Finite Group Databasebeta

← #27383887

Subgroups — #27383887

Order 13568.

1774 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D53C211yesmaximal, characteristic
2C2×#16810585C211yesmaximal, characteristic
3#20277097C211yesmaximal, characteristic
4#20274804C214yesmaximal, complemented
5#20274802C211yesmaximal, characteristic, complemented
6#20274801C211yesmaximal, characteristic, complemented
7#20274799C211yesmaximal, characteristic, complemented
8#20274433C214yesmaximal, complemented
9C53×#1823C211yesmaximal, characteristic, complemented
10C23×C4×D53—21no—
11C23×C4×D53C2211yescharacteristic
12C22×#11719912C2212yes—
13C2×#13133175C2212yes—
14C22×C4⋊Dic53C2211yescharacteristic
15C22×#11719919—21no—
16C2×C4⋊C4×D53—21no—
17C2×C4⋊C4×D53—21no—
18C2×C4⋊C4×D53—24no—
19C2×C4⋊C4×D53C418yescomplemented
20C2×C4⋊C4×D53C2212yes—
21C2×C4⋊C4×D53C2212yes—
22#16810585—24no—
23C2×C22⋊#88322C2211yescharacteristic
24C2×#13133129C2212yes—
25#16810485—21no—
26#16810483C2211yescharacteristic
27#16810480C2212yes—
28#16810478—21no—
29C22⋊C4×Dic53—21no—
30#16810470—21no—
31#16810386C2214yescomplemented
32#16810385C2214yescomplemented
33#16810385C2214yescomplemented
34C22×C4⋊C4×C53C2211yescharacteristic
35C2×C22⋊C8×C53C2211yescharacteristic
36C53×#380C2211yescharacteristic
37C53×#340C2214yescomplemented
38C24×D53D411yescharacteristic
39C23×Dic53—21no—
40C23×Dic53D411yescharacteristic
41C22×C4×D53—28no—
42C22×C4×D53—28no—
43C22×C4×D53—22no—
44C22×C4×D53—22no—
45C22×C4×D53—44no—
46C22×C4×D53—44no—
47C22×C4×D53C2×C414yes—
48C22×C4×D53D411yescharacteristic
49C22×C4×D53D411yescharacteristic
50C22×C4×D53D414yes—
51C22×C4×D53C2311yescharacteristic
52C22×C4×D53C2311yescharacteristic
53C2×#11719912—24no—
54C2×#11719912—28no—
55C22×#88322C2312yes—
56#13133175—28no—
57C2×C4⋊Dic53—22no—
58C2×C4⋊Dic53—21no—
59C2×C4⋊Dic53C2×C414yes—
60C2×C4⋊Dic53C2312yes—
61C2×C4⋊Dic53C2312yes—
62C2×#11719919—28no—
63C2×#11719919—22no—
64C2×#11719919—22no—
65C4⋊C4×D53—28no—
66C4⋊C4×D53—28no—
67C4⋊C4×D53—24no—
68C4⋊C4×D53—24no—
69C4⋊C4×D53C2×C418yes—
70C4⋊C4×D53C2×C418yes—
71C2×C22⋊Dic53—21no—
72C2×#11719921—22no—
73C2×#11719921—22no—
74C2×C4×Dic53—22no—
75C22⋊#88322—22no—
76#13133129—24no—
77#13133124—21no—
78#13133124—22no—
79C23×C4×C53—21no—
80C23×C4×C53C2311yescharacteristic
81C22×C8×C53C2312yes—
82C2×C4⋊C4×C53—22no—
83C2×C4⋊C4×C53—21no—
84C2×C4⋊C4×C53C2×C414yes—
85C2×C4⋊C4×C53C2312yes—
86C2×C4⋊C4×C53C2312yes—
87C2×C22⋊C4×C53—21no—
88C22⋊C8×C53—22no—
89#95×C53—21no—
90C24×C53C2×D411yescharacteristic
91C23×D53—22no—
92C23×D53—22no—
93C23×D53—24no—
94C23×D53—24no—
95C23×D53C22⋊C414yes—
96C23×D53C2×D411yescharacteristic
97C23×D53C2×D411yescharacteristic
98C22×C4×C53—24no—
99C22×C4×C53—24no—
100C22×C4×C53—22no—
101C22×C4×C53—22no—
102C22×C4×C53—42no—
103C22×C4×C53—42no—
104C22×C4×C53C22×C412yes—
