Finite Group Databasebeta

← #25251496

Subgroups — #25251496

Order 11008.

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#19144205C211yesmaximal, characteristic
2C2×#14582946C211yesmaximal, characteristic, complemented
3#19136816C214yesmaximal, complemented
4#19136811C214yesmaximal, complemented
5#19136367C211yesmaximal, characteristic, complemented
6#19136345C211yesmaximal, characteristic, complemented
7#19136342C211yesmaximal, characteristic, complemented
8C2×#14582680C211yesmaximal, characteristic, complemented
9C43×#2155C211yesmaximal, characteristic, complemented
10C2×#13063347C2211yescharacteristic
11C22×#10621872C2211yescharacteristic
12C2×M16×D43—21no—
13#14583411—21no—
14C2×#13063265C2211yescharacteristic
15#14583158C2212yes—
16#14583153C2212yes—
17#14583146—22no—
18#14582946—22no—
19#14582946C2214yes—
20C2×#13063261C2211yescharacteristic
21C2×#13063260—21no—
22#14582893—21no—
23#14582721C2214yescomplemented
24#14582716C2212yescomplemented
25#14582714C2212yescomplemented
26#14582713C2212yescomplemented
27#14582711C2212yescomplemented
28#14582680C2214yes—
29C2×M32×C43C2211yescharacteristic
30C2×#108×C43C2211yescharacteristic
31C43×#427C2211yescharacteristic
32C43×#365C2214yescomplemented
33C2×#10621898—21no—
34C22×C4×D43D411yescharacteristic
35#13063347—21no—
36#13063347—21no—
37#13063347—21no—
38#13063347—21no—
39C2×Dic172C2311yescharacteristic
40C2×#10621873—21no—
41C2×D344C2311yescharacteristic
42C2×#10621877—21no—
43C2×#10621872—21no—
44C2×#10621872—21no—
45C2×#10621872—21no—
46C2×#10621872—21no—
47C2×#10621872—41no—
48C2×#10621872D412yes—
49C2×#10621872C2312yes—
50M16×D43—24no—
51M16×D43—42no—
52C22×C43⋊C8D411yescharacteristic
53C2×C8×D43—41no—
54#13063309—41no—
55#13063305—22no—
56#13063300—22no—
57#13063295—41no—
58C4×D172—42no—
59#13063265—21no—
60#13063265C2312yes—
61C2×C43⋊C16—21no—
62#13063261—21no—
63#13063261C2312yes—
64#13063260—24no—
65#13063248—22no—
66#13063245—22no—
67C2×M16×C43—21no—
68C22×C8×C43C2311yescharacteristic
69#117×C43—21no—
70M32×C43—21no—
71M32×C43C2312yes—
72C2×C16×C43—21no—
73#108×C43—21no—
74#108×C43C2312yes—
75#107×C43—21no—
76#106×C43C2312yes—
77C23×D43SD1611yescharacteristic
78C22×C4×C43C2×D411yescharacteristic
79C2×C22⋊D43—41no—
80C22×Dic43SD1611yescharacteristic
81#10621898—41no—
82#10621898—41no—
83#10621898—41no—
84#10621898—41no—
85C2×D172—21no—
86C2×D172—42no—
87C2×C4×D43—21no—
88C2×C4×D43—21no—
89C2×C4×D43—21no—
90C2×C4×D43—21no—
91C2×C4×D43—41no—
92C2×C4×D43SD1612yes—
93C2×C4×D43C2×D412yes—
94C2×Dic86—21no—
95M16×C43—22no—
96M16×C43—41no—
97C2×C8×C43—41no—
98C2×C8×C43C2×D411yescharacteristic
99C2×C8×C43#4111yescharacteristic
100C2×C8×C43#4111yescharacteristic
101C2×C8×C43#4111yescharacteristic
102C2×C8×C43#4111yescharacteristic
103C2×C8×C43C2411yescharacteristic
104C4⋊C4×C43—22no—
105C22⋊C4×C43—41no—
106C42×C43—41no—
107(C2×D43)⋊C4—42no—
108(C2×D43)⋊C4—42no—
109C4⋊Dic43—42no—
110#10621877—22no—
111#10621877—41no—
112C2×C43⋊C8—21no—
113C2×C43⋊C8—21no—
114C2×C43⋊C8—41no—
115C2×C43⋊C8C2×D412yes—
116Dic172—21no—
117Dic172—21no—
118D344—21no—
119D344—21no—
120#10621873—41no—
121#10621873—41no—
