Finite Group Databasebeta

← #27371969

Subgroups — #27371969

Order 13568.

772 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285856C211yesmaximal, characteristic, complemented
2#20285843C211yesmaximal, characteristic
3#20285275C211yesmaximal, characteristic
4#20285174C211yesmaximal, characteristic
5C2×#16810973C211yesmaximal, characteristic, complemented
6#20274349C212yesmaximal, complemented
7#20274338C212yesmaximal, complemented
8#20274317C212yesmaximal, complemented
9#20274313C212yesmaximal, complemented
10M16×#88322C211yesmaximal, characteristic, complemented
11C53×#3165C211yesmaximal, characteristic, complemented
12C2×#13133250—22no—
13#16811173C2211yescharacteristic
14C2×#13133217C2211yescharacteristic
15#16811019C2211yescharacteristic
16#16811019C2212yes—
17#16811019C2212yes—
18#16811017—22no—
19#16811003C2212yes—
20#16810997C2211yescharacteristic
21#16810996C2212yes—
22#16810973—22no—
23#16810973C2211yescharacteristic
24#16810973C2212yes—
25#16810973C2212yes—
26#16810973C2212yes—
27#16810971C2211yescharacteristic
28#16810971C2212yes—
29#16810951—22no—
30#16810941—22no—
31#16810910—22no—
32#16810908—22no—
33#16810854—22no—
34C4×#11719924—22no—
35#16810676C2211yescharacteristic
36C2×#13133141—22no—
37C2×C4×#88322C2211yescharacteristic
38#16810344C2212yes—
39C8×#88322C2212yes—
40C53×#533C2211yescharacteristic
41C2×C4⋊Q8×C53C2211yescharacteristic
42C53×#501—22no—
43C53×#500C2212yes—
44C53×#498C2212yes—
45C53×#428—22no—
46C4×M16×C53C2211yescharacteristic
47C2×#11719945—21no—
48C22×Dic106—21no—
49#13133250—48no—
50C2×Dic212—24no—
51C2×#11719913—24no—
52C2×#11719924—22no—
53C22×#88322D412yes—
54#13133217—21no—
55#13133217—21no—
56#13133217—21no—
57#13133217C2311yescharacteristic
58#13133217C2312yes—
59#13133214C2311yescharacteristic
60#13133205—21no—
61#13133191—21no—
62#13133191—21no—
63C4×D212—21no—
64C4×D212—21no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—21no—
67#13133148—22no—
68#13133148—24no—
69#13133148—24no—
70C4⋊#88322D412yes—
71#13133141—22no—
72#13133141—24no—
73#13133141—24no—
74#13133141—24no—
75#13133137—22no—
76#13133137—24no—
77C22⋊#88322D412yes—
78C4×#88322D412yes—
79C4×#88322C2312yes—
80C4×#88322C2311yescharacteristic
81C4×#88322C2311yescharacteristic
82C22×Q8×C53—21no—
83C2×M16×C53D412yes—
84C4⋊Q8×C53—21no—
85C4⋊Q8×C53—21no—
86C4⋊Q8×C53—21no—
87C4⋊Q8×C53C2311yescharacteristic
88C4⋊Q8×C53C2311yescharacteristic
89C4⋊Q8×C53C2312yes—
90#125×C53—21no—
91C22⋊Q8×C53—21no—
92C22⋊Q8×C53—21no—
93C4×Q8×C53—21no—
94C4×Q8×C53—21no—
95#117×C53—21no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53—21no—
98C2×C42×C53C2311yescharacteristic
99#107×C53—24no—
100#106×C53—24no—
101#97×C53D412yes—
102C4×C8×C53C2312yes—
103C2×Q8×C53—21no—
104C2×Q8×C53—21no—
105C2×Q8×C53—22no—
106C2×Q8×C53—44no—
107C2×Q8×C53—41no—
108C22×C4×C53—41no—
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53C2×D411yescharacteristic
112C2×C22⋊D53—41no—
113C22×Dic53—41no—
114#11719945—44no—
115C2×D212—21no—
116C2×D212—21no—
117C2×D212—21no—
118C2×C4×D53—41no—
119C2×Dic106—21no—
120C2×Dic106—22no—
121C2×Dic106—44no—
122C2×Dic106—41no—
123M16×C53SD1618yescomplemented
124C2×C8×C53C2×D414yes—
125C4⋊C4×C53—21no—
126C4⋊C4×C53—22no—
127C4⋊C4×C53—21no—
128C4⋊C4×C53—22no—
129C4⋊C4×C53—21no—
