Finite Group Databasebeta

← #27383895

Subgroups — #27383895

Order 13568.

1470 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133175C211yesmaximal, characteristic
2C2×#16810586C211yesmaximal, characteristic
3#20277094C211yesmaximal, characteristic
4#20274809C214yesmaximal, complemented
5#20274808C214yesmaximal, complemented
6#20274436C211yesmaximal, characteristic, complemented
7#20274435C211yesmaximal, characteristic, complemented
8#20274434C211yesmaximal, characteristic, complemented
9C53×#1736C211yesmaximal, characteristic, complemented
10C23×C4×D53C2211yescharacteristic
11C22×#11719917C2211yescharacteristic
12C2×#13133176C2211yescharacteristic
13C2×#13133175—24no—
14C2×#13133175—21no—
15C2×#13133175—21no—
16C2×#13133175C418yescomplemented
17C2×#13133175C2212yes—
18C2×#13133175C2212yes—
19C22×C4⋊Dic53C2211yescharacteristic
20C22×#11719921—21no—
21C22×C4×Dic53—21no—
22#16810586—24no—
23C2×#13133129C2212yes—
24C2×#13133129C2212yes—
25#16810484—21no—
26#16810480C2212yes—
27#16810480C2212yes—
28#16810478—21no—
29C4×#11719921—21no—
30C4×#11719921—21no—
31#16810386C2214yescomplemented
32#16810385C2214yescomplemented
33#16810385C2214yescomplemented
34C22×M16×C53C2211yescharacteristic
35C2×#117×C53C2211yescharacteristic
36C22×C4⋊C4×C53C2211yescharacteristic
37C53×#340C2214yescomplemented
38C24×D53D411yescharacteristic
39C23×Dic53—21no—
40C23×Dic53D411yescharacteristic
41C22×C4×D53—22no—
42C22×C4×D53—22no—
43C22×C4×D53—22no—
44C22×C4×D53—22no—
45C22×C4×D53C2×C414yes—
46C22×C4×D53D411yescharacteristic
47C22×C4×D53D411yescharacteristic
48C22×C4×D53D414yes—
49C22×C4×D53C2311yescharacteristic
50C22×C4×D53C2311yescharacteristic
51C2×#11719917—22no—
52C2×#11719917—22no—
53C2×#11719917—22no—
54C22×#88322C2312yes—
55#13133176—22no—
56#13133176—22no—
57#13133175—28no—
58#13133175—28no—
59#13133175—24no—
60#13133175—24no—
61#13133175C2×C418yes—
62#13133175C2×C418yes—
63C2×C4⋊Dic53—22no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53C2×C414yes—
66C2×C4⋊Dic53C2312yes—
67C2×C4⋊Dic53C2312yes—
68C2×C22⋊Dic53—21no—
69C2×#11719921—28no—
70C2×#11719921—22no—
71C2×#11719921—22no—
72C2×#11719921—22no—
73C2×#11719921—22no—
74C2×C4×Dic53—21no—
75C2×C4×Dic53—28no—
76C2×C4×Dic53—22no—
77C2×C4×Dic53—22no—
78#13133129—24no—
79#13133129—24no—
80#13133124—22no—
81#13133124—22no—
82C23×C4×C53—21no—
83C23×C4×C53C2311yescharacteristic
84C2×M16×C53—22no—
85C2×M16×C53—22no—
86C2×M16×C53—22no—
87C22×C8×C53C2312yes—
88#117×C53—22no—
89#117×C53—22no—
90C2×C4⋊C4×C53—22no—
91C2×C4⋊C4×C53—21no—
92C2×C4⋊C4×C53C2×C414yes—
93C2×C4⋊C4×C53C2312yes—
94C2×C4⋊C4×C53C2312yes—
95C2×C22⋊C4×C53—21no—
96C2×C42×C53—21no—
97C24×C53C2×D411yescharacteristic
98C23×D53—22no—
99C23×D53—22no—
100C23×D53—42no—
101C23×D53—42no—
102C23×D53C22⋊C414yes—
103C23×D53C2×D411yescharacteristic
104C23×D53C2×D411yescharacteristic
105C22×C4×C53—24no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—44no—
109C22×C4×C53C22×C412yes—
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53C2×D412yes—
112C22×C4×C53#4112yes—
113C22×C4×C53#4112yes—
114C22×C4×C53#4112yes—
115C22×C4×C53#4112yes—
116C22×C4×C53C2411yescharacteristic
117C22×Dic53—22no—
118C22×Dic53—22no—
119C22×Dic53—24no—
120C22×Dic53—22no—