122#10621872—22no—
123#10621872—22no—
124#10621872—22no—
125#10621872—22no—
126#10621872—44no—
127C8×D43—44no—
128C16×C43—22no—
129C43⋊C16—22no—
130C23×C43C2×SD1611yescharacteristic
131C22×D43—21no—
132C22×D43—21no—
133C22×D43—41no—
134C22×D43—81no—
135C22×D43C2×SD1612yes—
136C2×C4×C43—42no—
137C2×C4×C43#12311yescharacteristic
138C2×C4×C43#12311yescharacteristic
139C2×C4×C43C2×SD1611yescharacteristic
140C2×C4×C43AGL(1,ℤ/8)11yescharacteristic
141C2×C4×C43AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C43C22×D411yescharacteristic
143C22⋊D43—81no—
144C22⋊D43—81no—
145C2×Dic43—21no—
146C2×Dic43—21no—
147C2×Dic43—41no—
148C2×Dic43C2×SD1612yes—
149D172—41no—
150D172—41no—
151D172—84no—
152C4×D43—22no—
153C4×D43—22no—
154C4×D43—22no—
155C4×D43—22no—
156C4×D43—81no—
157C4×D43—81no—
158Dic86—41no—
159Dic86—41no—
160C8×C43—42no—
161C8×C43#12311yescharacteristic
162C8×C43#12311yescharacteristic
163C8×C43C2×#4111yescharacteristic
164C8×C43C2×#4111yescharacteristic
165C43⋊C8—22no—
166C43⋊C8—42no—
167#33656—431nomaximal, Sylow
168C22×C43—41no—
169C22×C43#48111yescharacteristic
170C22×C43#48111yescharacteristic
171C22×C43C22×SD1611yescharacteristic
172C2×D43—22no—
173C2×D43—42no—
174C2×D43—42no—
175C2×D43—81no—
176C2×D43—82no—
177C2×D43—81no—
178C4×C43—82no—
179C4×C43#48111yescharacteristic
180C4×C43#48111yescharacteristic
181C4×C43C2×#12311yescharacteristic
182C4×C43C2×AGL(1,ℤ/8)11yescharacteristic
183Dic43—22no—
184Dic43—81no—
185#3054—431no—
186C2×#430—431no—
187#2239—434no—
188#2238—434no—
189#2155—431no—
190#2155D4311yescharacteristic, complemented
191#2148—431no—
192#2148—431no—
193C2×#349—431no—
194C2×C43—21no—
195C2×C43—21no—
196C2×C43—21no—
197C2×C43C2×#48111yescharacteristic
198D43—42no—
199D43—161no—
200C2×#135—431no—
201C22×M16—431no—
202C22×M16—861no—
203#525—861no—
204C2×M32—431no—
205C2×M32C2×D4311yescharacteristic
206#470—432no—
207#469—432no—
208#468—862no—
209#430—434no—
210#430—862no—
211C2×#108—431no—
212C2×#108—861no—
213C2×#108C2×D4311yescharacteristic
214#427C2×D4311yescharacteristic
215#416—861no—
216#365—434no—
217#365C2×D4314yescomplemented
218#362—432no—
219#362—432no—
220#361—432no—
221#361—432no—
222#349—434no—
223C43#3365611yescharacteristic, complemented, Sylow
224C2×#41—861no—
225C23×C4—431no—
226#135—861no—
227#135—861no—
228#135—861no—
229#135—861no—
230C2×Q16—431no—
231C2×SD16—861no—
232C2×D8—431no—
233C2×M16—21no—
234C2×M16—432no—
235C2×M16—432no—
236C2×M16—861no—
237C2×M16—861no—
238C2×M16—861no—
239C2×M16—861no—
240C2×M16—864no—
241C2×M16—861no—
242C2×M16—1721no—
243C2×M16—1722no—
244C22×C8—431no—
245C22×C8—1721no—
246C22×C8C22×D4311yescharacteristic
247#123—1721no—
248C22⋊Q8—862no—
249#121—862no—
250#120—1721no—
251C4×D4—1722no—
252#117—21no—
253M32—21no—
254M32—432no—
255M32—861no—
256M32C22×D4312yes—
257C2×C16—21no—
258C2×C16—861no—
259#108—21no—
260#108—432no—
261#108—864no—
262#108—861no—
263#108C22×D4312yes—
264#107—21no—
265#106C22×D4312yes—
266#103—862no—
267#102—862no—
268C24—431no—
269#41—1721no—
270#41—1721no—
271#41—1721no—
272#41—1721no—
273C2×Q8—861no—
274C2×D4—861no—
275C2×D4—1722no—
