Finite Group Databasebeta

← #27306322

Subgroups — #27306322

Order 13568.

2278 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D53C211yesmaximal, characteristic
2C2×#16810485C214yesmaximal
3C2×#16810480C211yesmaximal, characteristic
4C2×C4×#11719921C211yesmaximal, characteristic
5#20272981C214yesmaximal, complemented
6#20272945C211yesmaximal, characteristic, complemented
7#20272944C211yesmaximal, characteristic, complemented
8#20272944C211yesmaximal, characteristic, complemented
9C53×C4⋊#95C211yesmaximal, characteristic, complemented
10C23×C4×D53C2212yes—
11C23×C4×D53C2211yescharacteristic
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#16810485—216no—
22#16810480—24no—
23C4×#11719921—24no—
24C2×#13133124C2214yes—
25C2×#13133124C2214yes—
26C2×#13133124C2214yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C22×C4⋊C4×C53C2211yescharacteristic
30C22×C4⋊C4×C53C2211yescharacteristic
31C22×C42×C53C2211yescharacteristic
32C2×#95×C53C2214yes—
33C24×D53C2311yescharacteristic
34C23×Dic53C2312yes—
35C23×Dic53C2312yes—
36C23×Dic53C2312yes—
37C23×Dic53C2311yescharacteristic
38C22×C4×D53—22no—
39C22×C4×D53—22no—
40C22×C4×D53—22no—
41C22×C4×D53—22no—
42C22×C4×D53—24no—
43C22×C4×D53—24no—
44C22×C4×D53—28no—
45C22×C4×D53—28no—
46C22×C4×D53C2×C418yes—
47C22×C4×D53C2×C414yes—
48C22×C4×D53D414yes—
49C22×C4×D53D414yes—
50C2×C4⋊Dic53—22no—
51C2×C4⋊Dic53—22no—
52C2×C4⋊Dic53—22no—
53C2×C4⋊Dic53—22no—
54C2×C4⋊Dic53—22no—
55C2×C4⋊Dic53C2×C414yes—
56C2×#11719919—24no—
57C2×#11719919—24no—
58C2×#11719919C2×C418yes—
59C4⋊C4×D53—232no—
60C2×#11719921—24no—
61C2×#11719921—24no—
62C2×#11719921—28no—
63C2×#11719921—24no—
64C2×#11719921D418yes—
65C2×C4×Dic53—22no—
66C2×C4×Dic53—22no—
67C2×C4×Dic53—22no—
68#13133124—28no—
69#13133124—24no—
70#13133124—24no—
71#13133124—28no—
72#13133124—28no—
73C23×C4×C53C2312yes—
74C23×C4×C53C2312yes—
75C23×C4×C53C2312yes—
76C23×C4×C53C2311yescharacteristic
77C2×C4⋊C4×C53—22no—
78C2×C4⋊C4×C53—22no—
79C2×C4⋊C4×C53—22no—
80C2×C4⋊C4×C53—22no—
81C2×C4⋊C4×C53—22no—
82C2×C4⋊C4×C53C2×C414yes—
83C2×C42×C53—22no—
84C2×C42×C53—22no—
85C2×C42×C53—22no—
86#95×C53—28no—
87C24×C53C2411yescharacteristic
88C23×D53—21no—
89C23×D53—21no—
90C23×D53—21no—
91C23×D53—21no—
92C23×D53—22no—
93C23×D53—22no—
94C23×D53C22×C412yes—
95C23×D53C2×D412yes—
96C23×D53C2×D412yes—
97C23×D53C2×D414yes—
98C23×D53#4114yes—
99C22×C4×C53—24no—
100C22×C4×C53—22no—
101C22×C4×C53—22no—
102C22×C4×C53—22no—
103C22×C4×C53—22no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—24no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—24no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—24no—
114C22×C4×C53C22×C414yes—
115C22×C4×C53C22×C412yes—
116C22×C4×C53C2×D414yes—
117C22×C4×C53C2×D412yes—
118C22×C4×C53C2×D412yes—
119C22×C4×C53#4112yes—
120C22×C4×C53#4112yes—
121C22×C4×C53#4112yes—
122C22×C4×C53#4112yes—
123C22×Dic53—21no—
124C22×Dic53—21no—
125C22×Dic53—21no—
126C22×Dic53—21no—
127C22×Dic53—24no—
128C22×Dic53—22no—
129C22×Dic53—22no—
130C22×Dic53—22no—
131C22×Dic53—22no—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—24no—
135C22×Dic53—22no—
136C22×Dic53—22no—
137C22×Dic53—24no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—24no—
142C22×Dic53C22×C414yes—
