Finite Group Databasebeta

← #27299809

Subgroups — #27299809

Order 13568.

886 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20276451C211yesmaximal, characteristic, complemented
2#20276443C211yesmaximal, characteristic, complemented
3C2×#16810430C211yesmaximal, characteristic, complemented
4#20276422C211yesmaximal, characteristic, complemented
5C2×#16810407C211yesmaximal, characteristic, complemented
6#20274769C211yesmaximal, characteristic, complemented
7C53×#3584C211yesmaximal, characteristic
8C2×#13133157C2211yescharacteristic, complemented
9C22×#11719908C2211yescharacteristic, complemented
10C2×#13133153C2211yescharacteristic, complemented
11#16810661C2211yescharacteristic, complemented
12#16810640—21no—
13#16810433—22no—
14#16810430—22no—
15#16810430—22no—
16#16810427—21no—
17#16810424—21no—
18#16810407—22no—
19#16810395—21no—
20C53×#585—21no—
21C53×#578—21no—
22C53×#578C412yescomplemented
23C53×#577—22no—
24C53×#577—22no—
25C53×#576—21no—
26C53×#576C412yescomplemented
27C53×#575—22no—
28C53×#575—21no—
29C2×#135×C53—22no—
30C2×#135×C53C2211yescharacteristic
31C2×#135×C53C2211yescharacteristic
32C2×#131×C53C2211yescharacteristic
33C2×#11719917—21no—
34#13133176—21no—
35#13133157—21no—
36#13133157—21no—
37C2×#11719908—21no—
38C2×#11719908—21no—
39C2×#11719908—21no—
40C2×#11719908—21no—
41#13133153—21no—
42#13133153—21no—
43#13133151—21no—
44#13133149—21no—
45#13133144—42no—
462−1+4×C53—22no—
472−1+4×C53—22no—
482−1+4×C53—41no—
49H(5,𝔽2)×C53—22no—
50H(5,𝔽2)×C53—22no—
51H(5,𝔽2)×C53—42no—
52H(5,𝔽2)×C53—41no—
53C2×#41×C53—24no—
54C2×#41×C53—21no—
55C2×#41×C53—21no—
56C2×#41×C53—42no—
57C2×#41×C53—42no—
58C2×#41×C53D411yescharacteristic
59#137×C53—24no—
60#137×C53—22no—
61#137×C53—24no—
62#137×C53—22no—
63AGL(1,ℤ/8)×C53—24no—
64AGL(1,ℤ/8)×C53—22no—
65AGL(1,ℤ/8)×C53—24no—
66AGL(1,ℤ/8)×C53—22no—
67#135×C53—24no—
68#135×C53—24no—
69#135×C53—22no—
70#135×C53—22no—
71#135×C53—21no—
72#135×C53—21no—
73#135×C53—21no—
74#135×C53—21no—
75#135×C53C2×C412yes—
76#135×C53D412yes—
77#135×C53D412yes—
78#135×C53D412yes—
79C2×Q16×C53—22no—
80C2×Q16×C53C2×C411yescharacteristic
81C2×Q16×C53D411yescharacteristic
82C2×SD16×C53—24no—
83C2×SD16×C53—21no—
84C2×SD16×C53—21no—
85C2×D8×C53—22no—
86C2×D8×C53C2×C411yescharacteristic
87C2×D8×C53D411yescharacteristic
88#131×C53—21no—
89#131×C53—21no—
90#131×C53C2×C412yes—
91#131×C53D412yes—
92C2×M16×C53—21no—
93C2×M16×C53D411yescharacteristic
94C22×C8×C53—21no—
95C22×C8×C53C2311yescharacteristic
96#41×C53—21no—
97#41×C53—21no—
98#41×C53—24no—
99#41×C53—24no—
100#41×C53—22no—
101#41×C53—22no—
102#41×C53—22no—
103#41×C53—22no—
104#41×C53—44no—
105#41×C53—44no—
106#41×C53—44no—
107#41×C53—44no—
108#41×C53—42no—
109#41×C53—42no—
110#41×C53—42no—
111#41×C53—42no—
112#41×C53—41no—
113#41×C53—41no—
114#41×C53—41no—
115#41×C53—41no—
116#41×C53C22⋊C412yes—
117#41×C53D812yes—