121C22×Dic53—22no—
122C22×Dic53—44no—
123C22×Dic53C22⋊C414yes—
124C22×Dic53C2×D411yescharacteristic
125C22×Dic53C2×D411yescharacteristic
126C2×C4×D53—28no—
127C2×C4×D53—28no—
128C2×C4×D53—28no—
129C2×C4×D53—28no—
130C2×C4×D53—22no—
131C2×C4×D53—22no—
132C2×C4×D53—22no—
133C2×C4×D53—22no—
134C2×C4×D53—21no—
135C2×C4×D53—21no—
136C2×C4×D53—21no—
137C2×C4×D53—21no—
138C2×C4×D53—21no—
139C2×C4×D53—21no—
140C2×C4×D53—21no—
141C2×C4×D53—21no—
142C2×C4×D53C22⋊C414yes—
143C2×C4×D53C22⋊C414yes—
144C2×C4×D53C22×C412yes—
145C2×C4×D53C22×C412yes—
146C2×C4×D53C2×D412yes—
147C2×C4×D53C2×D412yes—
148M16×C53—44no—
149C2×C8×C53—22no—
150C2×C8×C53—22no—
151C2×C8×C53—22no—
152C4⋊C4×C53—24no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—22no—
155C4⋊C4×C53—22no—
156C4⋊C4×C53C22×C414yes—
157C22⋊C4×C53—42no—
158C42×C53—42no—
159C22⋊Dic53—42no—
160#11719921—28no—
161#11719921—28no—
162#11719921—44no—
163#11719921—44no—
164C4⋊Dic53—24no—
165C4⋊Dic53—22no—
166C4⋊Dic53—22no—
167C4⋊Dic53—22no—
168C4⋊Dic53C22×C414yes—
169C4×Dic53—28no—
170C4×Dic53—28no—
171C4×Dic53—42no—
172#11719917—44no—
173C2×#88322—22no—
174C2×#88322—22no—
175C2×#88322—22no—
176C23×C53—22no—
177C23×C53—22no—
178C23×C53C2×C22⋊C412yes—
179C23×C53#12312yes—
180C23×C53#12312yes—
181C23×C53C22×D411yescharacteristic
182C22×D53—21no—
183C22×D53—21no—
184C22×D53—21no—
185C22×D53—21no—
186C22×D53—22no—
187C22×D53—22no—
188C22×D53—22no—
189C22×D53—22no—
190C22×D53—28no—
191C22×D53—44no—
192C22×D53—44no—
193C22×D53—48no—
194C22×D53—48no—
195C22×D53C2×C22⋊C412yes—
196C22×D53C2×C22⋊C412yes—
197C2×C4×C53—24no—
198C2×C4×C53—48no—
199C2×C4×C53—44no—
200C2×C4×C53—42no—
201C2×C4×C53—42no—
202C2×C4×C53—42no—
203C2×C4×C53C2×C22⋊C412yes—
204C2×C4×C53#11714yes—
205C2×C4×C53#11714yes—
206C2×C4×C53#12314yes—
207C2×C4×C53#12312yes—
208C2×C4×C53#12314yes—
209C2×C4×C53#12312yes—
210C2×C4×C53C23×C411yescharacteristic
211C2×C4×C53C22×D411yescharacteristic
212C2×C4×C53C2×#4111yescharacteristic
213C2×C4×C53C2×#4111yescharacteristic
214C2×C4×C53C2×#4111yescharacteristic
215C2×C4×C53C2×#4111yescharacteristic
216C2×Dic53—21no—
217C2×Dic53—21no—
218C2×Dic53—21no—
219C2×Dic53—21no—
220C2×Dic53—28no—
221C2×Dic53—24no—
222C2×Dic53—48no—
223C2×Dic53—44no—
224C2×Dic53—42no—
225C2×Dic53—42no—
226C2×Dic53—42no—
227C2×Dic53C2×C22⋊C412yes—
228C2×Dic53C2×C22⋊C412yes—
229C4×D53—22no—
230C4×D53—22no—
231C4×D53—22no—
232C4×D53—22no—
233C4×D53—22no—
234C4×D53—22no—
235C4×D53—22no—
236C4×D53—22no—
237C4×D53—28no—
238C4×D53—28no—
239C8×C53—42no—
240#88322—42no—
241#20701—531nomaximal, Sylow
242C22×C53—24no—
243C22×C53—24no—
244C22×C53—22no—
245C22×C53—22no—
246C22×C53#38114yes—
247C22×C53C22×C22⋊C411yescharacteristic
248C22×C53C2×#12311yescharacteristic
249C22×C53C2×#12311yescharacteristic
250C22×C53#57512yes—
251C22×C53#57512yes—
252C2×D53—22no—
253C2×D53—22no—
254C2×D53—22no—
255C2×D53—22no—
256C2×D53—22no—
257C2×D53—22no—
258C2×D53—416no—
259C2×D53—42no—
260C2×D53—42no—
261C2×D53—42no—
262C2×D53—42no—
263C4×C53—44no—
264C4×C53—82no—
265C4×C53#38114yes—
266C4×C53C2×#11711yescharacteristic
267C4×C53C2×#11711yescharacteristic
268C4×C53C2×#12311yescharacteristic
269C4×C53C2×#12311yescharacteristic
270Dic53—22no—
271Dic53—22no—
