Finite Group Databasebeta

← #26371615

Subgroups — #26371615

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#20253874C211yesmaximal, characteristic
2#20245631C212yesmaximal, complemented
3#20245627C212yesmaximal, complemented
4#20245622C212yesmaximal, complemented
5#20245613C212yesmaximal, complemented
6#20245103C211yesmaximal, characteristic, complemented
7#20245098C211yesmaximal, characteristic, complemented
8#20245093C211yesmaximal, characteristic, complemented
9C2×#15688923C211yesmaximal, characteristic, complemented
10#20244985C211yesmaximal, characteristic, complemented
11C47×#3164C211yesmaximal, characteristic, complemented
12C2×#13094052—21no—
13C2×#13094046C2211yescharacteristic
14C22×D376C2211yescharacteristic
15#15689692—21no—
16C2×#13093964C2211yescharacteristic
17C22×C47⋊C16C2211yescharacteristic
18#15689512—22no—
19#15689507C2212yes—
20#15689500C2212yes—
21#15688926C2211yescharacteristic, complemented
22#15688926C2211yescharacteristic, complemented
23#15688926C2212yescomplemented
24#15688923C2212yes—
25#15688923C2212yes—
26#15688923C2212yes—
27#15688923C2212yes—
28#15688923C2211yescharacteristic, complemented
29#15688923C2211yescharacteristic, complemented
30#15688923C2212yescomplemented
31#15688899C2212yescomplemented
32C4×C47⋊C16C2212yescomplemented
33C47×#544—21no—
34C47×#501—22no—
35C47×#500C2212yescomplemented
36C47×#499C2212yescomplemented
37C47×#428—21no—
38C47×#427C2211yescharacteristic
39C2×#107×C47C2211yescharacteristic
40C47×#405C2211yescharacteristic
41C2×#10625350—21no—
42C22×D188—21no—
43#13094052—44no—
44#13094046—22no—
45#13094046—22no—
46C2×Dic188C2311yescharacteristic
47C2×#10625325—22no—
48C2×#10625325—21no—
49C2×D376—22no—
50C2×D376—22no—
51C2×D376—22no—
52C2×D376—21no—
53C2×D376C2311yescharacteristic
54C2×D376C2312yes—
55#13094021—22no—
56#13094012—22no—
57#13093999—41no—
58#13093999—41no—
59#13093994—42no—
60#13093964—22no—
61C2×C47⋊C16—22no—
62C2×C47⋊C16C2311yescharacteristic
63C2×C47⋊C16C2311yescharacteristic
64C2×C47⋊C16C2312yes—
65C2×M16×C47D411yescharacteristic
66C22×C8×C47C2311yescharacteristic
67C4⋊Q8×C47—22no—
68C4⋊Q8×C47—41no—
69#125×C47—22no—
70#125×C47—41no—
71C22⋊Q8×C47—42no—
72#117×C47—21no—
73#117×C47D411yescharacteristic
74C2×C4⋊C4×C47—21no—
75#107×C47—22no—
76#107×C47—21no—
77#107×C47C2311yescharacteristic
78#107×C47C2312yes—
79#107×C47C2311yescharacteristic
80#106×C47—22no—
81#106×C47—21no—
82#97×C47D412yes—
83C4×C8×C47C2312yes—
84C23×D47—41no—
85C2×Q8×C47—42no—
86C22×C4×C47—41no—
87C22×C4×C47C2×D411yescharacteristic
88C2×C22⋊D47—41no—
89#10625350—42no—
90#10625350—42no—
91C2×D188—22no—
92C2×D188—21no—
93C2×D188—42no—
94C2×D188—42no—
95C2×D188—41no—
96C2×D188—42no—
97C2×D188—42no—
98C2×C4×D47—41no—
99C2×Dic94—21no—
100M16×C47SD1614yes—
101C2×C8×C47C2×D411yescharacteristic
102C2×C8×C47C2×D412yes—
103C2×C8×C47#4112yes—
104C2×C8×C47#4112yes—
105C2×C8×C47C2411yescharacteristic
106C4⋊C4×C47—21no—
107C4⋊C4×C47—22no—
108C4⋊C4×C47—21no—
109C4⋊C4×C47—41no—
110C4⋊C4×C47—42no—
111C4⋊C4×C47—42no—
112C4⋊C4×C47—42no—
113C4⋊C4×C47D811yescharacteristic
