Finite Group Databasebeta

← #27537988

Subgroups — #27537988

Order 13568.

1132 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287849C212yesmaximal, complemented
2#20284324C211yesmaximal, characteristic, complemented
3#20276666C214yesmaximal, complemented
4#20276661C214yesmaximal, complemented
5#20276451C212yesmaximal, complemented
6C2×#16810405C211yesmaximal, characteristic, complemented
7C2×C53×#575C211yesmaximal, characteristic, complemented
8C2×#13133239C2211yescharacteristic, complemented
9#16811134—21no—
10#16811064—22no—
11#16811063C2214yescomplemented
12#16811059—22no—
13#16811058C2214yescomplemented
14#16811053—24no—
15C2×#13133157C2212yescomplemented
16#16810881C2212yescomplemented
17#16810876C2212yescomplemented
18#16810869—22no—
19#16810661C2211yescharacteristic, complemented
20#16810620—21no—
21#16810433C2218yescomplemented
22#16810405C2214yescomplemented
23C22×#41×C53—21no—
24C53×#575—24no—
25C53×#575—22no—
26C53×#575C2214yes—
27C2×#137×C53—22no—
28C2×AGL(1,ℤ/8)×C53—22no—
29C2×#135×C53—21no—
30C2×#135×C53C2212yes—
31C22×M16×C53C2211yescharacteristic
32C2×#11719945—21no—
33#13133239—22no—
34#13133239—22no—
35C2×Dic212C2311yescharacteristic, complemented
36C2×#11719913—21no—
37C2×D424C2311yescharacteristic, complemented
38#13133200—41no—
39#13133196—22no—
40#13133191—22no—
41#13133186—41no—
42C4×C22⋊D53—42no—
43#13133176—21no—
44#13133163—24no—
45#13133161—24no—
46#13133160—24no—
47#13133158—24no—
48#13133157—22no—
49#13133157C2314yescomplemented
50C2×#11719908—22no—
51#13133151—21no—
52#13133149C2312yescomplemented
53#13133142—22no—
54#13133139—22no—
55C2×#41×C53—22no—
56C2×#41×C53—24no—
57C2×#41×C53—22no—
58C2×#41×C53—24no—
59C2×#41×C53—41no—
60C2×#41×C53—42no—
61C2×#41×C53—42no—
62C2×#41×C53—41no—
63C22×Q8×C53—21no—
64C22×D4×C53—21no—
65C22×D4×C53—41no—
66C23×C4×C53—41no—
67C23×C4×C53D411yescharacteristic
68#137×C53—28no—
69#137×C53—28no—
70AGL(1,ℤ/8)×C53—28no—
71AGL(1,ℤ/8)×C53—28no—
72#135×C53—24no—
73#135×C53—24no—
74#135×C53—22no—
75#135×C53—22no—
76#135×C53D414yescomplemented
77#135×C53C2314yes—
78C2×Q16×C53—21no—
79C2×Q16×C53D412yescomplemented
80C2×SD16×C53—22no—
81C2×SD16×C53—22no—
82C2×D8×C53—21no—
83C2×D8×C53D412yescomplemented
84C2×M16×C53—22no—
85C2×M16×C53—22no—
86C2×M16×C53D412yes—
87C2×M16×C53C2312yes—
88C22×C8×C53D411yescharacteristic
89C22×C8×C53C2311yescharacteristic
90C24×C53—41no—
91C24×C53D811yescharacteristic
92#41×C53—28no—
93#41×C53—24no—
94#41×C53—24no—
95#41×C53—44no—
96#41×C53—44no—
97#41×C53—44no—
98#41×C53—44no—
99#41×C53—44no—
100#41×C53—42no—
101#41×C53—42no—
102C2×Q8×C53—24no—
103C2×Q8×C53—22no—
104C2×Q8×C53—41no—
105C2×Q8×C53—42no—
106C2×D4×C53—24no—
107C2×D4×C53—22no—
108C2×D4×C53—41no—
109C2×D4×C53—42no—
110C2×D4×C53—44no—
111C2×D4×C53—42no—