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53C2×D412yes—
107C22×C4×C53C2411yescharacteristic
108C22×Dic53—24no—
109C22×Dic53—24no—
110C22×Dic53—22no—
111C22×Dic53—22no—
112C22×Dic53—42no—
113C22×Dic53—42no—
114C22×Dic53C22⋊C414yes—
115C22×Dic53C2×D411yescharacteristic
116C22×Dic53C2×D411yescharacteristic
117C2×C4×D53—24no—
118C2×C4×D53—24no—
119C2×C4×D53—24no—
120C2×C4×D53—216no—
121C2×C4×D53—216no—
122C2×C4×D53—28no—
123C2×C4×D53—28no—
124C2×C4×D53—44no—
125C2×C4×D53—44no—
126C2×C4×D53—44no—
127C2×C4×D53—44no—
128C2×C4×D53—432no—
129C2×C4×D53C22⋊C414yes—
130C2×C4×D53C22⋊C414yes—
131C2×C4×D53C22×C412yes—
132C2×C4×D53C22×C412yes—
133C2×C4×D53C2×D412yes—
134C2×C4×D53C2×D412yes—
135C2×C4×D53#4118yes—
136C2×C8×C53—22no—
137C2×C8×C53—24no—
138C4⋊C4×C53—24no—
139C4⋊C4×C53—22no—
140C4⋊C4×C53—24no—
141C4⋊C4×C53C22×C414yes—
142C22⋊C4×C53—42no—
143C22⋊Dic53—42no—
144#11719921—44no—
145#11719921—44no—
146C4⋊Dic53—24no—
147C4⋊Dic53—22no—
148C4⋊Dic53—24no—
149C4⋊Dic53C22×C414yes—
150#11719919—28no—
151#11719919—28no—
152C4×Dic53—44no—
153C2×#88322—22no—
154C2×#88322—24no—
155#11719912—48no—
156C23×C53—22no—
157C23×C53—22no—
158C23×C53C2×C22⋊C412yes—
159C23×C53AGL(1,ℤ/8)12yes—
160C23×C53AGL(1,ℤ/8)12yes—
161C23×C53C22×D411yescharacteristic
162C22×D53—216no—
163C22×D53—216no—
164C22×D53—22no—
165C22×D53—21no—
166C22×D53—21no—
167C22×D53—21no—
168C22×D53—21no—
169C22×D53—22no—
170C22×D53—28no—
171C22×D53—28no—
172C22×D53—28no—
173C22×D53C2×C22⋊C412yes—
174C22×D53C2×C22⋊C412yes—
175C22×D53#12314yes—
176C22×D53#13714yes—
177C2×C4×C53—22no—
178C2×C4×C53—24no—
179C2×C4×C53—24no—
180C2×C4×C53—24no—
181C2×C4×C53—42no—
182C2×C4×C53—42no—
183C2×C4×C53—48no—
184C2×C4×C53—42no—
185C2×C4×C53—42no—
186C2×C4×C53—42no—
187C2×C4×C53C2×C22⋊C412yes—
188C2×C4×C53C23×C411yescharacteristic
189C2×C4×C53C22×D411yescharacteristic
190C2×C4×C53C2×#4114yes—
191C2×Dic53—22no—
192C2×Dic53—28no—
193C2×Dic53—24no—
194C2×Dic53—42no—
195C2×Dic53—42no—
196C2×Dic53—48no—
197C2×Dic53—42no—
198C2×Dic53—42no—
199C2×Dic53—42no—
200C2×Dic53C2×C22⋊C412yes—
201C2×Dic53C2×C22⋊C412yes—
202C2×Dic53#12314yes—
203C4×D53—28no—
204C4×D53—24no—
205C4×D53—24no—
206C4×D53—28no—
207C4×D53—416no—
208C4×D53—416no—
209C4×D53#11718yes—
210C4×D53#12318yes—
211C8×C53—42no—
212#88322—42no—
213#20700—531nomaximal, Sylow
214C22×C53—24no—
215C22×C53—24no—
216C22×C53—22no—
217C22×C53—22no—
218C22×C53#41814yes—
219C22×C53C22×C22⋊C411yescharacteristic
220C22×C53C2×#12312yes—
221C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
222C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
223C22×C53C2×#13712yes—
224C2×D53—232no—
225C2×D53—24no—
226C2×D53—24no—
227C2×D53—24no—
228C2×D53—24no—
229C2×D53—22no—
230C2×D53—22no—
231C2×D53—22no—
232C2×D53—22no—
233C2×D53#38114yes—
234C2×D53#41914yes—
235C4×C53—22no—
236C4×C53—44no—
237C4×C53—82no—