114C4⋊C4×C47D811yescharacteristic
115C22⋊C4×C47—41no—
116C22⋊C4×C47D812yes—
117C42×C47—41no—
118C42×C47C2×D412yes—
119(C2×D47)⋊C4—42no—
120(C2×D47)⋊C4—42no—
121Dic188—21no—
122Dic188—21no—
123D376—22no—
124D376—22no—
125D376—21no—
126D376—21no—
127D376—42no—
128D376—44no—
129#10625325—42no—
130#10625325—44no—
131C47⋊C16—21no—
132C47⋊C16—21no—
133C47⋊C16—22no—
134C23×C47C2×D811yescharacteristic
135C22×D47—42no—
136C22×D47—41no—
137C22×D47—82no—
138C22×D47—161no—
139C22×D47—161no—
140Q8×C47—162no—
141C2×C4×C47—41no—
142C2×C4×C47—41no—
143C2×C4×C47—42no—
144C2×C4×C47—161no—
145C2×C4×C47#12412yes—
146C2×C4×C47C2×D812yes—
147C2×C4×C47C2×D812yes—
148C2×C4×C47C2×D811yescharacteristic
149C2×C4×C47C2×SD1612yes—
150C2×C4×C47C22×D411yescharacteristic
151C22⋊D47—82no—
152C2×Dic47—41no—
153D188—41no—
154D188—42no—
155D188—42no—
156D188—41no—
157D188—82no—
158D188—162no—
159D188—162no—
160C4×D47—82no—
161Dic94—41no—
162Dic94—41no—
163C8×C47#12412yes—
164C8×C47C2×SD1614yes—
165C8×C47C2×#4111yescharacteristic
166C8×C47C2×#4111yescharacteristic
167#33702—471nomaximal, Sylow
168C22×C47—21no—
169C22×C47—21no—
170C22×C47#48612yes—
171C22×C47C22×D811yescharacteristic
172C2×D47—81no—
173C2×D47—82no—
174C2×D47—81no—
175C2×D47—82no—
176C2×D47—81no—
177C2×D47—82no—
178C2×D47—164no—
179C4×C47—22no—
180C4×C47—22no—
181C4×C47—82no—
182C4×C47—81no—
183C4×C47—81no—
184C4×C47#48612yes—
185C4×C47C2×#12411yescharacteristic
186C4×C47C22×SD1611yescharacteristic
187Dic47—81no—
188#3164D4711yescharacteristic, complemented
189#3155—471no—
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×C47C2×#48611yescharacteristic
201C2×C47#341511yescharacteristic
202C2×C47#341611yescharacteristic
203D47—161no—
204D47—162no—
205C2×AGL(1,ℤ/8)—941no—
206C2×#135—471no—
207C22×D8—471no—
208#544—21no—
209#535—941no—
210C2×M32—471no—
211C22×C16—471no—
212#501—22no—
213#500C2×D4712yescomplemented
214#499C2×D4712yescomplemented
215#496—942no—
216#495—472no—
217#493—472no—
218#428—21no—
219#427C2×D4711yescharacteristic
220C2×#107C2×D4711yescharacteristic
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—941no—
236C22×D4—941no—
237AGL(1,ℤ/8)—1884no—
238#135—942no—
239#135—942no—
240C2×Q16—471no—
241C2×SD16—941no—
242C2×SD16—942no—
243C2×D8—472no—
244C2×D8—471no—
245C2×D8—942no—
246C2×D8—942no—
247C2×D8—941no—
248C2×D8—942no—
249C2×M16C22⋊D4711yescharacteristic
250C22×C8C22×D4711yescharacteristic
251C4⋊Q8—22no—
252C4⋊Q8—41no—
253#127—942no—
254#125—22no—
255#125—41no—
256#124—942no—
257C22⋊Q8—42no—
258#121—1881no—
259#121—1881no—
260#120—1882no—
261#117—21no—
262#117C22⋊D4711yescharacteristic
263C2×C4⋊C4—21no—
264M32—942no—
265C2×C16—471no—
266C2×C16—472no—
267C2×C16—471no—
268C2×C16—942no—
269#107—21no—
270#107—22no—
271#107C22×D4712yes—
272#107C22×D4711yescharacteristic
273#107C22×D4711yescharacteristic
274#106—21no—
275#106—22no—
276#97C22⋊D4712yes—
277C4×C8C22×D4712yes—
278C24—1881no—
279#41—1882no—
280#41—1882no—
281C2×Q8—42no—
282C2×Q8—941no—
283C2×D4—942no—
284C2×D4—941no—
285C2×D4—1881no—
286C2×D4—1882no—
287C2×D4—1881no—
288C2×D4—1882no—
289C2×D4—1882no—
290C2×D4—1882no—
