Finite Group Databasebeta

← #26371153

Subgroups — #26371153

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#20253058C211yesmaximal, characteristic
2#20245596C212yesmaximal, complemented
3#20245585C212yesmaximal, complemented
4#20245579C212yesmaximal, complemented
5#20245573C212yesmaximal, complemented
6#20245177C211yesmaximal, characteristic, complemented
7C2×#15688950C211yesmaximal, characteristic, complemented
8#20245098C211yesmaximal, characteristic, complemented
9#20245093C211yesmaximal, characteristic, complemented
10#20245002C211yesmaximal, characteristic, complemented
11C47×#2913C211yesmaximal, characteristic, complemented
12C2×#13094046C2211yescharacteristic
13#15689701—21no—
14C2×#13093964C2211yescharacteristic
15C22×C47⋊C16C2211yescharacteristic
16#15689369—22no—
17#15689364C2212yes—
18#15689359C2212yes—
19C2×#13093958C2211yescharacteristic
20#15689125—21no—
21#15688950C2212yes—
22#15688950C2212yes—
23#15688950C2212yes—
24#15688948C2212yes—
25#15688926C2211yescharacteristic
26#15688926C2212yes—
27#15688926C2211yescharacteristic
28#15688923C2211yescharacteristic
29#15688923C2212yes—
30#15688923C2211yescharacteristic
31C22⋊(C47⋊C16)C2212yes—
32C22⋊(C47⋊C16)C2212yes—
33C2×#131×C47C2211yescharacteristic
34C47×#520—21no—
35C47×#428—22no—
36C47×#428—21no—
37C47×#427C2212yes—
38C47×#427C2211yescharacteristic
39C2×#107×C47C2212yes—
40C2×#107×C47C2211yescharacteristic
41C2×#10625350—21no—
42#13094046—22no—
43#13094046—22no—
44C2×Dic188C2311yescharacteristic
45C2×#10625325—21no—
46C2×D376C2311yescharacteristic
47#13094003—22no—
48#13094000—41no—
49#13093998—22no—
50#13093996—42no—
51#13093995—41no—
52C2×C4⋊Dic47—21no—
53#13093964—22no—
54C2×C47⋊C16—22no—
55C2×C47⋊C16C2311yescharacteristic
56C2×C47⋊C16C2311yescharacteristic
57C2×C47⋊C16C2312yes—
58#13093958—21no—
59#13093958C2312yes—
60#13093949—22no—
61#13093946—22no—
62C2×#41×C47D411yescharacteristic
63#131×C47—22no—
64#131×C47—22no—
65C2×M16×C47D412yes—
66C2×M16×C47D411yescharacteristic
67C22×C8×C47C2312yes—
68C22×C8×C47C2311yescharacteristic
69C4×Q8×C47—41no—
70C4×D4×C47—41no—
71C4×D4×C47—42no—
72#117×C47—22no—
73#117×C47—21no—
74C2×C4⋊C4×C47—22no—
75C2×C4⋊C4×C47—21no—
76#107×C47—22no—
77#107×C47—21no—
78#107×C47C2311yescharacteristic
79#107×C47C2312yes—
80#107×C47C2311yescharacteristic
81#106×C47—22no—
82#106×C47—21no—
83#41×C47—22no—
84#41×C47—22no—
85C2×Q8×C47D811yescharacteristic
86C2×D4×C47D811yescharacteristic
87C2×D4×C47D812yes—
88C22×C4×C47—42no—
89C22×C4×C47—41no—
90C22×C4×C47C2×D412yes—
91C22×C4×C47C2×D411yescharacteristic
92C2×C22⋊D47—41no—
93C2×C22⋊D47—42no—
94C22×Dic47—41no—
95#10625350—42no—
96#10625350—42no—
97C2×D188—21no—
98C2×C4×D47—41no—
99C2×Dic94—21no—
100M16×C47—22no—
101M16×C47—24no—
102C2×C8×C47—22no—
103C2×C8×C47—22no—
104C2×C8×C47C2×D412yes—
105C2×C8×C47C2×D411yescharacteristic
