Finite Group Databasebeta

← #27306492

Subgroups — #27306492

Order 13568.

2166 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D53C211yesmaximal, characteristic
2C2×#16810480C211yesmaximal, characteristic
3C2×C4×#11719921C211yesmaximal, characteristic
4#20272981C214yesmaximal, complemented
5#20272945C211yesmaximal, characteristic, complemented
6#20272944C211yesmaximal, characteristic, complemented
7#20272944C211yesmaximal, characteristic, complemented
8#20272936C214yesmaximal
9C53×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×C4×D53C2211yescharacteristic
11C23×C4×D53C2212yes—
12C22×C4⋊Dic53C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14C22×#11719919C2212yes—
15C2×C4⋊C4×D53—24no—
16C2×C4⋊C4×D53—24no—
17C2×C4⋊C4×D53C418yescomplemented
18C22×#11719921C2212yes—
19C22×#11719921C2212yes—
20C22×C4×Dic53C2211yescharacteristic
21#16810480—24no—
22C4×#11719921—24no—
23C2×#13133124C2212yes—
24C2×#13133124C2212yes—
25C2×#13133124C2214yes—
26C2×#13133124C2214yes—
27C2×#13133124C2214yes—
28C22×C4⋊C4×C53C2211yescharacteristic
29C22×C4⋊C4×C53C2211yescharacteristic
30C22×C42×C53C2211yescharacteristic
31C2×#95×C53C2214yes—
32C24×D53C2311yescharacteristic
33C23×Dic53C2312yes—
34C23×Dic53C2312yes—
35C23×Dic53C2312yes—
36C23×Dic53C2311yescharacteristic
37C22×C4×D53—22no—
38C22×C4×D53—22no—
39C22×C4×D53—22no—
40C22×C4×D53—22no—
41C22×C4×D53—24no—
42C22×C4×D53—24no—
43C22×C4×D53—44no—
44C22×C4×D53—44no—
45C22×C4×D53C2×C414yes—
46C22×C4×D53C2×C418yes—
47C22×C4×D53D414yes—
48C22×C4×D53D414yes—
49C2×C4⋊Dic53—22no—
50C2×C4⋊Dic53—22no—
51C2×C4⋊Dic53—22no—
52C2×C4⋊Dic53—22no—
53C2×C4⋊Dic53—22no—
54C2×C4⋊Dic53C2×C414yes—
55C2×#11719919—24no—
56C2×#11719919—24no—
57C2×#11719919C2×C418yes—
58C4⋊C4×D53—232no—
59C2×#11719921—24no—
60C2×#11719921—24no—
61C2×#11719921—24no—
62C2×#11719921—28no—
63C2×#11719921Q818yes—
64C2×C4×Dic53—22no—
65C2×C4×Dic53—22no—
66C2×C4×Dic53—22no—
67#13133124—24no—
68#13133124—24no—
69#13133124—44no—
70#13133124—44no—
71#13133124—44no—
72C23×C4×C53C2312yes—
73C23×C4×C53C2312yes—
74C23×C4×C53C2312yes—
75C23×C4×C53C2311yescharacteristic
76C2×C4⋊C4×C53—22no—
77C2×C4⋊C4×C53—22no—
78C2×C4⋊C4×C53—22no—
79C2×C4⋊C4×C53—22no—
80C2×C4⋊C4×C53—22no—
81C2×C4⋊C4×C53C2×C414yes—
82C2×C42×C53—22no—
83C2×C42×C53—22no—
84C2×C42×C53—22no—
85#95×C53—44no—
86C24×C53C2411yescharacteristic
87C23×D53—21no—
88C23×D53—21no—
89C23×D53—21no—
90C23×D53—21no—
91C23×D53—22no—
92C23×D53—22no—
93C23×D53C22×C412yes—
94C23×D53C2×D412yes—
95C23×D53C2×D412yes—
96C23×D53C2×Q814yes—
97C23×D53#4114yes—
98C22×C4×C53—22no—
99C22×C4×C53—22no—
100C22×C4×C53—22no—