276C2×D4—1721no—
277C22×C4—431no—
278C22×C4—432no—
279C22×C4—432no—
280C22×C4—861no—
281C22×C4—861no—
282C22×C4—861no—
283C22×C4—861no—
284C22×C4—1721no—
285C22×C4D4×D4311yescharacteristic
286Q16—861no—
287Q16—861no—
288SD16—1721no—
289SD16—1721no—
290D8—861no—
291D8—861no—
292M16—22no—
293M16—41no—
294M16—862no—
295M16—862no—
296M16—862no—
297M16—862no—
298M16—862no—
299M16—1724no—
300M16—1721no—
301C2×C8—41no—
302C2×C8—432no—
303C2×C8—861no—
304C2×C8—861no—
305C2×C8—1724no—
306C2×C8—1721no—
307C2×C8#1062189811yescharacteristic
308C2×C8#1062189811yescharacteristic
309C2×C8D4×D4311yescharacteristic
310C2×C8#1062190211yescharacteristic
311C2×C8#1062190211yescharacteristic
312C2×C8C23×D4311yescharacteristic
313C4⋊C4—22no—
314C4⋊C4—1722no—
315C22⋊C4—41no—
316C22⋊C4—1722no—
317C22⋊C4—1722no—
318C42—41no—
319C16—22no—
320C16—862no—
321C23—432no—
322C23—861no—
323C23—861no—
324C23—3441no—
325C23—1721no—
326C23SD16×D4311yescharacteristic
327Q8—1721no—
328Q8—1721no—
329D4—3444no—
330D4—3441no—
331D4—3441no—
332D4—1721no—
333D4—1721no—
334C2×C4—42no—
335C2×C4—432no—
336C2×C4—861no—
337C2×C4—861no—
338C2×C4—862no—
339C2×C4—862no—
340C2×C4—862no—
341C2×C4—862no—
342C2×C4—3441no—
343C2×C4—3441no—
344C2×C4—1721no—
345C2×C4#1306331011yescharacteristic
346C2×C4#1306331011yescharacteristic
347C2×C4SD16×D4311yescharacteristic
348C2×C4#1306335211yescharacteristic
349C2×C4#1306335211yescharacteristic
350C2×C4C2×D4×D4311yescharacteristic
351C8—42no—
352C8—862no—
353C8—1722no—
354C8#1306331011yescharacteristic
355C8#1306331011yescharacteristic
356C8C2×#1062189811yescharacteristic
357C8C2×#1062190211yescharacteristic
358C22—41no—
359C22—862no—
360C22—3441no—
361C22—3442no—
362C22—3441no—
363C22—1722no—
364C22—1722no—
365C22#1458321911yescharacteristic
366C22#1458321911yescharacteristic
367C22C2×SD16×D4311yescharacteristic
368C4—82no—
369C4—862no—
370C4—3441no—
371C4#1458321911yescharacteristic
372C4#1458321911yescharacteristic
373C4C2×#1306331011yescharacteristic
374C4C2×#1306335211yescharacteristic
375C2—21no—
376C2—21no—
377C2—21no—
378C2—1722no—
379C2—6881no—
380C2C2×#1458321911yescharacteristic
subgroupG/H
centerC4#14583219
commutator, cosocleC2×C8×C43C24
FrattiniC2×C8C23×D43
FittingC43×#2155C2
socleC2×C43C2×#481
prime radical#19144205C2
hypercenter#2155D43
hypocenterC43#33656
LCSG ▷ C2×C8×C43 ▷ C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #2155

Complements

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

NH ≅ G/Ntype
2. C2×#14582946C2semidirect
3. #19136816C2semidirect
4. #19136811C2semidirect
5. #19136367C2semidirect
6. #19136345C2semidirect
7. #19136342C2semidirect
8. C2×#14582680C2semidirect
9. C43×#2155C2semidirect
23. #14582721C22semidirect
24. #14582716C22semidirect
25. #14582714C22semidirect
26. #14582713C22semidirect
27. #14582711C22semidirect
32. C43×#365C22semidirect
190. #2155D43semidirect
217. #365C2×D43semidirect
223. C43#33656semidirect

Maximal subgroups

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

subgroupindex
#191442052
C2×#145829462
#191368162
#191368112
#191363672
#191363452
#191363422
C2×#145826802
C43×#21552
#3365643