Finite Group Databasebeta

← #26190180

Subgroups — #26190180

Order 12032.

1282 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254136C212yesmaximal, complemented
2#20249130C211yesmaximal, characteristic, complemented
3#20242391C214yesmaximal, complemented
4#20242323C214yesmaximal, complemented
5#20241956C212yesmaximal, complemented
6C2×#15688849C211yesmaximal, characteristic, complemented
7C2×C47×#537C211yesmaximal, characteristic, complemented
8C2×#13094007—21no—
9C2×#13093998C2211yescharacteristic
10#15689436—22no—
11#15689434C2214yescomplemented
12#15689357—24no—
13#15689257—22no—
14#15689256C2214yescomplemented
15C2×C22⋊(C47⋊C8)C2212yes—
16#15689053C2212yescomplemented
17#15689045—22no—
18#15689040C2212yescomplemented
19#15689002—21no—
20#15688983C2211yescharacteristic
21#15688857C2218yescomplemented
22#15688849C2214yescomplemented
23C22×#41×C47—21no—
24C47×#537—24no—
25C47×#537—22no—
26C47×#537C2214yes—
27C2×C22⋊Q8×C47—22no—
28C2×#121×C47—22no—
29C2×#121×C47—21no—
30C2×#121×C47C2212yes—
31C22×C4⋊C4×C47C2211yescharacteristic
32C22×D188C2311yescharacteristic
33C22×C22⋊D47—21no—
34C2×#10625334—24no—
35C22×C47⋊C8—22no—
36#13094007—44no—
37#13093998—22no—
38#13093998—22no—
39C2×C4⋊Dic47—21no—
40C2×C4⋊Dic47C2311yescharacteristic
41C2×Dic47⋊C4—22no—
42C2×C22⋊Dic47—21no—
43C2×(C2×D47)⋊C4—22no—
44C2×(C2×D47)⋊C4—21no—
45C2×(C2×D47)⋊C4—22no—
46#13093957—24no—
47#13093945—24no—
48#13093944—24no—
49C22⋊(C47⋊C8)—22no—
50C22⋊(C47⋊C8)C2314yes—
51C47⋊#95—22no—
52C47⋊#95C2312yes—
53C2×#41×C47—24no—
54C2×#41×C47—24no—
55C2×#41×C47—44no—
56C2×#41×C47—44no—
57C22×Q8×C47—21no—
58C22×D4×C47—21no—
59C22×D4×C47—22no—
60C22×D4×C47—21no—
61C22×D4×C47D412yes—
62C23×C4×C47—22no—
63C23×C4×C47D411yescharacteristic
64C23×C4×C47D411yescharacteristic
65C23×C4×C47C2311yescharacteristic
66C22⋊Q8×C47—28no—
67C22⋊Q8×C47—28no—
68#121×C47—24no—
69#121×C47—28no—
70#121×C47—28no—
71#121×C47—24no—
72#121×C47—22no—
73#121×C47—22no—
74#121×C47D414yes—
75#121×C47C2314yes—
76C2×C4⋊C4×C47—21no—
77C2×C4⋊C4×C47—22no—
78C2×C4⋊C4×C47—22no—
79C2×C4⋊C4×C47D412yes—
80C2×C4⋊C4×C47D412yes—
81C2×C4⋊C4×C47C2312yes—
82C2×C22⋊C4×C47—22no—
83C2×C22⋊C4×C47—22no—
84C24×C47—22no—
85C24×C47C2×D411yescharacteristic
86C23×D47—21no—
87#41×C47—48no—
88#41×C47—416no—
89#41×C47—48no—
90C2×Q8×C47—24no—
91C2×Q8×C47—44no—
92C2×D4×C47—24no—
93C2×D4×C47—24no—
94C2×D4×C47—24no—
95C2×D4×C47—24no—
96C2×D4×C47—22no—
97C2×D4×C47—44no—
98C2×D4×C47—44no—
99C2×D4×C47—44no—
100C2×D4×C47—82no—
101C2×D4×C47—82no—
102C2×D4×C47C2×D414yes—
103C22×C4×C47—21no—
104C22×C4×C47—21no—
105C22×C4×C47—22no—
106C22×C4×C47—28no—
107C22×C4×C47—21no—
108C22×C4×C47—22no—
109C22×C4×C47—21no—
110C22×C4×C47—48no—
111C22×C4×C47—81no—
112C22×C4×C47—81no—
113C22×C4×C47D814yescomplemented
114C22×C4×C47C2×D412yes—
115C22×C4×C47C2×D412yes—
116C22×C4×C47C2×D411yescharacteristic
117C22×C4×C47C2×D412yes—
118C22×C4×C47C2×D412yes—
119C22×C4×C47C2411yescharacteristic
120C2×C22⋊D47—41no—
121C2×C22⋊D47—41no—
122C2×C22⋊D47—42no—
123C2×C22⋊D47—42no—
124C22×Dic47—21no—
125C22×Dic47—22no—
126C2×D188—22no—
127C2×D188—81no—
128C4⋊C4×C47—24no—
129C4⋊C4×C47—24no—
130C4⋊C4×C47—22no—
131C4⋊C4×C47—24no—
132C4⋊C4×C47C2×D414yes—
133C22⋊C4×C47—28no—
134C22⋊C4×C47—24no—
135C22⋊C4×C47—24no—
136C22⋊Dic47—42no—
137#10625334—84no—
138(C2×D47)⋊C4—42no—
139(C2×D47)⋊C4—44no—
140(C2×D47)⋊C4—82no—
141C4⋊Dic47—22no—
142C4⋊Dic47—81no—
