Finite Group Databasebeta

← #27539028

Subgroups — #27539028

Order 13568.

558 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133239C211yesmaximal, characteristic, complemented
2#20276920C211yesmaximal, characteristic, complemented
3#20276903C211yesmaximal, characteristic, complemented
4#20276876C211yesmaximal, characteristic, complemented
5#20276869C211yesmaximal, characteristic, complemented
6#20276774C211yesmaximal, characteristic, complemented
7#20276767C211yesmaximal, characteristic, complemented
8#20276653C211yesmaximal, characteristic, complemented
9#20276636C211yesmaximal, characteristic
10C4×#13133163C211yesmaximal, characteristic, complemented
11C4×#13133160C211yesmaximal, characteristic, complemented
12C4×#13133161C211yesmaximal, characteristic, complemented
13C4×#13133158C211yesmaximal, characteristic, complemented
14#20276514C211yesmaximal, characteristic, complemented
15C53×#3168C211yesmaximal, characteristic, complemented
16C2×#13133239C2211yescharacteristic
17C4×#11719945—21no—
18C2×#13133163C2211yescharacteristic
19C2×#13133161C2211yescharacteristic
20C2×#13133160C2211yescharacteristic
21C2×#13133158C2211yescharacteristic
22C4×Dic212C2211yescharacteristic
23C4×Dic212C2211yescharacteristic
24C4×#11719913—21no—
25C4×#11719913—21no—
26C4×D424C2211yescharacteristic
27C4×D424C2211yescharacteristic
28#16810661C2211yescharacteristic
29#16810620—21no—
30#16810450C2211yescharacteristic
31#16810450C2211yescharacteristic
32#16810448C2211yescharacteristic
33#16810448C2211yescharacteristic
34C4⋊#11719908C2211yescharacteristic
35C4⋊#11719908C2211yescharacteristic
36#16810427C2211yescharacteristic
37#16810427C2211yescharacteristic
38#16810426C2211yescharacteristic
39#16810426C2211yescharacteristic
40#16810424C2211yescharacteristic
41#16810424C2211yescharacteristic
42#16810423C2211yescharacteristic
43#16810423C2211yescharacteristic
44C22⋊#11719908C2211yescharacteristic
45C22⋊#11719908C2211yescharacteristic
46C4×#11719908C2211yescharacteristic
47C4×#11719908C2211yescharacteristic
48C53×#529—21no—
49C53×#499C2211yescharacteristic
50C53×#495C2211yescharacteristic
51C53×#488—21no—
52C53×#467C2211yescharacteristic
53C53×#466C2211yescharacteristic
54C53×#465—21no—
55C4×Q16×C53C2211yescharacteristic
56C4×SD16×C53—21no—
57C4×D8×C53C2211yescharacteristic
58C2×C4×C8×C53C2211yescharacteristic
59C2×#11719945—21no—
60#13133239—22no—
61#13133239—22no—
62C2×Dic212C2311yescharacteristic
63C2×#11719913—21no—
64C2×D424C2311yescharacteristic
65C4×Dic106—21no—
66C4×Dic106—21no—
67C4×D212—21no—
68C4×D212—21no—
69C4×C22⋊D53—41no—
70C4×C22⋊D53—41no—
71#13133176—21no—
72#13133174—41no—
73C42×D53—41no—
74#13133163—22no—
75#13133161—22no—
76#13133160—22no—
77#13133158—22no—
78C2×#11719908C2311yescharacteristic
79C2×#11719908C2311yescharacteristic
80C2×#11719908C2311yescharacteristic
81C2×#11719908C2311yescharacteristic
82#13133151C2311yescharacteristic
83#13133151C2311yescharacteristic
84#13133149—21no—
85#13133142—21no—
86#13133142—21no—
87#13133139—21no—
88#13133139—21no—
