Finite Group Databasebeta

← #27391044

Subgroups — #27391044

Order 13568.

868 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284436C211yesmaximal, characteristic
2#20283619C212yesmaximal, complemented
3C2×#16810746C211yesmaximal, characteristic, complemented
4#20283140C211yesmaximal, characteristic, complemented
5#20275611C211yesmaximal, characteristic, complemented
6#20275506C211yesmaximal, characteristic, complemented
7#20275385C212yesmaximal, complemented
8#20275215C212yesmaximal, complemented
9#20274766C211yesmaximal, characteristic, complemented
10#20274516C212yesmaximal, complemented
11C53×#1773C211yesmaximal, characteristic, complemented
12#16811423C2212yes—
13C2×#13133239C2211yescharacteristic
14C2×#13133223C2212yes—
15C2×#13133223C2211yescharacteristic
16C2×#13133223C2211yescharacteristic
17C4×#11719945C2211yescharacteristic
18#16810899C412yescomplemented
19#16810894C412yescomplemented
20#16810887C414yescomplemented
21#16810746—24no—
22#16810746—21no—
23#16810746—21no—
24#16810746—21no—
25#16810746—21no—
26#16810741—24no—
27#16810672C2212yes—
28C2×#13133153C2211yescharacteristic
29#16810641C2212yes—
30#16810641C2212yes—
31C2×#13133145C2212yes—
32#16810605C2212yes—
33#16810596C2212yes—
34#16810580—21no—
35#16810580—21no—
36#16810578C2211yescharacteristic
37#16810578C2211yescharacteristic
38C4×#11719917C2211yescharacteristic
39#16810575—21no—
40#97×D53—21no—
41#16810395C2212yes—
42#16810393C2212yes—
43C2×#108×C53C2211yescharacteristic
44C53×#427C2211yescharacteristic
45C53×#405C2212yes—
46C2×#97×C53C2211yescharacteristic
47C53×#339C2212yes—
48C2×#11719945C2312yes—
49C2×#11719945C2311yescharacteristic
50#13133250—22no—
51#13133250C2×C414yes—
52#13133245—22no—
53#13133245C2×C414yes—
54#13133239—21no—
55#13133239—21no—
56#13133239—21no—
57#13133239—21no—
58#13133239—24no—
59C2×Dic212C2×C411yescharacteristic, complemented
60C2×#11719913C2×C412yescomplemented
61C2×D424C2×C411yescharacteristic, complemented
62#13133223—21no—
63#13133223—21no—
64#13133223—22no—
65#13133223—22no—
66#13133223—22no—
67#13133223—22no—
68#13133223—22no—
69#13133223C2×C412yes—
70#13133223C2×C412yes—
71#13133223D414yes—
72C2×#11719917C2312yes—
73C2×#11719917C2311yescharacteristic
74C2×#11719917C2311yescharacteristic
75C2×#11719912—21no—
76C2×#11719912—21no—
77C2×#11719912—22no—
78C2×C8×D53—21no—
79C2×C8×D53—21no—
80C2×C8×D53—22no—
81C4×Dic106—21no—
82C4×D212—21no—
83C4×C22⋊D53—22no—
84#13133176C2311yescharacteristic
85#13133174—21no—
86C42×D53—21no—
87#13133153—21no—
88#13133153—21no—
89#13133153—24no—
90#13133145—22no—
91#13133145—22no—
92#13133145—24no—
93#13133145—24no—
94#13133135C2×C414yescomplemented
95#13133133C2×C414yescomplemented
96#13133131—24no—
97#13133128—21no—
98#13133127—21no—
99#13133127—21no—
100C4×#88322—21no—
101C8×Dic53—21no—
102C8×Dic53—21no—
103C2×M16×C53C2312yes—
104C22×C8×C53C2311yescharacteristic
105C22×C8×C53C2311yescharacteristic
106#117×C53C2312yes—
107C2×C42×C53C2311yescharacteristic
108#108×C53C2×C412yescomplemented
109#108×C53C2×C412yescomplemented
110#107×C53—21no—
111#106×C53—21no—
112#97×C53—22no—
113#97×C53D411yescharacteristic
114#97×C53D411yescharacteristic