101C22×C4×C53—22no—
102C22×C4×C53—22no—
103C22×C4×C53—22no—
104C22×C4×C53—22no—
105C22×C4×C53—24no—
106C22×C4×C53—22no—
107C22×C4×C53—24no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—24no—
112C22×C4×C53—24no—
113C22×C4×C53C22×C414yes—
114C22×C4×C53C22×C412yes—
115C22×C4×C53C2×D412yes—
116C22×C4×C53C2×D412yes—
117C22×C4×C53C2×Q814yes—
118C22×C4×C53#4112yes—
119C22×C4×C53#4112yes—
120C22×C4×C53#4112yes—
121C22×C4×C53#4112yes—
122C22×Dic53—21no—
123C22×Dic53—21no—
124C22×Dic53—21no—
125C22×Dic53—21no—
126C22×Dic53—22no—
127C22×Dic53—22no—
128C22×Dic53—22no—
129C22×Dic53—22no—
130C22×Dic53—22no—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—24no—
135C22×Dic53—24no—
136C22×Dic53—22no—
137C22×Dic53—24no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—24no—
141C22×Dic53C22×C414yes—
142C22×Dic53C22×C412yes—
143C22×Dic53C2×D412yes—
144C22×Dic53C2×D412yes—
145C22×Dic53C2×Q814yes—
146C2×C4×D53—28no—
147C2×C4×D53—28no—
148C2×C4×D53—28no—
149C2×C4×D53—28no—
150C2×C4×D53—28no—
151C2×C4×D53—28no—
152C2×C4×D53—216no—
153C2×C4×D53—416no—
154C2×C4×D53—48no—
155C2×C4×D53—416no—
156C2×C4×D53—48no—
157C2×C4×D53C22⋊C4116yes—
158C4⋊C4×C53—28no—
159C4⋊C4×C53—44no—
160C42×C53—44no—
161#11719921—216no—
162#11719921—48no—
163C4⋊Dic53—28no—
164C4⋊Dic53—44no—
165#11719919—216no—
166C4×Dic53—44no—
167C23×C53C23×C411yescharacteristic
168C23×C53C22×D411yescharacteristic
169C23×C53C22×D411yescharacteristic
170C23×C53C22×Q812yes—
171C23×C53C2×#4112yes—
172C23×C53C2×#4111yescharacteristic
173C23×C53C2×#4111yescharacteristic
174C23×C53C2×#4111yescharacteristic
175C23×C53C2×#4111yescharacteristic
176C23×C53H(5,𝔽2)12yes—
177C23×C53H(5,𝔽2)12yes—
178C22×D53—22no—
179C22×D53—22no—
180C22×D53—24no—
181C22×D53—22no—
182C22×D53—22no—
183C22×D53—22no—
184C22×D53—22no—
185C22×D53—22no—
186C22×D53—24no—
187C22×D53—22no—
188C22×D53—24no—
189C22×D53—24no—
190C22×D53—24no—
191C22×D53—24no—
192C22×D53—24no—
193C22×D53—24no—
194C22×D53—24no—
195C22×D53—24no—
196C22×D53C2×C22⋊C414yes—
197C22×D53C4×Q818yes—
198C22×D53C22⋊Q818yes—
199C22×D53#12414yes—
200C22×D53C4⋊Q814yes—
201C2×C4×C53—24no—
202C2×C4×C53—28no—
203C2×C4×C53—24no—
204C2×C4×C53—24no—
205C2×C4×C53—24no—
206C2×C4×C53—42no—
207C2×C4×C53—42no—
208C2×C4×C53—44no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—44no—
217C2×C4×C53—44no—
218C2×C4×C53C2×C22⋊C414yes—
219C2×C4×C53#11714yes—
220C2×C4×C53#11714yes—
221C2×C4×C53#12314yes—
222C2×C4×C53#12314yes—
223C2×C4×C53#12314yes—
224C2×C4×C53#12314yes—
225C2×Dic53—22no—
226C2×Dic53—22no—
227C2×Dic53—22no—
228C2×Dic53—22no—
229C2×Dic53—22no—
230C2×Dic53—22no—
231C2×Dic53—24no—
