Finite Group Databasebeta

← #27537444

Subgroups — #27537444

Order 13568.

660 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811065C211yesmaximal, characteristic
2#20283590C211yesmaximal, characteristic
3#20283578C211yesmaximal, characteristic, complemented
4#20276583C211yesmaximal, characteristic, complemented
5#20276582C211yesmaximal, characteristic, complemented
6#20276581C211yesmaximal, characteristic, complemented
7Q32×Dic53C211yesmaximal, characteristic, complemented
8C4×#13133163C211yesmaximal, characteristic, complemented
9C4×#13133160C211yesmaximal, characteristic, complemented
10#20276566C211yesmaximal, characteristic, complemented
11#20276502C211yesmaximal, characteristic, complemented
12#20276421C211yesmaximal, characteristic
13#358×D53C211yesmaximal, characteristic, complemented
14#20276313C211yesmaximal, characteristic, complemented
15C53×#2176C211yesmaximal, characteristic, complemented
16C2×#13133240C2211yescharacteristic
17C2×Q16×D53C2211yescharacteristic
18#16811105—21no—
19#16811099—21no—
20#16811065C418yescomplemented
21C2×#13133163C2211yescharacteristic
22C2×#13133162C2211yescharacteristic
23C2×#13133160C2211yescharacteristic
24C2×#13133156C2211yescharacteristic
25C2×C16×D53C2211yescharacteristic
26#16810728—21no—
27#16810727—21no—
28#16810726—21no—
29#16810721—21no—
30C4×Dic212C2211yescharacteristic
31Q16×Dic53C2211yescharacteristic
32Q16×Dic53C2211yescharacteristic
33#16810717C2211yescharacteristic
34C4×D424C2211yescharacteristic
35#16810705C2211yescharacteristic
36#16810660C2211yescharacteristic
37#107×D53C2211yescharacteristic
38#16810630—21no—
39#16810623—21no—
40#16810577C2211yescharacteristic
41#16810450C2211yescharacteristic
42#16810449C2211yescharacteristic
43#16810448C2211yescharacteristic
44#16810428C2211yescharacteristic
45#16810427C2211yescharacteristic
46#16810427C2211yescharacteristic
47#16810426C2211yescharacteristic
48#16810425C2211yescharacteristic
49#16810423C2211yescharacteristic
50#16810398C2211yescharacteristic
51C4×#11719908C2211yescharacteristic
52C16×Dic53C2211yescharacteristic
53C2×Q32×C53C2211yescharacteristic
54C4×Q16×C53C2211yescharacteristic
55C4×Q16×C53C2211yescharacteristic
56C53×#367C2211yescharacteristic
57C53×#358C2211yescharacteristic
58C53×#358C2211yescharacteristic
59C53×#346C2211yescharacteristic
60C2×#11719949—21no—
61C2×Q8×D53—21no—
62#13133240—22no—
63#13133240C2×C414yes—
64C2×Dic212C2311yescharacteristic
65C2×#11719925—21no—
66Q16×D53—22no—
67Q16×D53C2×C414yescomplemented
68C2×#11719924—21no—
69C2×D424C2311yescharacteristic
70C2×#11719912D411yescharacteristic
71C2×C8×D53C2311yescharacteristic
72Q8×Dic53—21no—
73Q8×Dic53—21no—
74C4×Dic106—21no—
75C4×Dic106—41no—
76#13133182—21no—
77#13133182—41no—
78C4×D212—21no—
79C4×D212—41no—
80#13133179—21no—
81#13133179—41no—
82#13133175—21no—
83#13133175—41no—
84#13133174—41no—
85#13133174—41no—
86#13133174D411yescharacteristic
87C4⋊C4×D53—21no—
88C4⋊C4×D53—41no—
89#13133163C2×C414yescomplemented
90#13133162C2×C414yescomplemented