130C4⋊C4×C53—21no—
131C4⋊C4×C53—41no—
132C4⋊C4×C53—41no—
133C4⋊C4×C53—41no—
134C4⋊C4×C53—41no—
135C22⋊C4×C53—41no—
136C42×C53—41no—
137C42×C53C2×D411yescharacteristic
138C42×C53C2×D411yescharacteristic
139C42×C53C2×D411yescharacteristic
140C42×C53C2411yescharacteristic
141#11719924—44no—
142#11719921—41no—
143#11719921—41no—
144C4⋊Dic53—22no—
145C4⋊Dic53—22no—
146C4⋊Dic53—21no—
147C4⋊Dic53—21no—
148C4⋊Dic53—41no—
149C4⋊Dic53—41no—
150C2×#88322—42no—
151C2×#88322C2×D412yes—
152C2×#88322C2×D412yes—
153C2×#88322C2×D414yes—
154Dic212—48no—
155#11719913—48no—
156C23×C53#12711yescharacteristic
157C22×D53—41no—
158Q8×C53—44no—
159Q8×C53—42no—
160Q8×C53—42no—
161Q8×C53—82no—
162C2×C4×C53—22no—
163C2×C4×C53—22no—
164C2×C4×C53—42no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—82no—
168C2×C4×C53#12711yescharacteristic
169C2×C4×C53#12711yescharacteristic
170C2×C4×C53#12711yescharacteristic
171C2×C4×C53C2×SD1614yes—
172C2×C4×C53C22×D411yescharacteristic
173C2×C4×C53C22×D411yescharacteristic
174C2×C4×C53C22×D411yescharacteristic
175C22⋊D53—82no—
176C2×Dic53—42no—
177C2×Dic53—41no—
178C2×Dic53—82no—
179D212—42no—
180D212—42no—
181D212—42no—
182C4×D53—82no—
183Dic106—44no—
184Dic106—42no—
185Dic106—82no—
186C8×C53C2×SD1618yes—
187#88322—22no—
188#88322—22no—
189#88322—24no—
190#19624—531nomaximal, Sylow
191C22×C53—21no—
192C22×C53—21no—
193C22×C53#49212yes—
194C22×C53C2×#12711yescharacteristic
195C2×D53—81no—
196C2×D53—81no—
197C2×D53—81no—
198C4×C53—21no—
199C4×C53—21no—
200C4×C53—82no—
201C4×C53—81no—
202C4×C53—81no—
203C4×C53#49212yes—
204C4×C53C22×SD1612yes—
205C4×C53#57512yes—
206C4×C53#57812yes—
207Dic53—82no—
208Dic53—81no—
209#3229—531no—
210#3227—531no—
211#3167—531no—
212#3165D5311yescharacteristic, complemented
213#3152—531no—
214C2×#487—531no—
215#1716—532no—
216#1714—532no—
217#1708—532no—
218#1707—532no—
219C8×M16—531no—
220C2×C53—22no—
221C2×C53C2×#49211yescharacteristic
222C2×C53#315311yescharacteristic
223C2×C53#315611yescharacteristic
224D53—161no—
225C2×#137—1062no—
226#533C2×D5311yescharacteristic
227#532—531no—
228C2×C4⋊Q8—531no—
229C2×C4⋊Q8C2×D5311yescharacteristic
230#501—22no—
231#500C2×D5312yes—
232#498—531no—
233#498—532no—
234#498—532no—
235#498C2×D5312yes—
236#497—1062no—
237#494—532no—
238#492—531no—
239#492—532no—
240#487—532no—
241#487—532no—
242#487—531no—
243#487—532no—
244#487—1062no—
245#486—531no—
246#486—532no—
247#483—1062no—
248#481—1062no—
249#476—1062no—
250#475—1062no—
251#465—1062no—
252C4×SD16—1062no—
253#431—531no—
254#428—22no—
255C2×#103—1062no—
256C4×M16C2×D5311yescharacteristic
257C2×C4×C8—531no—
258#322—532no—
259C82—532no—
260C53#1962411yescharacteristic, complemented, Sylow
261C2×#41—1061no—
262C22×Q8—21no—
263C22×Q8—1061no—
264#137—2128no—
265C2×Q16—1064no—
266C2×SD16—1062no—
267C2×SD16—1064no—
268C2×M16C22⋊D5312yes—
269C22×C8—532no—
270C4⋊Q8—21no—
271C4⋊Q8—21no—
272C4⋊Q8—21no—
273C4⋊Q8—532no—
274C4⋊Q8—531no—
275C4⋊Q8—1061no—
276C4⋊Q8—1061no—
277C4⋊Q8—1061no—
278C4⋊Q8C22×D5312yes—
279C4⋊Q8C22×D5311yescharacteristic
280C4⋊Q8C22×D5311yescharacteristic
281#127—531no—
282#125—21no—
283#124—1061no—
284C22⋊Q8—21no—
285C22⋊Q8—21no—
286#121—1061no—
287#121—1061no—
288C4×Q8—21no—
289C4×Q8—21no—
290C4×D4—1061no—
291C4×D4—1061no—
