Finite Group Databasebeta

← #27537480

Subgroups — #27537480

Order 13568.

674 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811054C211yesmaximal, characteristic
2#20283590C211yesmaximal, characteristic, complemented
3#20283585C211yesmaximal, characteristic, complemented
4#20276589C211yesmaximal, characteristic, complemented
5#20276588C211yesmaximal, characteristic, complemented
6#20276584C211yesmaximal, characteristic, complemented
7#20276583C211yesmaximal, characteristic, complemented
8C4×#13133162C211yesmaximal, characteristic, complemented
9SD32×Dic53C211yesmaximal, characteristic, complemented
10#20276566C211yesmaximal, characteristic, complemented
11#367×D53C211yesmaximal, characteristic, complemented
12#20276426C211yesmaximal, characteristic, complemented
13#20276424C211yesmaximal, characteristic, complemented
14#20276310C211yesmaximal, characteristic, complemented
15C4×SD32×C53C211yesmaximal, characteristic, complemented
16C2×#13133240C2211yescharacteristic
17C2×#13133236C2211yescharacteristic
18#16811105—21no—
19#16811102—21no—
20#16811054C418yescomplemented
21C2×#13133163C2211yescharacteristic
22C2×#13133162C2211yescharacteristic
23C2×#13133158C2211yescharacteristic
24C2×#13133156C2211yescharacteristic
25C2×C16×D53C2211yescharacteristic
26#16810726—21no—
27#16810723—21no—
28#16810721—21no—
29#16810720—21no—
30C4×Dic212C2211yescharacteristic
31Q16×Dic53C2211yescharacteristic
32#16810717C2211yescharacteristic
33C4×D424C2211yescharacteristic
34D8×Dic53C2211yescharacteristic
35#16810705C2211yescharacteristic
36#107×D53C2211yescharacteristic
37#107×D53C2211yescharacteristic
38#16810623—21no—
39#16810621—21no—
40#16810577C2211yescharacteristic
41#16810451C2211yescharacteristic
42#16810450C2211yescharacteristic
43#16810450C2211yescharacteristic
44#16810428C2211yescharacteristic
45#16810427C2211yescharacteristic
46#16810426C2211yescharacteristic
47#16810425C2211yescharacteristic
48#16810424C2211yescharacteristic
49#16810423C2211yescharacteristic
50#16810399C2211yescharacteristic
51#16810398C2211yescharacteristic
52C16×Dic53C2211yescharacteristic
53C2×SD32×C53C2211yescharacteristic
54C4×Q16×C53C2211yescharacteristic
55C4×D8×C53C2211yescharacteristic
56C53×#367C2211yescharacteristic
57C53×#358C2211yescharacteristic
58C53×#357C2211yescharacteristic
59C4×C16×C53C2211yescharacteristic
60C2×#11719949—21no—
61C2×#11719947—21no—
62#13133240—22no—
63#13133240C2×C414yescomplemented
64#13133236—22no—
65#13133236C2×C414yescomplemented
66C2×Dic212C2311yescharacteristic
67C2×#11719923—21no—
68C2×#11719924—21no—
69C2×D424C2311yescharacteristic
70C2×#11719912D411yescharacteristic
71C2×C8×D53C2311yescharacteristic
72Q8×Dic53—21no—
73C4×Dic106—21no—
74#13133182—21no—
75C4×D212—21no—
76C4×D212—41no—
77D4×Dic53—21no—
78#13133179—21no—
79#13133179—41no—
80C4×C22⋊D53—41no—
81#13133177—41no—
82#13133174—41no—
83#13133174D411yescharacteristic
84#13133173—41no—
85C2×#11719919—41no—
86C4⋊C4×D53—21no—
87C4⋊C4×D53—21no—
88C4⋊C4×D53—41no—
89#13133163C2×C414yescomplemented
90#13133162C2×C414yescomplemented
91#13133158C2×C414yescomplemented
92#13133156—22no—
93C2×#11719908C2311yescharacteristic
94C2×#11719908C2311yescharacteristic