112C2×D4×C53—44no—
113C2×D4×C53—42no—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—42no—
119C22×C4×C53—44no—
120C22×C4×C53—44no—
121C22×C4×C53—41no—
122C22×C4×C53—42no—
123C22×C4×C53D812yes—
124C22×C4×C53D811yescharacteristic
125C22×C4×C53C2×D412yes—
126C22×C4×C53C2×D411yescharacteristic
127C2×C22⋊D53—42no—
128C2×C22⋊D53—41no—
129#11719945—41no—
130#11719945—41no—
131#11719945—41no—
132#11719945—41no—
133C2×D212—21no—
134C2×C4×D53—41no—
135C2×Dic106—21no—
136Q16×C53—24no—
137Q16×C53—22no—
138Q16×C53C2×D414yes—
139SD16×C53—28no—
140SD16×C53—24no—
141SD16×C53—24no—
142D8×C53—24no—
143D8×C53—22no—
144D8×C53C2×D414yes—
145M16×C53—24no—
146M16×C53—24no—
147C2×C8×C53—21no—
148C2×C8×C53—21no—
149C2×C8×C53C2×D412yes—
150C2×C8×C53C2×D411yescharacteristic
151C2×C8×C53C2×D412yes—
152C2×C8×C53C2×D412yes—
153C2×C8×C53C2411yescharacteristic
154C22⋊Dic53—41no—
155C22⋊Dic53—42no—
156#11719921—42no—
157C4⋊Dic53—22no—
158#11719919—42no—
159C4×Dic53—41no—
160Dic212—22no—
161D424—22no—
162#11719913—42no—
163#11719908—24no—
164C23×C53—21no—
165C23×C53—21no—
166C23×C53—44no—
167C23×C53—42no—
168C23×C53—81no—
169C23×C53—84no—
170C23×C53C2×D812yes—
171C23×C53C2×D811yescharacteristic
172C22×D53—41no—
173Q8×C53—24no—
174Q8×C53—44no—
175Q8×C53—42no—
176D4×C53—24no—
177D4×C53—44no—
178D4×C53—48no—
179D4×C53—44no—
180D4×C53—44no—
181D4×C53—42no—
182C2×C4×C53—21no—
183C2×C4×C53—21no—
184C2×C4×C53—22no—
185C2×C4×C53—22no—
186C2×C4×C53—22no—
187C2×C4×C53—22no—
188C2×C4×C53—48no—
189C2×C4×C53—44no—
190C2×C4×C53—42no—
191C2×C4×C53—44no—
192C2×C4×C53—44no—
193C2×C4×C53#12011yescharacteristic
194C2×C4×C53#12011yescharacteristic
195C2×C4×C53C2×D812yes—
196C2×C4×C53C2×D811yescharacteristic
197C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
198C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
199C2×C4×C53C22×D411yescharacteristic
200C22⋊D53—84no—
201C22⋊D53—81no—
202C22⋊D53—81no—
203C2×Dic53—41no—
204C2×Dic53—42no—
205D212—41no—
206D212—41no—
207C4×D53—81no—
208C4×D53—81no—
209Dic106—41no—
210Dic106—41no—
211C8×C53—22no—
212C8×C53#12012yes—
213C8×C53C22×D412yes—
214#33530—531nomaximal, Sylow
215C22×C53—22no—
216C22×C53—41no—
217C22×C53—42no—
218C22×C53—42no—
219C22×C53—44no—
220C22×C53—81no—
221C22×C53—84no—
222C22×C53—88no—
223C22×C53#44711yescharacteristic
224C22×C53#44711yescharacteristic
225C22×C53C22×D811yescharacteristic
226C2×D53—82no—
227C2×D53—81no—
228C4×C53—22no—
229C4×C53—44no—
230C4×C53#44711yescharacteristic
231C4×C53#44711yescharacteristic
232C4×C53C2×#12011yescharacteristic
233C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
234Dic53—82no—
235Dic53—81no—
236C2×#575D5311yescharacteristic, complemented
237#3414—532no—
238#3054—531no—
239#2191—534no—
240#2190—534no—
241#2145—532no—
242C2×#349—531no—