115#97×C53D411yescharacteristic
116#97×C53D411yescharacteristic
117C4×C8×C53—22no—
118C22×C4×C53—21no—
119C22×C4×C53C2411yescharacteristic
120C2×C22⋊D53—21no—
121C2×C22⋊D53—22no—
122#11719945—21no—
123#11719945—21no—
124#11719945—22no—
125#11719945—22no—
126#11719945—22no—
127#11719945—22no—
128#11719945C22×C412yes—
129#11719945C2×D412yes—
130C2×D212C22×C412yes—
131C2×D212#4111yescharacteristic
132C2×C4×D53—21no—
133C2×C4×D53—22no—
134C2×C4×D53—22no—
135C2×Dic106C22×C412yes—
136C2×Dic106#4111yescharacteristic
137M16×C53—22no—
138M16×C53C22×C414yes—
139C2×C8×C53—22no—
140C2×C8×C53C22×C411yescharacteristic
141C2×C8×C53C22×C411yescharacteristic
142C2×C8×C53C2×D411yescharacteristic
143C2×C8×C53C2×D411yescharacteristic
144C2×C8×C53C2×D411yescharacteristic
145C2×C8×C53C2×D411yescharacteristic
146C2×C8×C53C2×D412yes—
147C2×C8×C53#4111yescharacteristic
148C2×C8×C53#4111yescharacteristic
149C2×C8×C53#4112yes—
150C4⋊C4×C53—22no—
151C22⋊C4×C53—22no—
152C42×C53—21no—
153C42×C53—22no—
154C42×C53C2×D411yescharacteristic
155C42×C53C2×D411yescharacteristic
156C22⋊Dic53#4112yes—
157#11719921—22no—
158C4⋊Dic53—21no—
159#11719919—22no—
160C4×Dic53—22no—
161C4×Dic53—21no—
162#11719917—21no—
163#11719917—21no—
164#11719917—22no—
165#11719917—22no—
166#11719917C22×C412yes—
167#11719917C2×D412yes—
168C2×#88322—21no—
169C2×#88322—21no—
170C2×#88322—22no—
171Dic212—24no—
172Dic212—21no—
173Dic212—21no—
174D424—24no—
175D424—21no—
176D424—21no—
177#11719913—24no—
178#11719913—24no—
179#11719913—22no—
180#11719913—22no—
181#11719912—22no—
182#11719912—22no—
183#11719912—24no—
184#11719912—42no—
185#11719912—42no—
186C8×D53—22no—
187C8×D53—22no—
188C8×D53—24no—
189C8×D53—42no—
190C8×D53—42no—
191C23×C53C2×#4111yescharacteristic
192C22×D53—21no—
193C22×D53—22no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—42no—
197C2×C4×C53—81no—
198C2×C4×C53C23×C411yescharacteristic
199C2×C4×C53C22×D411yescharacteristic
200C2×C4×C53C22×D411yescharacteristic
201C2×C4×C53C2×#4111yescharacteristic
202C2×C4×C53C2×#4111yescharacteristic
203C2×C4×C53C2×#4111yescharacteristic
204C22⋊D53—22no—
205C22⋊D53—22no—
206C22⋊D53—42no—
207C22⋊D53—42no—
208C2×Dic53—21no—
209C2×Dic53—22no—
210C2×Dic53—22no—
211D212—22no—
212D212—22no—
213D212—21no—
214D212C4×D412yes—
215C4×D53—22no—
216C4×D53—22no—
217C4×D53—42no—
218C4×D53—42no—
219C4×D53—82no—
220Dic106—21no—
221Dic106—22no—
222Dic106—22no—
223Dic106C4×D412yes—
224C8×C53—24no—
225C8×C53C4×D411yescharacteristic
226C8×C53C4×D411yescharacteristic
227C8×C53C4×D411yescharacteristic
228C8×C53C4×D411yescharacteristic
229C8×C53#12112yes—
230C8×C53#12112yes—
231#88322—22no—
232#88322—22no—
233#88322—42no—
234#21559—531nomaximal, Sylow
235C22×C53—41no—
236C22×C53C4×#4111yescharacteristic
237C22×C53#52511yescharacteristic
238C22×C53#53211yescharacteristic
239C2×D53—22no—
240C2×D53—42no—
241C2×D53—42no—
242C2×D53—82no—
243C4×C53—41no—
244C4×C53—41no—
245C4×C53—82no—
246C4×C53C2×C4×D411yescharacteristic
247C4×C53C2×C4×D411yescharacteristic
248C4×C53C2×#12111yescharacteristic
249C4×C53#52911yescharacteristic
250Dic53—22no—
251Dic53—42no—
252Dic53—42no—
253#3067—531no—
254#2955—532no—