95C16×D53C2×C414yescomplemented
96#13133151C2311yescharacteristic
97#13133151C2311yescharacteristic
98#13133150—21no—
99#13133150—21no—
100#13133142—21no—
101#13133141—21no—
102#13133140—21no—
103#13133139—21no—
104#13133138—21no—
105#13133137—21no—
106#13133128D411yescharacteristic
107#13133127D411yescharacteristic
108C8×Dic53C2311yescharacteristic
109C2×Q16×C53C2311yescharacteristic
110C2×D8×C53C2311yescharacteristic
111C4×Q8×C53—21no—
112C4×D4×C53—21no—
113SD32×C53C2×C414yescomplemented
114C2×C16×C53C2311yescharacteristic
115C2×C16×C53C2311yescharacteristic
116#107×C53C2311yescharacteristic
117#107×C53C2311yescharacteristic
118#103×C53—21no—
119#102×C53—21no—
120C4×C8×C53C2311yescharacteristic
121C2×Q8×C53—21no—
122C2×D4×C53—21no—
123C22×C4×C53—41no—
124C2×C22⋊D53—41no—
125C22×Dic53—41no—
126C22×Dic53—41no—
127#11719949—24no—
128#11719949—42no—
129#11719947—24no—
130#11719947—42no—
131C2×D212—21no—
132C2×D212—41no—
133C2×C4×D53—41no—
134C2×C4×D53—41no—
135C2×C4×D53—41no—
136C2×C4×D53—41no—
137C2×C4×D53—41no—
138C2×C4×D53C2×D411yescharacteristic
139C2×Dic106—21no—
140Q16×C53—21no—
141Q16×C53C22×C412yes—
142D8×C53—21no—
143D8×C53C22×C412yes—
144C2×C8×C53C2×D411yescharacteristic
145C2×C8×C53C2411yescharacteristic
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—41no—
149C22⋊C4×C53—41no—
150C42×C53—41no—
151C42×C53C2×D411yescharacteristic
152C22⋊Dic53—41no—
153#11719924—24no—
154#11719923—24no—
155#11719921—41no—
156#11719921—41no—
157#11719921D811yescharacteristic
158#11719921D811yescharacteristic
159C4⋊Dic53—21no—
160C4⋊Dic53—21no—
161C4⋊Dic53—41no—
162#11719919—41no—
163#11719919—41no—
164#11719919—41no—
165#11719919—41no—
166#11719919D811yescharacteristic
167#11719919D811yescharacteristic
168C4×Dic53—41no—
169C4×Dic53—41no—
170C4×Dic53—41no—
171C4×Dic53C2×D411yescharacteristic
172C2×#88322C2×D411yescharacteristic
173C2×#88322C2×D411yescharacteristic
174Dic212—21no—
175Dic212C22×C412yes—
176D424—21no—
177D424C22×C412yes—
178#11719912—22no—
179C8×D53C22×C412yes—
180C8×D53#4112yes—
181C16×C53—21no—
182C16×C53C22×C412yes—
183#11719908—21no—
184#11719908C22×C412yes—
185C23×C53—41no—
186C22×D53—41no—
187C22×D53C2×D811yescharacteristic
188Q8×C53—22no—
189Q8×C53—41no—
190D4×C53—22no—
191D4×C53—41no—
192C2×C4×C53—41no—
193C2×C4×C53—41no—
194C2×C4×C53—41no—
195C2×C4×C53—161no—
196C2×C4×C53C2×D811yescharacteristic
197C2×C4×C53C2×D811yescharacteristic
198C2×C4×C53C22×D411yescharacteristic
199C22⋊D53—44no—
200C2×Dic53—41no—
201C2×Dic53—44no—
202C2×Dic53—41no—
203C2×Dic53—41no—
204C2×Dic53—161no—
205C2×Dic53C2×D811yescharacteristic
206C2×Dic53C2×D811yescharacteristic
207C2×Dic53C2×D811yescharacteristic
208D212—22no—
209D212—44no—
210D212—41no—
211C4×D53—44no—
212C4×D53—161no—
213C4×D53—161no—
214C4×D53—161no—
215C4×D53—161no—
216C4×D53C4×D412yes—
217C4×D53#13512yes—
218Dic106—22no—
219Dic106—41no—
220C8×C53—21no—
221C8×C53C23×C411yescharacteristic
222C8×C53C2×#4111yescharacteristic
223#88322—21no—
224#88322C4×D412yes—
225#33460—531nomaximal, Sylow
226C22×C53—42no—
227C22×C53—81no—
228C22×C53—81no—
229C22×C53C22×D811yescharacteristic
230C2×D53—21no—
231C2×D53—21no—
232C2×D53—42no—