118C2×Q8×C53—24no—
119C2×Q8×C53—21no—
120C2×Q8×C53—21no—
121C2×Q8×C53—42no—
122C2×Q8×C53—42no—
123C2×Q8×C53SD1611yescharacteristic
124C2×D4×C53—21no—
125C2×D4×C53—24no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—44no—
129C2×D4×C53—44no—
130C2×D4×C53—42no—
131C2×D4×C53—44no—
132C2×D4×C53—41no—
133C2×D4×C53—41no—
134C2×D4×C53—42no—
135C2×D4×C53SD1611yescharacteristic
136C22×C4×C53—21no—
137C22×C4×C53—44no—
138C22×C4×C53—41no—
139C22×C4×C53—41no—
140C22×C4×C53C2×D411yescharacteristic
141Q16×C53—24no—
142Q16×C53—21no—
143Q16×C53—21no—
144Q16×C53C22⋊C412yes—
145Q16×C53C22⋊C412yes—
146SD16×C53—24no—
147SD16×C53—24no—
148SD16×C53—22no—
149SD16×C53—22no—
150SD16×C53—22no—
151SD16×C53—22no—
152D8×C53—24no—
153D8×C53—21no—
154D8×C53—21no—
155D8×C53C22⋊C412yes—
156D8×C53C22⋊C412yes—
157M16×C53—21no—
158M16×C53—22no—
159M16×C53—22no—
160M16×C53C22⋊C412yes—
161C2×C8×C53—21no—
162C2×C8×C53—22no—
163C2×C8×C53—22no—
164C2×C8×C53C22×C411yescharacteristic
165C2×C8×C53C2×D411yescharacteristic
166C2×C8×C53C2×D411yescharacteristic
167C2×C8×C53C2×D411yescharacteristic
168C2×C8×C53C2×D411yescharacteristic
169C2×C8×C53C2×D411yescharacteristic
170C22⋊Dic53—41no—
171C4⋊Dic53—41no—
172C4×Dic53—41no—
173#11719917—41no—
174#11719917—41no—
175C2×#88322—41no—
176#11719908—41no—
177#11719908—41no—
178C23×C53—21no—
179C23×C53—44no—
180C23×C53—41no—
181C23×C53—41no—
182C23×C53AGL(1,ℤ/8)11yescharacteristic
183Q8×C53—21no—
184Q8×C53—22no—
185Q8×C53—22no—
186Q8×C53—44no—
187Q8×C53—41no—
188Q8×C53—41no—
189Q8×C53#10212yes—
190D4×C53—22no—
191D4×C53—22no—
192D4×C53—22no—
193D4×C53—22no—
194D4×C53—21no—
195D4×C53—44no—
196D4×C53—44no—
197D4×C53—44no—
198D4×C53—42no—
199D4×C53—42no—
200D4×C53—42no—
201D4×C53—42no—
202D4×C53—41no—
203D4×C53—41no—
204D4×C53#10212yes—
205C2×C4×C53—21no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—44no—
209C2×C4×C53—44no—
210C2×C4×C53—44no—
211C2×C4×C53—41no—
212C2×C4×C53—41no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53C2×C22⋊C411yescharacteristic
218C2×C4×C53#12011yescharacteristic
219C2×C4×C53#12011yescharacteristic
220C2×C4×C53C2×D811yescharacteristic
221C2×C4×C53C2×SD1611yescharacteristic
222C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
223C2×Dic53—81no—
224C8×C53—22no—
225C8×C53C2×C22⋊C411yescharacteristic
226C8×C53C2×C22⋊C411yescharacteristic
227C8×C53#12011yescharacteristic
228C8×C53#12011yescharacteristic
229#88322—81no—
230#12747—531nomaximal, Sylow
231C22×C53—22no—
232C22×C53—41no—
233C22×C53—42no—
234C22×C53—42no—
235C22×C53—81no—
236C22×C53—161no—
237C22×C53—161no—
238C22×C53—164no—
239C22×C53#41811yescharacteristic
240C22×C53#44711yescharacteristic
241C22×C53#45011yescharacteristic
242C4×C53—22no—
243C4×C53—42no—
244C4×C53—42no—
245C4×C53#37911yescharacteristic
246C4×C53C2×#10211yescharacteristic
247C4×C53#44711yescharacteristic
248C4×C53#45011yescharacteristic
