Finite Group Databasebeta

← #26371173

Subgroups — #26371173

Order 12032.

556 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252993C211yesmaximal, characteristic
2C2×#15689170C211yesmaximal, characteristic, complemented
3#20245606C214yesmaximal, complemented
4#20245601C214yesmaximal, complemented
5#20245162C211yesmaximal, characteristic, complemented
6#20245161C211yesmaximal, characteristic, complemented
7#20245137C211yesmaximal, characteristic, complemented
8C2×#15688904C211yesmaximal, characteristic, complemented
9C47×#2148C211yesmaximal, characteristic, complemented
10C22×#10625324C2211yescharacteristic
11C2×M16×D47—21no—
12#15689630—21no—
13C2×#13093964C2211yescharacteristic
14#15689378C2212yes—
15#15689373C2212yes—
16#15689366—22no—
17#15689170—22no—
18#15689170C2214yes—
19C2×#13093960C2211yescharacteristic
20C2×#13093959—21no—
21#15689161C2211yescharacteristic
22#15689119—21no—
23#15688947C2214yescomplemented
24#15688946C2214yescomplemented
25#15688939C2212yescomplemented
26#15688936C2212yescomplemented
27#15688904C2214yes—
28C2×#135×C47C2211yescharacteristic
29C2×M32×C47C2211yescharacteristic
30C2×#108×C47C2211yescharacteristic
31C47×#362C2212yescomplemented
32C47×#361C2212yescomplemented
33C22×C4×D47D411yescharacteristic
34C2×#10625329—21no—
35C2×#10625324—21no—
36C2×#10625324—21no—
37C2×#10625324—21no—
38C2×#10625324—21no—
39C2×#10625324—41no—
40C2×#10625324D412yes—
41C2×#10625324C2312yes—
42M16×D47—24no—
43M16×D47—42no—
44C22×C47⋊C8D411yescharacteristic
45C2×C8×D47—41no—
46#13094005—41no—
47#13094000—22no—
48#13093995—22no—
49#13093992—41no—
50C4×C22⋊D47—42no—
51#13093983—21no—
52#13093964—21no—
53#13093964C2312yes—
54C2×C47⋊C16—21no—
55#13093960—21no—
56#13093960C2312yes—
57#13093959—24no—
58#13093958—21no—
59#13093956C2312yes—
60#13093948—22no—
61#13093945—22no—
62C2×#41×C47—21no—
63#135×C47—21no—
64#135×C47—21no—
65#135×C47—21no—
66#135×C47—21no—
67C2×Q16×C47C2311yescharacteristic
68C2×SD16×C47—21no—
69C2×D8×C47C2311yescharacteristic
70C2×M16×C47—21no—
71C22×C8×C47C2311yescharacteristic
72M32×C47—21no—
73M32×C47C2312yes—
74C2×C16×C47—21no—
75#108×C47—21no—
76#108×C47C2312yes—
77C23×D47SD1611yescharacteristic
78#41×C47—41no—
79#41×C47—41no—
80#41×C47—41no—
81#41×C47—41no—
82C2×Q8×C47—21no—
83C2×D4×C47—21no—
84C2×D4×C47—41no—
85C22×C4×C47—41no—
86C22×C4×C47C2×D411yescharacteristic
87C2×C22⋊D47—42no—
88C22×Dic47SD1611yescharacteristic
89C2×C4×D47—21no—
90C2×C4×D47—21no—
91C2×C4×D47—21no—
92C2×C4×D47—21no—
93C2×C4×D47SD1612yes—
94C2×C4×D47C2×D412yes—
95Q16×C47—21no—
96Q16×C47—21no—
97SD16×C47—41no—
98SD16×C47—41no—
99D8×C47—21no—
100D8×C47—21no—
101M16×C47—22no—
102M16×C47—41no—
103C2×C8×C47—41no—
104C2×C8×C47C2×D411yescharacteristic
105C2×C8×C47#4111yescharacteristic
106C2×C8×C47#4111yescharacteristic
107C2×C8×C47#4111yescharacteristic
108C2×C8×C47#4111yescharacteristic
109C2×C8×C47C2411yescharacteristic
110C22⋊Dic47—41no—
111C22⋊Dic47—42no—
112(C2×D47)⋊C4—42no—
113C4⋊Dic47—22no—
114Dic47⋊C4—42no—
115C4×Dic47—41no—
116#10625329—22no—
117#10625329—41no—
118C2×C47⋊C8—21no—
119C2×C47⋊C8—21no—
120C2×C47⋊C8—41no—
