Finite Group Databasebeta

← #26371149

Subgroups — #26371149

Order 12032.

568 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251988C211yesmaximal, characteristic, complemented
2#20245593C212yesmaximal, complemented
3#20245583C212yesmaximal, complemented
4#20245576C212yesmaximal, complemented
5#20245571C212yesmaximal, complemented
6#20245174C211yesmaximal, characteristic, complemented
7C2×#15688950C211yesmaximal, characteristic, complemented
8#20245100C211yesmaximal, characteristic, complemented
9#20245095C211yesmaximal, characteristic, complemented
10#20245002C211yesmaximal, characteristic, complemented
11C47×#3061C211yesmaximal, characteristic
12#15689606—21no—
13C2×#13093964C2211yescharacteristic
14C22×C47⋊C16C2211yescharacteristic
15#15689165—22no—
16#15689165—21no—
17#15689161C2212yes—
18#15689161C2211yescharacteristic
19C2×#13093958C2212yes—
20C2×#13093958C2211yescharacteristic
21#15688950C2212yes—
22#15688950C2212yes—
23#15688950C2212yes—
24#15688948C2212yes—
25#15688927C2211yescharacteristic
26#15688927C2212yes—
27#15688927C2211yescharacteristic
28#15688924C2211yescharacteristic
29#15688924C2212yes—
30#15688924C2211yescharacteristic
31C22⋊(C47⋊C16)C2212yes—
32C22⋊(C47⋊C16)C2212yes—
33C2×#135×C47C2211yescharacteristic
34C2×#131×C47C2211yescharacteristic
35C47×#537—21no—
36C47×#468—22no—
37C47×#467C2212yes—
38C47×#466C2212yes—
39C2×#107×C47C2211yescharacteristic
40C47×#417—21no—
41Q8×Dic47—41no—
42D4×Dic47—42no—
43D4×Dic47—41no—
44#13093983—22no—
45#13093983—21no—
46C2×C4⋊Dic47—22no—
47C2×C4⋊Dic47—21no—
48#13093964—22no—
49C2×C47⋊C16—22no—
50C2×C47⋊C16C2311yescharacteristic
51C2×C47⋊C16C2311yescharacteristic
52C2×C47⋊C16C2312yes—
53#13093958—22no—
54#13093958—21no—
55#13093958C2311yescharacteristic
56#13093958C2312yes—
57#13093958C2311yescharacteristic
58#13093956—22no—
59#13093956—21no—
60C2×#41×C47—21no—
61C2×#41×C47D411yescharacteristic
62#135×C47—22no—
63#135×C47—22no—
64C2×Q16×C47C2311yescharacteristic
65C2×SD16×C47—21no—
66C2×D8×C47C2311yescharacteristic
67#131×C47—22no—
68#131×C47—22no—
69C2×M16×C47D412yes—
70C2×M16×C47D411yescharacteristic
71C22×C8×C47C2312yes—
72C22×C8×C47C2311yescharacteristic
73C22⋊Q8×C47—22no—
74C22⋊Q8×C47—41no—
75#121×C47—22no—
76#121×C47—41no—
77#121×C47—42no—
78C2×C4⋊C4×C47—21no—
79#107×C47—21no—
80#107×C47C2312yes—
81#103×C47—22no—
82#102×C47—22no—
83#41×C47—22no—
84#41×C47—22no—
85#41×C47—42no—
86#41×C47—42no—
87C2×Q8×C47—21no—
88C2×Q8×C47D811yescharacteristic
89C2×D4×C47—21no—
90C2×D4×C47—41no—
91C2×D4×C47—42no—
92C2×D4×C47D812yes—
93C2×D4×C47D811yescharacteristic
94C22×C4×C47—41no—
95C22×C4×C47—41no—
96C22×C4×C47C2×D412yes—
97C22×C4×C47C2×D411yescharacteristic
98C22×Dic47—42no—
99C22×Dic47—41no—
100Q16×C47—21no—
101Q16×C47—21no—
102SD16×C47—42no—
103D8×C47—21no—
104D8×C47—21no—
105M16×C47—22no—
106M16×C47—24no—
107C2×C8×C47—22no—
108C2×C8×C47—22no—
109C2×C8×C47C2×D412yes—
110C2×C8×C47C2×D411yescharacteristic
111C2×C8×C47#4112yes—
112C2×C8×C47#4112yes—