238C4×C53C2×#11712yes—
239C4×C53C2×#12312yes—
240Dic53—22no—
241Dic53—44no—
242Dic53—82no—
243Dic53#38114yes—
244C23×C4⋊C4—531no—
245C2×#407—531no—
246#2280—531no—
247#1824—534no—
248#1823—531no—
249#1823—531no—
250#1823—531no—
251#1823D5311yescharacteristic, complemented
252#1736—534no—
253C2×C53—24no—
254C2×C53C2×#38111yescharacteristic
255C2×C53C2×#41811yescharacteristic
256C2×C53C2×#41911yescharacteristic
257C2×C53#309212yes—
258C2×C53#309212yes—
259D53—24no—
260D53—24no—
261C24×C4—531no—
262C24×C4—1061no—
263C22×M16—532no—
264C2×#117—532no—
265C22×C4⋊C4—531no—
266C22×C4⋊C4—538no—
267C22×C4⋊C4—532no—
268C22×C4⋊C4—532no—
269C22×C4⋊C4—1061no—
270C22×C4⋊C4—1061no—
271C22×C4⋊C4—1064no—
272C22×C4⋊C4—1061no—
273C22×C4⋊C4C2×D5311yescharacteristic
274#407—1064no—
275C2×C22⋊C8—531no—
276C2×C22⋊C8—532no—
277C2×C22⋊C8C2×D5311yescharacteristic
278#381—1061no—
279#380—531no—
280#380—532no—
281#380C2×D5311yescharacteristic
282#379—1061no—
283C4×C22⋊C4—1061no—
284C4×C22⋊C4—1061no—
285#340—534no—
286#340—534no—
287#340—534no—
288#340C2×D5314yescomplemented
289C53#2070011yescharacteristic, complemented, Sylow
290C25—531no—
291C23×C4—21no—
292C23×C4—531no—
293C23×C4—531no—
294C23×C4—534no—
295C23×C4—531no—
296C23×C4—531no—
297C23×C4—534no—
298C23×C4—531no—
299C23×C4—1068no—
300C23×C4—1068no—
301C23×C4—1062no—
302C23×C4—1062no—
303C23×C4—1061no—
304C23×C4—2124no—
305C23×C4—2124no—
306C23×C4C22×D5311yescharacteristic
307C2×M16—1064no—
308C2×M16—1068no—
309C22×C8—532no—
310C22×C8C22×D5312yes—
311#117—1068no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—534no—
315C2×C4⋊C4—532no—
316C2×C4⋊C4—538no—
317C2×C4⋊C4—538no—
318C2×C4⋊C4—532no—
319C2×C4⋊C4—1068no—
320C2×C4⋊C4—1068no—
321C2×C4⋊C4—1064no—
322C2×C4⋊C4—1064no—
323C2×C4⋊C4—1068no—
324C2×C4⋊C4—1062no—
325C2×C4⋊C4—1062no—
326C2×C4⋊C4—1061no—
327C2×C4⋊C4—1062no—
328C2×C4⋊C4C4×D5314yes—
329C2×C4⋊C4C22×D5312yes—
330C2×C4⋊C4C22×D5312yes—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—1062no—
333C2×C22⋊C4—1062no—
334C2×C22⋊C4—1061no—
335C2×C42—1062no—
336C22⋊C8—22no—
337C22⋊C8—1064no—
338C22⋊C8—1062no—
339#95—21no—
340#95—1062no—
341#95—1061no—
342C24—531no—
343C24—531no—
344C24—534no—
345C24—1064no—
346C24—1064no—
347C24—1062no—
348C24—1062no—
349C24D4×D5311yescharacteristic
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—531no—
357C22×C4—532no—
358C22×C4—532no—
359C22×C4—538no—
360C22×C4—534no—
361C22×C4—531no—
362C22×C4—534no—
363C22×C4—534no—
364C22×C4—532no—
365C22×C4—532no—
366C22×C4—1064no—
367C22×C4—10616no—
368C22×C4—10616no—
369C22×C4—1064no—
370C22×C4—1068no—
371C22×C4—1068no—
372C22×C4—1064no—
373C22×C4—1064no—
374C22×C4—1064no—
375C22×C4—1062no—
376C22×C4—1062no—
377C22×C4—21232no—
378C22×C4—2124no—
379C22×C4—2124no—
380C22×C4—2124no—
381C22×C4—2124no—
382C22×C4—2122no—
383C22×C4—2122no—
384C22×C4C2×C4×D5312yes—
