Finite Group Databasebeta

← #26190459

Subgroups — #26190459

Order 12032.

1230 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1230 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#20254194C212yesmaximal, complemented
2#20248768C211yesmaximal, characteristic, complemented
3#20242391C214yesmaximal, complemented
4#20242357C214yesmaximal, complemented
5#20241904C212yesmaximal, complemented
6#20241718C211yesmaximal, characteristic, complemented
7C2×C47×#537C211yesmaximal, characteristic
8C2×#13094003C2211yescharacteristic
9#15689458—22no—
10#15689434C2214yescomplemented
11#15689367—24no—
12#15689274—22no—
13#15689261C2214yescomplemented
14#15689087C2212yes—
15#15689053C2212yescomplemented
16#15689043C2212yescomplemented
17#15689018—22no—
18#15688996—21no—
19#15688996—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×Dic94C2311yescharacteristic
33C2×#10625335—24no—
34C2×#10625329—22no—
35#13094003—22no—
36#13094003—22no—
37C2×C4⋊Dic47C2311yescharacteristic
38C2×Dic47⋊C4—22no—
39C2×Dic47⋊C4—21no—
40C2×Dic47⋊C4—22no—
41C2×C22⋊Dic47—21no—
42C2×C22⋊Dic47—21no—
43C2×C4×Dic47—21no—
44#13093957—24no—
45#13093947—24no—
46#13093945—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
84#41×C47—416no—
85#41×C47—48no—
86#41×C47—48no—
87C2×Q8×C47—24no—
88C2×Q8×C47—44no—
89C2×D4×C47—24no—
90C2×D4×C47—24no—
91C2×D4×C47—24no—
92C2×D4×C47—24no—
93C2×D4×C47—22no—
94C2×D4×C47—44no—
95C2×D4×C47—48no—
96C2×D4×C47—82no—
97C2×D4×C47—82no—
98C2×D4×C47C2×D414yes—
99C22×C4×C47—21no—
100C22×C4×C47—21no—
101C22×C4×C47—24no—
102C22×C4×C47—28no—
103C22×C4×C47—21no—
104C22×C4×C47—22no—
105C22×C4×C47—21no—
106C22×C4×C47—48no—
107C22×C4×C47—81no—
108C22×C4×C47—81no—
109C22×C4×C47C2×D412yes—
110C22×C4×C47C2×D412yes—
111C22×C4×C47C2×D411yescharacteristic
112C22×C4×C47C2×D412yes—
113C22×C4×C47C2×D412yes—
114C22×C4×C47C2411yescharacteristic
115C22×Dic47—22no—
116C22×Dic47—21no—
117C22×Dic47—21no—
118C2×Dic94—22no—
119C2×Dic94—81no—
120C4⋊C4×C47—24no—
121C4⋊C4×C47—24no—
122C4⋊C4×C47—22no—
123C4⋊C4×C47—24no—
124C4⋊C4×C47C2×D414yes—
125C22⋊C4×C47—28no—
126C22⋊C4×C47—24no—
127C22⋊C4×C47—24no—
128C22⋊Dic47—24no—
129C22⋊Dic47—41no—
130C22⋊Dic47—41no—
131#10625335—84no—
132C4⋊Dic47—22no—
133Dic47⋊C4—42no—
134Dic47⋊C4—44no—
135Dic47⋊C4—82no—
136C4×Dic47—81no—
137#10625329—82no—
138C2×C47⋊C8—24no—
139C23×C47—22no—
140C23×C47—22no—
141C23×C47—24no—
142C23×C47—41no—
143C23×C47—44no—
144C23×C47—44no—
145C23×C47—44no—
146C23×C47#12011yescharacteristic
147C23×C47#12011yescharacteristic
148C23×C47C22×D411yescharacteristic
149Q8×C47—48no—
150D4×C47—44no—
151D4×C47—48no—
152D4×C47—48no—
153D4×C47—48no—
154D4×C47—82no—
155D4×C47—84no—
156C2×C4×C47—24no—
157C2×C4×C47—24no—
158C2×C4×C47—22no—
159C2×C4×C47—22no—
