Finite Group Databasebeta

← #26190452

Subgroups — #26190452

Order 12032.

1256 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1256 conjugacy classes of subgroups — too many to draw. Only its 183 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#20254185C212yesmaximal, complemented
2#20248868C211yesmaximal, characteristic, complemented
3#20242425C214yesmaximal, complemented
4#20242323C214yesmaximal, complemented
5#20241904C212yesmaximal, complemented
6#20241718C211yesmaximal, characteristic, complemented
7C2×C47×#537C211yesmaximal, characteristic, complemented
8C2×#13093998C2211yescharacteristic
9#15689441—22no—
10#15689439C2214yescomplemented
11#15689374—24no—
12#15689257—22no—
13#15689256C2214yescomplemented
14#15689087C2212yes—
15#15689062C2212yescomplemented
16#15689040C2212yescomplemented
17#15689013—22no—
18#15689002—21no—
19#15689002—21no—
20C4×C22⋊Dic47C2211yescharacteristic
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—
35C2×#10625329—22no—
36#13093998—22no—
37#13093998—22no—
38C2×C4⋊Dic47C2311yescharacteristic
39C2×C22⋊Dic47—21no—
40C2×(C2×D47)⋊C4—22no—
41C2×(C2×D47)⋊C4—21no—
42C2×(C2×D47)⋊C4—22no—
43C2×C4×Dic47—21no—
44#13093955—24no—
45#13093945—24no—
46#13093944—24no—
47C22⋊(C47⋊C8)—22no—
48C22⋊(C47⋊C8)C2314yes—
49C47⋊#95—22no—
50C47⋊#95—22no—
51C47⋊#95C2312yes—
52C2×#41×C47—24no—
53C2×#41×C47—24no—
54C2×#41×C47—48no—
55C22×Q8×C47—21no—
56C22×D4×C47—21no—
57C22×D4×C47—22no—
58C22×D4×C47—21no—
59C22×D4×C47D412yes—
60C23×C4×C47—22no—
61C23×C4×C47D411yescharacteristic
62C23×C4×C47D411yescharacteristic
63C23×C4×C47C2311yescharacteristic
64C22⋊Q8×C47—28no—
65C22⋊Q8×C47—28no—
66#121×C47—24no—
67#121×C47—28no—
68#121×C47—28no—
69#121×C47—24no—
70#121×C47—22no—
71#121×C47—22no—
72#121×C47D414yes—
73#121×C47C2314yes—
74C2×C4⋊C4×C47—21no—
75C2×C4⋊C4×C47—22no—
76C2×C4⋊C4×C47—22no—
77C2×C4⋊C4×C47D412yes—
78C2×C4⋊C4×C47D412yes—
79C2×C4⋊C4×C47C2312yes—
80C2×C22⋊C4×C47—22no—
81C2×C22⋊C4×C47—22no—
82C24×C47—22no—
83C24×C47C2×D411yescharacteristic
84C23×D47—21no—
85#41×C47—416no—
86#41×C47—48no—
87#41×C47—48no—
88C2×Q8×C47—24no—
89C2×Q8×C47—44no—
90C2×D4×C47—24no—
91C2×D4×C47—24no—
92C2×D4×C47—24no—
93C2×D4×C47—24no—
94C2×D4×C47—22no—
95C2×D4×C47—44no—
96C2×D4×C47—48no—
97C2×D4×C47—82no—
98C2×D4×C47—82no—
99C2×D4×C47C2×D414yes—
100C22×C4×C47—21no—
101C22×C4×C47—21no—
102C22×C4×C47—24no—
103C22×C4×C47—28no—
104C22×C4×C47—21no—
105C22×C4×C47—22no—
106C22×C4×C47—21no—
107C22×C4×C47—48no—
108C22×C4×C47—81no—
109C22×C4×C47—81no—
110C22×C4×C47C2×D412yes—
111C22×C4×C47C2×D412yes—
112C22×C4×C47C2×D411yescharacteristic
113C22×C4×C47C2×D412yes—
114C22×C4×C47C2×D412yes—
115C22×C4×C47C2411yescharacteristic
116C2×C22⋊D47—44no—
117C2×C22⋊D47—41no—
118C2×C22⋊D47—41no—
119C22×Dic47—21no—
120C22×Dic47—22no—
121C2×D188—22no—
122C2×D188—81no—
123C4⋊C4×C47—24no—
124C4⋊C4×C47—24no—
125C4⋊C4×C47—22no—
126C4⋊C4×C47—24no—
127C4⋊C4×C47C2×D414yes—
128C22⋊C4×C47—28no—
129C22⋊C4×C47—24no—
130C22⋊C4×C47—24no—
131C22⋊Dic47—42no—
132#10625334—84no—
133(C2×D47)⋊C4—42no—
134(C2×D47)⋊C4—44no—
135(C2×D47)⋊C4—82no—
136C4⋊Dic47—22no—