121C2×C47⋊C8C2×D412yes—
122#10625324—22no—
123#10625324—22no—
124#10625324—22no—
125#10625324—22no—
126#10625324—44no—
127C8×D47—44no—
128C16×C47—22no—
129C47⋊C16—22no—
130C23×C47—41no—
131C23×C47C2×SD1611yescharacteristic
132C22×D47—21no—
133C22×D47—21no—
134C22×D47—81no—
135C22×D47C2×SD1612yes—
136Q8×C47—41no—
137Q8×C47—41no—
138D4×C47—41no—
139D4×C47—41no—
140D4×C47—81no—
141D4×C47—81no—
142C2×C4×C47—41no—
143C2×C4×C47—81no—
144C2×C4×C47—81no—
145C2×C4×C47#12311yescharacteristic
146C2×C4×C47#12311yescharacteristic
147C2×C4×C47C2×SD1611yescharacteristic
148C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C47C22×D411yescharacteristic
151C22⋊D47—84no—
152C2×Dic47—21no—
153C2×Dic47—21no—
154C2×Dic47—42no—
155C2×Dic47C2×SD1612yes—
156C4×D47—22no—
157C4×D47—22no—
158C4×D47—22no—
159C4×D47—22no—
160C8×C47—42no—
161C8×C47#12311yescharacteristic
162C8×C47#12311yescharacteristic
163C8×C47C2×#4111yescharacteristic
164C8×C47C2×#4111yescharacteristic
165C47⋊C8—22no—
166C47⋊C8—42no—
167#33656—471nomaximal, Sylow
168C22×C47—41no—
169C22×C47—82no—
170C22×C47—81no—
171C22×C47#48111yescharacteristic
172C22×C47#48111yescharacteristic
173C22×C47C22×SD1611yescharacteristic
174C2×D47—22no—
175C2×D47—42no—
176C2×D47—42no—
177C2×D47—81no—
178C4×C47—81no—
179C4×C47#48111yescharacteristic
180C4×C47#48111yescharacteristic
181C4×C47C2×#12311yescharacteristic
182C4×C47C2×AGL(1,ℤ/8)11yescharacteristic
183Dic47—22no—
184Dic47—82no—
185#3054—471no—
186C2×#430—471no—
187#2239—474no—
188#2238—474no—
189#2155—471no—
190#2155—471no—
191#2148—471no—
192#2148D4711yescharacteristic, complemented
193C2×#349—471no—
194C2×C47—21no—
195C2×C47—21no—
196C2×C47—21no—
197C2×C47—161no—
198C2×C47C2×#48111yescharacteristic
199D47—42no—
200C2×#135C2×D4711yescharacteristic
201C22×M16—471no—
202C22×M16—941no—
203#525—941no—
204C2×M32—471no—
205C2×M32C2×D4711yescharacteristic
206#470—472no—
207#469—472no—
208#468—942no—
209#430—474no—
210#430—942no—
211C2×#108—471no—
212C2×#108—941no—
213C2×#108C2×D4711yescharacteristic
214#427—471no—
215#416—941no—
216#365—474no—
217#365—474no—
218#362—472no—
219#362C2×D4712yescomplemented
220#361—472no—
221#361C2×D4712yescomplemented
222#349—474no—
223C47#3365611yescharacteristic, complemented, Sylow
224C2×#41—21no—
225C23×C4—471no—
226#135—21no—
227#135—21no—
228#135—21no—
229#135—21no—
230C2×Q16C22×D4711yescharacteristic
231C2×SD16—21no—
232C2×D8C22×D4711yescharacteristic
233C2×M16—21no—
234C2×M16—472no—
235C2×M16—472no—
236C2×M16—941no—
237C2×M16—941no—
238C2×M16—941no—
239C2×M16—941no—
240C2×M16—944no—
241C2×M16—941no—
242C2×M16—1881no—
243C2×M16—1882no—
244C22×C8—471no—
245C22×C8—1881no—
246C22×C8C22×D4711yescharacteristic
247#123—1881no—
248C22⋊Q8—942no—
249#121—942no—
250#120—1881no—
251C4×D4—1882no—
252#117—941no—
253M32—21no—
254M32—472no—
255M32—941no—
256M32C22×D4712yes—
257C2×C16—21no—
258C2×C16—941no—
259#108—21no—
260#108—472no—
261#108—944no—
262#108—941no—
263#108C22×D4712yes—
264#107—941no—
265#106—472no—
266#103—942no—
267#102—942no—
268C24—471no—
269#41—41no—
270#41—41no—
271#41—41no—
272#41—41no—
273C2×Q8—21no—
274C2×D4—21no—
275C2×D4—41no—
