Finite Group Databasebeta

← #27371958

Subgroups — #27371958

Order 13568.

848 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285653C211yesmaximal, characteristic, complemented
2#20285640C211yesmaximal, characteristic, complemented
3#20285275C211yesmaximal, characteristic, complemented
4#20285167C211yesmaximal, characteristic
5C2×#16810971C211yesmaximal, characteristic, complemented
6#20274344C212yesmaximal, complemented
7#20274338C212yesmaximal, complemented
8#20274317C212yesmaximal, complemented
9#20274308C212yesmaximal, complemented
10M16×#88322C211yesmaximal, characteristic, complemented
11C53×#3165C211yesmaximal, characteristic, complemented
12C2×#13133245—22no—
13#16811173C2211yescharacteristic
14C2×#13133214C2211yescharacteristic
15#16811019C2211yescharacteristic, complemented
16#16811012—22no—
17#16811000C2212yes—
18#16810997C2212yes—
19#16810997C2211yescharacteristic, complemented
20#16810997C2212yescomplemented
21#16810996C2212yes—
22#16810973C2211yescharacteristic, complemented
23#16810973C2212yescomplemented
24#16810971—22no—
25#16810971C2212yes—
26#16810971C2212yes—
27#16810971C2212yes—
28#16810971C2211yescharacteristic, complemented
29#16810854—22no—
30#16810825—22no—
31#16810823—22no—
32#16810773—22no—
33#16810762—22no—
34C4×#11719924—22no—
35#16810676C2211yescharacteristic
36C2×#13133137—22no—
37C2×C4×#88322C2211yescharacteristic
38#16810344C2212yescomplemented
39C8×#88322C2212yescomplemented
40C53×#533C2211yescharacteristic
41C2×C4⋊Q8×C53C2211yescharacteristic
42C53×#501—22no—
43C53×#500C2212yescomplemented
44C53×#498C2212yescomplemented
45C53×#428—22no—
46C4×M16×C53C2211yescharacteristic
47C2×#11719945—21no—
48C22×D212—21no—
49C22×D212—21no—
50C22×D212—21no—
51#13133245—48no—
52C2×#11719913—24no—
53C2×#11719924—22no—
54C2×#11719924—24no—
55C2×#11719924—24no—
56C2×D424—24no—
57C22×#88322D412yes—
58#13133217C2311yescharacteristic
59#13133214—21no—
60#13133214—21no—
61#13133214—21no—
62#13133214C2312yes—
63#13133214C2311yescharacteristic
64#13133205—21no—
65#13133191—21no—
66#13133191—21no—
67C4×D212—21no—
68C4×D212—21no—
69#13133148—22no—
70C4⋊#88322D412yes—
71#13133141—24no—
72#13133141—22no—
73#13133137—24no—
74#13133137—22no—
75#13133137—24no—
76#13133137—24no—
77C22⋊#88322D412yes—
78C4×#88322D412yes—
79C4×#88322C2311yescharacteristic
80C4×#88322C2311yescharacteristic
81C4×#88322C2312yes—
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—
103C23×D53—41no—
104C2×Q8×C53—21no—
105C2×Q8×C53—21no—
106C2×Q8×C53—22no—
107C2×Q8×C53—41no—
108C2×Q8×C53—82no—
109C22×C4×C53—41no—
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53C2×D411yescharacteristic
113C2×C22⋊D53—41no—
114#11719945—44no—
115C2×D212—22no—
116C2×D212—21no—
117C2×D212—22no—
118C2×D212—21no—
119C2×D212—21no—
120C2×D212—22no—
121C2×D212—44no—
122C2×D212—44no—
123C2×D212—41no—
124C2×D212—41no—
125C2×D212—41no—
126C2×D212—82no—
127C2×C4×D53—41no—
128C2×Dic106—21no—
129M16×C53SD1618yescomplemented
130C2×C8×C53C2×D414yes—
131C4⋊C4×C53—21no—
132C4⋊C4×C53—22no—
133C4⋊C4×C53—21no—
134C4⋊C4×C53—22no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—41no—
138C4⋊C4×C53—41no—
139C4⋊C4×C53—41no—
140C4⋊C4×C53—41no—
141C22⋊C4×C53—41no—
142C42×C53—41no—
143C42×C53C2×D411yescharacteristic
144C42×C53C2×D411yescharacteristic
145C42×C53C2×D411yescharacteristic
146C42×C53C2411yescharacteristic
147#11719924—44no—
148#11719924—84no—
149#11719924—84no—
150#11719921—41no—
151#11719921—41no—
