Finite Group Databasebeta

← #26371610

Subgroups — #26371610

Order 12032.

602 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253927C211yesmaximal, characteristic, complemented
2#20245629C212yesmaximal, complemented
3#20245624C212yesmaximal, complemented
4#20245619C212yesmaximal, complemented
5#20245611C212yesmaximal, complemented
6#20245104C211yesmaximal, characteristic, complemented
7#20245100C211yesmaximal, characteristic, complemented
8#20245095C211yesmaximal, characteristic, complemented
9C2×#15688924C211yesmaximal, characteristic, complemented
10#20244985C211yesmaximal, characteristic, complemented
11C47×#3155C211yesmaximal, characteristic
12#15689751—21no—
13C2×#13093964C2211yescharacteristic
14C22×C47⋊C16C2211yescharacteristic
15#15689532—22no—
16#15689528C2212yescomplemented
17#15689525C2212yescomplemented
18#15689165—21no—
19#15689161C2211yescharacteristic
20C2×#13093958C2211yescharacteristic
21#15688927C2211yescharacteristic, complemented
22#15688927C2211yescharacteristic, complemented
23#15688927C2212yescomplemented
24#15688924C2212yes—
25#15688924C2212yes—
26#15688924C2212yes—
27#15688924C2212yes—
28#15688924C2211yescharacteristic, complemented
29#15688924C2211yescharacteristic, complemented
30#15688924C2212yescomplemented
31#15688899C2212yescomplemented
32C4×C47⋊C16C2212yescomplemented
33C2×AGL(1,ℤ/8)×C47—21no—
34C2×#135×C47C2211yescharacteristic
35C22×D8×C47C2211yescharacteristic
36C47×#535—21no—
37C47×#496—22no—
38C47×#495C2212yes—
39C47×#493C2212yes—
40C47×#405C2211yescharacteristic
41#13094024—22no—
42#13094023—41no—
43#13094016—22no—
44#13094015—41no—
45#13094001—42no—
46#13093983—21no—
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—
61C22×D4×C47—21no—
62AGL(1,ℤ/8)×C47—44no—
63#135×C47—22no—
64#135×C47—22no—
65C2×Q16×C47C2311yescharacteristic
66C2×SD16×C47—22no—
67C2×SD16×C47—21no—
68C2×D8×C47—22no—
69C2×D8×C47—21no—
70C2×D8×C47—22no—
71C2×D8×C47—22no—
72C2×D8×C47C2311yescharacteristic
73C2×D8×C47C2312yes—
74C2×M16×C47D411yescharacteristic
75C22×C8×C47C2311yescharacteristic
76#127×C47—22no—
77#124×C47—22no—
78#121×C47—41no—
79#121×C47—41no—
80#120×C47—42no—
81#117×C47D411yescharacteristic
82#97×C47D412yes—
83C4×C8×C47C2312yes—
84C24×C47—41no—
85#41×C47—42no—
86#41×C47—42no—
87C2×Q8×C47—21no—
88C2×D4×C47—22no—
89C2×D4×C47—21no—
90C2×D4×C47—41no—
91C2×D4×C47—42no—
92C2×D4×C47—41no—
93C2×D4×C47—42no—
94C2×D4×C47—42no—
95C2×D4×C47—42no—
96C22×C4×C47—41no—
97C22×C4×C47C2×D411yescharacteristic
98C22×Dic47—41no—
99C2×Dic94—42no—
100Q16×C47—21no—
101Q16×C47—21no—
102SD16×C47—42no—
103SD16×C47—44no—
104D8×C47—22no—
105D8×C47—22no—
106D8×C47—21no—
107D8×C47—21no—
108D8×C47—42no—
109D8×C47—44no—
110M16×C47SD1614yes—
111C2×C8×C47C2×D411yescharacteristic
112C2×C8×C47C2×D412yes—
113C2×C8×C47#4112yes—
114C2×C8×C47#4112yes—
115C2×C8×C47C2411yescharacteristic
116C4⋊C4×C47D811yescharacteristic
117C4⋊C4×C47D811yescharacteristic
118C22⋊C4×C47—42no—
119C22⋊C4×C47—42no—
120C22⋊C4×C47D812yes—
121C42×C47C2×D412yes—
122C22⋊Dic47—41no—
123C4⋊Dic47—21no—
124C4⋊Dic47—22no—
125C4⋊Dic47—21no—