89C2×Q16×C53C2311yescharacteristic
90C2×SD16×C53—21no—
91C2×D8×C53C2311yescharacteristic
92C22×C8×C53D411yescharacteristic
93C22×C8×C53C2311yescharacteristic
94#126×C53—41no—
95#125×C53—21no—
96#124×C53—21no—
97#123×C53—41no—
98C22⋊Q8×C53—21no—
99#121×C53—21no—
100C4×Q8×C53—21no—
101C4×D4×C53—21no—
102C4×D4×C53—41no—
103#117×C53—41no—
104C2×C42×C53D411yescharacteristic
105#107×C53C2311yescharacteristic
106#107×C53C2311yescharacteristic
107#106×C53—21no—
108#103×C53—21no—
109#103×C53—21no—
110#102×C53—21no—
111#102×C53—21no—
112C4×C8×C53D411yescharacteristic
113C4×C8×C53D411yescharacteristic
114C4×C8×C53C2311yescharacteristic
115C4×C8×C53C2311yescharacteristic
116C2×Q8×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—41no—
119C22×C4×C53—41no—
120C22×C4×C53D811yescharacteristic
121C22×C4×C53D811yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C2×C22⋊D53—41no—
124#11719945—24no—
125#11719945—42no—
126C2×D212—21no—
127C2×C4×D53—41no—
128C2×C4×D53—41no—
129C2×C4×D53—41no—
130C2×Dic106—21no—
131Q16×C53—21no—
132Q16×C53—21no—
133SD16×C53—42no—
134D8×C53—21no—
135D8×C53—21no—
136C2×C8×C53—21no—
137C2×C8×C53—21no—
138C2×C8×C53—41no—
139C2×C8×C53C2×D411yescharacteristic
140C2×C8×C53C2×D411yescharacteristic
141C2×C8×C53C2×D411yescharacteristic
142C2×C8×C53C2411yescharacteristic
143C4⋊C4×C53—21no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—41no—
146C4⋊C4×C53—41no—
147C4⋊C4×C53—41no—
148C4⋊C4×C53—41no—
149C22⋊C4×C53—41no—
150C22⋊C4×C53—41no—
151C22⋊C4×C53—41no—
152C22⋊C4×C53—41no—
153C22⋊C4×C53—41no—
154C42×C53—41no—
155C42×C53D811yescharacteristic
156C42×C53D811yescharacteristic
157C42×C53C2×D411yescharacteristic
158C42×C53C2×D411yescharacteristic
159C22⋊Dic53—41no—
160#11719921—41no—
161#11719921—41no—
162C4⋊Dic53—21no—
163C4⋊Dic53—21no—
164#11719919—41no—
165#11719919—41no—
166C4×Dic53—41no—
167C4×Dic53—41no—
168C4×Dic53—41no—
169Dic212—21no—
170Dic212#4112yes—
171D424—21no—
172D424#4112yes—
173#11719913—24no—
174#11719908—21no—
175#11719908—21no—
176#11719908—21no—
177#11719908—21no—
178C23×C53—41no—
179C23×C53C2×D811yescharacteristic
180C22×D53—41no—
181Q8×C53—41no—
182Q8×C53—41no—
183D4×C53—41no—
184D4×C53—41no—
185D4×C53—82no—
186C2×C4×C53—21no—
187C2×C4×C53—21no—
188C2×C4×C53—22no—
189C2×C4×C53—41no—
190C2×C4×C53—41no—
191C2×C4×C53—41no—
192C2×C4×C53—41no—
193C2×C4×C53—82no—
194C2×C4×C53C2×D811yescharacteristic
195C2×C4×C53C2×D811yescharacteristic
196C2×C4×C53C2×D811yescharacteristic
197C2×C4×C53C2×D811yescharacteristic
198C2×C4×C53C2×D811yescharacteristic
199C2×C4×C53C22×D411yescharacteristic
200C22⋊D53—44no—
201C2×Dic53—41no—
202C2×Dic53—41no—
203C2×Dic53—41no—
204D212—22no—
205D212—41no—
206C4×D53—44no—
207C4×D53—82no—
208C4×D53—82no—
209Dic106—22no—
210Dic106—41no—
211C8×C53—21no—
212C8×C53—21no—
213C8×C53—21no—
214C8×C53C2×#4111yescharacteristic
215C8×C53C2×#4111yescharacteristic
216#33662—531nomaximal, Sylow
217C22×C53—21no—
218C22×C53—21no—
219C22×C53—21no—
220C22×C53—81no—
221C22×C53—81no—
222C22×C53—81no—
223C22×C53C22×D811yescharacteristic
224C2×D53—42no—