232C2×Dic53—28no—
233C2×Dic53—24no—
234C2×Dic53—24no—
235C2×Dic53—24no—
236C2×Dic53—42no—
237C2×Dic53—42no—
238C2×Dic53—44no—
239C2×Dic53—42no—
240C2×Dic53—42no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53—44no—
247C2×Dic53—44no—
248C2×Dic53C2×C22⋊C414yes—
249C4×D53—232no—
250C4×D53—432no—
251#13924—531nomaximal, Sylow
252C22×C53C22×C22⋊C411yescharacteristic
253C22×C53C2×#11711yescharacteristic
254C22×C53C2×#11711yescharacteristic
255C22×C53C2×C4×Q812yes—
256C22×C53#51812yes—
257C22×C53C2×C22⋊Q812yes—
258C22×C53C2×#12311yescharacteristic
259C22×C53C2×#12311yescharacteristic
260C22×C53C2×#12311yescharacteristic
261C22×C53C2×#12311yescharacteristic
262C22×C53C2×#12411yescharacteristic
263C22×C53C2×C4⋊Q811yescharacteristic
264C22×C53#53511yescharacteristic
265C22×C53#53512yes—
266C22×C53#53511yescharacteristic
267C22×C53#55312yes—
268C22×C53#55312yes—
269C22×C53#55312yes—
270C22×C53#55312yes—
271C22×C53#55412yes—
272C22×C53#55412yes—
273C22×C53#55412yes—
274C22×C53#55412yes—
275C2×D53—24no—
276C2×D53—28no—
277C2×D53—28no—
278C2×D53—28no—
279C2×D53—28no—
280C2×D53—24no—
281C2×D53—24no—
282C2×D53—24no—
283C2×D53—24no—
284C2×D53—24no—
285C2×D53#39118yes—
286C4×C53—44no—
287C4×C53—44no—
288C4×C53—82no—
289C4×C53#38118yes—
290Dic53—24no—
291Dic53—44no—
292Dic53—44no—
293Dic53—82no—
294C23×C4⋊C4—531no—
295C2×#380—531no—
296C2×C4×C22⋊C4—531no—
297#1445—534no—
298C4⋊#95—531no—
299C4⋊#95—531no—
300C4⋊#95—531no—
301C4⋊#95D5311yescharacteristic, complemented
302C22⋊#95—534no—
303C2×C53C2×#38111yescharacteristic
304C2×C53C2×#39111yescharacteristic
305C2×C53#230911yescharacteristic
306C2×C53#234512yes—
307C2×C53#234512yes—
308C2×C53#248712yes—
309C2×C53#248712yes—
310C2×C53#250211yescharacteristic
311C2×C53#250211yescharacteristic
312C2×C53#250411yescharacteristic
313C2×C53#250411yescharacteristic
314D53—28no—
315C24×C4—531no—
316C24×C4—532no—
317C22×C4⋊C4—532no—
318C22×C4⋊C4—531no—
319C22×C4⋊C4—538no—
320C22×C4⋊C4—531no—
321C22×C4⋊C4—1064no—
322C22×C4⋊C4—1064no—
323C22×C4⋊C4C2×D5311yescharacteristic
324C22×C4⋊C4C2×D5311yescharacteristic
325C22×C22⋊C4—532no—
326C22×C22⋊C4—532no—
327C22×C42—531no—
328C22×C42C2×D5311yescharacteristic
329#380—1064no—
330C4×C22⋊C4—1064no—
331C2×#95—532no—
332C2×#95—532no—
333C2×#95—534no—
334C2×#95—534no—
335C2×#95—534no—
336C2×#95C2×D5314yes—
337C53#1392411yescharacteristic, complemented, Sylow
338C25—531no—
339C23×C4—531no—
340C23×C4—534no—
341C23×C4—534no—
342C23×C4—534no—
343C23×C4—532no—
344C23×C4—532no—
345C23×C4—538no—
346C23×C4—532no—
347C23×C4—1064no—
348C23×C4—1064no—
349C23×C4—1062no—
350C23×C4—1062no—
351C23×C4—1062no—