292#117—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—1061no—
296C2×C4⋊C4—1061no—
297C2×C42C22×D5311yescharacteristic
298#107—24no—
299#106—24no—
300#106—1062no—
301#106—1064no—
302#106—1064no—
303C4⋊C8—532no—
304#103—1064no—
305#103—1064no—
306#103—1062no—
307#103—1064no—
308#102—1062no—
309#102—1064no—
310C22⋊C8—532no—
311#97C22⋊D5312yes—
312C4×C8—532no—
313C4×C8—531no—
314C4×C8—532no—
315C4×C8—531no—
316C4×C8C22×D5312yes—
317#41—2124no—
318C2×Q8—21no—
319C2×Q8—22no—
320C2×Q8—21no—
321C2×Q8—44no—
322C2×Q8—41no—
323C2×Q8—1061no—
324C2×Q8—1062no—
325C2×Q8—2124no—
326C2×Q8—2121no—
327C2×D4—1061no—
328C2×D4—1061no—
329C2×D4—1061no—
330C2×D4—2121no—
331C22×C4—41no—
332C22×C4—2121no—
333C22×C4—2121no—
334C22×C4D4×D5311yescharacteristic
335C22×C4D4×D5311yescharacteristic
336C22×C4C2×C22⋊D5311yescharacteristic
337Q16—2128no—
338SD16—2124no—
339SD16—2128no—
340M16#1171992418yescomplemented
341C2×C8—532no—
342C2×C8—532no—
343C2×C8—534no—
344C2×C8—2122no—
345C2×C8C2×C22⋊D5314yes—
346C4⋊C4—22no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—22no—
351C4⋊C4—21no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—1062no—
357C4⋊C4—1062no—
358C4⋊C4—1061no—
359C4⋊C4—1061no—
360C4⋊C4—2121no—
361C4⋊C4—2121no—
362C22⋊C4—41no—
363C22⋊C4—2121no—
364C22⋊C4—2121no—
365C42—41no—
366C42D4×D5311yescharacteristic
367C42D4×D5311yescharacteristic
368C42C2×C22⋊D5311yescharacteristic
369C42C23×D5311yescharacteristic
370C23—2121no—
371C23#1313321311yescharacteristic
372Q8—42no—
373Q8—44no—
374Q8—42no—
375Q8—82no—
376Q8—4242no—
377Q8—2124no—
378Q8—2122no—
379D4—4242no—
380D4—2122no—
381D4—2122no—
382D4—2122no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—82no—
389C2×C4—4242no—
390C2×C4—4242no—
391C2×C4—2121no—
392C2×C4—2122no—
393C2×C4#1313321311yescharacteristic
394C2×C4#1313321311yescharacteristic
395C2×C4#1313321311yescharacteristic
396C2×C4C2×#1171992414yes—
397C2×C4C2×D4×D5311yescharacteristic
398C2×C4C2×D4×D5311yescharacteristic
399C2×C4C22×C22⋊D5311yescharacteristic
400C8—1062no—
401C8—1062no—
402C8—1064no—
403C8C2×#1171992418yes—
404C22—21no—
405C22—21no—
406C22—4241no—
407C22—4241no—
408C22—4241no—
409C22#1681099712yes—
410C22C2×#1313321311yescharacteristic
411C4—21no—
412C4—21no—
413C4—81no—
414C4—82no—
415C4—81no—
416C4—4241no—
417C4—4242no—
418C4#1681099712yes—
419C4C22×#1171992412yes—
420C4#1681141912yes—
421C4#1681143912yes—
422C2—22no—
423C2—8481no—
424C2C2×#1681099711yescharacteristic
425C2#2028517711yescharacteristic
426C2#2028520811yescharacteristic
subgroupG/H
centerC22C2×#13133213
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3165C2
socleC22×C53C2×#127
prime radical#16811173C22
hypercenter#3165D53
hypocenterC53#19624
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
1. #20285856C2semidirect
5. C2×#16810973C2semidirect
6. #20274349C2semidirect
7. #20274338C2semidirect
8. #20274317C2semidirect
9. #20274313C2semidirect
10. M16×#88322C2semidirect
11. C53×#3165C2semidirect
123. M16×C53SD16semidirect
212. #3165D53semidirect
260. C53#19624semidirect
340. M16#11719924semidirect

Maximal subgroups

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

subgroupindex
#202858562
#202858432
#202852752
#202851742
C2×#168109732
#202743492
#202743382
#202743172
#202743132
M16×#883222
C53×#31652
#1962453