385C22×C4D4×D5311yescharacteristic
386C22×C4D4×D5312yes—
387C22×C4C23×D5311yescharacteristic
388M16—2128no—
389C2×C8—22no—
390C2×C8—24no—
391C2×C8—1062no—
392C2×C8—1064no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—24no—
396C4⋊C4—534no—
397C4⋊C4—1068no—
398C4⋊C4—1068no—
399C4⋊C4—1064no—
400C4⋊C4—1062no—
401C4⋊C4—1064no—
402C4⋊C4C2×C4×D5314yes—
403C22⋊C4—42no—
404C22⋊C4—2124no—
405C22⋊C4—2124no—
406C22⋊C4—2122no—
407C42—2124no—
408C23—22no—
409C23—22no—
410C23—532no—
411C23—532no—
412C23—534no—
413C23—534no—
414C23—1068no—
415C23—1068no—
416C23—10616no—
417C23—10616no—
418C23—1062no—
419C23—1068no—
420C23—1061no—
421C23—1061no—
422C23—1061no—
423C23—1061no—
424C23—1062no—
425C23C22⋊C4×D5312yes—
426C23#1313324112yes—
427C23#1313324412yes—
428C23C2×D4×D5311yescharacteristic
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—48no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—532no—
440C2×C4—534no—
441C2×C4—538no—
442C2×C4—538no—
443C2×C4—532no—
444C2×C4—1068no—
445C2×C4—1068no—
446C2×C4—1068no—
447C2×C4—1062no—
448C2×C4—1064no—
449C2×C4—1064no—
450C2×C4—1064no—
451C2×C4—21216no—
452C2×C4—21216no—
453C2×C4—2122no—
454C2×C4—2122no—
455C2×C4—2122no—
456C2×C4—2128no—
457C2×C4—2122no—
458C2×C4—2122no—
459C2×C4C22⋊C4×D5312yes—
460C2×C4C22×C4×D5311yescharacteristic
461C2×C4C2×D4×D5311yescharacteristic
462C2×C4#41×D5314yes—
463C8—42no—
464C8—2122no—
465C22—22no—
466C22—22no—
467C22—24no—
468C22—24no—
469C22—534no—
470C22—534no—
471C22—10632no—
472C22—1062no—
473C22—1062no—
474C22—1062no—
475C22—1062no—
476C22—1064no—
477C22—1064no—
478C22—1064no—
479C22—1064no—
480C22#1681062614yes—
481C22C2×C22⋊C4×D5311yescharacteristic
482C22#123×D5312yes—
483C22C2×#1313324111yescharacteristic
484C22C2×#1313324411yescharacteristic
485C22#137×D5312yes—
486C4—22no—
487C4—44no—
488C4—82no—
489C4—534no—
490C4—1062no—
491C4—4242no—
492C4—2124no—
493C4#117×D5312yes—
494C4#123×D5312yes—
495C2—24no—
496C2—1064no—
497C2—1064no—
498C2#381×D5311yescharacteristic
499C2C2×#1681062611yescharacteristic
500C2#419×D5311yescharacteristic
501C2#2028463412yes—
502C2#2028463912yes—
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1823C2
socleC23×C53C22×D4
prime radicalC24×D53D4
cosocleC22×C4×C53C24
hypercenter#1823D53
hypocenterC53#20700
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
4. #20274804C2semidirect
5. #20274802C2semidirect
6. #20274801C2semidirect
7. #20274799C2semidirect
8. #20274433C2semidirect
9. C53×#1823C2semidirect
19. C2×C4⋊C4×D53C4semidirect
31. #16810386C22semidirect
32. #16810385C22semidirect
33. #16810385C22semidirect
37. C53×#340C22semidirect
251. #1823D53semidirect
288. #340C2×D53semidirect
289. C53#20700semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D532
C2×#168105852
#202770972
#202748042
#202748022
#202748012
#202747992
#202744332
C53×#18232
#2070053