243C2×C53—21no—
244C2×C53—21no—
245C2×C53—21no—
246C2×C53—42no—
247C2×C53—84no—
248C2×C53C2×#44711yescharacteristic
249D53—161no—
250C22×#41—21no—
251#575—24no—
252#575—22no—
253#575C2×D5314yes—
254C2×#137—22no—
255C2×AGL(1,ℤ/8)—22no—
256C2×#135—21no—
257C2×#135—531no—
258C2×#135C2×D5312yes—
259C22×M16C2×D5311yescharacteristic
260#525—1061no—
261#510—534no—
262#510—1062no—
263#509—534no—
264#509—1062no—
265#508—1064no—
266C2×M32—532no—
267#470—532no—
268#469—532no—
269#468—1062no—
270#427—531no—
271#416—1061no—
272#360—538no—
273#349—534no—
274C53#3353011yescharacteristic, complemented, Sylow
275C2×#41—22no—
276C2×#41—24no—
277C2×#41—24no—
278C2×#41—22no—
279C2×#41—41no—
280C2×#41—42no—
281C2×#41—42no—
282C2×#41—41no—
283C2×#41—1061no—
284C22×Q8—21no—
285C22×D4—21no—
286C22×D4—41no—
287C23×C4—41no—
288C23×C4C22⋊D5311yescharacteristic
289#137—28no—
290#137—28no—
291AGL(1,ℤ/8)—28no—
292AGL(1,ℤ/8)—28no—
293#135—24no—
294#135—24no—
295#135—22no—
296#135—22no—
297#135—1062no—
298#135—1062no—
299#135C22⋊D5314yescomplemented
300#135C22×D5314yes—
301C2×Q16—21no—
302C2×Q16—531no—
303C2×Q16C22⋊D5312yescomplemented
304C2×SD16—22no—
305C2×SD16—22no—
306C2×SD16—1061no—
307C2×D8—21no—
308C2×D8—531no—
309C2×D8C22⋊D5312yescomplemented
310C2×M16—22no—
311C2×M16—22no—
312C2×M16C22⋊D5312yes—
313C2×M16C22×D5312yes—
314C22×C8C22⋊D5311yescharacteristic
315C22×C8C22×D5311yescharacteristic
316#123—2121no—
317C22⋊Q8—1062no—
318#121—1062no—
319#120—2121no—
320C4×D4—2122no—
321#117—1061no—
322Q32—1064no—
323SD32—1064no—
324SD32—1064no—
325D16—1064no—
326M32—534no—
327M32—1062no—
328C2×C16—1062no—
329#107—1061no—
330#106—532no—
331#103—1062no—
332#102—1062no—
333C24—41no—
334C24#1171992211yescharacteristic
335#41—28no—
336#41—24no—
337#41—24no—
338#41—44no—
339#41—44no—
340#41—44no—
341#41—44no—
342#41—44no—
343#41—42no—
344#41—42no—
345#41—2121no—
346#41—2121no—
347#41—2121no—
348#41—2121no—
349C2×Q8—24no—
350C2×Q8—22no—
351C2×Q8—41no—
352C2×Q8—42no—
353C2×Q8—1061no—
354C2×D4—24no—
355C2×D4—22no—
356C2×D4—41no—
357C2×D4—42no—
358C2×D4—44no—
359C2×D4—42no—
360C2×D4—44no—
361C2×D4—42no—
362C2×D4—1061no—
363C2×D4—2122no—
364C2×D4—2121no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—42no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—2121no—
375C22×C4#1171992212yes—
376C22×C4#1171992211yescharacteristic
377C22×C4C2×C22⋊D5312yes—
378C22×C4C2×C22⋊D5311yescharacteristic
379Q16—24no—
380Q16—22no—
381Q16—1062no—
382Q16C2×C22⋊D5314yes—
383SD16—28no—
384SD16—24no—
385SD16—24no—
386SD16—2122no—
387D8—24no—
388D8—22no—
389D8—1062no—
390D8C2×C22⋊D5314yes—
391M16—24no—
392M16—24no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8C2×C22⋊D5312yes—
396C2×C8C2×C22⋊D5311yescharacteristic
397C2×C8C2×C22⋊D5312yes—
