Finite Group Databasebeta

← #26126509

Subgroups — #26126509

Order 12032.

504 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252969C211yesmaximal, characteristic
2#20244999C211yesmaximal, characteristic, complemented
3#20241399C212yesmaximal, complemented
4#20241396C212yesmaximal, complemented
5C2×C47×#340C211yesmaximal, characteristic, complemented
6C2×#13094046C411yescharacteristic, complemented
7#15689634—21no—
8C2×#13093964—21no—
9#15689364C411yescharacteristic, complemented
10#15689364C2211yescharacteristic
11#15689359C411yescharacteristic, complemented
12#15689359C2211yescharacteristic
13#15689352—21no—
14#15689352—21no—
15#15689161C411yescharacteristic, complemented
16#15689120—21no—
17C22⋊(C47⋊C16)—21no—
18C22⋊(C47⋊C16)C2212yes—
19C23×C8×C47C2211yescharacteristic
20C22×C4⋊C4×C47—21no—
21C47×#340—21no—
22C47×#340C2212yes—
23C2×#10625350—21no—
24#13094046—42no—
25C2×Dic188C2×C411yescharacteristic, complemented
26C2×#10625325—21no—
27C2×D376C2×C411yescharacteristic, complemented
28#13094007—41no—
29#13094003—21no—
30#13094003—21no—
31#13093998—21no—
32#13093998—21no—
33#13093993—41no—
34C4×C22⋊D47—41no—
35C4×C22⋊D47—41no—
36#13093983—21no—
37#13093964—42no—
38C2×C47⋊C16—22no—
39#13093958C2×C411yescharacteristic, complemented
40#13093958C2×C411yescharacteristic, complemented
41#13093956—21no—
42#13093949—21no—
43#13093949—21no—
44#13093946—21no—
45#13093946—21no—
46C23×C4×C47—41no—
47C23×C4×C47D411yescharacteristic
48C22×C8×C47—42no—
49C22×C8×C47C2×C411yescharacteristic
50C22×C8×C47C2×C411yescharacteristic
51C22×C8×C47D411yescharacteristic
52C22×C8×C47D411yescharacteristic
53C22×C8×C47D411yescharacteristic
54C22×C8×C47C2311yescharacteristic
55C2×C4⋊C4×C47—24no—
56C2×C4⋊C4×C47—22no—
57C2×C4⋊C4×C47—42no—
58C2×C4⋊C4×C47—41no—
59C24×C47D811yescharacteristic
60C22×C4×C47—22no—
61C22×C4×C47—22no—
62C22×C4×C47—44no—
63C22×C4×C47—42no—
64C22×C4×C47—82no—
65C22×C4×C47—82no—
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47D811yescharacteristic
69C22×C4×C47SD1611yescharacteristic
70C22×C4×C47SD1611yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C2×C22⋊D47—41no—
73C2×C22⋊D47—41no—
74C2×C22⋊D47—41no—
75#10625350—42no—
76#10625350—42no—
77C2×D188—21no—
78C2×C4×D47—41no—
79C2×Dic94—21no—
80C2×C8×C47—22no—
81C2×C8×C47—22no—
82C2×C8×C47—41no—
83C2×C8×C47—41no—
84C2×C8×C47—41no—
85C2×C8×C47—41no—
86C2×C8×C47C22⋊C411yescharacteristic
87C2×C8×C47C22⋊C411yescharacteristic
88C2×C8×C47C22×C411yescharacteristic
89C2×C8×C47C2×D411yescharacteristic
90C4⋊C4×C47—24no—
91C4⋊C4×C47—44no—
92C4⋊C4×C47—42no—
93C22⋊Dic47—41no—
94C22⋊Dic47—41no—
95C22⋊Dic47—41no—
96(C2×D47)⋊C4—41no—
97(C2×D47)⋊C4—41no—
98C4⋊Dic47—21no—
99C4⋊Dic47—21no—
100Dic47⋊C4—41no—
101Dic47⋊C4—41no—
102C4×Dic47—41no—
103Dic188—41no—
104D376—41no—
105#10625325—42no—
106C47⋊C16—42no—
107C23×C47—22no—
108C23×C47—22no—
109C23×C47—21no—
110C23×C47—21no—
