Finite Group Databasebeta

← #26371079

Subgroups — #26371079

Order 12032.

590 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252969C211yesmaximal, characteristic
2C2×#15689161C211yesmaximal, characteristic, complemented
3#20245597C212yesmaximal, complemented
4#20245591C211yesmaximal, characteristic, complemented
5#20245586C211yesmaximal, characteristic, complemented
6#20245580C212yesmaximal, complemented
7#20245574C211yesmaximal, characteristic, complemented
8#20245569C211yesmaximal, characteristic, complemented
9#20245172C212yesmaximal, complemented
10#20245096C212yesmaximal, complemented
11C2×C47×C22⋊C16C211yesmaximal, characteristic, complemented
12C2×#13094046C2211yescharacteristic
13#15689634—21no—
14C2×#13093983—21no—
15#15689364C2211yescharacteristic
16#15689364C2211yescharacteristic
17#15689359C2211yescharacteristic
18#15689359C2211yescharacteristic
19#15689352—21no—
20#15689352—21no—
21#15689161—21no—
22#15689161—21no—
23#15689161C2211yescharacteristic
24#15689161C2212yes—
25#15689161C2212yes—
26C2×#13093958C2211yescharacteristic
27C2×#13093958C2211yescharacteristic
28C2×#13093956—21no—
29#15689120—21no—
30#15688951C2212yescomplemented
31#15688951C2212yescomplemented
32#15688949C2212yescomplemented
33#15688949C2212yescomplemented
34#15688928C2212yescomplemented
35#15688928C2212yescomplemented
36#15688925C2212yescomplemented
37#15688925C2212yescomplemented
38C23×C8×C47C2211yescharacteristic
39C22×C16×C47C2212yes—
40C47×C22⋊C16C2211yescharacteristic
41C47×C22⋊C16C2211yescharacteristic
42C47×C22⋊C16C2212yes—
43C2×#10625350—21no—
44#13094046—24no—
45C2×Dic188C2311yescharacteristic
46C2×#10625325—21no—
47C2×D376C2311yescharacteristic
48#13094007—41no—
49#13094003—21no—
50#13094003—21no—
51#13093998—21no—
52#13093998—21no—
53#13093993—41no—
54C4×C22⋊D47—41no—
55C4×C22⋊D47—41no—
56#13093983—22no—
57#13093983—22no—
58#13093983—21no—
59#13093983—41no—
60#13093983—41no—
61C2×C4⋊Dic47—21no—
62C2×C4⋊Dic47—21no—
63C2×C22⋊Dic47—41no—
64C2×C4×Dic47—41no—
65#13093958—21no—
66#13093958—21no—
67#13093958C2311yescharacteristic
68#13093958C2311yescharacteristic
69#13093958C2312yes—
70#13093958C2312yes—
71#13093956—21no—
72#13093956—22no—
73#13093956—22no—
74#13093949—21no—
75#13093949—21no—
76#13093946—21no—
77#13093946—21no—
78C23×C4×C47D411yescharacteristic
79C22×C8×C47—24no—
80C22×C8×C47D411yescharacteristic
81C22×C8×C47D411yescharacteristic
82C22×C8×C47D411yescharacteristic
83C22×C8×C47C2311yescharacteristic
84C22×C8×C47C2311yescharacteristic
85C22×C8×C47C2311yescharacteristic
86C2×C16×C47—24no—
87C2×C16×C47C2312yes—
88C2×C16×C47C2312yes—
89C24×C47D811yescharacteristic
90C22×C4×C47—22no—
91C22×C4×C47—22no—
92C22×C4×C47D811yescharacteristic
93C22×C4×C47D811yescharacteristic
94C22×C4×C47D811yescharacteristic
95C22×C4×C47C2×D411yescharacteristic
96C22×C4×C47C2×D411yescharacteristic
97C22×C4×C47C2×D411yescharacteristic
98C2×C22⋊D47—41no—
99C2×C22⋊D47—41no—
100C2×C22⋊D47—41no—
101C22×Dic47—41no—
102C22×Dic47—41no—
103#10625350—42no—
104#10625350—42no—
105C2×D188—21no—
106C2×C4×D47—41no—
107C2×Dic94—21no—
108C2×C8×C47—22no—
109C2×C8×C47—22no—
110C2×C8×C47—22no—
111C2×C8×C47—22no—
112C2×C8×C47—22no—
113C2×C8×C47C2×D411yescharacteristic
114C2×C8×C47C2×D411yescharacteristic
115C2×C8×C47C2×D411yescharacteristic
116C2×C8×C47#4114yes—
117C2×C8×C47C2411yescharacteristic
118C22⋊Dic47—42no—
119C22⋊Dic47—42no—
120C22⋊Dic47—41no—
121C22⋊Dic47—41no—
122C22⋊Dic47—41no—