160C2×C4×C47—22no—
161C2×C4×C47—24no—
162C2×C4×C47—42no—
163C2×C4×C47—41no—
164C2×C4×C47—41no—
165C2×C4×C47—42no—
166C2×C4×C47—416no—
167C2×C4×C47—44no—
168C2×C4×C47—42no—
169C2×C4×C47—42no—
170C2×C4×C47—81no—
171C2×C4×C47#12012yes—
172C2×C4×C47AGL(1,ℤ/8)14yes—
173C2×C4×C47C22×D412yes—
174C2×Dic47—24no—
175C2×Dic47—42no—
176C2×Dic47—42no—
177C2×Dic47—42no—
178C2×Dic47—41no—
179C2×Dic47—41no—
180C2×Dic47—42no—
181Dic94—44no—
182C47⋊C8—44no—
183#18325—471nomaximal, Sylow
184C22×C47—21no—
185C22×C47—21no—
186C22×C47—21no—
187C22×C47—21no—
188C22×C47—21no—
189C22×C47—21no—
190C22×C47—21no—
191C22×C47—44no—
192C22×C47—44no—
193C22×C47—44no—
194C22×C47—44no—
195C22×C47—88no—
196C22×C47#45612yes—
197C22×C47C2×#12011yescharacteristic
198C22×C47C2×AGL(1,ℤ/8)12yes—
199C4×C47—24no—
200C4×C47—22no—
201C4×C47—22no—
202C4×C47—44no—
203C4×C47—42no—
204Dic47—44no—
205Dic47—81no—
206Dic47—81no—
207C2×#537D4711yescharacteristic
208#3193—472no—
209#2583—471no—
210#1610—474no—
211#1604—474no—
212#1507—472no—
213#1463—471no—
214C2×C47—22no—
215C2×C47—22no—
216C2×C47—41no—
217C2×C47—84no—
218C2×C47#300311yescharacteristic
219C2×C47C2×#45611yescharacteristic
220C2×C47#302611yescharacteristic
221C22×#41—21no—
222#537—24no—
223#537—22no—
224#537C2×D4714yes—
225C2×C22⋊Q8—22no—
226C2×C22⋊Q8—471no—
227C2×#121—22no—
228C2×#121—21no—
229C2×#121C2×D4712yes—
230C22×C4⋊C4C2×D4711yescharacteristic
231#484—942no—
232#480—474no—
233#468—944no—
234#450—942no—
235#448—474no—
236#407—472no—
237#398—472no—
238#395—472no—
239#389—942no—
240#385—941no—
241#385—941no—
242C4×C22⋊C4—471no—
243#327—478no—
244#324—474no—
245C47#1832511yescharacteristic, complemented, Sylow
246C2×#41—24no—
247C2×#41—24no—
248C2×#41—48no—
249C22×Q8—21no—
250C22×Q8—471no—
251C22×D4—21no—
252C22×D4—22no—
253C22×D4—21no—
254C22×D4C22⋊D4712yes—
255C23×C4—22no—
256C23×C4C22⋊D4711yescharacteristic
257C23×C4C22⋊D4711yescharacteristic
258C23×C4C22×D4711yescharacteristic
259C2×SD16—944no—
260C2×M16—942no—
261C22⋊Q8—28no—
262C22⋊Q8—28no—
263C22⋊Q8—942no—
264C22⋊Q8—942no—
265#121—24no—
266#121—28no—
267#121—28no—
268#121—24no—
269#121—22no—
270#121—22no—
271#121C22⋊D4714yes—
272#121C22×D4714yes—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—471no—
277C2×C4⋊C4—942no—
278C2×C4⋊C4—941no—
279C2×C4⋊C4—942no—
280C2×C4⋊C4C22⋊D4712yes—
281C2×C4⋊C4C22⋊D4712yes—
282C2×C4⋊C4C22×D4712yes—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—941no—
286C2×C22⋊C4—941no—
287C2×C42—941no—
288#107—944no—
289#103—944no—
290#102—944no—
291C22⋊C8—474no—
292C22⋊C8—942no—
293#95—472no—
294#95—942no—
295#95—942no—
296C24—22no—
297C24C2×C22⋊D4711yescharacteristic
298#41—416no—
299#41—48no—
300#41—48no—
301C2×Q8—24no—
302C2×Q8—44no—
303C2×Q8—942no—