272Dic53—44no—
273Dic53—82no—
274C22×#117—531no—
275C2×#407—531no—
276#2280—531no—
277#1825—534no—
278#1825—534no—
279#1736—531no—
280#1736—531no—
281#1736—531no—
282#1736D5311yescharacteristic, complemented
283C2×C53—24no—
284C2×C53C2×#38111yescharacteristic
285C2×C53#289611yescharacteristic
286C2×C53#289611yescharacteristic
287C2×C53#309312yes—
288C2×C53#309312yes—
289D53—42no—
290D53—42no—
291C24×C4—531no—
292C22×M16—531no—
293C22×M16C2×D5311yescharacteristic
294C2×#117—531no—
295C2×#117—538no—
296C2×#117—532no—
297C2×#117—532no—
298C2×#117—1061no—
299C2×#117—1061no—
300C2×#117—1064no—
301C2×#117C2×D5311yescharacteristic
302C22×C4⋊C4—531no—
303C22×C4⋊C4C2×D5311yescharacteristic
304C22×C22⋊C4—1061no—
305C22×C42—1061no—
306#407—1064no—
307C2×C22⋊C8—532no—
308C2×C22⋊C8—532no—
309#381—1061no—
310#380—532no—
311#380—532no—
312#379—1061no—
313C4×C22⋊C4—1061no—
314C4×C22⋊C4—1061no—
315#340—534no—
316#340—534no—
317#340—534no—
318#340C2×D5314yescomplemented
319C53#2070111yescharacteristic, complemented, Sylow
320C25—531no—
321C23×C4—21no—
322C23×C4—531no—
323C23×C4—534no—
324C23×C4—531no—
325C23×C4—531no—
326C23×C4—534no—
327C23×C4—531no—
328C23×C4—531no—
329C23×C4—1062no—
330C23×C4—1062no—
331C23×C4—1062no—
332C23×C4—1062no—
333C23×C4—1061no—
334C23×C4C22×D5311yescharacteristic
335C2×M16—22no—
336C2×M16—22no—
337C2×M16—22no—
338C2×M16—1062no—
339C2×M16—1062no—
340C2×M16—1062no—
341C22×C8—532no—
342C22×C8C22×D5312yes—
343#117—22no—
344#117—22no—
345#117—538no—
346#117—538no—
347#117—1068no—
348#117—1068no—
349#117—1064no—
350#117—1064no—
351#117—1062no—
352#117—1062no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—534no—
356C2×C4⋊C4—532no—
357C2×C4⋊C4—532no—
358C2×C4⋊C4—1061no—
359C2×C4⋊C4—1062no—
360C2×C4⋊C4C4×D5314yes—
361C2×C4⋊C4C22×D5312yes—
362C2×C4⋊C4C22×D5312yes—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—1068no—
365C2×C22⋊C4—1062no—
366C2×C22⋊C4—1062no—
367C2×C22⋊C4—1062no—
368C2×C22⋊C4—1062no—
369C2×C22⋊C4—1061no—
370C2×C42—21no—
371C2×C42—1068no—
372C2×C42—1062no—
373C2×C42—1062no—
374C2×C42—1061no—
375C22⋊C8—1064no—
376C22⋊C8—1064no—
377#95—1062no—
378#95—1062no—
379C24—531no—
380C24—531no—
381C24—534no—
382C24—1062no—
383C24—1062no—
384C24—2122no—
385C24—2122no—
386C24D4×D5311yescharacteristic
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—44no—
391C22×C4—534no—
392C22×C4—531no—
393C22×C4—531no—
394C22×C4—534no—
395C22×C4—534no—
396C22×C4—532no—
397C22×C4—532no—
398C22×C4—532no—
399C22×C4—532no—
400C22×C4—1062no—
401C22×C4—1062no—
402C22×C4—1068no—
403C22×C4—1068no—
404C22×C4—1062no—
405C22×C4—1062no—
406C22×C4—1068no—
407C22×C4—1062no—
408C22×C4—1062no—
409C22×C4—1068no—
410C22×C4—1061no—
411C22×C4—1061no—
412C22×C4—1061no—
413C22×C4—1061no—
414C22×C4—1061no—
415C22×C4—1061no—
416C22×C4—1061no—
417C22×C4—1061no—
418C22×C4—1064no—
419C22×C4—1062no—
420C22×C4—1062no—
421C22×C4—2124no—
422C22×C4C2×C4×D5312yes—
423C22×C4#1171994512yes—
424C22×C4#1171994512yes—
425C22×C4D4×D5311yescharacteristic
426C22×C4D4×D5312yes—
427C22×C4#1171994712yes—
428C22×C4#1171994912yes—
429C22×C4C23×D5311yescharacteristic
430M16—44no—
431M16—2124no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—1062no—
436C2×C8—1062no—