111C23×C47#10211yescharacteristic
112C23×C47#10211yescharacteristic
113C23×C47C2×D811yescharacteristic
114C22×D47—41no—
115C2×C4×C47—22no—
116C2×C4×C47—22no—
117C2×C4×C47—22no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—44no—
123C2×C4×C47—88no—
124C2×C4×C47—82no—
125C2×C4×C47—82no—
126C2×C4×C47#9912yes—
127C2×C4×C47#10211yescharacteristic
128C2×C4×C47#10211yescharacteristic
129C2×C4×C47#10412yes—
130C2×C4×C47C2×C22⋊C411yescharacteristic
131C2×C4×C47C2×SD1611yescharacteristic
132C22⋊D47—82no—
133C22⋊D47—82no—
134C22⋊D47—81no—
135C22⋊D47—81no—
136C2×Dic47—41no—
137C2×Dic47—41no—
138C2×Dic47—41no—
139D188—41no—
140D188—41no—
141C4×D47—82no—
142Dic94—41no—
143Dic94—41no—
144C8×C47—21no—
145C8×C47—21no—
146C8×C47—21no—
147C8×C47—21no—
148#11704—471nomaximal, Sylow
149C22×C47—24no—
150C22×C47—24no—
151C22×C47—21no—
152C22×C47—21no—
153C22×C47—21no—
154C22×C47—21no—
155C22×C47—21no—
156C22×C47—21no—
157C22×C47#32712yes—
158C22×C47#36012yes—
159C22×C47C2×#10211yescharacteristic
160C22×C47#50911yescharacteristic
161C22×C47#51011yescharacteristic
162C2×D47—81no—
163C2×D47—81no—
164C2×D47—81no—
165C4×C47—21no—
166C4×C47—21no—
167C4×C47—84no—
168C4×C47#32712yes—
169C4×C47C2×#9911yescharacteristic
170C4×C47C2×#10411yescharacteristic
171Dic47—81no—
172Dic47—81no—
173Dic47—81no—
174#3050—471no—
175#2112—471no—
176C2×#340D4711yescharacteristic, complemented
177#1348—472no—
178#1347—472no—
179C2×C47—21no—
180C2×C47—22no—
181C2×C47—22no—
182C2×C47—21no—
183C2×C47C2×#32711yescharacteristic
184C2×C47#213911yescharacteristic
185C2×C47C2×#36011yescharacteristic
186D47—161no—
187C2×#135—471no—
188C23×C8C2×D4711yescharacteristic
189#525—941no—
190C22×C4⋊C4—21no—
191C2×M32—941no—
192#467—471no—
193#467—471no—
194#466—471no—
195#466—471no—
196#465—941no—
197#465—941no—
198#427—471no—
199#416—941no—
200C22⋊C16—472no—
201C22⋊C16—941no—
202#340—21no—
203#340C2×D4712yes—
204C47#1170411yescharacteristic, complemented, Sylow
205C2×#41—941no—
206C23×C4—41no—
207C23×C4C22⋊D4711yescharacteristic
208#135—1882no—
209C2×Q16—471no—
210C2×SD16—941no—
211C2×D8—471no—
212C22×C8—42no—
213C22×C8C4×D4711yescharacteristic
214C22×C8C4×D4711yescharacteristic
215C22×C8D18811yescharacteristic
216C22×C8D18811yescharacteristic
217C22×C8C22⋊D4711yescharacteristic
218C22×C8C22×D4711yescharacteristic
219#123—1881no—
220C22⋊Q8—941no—
221C22⋊Q8—941no—
222#121—941no—
223#121—941no—
224#120—1881no—
225C4×D4—1881no—
226C4×D4—1881no—
227#117—941no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—41no—
232M32—1882no—
233C2×C16—942no—
234#107—471no—
235#107—471no—
236#106—941no—
237#103—941no—
238#103—941no—
239#102—941no—
240#102—941no—
241C24#1062533411yescharacteristic
242#41—1882no—
243#41—1882no—
244C2×Q8—941no—
245C2×D4—941no—
246C2×D4—1881no—
247C2×D4—1881no—
248C2×D4—1881no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—44no—
252C22×C4—42no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4—1881no—