304C2×Q8—3761no—
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—24no—
309C2×D4—22no—
310C2×D4—44no—
311C2×D4—48no—
312C2×D4—82no—
313C2×D4—82no—
314C2×D4C2×C22⋊D4714yes—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—24no—
318C22×C4—28no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—48no—
323C22×C4—81no—
324C22×C4—81no—
325C22×C4—942no—
326C22×C4—941no—
327C22×C4—941no—
328C22×C4C2×C22⋊D4712yes—
329C22×C4C2×C22⋊D4712yes—
330C22×C4C2×C22⋊D4711yescharacteristic
331C22×C4C2×C22⋊D4712yes—
332C22×C4C2×C22⋊D4712yes—
333C22×C4C23×D4711yescharacteristic
334SD16—3764no—
335M16—3762no—
336C2×C8—944no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—24no—
341C4⋊C4—942no—
342C4⋊C4—3762no—
343C4⋊C4—1882no—
344C4⋊C4—1884no—
345C4⋊C4C2×C22⋊D4714yes—
346C22⋊C4—28no—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—944no—
350C22⋊C4—1881no—
351C22⋊C4—1881no—
352C42—3761no—
353C23—22no—
354C23—22no—
355C23—24no—
356C23—41no—
357C23—44no—
358C23—44no—
359C23—44no—
360C23#1309399311yescharacteristic
361C23#1309399311yescharacteristic
362C23C22×C22⋊D4711yescharacteristic
363Q8—48no—
364Q8—1884no—
365D4—44no—
366D4—48no—
367D4—48no—
368D4—48no—
369D4—82no—
370D4—84no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—416no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—81no—
386C2×C4—944no—
387C2×C4—1882no—
388C2×C4—1881no—
389C2×C4—1881no—
390C2×C4—1882no—
391C2×C4—1882no—
392C2×C4—1882no—
393C2×C4#1309399312yes—
394C2×C4#1309405014yes—
395C2×C4C22×C22⋊D4712yes—
396C8—1884no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—44no—
405C22—44no—
406C22—44no—
407C22—44no—
408C22—88no—
409C22#1568930612yes—
410C22C2×#1309399311yescharacteristic
411C22C2×#1309405012yes—
412C4—22no—
413C4—22no—
414C4—24no—
415C4—42no—
416C4—44no—
417C4—3761no—
418C4—3761no—
419C4—1884no—
420C2—22no—
421C2—22no—
422C2—41no—
423C2—84no—
424C2#2025261511yescharacteristic
425C2C2×#1568930611yescharacteristic
426C2#2025278511yescharacteristic
subgroupG/H
centerC22C2×#13093993
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
Fitting, prime radicalC2×C47×#537C2
socleC22×C47C2×#120
hypercenterC2×#537D47
hypocenterC47#18325
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. #20254194C2semidirect
2. #20248768C2semidirect
3. #20242391C2semidirect
4. #20242357C2semidirect
5. #20241904C2semidirect
6. #20241718C2semidirect
10. #15689434C22semidirect
13. #15689261C22semidirect
15. #15689053C22semidirect
16. #15689043C22semidirect
21. #15688857C22semidirect
22. #15688849C22semidirect
245. C47#18325semidirect

Maximal subgroups

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

subgroupindex
#202541942
#202487682
#202423912
#202423572
#202419042
#202417182
C2×C47×#5372
#1832547