437C2×C8—1062no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—534no—
443C4⋊C4—1064no—
444C4⋊C4—1062no—
445C4⋊C4—1062no—
446C4⋊C4—1062no—
447C4⋊C4C2×C4×D5314yes—
448C22⋊C4—42no—
449C22⋊C4—1068no—
450C22⋊C4—1068no—
451C22⋊C4—2124no—
452C22⋊C4—2124no—
453C22⋊C4—2122no—
454C42—42no—
455C42—1068no—
456C42—1068no—
457C42—2122no—
458C23—22no—
459C23—22no—
460C23—532no—
461C23—532no—
462C23—1061no—
463C23—1061no—
464C23—1061no—
465C23—1061no—
466C23—1062no—
467C23—1062no—
468C23—1062no—
469C23—1062no—
470C23—1068no—
471C23—2124no—
472C23—2124no—
473C23—2128no—
474C23—2128no—
475C23C22⋊C4×D5312yes—
476C23#1313319912yes—
477C23#1313320212yes—
478C23C2×D4×D5311yescharacteristic
479C2×C4—24no—
480C2×C4—48no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—532no—
486C2×C4—532no—
487C2×C4—1061no—
488C2×C4—1061no—
489C2×C4—1061no—
490C2×C4—1061no—
491C2×C4—1068no—
492C2×C4—1062no—
493C2×C4—1062no—
494C2×C4—1062no—
495C2×C4—1062no—
496C2×C4—1062no—
497C2×C4—1062no—
498C2×C4—1062no—
499C2×C4—1062no—
500C2×C4—1068no—
501C2×C4—1068no—
502C2×C4—1064no—
503C2×C4—2122no—
504C2×C4—2122no—
505C2×C4—2122no—
506C2×C4—2128no—
507C2×C4—2124no—
508C2×C4C22⋊C4×D5312yes—
509C2×C4#1313317414yes—
510C2×C4#1313317514yes—
511C2×C4#1313319914yes—
512C2×C4#1313319912yes—
513C2×C4#1313320214yes—
514C2×C4#1313320212yes—
515C2×C4C22×C4×D5311yescharacteristic
516C2×C4C2×D4×D5311yescharacteristic
517C2×C4C2×#1171994711yescharacteristic
518C2×C4C2×#1171994511yescharacteristic
519C2×C4C2×#1171994511yescharacteristic
520C2×C4C2×#1171994911yescharacteristic
521C8—42no—
522C8—2122no—
523C22—22no—
524C22—22no—
525C22—24no—
526C22—24no—
527C22—1062no—
528C22—1062no—
529C22—1062no—
530C22—1062no—
531C22—1062no—
532C22—1062no—
533C22—21216no—
534C22—2122no—
535C22—2122no—
536C22—2122no—
537C22—2122no—
538C22#1681048514yes—
539C22C2×C22⋊C4×D5311yescharacteristic
540C22C2×#1313319911yescharacteristic
541C22C2×#1313320211yescharacteristic
542C22#1681141912yes—
543C22#1681142212yes—
544C4—44no—
545C4—82no—
546C4—1062no—
547C4—1062no—
548C4—4242no—
549C4—2124no—
550C4#1681048514yes—
551C4C2×#1313317411yescharacteristic
552C4C2×#1313317511yescharacteristic
553C4C2×#1313319911yescharacteristic
554C4C2×#1313320211yescharacteristic
555C2—24no—
556C2—2122no—
557C2—2122no—
558C2C2×#1681048511yescharacteristic
559C2#2028322911yescharacteristic
560C2#2028323311yescharacteristic
561C2#2028464512yes—
562C2#2028464912yes—
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1736C2
socleC23×C53C22×D4
prime radicalC24×D53D4
cosocleC22×C4×C53C24
hypercenter#1736D53
hypocenterC53#20701
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
4. #20274809C2semidirect
5. #20274808C2semidirect
6. #20274436C2semidirect
7. #20274435C2semidirect
8. #20274434C2semidirect
9. C53×#1736C2semidirect
16. C2×#13133175C4semidirect
31. #16810386C22semidirect
32. #16810385C22semidirect
33. #16810385C22semidirect
37. C53×#340C22semidirect
282. #1736D53semidirect
318. #340C2×D53semidirect
319. C53#20701semidirect

Maximal subgroups

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

subgroupindex
C22×#131331752
C2×#168105862
#202770942
#202748092
#202748082
#202744362
#202744352
#202744342
C53×#17362
#2070153