143C22×Dic53C22×C412yes—
144C22×Dic53C2×D414yes—
145C22×Dic53C2×D412yes—
146C22×Dic53C2×D412yes—
147C2×C4×D53—28no—
148C2×C4×D53—28no—
149C2×C4×D53—28no—
150C2×C4×D53—28no—
151C2×C4×D53—28no—
152C2×C4×D53—28no—
153C2×C4×D53—216no—
154C2×C4×D53—232no—
155C2×C4×D53—416no—
156C2×C4×D53—48no—
157C2×C4×D53—48no—
158C2×C4×D53C22⋊C4116yes—
159C4⋊C4×C53—28no—
160C4⋊C4×C53—44no—
161C42×C53—44no—
162#11719921—216no—
163#11719921—48no—
164C4⋊Dic53—28no—
165C4⋊Dic53—44no—
166#11719919—216no—
167C4×Dic53—44no—
168C23×C53C23×C411yescharacteristic
169C23×C53C22×D411yescharacteristic
170C23×C53C22×D411yescharacteristic
171C23×C53C22×D412yes—
172C23×C53C2×#4112yes—
173C23×C53C2×#4111yescharacteristic
174C23×C53C2×#4111yescharacteristic
175C23×C53C2×#4111yescharacteristic
176C23×C53C2×#4111yescharacteristic
177C23×C532−1+412yes—
178C23×C532−1+412yes—
179C22×D53—22no—
180C22×D53—22no—
181C22×D53—24no—
182C22×D53—22no—
183C22×D53—22no—
184C22×D53—22no—
185C22×D53—22no—
186C22×D53—22no—
187C22×D53—24no—
188C22×D53—22no—
189C22×D53—24no—
190C22×D53—24no—
191C22×D53—24no—
192C22×D53—24no—
193C22×D53—24no—
194C22×D53—24no—
195C22×D53—24no—
196C22×D53—24no—
197C22×D53C2×C22⋊C414yes—
198C22×D53C4×D418yes—
199C22×D53#12118yes—
200C22×D53#12414yes—
201C22×D53#12714yes—
202C2×C4×C53—24no—
203C2×C4×C53—28no—
204C2×C4×C53—24no—
205C2×C4×C53—24no—
206C2×C4×C53—24no—
207C2×C4×C53—44no—
208C2×C4×C53—42no—
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—42no—
217C2×C4×C53—44no—
218C2×C4×C53—44no—
219C2×C4×C53C2×C22⋊C414yes—
220C2×C4×C53#11714yes—
221C2×C4×C53#11714yes—
222C2×C4×C53#12314yes—
223C2×C4×C53#12314yes—
224C2×C4×C53#12314yes—
225C2×C4×C53#12314yes—
226C2×Dic53—22no—
227C2×Dic53—22no—
228C2×Dic53—22no—
229C2×Dic53—22no—
230C2×Dic53—22no—
231C2×Dic53—22no—
232C2×Dic53—24no—
233C2×Dic53—28no—
234C2×Dic53—24no—
235C2×Dic53—24no—
236C2×Dic53—24no—
237C2×Dic53—44no—
238C2×Dic53—42no—
239C2×Dic53—42no—
240C2×Dic53—42no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—44no—
248C2×Dic53—44no—
249C2×Dic53C2×C22⋊C414yes—
250C4×D53—232no—
251C4×D53—432no—
252#13904—531nomaximal, Sylow
253C22×C53C22×C22⋊C411yescharacteristic
254C22×C53C2×#11711yescharacteristic
255C22×C53C2×#11711yescharacteristic
256C22×C53C2×C4×D412yes—
257C22×C53#51912yes—
258C22×C53C2×#12112yes—
259C22×C53C2×#12311yescharacteristic
260C22×C53C2×#12311yescharacteristic
261C22×C53C2×#12311yescharacteristic
262C22×C53C2×#12311yescharacteristic
263C22×C53C2×#12411yescharacteristic
264C22×C53C2×#12711yescharacteristic
265C22×C53#53611yescharacteristic
266C22×C53#53612yes—
267C22×C53#53611yescharacteristic
268C22×C53#54712yes—
269C22×C53#54712yes—
270C22×C53#54712yes—
271C22×C53#54712yes—
272C22×C53#55212yes—
273C22×C53#55212yes—
274C22×C53#55212yes—
275C22×C53#55212yes—
276C2×D53—24no—
277C2×D53—28no—
278C2×D53—28no—
279C2×D53—28no—
280C2×D53—28no—
281C2×D53—24no—
282C2×D53—24no—
283C2×D53—24no—
284C2×D53—24no—
285C2×D53—24no—
286C2×D53#39018yes—
287C4×C53—44no—
288C4×C53—44no—
289C4×C53—82no—
290C4×C53#38118yescomplemented
291Dic53—24no—
292Dic53—44no—
293Dic53—44no—
294Dic53—82no—
295C23×C4⋊C4—531no—
296C2×#381—534no—
297C2×#380—531no—
298C2×C4×C22⋊C4—531no—
299#1445—534no—
300C4⋊#95—531no—
301C4⋊#95—531no—
302C4⋊#95—531no—