91#13133160C2×C414yescomplemented
92#13133156C2×C414yescomplemented
93C2×#11719908C2311yescharacteristic
94C2×#11719908C2311yescharacteristic
95C16×D53—22no—
96#13133151C2311yescharacteristic
97#13133151C2311yescharacteristic
98#13133150—21no—
99#13133148—21no—
100#13133142—21no—
101#13133141—21no—
102#13133141—21no—
103#13133140—21no—
104#13133139—21no—
105#13133137—21no—
106#13133128D411yescharacteristic
107#13133127D411yescharacteristic
108C8×Dic53C2311yescharacteristic
109C2×Q16×C53C2311yescharacteristic
110C2×Q16×C53C2311yescharacteristic
111C4×Q8×C53—21no—
112C4×Q8×C53—21no—
113Q32×C53C2×C414yescomplemented
114C2×C16×C53C2311yescharacteristic
115C2×C16×C53C2311yescharacteristic
116#107×C53C2311yescharacteristic
117#107×C53C2311yescharacteristic
118#103×C53—21no—
119#103×C53—21no—
120C4×C8×C53C2311yescharacteristic
121C2×Q8×C53—21no—
122C2×Q8×C53—21no—
123#11719949—24no—
124#11719949—42no—
125Q8×D53—24no—
126Q8×D53—42no—
127C2×D212—21no—
128C2×D212—41no—
129C2×C4×D53—41no—
130C2×C4×D53—41no—
131C2×C4×D53—41no—
132C2×C4×D53—41no—
133C2×C4×D53—41no—
134C2×C4×D53C2×D411yescharacteristic
135C2×Dic106—21no—
136C2×Dic106—41no—
137Q16×C53—21no—
138Q16×C53—21no—
139Q16×C53C22×C412yes—
140Q16×C53C22×C412yes—
141C2×C8×C53C2×D411yescharacteristic
142C2×C8×C53C2411yescharacteristic
143C4⋊C4×C53—21no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—41no—
146C4⋊C4×C53—41no—
147C42×C53—41no—
148C42×C53—41no—
149C42×C53C2×D411yescharacteristic
150#11719925—24no—
151#11719924—24no—
152#11719921—41no—
153#11719921—41no—
154#11719921—41no—
155#11719921D811yescharacteristic
156#11719921D811yescharacteristic
157C4⋊Dic53—21no—
158C4⋊Dic53—21no—
159C4⋊Dic53—41no—
160C4⋊Dic53—41no—
161#11719919—41no—
162#11719919—41no—
163#11719919—41no—
164#11719919D811yescharacteristic
165#11719919D811yescharacteristic
166C4×Dic53—41no—
167C4×Dic53—41no—
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—
185C22×D53—41no—
186C22×D53C2×D811yescharacteristic
187Q8×C53—22no—
188Q8×C53—22no—
189Q8×C53—41no—
190Q8×C53—41no—
191C2×C4×C53—41no—
192C2×C4×C53—41no—
193C2×C4×C53—41no—
194C2×C4×C53—41no—
195C2×C4×C53C2×D811yescharacteristic
196C2×C4×C53C2×D811yescharacteristic
197C2×C4×C53C22×D411yescharacteristic
198C2×Dic53—41no—
199C2×Dic53—41no—
200C2×Dic53—41no—
201C2×Dic53C2×D811yescharacteristic
202C2×Dic53C2×D811yescharacteristic
203C2×Dic53C2×D811yescharacteristic
204D212—22no—
205D212—44no—
206D212—41no—
207C4×D53—44no—
208C4×D53—44no—
209C4×D53—161no—
210C4×D53—161no—
211C4×D53—161no—
212C4×D53C4×D412yes—
213C4×D53#13512yes—
214Dic106—22no—
215Dic106—41no—
216Dic106—44no—
217C8×C53—21no—
218C8×C53C23×C411yescharacteristic
219C8×C53C2×#4111yescharacteristic
220#88322—21no—
221#88322C4×D412yes—
222#33455—531nomaximal, Sylow
223C22×C53C22×D811yescharacteristic
224C2×D53—21no—
225C2×D53—21no—
226C2×D53—42no—
227C2×D53—81no—
228C2×D53—81no—
229C2×D53C4×D812yes—
230C4×C53—21no—
231C4×C53—21no—
232C4×C53—42no—
233C4×C53—42no—
234C4×C53—81no—
