Finite Group Databasebeta

← #27537418

Subgroups — #27537418

Order 13568.

674 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811065C211yesmaximal, characteristic
2C4×#13133240C211yesmaximal, characteristic
3C4×Q16×D53C211yesmaximal, characteristic, complemented
4#20276584C211yesmaximal, characteristic, complemented
5#20276581C211yesmaximal, characteristic, complemented
6#20276577C211yesmaximal, characteristic, complemented
7Q32×Dic53C211yesmaximal, characteristic, complemented
8#20276571C211yesmaximal, characteristic, complemented
9#20276566C211yesmaximal, characteristic, complemented
10#20276564C211yesmaximal, characteristic, complemented
11#20276502C211yesmaximal, characteristic, complemented
12#20276433C211yesmaximal, characteristic, complemented
13#20276426C211yesmaximal, characteristic
14#20276308C211yesmaximal, characteristic, complemented
15C53×#2176C211yesmaximal, characteristic, complemented
16C2×#13133240C2211yescharacteristic
17C2×Q16×D53C2211yescharacteristic
18C4×#11719949—21no—
19C4×Q8×D53—21no—
20#16811065C418yescomplemented
21C2×#13133163C2211yescharacteristic
22C2×#13133162C2211yescharacteristic
23C2×#13133160C2211yescharacteristic
24C2×#13133156C2211yescharacteristic
25C2×#13133156C2211yescharacteristic
26C4×Dic212C2211yescharacteristic
27Q16×Dic53C2211yescharacteristic
28Q16×Dic53C2211yescharacteristic
29#16810717C2211yescharacteristic
30C4×#11719925—21no—
31#16810715—21no—
32C4×#11719924—21no—
33#16810708—21no—
34C4×D424C2211yescharacteristic
35#16810705C2211yescharacteristic
36#16810660C2211yescharacteristic
37#107×D53C2211yescharacteristic
38#16810618—21no—
39#103×D53—21no—
40C4×C8×D53C2211yescharacteristic
41#16810450C2211yescharacteristic
42#16810449C2211yescharacteristic
43#16810448C2211yescharacteristic
44#16810428C2211yescharacteristic
45#16810427C2211yescharacteristic
46#16810427C2211yescharacteristic
47#16810426C2211yescharacteristic
48#16810425C2211yescharacteristic
49#16810423C2211yescharacteristic
50#16810399C2211yescharacteristic
51C16×Dic53C2211yescharacteristic
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×C8×D53D411yescharacteristic
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—
84C4⋊C4×D53—21no—
85C4⋊C4×D53—41no—
86C42×D53—41no—
87C42×D53—41no—
88C42×D53D411yescharacteristic
89#13133163C2×C414yescomplemented
90#13133162C2×C414yescomplemented
91#13133160C2×C414yescomplemented
92#13133156—22no—
93#13133156C2×C414yescomplemented
94C2×#11719908C2311yescharacteristic
95C2×#11719908C2311yescharacteristic
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—
106C4×#88322D411yescharacteristic
107C8×Dic53D411yescharacteristic
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×D53D811yescharacteristic
135C2×C4×D53D811yescharacteristic
136C2×C4×D53C2×D411yescharacteristic
137C2×Dic106—21no—
138C2×Dic106—41no—
139Q16×C53—21no—
140Q16×C53—21no—
141Q16×C53C22×C412yes—
142Q16×C53C22×C412yes—
143C2×C8×C53C2×D411yescharacteristic
144C2×C8×C53C2411yescharacteristic
145C4⋊C4×C53—21no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—41no—
148C4⋊C4×C53—41no—
149C42×C53—41no—
150C42×C53—41no—
151C42×C53C2×D411yescharacteristic
152#11719925—24no—
153#11719924—24no—
154#11719921—41no—
155#11719921—41no—
156#11719921—41no—
157C4⋊Dic53—21no—
158C4⋊Dic53—21no—
159C4⋊Dic53—41no—
160C4⋊Dic53—41no—
161#11719919—41no—
162#11719919—41no—
163#11719919—41no—
164C4×Dic53—41no—
165C4×Dic53—41no—
166C4×Dic53—41no—
167C4×Dic53—41no—