276C2×D4—1882no—
277C22×C4—41no—
278C22×C4—471no—
279C22×C4—472no—
280C22×C4—472no—
281C22×C4—941no—
282C22×C4—941no—
283C22×C4—941no—
284C22×C4—941no—
285C22×C4D4×D4711yescharacteristic
286Q16—21no—
287Q16—21no—
288SD16—41no—
289SD16—41no—
290D8—21no—
291D8—21no—
292M16—22no—
293M16—41no—
294M16—942no—
295M16—942no—
296M16—942no—
297M16—942no—
298M16—942no—
299M16—1884no—
300M16—1881no—
301C2×C8—41no—
302C2×C8—472no—
303C2×C8—941no—
304C2×C8—941no—
305C2×C8—1884no—
306C2×C8—1881no—
307C2×C8#1062535011yescharacteristic
308C2×C8#1062535011yescharacteristic
309C2×C8D4×D4711yescharacteristic
310C2×C8#1062535211yescharacteristic
311C2×C8#1062535211yescharacteristic
312C2×C8C23×D4711yescharacteristic
313C4⋊C4—942no—
314C4⋊C4—1882no—
315C22⋊C4—1882no—
316C22⋊C4—1882no—
317C22⋊C4—1881no—
318C42—1881no—
319C16—22no—
320C16—942no—
321C23—41no—
322C23—472no—
323C23—941no—
324C23—941no—
325C23—3761no—
326C23SD16×D4711yescharacteristic
327Q8—41no—
328Q8—41no—
329D4—41no—
330D4—41no—
331D4—81no—
332D4—81no—
333D4—3764no—
334C2×C4—41no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—472no—
338C2×C4—941no—
339C2×C4—941no—
340C2×C4—942no—
341C2×C4—942no—
342C2×C4—942no—
343C2×C4—942no—
344C2×C4—1882no—
345C2×C4#1309400611yescharacteristic
346C2×C4#1309400611yescharacteristic
347C2×C4SD16×D4711yescharacteristic
348C2×C4#1309404811yescharacteristic
349C2×C4#1309404811yescharacteristic
350C2×C4C2×D4×D4711yescharacteristic
351C8—42no—
352C8—942no—
353C8—1882no—
354C8#1309400611yescharacteristic
355C8#1309400611yescharacteristic
356C8C2×#1062535211yescharacteristic
357C8C2×#1062535011yescharacteristic
358C22—41no—
359C22—82no—
360C22—81no—
361C22—942no—
362C22—3761no—
363C22—1882no—
364C22—1882no—
365C22#1568944011yescharacteristic
366C22#1568944011yescharacteristic
367C22C2×SD16×D4711yescharacteristic
368C4—81no—
369C4—942no—
370C4—3762no—
371C4#1568944011yescharacteristic
372C4#1568944011yescharacteristic
373C4C2×#1309400611yescharacteristic
374C4C2×#1309404811yescharacteristic
375C2—21no—
376C2—21no—
377C2—21no—
378C2—161no—
379C2—1882no—
380C2C2×#1568944011yescharacteristic
subgroupG/H
centerC4#15689440
commutator, cosocleC2×C8×C47C24
FrattiniC2×C8C23×D47
FittingC47×#2148C2
socleC2×C47C2×#481
prime radical#20252993C2
hypercenter#2148D47
hypocenterC47#33656
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #2148

Complements

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

NH ≅ G/Ntype
2. C2×#15689170C2semidirect
3. #20245606C2semidirect
4. #20245601C2semidirect
5. #20245162C2semidirect
6. #20245161C2semidirect
7. #20245137C2semidirect
8. C2×#15688904C2semidirect
9. C47×#2148C2semidirect
23. #15688947C22semidirect
24. #15688946C22semidirect
25. #15688939C22semidirect
26. #15688936C22semidirect
31. C47×#362C22semidirect
32. C47×#361C22semidirect
192. #2148D47semidirect
219. #362C2×D47semidirect
221. #361C2×D47semidirect
223. C47#33656semidirect

Maximal subgroups

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

subgroupindex
#202529932
C2×#156891702
#202456062
#202456012
#202451622
#202451612
#202451372
C2×#156889042
C47×#21482
#3365647