255C2×#444—531no—
256#2875—531no—
257#1976—531no—
258#1953—531no—
259#1932—532no—
260#1906—532no—
261#1814—531no—
262#1773D5311yescharacteristic, complemented
263#1757—532no—
264C2×C53—21no—
265C2×C53—21no—
266C2×C53—21no—
267C2×C53C4×#12111yescharacteristic
268D53—42no—
269D53—82no—
270#575—532no—
271C2×#135—531no—
272C2×#131—531no—
273C2×#131—531no—
274C2×#131—532no—
275C4×#41—531no—
276#473—532no—
277#472—532no—
278#471—534no—
279#444—1064no—
280#444—1061no—
281#444—1061no—
282#444—1061no—
283#444—1061no—
284#443—1064no—
285#430—532no—
286C2×#108—531no—
287C2×#108C2×D5311yescharacteristic
288#427C2×D5311yescharacteristic
289#421—532no—
290#421—532no—
291C2×#104—532no—
292#413—532no—
293#410—532no—
294#405—531no—
295#405—531no—
296#405—1061no—
297#405—1061no—
298#405C2×D5312yes—
299C4×M16—531no—
300C4×M16—1061no—
301C2×#97—1061no—
302C2×#97C2×D5311yescharacteristic
303#344—532no—
304#344—532no—
305#339C2×D5312yes—
306C53#2155911yescharacteristic, complemented, Sylow
307C2×#41—531no—
308C2×#41—532no—
309#137—534no—
310#137—1062no—
311AGL(1,ℤ/8)—534no—
312AGL(1,ℤ/8)—1062no—
313#135—1064no—
314#135—1061no—
315#135—1061no—
316#135—1061no—
317#135—1061no—
318C2×Q16—531no—
319C2×SD16—532no—
320C2×D8—531no—
321#131—532no—
322#131—532no—
323#131—534no—
324#131—1061no—
325#131—1061no—
326#131—1062no—
327#131—1062no—
328#131—1062no—
329#131—1062no—
330#131—1062no—
331C2×M16—531no—
332C2×M16—531no—
333C2×M16—532no—
334C2×M16—1061no—
335C2×M16—1061no—
336C2×M16—1062no—
337C2×M16C22×D5312yes—
338C22×C8—1061no—
339C22×C8—1061no—
340C22×C8—1062no—
341C22×C8C22×D5311yescharacteristic
342C22×C8C22×D5311yescharacteristic
343C4×Q8—1061no—
344C4×D4—1061no—
345C4×D4—1062no—
346#117—531no—
347#117—1061no—
348#117C22×D5312yes—
349C2×C42—1061no—
350C2×C42C22×D5311yescharacteristic
351#108—1064no—
352#108—1061no—
353#108—1061no—
354#108C4×D5312yescomplemented
355#108C4×D5312yescomplemented
356#107—21no—
357#106—21no—
358#104—1064no—
359#104—1064no—
360#104—1062no—
361#104—1062no—
362#101—534no—
363#100—534no—
364#99—1064no—
365#97—22no—
366#97—1061no—
367#97—1061no—
368#97—1061no—
369#97D21211yescharacteristic
370#97D21211yescharacteristic
371#97D21211yescharacteristic
372#97D21211yescharacteristic
373C4×C8—22no—
374C4×C8—1061no—
375C4×C8—1061no—
376C4×C8—1061no—
377#41—532no—
378#41—532no—
379#41—1061no—
380#41—1061no—
381#41—1062no—
382#41—1062no—
383#41—1062no—
384#41—1062no—
385C2×Q8—532no—
386C2×Q8—531no—
387C2×D4—532no—
388C2×D4—531no—
389C2×D4—1061no—
390C2×D4—1062no—
391C22×C4—21no—
392C22×C4—1062no—
393C22×C4—1061no—
394C22×C4—1062no—
395C22×C4C23×D5311yescharacteristic
396Q16—1064no—
397Q16—1061no—
398Q16—1061no—
399SD16—1064no—
400SD16—1064no—
401SD16—1062no—
402SD16—1062no—
403D8—1064no—
404D8—1061no—
405D8—1061no—
406M16—22no—
407M16—532no—
408M16—532no—
409M16—1061no—
410M16—1061no—
411M16—1062no—
412M16—1062no—
413M16—1064no—
414M16—1062no—
415M16—1062no—
416M16—2122no—
417M16—2122no—
418M16C2×C4×D5314yes—
419C2×C8—22no—
420C2×C8—1061no—
421C2×C8—1061no—
422C2×C8—1062no—
423C2×C8—1062no—
424C2×C8—1064no—
425C2×C8—1062no—
426C2×C8—2122no—
427C2×C8—2122no—