233C2×D53—81no—
234C2×D53—81no—
235C2×D53C4×D812yes—
236C4×C53—21no—
237C4×C53—21no—
238C4×C53—42no—
239C4×C53—81no—
240C4×C53—81no—
241C4×C53C2×C4×D411yescharacteristic
242C4×C53C2×#13511yescharacteristic
243Dic53—21no—
244Dic53—21no—
245Dic53—42no—
246Dic53—81no—
247Dic53—81no—
248Dic53C4×D812yes—
249C2×#508—531no—
250#2951—531no—
251#2951—531no—
252#2177—531no—
253#2177—531no—
254#2176—531no—
255#2176—531no—
256C4×SD32—531no—
257C4×SD32—531no—
258C4×SD32—531no—
259C4×SD32D5311yescharacteristic, complemented
260C2×#367—531no—
261#2139—531no—
262#2139—531no—
263#2108—531no—
264C2×C53—82no—
265C2×C53C2×C4×D811yescharacteristic
266C2×C53#341611yescharacteristic
267C2×C53#341611yescharacteristic
268D53—22no—
269D53—82no—
270C2×#135—531no—
271C2×#135—531no—
272#520—1061no—
273#520—1061no—
274#508—538no—
275C2×Q32—531no—
276C2×SD32—531no—
277C2×SD32C2×D5311yescharacteristic
278C2×D16—531no—
279C2×M32—531no—
280C22×C16—531no—
281#440—1061no—
282#440—1061no—
283#440—1061no—
284#440—1061no—
285C4×Q16—531no—
286C4×Q16—531no—
287C4×Q16—531no—
288C4×Q16C2×D5311yescharacteristic
289C4×D8—531no—
290C4×D8—531no—
291C4×D8—531no—
292C4×D8C2×D5311yescharacteristic
293C2×#107—531no—
294C2×#107—531no—
295#417—1061no—
296#417—1061no—
297#405—531no—
298#367—531no—
299#367—531no—
300#367—531no—
301#367C2×D5311yescharacteristic
302#358—531no—
303#358—531no—
304#358—531no—
305#358C2×D5311yescharacteristic
306#357—531no—
307#357—531no—
308#357—531no—
309#357C2×D5311yescharacteristic
310#346—531no—
311#346—531no—
312C4×C16—531no—
313C4×C16C2×D5311yescharacteristic
314C53#3346011yescharacteristic, complemented, Sylow
315C2×#41—1061no—
316C2×#41—1061no—
317#135—534no—
318#135—534no—
319#135—1062no—
320#135—1062no—
321C2×Q16—531no—
322C2×Q16C22×D5311yescharacteristic
323C2×SD16—1061no—
324C2×SD16—1061no—
325C2×D8—531no—
326C2×D8C22×D5311yescharacteristic
327C2×M16—531no—
328C22×C8—531no—
329C4×Q8—21no—
330C4×Q8—1061no—
331C4×Q8—1061no—
332C4×Q8—1061no—
333C4×D4—21no—
334C4×D4—1061no—
335C4×D4—1061no—
336C4×D4—1061no—
337C4×D4—2121no—
338C4×D4—2121no—
339C4×D4—2121no—
340C4×D4—2121no—
341#117—531no—
342#117—2121no—
343#117—2121no—
344C2×C4⋊C4—1061no—
345C2×C4⋊C4—1061no—
346C2×C4⋊C4—2121no—
347C2×C4⋊C4—2121no—
348Q32—534no—
349SD32—534no—
350SD32C4×D5314yescomplemented
351D16—534no—
352M32—1062no—
353C2×C16—531no—
354C2×C16—534no—
355C2×C16—531no—
356C2×C16C22×D5311yescharacteristic
357C2×C16C22×D5311yescharacteristic
358#107—531no—
359#107—531no—
360#107—1061no—
361#107—1061no—
362#107C22×D5311yescharacteristic
363#107C22×D5311yescharacteristic
364#103—21no—
365#103—1061no—
366#103—1061no—
367#103—1061no—
368#102—21no—
369#102—1061no—
370#102—1061no—
371#102—1061no—
372#97—531no—
373#97—531no—
374C4×C8—531no—
375C4×C8C22×D5311yescharacteristic
376#41—1064no—
377#41—1064no—
378#41—2122no—
379#41—2122no—
380C2×Q8—21no—
381C2×Q8—1061no—
382C2×D4—21no—
383C2×D4—1061no—
384C2×D4—2121no—
385C2×D4—2121no—
386C22×C4—41no—
387C22×C4—531no—
388C22×C4—2121no—
389C22×C4—2121no—
390C22×C4—2121no—
391C22×C4—2121no—
392C22×C4—2121no—
393C22×C4—2121no—
394C22×C4—2121no—
395Q16—21no—
396Q16—532no—
397Q16—1061no—
398Q16C2×C4×D5312yes—
399SD16—1064no—