152C4⋊Dic53—21no—
153C4⋊Dic53—21no—
154C2×#88322—42no—
155C2×#88322C2×D412yes—
156C2×#88322C2×D412yes—
157C2×#88322C2×D414yes—
158D424—48no—
159#11719913—48no—
160C23×C53#12711yescharacteristic
161C22×D53—41no—
162C22×D53—42no—
163C22×D53—82no—
164C22×D53—82no—
165C22×D53—82no—
166Q8×C53—42no—
167Q8×C53—44no—
168Q8×C53—42no—
169Q8×C53—82no—
170C2×C4×C53—22no—
171C2×C4×C53—22no—
172C2×C4×C53—42no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53—82no—
176C2×C4×C53#12711yescharacteristic
177C2×C4×C53#12711yescharacteristic
178C2×C4×C53#12711yescharacteristic
179C2×C4×C53C2×SD1614yes—
180C2×C4×C53C22×D411yescharacteristic
181C2×C4×C53C22×D411yescharacteristic
182C2×C4×C53C22×D411yescharacteristic
183C22⋊D53—82no—
184C2×Dic53—41no—
185D212—44no—
186D212—42no—
187D212—44no—
188D212—42no—
189D212—44no—
190D212—42no—
191D212—82no—
192D212—82no—
193D212—82no—
194C4×D53—82no—
195Dic106—42no—
196C8×C53C2×SD1618yes—
197#88322—22no—
198#88322—22no—
199#88322—24no—
200#19623—531nomaximal, Sylow
201C22×C53—21no—
202C22×C53—21no—
203C22×C53#49212yes—
204C22×C53C2×#12711yescharacteristic
205C2×D53—81no—
206C2×D53—82no—
207C2×D53—82no—
208C2×D53—81no—
209C2×D53—82no—
210C2×D53—81no—
211C2×D53—164no—
212C4×C53—21no—
213C4×C53—21no—
214C4×C53—81no—
215C4×C53—82no—
216C4×C53—81no—
217C4×C53#49212yes—
218C4×C53C22×SD1612yes—
219C4×C53#57512yes—
220C4×C53#57612yes—
221Dic53—81no—
222#3208—531no—
223#3206—531no—
224#3167—531no—
225#3165D5311yescharacteristic, complemented
226#3151—531no—
227C2×#486—531no—
228#1715—532no—
229#1714—532no—
230#1708—532no—
231#1706—532no—
232C8×M16—531no—
233C2×C53—22no—
234C2×C53C2×#49211yescharacteristic
235C2×C53#315311yescharacteristic
236C2×C53#315511yescharacteristic
237D53—161no—
238D53—162no—
239C2×AGL(1,ℤ/8)—1062no—
240#533C2×D5311yescharacteristic
241#532—531no—
242C2×C4⋊Q8C2×D5311yescharacteristic
243C2×#127—531no—
244#501—22no—
245#500C2×D5312yescomplemented
246#498—531no—
247#498C2×D5312yescomplemented
248#496—1062no—
249#493—532no—
250#492—531no—
251#492—532no—
252#492—532no—
253#492—532no—
254#487—531no—
255#487—532no—
256#486—532no—
257#486—532no—
258#486—531no—
259#486—532no—
260#486—1062no—
261#465—1062no—
262#461—1062no—
263#460—1062no—
264#450—1062no—
265#448—1062no—
266C4×SD16—1062no—
267#431—531no—
268#428—22no—
269C2×#102—1062no—
270C4×M16C2×D5311yescharacteristic
271C2×C4×C8—531no—
272#322—532no—
273C82—532no—
274C53#1962311yescharacteristic, complemented, Sylow
275C2×#41—1061no—
276C22×Q8—21no—
277C22×D4—1061no—
278C22×D4—1061no—
279C22×D4—1061no—
280AGL(1,ℤ/8)—2128no—
281C2×SD16—1064no—
282C2×SD16—1064no—
283C2×SD16—1062no—
284C2×SD16—1064no—
285C2×D8—1064no—
286C2×M16C22⋊D5312yes—
287C22×C8—532no—
288C4⋊Q8—21no—
289C4⋊Q8—21no—
290C4⋊Q8—21no—
291C4⋊Q8—531no—
292C4⋊Q8C22×D5312yes—
293C4⋊Q8C22×D5311yescharacteristic
294C4⋊Q8C22×D5311yescharacteristic
295#127—532no—
296#127—531no—
297#127—1061no—
298#127—1061no—
299#127—1061no—
300#125—21no—
301#124—1061no—
302C22⋊Q8—21no—
303C22⋊Q8—21no—
304#121—1061no—
305#121—1061no—
306C4×Q8—21no—
307C4×Q8—21no—
308C4×D4—1061no—
309C4×D4—1061no—
310#117—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C42C22×D5311yescharacteristic
314#107—24no—
315#106—24no—
316#106—1062no—
317C4⋊C8—532no—
318#103—1062no—
319#103—1064no—
320#102—1064no—
321#102—1064no—
322#102—1062no—
323#102—1064no—
324C22⋊C8—532no—
325#97C22⋊D5312yes—
326C4×C8—532no—
327C4×C8—531no—
328C4×C8—532no—