113C2×C8×C47C2411yescharacteristic
114C4⋊C4×C47—22no—
115C4⋊C4×C47—41no—
116C4⋊C4×C47—42no—
117C22⋊C4×C47—42no—
118C22⋊C4×C47—42no—
119C22⋊Dic47—42no—
120C22⋊Dic47—41no—
121C4⋊Dic47—21no—
122C4⋊Dic47—22no—
123C4⋊Dic47—21no—
124C4⋊Dic47—42no—
125C4⋊Dic47—41no—
126C4×Dic47—42no—
127C4×Dic47—41no—
128C47⋊C16—21no—
129C47⋊C16—21no—
130C47⋊C16—22no—
131C23×C47—41no—
132C23×C47C2×D812yes—
133C23×C47C2×D811yescharacteristic
134Q8×C47—22no—
135Q8×C47—41no—
136Q8×C47—41no—
137D4×C47—24no—
138D4×C47—22no—
139D4×C47—41no—
140D4×C47—41no—
141D4×C47—82no—
142D4×C47—162no—
143C2×C4×C47—22no—
144C2×C4×C47—22no—
145C2×C4×C47—42no—
146C2×C4×C47—41no—
147C2×C4×C47—82no—
148C2×C4×C47—161no—
149C2×C4×C47#12312yes—
150C2×C4×C47C2×D812yes—
151C2×C4×C47C2×D811yescharacteristic
152C2×C4×C47#13712yes—
153C2×C4×C47C22×D411yescharacteristic
154C2×Dic47—41no—
155C2×Dic47—42no—
156C2×Dic47—41no—
157C2×Dic47—161no—
158C2×Dic47—162no—
159C8×C47—22no—
160C8×C47#12312yes—
161C8×C47C2×#4111yescharacteristic
162C8×C47C2×#4111yescharacteristic
163#33653—471nomaximal, Sylow
164C22×C47—21no—
165C22×C47—21no—
166C22×C47—22no—
167C22×C47—22no—
168C22×C47—22no—
169C22×C47—81no—
170C22×C47—81no—
171C22×C47—81no—
172C22×C47#48012yes—
173C22×C47C22×D811yescharacteristic
174C4×C47—22no—
175C4×C47—81no—
176C4×C47—82no—
177C4×C47#48012yes—
178C4×C47C2×#12311yescharacteristic
179C4×C47C2×#13711yescharacteristic
180Dic47—81no—
181Dic47—82no—
182Dic47—81no—
183#3061D4711yescharacteristic
184#2913—471no—
185#2237—472no—
186#2235—472no—
187#2234—472no—
188#2233—472no—
189#2159—471no—
190C2×#367—471no—
191#2139—471no—
192#2138—471no—
193#2113—471no—
194C2×C47—22no—
195C2×C47—42no—
196C2×C47—161no—
197C2×C47C2×#48011yescharacteristic
198C2×C47#341611yescharacteristic
199C2×C47#341611yescharacteristic
200C2×#135C2×D4711yescharacteristic
201C2×#131C2×D4711yescharacteristic
202#537—21no—
203#520—941no—
204C2×M32—471no—
205C22×C16—471no—
206#468—22no—
207#467C2×D4712yes—
208#466C2×D4712yes—
209#428—941no—
210#428—942no—
211#427—471no—
212#427—472no—
213C2×#107—471no—
214C2×#107—472no—
215C2×#107C2×D4711yescharacteristic
216#417—21no—
217#367—472no—
218#367—472no—
219#367—472no—
220#366—472no—
221#358—471no—
222#358—471no—
223#358—472no—
224#357—471no—
225#357—471no—
226#357—472no—
227C22⋊C16—472no—
228C22⋊C16—472no—
229C47#3365311yescharacteristic, complemented, Sylow
230C2×#41—21no—
231C2×#41C22⋊D4711yescharacteristic
232#135—22no—
233#135—22no—
234C2×Q16C22×D4711yescharacteristic
235C2×SD16—21no—
236C2×D8C22×D4711yescharacteristic
237#131—22no—
238#131—22no—
239C2×M16C22⋊D4711yescharacteristic
240C2×M16C22⋊D4712yes—
241C22×C8C22×D4712yes—
242C22×C8C22×D4711yescharacteristic
243C22⋊Q8—22no—
244C22⋊Q8—41no—
245#121—22no—
246#121—41no—
247#121—42no—
248C4×Q8—1881no—
249C4×D4—1882no—
250C4×D4—1881no—
251#117—942no—
252#117—941no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—942no—
255C2×C4⋊C4—941no—
256M32—942no—
257C2×C16—472no—
258C2×C16—471no—
259C2×C16—471no—
260C2×C16—942no—