137C4×Dic47—81no—
138#10625329—82no—
139C2×C47⋊C8—24no—
140C23×C47—22no—
141C23×C47—22no—
142C23×C47—24no—
143C23×C47—41no—
144C23×C47—44no—
145C23×C47—44no—
146C23×C47—44no—
147C23×C47#12011yescharacteristic
148C23×C47#12011yescharacteristic
149C23×C47C22×D411yescharacteristic
150C22×D47—42no—
151C22×D47—42no—
152C22×D47—41no—
153C22×D47—42no—
154Q8×C47—48no—
155D4×C47—44no—
156D4×C47—48no—
157D4×C47—48no—
158D4×C47—48no—
159D4×C47—82no—
160D4×C47—84no—
161C2×C4×C47—24no—
162C2×C4×C47—24no—
163C2×C4×C47—22no—
164C2×C4×C47—22no—
165C2×C4×C47—22no—
166C2×C4×C47—24no—
167C2×C4×C47—42no—
168C2×C4×C47—41no—
169C2×C4×C47—41no—
170C2×C4×C47—42no—
171C2×C4×C47—416no—
172C2×C4×C47—44no—
173C2×C4×C47—42no—
174C2×C4×C47—42no—
175C2×C4×C47—81no—
176C2×C4×C47#12012yes—
177C2×C4×C47AGL(1,ℤ/8)14yes—
178C2×C4×C47C22×D412yes—
179C22⋊D47—82no—
180C22⋊D47—82no—
181C2×Dic47—42no—
182C2×Dic47—42no—
183C2×Dic47—42no—
184C2×Dic47—41no—
185C2×Dic47—42no—
186D188—44no—
187C47⋊C8—44no—
188#18324—471nomaximal, Sylow
189C22×C47—21no—
190C22×C47—21no—
191C22×C47—21no—
192C22×C47—21no—
193C22×C47—21no—
194C22×C47—21no—
195C22×C47—21no—
196C22×C47—44no—
197C22×C47—44no—
198C22×C47—44no—
199C22×C47—44no—
200C22×C47—88no—
201C22×C47#45412yes—
202C22×C47C2×#12011yescharacteristic
203C22×C47C2×AGL(1,ℤ/8)12yes—
204C2×D47—84no—
205C2×D47—81no—
206C2×D47—81no—
207C2×D47—81no—
208C4×C47—24no—
209C4×C47—22no—
210C4×C47—22no—
211C4×C47—44no—
212C4×C47—42no—
213Dic47—82no—
214Dic47—81no—
215C2×#537D4711yescharacteristic, complemented
216#3192—472no—
217#2591—471no—
218#1616—474no—
219#1598—474no—
220#1507—472no—
221#1463—471no—
222C2×C47—22no—
223C2×C47—22no—
224C2×C47—41no—
225C2×C47—84no—
226C2×C47#300311yescharacteristic
227C2×C47C2×#45411yescharacteristic
228C2×C47#302611yescharacteristic
229D47—161no—
230C22×#41—21no—
231#537—24no—
232#537—22no—
233#537C2×D4714yes—
234C2×C22⋊Q8—22no—
235C2×#121—22no—
236C2×#121—21no—
237C2×#121—471no—
238C2×#121C2×D4712yes—
239C22×C4⋊C4C2×D4711yescharacteristic
240#481—474no—
241#481—942no—
242#469—944no—
243#447—474no—
244#447—942no—
245#407—472no—
246#400—472no—
247#394—472no—
248#388—942no—
249#386—941no—
250#386—941no—
251C4×C22⋊C4—471no—
252#327—478no—
253#324—474no—
254C47#1832411yescharacteristic, complemented, Sylow
255C2×#41—24no—
256C2×#41—24no—
257C2×#41—48no—
258C22×Q8—21no—
259C22×D4—21no—
260C22×D4—22no—
261C22×D4—21no—
262C22×D4—471no—
263C22×D4—941no—
264C22×D4C22⋊D4712yes—
265C23×C4—22no—
266C23×C4C22⋊D4711yescharacteristic
267C23×C4C22⋊D4711yescharacteristic
268C23×C4C22×D4711yescharacteristic
269C2×D8—944no—
270C2×M16—942no—
271C22⋊Q8—28no—
272C22⋊Q8—28no—
273#121—24no—
274#121—28no—
275#121—28no—
276#121—24no—
277#121—22no—
278#121—22no—
279#121—942no—
280#121—942no—
281#121C22⋊D4714yes—
282#121C22×D4714yes—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—471no—
287C2×C4⋊C4C22⋊D4712yes—
288C2×C4⋊C4C22⋊D4712yes—
289C2×C4⋊C4C22×D4712yes—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—942no—
293C2×C22⋊C4—941no—
294C2×C22⋊C4—941no—
295C2×C22⋊C4—942no—
296C2×C42—941no—
297#106—944no—
298#102—944no—