225C2×D53—81no—
226C2×D53—81no—
227C4×C53—21no—
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—81no—
232C4×C53—81no—
233C4×C53—81no—
234C4×C53#50812yes—
235C4×C53#52511yescharacteristic
236C4×C53#57511yescharacteristic
237Dic53—42no—
238Dic53—81no—
239Dic53—81no—
240#3168D5311yescharacteristic, complemented
241C4×#135—531no—
242#2237—531no—
243#2234—531no—
244#2229—531no—
245#2228—531no—
246#2211—531no—
247#2210—531no—
248#2189—531no—
249#2186—531no—
250C4×Q32—531no—
251C4×SD32—531no—
252C4×SD32—531no—
253C4×D16—531no—
254#2162—531no—
255C2×C53—41no—
256C2×C53—161no—
257C2×C53#310011yescharacteristic
258C2×C53C2×#50811yescharacteristic
259C2×C53#341411yescharacteristic
260D53—82no—
261C2×#135—531no—
262#529—21no—
263C4×#41—1061no—
264C2×Q32—531no—
265C2×SD32—531no—
266C2×SD32—531no—
267C2×D16—531no—
268#499C2×D5311yescharacteristic
269#495C2×D5311yescharacteristic
270#488—21no—
271#467C2×D5311yescharacteristic
272#466C2×D5311yescharacteristic
273#465—21no—
274C4×Q16—531no—
275C4×Q16—531no—
276C4×Q16C2×D5311yescharacteristic
277C4×SD16—21no—
278C4×SD16—1061no—
279C4×SD16—1061no—
280C4×D8—531no—
281C4×D8—531no—
282C4×D8C2×D5311yescharacteristic
283#427—531no—
284#416—1061no—
285C2×C4×C8C2×D5311yescharacteristic
286#367—531no—
287#367—531no—
288#366—531no—
289#366—531no—
290C4⋊C16—531no—
291C4⋊C16—531no—
292#358—531no—
293#358—531no—
294#358—531no—
295#358—531no—
296#357—531no—
297#357—531no—
298#357—531no—
299#357—531no—
300C22⋊C16—531no—
301C22⋊C16—531no—
302C4×C16—531no—
303C4×C16—531no—
304C53#3366211yescharacteristic, complemented, Sylow
305C2×#41—1061no—
306#135—1062no—
307#135—1062no—
308C2×Q16—531no—
309C2×Q16C22×D5311yescharacteristic
310C2×SD16—21no—
311C2×SD16—1061no—
312C2×D8—531no—
313C2×D8C22×D5311yescharacteristic
314C22×C8C22⋊D5311yescharacteristic
315C22×C8C22×D5311yescharacteristic
316#126—41no—
317#125—21no—
318#124—21no—
319#123—41no—
320C22⋊Q8—21no—
321#121—21no—
322C4×Q8—21no—
323C4×Q8—1061no—
324C4×Q8—1061no—
325C4×D4—21no—
326C4×D4—41no—
327C4×D4—1061no—
328C4×D4—1061no—
329C4×D4—2121no—
330C4×D4—2121no—
331#117—41no—
332#117—1061no—
333#117—2121no—
334C2×C42—2121no—
335C2×C42C22⋊D5311yescharacteristic
336Q32—1062no—
337SD32—1062no—
338SD32—1062no—
339D16—1062no—
340C2×C16—531no—
341C2×C16—531no—
342C2×C16—531no—
343C2×C16—531no—
344#107—531no—
345#107—531no—
346#107C22×D5311yescharacteristic
347#107C22×D5311yescharacteristic
348#106—21no—
349#106—1061no—
350#103—21no—
351#103—21no—
352#103—1061no—
353#103—1061no—
354#102—21no—
355#102—21no—
356#102—1061no—
357#102—1061no—
358C4×C8C22⋊D5311yescharacteristic
359C4×C8C22⋊D5311yescharacteristic
360C4×C8C22×D5311yescharacteristic
361C4×C8C22×D5311yescharacteristic
362#41—1064no—
363#41—2122no—
364C2×Q8—21no—
365C2×Q8—1061no—
366C2×D4—21no—
367C2×D4—41no—
368C2×D4—1061no—
369C2×D4—2121no—
370C22×C4—41no—
371C22×C4—2121no—
372C22×C4—2121no—
373C22×C4—2121no—
374C22×C4#1171992211yescharacteristic
375C22×C4#1171992211yescharacteristic
376C22×C4C2×C22⋊D5311yescharacteristic
377Q16—21no—
378Q16—21no—