168C4×Dic53—41no—
169C4×Dic53D811yescharacteristic
170C4×Dic53D811yescharacteristic
171C4×Dic53C2×D411yescharacteristic
172C2×#88322C2×D411yescharacteristic
173C2×#88322C2×D411yescharacteristic
174Dic212—21no—
175Dic212C22×C412yes—
176D424—21no—
177D424C22×C412yes—
178C8×D53—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—22no—
208C4×D53—41no—
209C4×D53—44no—
210C4×D53—44no—
211C4×D53—82no—
212C4×D53—82no—
213C4×D53—82no—
214C4×D53C4×D412yes—
215C4×D53#13512yes—
216Dic106—22no—
217Dic106—41no—
218Dic106—44no—
219C8×C53—21no—
220C8×C53C23×C411yescharacteristic
221C8×C53C2×#4111yescharacteristic
222#88322—21no—
223#88322C4×D412yes—
224C4×#510—531nomaximal, Sylow
225C22×C53C22×D811yescharacteristic
226C2×D53—21no—
227C2×D53—21no—
228C2×D53—42no—
229C2×D53—81no—
230C2×D53—81no—
231C2×D53C4×D812yes—
232C4×C53—21no—
233C4×C53—21no—
234C4×C53—42no—
235C4×C53—42no—
236C4×C53—81no—
237C4×C53—81no—
238C4×C53C2×C4×D411yescharacteristic
239C4×C53C2×#13511yescharacteristic
240Dic53—21no—
241Dic53—42no—
242Dic53—81no—
243Dic53—81no—
244Dic53C4×D812yes—
245Dic53#51012yescomplemented
246C2×#510—531no—
247C4×#135—531no—
248C2×C4×Q16—531no—
249#2176—531no—
250#2176D5311yescharacteristic, complemented
251#2175—531no—
252#2175—531no—
253C4×Q32—531no—
254C4×Q32—531no—
255C4×SD32—531no—
256C4×SD32—531no—
257#2159—531no—
258#2141—531no—
259#2139—531no—
260C4×M32—531no—
261C2×C53C2×C4×D811yescharacteristic
262C2×C53C2×#51011yescharacteristic
263C2×C53#341411yescharacteristic
264D53—22no—
265D53—82no—
266C2×#135—531no—
267C22×Q16—531no—
268C4×#41—1061no—
269C2×C4×Q8—1061no—
270#510—538no—
271C2×Q32—531no—
272C2×Q32C2×D5311yescharacteristic
273C2×SD32—531no—
274C2×SD32—531no—
275C2×M32—531no—
276C2×M32—531no—
277C4×Q16—531no—
278C4×Q16—531no—
279C4×Q16—531no—
280C4×Q16—531no—
281C4×Q16—1061no—
282C4×Q16—1061no—
283C4×Q16C2×D5311yescharacteristic
284C4×Q16C2×D5311yescharacteristic
285C4×SD16—1061no—
286C4×SD16—1061no—
287C4×D8—531no—
288C4×D8—531no—
289#427—531no—
290C2×#107—531no—
291#416—1061no—
292C2×#103—1061no—
293C2×C4×C8—531no—
294#367—531no—
295#367C2×D5311yescharacteristic
296#366—531no—
297#366—531no—
298#358—531no—
299#358—531no—
300#358—531no—
301#358—531no—
302#358C2×D5311yescharacteristic
303#358C2×D5311yescharacteristic
304#357—531no—
305#357—531no—
306#346—531no—
307#346C2×D5311yescharacteristic
308C4×C16—531no—
309C4×C16—531no—
310C53C4×#51011yescharacteristic, complemented, Sylow
311C2×#41—1061no—
312C22×Q8—1061no—
313#135—534no—
314#135—1062no—
315C2×Q16—534no—
316C2×Q16—531no—
317C2×Q16—1062no—
318C2×Q16—1061no—
319C2×Q16C22×D5311yescharacteristic
320C2×Q16C22×D5311yescharacteristic
321C2×SD16—1061no—
322C2×D8—531no—
323C22×C8—531no—
324C22×C8—531no—
325C4×Q8—21no—
326C4×Q8—21no—
327C4×Q8—1061no—
328C4×Q8—1061no—
329C4×Q8—1061no—
330C4×Q8—1061no—
331C4×Q8—2121no—
332C4×Q8—2121no—
333C4×D4—1061no—
334C4×D4—1061no—
335C4×D4—2121no—
336C4×D4—2121no—
337#117—1061no—
338#117—2121no—
339C2×C4⋊C4—1061no—
340C2×C4⋊C4—2121no—
341C2×C42—531no—
342C2×C42—2121no—
343C2×C42—2121no—
344Q32—534no—
345Q32C4×D5314yescomplemented
346SD32—534no—
347SD32—534no—
348M32—534no—
349M32—1062no—
350C2×C16—531no—
351C2×C16—531no—
352C2×C16C22×D5311yescharacteristic
353C2×C16C22×D5311yescharacteristic
354#107—531no—
355#107—531no—
356#107—1061no—
357#107C22×D5311yescharacteristic
358#107C22×D5311yescharacteristic
359#106—1061no—
360#103—21no—
361#103—21no—
362#103—1061no—
363#103—1061no—
364#103—1061no—
