44no—
89C22×C4×C47—82no—
90C22×C4×C47—82no—
91C22×C4×C47C2×D411yescharacteristic
92C22×C4×C47C2×D411yescharacteristic
93C22×C4×C47C2×D411yescharacteristic
94C2×C22⋊D47—41no—
95#10625350—41no—
96#10625350—41no—
97#10625350—41no—
98#10625350—41no—
99C2×D188—21no—
100C2×D188—21no—
101C2×D188—21no—
102C2×C4×D47—41no—
103C2×Dic94—21no—
104C4⋊C4×C47—21no—
105C4⋊C4×C47—42no—
106C4⋊C4×C47—41no—
107C4⋊C4×C47—41no—
108C4⋊C4×C47—82no—
109C4⋊C4×C47C22⋊C412yes—
110C22⋊C4×C47—22no—
111C22⋊C4×C47—22no—
112C22⋊C4×C47—44no—
113C42×C47—41no—
114C42×C47—41no—
115C42×C47C22×C411yescharacteristic
116C42×C47C2×D411yescharacteristic
117C42×C47#4111yescharacteristic
118C42×C47#4111yescharacteristic
119#10625335—44no—
120#10625334—44no—
121(C2×D47)⋊C4—41no—
122(C2×D47)⋊C4—41no—
123C4⋊Dic47—21no—
124C4⋊Dic47—21no—
125#10625329—41no—
126#10625329—41no—
127C2×C47⋊C8—22no—
128C2×C47⋊C8—41no—
129C2×C47⋊C8—41no—
130C23×C47—22no—
131C23×C47—22no—
132C23×C47—22no—
133C23×C47—42no—
134C23×C47—42no—
135C23×C47#12011yescharacteristic
136C22×D47—41no—
137D4×C47—22no—
138D4×C47—22no—
139D4×C47—22no—
140D4×C47#10214yes—
141C2×C4×C47—22no—
142C2×C4×C47—22no—
143C2×C4×C47—41no—
144C2×C4×C47—41no—
145C2×C4×C47—42no—
146C2×C4×C47—41no—
147C2×C4×C47—41no—
148C2×C4×C47—41no—
149C2×C4×C47—41no—
150C2×C4×C47—42no—
151C2×C4×C47—42no—
152C2×C4×C47—41no—
153C2×C4×C47—42no—
154C2×C4×C47—82no—
155C2×C4×C47—82no—
156C2×C4×C47—88no—
157C2×C4×C47C2×C22⋊C411yescharacteristic
158C2×C4×C47C4×D411yescharacteristic
159C2×C4×C47C4×D411yescharacteristic
160C2×C4×C47#12111yescharacteristic
161C2×C4×C47#12111yescharacteristic
162C2×C4×C47#12311yescharacteristic
163C2×C4×C47C2×D811yescharacteristic
164C2×C4×C47C2×SD1611yescharacteristic
165C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
166C2×C4×C47#13711yescharacteristic
167C22⋊D47—81no—
168C22⋊D47—81no—
169C2×Dic47—41no—
170D188—41no—
171D188—41no—
172D188—81no—
173D188—81no—
174C4×D47—81no—
175C4×D47—81no—
176Dic94—41no—
177Dic94—41no—
178C47⋊C8—42no—
179C47⋊C8—81no—
180C47⋊C8—81no—
181#9179—471nomaximal, Sylow
182C22×C47—21no—
183C22×C47—21no—
184C22×C47—24no—
185C22×C47—44no—
186C22×C47—44no—
187C22×C47—42no—
188C22×C47—42no—
189C22×C47#38611yescharacteristic
190C22×C47#44711yescharacteristic
191C22×C47#44811yescharacteristic
192C2×D47—81no—
193C2×D47—81no—
194C2×D47—81no—
195C4×C47—21no—
196C4×C47—21no—
197C4×C47—21no—
198C4×C47—21no—
199C4×C47—42no—
200C4×C47—42no—
201C4×C47—81no—
202C4×C47—82no—
203C4×C47C2×#10211yescharacteristic
204C4×C47#41711yescharacteristic
205C4×C47#48011yescharacteristic
206C4×C47#48111yescharacteristic
207Dic47—81no—
208#3036—471no—
209#2349D4711yescharacteristic, complemented
210#1517—471no—
211#1502—471no—
212#1478—471no—
213#1303—471no—
214#1286—471no—
215C2×C47—21no—
216C2×C47—21no—
217C2×C47—44no—
218C2×C47#187411yescharacteristic
219C2×C47#190411yescharacteristic
220C2×C47#190611yescharacteristic
221D47—161no—
222C2×AGL(1,ℤ/8)—941no—
223#532—471no—
224C2×C4×D4—21no—
225C2×C4×D4C2×D4711yescharacteristic
226#463—942no—
227#461—472no—
228#459—472no—
229#431—471no—
230#431—471no—
231#423—471no—
232#417—941no—
233#389—21no—
234#385—22no—
235#382C2×D4711yescharacteristic
236#380—21no—
237C4×C4⋊C4C2×D4711yescharacteristic
238C2×#95—21no—
239#327—941no—
240#327—941no—
241#327—942no—
242D4⋊C8—942no—
243C47#917911yescharacteristic, complemented, Sylow
244C2×#41—941no—
245C22×D4C22⋊D4711yescharacteristic
246C23×C4—21no—
247C23×C4—22no—
248AGL(1,ℤ/8)—1882no—
249AGL(1,ℤ/8)—1882no—
250C2×SD16—942no—
251C2×D8—942no—
252C2×M16—941no—
253C4⋊Q8—471no—
254#127—471no—
255#124—941no—
256#121—941no—
257#121—941no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4—42no—
261C4×D4—42no—
262C4×D4—941no—
263C4×D4—941no—