235C4×C53—81no—
236C4×C53C2×C4×D411yescharacteristic
237C4×C53C2×#13511yescharacteristic
238Dic53—21no—
239Dic53—21no—
240Dic53—42no—
241Dic53—81no—
242Dic53—81no—
243Dic53C4×D812yes—
244C2×#510—531no—
245#2951—531no—
246#2950—531no—
247#2176—531no—
248#2176D5311yescharacteristic, complemented
249#2175—531no—
250#2175—531no—
251C4×Q32—531no—
252C4×Q32—531no—
253C4×SD32—531no—
254C4×SD32—531no—
255#2159—531no—
256#2138—531no—
257C2×#358—531no—
258#2108—531no—
259C2×C53C2×C4×D811yescharacteristic
260C2×C53#341611yescharacteristic
261C2×C53#341711yescharacteristic
262D53—22no—
263D53—82no—
264C2×#135—531no—
265C22×Q16—531no—
266#520—1061no—
267#519—1061no—
268#510—538no—
269C2×Q32—531no—
270C2×Q32C2×D5311yescharacteristic
271C2×SD32—531no—
272C2×SD32—531no—
273C2×M32—531no—
274C22×C16—531no—
275#441—1061no—
276#441—1061no—
277#440—1061no—
278#440—1061no—
279C4×Q16—531no—
280C4×Q16—531no—
281C4×Q16—531no—
282C4×Q16—531no—
283C4×Q16C2×D5311yescharacteristic
284C4×Q16C2×D5311yescharacteristic
285C4×D8—531no—
286C4×D8—531no—
287#427—531no—
288C2×#107—531no—
289#419—1061no—
290#417—1061no—
291#405—531no—
292#367—531no—
293#367C2×D5311yescharacteristic
294#366—531no—
295#366—531no—
296#358—531no—
297#358—531no—
298#358—531no—
299#358—531no—
300#358C2×D5311yescharacteristic
301#358C2×D5311yescharacteristic
302#357—531no—
303#357—531no—
304#346—531no—
305#346C2×D5311yescharacteristic
306C4×C16—531no—
307C4×C16—531no—
308C53#3345511yescharacteristic, complemented, Sylow
309C2×#41—1061no—
310C22×Q8—1061no—
311#135—534no—
312#135—1062no—
313C2×Q16—534no—
314C2×Q16—531no—
315C2×Q16—1062no—
316C2×Q16—1061no—
317C2×Q16C22×D5311yescharacteristic
318C2×Q16C22×D5311yescharacteristic
319C2×SD16—1061no—
320C2×D8—531no—
321C2×M16—531no—
322C22×C8—531no—
323C4×Q8—21no—
324C4×Q8—21no—
325C4×Q8—1061no—
326C4×Q8—1061no—
327C4×Q8—1061no—
328C4×Q8—1061no—
329C4×Q8—2121no—
330C4×Q8—2121no—
331C4×D4—1061no—
332C4×D4—1061no—
333C4×D4—2121no—
334C4×D4—2121no—
335#117—531no—
336#117—1061no—
337#117—2121no—
338#117—2121no—
339#117—2121no—
340C2×C4⋊C4—1061no—
341C2×C4⋊C4—2121no—
342Q32—534no—
343Q32C4×D5314yescomplemented
344SD32—534no—
345SD32—534no—
346M32—534no—
347C2×C16—531no—
348C2×C16—531no—
349C2×C16—1062no—
350C2×C16C22×D5311yescharacteristic
351C2×C16C22×D5311yescharacteristic
352#107—531no—
353#107—531no—
354#107—1061no—
355#107C22×D5311yescharacteristic
356#107C22×D5311yescharacteristic
357#106—1061no—
358#103—21no—
359#103—21no—
360#103—1061no—
361#103—1061no—
362#103—1061no—
363#103—1061no—
364#102—1061no—
365#102—1061no—
366#97—531no—
367#97—531no—
368C4×C8—531no—
369C4×C8C22×D5311yescharacteristic
370#41—1064no—
371#41—2122no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—1064no—
375C2×Q8—1061no—
376C2×Q8—2121no—
377C2×Q8—2122no—
378C2×D4—1061no—
379C2×D4—2121no—
380C22×C4—531no—
381C22×C4—2121no—
382C22×C4—2121no—
383C22×C4—2121no—
384C22×C4—2121no—
385C22×C4—2121no—
386Q16—21no—
387Q16—21no—
388Q16—532no—
389Q16—1064no—
390Q16—1061no—