400SD16—1064no—
401D8—21no—
402D8—532no—
403D8—1061no—
404D8C2×C4×D5312yes—
405M16—1062no—
406C2×C8—531no—
407C2×C8—532no—
408C2×C8—532no—
409C2×C8—531no—
410C2×C8D4×D5311yescharacteristic
411C2×C8C23×D5311yescharacteristic
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—41no—
415C4⋊C4—531no—
416C4⋊C4—531no—
417C4⋊C4—1061no—
418C4⋊C4—1061no—
419C4⋊C4—2121no—
420C4⋊C4—2121no—
421C4⋊C4—2121no—
422C4⋊C4—2121no—
423C4⋊C4—2121no—
424C22⋊C4—41no—
425C22⋊C4—531no—
426C22⋊C4—531no—
427C22⋊C4—2121no—
428C22⋊C4—2121no—
429C22⋊C4—2121no—
430C42—41no—
431C42—531no—
432C42—2121no—
433C42—2121no—
434C42—2121no—
435C42D4×D5311yescharacteristic
436C16—21no—
437C16—532no—
438C16—1061no—
439C16C2×C4×D5312yes—
440C23—41no—
441C23—531no—
442C23—2121no—
443Q8—22no—
444Q8—41no—
445Q8—1062no—
446Q8—2121no—
447D4—22no—
448D4—41no—
449D4—1062no—
450D4—2124no—
451D4—2124no—
452D4—2121no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—161no—
457C2×C4—531no—
458C2×C4—532no—
459C2×C4—532no—
460C2×C4—531no—
461C2×C4—531no—
462C2×C4—8481no—
463C2×C4—8481no—
464C2×C4—8481no—
465C2×C4—8481no—
466C2×C4—8481no—
467C2×C4—2124no—
468C2×C4—2124no—
469C2×C4—2121no—
470C2×C4—2121no—
471C2×C4—2121no—
472C2×C4D8×D5311yescharacteristic
473C2×C4D8×D5311yescharacteristic
474C2×C4C2×D4×D5311yescharacteristic
475C8—21no—
476C8—532no—
477C8—1061no—
478C8C22×C4×D5311yescharacteristic
479C8#41×D5311yescharacteristic
480C22—42no—
481C22—81no—
482C22—81no—
483C22—532no—
484C22—1061no—
485C22—1061no—
486C22—4241no—
487C22—4241no—
488C22—2122no—
489C22C2×D8×D5311yescharacteristic
490C4—21no—
491C4—21no—
492C4—42no—
493C4—81no—
494C4—81no—
495C4—532no—
496C4—1061no—
497C4—1061no—
498C4—4241no—
499C4—4241no—
500C4—2122no—
501C4C4×D4×D5311yescharacteristic
502C4#135×D5311yescharacteristic
503C2—82no—
504C2—1062no—
505C2—4242no—
506C2C4×D8×D5311yescharacteristic
507C2#2028785711yescharacteristic
508C2#2028785811yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutatorC8×C53C23×C4
FrattiniC2×C8C23×D53
FittingC4×SD32×C53C2
socleC22×C53C22×D8
prime radicalC2×#16811054C2
cosocleC2×C8×C53C24
hypercenterC4×SD32D53
hypocenterC53#33460
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ C4×SD32

Complements

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

NH ≅ G/Ntype
2. #20283590C2semidirect
3. #20283585C2semidirect
4. #20276589C2semidirect
5. #20276588C2semidirect
6. #20276584C2semidirect
7. #20276583C2semidirect
8. C4×#13133162C2semidirect
9. SD32×Dic53C2semidirect
10. #20276566C2semidirect
11. #367×D53C2semidirect
12. #20276426C2semidirect
13. #20276424C2semidirect
14. #20276310C2semidirect
15. C4×SD32×C53C2semidirect
20. #16811054C4semidirect
63. #13133240C2×C4semidirect
65. #13133236C2×C4semidirect
89. #13133163C2×C4semidirect
90. #13133162C2×C4semidirect
91. #13133158C2×C4semidirect
95. C16×D53C2×C4semidirect
113. SD32×C53C2×C4semidirect
259. C4×SD32D53semidirect
314. C53#33460semidirect
350. SD32C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168110542
#202835902
#202835852
#202765892
#202765882
#202765842
#202765832
C4×#131331622
SD32×Dic532
#202765662
#367×D532
#202764262
#202764242
#202763102
C4×SD32×C532
#3346053