126C4⋊Dic47—41no—
127C4⋊Dic47—42no—
128Dic47⋊C4—42no—
129Dic47⋊C4—42no—
130C4×Dic47—41no—
131C47⋊C16—21no—
132C47⋊C16—21no—
133C47⋊C16—22no—
134C23×C47—42no—
135C23×C47—41no—
136C23×C47—82no—
137C23×C47—161no—
138C23×C47—161no—
139C23×C47C2×D811yescharacteristic
140Q8×C47—41no—
141Q8×C47—41no—
142D4×C47—42no—
143D4×C47—41no—
144D4×C47—41no—
145D4×C47—42no—
146D4×C47—82no—
147D4×C47—82no—
148D4×C47—162no—
149D4×C47—162no—
150C2×C4×C47—41no—
151C2×C4×C47—82no—
152C2×C4×C47#12412yes—
153C2×C4×C47C2×D812yes—
154C2×C4×C47C2×D812yes—
155C2×C4×C47C2×D811yescharacteristic
156C2×C4×C47C2×SD1612yes—
157C2×C4×C47C22×D411yescharacteristic
158C2×Dic47—41no—
159C2×Dic47—41no—
160C2×Dic47—42no—
161C2×Dic47—161no—
162Dic94—162no—
163C8×C47#12412yes—
164C8×C47C2×SD1614yes—
165C8×C47C2×#4111yescharacteristic
166C8×C47C2×#4111yescharacteristic
167#33702—471nomaximal, Sylow
168C22×C47—21no—
169C22×C47—21no—
170C22×C47—81no—
171C22×C47—82no—
172C22×C47—81no—
173C22×C47—82no—
174C22×C47—81no—
175C22×C47—82no—
176C22×C47—164no—
177C22×C47#48612yes—
178C22×C47C22×D811yescharacteristic
179C4×C47—22no—
180C4×C47—22no—
181C4×C47—81no—
182C4×C47#48612yes—
183C4×C47C2×#12411yescharacteristic
184C4×C47C22×SD1611yescharacteristic
185Dic47—81no—
186Dic47—82no—
187Dic47—81no—
188#3164—471no—
189#3155D4711yescharacteristic
190#2244—472no—
191#2243—472no—
192#2242—472no—
193#2240—472no—
194#2140—471no—
195#2139—471no—
196#2138—471no—
197C2×#357—471no—
198#2108—471no—
199C2×C47—22no—
200C2×C47—161no—
201C2×C47—162no—
202C2×C47C2×#48611yescharacteristic
203C2×C47#341511yescharacteristic
204C2×C47#341611yescharacteristic
205C2×AGL(1,ℤ/8)—21no—
206C2×#135C2×D4711yescharacteristic
207C22×D8C2×D4711yescharacteristic
208#544—941no—
209#535—21no—
210C2×M32—471no—
211C22×C16—471no—
212#501—942no—
213#500—472no—
214#499—472no—
215#496—22no—
216#495C2×D4712yes—
217#493C2×D4712yes—
218#428—941no—
219#427—471no—
220C2×#107—471no—
221#405C2×D4711yescharacteristic
222#358—471no—
223#358—471no—
224#358—472no—
225#357—472no—
226#357—472no—
227#357—471no—
228#357—471no—
229#357—472no—
230#357—472no—
231#357—472no—
232#346—472no—
233C4×C16—472no—
234C47#3370211yescharacteristic, complemented, Sylow
235C2×#41—21no—
236C22×D4—21no—
237AGL(1,ℤ/8)—44no—
238#135—22no—
239#135—22no—
240C2×Q16C22×D4711yescharacteristic
241C2×SD16—21no—
242C2×SD16—22no—
243C2×D8—22no—
244C2×D8—22no—
245C2×D8—21no—
246C2×D8—22no—
247C2×D8C22×D4712yes—
248C2×D8C22×D4711yescharacteristic
249C2×M16C22⋊D4711yescharacteristic
250C22×C8C22×D4711yescharacteristic
251C4⋊Q8—942no—
252C4⋊Q8—1881no—
253#127—22no—
254#125—942no—
255#125—1881no—
256#124—22no—
257C22⋊Q8—1882no—
258#121—41no—
259#121—41no—
260#120—42no—
261#117—941no—
262#117C22⋊D4711yescharacteristic
263C2×C4⋊C4—941no—
264M32—942no—
265C2×C16—471no—
266C2×C16—472no—
267C2×C16—471no—
268C2×C16—942no—
269#107—472no—
270#107—471no—
271#107—471no—
272#107—941no—
273#107—942no—
274#106—941no—
275#106—942no—
276#97C22⋊D4712yes—
277C4×C8C22×D4712yes—
278C24—41no—
279#41—42no—
280#41—42no—
281C2×Q8—21no—
282C2×Q8—1882no—
283C2×D4—22no—