123(C2×D47)⋊C4—41no—
124(C2×D47)⋊C4—41no—
125C4⋊Dic47—21no—
126C4⋊Dic47—21no—
127C4⋊Dic47—22no—
128C4⋊Dic47—22no—
129C4⋊Dic47—41no—
130C4⋊Dic47—41no—
131Dic47⋊C4—41no—
132Dic47⋊C4—41no—
133C4×Dic47—42no—
134C4×Dic47—42no—
135C4×Dic47—41no—
136Dic188—22no—
137D376—22no—
138#10625325—42no—
139C16×C47—22no—
140C16×C47—22no—
141C23×C47—22no—
142C23×C47—22no—
143C23×C47—41no—
144C23×C47C2×D811yescharacteristic
145C23×C47C2×D811yescharacteristic
146C23×C47C2×D811yescharacteristic
147C22×D47—41no—
148C2×C4×C47—22no—
149C2×C4×C47—21no—
150C2×C4×C47—21no—
151C2×C4×C47—21no—
152C2×C4×C47—21no—
153C2×C4×C47—22no—
154C2×C4×C47—22no—
155C2×C4×C47#12312yes—
156C2×C4×C47C2×D811yescharacteristic
157C2×C4×C47C2×D811yescharacteristic
158C2×C4×C47C2×D811yescharacteristic
159C2×C4×C47#13712yes—
160C2×C4×C47C22×D411yescharacteristic
161C22⋊D47—82no—
162C22⋊D47—82no—
163C22⋊D47—82no—
164C2×Dic47—42no—
165C2×Dic47—42no—
166C2×Dic47—41no—
167C2×Dic47—41no—
168C2×Dic47—41no—
169C2×Dic47—82no—
170C2×Dic47—82no—
171D188—41no—
172D188—41no—
173C4×D47—82no—
174Dic94—41no—
175Dic94—41no—
176C8×C47—21no—
177C8×C47—21no—
178C8×C47#12312yes—
179C8×C47C2×#4112yes—
180#33645—471nomaximal, Sylow
181C22×C47—21no—
182C22×C47—21no—
183C22×C47—21no—
184C22×C47—21no—
185C22×C47—21no—
186C22×C47—21no—
187C22×C47—21no—
188C22×C47—44no—
189C22×C47#48012yes—
190C22×C47#50812yes—
191C22×C47C22×D811yescharacteristic
192C2×D47—81no—
193C2×D47—81no—
194C2×D47—81no—
195C4×C47—21no—
196C4×C47—21no—
197C4×C47#48012yes—
198C4×C47C2×#12311yescharacteristic
199C4×C47C2×#13711yescharacteristic
200Dic47—81no—
201Dic47—82no—
202Dic47—81no—
203Dic47—82no—
204Dic47—81no—
205#3050—471no—
206C2×#427—471no—
207#2237—472no—
208#2236—471no—
209#2235—471no—
210#2234—472no—
211#2233—471no—
212#2232—471no—
213#2158—472no—
214#2138—472no—
215C2×C22⋊C16D4711yescharacteristic, complemented
216C2×C47—22no—
217C2×C47—22no—
218C2×C47—41no—
219C2×C47C2×#48011yescharacteristic
220C2×C47C2×#50811yescharacteristic
221C2×C47#341411yescharacteristic
222D47—161no—
223C2×#135—471no—
224C23×C8C2×D4711yescharacteristic
225#525—941no—
226C2×#117—941no—
227C22×C16C2×D4712yes—
228#467—471no—
229#467—471no—
230#466—471no—
231#466—471no—
232#465—941no—
233#465—941no—
234#427—472no—
235#427—472no—
236#427—471no—
237#427—941no—
238#427—941no—
239C2×#107—471no—
240C2×#107—471no—
241C2×#106—941no—
242#416—941no—
243#367—472no—
244#367—472no—
245#366—472no—
246#366—472no—
247#358—472no—
248#358—472no—
249#357—472no—
250#357—472no—
251C22⋊C16C2×D4711yescharacteristic
252C22⋊C16C2×D4711yescharacteristic
253C22⋊C16C2×D4712yes—
254C47#3364511yescharacteristic, complemented, Sylow
255C2×#41—941no—
256C23×C4D18811yescharacteristic
257#135—944no—
258C2×Q16—471no—
259C2×SD16—941no—
260C2×D8—471no—
261C22×C8—24no—
262C22×C8D18811yescharacteristic
263C22×C8D18811yescharacteristic
264C22×C8D18811yescharacteristic
265C22×C8C22×D4711yescharacteristic
266C22×C8C22×D4711yescharacteristic
267C22×C8C22×D4711yescharacteristic
268#123—1881no—
269C22⋊Q8—941no—
270C22⋊Q8—941no—
271#121—941no—
272#121—941no—
273#120—1881no—
274C4×D4—1881no—
275C4×D4—1881no—
276#117—942no—
277#117—942no—
278#117—941no—
279#117—1881no—
280#117—1881no—
281C2×C4⋊C4—941no—
282C2×C4⋊C4—941no—
283C2×C22⋊C4—1881no—
284C2×C42—1881no—
285C2×C16—24no—
286C2×C16C22×D4712yes—
287C2×C16C22×D4712yes—
288#107—472no—
289#107—472no—
290#107—471no—
291#107—471no—
292#107—941no—
293#107—941no—
294#106—942no—