106C2×C8×C47#4112yes—
107C2×C8×C47#4112yes—
108C2×C8×C47C2411yescharacteristic
109C4⋊C4×C47—21no—
110C4⋊C4×C47—22no—
111C4⋊C4×C47—21no—
112C4⋊C4×C47—42no—
113C4⋊C4×C47—41no—
114C22⋊C4×C47—42no—
115C22⋊C4×C47—41no—
116C42×C47—42no—
117C42×C47—41no—
118C22⋊Dic47—42no—
119(C2×D47)⋊C4—42no—
120C4⋊Dic47—22no—
121C4⋊Dic47—41no—
122Dic47⋊C4—42no—
123Dic188—21no—
124Dic188—21no—
125D376—21no—
126D376—21no—
127#10625325—42no—
128C47⋊C16—21no—
129C47⋊C16—21no—
130C47⋊C16—22no—
131C23×C47C2×D812yes—
132C23×C47C2×D811yescharacteristic
133C22×D47—41no—
134Q8×C47—22no—
135D4×C47—24no—
136D4×C47—22no—
137C2×C4×C47—22no—
138C2×C4×C47—22no—
139C2×C4×C47—41no—
140C2×C4×C47—42no—
141C2×C4×C47—41no—
142C2×C4×C47—161no—
143C2×C4×C47—162no—
144C2×C4×C47#12312yes—
145C2×C4×C47C2×D812yes—
146C2×C4×C47C2×D811yescharacteristic
147C2×C4×C47#13712yes—
148C2×C4×C47C22×D411yescharacteristic
149C22⋊D47—82no—
150C22⋊D47—162no—
151C2×Dic47—41no—
152C2×Dic47—42no—
153C2×Dic47—161no—
154D188—41no—
155D188—41no—
156C4×D47—82no—
157Dic94—41no—
158Dic94—41no—
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#48012yes—
170C22×C47C22×D811yescharacteristic
171C2×D47—81no—
172C2×D47—81no—
173C2×D47—81no—
174C4×C47—22no—
175C4×C47—81no—
176C4×C47—82no—
177C4×C47—81no—
178C4×C47#48012yes—
179C4×C47C2×#12311yescharacteristic
180C4×C47C2×#13711yescharacteristic
181Dic47—81no—
182Dic47—82no—
183#3061—471no—
184#2913D4711yescharacteristic, complemented
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×C47C2×#48011yescharacteristic
197C2×C47#341611yescharacteristic
198C2×C47#341611yescharacteristic
199D47—161no—
200C2×#135—471no—
201C2×#131C2×D4711yescharacteristic
202#537—941no—
203#520—21no—
204C2×M32—471no—
205C22×C16—471no—
206#468—942no—
207#467—472no—
208#466—472no—
209#428—21no—
210#428—22no—
211#427C2×D4711yescharacteristic
212#427C2×D4712yes—
213C2×#107—471no—
214C2×#107C2×D4711yescharacteristic
215C2×#107C2×D4712yes—
216#417—941no—
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—941no—
231C2×#41C22⋊D4711yescharacteristic
232#135—942no—
233#135—942no—
234C2×Q16—471no—
235C2×SD16—941no—
236C2×D8—471no—
237#131—22no—
238#131—22no—
239C2×M16C22⋊D4711yescharacteristic
240C2×M16C22⋊D4712yes—
241C22×C8C22×D4712yes—
242C22×C8C22×D4711yescharacteristic
243C22⋊Q8—942no—
244C22⋊Q8—1881no—
245#121—942no—
246#121—1881no—
247#121—1882no—
248C4×Q8—41no—
249C4×D4—42no—
250C4×D4—41no—
251#117—22no—
252#117—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—941no—
256M32—942no—
257C2×C16—472no—
258C2×C16—471no—
259C2×C16—471no—
260C2×C16—942no—
261#107—22no—
262#107—21no—
263#107—472no—
264#107—941no—
265#107C22×D4711yescharacteristic
266#107C22×D4711yescharacteristic
267#107C22×D4712yes—
268#106—22no—
269#106—21no—
270#103—942no—
271#102—942no—
272#41—22no—
273#41—22no—