352C23×C4—1062no—
353C23×C4—2124no—
354C23×C4—2124no—
355C23×C4C22×D5312yes—
356C23×C4C22×D5312yes—
357C23×C4C22×D5312yes—
358C23×C4C22×D5311yescharacteristic
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—538no—
365C2×C4⋊C4—534no—
366C2×C4⋊C4—10632no—
367C2×C4⋊C4—1064no—
368C2×C4⋊C4—1064no—
369C2×C4⋊C4—1062no—
370C2×C4⋊C4—1062no—
371C2×C4⋊C4—1062no—
372C2×C4⋊C4—1062no—
373C2×C4⋊C4—1062no—
374C2×C4⋊C4C4×D5314yes—
375C2×C22⋊C4—538no—
376C2×C22⋊C4—1064no—
377C2×C22⋊C4—1064no—
378C2×C22⋊C4—1068no—
379C2×C22⋊C4—1064no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—1062no—
384C2×C42—1062no—
385C2×C42—1062no—
386#95—44no—
387#95—1064no—
388#95—1064no—
389#95—2124no—
390#95—2124no—
391#95—2124no—
392C24—532no—
393C24—532no—
394C24—532no—
395C24—534no—
396C24—534no—
397C24—1061no—
398C24—1061no—
399C24—1061no—
400C24—1061no—
401C24—1062no—
402C24—1062no—
403C24C23×D5311yescharacteristic
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—532no—
420C22×C4—532no—
421C22×C4—532no—
422C22×C4—5316no—
423C22×C4—534no—
424C22×C4—534no—
425C22×C4—10616no—
426C22×C4—1061no—
427C22×C4—1061no—
428C22×C4—1061no—
429C22×C4—1061no—
430C22×C4—1068no—
431C22×C4—1068no—
432C22×C4—1068no—
433C22×C4—1068no—
434C22×C4—1068no—
435C22×C4—1068no—
436C22×C4—1062no—
437C22×C4—1062no—
438C22×C4—1062no—
439C22×C4—1062no—
440C22×C4—1062no—
441C22×C4—1062no—
442C22×C4—1062no—
443C22×C4—1064no—
444C22×C4—1062no—
445C22×C4—1064no—
446C22×C4—1062no—
447C22×C4—1064no—
448C22×C4—1062no—
449C22×C4—1062no—
450C22×C4—1064no—
451C22×C4—21216no—
452C22×C4—2128no—
453C22×C4—21216no—
454C22×C4—2128no—
455C22×C4C2×C4×D5314yes—
456C22×C4C2×C4×D5312yes—
457C22×C4#1171994512yes—
458C22×C4#1171994512yes—
459C22×C4D4×D5312yes—
460C22×C4D4×D5312yes—
461C22×C4#1171994712yes—
462C22×C4Q8×D5314yes—
463C22×C4#1171994912yes—
464C4⋊C4—28no—
465C4⋊C4—44no—
466C4⋊C4—10616no—
467C4⋊C4—1068no—
468C4⋊C4—2124no—
469C22⋊C4—10616no—
470C22⋊C4—2128no—
471C42—44no—
472C42—2124no—
473C23—534no—
474C23—538no—
475C23—538no—
476C23—534no—
477C23—534no—
478C23—1064no—
479C23—1064no—
480C23—1064no—
481C23—1064no—
482C23—1064no—
483C23—1064no—
484C23—1062no—
485C23—1064no—
486C23—1062no—
487C23—1064no—
488C23—1064no—
489C23—1062no—
490C23—1062no—
491C23—1062no—
492C23—1062no—
493C23—1062no—
494C23—1064no—
495C23—1062no—
496C23C22×C4×D5311yescharacteristic
497C23C2×D4×D5311yescharacteristic
498C23C2×D4×D5311yescharacteristic
499C23C2×Q8×D5312yes—
500C23#41×D5312yes—
501C23C2×#1171994711yescharacteristic
502C23C2×#1171994511yescharacteristic
503C23C2×#1171994511yescharacteristic
504C23C2×#1171994911yescharacteristic