143Dic47⋊C4—44no—
144C2×C47⋊C8—24no—
145C2×C47⋊C8—82no—
146C23×C47—24no—
147C23×C47—41no—
148C23×C47—41no—
149C23×C47—44no—
150C23×C47—44no—
151C23×C47—44no—
152C23×C47#12011yescharacteristic
153C23×C47#12011yescharacteristic
154C23×C47C2×D812yes—
155C23×C47AGL(1,ℤ/8)12yes—
156C23×C47C22×D411yescharacteristic
157C22×D47—42no—
158C22×D47—42no—
159C22×D47—41no—
160C22×D47—41no—
161C22×D47—41no—
162Q8×C47—48no—
163D4×C47—44no—
164D4×C47—48no—
165D4×C47—44no—
166D4×C47—44no—
167D4×C47—48no—
168D4×C47—82no—
169D4×C47—84no—
170C2×C4×C47—22no—
171C2×C4×C47—22no—
172C2×C4×C47—24no—
173C2×C4×C47—24no—
174C2×C4×C47—22no—
175C2×C4×C47—24no—
176C2×C4×C47—42no—
177C2×C4×C47—41no—
178C2×C4×C47—41no—
179C2×C4×C47—42no—
180C2×C4×C47—416no—
181C2×C4×C47—42no—
182C2×C4×C47—44no—
183C2×C4×C47—42no—
184C2×C4×C47—81no—
185C2×C4×C47#12012yes—
186C2×C4×C47C2×D814yes—
187C2×C4×C47C22×D412yes—
188C22⋊D47—82no—
189C22⋊D47—82no—
190C2×Dic47—42no—
191C2×Dic47—42no—
192C2×Dic47—41no—
193C2×Dic47—41no—
194C2×Dic47—42no—
195C2×Dic47—41no—
196D188—44no—
197C47⋊C8—44no—
198#18292—471nomaximal, Sylow
199C22×C47—21no—
200C22×C47—21no—
201C22×C47—21no—
202C22×C47—21no—
203C22×C47—21no—
204C22×C47—21no—
205C22×C47—42no—
206C22×C47—42no—
207C22×C47—44no—
208C22×C47—44no—
209C22×C47—44no—
210C22×C47—88no—
211C22×C47#44712yes—
212C22×C47#44712yes—
213C22×C47C2×#12011yescharacteristic
214C22×C47C22×D811yescharacteristic
215C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
216C2×D47—82no—
217C2×D47—82no—
218C2×D47—81no—
219C2×D47—81no—
220C2×D47—81no—
221C4×C47—24no—
222C4×C47—22no—
223C4×C47—22no—
224C4×C47—44no—
225C4×C47—42no—
226Dic47—82no—
227Dic47—81no—
228C2×#537D4711yescharacteristic, complemented
229#3186—472no—
230#2618—471no—
231#1610—474no—
232#1598—474no—
233#1519—472no—
234C2×#324—471no—
235C2×C47—21no—
236C2×C47—21no—
237C2×C47—22no—
238C2×C47—22no—
239C2×C47—84no—
240C2×C47C2×#44711yescharacteristic
241C2×C47#302011yescharacteristic
242C2×C47#302611yescharacteristic
243D47—161no—
244C22×#41—21no—
245#537—24no—
246#537—22no—
247#537C2×D4714yes—
248C2×#123—941no—
249C2×C22⋊Q8—22no—
250C2×#121—22no—
251C2×#121—21no—
252C2×#121—471no—
253C2×#121C2×D4712yes—
254C22×C4⋊C4C2×D4711yescharacteristic
255#480—474no—
256#480—942no—
257#466—944no—
258#447—474no—
259#447—942no—
260C2×C22⋊C8—472no—
261#398—472no—
262#396—942no—
263#394—472no—
264#386—941no—
265#380—471no—
266#327—478no—
267#324—474no—
268C47#1829211yescharacteristic, complemented, Sylow
269C2×#41—24no—
270C2×#41—24no—
271C2×#41—44no—
272C2×#41—44no—
273C22×Q8—21no—
274C22×D4—21no—
275C22×D4—22no—
276C22×D4—21no—
277C22×D4—471no—
278C22×D4—941no—
279C22×D4C22⋊D4712yes—
280C23×C4—22no—
281C23×C4C22⋊D4711yescharacteristic
282C23×C4C22⋊D4711yescharacteristic
283C23×C4C22×D4711yescharacteristic
284C2×D8—944no—
285C22×C8—942no—
286#123—1884no—
287C22⋊Q8—28no—
288C22⋊Q8—28no—
289#121—24no—
290#121—28no—
291#121—28no—
292#121—24no—
293#121—22no—
294#121—22no—
295#121—942no—
296#121—942no—
297#121C22⋊D4714yes—
298#121C22×D4714yes—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—471no—
303C2×C4⋊C4—941no—
304C2×C4⋊C4—942no—
305C2×C4⋊C4C22⋊D4712yes—
306C2×C4⋊C4C22⋊D4712yes—
307C2×C4⋊C4C22×D4712yes—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—942no—
311C2×C22⋊C4—941no—
312C2×C22⋊C4—941no—
313C2×C22⋊C4—942no—
314#107—944no—
315#102—944no—
316#102—944no—