428C2×C8C2×C4×D5311yescharacteristic
429C2×C8C2×C4×D5311yescharacteristic
430C2×C8C2×D21211yescharacteristic
431C2×C8C2×D21211yescharacteristic
432C2×C8C2×D21211yescharacteristic
433C2×C8C2×D21211yescharacteristic
434C2×C8#1171994511yescharacteristic
435C2×C8#1171994511yescharacteristic
436C2×C8D4×D5312yes—
437C2×C8#1171994912yes—
438C4⋊C4—22no—
439C4⋊C4—1062no—
440C4⋊C4—1061no—
441C22⋊C4—22no—
442C22⋊C4—532no—
443C22⋊C4—1062no—
444C42—21no—
445C42—22no—
446C42—1062no—
447C42—1061no—
448C42C2×D21211yescharacteristic
449C42C2×D21211yescharacteristic
450C23—1061no—
451C23—1062no—
452C23#41×D5311yescharacteristic
453Q8—532no—
454Q8—1061no—
455Q8—1062no—
456Q8—1062no—
457D4—532no—
458D4—1062no—
459D4—1062no—
460D4—1062no—
461D4—1062no—
462D4—1061no—
463D4—2122no—
464D4—2122no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—42no—
468C2×C4—81no—
469C2×C4—1061no—
470C2×C4—1062no—
471C2×C4—1062no—
472C2×C4—1062no—
473C2×C4—1062no—
474C2×C4—4242no—
475C2×C4—2122no—
476C2×C4—2122no—
477C2×C4C22×C4×D5311yescharacteristic
478C2×C4C2×D4×D5311yescharacteristic
479C2×C4C22×D21211yescharacteristic
480C2×C4#41×D5311yescharacteristic
481C2×C4C2×#1171994511yescharacteristic
482C2×C4C2×#1171994911yescharacteristic
483C8—24no—
484C8—1062no—
485C8—1062no—
486C8—2122no—
487C8C4×D21211yescharacteristic
488C8C4×D21211yescharacteristic
489C8C4×D21211yescharacteristic
490C8C4×D21211yescharacteristic
491C8#1313319212yes—
492C8#1313319212yes—
493C22—41no—
494C22—1062no—
495C22—4242no—
496C22—2122no—
497C22—2122no—
498C22C4×#1171994911yescharacteristic
499C22#1681113511yescharacteristic
500C22#1681117011yescharacteristic
501C4—41no—
502C4—41no—
503C4—82no—
504C4—1062no—
505C4—2122no—
506C4—2122no—
507C4C4×D4×D5311yescharacteristic
508C4C2×C4×D21211yescharacteristic
509C4C2×#1313319211yescharacteristic
510C4#1681116011yescharacteristic
511C2—21no—
512C2—21no—
513C2—21no—
514C2—4242no—
515C2—2122no—
516C2C4×#1313319211yescharacteristic
subgroupG/H
centerC4C2×#13133192
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1773C2
socleC2×C53C4×#121
prime radical#20284436C2
cosocleC22×C4×C53C24
hypercenter#1773D53
hypocenterC53#21559
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C8 ◁ #1773

Complements

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

NH ≅ G/Ntype
2. #20283619C2semidirect
3. C2×#16810746C2semidirect
4. #20283140C2semidirect
5. #20275611C2semidirect
6. #20275506C2semidirect
7. #20275385C2semidirect
8. #20275215C2semidirect
9. #20274766C2semidirect
10. #20274516C2semidirect
11. C53×#1773C2semidirect
18. #16810899C4semidirect
19. #16810894C4semidirect
20. #16810887C4semidirect
59. C2×Dic212C2×C4semidirect
60. C2×#11719913C2×C4semidirect
61. C2×D424C2×C4semidirect
94. #13133135C2×C4semidirect
95. #13133133C2×C4semidirect
108. #108×C53C2×C4semidirect
109. #108×C53C2×C4semidirect
262. #1773D53semidirect
306. C53#21559semidirect
354. #108C4×D53semidirect
355. #108C4×D53semidirect

Maximal subgroups

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

subgroupindex
#202844362
#202836192
C2×#168107462
#202831402
#202756112
#202755062
#202753852
#202752152
#202747662
#202745162
C53×#17732
#2155953