249Dic53—161no—
250#3584D5311yescharacteristic
251#2145—531no—
252#2143—531no—
253C2×#359—531no—
254#2138—531no—
255C2×#350—531no—
256#1814—531no—
257C2×C53—21no—
258C2×C53—21no—
259C2×C53—21no—
260C2×C53—42no—
261C2×C53—82no—
262C2×C53—82no—
263C2×C53#178611yescharacteristic
264#585—21no—
265#578—21no—
266#578Dic5312yescomplemented
267#577—22no—
268#577—22no—
269#576—21no—
270#576Dic5312yescomplemented
271#575—22no—
272#575—21no—
273C2×#135—22no—
274C2×#135C2×D5311yescharacteristic
275C2×#135C2×D5311yescharacteristic
276C2×#131C2×D5311yescharacteristic
277C2×M32—531no—
278C22×C16—531no—
279C2×#108—531no—
280#427—531no—
281#421—1061no—
282#360—1062no—
283#359—1062no—
284#359—1062no—
285#358—1061no—
286#357—1061no—
287#350—1062no—
288#344—1061no—
289C53#1274711yescharacteristic, complemented, Sylow
2902−1+4—22no—
2912−1+4—22no—
2922−1+4—41no—
293H(5,𝔽2)—22no—
294H(5,𝔽2)—22no—
295H(5,𝔽2)—42no—
296H(5,𝔽2)—41no—
297C2×#41—24no—
298C2×#41—21no—
299C2×#41—21no—
300C2×#41—42no—
301C2×#41—42no—
302C2×#41C22⋊D5311yescharacteristic
303#137—22no—
304#137—24no—
305#137—24no—
306#137—22no—
307AGL(1,ℤ/8)—22no—
308AGL(1,ℤ/8)—24no—
309AGL(1,ℤ/8)—24no—
310AGL(1,ℤ/8)—22no—
311#135—24no—
312#135—24no—
313#135—22no—
314#135—22no—
315#135—21no—
316#135—21no—
317#135—21no—
318#135—21no—
319#135C2×Dic5312yes—
320#135C22⋊D5312yes—
321#135C22⋊D5312yes—
322#135C22⋊D5312yes—
323C2×Q16—22no—
324C2×Q16C2×Dic5311yescharacteristic
325C2×Q16C22⋊D5311yescharacteristic
326C2×SD16—24no—
327C2×SD16—21no—
328C2×SD16—21no—
329C2×D8—22no—
330C2×D8C2×Dic5311yescharacteristic
331C2×D8C22⋊D5311yescharacteristic
332#131—21no—
333#131—21no—
334#131C2×Dic5312yes—
335#131C22⋊D5312yes—
336C2×M16—21no—
337C2×M16—1061no—
338C2×M16C22⋊D5311yescharacteristic
339C22×C8—21no—
340C22×C8C22×D5311yescharacteristic
341#117—1061no—
342M32—1061no—
343M32—1061no—
344C2×C16—1061no—
345C2×C16—1061no—
346C2×C16—1061no—
347C2×C16—1061no—
348#108—1061no—
349#108—1061no—
350#107—1061no—
351#106—1061no—
352#104—2122no—
353#41—21no—
354#41—21no—
355#41—24no—
356#41—24no—
357#41—22no—
358#41—22no—
359#41—22no—
360#41—22no—
361#41—44no—
362#41—44no—
363#41—44no—
364#41—44no—
365#41—42no—
366#41—42no—
367#41—42no—
368#41—42no—
369#41—41no—
370#41—41no—
371#41—41no—
372#41—41no—
373#41#1171992212yes—
374#41C22⋊Dic5312yes—
375C2×Q8—24no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—42no—
379C2×Q8—42no—
380C2×Q8#1171992311yescharacteristic
381C2×D4—21no—
382C2×D4—24no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—42no—
388C2×D4—44no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—42no—
392C2×D4#1171992311yescharacteristic
393C22×C4—21no—
394C22×C4—44no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4C2×C22⋊D5311yescharacteristic
398Q16—24no—
399Q16—21no—
400Q16—21no—
401Q16C22⋊Dic5312yes—
402Q16C22⋊Dic5312yes—
403SD16—24no—
404SD16—24no—
405SD16—22no—