379Q16—532no—
380Q16—1061no—
381SD16—42no—
382SD16—1064no—
383D8—21no—
384D8—21no—
385D8—532no—
386D8—1061no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—41no—
390C2×C8C2×C22⋊D5311yescharacteristic
391C2×C8C2×C22⋊D5311yescharacteristic
392C2×C8C2×C22⋊D5311yescharacteristic
393C2×C8C23×D5311yescharacteristic
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—1061no—
401C4⋊C4—1061no—
402C4⋊C4—2121no—
403C4⋊C4—2121no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—2121no—
410C22⋊C4—2121no—
411C22⋊C4—2121no—
412C42—41no—
413C42—2121no—
414C42—2121no—
415C42—2121no—
416C42#1171992211yescharacteristic
417C42#1171992211yescharacteristic
418C42C2×C22⋊D5311yescharacteristic
419C42C2×C22⋊D5311yescharacteristic
420C16—1061no—
421C16—1061no—
422C16—1061no—
423C16—1061no—
424C23—41no—
425C23—2121no—
426C23C2×#1171992211yescharacteristic
427Q8—41no—
428Q8—41no—
429Q8—1062no—
430Q8—2121no—
431D4—41no—
432D4—41no—
433D4—82no—
434D4—1062no—
435D4—2124no—
436D4—2121no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—82no—
445C2×C4—4242no—
446C2×C4—4242no—
447C2×C4—2124no—
448C2×C4—2121no—
449C2×C4—2121no—
450C2×C4—2121no—
451C2×C4C2×#1171992211yescharacteristic
452C2×C4C2×#1171992211yescharacteristic
453C2×C4C2×#1171992211yescharacteristic
454C2×C4C2×#1171992211yescharacteristic
455C2×C4C2×#1171992211yescharacteristic
456C2×C4C22×C22⋊D5311yescharacteristic
457C8—21no—
458C8—21no—
459C8—21no—
460C8#41×D5311yescharacteristic
461C8#41×D5311yescharacteristic
462C22—21no—
463C22—21no—
464C22—21no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—4241no—
469C22—4241no—
470C22—2122no—
471C22C22×#1171992211yescharacteristic
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—81no—
477C4—81no—
478C4—81no—
479C4—4241no—
480C4—4241no—
481C4—2122no—
482C4#1681105312yes—
483C4#1681113011yescharacteristic
484C4#1681142111yescharacteristic
485C2—41no—
486C2—161no—
487C2—4242no—
488C2#2028471111yescharacteristic
489C2C2×#1681105311yescharacteristic
490C2#2028784911yescharacteristic
subgroupG/H
centerC2×C4C2×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#3168C2
socleC22×C53C22×D8
prime radical#20276636C2
hypercenter#3168D53
hypocenterC53#33662
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3168

Complements

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

NH ≅ G/Ntype
1. C4×#13133239C2semidirect
2. #20276920C2semidirect
3. #20276903C2semidirect
4. #20276876C2semidirect
5. #20276869C2semidirect
6. #20276774C2semidirect
7. #20276767C2semidirect
8. #20276653C2semidirect
10. C4×#13133163C2semidirect
11. C4×#13133160C2semidirect
12. C4×#13133161C2semidirect
13. C4×#13133158C2semidirect
14. #20276514C2semidirect
15. C53×#3168C2semidirect
240. #3168D53semidirect
304. C53#33662semidirect

Maximal subgroups

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

subgroupindex
C4×#131332392
#202769202
#202769032
#202768762
#202768692
#202767742
#202767672
#202766532
#202766362
C4×#131331632
C4×#131331602
C4×#131331612
C4×#131331582
#202765142
C53×#31682
#3366253