274#41—1882no—
275#41—1882no—
276C2×Q8—941no—
277C2×Q8#1062533411yescharacteristic
278C2×D4—941no—
279C2×D4—1881no—
280C2×D4—1882no—
281C2×D4#1062533412yes—
282C2×D4#1062533411yescharacteristic
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—1881no—
286C22×C4—1881no—
287C22×C4C2×C22⋊D4712yes—
288C22×C4C2×C22⋊D4711yescharacteristic
289Q16—941no—
290Q16—941no—
291SD16—1882no—
292D8—941no—
293D8—941no—
294M16—22no—
295M16—24no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8#1062535012yes—
299C2×C8#1062535012yes—
300C2×C8C2×C22⋊D4712yes—
301C2×C8C2×C22⋊D4711yescharacteristic
302C2×C8C23×D4711yescharacteristic
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—942no—
309C4⋊C4—1882no—
310C4⋊C4—1881no—
311C22⋊C4—42no—
312C22⋊C4—41no—
313C22⋊C4—1882no—
314C22⋊C4—1882no—
315C42—42no—
316C42—41no—
317C16—941no—
318C16—941no—
319C16—942no—
320C23—1881no—
321C23C2×#1062533412yes—
322C23C2×#1062533411yescharacteristic
323Q8—22no—
324Q8—1881no—
325Q8—1881no—
326D4—24no—
327D4—22no—
328D4—3762no—
329D4—1881no—
330D4—1881no—
331D4—7522no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—161no—
338C2×C4—162no—
339C2×C4—3762no—
340C2×C4—1882no—
341C2×C4—1881no—
342C2×C4—7521no—
343C2×C4#1309400712yes—
344C2×C4C2×#1062533412yes—
345C2×C4C2×#1062533411yescharacteristic
346C2×C4#1309405512yes—
347C2×C4C22×C22⋊D4711yescharacteristic
348C8—22no—
349C8#1309400712yes—
350C8C2×#1062535011yescharacteristic
351C8C2×#1062535011yescharacteristic
352C22—21no—
353C22—21no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—3761no—
358C22—3761no—
359C22—3761no—
360C22#1568943612yes—
361C22C22×#1062533411yescharacteristic
362C4—22no—
363C4—81no—
364C4—82no—
365C4—81no—
366C4—3762no—
367C4—3761no—
368C4#1568943612yes—
369C4C2×#1309400711yescharacteristic
370C4C2×#1309405511yescharacteristic
371C2—22no—
372C2—42no—
373C2—7521no—
374C2C2×#1568943611yescharacteristic
375C2#2025653211yescharacteristic
376C2#2025653511yescharacteristic
subgroupG/H
centerC22C22×#10625334
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC47×#2913C2
socleC22×C47C22×D8
prime radical#20253058C2
hypercenter#2913D47
hypocenterC47#33653
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ #2913

Complements

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

NH ≅ G/Ntype
2. #20245596C2semidirect
3. #20245585C2semidirect
4. #20245579C2semidirect
5. #20245573C2semidirect
6. #20245177C2semidirect
7. C2×#15688950C2semidirect
8. #20245098C2semidirect
9. #20245093C2semidirect
10. #20245002C2semidirect
11. C47×#2913C2semidirect
184. #2913D47semidirect
229. C47#33653semidirect

Maximal subgroups

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

subgroupindex
#202530582
#202455962
#202455852
#202455792
#202455732
#202451772
C2×#156889502
#202450982
#202450932
#202450022
C47×#29132
#3365347