303C4⋊#95D5311yescharacteristic, complemented
304C2×C53C2×#38111yescharacteristic
305C2×C53C2×#39011yescharacteristic
306C2×C53#231011yescharacteristic
307C2×C53#234412yes—
308C2×C53#234412yes—
309C2×C53#248612yes—
310C2×C53#248612yes—
311C2×C53#250111yescharacteristic
312C2×C53#250111yescharacteristic
313C2×C53#250311yescharacteristic
314C2×C53#250311yescharacteristic
315D53—28no—
316C24×C4—531no—
317C24×C4—532no—
318C22×C4⋊C4—532no—
319C22×C4⋊C4—531no—
320C22×C4⋊C4—538no—
321C22×C4⋊C4—531no—
322C22×C4⋊C4—1064no—
323C22×C4⋊C4—1064no—
324C22×C4⋊C4C2×D5311yescharacteristic
325C22×C4⋊C4C2×D5311yescharacteristic
326C22×C22⋊C4—532no—
327C22×C22⋊C4—532no—
328C22×C42—531no—
329C22×C42C2×D5311yescharacteristic
330#381—10616no—
331#380—1064no—
332C4×C22⋊C4—1064no—
333C2×#95—534no—
334C2×#95—534no—
335C2×#95—534no—
336C2×#95—532no—
337C2×#95—532no—
338C2×#95C2×D5314yes—
339C53#1390411yescharacteristic, complemented, Sylow
340C25—531no—
341C23×C4—531no—
342C23×C4—534no—
343C23×C4—534no—
344C23×C4—534no—
345C23×C4—532no—
346C23×C4—532no—
347C23×C4—538no—
348C23×C4—532no—
349C23×C4—1064no—
350C23×C4—1064no—
351C23×C4—1068no—
352C23×C4—1068no—
353C23×C4—1062no—
354C23×C4—1062no—
355C23×C4—1062no—
356C23×C4—1062no—
357C23×C4C22×D5312yes—
358C23×C4C22×D5312yes—
359C23×C4C22×D5312yes—
360C23×C4C22×D5311yescharacteristic
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—538no—
367C2×C4⋊C4—534no—
368C2×C4⋊C4—10632no—
369C2×C4⋊C4—1064no—
370C2×C4⋊C4—1064no—
371C2×C4⋊C4—1062no—
372C2×C4⋊C4—1062no—
373C2×C4⋊C4—1062no—
374C2×C4⋊C4—1062no—
375C2×C4⋊C4—1062no—
376C2×C4⋊C4C4×D5314yes—
377C2×C22⋊C4—538no—
378C2×C22⋊C4—1064no—
379C2×C22⋊C4—1064no—
380C2×C22⋊C4—1064no—
381C2×C22⋊C4—1068no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—1062no—
386C2×C42—1062no—
387C2×C42—1062no—
388#95—28no—
389#95—1064no—
390#95—1064no—
391#95—1068no—
392#95—1068no—
393#95—1068no—
394C24—532no—
395C24—534no—
396C24—532no—
397C24—532no—
398C24—534no—
399C24—1061no—
400C24—1061no—
401C24—1061no—
402C24—1061no—
403C24—1062no—
404C24—1062no—
405C24C23×D5311yescharacteristic
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—532no—
422C22×C4—532no—
423C22×C4—532no—
424C22×C4—5316no—
425C22×C4—534no—
426C22×C4—534no—
427C22×C4—10616no—
428C22×C4—10632no—
429C22×C4—1061no—
430C22×C4—1061no—
431C22×C4—1061no—
432C22×C4—1061no—
433C22×C4—1068no—
434C22×C4—1068no—
435C22×C4—1068no—
436C22×C4—1068no—
437C22×C4—1068no—
438C22×C4—1068no—
439C22×C4—1064no—
440C22×C4—1062no—
441C22×C4—1062no—
442C22×C4—1062no—
443C22×C4—1062no—
444C22×C4—1064no—
445C22×C4—1062no—
446C22×C4—1062no—
447C22×C4—1062no—
448C22×C4—1062no—
449C22×C4—1064no—
450C22×C4—1062no—
451C22×C4—1062no—
452C22×C4—1062no—
453C22×C4—1064no—
454C22×C4—21216no—
455C22×C4—2128no—
456C22×C4—2128no—
457C22×C4C2×C4×D5314yes—
458C22×C4C2×C4×D5312yes—
459C22×C4#1171994512yes—
460C22×C4#1171994512yes—
461C22×C4D4×D5314yes—
462C22×C4D4×D5312yes—
463C22×C4D4×D5312yes—
464C22×C4#1171994712yes—
465C22×C4#1171994912yes—
466C4⋊C4—28no—
467C4⋊C4—44no—
468C4⋊C4—10616no—
469C4⋊C4—1068no—
470C4⋊C4—2124no—
471C22⋊C4—10616no—
472C22⋊C4—2128no—
473C42—44no—
474C42—2124no—
475C23—534no—
476C23—538no—
477C23—538no—
478C23—534no—
479C23—534no—
480C23—1064no—