256C22×C4(C2×D47)⋊C411yescharacteristic
257C22×C4(C2×D47)⋊C411yescharacteristic
258C22×C4#1062533411yescharacteristic
259C22×C4#1062533611yescharacteristic
260C22×C4#1062533611yescharacteristic
261C22×C4C2×C22⋊D4711yescharacteristic
262Q16—1881no—
263SD16—1882no—
264D8—1881no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8(C2×D47)⋊C411yescharacteristic
272C2×C8(C2×D47)⋊C411yescharacteristic
273C2×C8C2×C4×D4711yescharacteristic
274C2×C8C2×D18811yescharacteristic
275C4⋊C4—24no—
276C4⋊C4—42no—
277C4⋊C4—44no—
278C4⋊C4—941no—
279C4⋊C4—941no—
280C4⋊C4—1881no—
281C4⋊C4—1881no—
282C22⋊C4—1881no—
283C22⋊C4—1881no—
284C22⋊C4—1881no—
285C22⋊C4—1881no—
286C22⋊C4—1881no—
287C42—1881no—
288C16—1882no—
289C23—21no—
290C23—21no—
291C23—22no—
292C23—22no—
293C23—1881no—
294C23#1309394411yescharacteristic
295C23#1309394411yescharacteristic
296C23C2×#1062533411yescharacteristic
297Q8—1881no—
298Q8—1881no—
299D4—3761no—
300D4—3761no—
301D4—3762no—
302D4—3762no—
303D4—1881no—
304D4—1881no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—44no—
313C2×C4—88no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—3762no—
317C2×C4—1881no—
318C2×C4—1881no—
319C2×C4—1881no—
320C2×C4#1309393812yes—
321C2×C4#1309394411yescharacteristic
322C2×C4#1309394411yescharacteristic
323C2×C4#1309395212yes—
324C2×C4C2×(C2×D47)⋊C411yescharacteristic
325C2×C4C2×#1062533611yescharacteristic
326C8—21no—
327C8—21no—
328C8—21no—
329C8—21no—
330C22—24no—
331C22—24no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—3761no—
339C22—3761no—
340C22—3761no—
341C22#1568885612yes—
342C22#1568893212yes—
343C22C2×#1309394411yescharacteristic
344C22#1568955911yescharacteristic
345C22#1568956411yescharacteristic
346C4—21no—
347C4—21no—
348C4—84no—
349C4—3761no—
350C4—3761no—
351C4—3761no—
352C4#1568885612yes—
353C4C2×#1309393811yescharacteristic
354C4C2×#1309395211yescharacteristic
355C2—21no—
356C2—21no—
357C2—22no—
358C2—22no—
359C2—7521no—
360C2C2×#1568885611yescharacteristic
361C2#2024509811yescharacteristic
362C2C2×#1568893211yescharacteristic
subgroupG/H
centerC22C2×#13093944
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC2×C47×#340C2
socleC22×C47C2×#102
prime radical#20252969C2
cosocleC22×C8×C47C23
hypercenterC2×#340D47
hypocenterC47#11704
LCSG ▷ C2×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
2. #20244999C2semidirect
3. #20241399C2semidirect
4. #20241396C2semidirect
5. C2×C47×#340C2semidirect
6. C2×#13094046C4semidirect
9. #15689364C4semidirect
11. #15689359C4semidirect
15. #15689161C4semidirect
25. C2×Dic188C2×C4semidirect
27. C2×D376C2×C4semidirect
39. #13093958C2×C4semidirect
40. #13093958C2×C4semidirect
176. C2×#340D47semidirect
204. C47#11704semidirect

Maximal subgroups

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

subgroupindex
#202529692
#202449992
#202413992
#202413962
C2×C47×#3402
#1170447