365#103—1061no—
366#102—1061no—
367#102—1061no—
368C4×C8—531no—
369C4×C8—531no—
370C4×C8—531no—
371C4×C8C22×D5311yescharacteristic
372#41—1064no—
373#41—2122no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—1064no—
377C2×Q8—1061no—
378C2×Q8—2121no—
379C2×Q8—2122no—
380C2×D4—1061no—
381C2×D4—2121no—
382C22×C4—531no—
383C22×C4—531no—
384C22×C4—531no—
385C22×C4—2121no—
386C22×C4—2121no—
387C22×C4—2121no—
388C22×C4—2121no—
389C22×C4—2121no—
390Q16—21no—
391Q16—21no—
392Q16—532no—
393Q16—1064no—
394Q16—1061no—
395Q16C2×C4×D5312yes—
396Q16C2×C4×D5312yes—
397SD16—1064no—
398D8—532no—
399D8—1061no—
400C2×C8—531no—
401C2×C8—532no—
402C2×C8—532no—
403C2×C8—531no—
404C2×C8—1062no—
405C2×C8D4×D5311yescharacteristic
406C2×C8C23×D5311yescharacteristic
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—1061no—
412C4⋊C4—1061no—
413C4⋊C4—2121no—
414C4⋊C4—2121no—
415C4⋊C4—2121no—
416C4⋊C4—2121no—
417C4⋊C4—2121no—
418C22⋊C4—2121no—
419C22⋊C4—2121no—
420C22⋊C4—2121no—
421C42—41no—
422C42—41no—
423C42—531no—
424C42—531no—
425C42—531no—
426C42—2121no—
427C42—2121no—
428C42—2121no—
429C42—2121no—
430C42—2121no—
431C42D4×D5311yescharacteristic
432C16—21no—
433C16—532no—
434C16—1061no—
435C16C2×C4×D5312yes—
436C23—531no—
437C23—2121no—
438Q8—22no—
439Q8—22no—
440Q8—41no—
441Q8—41no—
442Q8—1062no—
443Q8—2124no—
444Q8—2121no—
445D4—1062no—
446D4—2124no—
447D4—2121no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—531no—
453C2×C4—531no—
454C2×C4—532no—
455C2×C4—532no—
456C2×C4—531no—
457C2×C4—1062no—
458C2×C4—4242no—
459C2×C4—4242no—
460C2×C4—4242no—
461C2×C4—2121no—
462C2×C4—2124no—
463C2×C4—2124no—
464C2×C4—2121no—
465C2×C4—2121no—
466C2×C4—2121no—
467C2×C4D8×D5311yescharacteristic
468C2×C4D8×D5311yescharacteristic
469C2×C4C2×D4×D5311yescharacteristic
470C8—21no—
471C8—532no—
472C8—1061no—
473C8C22×C4×D5311yescharacteristic
474C8#41×D5311yescharacteristic
475C22—532no—
476C22—1061no—
477C22—1061no—
478C22—4241no—
479C22—4241no—
480C22—2122no—
481C22C2×D8×D5311yescharacteristic
482C4—21no—
483C4—21no—
484C4—42no—
485C4—42no—
486C4—81no—
487C4—81no—
488C4—532no—
489C4—532no—
490C4—1061no—
491C4—4241no—
492C4—4241no—
493C4—2122no—
494C4C4×D4×D5311yescharacteristic
495C4#135×D5311yescharacteristic
496C2—1062no—
497C2—4242no—
498C2C4×D8×D5311yescharacteristic
499C2#510×D5311yescharacteristic
500C2#2028785011yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutatorC8×C53C23×C4
FrattiniC2×C8C23×D53
FittingC53×#2176C2
socleC22×C53C22×D8
prime radicalC2×#13133240C22
cosocleC2×C8×C53C24
hypercenter#2176D53
hypocenterC53C4×#510
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. C4×Q16×D53C2semidirect
4. #20276584C2semidirect
5. #20276581C2semidirect
6. #20276577C2semidirect
7. Q32×Dic53C2semidirect
8. #20276571C2semidirect
9. #20276566C2semidirect
10. #20276564C2semidirect
11. #20276502C2semidirect
12. #20276433C2semidirect
14. #20276308C2semidirect
15. C53×#2176C2semidirect
20. #16811065C4semidirect
67. Q16×D53C2×C4semidirect
89. #13133163C2×C4semidirect
90. #13133162C2×C4semidirect
91. #13133160C2×C4semidirect
93. #13133156C2×C4semidirect
113. Q32×C53C2×C4semidirect
245. Dic53#510semidirect
250. #2176D53semidirect
310. C53C4×#510semidirect
345. Q32C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168110652
C4×#131332402
C4×Q16×D532
#202765842
#202765812
#202765772
Q32×Dic532
#202765712
#202765662
#202765642
#202765022
#202764332
#202764262
#202763082
C53×#21762
C4×#51053