295#106—942no—
296#106—941no—
297#103—941no—
298#103—941no—
299#102—941no—
300#102—941no—
301C24D37611yescharacteristic
302#41—1882no—
303#41—1882no—
304C2×Q8—941no—
305C2×D4—941no—
306C2×D4—1881no—
307C2×D4—1881no—
308C2×D4—1881no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—1881no—
312C22×C4—1881no—
313C22×C4—1881no—
314C22×C4D37611yescharacteristic
315C22×C4D37611yescharacteristic
316C22×C4D37611yescharacteristic
317C22×C4C2×D18811yescharacteristic
318C22×C4C2×D18811yescharacteristic
319C22×C4C2×D18811yescharacteristic
320Q16—942no—
321SD16—1882no—
322D8—942no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8C2×D18811yescharacteristic
329C2×C8C2×D18811yescharacteristic
330C2×C8C2×D18811yescharacteristic
331C2×C8#1062535214yes—
332C2×C8C23×D4711yescharacteristic
333C4⋊C4—941no—
334C4⋊C4—941no—
335C4⋊C4—942no—
336C4⋊C4—942no—
337C4⋊C4—1881no—
338C4⋊C4—1881no—
339C4⋊C4—1881no—
340C4⋊C4—1881no—
341C22⋊C4—1882no—
342C22⋊C4—1882no—
343C22⋊C4—1881no—
344C22⋊C4—1881no—
345C22⋊C4—1881no—
346C22⋊C4—1881no—
347C22⋊C4—1881no—
348C42—1882no—
349C42—1882no—
350C42—1881no—
351C16—22no—
352C16—22no—
353C23—22no—
354C23—22no—
355C23—41no—
356C23—1881no—
357C23C2×D37611yescharacteristic
358C23C2×D37611yescharacteristic
359C23C2×D37611yescharacteristic
360Q8—1881no—
361Q8—1881no—
362D4—3762no—
363D4—3762no—
364D4—3762no—
365D4—1881no—
366D4—1881no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—3762no—
375C2×C4—3762no—
376C2×C4—3762no—
377C2×C4—1882no—
378C2×C4—1882no—
379C2×C4—1881no—
380C2×C4—1881no—
381C2×C4—1881no—
382C2×C4#1309400812yes—
383C2×C4C2×D37611yescharacteristic
384C2×C4C2×D37611yescharacteristic
385C2×C4C2×D37611yescharacteristic
386C2×C4#1309405712yes—
387C2×C4C22×D18811yescharacteristic
388C8—21no—
389C8—21no—
390C8#1309400812yes—
391C8C2×#1062535212yes—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—44no—
400C22—3761no—
401C22—3761no—
402C22—3761no—
403C22#1568943712yes—
404C22#1568955512yes—
405C22C22×D37611yescharacteristic
406C4—21no—
407C4—21no—
408C4—3762no—
409C4—3761no—
410C4—3761no—
411C4—3762no—
412C4—3761no—
413C4#1568943712yes—
414C4C2×#1309400811yescharacteristic
415C4C2×#1309405711yescharacteristic
416C2—22no—
417C2—22no—
418C2—41no—
419C2—7521no—
420C2C2×#1568943711yescharacteristic
421C2C2×#1568955511yescharacteristic
422C2#2025652411yescharacteristic
subgroupG/H
centerC2×C4C2×D376
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC2×C47×C22⋊C16C2
socleC22×C47C22×D8
prime radical#20252969C2
hypercenterC2×C22⋊C16D47
hypocenterC47#33645
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ C2×C22⋊C16

Complements

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

NH ≅ G/Ntype
2. C2×#15689161C2semidirect
3. #20245597C2semidirect
4. #20245591C2semidirect
5. #20245586C2semidirect
6. #20245580C2semidirect
7. #20245574C2semidirect
8. #20245569C2semidirect
9. #20245172C2semidirect
10. #20245096C2semidirect
11. C2×C47×C22⋊C16C2semidirect
30. #15688951C22semidirect
31. #15688951C22semidirect
32. #15688949C22semidirect
33. #15688949C22semidirect
34. #15688928C22semidirect
35. #15688928C22semidirect
36. #15688925C22semidirect
37. #15688925C22semidirect
215. C2×C22⋊C16D47semidirect
254. C47#33645semidirect

Maximal subgroups

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

subgroupindex
#202529692
C2×#156891612
#202455972
#202455912
#202455862
#202455802
#202455742
#202455692
#202451722
#202450962
C2×C47×C22⋊C162
#3364547