291C22×C4—41no—
292C22×C4—1881no—
293C22×C4C2×C22⋊D4711yescharacteristic
294Q16—941no—
295Q16—941no—
296SD16—1882no—
297SD16—1884no—
298D8—942no—
299D8—942no—
300D8—941no—
301D8—941no—
302D8—1882no—
303D8—1884no—
304M16#1062533614yes—
305C2×C8#1062535412yes—
306C2×C8#1062535412yes—
307C2×C8C2×C22⋊D4711yescharacteristic
308C2×C8C2×C22⋊D4712yes—
309C2×C8C23×D4711yescharacteristic
310C4⋊C4—21no—
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4—41no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4#1062533411yescharacteristic
318C4⋊C4#1062533411yescharacteristic
319C22⋊C4—41no—
320C22⋊C4—1882no—
321C22⋊C4—1882no—
322C22⋊C4#1062533412yes—
323C42—41no—
324C42C2×C22⋊D4712yes—
325C16—941no—
326C16—941no—
327C16—942no—
328C23—3762no—
329C23—1882no—
330C23—1881no—
331C23—7521no—
332C23—7521no—
333C23C2×#1062533411yescharacteristic
334Q8—162no—
335Q8—1881no—
336Q8—1881no—
337D4—3762no—
338D4—3762no—
339D4—1881no—
340D4—1882no—
341D4—1882no—
342D4—1881no—
343D4—7522no—
344D4—7522no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—161no—
349C2×C4—3762no—
350C2×C4—1881no—
351C2×C4#1309401312yes—
352C2×C4C2×#1062533412yes—
353C2×C4C2×#1062533412yes—
354C2×C4C2×#1062533411yescharacteristic
355C2×C4C2×#1062533612yes—
356C2×C4C22×C22⋊D4711yescharacteristic
357C8#1309401312yes—
358C8C2×#1062533614yes—
359C8C2×#1062535411yescharacteristic
360C8C2×#1062535411yescharacteristic
361C22—21no—
362C22—21no—
363C22—3761no—
364C22—3762no—
365C22—3761no—
366C22—3762no—
367C22—3761no—
368C22—3762no—
369C22—7524no—
370C22#1568947112yes—
371C22C22×#1062533411yescharacteristic
372C4—22no—
373C4—22no—
374C4—82no—
375C4—81no—
376C4—81no—
377C4—3761no—
378C4#1568947112yes—
379C4C2×#1309401311yescharacteristic
380C4C22×#1062533611yescharacteristic
381C2—22no—
382C2—7522no—
383C2—7521no—
384C2C2×#1568947111yescharacteristic
385C2#2025652711yescharacteristic
386C2#2025653211yescharacteristic
subgroupG/H
centerC22C22×#10625334
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC47×#3164C2
socleC22×C47C22×D8
prime radical#20253874C2
hypercenter#3164D47
hypocenterC47#33702
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3164

Complements

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

NH ≅ G/Ntype
2. #20245631C2semidirect
3. #20245627C2semidirect
4. #20245622C2semidirect
5. #20245613C2semidirect
6. #20245103C2semidirect
7. #20245098C2semidirect
8. #20245093C2semidirect
9. C2×#15688923C2semidirect
10. #20244985C2semidirect
11. C47×#3164C2semidirect
21. #15688926C22semidirect
22. #15688926C22semidirect
23. #15688926C22semidirect
28. #15688923C22semidirect
29. #15688923C22semidirect
30. #15688923C22semidirect
31. #15688899C22semidirect
32. C4×C47⋊C16C22semidirect
35. C47×#500C22semidirect
36. C47×#499C22semidirect
188. #3164D47semidirect
213. #500C2×D47semidirect
214. #499C2×D47semidirect
234. C47#33702semidirect

Maximal subgroups

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

subgroupindex
#202538742
#202456312
#202456272
#202456222
#202456132
#202451032
#202450982
#202450932
C2×#156889232
#202449852
C47×#31642
#3370247