284C2×D4—21no—
285C2×D4—41no—
286C2×D4—42no—
287C2×D4—41no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C22×C4—41no—
292C22×C4—1881no—
293C22×C4C2×C22⋊D4711yescharacteristic
294Q16—21no—
295Q16—21no—
296SD16—42no—
297SD16—44no—
298D8—22no—
299D8—22no—
300D8—21no—
301D8—21no—
302D8—42no—
303D8—44no—
304M16#1062533514yes—
305C2×C8#1062535212yes—
306C2×C8#1062535212yes—
307C2×C8C2×C22⋊D4711yescharacteristic
308C2×C8C2×C22⋊D4712yes—
309C2×C8C23×D4711yescharacteristic
310C4⋊C4—941no—
311C4⋊C4—942no—
312C4⋊C4—941no—
313C4⋊C4—1882no—
314C4⋊C4—1882no—
315C4⋊C4—1881no—
316C4⋊C4—1882no—
317C4⋊C4#1062533411yescharacteristic
318C4⋊C4#1062533411yescharacteristic
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—1881no—
322C22⋊C4#1062533412yes—
323C42—1881no—
324C42C2×C22⋊D4712yes—
325C16—941no—
326C16—941no—
327C16—942no—
328C23—42no—
329C23—41no—
330C23—82no—
331C23—161no—
332C23—161no—
333C23C2×#1062533411yescharacteristic
334Q8—41no—
335Q8—41no—
336Q8—7522no—
337D4—42no—
338D4—41no—
339D4—41no—
340D4—42no—
341D4—82no—
342D4—82no—
343D4—162no—
344D4—162no—
345C2×C4—41no—
346C2×C4—82no—
347C2×C4—1881no—
348C2×C4—1881no—
349C2×C4—1882no—
350C2×C4—7521no—
351C2×C4#1309401112yes—
352C2×C4C2×#1062533412yes—
353C2×C4C2×#1062533412yes—
354C2×C4C2×#1062533411yescharacteristic
355C2×C4C2×#1062533512yes—
356C2×C4C22×C22⋊D4711yescharacteristic
357C8#1309401112yes—
358C8C2×#1062533514yes—
359C8C2×#1062535211yescharacteristic
360C8C2×#1062535211yescharacteristic
361C22—21no—
362C22—21no—
363C22—81no—
364C22—82no—
365C22—81no—
366C22—82no—
367C22—81no—
368C22—82no—
369C22—164no—
370C22#1568946912yes—
371C22C22×#1062533411yescharacteristic
372C4—22no—
373C4—22no—
374C4—81no—
375C4—3762no—
376C4—3761no—
377C4—3761no—
378C4#1568946912yes—
379C4C2×#1309401111yescharacteristic
380C4C22×#1062533511yescharacteristic
381C2—22no—
382C2—162no—
383C2—161no—
384C2C2×#1568946911yescharacteristic
385C2#2025652711yescharacteristic
386C2#2025653511yescharacteristic
subgroupG/H
centerC22C22×#10625334
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
Fitting, prime radicalC47×#3155C2
socleC22×C47C22×D8
hypercenter#3155D47
hypocenterC47#33702
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3155

Complements

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

NH ≅ G/Ntype
1. #20253927C2semidirect
2. #20245629C2semidirect
3. #20245624C2semidirect
4. #20245619C2semidirect
5. #20245611C2semidirect
6. #20245104C2semidirect
7. #20245100C2semidirect
8. #20245095C2semidirect
9. C2×#15688924C2semidirect
10. #20244985C2semidirect
16. #15689528C22semidirect
17. #15689525C22semidirect
21. #15688927C22semidirect
22. #15688927C22semidirect
23. #15688927C22semidirect
28. #15688924C22semidirect
29. #15688924C22semidirect
30. #15688924C22semidirect
31. #15688899C22semidirect
32. C4×C47⋊C16C22semidirect
234. C47#33702semidirect

Maximal subgroups

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

subgroupindex
#202539272
#202456292
#202456242
#202456192
#202456112
#202451042
#202451002
#202450952
C2×#156889242
#202449852
C47×#31552
#3370247