406SD16—22no—
407SD16—22no—
408SD16—22no—
409D8—24no—
410D8—21no—
411D8—21no—
412D8C22⋊Dic5312yes—
413D8C22⋊Dic5312yes—
414M16—21no—
415M16—22no—
416M16—22no—
417M16—2121no—
418M16—2121no—
419M16C22⋊Dic5312yes—
420C2×C8—21no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—2121no—
424C2×C8C22×Dic5311yescharacteristic
425C2×C8C2×C22⋊D5311yescharacteristic
426C2×C8C2×C22⋊D5311yescharacteristic
427C2×C8C2×C22⋊D5311yescharacteristic
428C2×C8C2×C22⋊D5311yescharacteristic
429C2×C8C2×C22⋊D5311yescharacteristic
430C4⋊C4—2121no—
431C22⋊C4—2121no—
432C42—2121no—
433C16—2121no—
434C16—2121no—
435C23—21no—
436C23—44no—
437C23—41no—
438C23—41no—
439C23#1313324311yescharacteristic
440Q8—21no—
441Q8—22no—
442Q8—22no—
443Q8—44no—
444Q8—41no—
445Q8—41no—
446Q8#1313313812yes—
447D4—22no—
448D4—22no—
449D4—21no—
450D4—22no—
451D4—22no—
452D4—44no—
453D4—44no—
454D4—44no—
455D4—42no—
456D4—42no—
457D4—42no—
458D4—42no—
459D4—41no—
460D4—41no—
461D4#1313313812yes—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—4241no—
475C2×C4C2×C22⋊Dic5311yescharacteristic
476C2×C4#1313318611yescharacteristic
477C2×C4#1313318611yescharacteristic
478C2×C4C2×#1171992211yescharacteristic
479C2×C4C2×#1171992311yescharacteristic
480C2×C4#1313324311yescharacteristic
481C8—22no—
482C8—4241no—
483C8C2×C22⋊Dic5311yescharacteristic
484C8C2×C22⋊Dic5311yescharacteristic
485C8#1313318611yescharacteristic
486C8#1313318611yescharacteristic
487C22—22no—
488C22—41no—
489C22—42no—
490C22—42no—
491C22—81no—
492C22—161no—
493C22—161no—
494C22—164no—
495C22#1681062811yescharacteristic
496C22#1681075711yescharacteristic
497C22#1681077411yescharacteristic
498C4—22no—
499C4—42no—
500C4—42no—
501C4—8481no—
502C4#1681047911yescharacteristic
503C4C2×#1313313811yescharacteristic
504C4#1681075711yescharacteristic
505C4#1681077411yescharacteristic
506C2—21no—
507C2—21no—
508C2—21no—
509C2—42no—
510C2—82no—
511C2—82no—
512C2#2027464611yescharacteristic
subgroupG/H
centerC4#16810757
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
Fitting, prime radicalC53×#3584C2
socleC2×C53#1786
cosocleC22×C8×C53C23
hypercenter#3584D53
hypocenterC53#12747
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ #3584

Complements

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

NH ≅ G/Ntype
1. #20276451C2semidirect
2. #20276443C2semidirect
3. C2×#16810430C2semidirect
4. #20276422C2semidirect
5. C2×#16810407C2semidirect
6. #20274769C2semidirect
8. C2×#13133157C22semidirect
9. C22×#11719908C22semidirect
10. C2×#13133153C22semidirect
11. #16810661C22semidirect
22. C53×#578C4semidirect
26. C53×#576C4semidirect
266. #578Dic53semidirect
270. #576Dic53semidirect
289. C53#12747semidirect

Maximal subgroups

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

subgroupindex
#202764512
#202764432
C2×#168104302
#202764222
C2×#168104072
#202747692
C53×#35842
#1274753