398C2×C8C2×C22⋊D5312yes—
399C2×C8C23×D5311yescharacteristic
400C4⋊C4—1062no—
401C4⋊C4—2122no—
402C22⋊C4—2121no—
403C22⋊C4—2122no—
404C22⋊C4—2122no—
405C42—2121no—
406C16—1064no—
407C23—21no—
408C23—21no—
409C23—44no—
410C23—42no—
411C23—81no—
412C23—84no—
413C23—2121no—
414C23C2×#1171992212yes—
415C23C2×#1171992211yescharacteristic
416Q8—24no—
417Q8—44no—
418Q8—42no—
419Q8—2121no—
420Q8—2121no—
421D4—24no—
422D4—44no—
423D4—48no—
424D4—44no—
425D4—44no—
426D4—42no—
427D4—4244no—
428D4—4241no—
429D4—4241no—
430D4—2121no—
431D4—2121no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—48no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—4241no—
444C2×C4—4241no—
445C2×C4—2122no—
446C2×C4—2121no—
447C2×C4#1313318611yescharacteristic
448C2×C4#1313318611yescharacteristic
449C2×C4C2×#1171992212yes—
450C2×C4C2×#1171992211yescharacteristic
451C2×C4#1313324311yescharacteristic
452C2×C4#1313324311yescharacteristic
453C2×C4C22×C22⋊D5311yescharacteristic
454C8—22no—
455C8#1313318612yes—
456C8C22×C22⋊D5312yes—
457C22—22no—
458C22—41no—
459C22—42no—
460C22—42no—
461C22—44no—
462C22—81no—
463C22—84no—
464C22—88no—
465C22—4242no—
466C22—4241no—
467C22#1681075711yescharacteristic
468C22#1681075711yescharacteristic
469C22C22×#1171992211yescharacteristic
470C4—22no—
471C4—44no—
472C4—4242no—
473C4—4241no—
474C4#1681075711yescharacteristic
475C4#1681075711yescharacteristic
476C4C2×#1313318611yescharacteristic
477C4C2×#1313324311yescharacteristic
478C2—21no—
479C2—21no—
480C2—21no—
481C2—42no—
482C2—84no—
483C2—8481no—
484C2C2×#1681075711yescharacteristic
subgroupG/H
centerC4#16810757
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC2×C53×#575C2
socleC2×C53C2×#447
hypercenterC2×#575D53
hypocenterC53#33530
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×M16 ◁ C2×#575

Complements

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

NH ≅ G/Ntype
1. #20287849C2semidirect
2. #20284324C2semidirect
3. #20276666C2semidirect
4. #20276661C2semidirect
5. #20276451C2semidirect
6. C2×#16810405C2semidirect
7. C2×C53×#575C2semidirect
8. C2×#13133239C22semidirect
11. #16811063C22semidirect
13. #16811058C22semidirect
15. C2×#13133157C22semidirect
16. #16810881C22semidirect
17. #16810876C22semidirect
19. #16810661C22semidirect
21. #16810433C22semidirect
22. #16810405C22semidirect
35. C2×Dic212C23semidirect
37. C2×D424C23semidirect
49. #13133157C23semidirect
52. #13133149C23semidirect
76. #135×C53D4semidirect
79. C2×Q16×C53D4semidirect
83. C2×D8×C53D4semidirect
236. C2×#575D53semidirect
274. C53#33530semidirect
299. #135C22⋊D53semidirect
303. C2×Q16C22⋊D53semidirect
309. C2×D8C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
#202878492
#202843242
#202766662
#202766612
#202764512
C2×#168104052
C2×C53×#5752
#3353053