505C23#1313326212yes—
506C23#1313326312yes—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—24no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—534no—
525C2×C4—1062no—
526C2×C4—1062no—
527C2×C4—1062no—
528C2×C4—1062no—
529C2×C4—1062no—
530C2×C4—1062no—
531C2×C4—10632no—
532C2×C4—1064no—
533C2×C4—1064no—
534C2×C4—1064no—
535C2×C4—1068no—
536C2×C4—1064no—
537C2×C4—21232no—
538C2×C4—2124no—
539C2×C4—2124no—
540C2×C4—2122no—
541C2×C4—2122no—
542C2×C4—2122no—
543C2×C4—2122no—
544C2×C4—2122no—
545C2×C4—2122no—
546C2×C4—2122no—
547C2×C4—2122no—
548C2×C4—2124no—
549C2×C4—2122no—
550C2×C4C22⋊C4×D5314yes—
551C2×C4#1313317414yes—
552C2×C4#1313317514yes—
553C2×C4#1313319914yes—
554C2×C4#1313319914yes—
555C2×C4#1313320214yes—
556C2×C4#1313320214yes—
557C22—538no—
558C22—1064no—
559C22—1068no—
560C22—1068no—
561C22—1064no—
562C22—1064no—
563C22—1064no—
564C22—1064no—
565C22—1064no—
566C22—1068no—
567C22—1068no—
568C22C2×C22⋊C4×D5311yescharacteristic
569C22C2×#1313317411yescharacteristic
570C22C2×#1313317511yescharacteristic
571C22C4×Q8×D5312yes—
572C22#1681109512yes—
573C22C22⋊Q8×D5312yes—
574C22C2×#1313319911yescharacteristic
575C22C2×#1313319911yescharacteristic
576C22C2×#1313320211yescharacteristic
577C22C2×#1313320211yescharacteristic
578C22#124×D5311yescharacteristic
579C22C4⋊Q8×D5311yescharacteristic
580C22#1681118611yescharacteristic
581C22#1681118912yes—
582C22#1681119111yescharacteristic
583C22#1681131712yes—
584C22#1681132012yes—
585C22#1681132312yes—
586C22#1681132612yes—
587C22#1681132812yes—
588C22#1681133112yes—
589C22#1681133312yes—
590C22#1681133412yes—
591C4—44no—
592C4—44no—
593C4—82no—
594C4—1064no—
595C4—4242no—
596C4—2124no—
597C4—2124no—
598C4#1681048518yes—
599C2—1068no—
600C2C2×#1681048511yescharacteristic
601C2#391×D5311yescharacteristic
602C2#2027729411yescharacteristic
603C2#2027758812yes—
604C2#2027759112yes—
605C2#2027902812yes—
606C2#2027903012yes—
607C2#2027919011yescharacteristic
608C2#2027919211yescharacteristic
609C2#2027920411yescharacteristic
610C2#2027920611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×C4⋊#95C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenterC4⋊#95D53
hypocenterC53#13924
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
4. #20272981C2semidirect
5. #20272945C2semidirect
6. #20272944C2semidirect
7. #20272944C2semidirect
9. C53×C4⋊#95C2semidirect
17. C2×C4⋊C4×D53C4semidirect
301. C4⋊#95D53semidirect
337. C53#13924semidirect

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×#168104802
C2×C4×#117199212
#202729812
#202729452
#202729442
#202729362
C53×C4⋊#952
#1392453