264C4×D4C4×D4712yes—
265C4×D4D18812yes—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4C22⋊D4711yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—22no—
272C2×C42—21no—
273C2×C42C22×D4711yescharacteristic
274C4⋊C8—472no—
275C4⋊C8—941no—
276C4⋊C8—941no—
277C4⋊C8—941no—
278#103—942no—
279#102—942no—
280C22⋊C8—941no—
281C22⋊C8—941no—
282C4×C8—941no—
283C4×C8—941no—
284#95—22no—
285#95—21no—
286#95C22⋊D4711yescharacteristic
287#95C22⋊D4711yescharacteristic
288#95C22⋊D4712yes—
289C24—21no—
290#41—1881no—
291#41—1881no—
292#41—1881no—
293#41—1881no—
294C2×Q8—941no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—941no—
299C2×D4—941no—
300C2×D4—941no—
301C2×D4—1881no—
302C2×D4(C2×D47)⋊C412yes—
303C2×D4#1062533412yes—
304C2×D4#1062533612yes—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—1881no—
316C22×C4C2×C22⋊D4711yescharacteristic
317C22×C4C2×C22⋊D4711yescharacteristic
318C22×C4C2×C22⋊D4711yescharacteristic
319SD16—1884no—
320D8—1884no—
321M16—1881no—
322M16—1881no—
323C2×C8—942no—
324C2×C8—1881no—
325C2×C8—1881no—
326C4⋊C4—21no—
327C4⋊C4—42no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—82no—
331C4⋊C4—941no—
332C4⋊C4—941no—
333C4⋊C4(C2×D47)⋊C412yes—
334C22⋊C4—22no—
335C22⋊C4—22no—
336C22⋊C4—44no—
337C22⋊C4—1881no—
338C22⋊C4—1881no—
339C42—41no—
340C42—41no—
341C42C2×C4×D4711yescharacteristic
342C42C2×D18811yescharacteristic
343C42#1062535011yescharacteristic
344C42#1062535011yescharacteristic
345C23—22no—
346C23—22no—
347C23—22no—
348C23—42no—
349C23—42no—
350C23—1881no—
351C23#1309399311yescharacteristic
352Q8—1881no—
353Q8—1881no—
354D4—22no—
355D4—22no—
356D4—22no—
357D4—3761no—
358D4—3761no—
359D4—3761no—
360D4—3761no—
361D4—1881no—
362D4—1881no—
363D4#1309394414yes—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—88no—
380C2×C4—3761no—
381C2×C4—3761no—
382C2×C4—1881no—
383C2×C4C2×(C2×D47)⋊C411yescharacteristic
384C2×C4C4×C22⋊D4711yescharacteristic
385C2×C4C4×C22⋊D4711yescharacteristic
386C2×C4#1309399811yescharacteristic
387C2×C4#1309399811yescharacteristic
388C2×C4#1309400711yescharacteristic
389C2×C4C2×#1062533411yescharacteristic
390C2×C4C2×#1062533611yescharacteristic
391C2×C4#1309405011yescharacteristic
392C2×C4#1309405511yescharacteristic
393C8—3761no—
394C8—3761no—
395C8—1882no—
396C22—21no—
397C22—21no—
398C22—24no—
399C22—44no—
400C22—44no—
401C22—42no—
402C22—42no—
403C22—3761no—
404C22—3761no—
405C22—3761no—
406C22#1568900211yescharacteristic
407C22#1568925711yescharacteristic
408C22#1568926211yescharacteristic
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—42no—
414C4—42no—
415C4—82no—
416C4—81no—
417C4—3761no—
418C4C2×#1309394411yescharacteristic
419C4#1568912211yescharacteristic
420C4#1568943611yescharacteristic
421C4#1568944111yescharacteristic
422C2—21no—
423C2—21no—
424C2—44no—
425C2—7521no—
426C2#2024370511yescharacteristic
427C2#2024387511yescharacteristic
428C2#2024388911yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2349C2
socleC22×C47#386
prime radical#20252852C2
cosocleC2×C42×C47C23
hypercenter#2349D47
hypocenterC47#9179
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2349

Complements

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

NH ≅ G/Ntype
2. #20241947C2semidirect
3. #20241882C2semidirect
4. #20241779C2semidirect
5. #20241291C2semidirect
6. #20241246C2semidirect
7. C47×#2349C2semidirect
11. #15689328C4semidirect
12. #15689318C4semidirect
35. #13094024C2×C4semidirect
36. #13094021C2×C4semidirect
45. C4⋊(C47⋊C8)C2×C4semidirect
209. #2349D47semidirect
243. C47#9179semidirect

Maximal subgroups

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

subgroupindex
#202528522
#202419472
#202418822
#202417792
#202412912
#202412462
C47×#23492
#917947