329C4×C8—531no—
330C4×C8C22×D5312yes—
331C24—2121no—
332#41—2124no—
333C2×Q8—21no—
334C2×Q8—22no—
335C2×Q8—21no—
336C2×Q8—41no—
337C2×Q8—82no—
338C2×Q8—1061no—
339C2×D4—1062no—
340C2×D4—1061no—
341C2×D4—1062no—
342C2×D4—1061no—
343C2×D4—1061no—
344C2×D4—1062no—
345C2×D4—4242no—
346C2×D4—2124no—
347C2×D4—2121no—
348C2×D4—2121no—
349C2×D4—2121no—
350C2×D4—2121no—
351C2×D4—2124no—
352C22×C4—41no—
353C22×C4—2121no—
354C22×C4D4×D5311yescharacteristic
355C22×C4D4×D5311yescharacteristic
356C22×C4C2×C22⋊D5311yescharacteristic
357SD16—4244no—
358SD16—4244no—
359SD16—2124no—
360SD16—2128no—
361D8—2128no—
362M16#1171992418yescomplemented
363C2×C8—532no—
364C2×C8—532no—
365C2×C8—534no—
366C2×C8—2122no—
367C2×C8C2×C22⋊D5314yes—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—21no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—1061no—
379C4⋊C4—1061no—
380C22⋊C4—41no—
381C22⋊C4—2121no—
382C22⋊C4—2121no—
383C42—41no—
384C42D4×D5311yescharacteristic
385C42D4×D5311yescharacteristic
386C42C2×C22⋊D5311yescharacteristic
387C42C23×D5311yescharacteristic
388C23—4242no—
389C23—4242no—
390C23—4242no—
391C23—2121no—
392C23—2122no—
393C23#1313321311yescharacteristic
394Q8—42no—
395Q8—42no—
396Q8—44no—
397Q8—82no—
398Q8—2122no—
399D4—4242no—
400D4—4242no—
401D4—4242no—
402D4—4242no—
403D4—2122no—
404D4—2124no—
405D4—2122no—
406D4—2124no—
407D4—2122no—
408D4—2124no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—82no—
415C2×C4—4242no—
416C2×C4—2121no—
417C2×C4#1313321311yescharacteristic
418C2×C4#1313321311yescharacteristic
419C2×C4#1313321311yescharacteristic
420C2×C4C2×#1171992414yes—
421C2×C4C2×D4×D5311yescharacteristic
422C2×C4C2×D4×D5311yescharacteristic
423C2×C4C22×C22⋊D5311yescharacteristic
424C8—1062no—
425C8—1062no—
426C8—1064no—
427C8C2×#1171992418yes—
428C22—21no—
429C22—21no—
430C22—8484no—
431C22—4242no—
432C22—4241no—
433C22—4241no—
434C22—4242no—
435C22—4242no—
436C22—4241no—
437C22#1681099712yes—
438C22C2×#1313321311yescharacteristic
439C4—21no—
440C4—21no—
441C4—82no—
442C4—81no—
443C4—81no—
444C4—4241no—
445C4#1681099712yes—
446C4C22×#1171992412yes—
447C4#1681142212yes—
448C4#1681142712yes—
449C2—22no—
450C2—8482no—
451C2—8481no—
452C2C2×#1681099711yescharacteristic
453C2#2028518211yescharacteristic
454C2#2028520011yescharacteristic
subgroupG/H
centerC22C2×#13133213
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3165C2
socleC22×C53C2×#127
prime radical#20285167C2
hypercenter#3165D53
hypocenterC53#19623
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. #20285653C2semidirect
2. #20285640C2semidirect
3. #20285275C2semidirect
5. C2×#16810971C2semidirect
6. #20274344C2semidirect
7. #20274338C2semidirect
8. #20274317C2semidirect
9. #20274308C2semidirect
10. M16×#88322C2semidirect
11. C53×#3165C2semidirect
15. #16811019C22semidirect
19. #16810997C22semidirect
20. #16810997C22semidirect
22. #16810973C22semidirect
23. #16810973C22semidirect
28. #16810971C22semidirect
38. #16810344C22semidirect
39. C8×#88322C22semidirect
43. C53×#500C22semidirect
44. C53×#498C22semidirect
129. M16×C53SD16semidirect
225. #3165D53semidirect
245. #500C2×D53semidirect
247. #498C2×D53semidirect
274. C53#19623semidirect
362. M16#11719924semidirect

Maximal subgroups

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

subgroupindex
#202856532
#202856402
#202852752
#202851672
C2×#168109712
#202743442
#202743382
#202743172
#202743082
M16×#883222
C53×#31652
#1962353