317C22⋊C8—474no—
318C22⋊C8—942no—
319#95—472no—
320#95—942no—
321C24—22no—
322C24—941no—
323C24C2×C22⋊D4711yescharacteristic
324#41—416no—
325#41—48no—
326#41—48no—
327C2×Q8—24no—
328C2×Q8—44no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—22no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—82no—
338C2×D4—82no—
339C2×D4—942no—
340C2×D4—3761no—
341C2×D4—1881no—
342C2×D4—1881no—
343C2×D4—1882no—
344C2×D4—1882no—
345C2×D4C2×C22⋊D4714yes—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—28no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—48no—
354C22×C4—81no—
355C22×C4—81no—
356C22×C4—942no—
357C22×C4—941no—
358C22×C4#1062533414yescomplemented
359C22×C4C2×C22⋊D4712yes—
360C22×C4C2×C22⋊D4712yes—
361C22×C4C2×C22⋊D4711yescharacteristic
362C22×C4C2×C22⋊D4712yes—
363C22×C4C2×C22⋊D4712yes—
364C22×C4C23×D4711yescharacteristic
365D8—3764no—
366C2×C8—944no—
367C2×C8—3762no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—942no—
373C4⋊C4—3761no—
374C4⋊C4—1884no—
375C4⋊C4C2×C22⋊D4714yes—
376C22⋊C4—28no—
377C22⋊C4—24no—
378C22⋊C4—24no—
379C22⋊C4—3762no—
380C22⋊C4—1882no—
381C22⋊C4—1882no—
382C22⋊C4—1884no—
383C23—24no—
384C23—41no—
385C23—41no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—1882no—
390C23—1882no—
391C23—1881no—
392C23—1881no—
393C23—1881no—
394C23#1309399311yescharacteristic
395C23#1309399311yescharacteristic
396C23C2×#1062533412yes—
397C23#1309405012yes—
398C23C22×C22⋊D4711yescharacteristic
399Q8—48no—
400D4—44no—
401D4—48no—
402D4—48no—
403D4—44no—
404D4—44no—
405D4—82no—
406D4—84no—
407D4—3762no—
408D4—3762no—
409D4—1884no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—416no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—81no—
425C2×C4—1882no—
426C2×C4—1882no—
427C2×C4—1881no—
428C2×C4—1882no—
429C2×C4—1881no—
430C2×C4—1881no—
431C2×C4#1309399312yes—
432C2×C4C2×#1062533414yes—
433C2×C4C22×C22⋊D4712yes—
434C8—1884no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—42no—
442C22—42no—
443C22—44no—
444C22—44no—
445C22—44no—
446C22—88no—
447C22—3762no—
448C22—3762no—
449C22—3761no—
450C22—3761no—
451C22—3761no—
452C22#1568925712yes—
453C22#1568925712yes—
454C22C2×#1309399311yescharacteristic
455C22C22×#1062533411yescharacteristic
456C22C2×#1309405011yescharacteristic
457C4—22no—
458C4—22no—
459C4—24no—
460C4—42no—
461C4—44no—
462C4—3762no—
463C4—3761no—
464C2—21no—
465C2—22no—
466C2—21no—
467C2—22no—
468C2—84no—
469C2—7521no—
470C2C2×#1568925711yescharacteristic
471C2#2025275211yescharacteristic
472C2#2025278511yescharacteristic
subgroupG/H
centerC22C2×#13093993
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#537C2
socleC22×C47C2×#120
hypercenterC2×#537D47
hypocenterC47#18292
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. #20254136C2semidirect
2. #20249130C2semidirect
3. #20242391C2semidirect
4. #20242323C2semidirect
5. #20241956C2semidirect
6. C2×#15688849C2semidirect
7. C2×C47×#537C2semidirect
11. #15689434C22semidirect
14. #15689256C22semidirect
16. #15689053C22semidirect
18. #15689040C22semidirect
21. #15688857C22semidirect
22. #15688849C22semidirect
113. C22×C4×C47D8semidirect
228. C2×#537D47semidirect
268. C47#18292semidirect
358. C22×C4#10625334semidirect

Maximal subgroups

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

subgroupindex
#202541362
#202491302
#202423912
#202423232
#202419562
C2×#156888492
C2×C47×#5372
#1829247