299#102—944no—
300C22⋊C8—474no—
301C22⋊C8—942no—
302#95—472no—
303#95—942no—
304#95—942no—
305C24—22no—
306C24—941no—
307C24C2×C22⋊D4711yescharacteristic
308#41—416no—
309#41—48no—
310#41—48no—
311C2×Q8—24no—
312C2×Q8—44no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—22no—
318C2×D4—44no—
319C2×D4—48no—
320C2×D4—82no—
321C2×D4—82no—
322C2×D4—942no—
323C2×D4—3761no—
324C2×D4—1884no—
325C2×D4—1881no—
326C2×D4—1881no—
327C2×D4C2×C22⋊D4714yes—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—24no—
331C22×C4—28no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—48no—
336C22×C4—81no—
337C22×C4—81no—
338C22×C4—942no—
339C22×C4—941no—
340C22×C4C2×C22⋊D4712yes—
341C22×C4C2×C22⋊D4712yes—
342C22×C4C2×C22⋊D4711yescharacteristic
343C22×C4C2×C22⋊D4712yes—
344C22×C4C2×C22⋊D4712yes—
345C22×C4C23×D4711yescharacteristic
346D8—3764no—
347M16—3762no—
348C2×C8—944no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—22no—
352C4⋊C4—24no—
353C4⋊C4—942no—
354C4⋊C4C2×C22⋊D4714yes—
355C22⋊C4—28no—
356C22⋊C4—24no—
357C22⋊C4—24no—
358C22⋊C4—3762no—
359C22⋊C4—1882no—
360C22⋊C4—1882no—
361C22⋊C4—1884no—
362C42—3761no—
363C23—22no—
364C23—22no—
365C23—24no—
366C23—41no—
367C23—44no—
368C23—44no—
369C23—44no—
370C23—1882no—
371C23—1882no—
372C23—1881no—
373C23—1882no—
374C23#1309399311yescharacteristic
375C23#1309399311yescharacteristic
376C23C22×C22⋊D4711yescharacteristic
377Q8—48no—
378D4—44no—
379D4—48no—
380D4—48no—
381D4—48no—
382D4—82no—
383D4—84no—
384D4—3762no—
385D4—3762no—
386D4—1884no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—416no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—81no—
402C2×C4—1882no—
403C2×C4—1882no—
404C2×C4—1882no—
405C2×C4—1881no—
406C2×C4—1882no—
407C2×C4#1309399312yes—
408C2×C4#1309405014yes—
409C2×C4C22×C22⋊D4712yes—
410C8—1884no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—44no—
419C22—44no—
420C22—44no—
421C22—44no—
422C22—88no—
423C22—3764no—
424C22—3761no—
425C22—3761no—
426C22—3761no—
427C22#1568929612yes—
428C22C2×#1309399311yescharacteristic
429C22C2×#1309405012yes—
430C4—22no—
431C4—22no—
432C4—24no—
433C4—42no—
434C4—44no—
435C4—3762no—
436C4—3761no—
437C2—22no—
438C2—22no—
439C2—41no—
440C2—84no—
441C2—7521no—
442C2#2025261511yescharacteristic
443C2C2×#1568929611yescharacteristic
444C2#2025278511yescharacteristic
subgroupG/H
centerC22C2×#13093993
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#537C2
socleC22×C47C2×#120
hypercenterC2×#537D47
hypocenterC47#18324
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. #20254185C2semidirect
2. #20248868C2semidirect
3. #20242425C2semidirect
4. #20242323C2semidirect
5. #20241904C2semidirect
6. #20241718C2semidirect
7. C2×C47×#537C2semidirect
10. #15689439C22semidirect
13. #15689256C22semidirect
15. #15689062C22semidirect
16. #15689040C22semidirect
21. #15688857C22semidirect
22. #15688849C22semidirect
215. C2×#537D47semidirect
254. C47#18324semidirect

Maximal subgroups

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

subgroupindex
#202541852
#202488682
#202424252
#202423232
#202419042
#202417182
C2×C47×#5372
#1832447