261#107—21no—
262#107—471no—
263#107—471no—
264#107—472no—
265#107—942no—
266#107—941no—
267#107C22×D4712yes—
268#106—942no—
269#106—941no—
270#103—22no—
271#102—22no—
272#41—22no—
273#41—22no—
274#41—42no—
275#41—42no—
276C2×Q8—21no—
277C2×Q8#1062533411yescharacteristic
278C2×D4—21no—
279C2×D4—41no—
280C2×D4—42no—
281C2×D4#1062533412yes—
282C2×D4#1062533411yescharacteristic
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—1882no—
286C22×C4—1881no—
287C22×C4C2×C22⋊D4712yes—
288C22×C4C2×C22⋊D4711yescharacteristic
289Q16—21no—
290Q16—21no—
291SD16—42no—
292D8—21no—
293D8—21no—
294M16—22no—
295M16—24no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8#1062535212yes—
299C2×C8#1062535212yes—
300C2×C8C2×C22⋊D4712yes—
301C2×C8C2×C22⋊D4711yescharacteristic
302C2×C8C23×D4711yescharacteristic
303C4⋊C4—22no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—941no—
307C4⋊C4—942no—
308C4⋊C4—941no—
309C4⋊C4—1882no—
310C4⋊C4—1881no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—1882no—
314C22⋊C4—1881no—
315C42—1882no—
316C42—1881no—
317C16—941no—
318C16—941no—
319C16—942no—
320C23—41no—
321C23C2×#1062533412yes—
322C23C2×#1062533411yescharacteristic
323Q8—22no—
324Q8—41no—
325Q8—41no—
326D4—24no—
327D4—22no—
328D4—41no—
329D4—41no—
330D4—82no—
331D4—162no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—82no—
337C2×C4—161no—
338C2×C4—1881no—
339C2×C4—1882no—
340C2×C4—1881no—
341C2×C4—7521no—
342C2×C4—7522no—
343C2×C4#1309400512yes—
344C2×C4C2×#1062533412yes—
345C2×C4C2×#1062533411yescharacteristic
346C2×C4#1309405412yes—
347C2×C4C22×C22⋊D4711yescharacteristic
348C8—22no—
349C8#1309400512yes—
350C8C2×#1062535211yescharacteristic
351C8C2×#1062535211yescharacteristic
352C22—21no—
353C22—21no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22#1568943412yes—
361C22C22×#1062533411yescharacteristic
362C4—22no—
363C4—82no—
364C4—81no—
365C4—3762no—
366C4—3761no—
367C4—3761no—
368C4#1568943412yes—
369C4C2×#1309400511yescharacteristic
370C4C2×#1309405411yescharacteristic
371C2—22no—
372C2—42no—
373C2—161no—
374C2C2×#1568943411yescharacteristic
375C2#2025653211yescharacteristic
376C2#2025653511yescharacteristic
subgroupG/H
centerC22C22×#10625334
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
Fitting, prime radicalC47×#3061C2
socleC22×C47C22×D8
hypercenter#3061D47
hypocenterC47#33653
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ #3061

Complements

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

NH ≅ G/Ntype
1. #20251988C2semidirect
2. #20245593C2semidirect
3. #20245583C2semidirect
4. #20245576C2semidirect
5. #20245571C2semidirect
6. #20245174C2semidirect
7. C2×#15688950C2semidirect
8. #20245100C2semidirect
9. #20245095C2semidirect
10. #20245002C2semidirect
229. C47#33653semidirect

Maximal subgroups

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

subgroupindex
#202519882
#202455932
#202455832
#202455762
#202455712
#202451742
C2×#156889502
#202451002
#202450952
#202450022
C47×#30612
#3365347