391Q16C2×C4×D5312yes—
392Q16C2×C4×D5312yes—
393SD16—1064no—
394D8—532no—
395D8—1061no—
396M16—1062no—
397C2×C8—531no—
398C2×C8—532no—
399C2×C8—532no—
400C2×C8—531no—
401C2×C8D4×D5311yescharacteristic
402C2×C8C23×D5311yescharacteristic
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—531no—
408C4⋊C4—531no—
409C4⋊C4—1061no—
410C4⋊C4—1061no—
411C4⋊C4—2121no—
412C4⋊C4—2121no—
413C4⋊C4—2121no—
414C4⋊C4—2121no—
415C4⋊C4—2121no—
416C22⋊C4—531no—
417C22⋊C4—531no—
418C22⋊C4—2121no—
419C22⋊C4—2121no—
420C22⋊C4—2121no—
421C42—41no—
422C42—41no—
423C42—531no—
424C42—2121no—
425C42—2121no—
426C42—2121no—
427C42—2121no—
428C42—2121no—
429C42D4×D5311yescharacteristic
430C16—21no—
431C16—532no—
432C16—1061no—
433C16C2×C4×D5312yes—
434C23—531no—
435C23—2121no—
436Q8—22no—
437Q8—22no—
438Q8—41no—
439Q8—41no—
440Q8—1062no—
441Q8—2124no—
442Q8—2121no—
443D4—1062no—
444D4—2124no—
445D4—2121no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—531no—
451C2×C4—532no—
452C2×C4—532no—
453C2×C4—531no—
454C2×C4—531no—
455C2×C4—8481no—
456C2×C4—8481no—
457C2×C4—8481no—
458C2×C4—2124no—
459C2×C4—2124no—
460C2×C4—2121no—
461C2×C4—2121no—
462C2×C4—2121no—
463C2×C4D8×D5311yescharacteristic
464C2×C4D8×D5311yescharacteristic
465C2×C4C2×D4×D5311yescharacteristic
466C8—21no—
467C8—532no—
468C8—1061no—
469C8C22×C4×D5311yescharacteristic
470C8#41×D5311yescharacteristic
471C22—532no—
472C22—1061no—
473C22—1061no—
474C22—4241no—
475C22—4241no—
476C22—2122no—
477C22C2×D8×D5311yescharacteristic
478C4—21no—
479C4—21no—
480C4—42no—
481C4—42no—
482C4—81no—
483C4—81no—
484C4—532no—
485C4—1061no—
486C4—1061no—
487C4—4241no—
488C4—4241no—
489C4—2122no—
490C4C4×D4×D5311yescharacteristic
491C4#135×D5311yescharacteristic
492C2—1062no—
493C2—4242no—
494C2C4×D8×D5311yescharacteristic
495C2#2028786011yescharacteristic
496C2#2028786711yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutatorC8×C53C23×C4
FrattiniC2×C8C23×D53
FittingC53×#2176C2
socleC22×C53C22×D8
prime radicalC2×#13133240C22
cosocleC2×C8×C53C24
hypercenter#2176D53
hypocenterC53#33455
LCSG ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2176

Complements

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

NH ≅ G/Ntype
3. #20283578C2semidirect
4. #20276583C2semidirect
5. #20276582C2semidirect
6. #20276581C2semidirect
7. Q32×Dic53C2semidirect
8. C4×#13133163C2semidirect
9. C4×#13133160C2semidirect
10. #20276566C2semidirect
11. #20276502C2semidirect
13. #358×D53C2semidirect
14. #20276313C2semidirect
15. C53×#2176C2semidirect
20. #16811065C4semidirect
67. Q16×D53C2×C4semidirect
89. #13133163C2×C4semidirect
90. #13133162C2×C4semidirect
91. #13133160C2×C4semidirect
92. #13133156C2×C4semidirect
113. Q32×C53C2×C4semidirect
248. #2176D53semidirect
308. C53#33455semidirect
343. Q32C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168110652
#202835902
#202835782
#202765832
#202765822
#202765812
Q32×Dic532
C4×#131331632
C4×#131331602
#202765662
#202765022
#202764212
#358×D532
#202763132
C53×#21762
#3345553