481C23—1064no—
482C23—1064no—
483C23—1064no—
484C23—1064no—
485C23—1064no—
486C23—1064no—
487C23—1064no—
488C23—1062no—
489C23—1062no—
490C23—1064no—
491C23—1062no—
492C23—1062no—
493C23—1062no—
494C23—1062no—
495C23—1062no—
496C23—1064no—
497C23—1062no—
498C23C22×C4×D5311yescharacteristic
499C23C2×D4×D5312yes—
500C23C2×D4×D5311yescharacteristic
501C23C2×D4×D5311yescharacteristic
502C23#41×D5312yes—
503C23C2×#1171994711yescharacteristic
504C23C2×#1171994511yescharacteristic
505C23C2×#1171994511yescharacteristic
506C23C2×#1171994911yescharacteristic
507C23#1313326412yes—
508C23#1313326512yes—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—28no—
513C2×C4—24no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—534no—
527C2×C4—1062no—
528C2×C4—1062no—
529C2×C4—1062no—
530C2×C4—1062no—
531C2×C4—1062no—
532C2×C4—1062no—
533C2×C4—10632no—
534C2×C4—1064no—
535C2×C4—1064no—
536C2×C4—1064no—
537C2×C4—1068no—
538C2×C4—1064no—
539C2×C4—21232no—
540C2×C4—2124no—
541C2×C4—2124no—
542C2×C4—2122no—
543C2×C4—2122no—
544C2×C4—2122no—
545C2×C4—2122no—
546C2×C4—2122no—
547C2×C4—2124no—
548C2×C4—2122no—
549C2×C4—2122no—
550C2×C4—2122no—
551C2×C4—2122no—
552C2×C4C22⋊C4×D5314yes—
553C2×C4#1313317414yes—
554C2×C4#1313317514yes—
555C2×C4#1313319914yes—
556C2×C4#1313319914yes—
557C2×C4#1313320214yes—
558C2×C4#1313320214yes—
559C22—538no—
560C22—1064no—
561C22—1068no—
562C22—1068no—
563C22—1064no—
564C22—1064no—
565C22—1064no—
566C22—1064no—
567C22—1064no—
568C22—1068no—
569C22—1068no—
570C22C2×C22⋊C4×D5311yescharacteristic
571C22C2×#1313317411yescharacteristic
572C22C2×#1313317511yescharacteristic
573C22C4×D4×D5312yes—
574C22#1681109912yes—
575C22#121×D5312yes—
576C22C2×#1313319911yescharacteristic
577C22C2×#1313319911yescharacteristic
578C22C2×#1313320211yescharacteristic
579C22C2×#1313320211yescharacteristic
580C22#124×D5311yescharacteristic
581C22#127×D5311yescharacteristic
582C22#1681119311yescharacteristic
583C22#1681119612yes—
584C22#1681119811yescharacteristic
585C22#1681127312yes—
586C22#1681127412yes—
587C22#1681127612yes—
588C22#1681127812yes—
589C22#1681130712yes—
590C22#1681131112yes—
591C22#1681131212yes—
592C22#1681131412yes—
593C4—44no—
594C4—44no—
595C4—82no—
596C4—1064no—
597C4—4242no—
598C4—2124no—
599C4—2124no—
600C4#1681048518yescomplemented
601C2—1068no—
602C2C2×#1681048511yescharacteristic
603C2#390×D5311yescharacteristic
604C2#2027729811yescharacteristic
605C2#2027758012yes—
606C2#2027758312yes—
607C2#2027901912yes—
608C2#2027902112yes—
609C2#2027918311yescharacteristic
610C2#2027918511yescharacteristic
611C2#2027919711yescharacteristic
612C2#2027919911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×C4⋊#95C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenterC4⋊#95D53
hypocenterC53#13904
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
5. #20272981C2semidirect
6. #20272945C2semidirect
7. #20272944C2semidirect
8. #20272944C2semidirect
9. C53×C4⋊#95C2semidirect
17. C2×C4⋊C4×D53C4semidirect
290. C4×C53#381semidirect
303. C4⋊#95D53semidirect
339. C53#13904semidirect
600. C4#16810485semidirect

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×#168104852
C2×#168104802
C2×C4×#117199212
#202729812
#202729452
#202729442
C53×C4⋊#952
#1390453