Finite Group Databasebeta

← #27306083

Subgroups — #27306083

Order 13568.

2222 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D53C211yesmaximal, characteristic
2C2×#16810489C211yesmaximal, characteristic, complemented
3C2×#16810488C211yesmaximal, characteristic, complemented
4C2×#16810480C211yesmaximal, characteristic
5C2×C4×#11719921C211yesmaximal, characteristic
6#20272970C214yesmaximal, complemented
7#20272944C211yesmaximal, characteristic, complemented
8#20272936C214yesmaximal
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×D53—24no—
18C22×#11719921C2212yes—
19C22×#11719921C2212yes—
20C22×C4×Dic53C2211yescharacteristic
21#16810489—24no—
22#16810488—24no—
23#16810480—24no—
24C4×#11719921C418yescomplemented
25C2×#13133124C2214yes—
26C2×#13133124C2214yes—
27C2×#13133124C2212yes—
28C2×#13133124C2214yes—
29C2×#13133124C2212yes—
30C22×C4⋊C4×C53C2211yescharacteristic
31C22×C4⋊C4×C53C2211yescharacteristic
32C22×C42×C53C2211yescharacteristic
33C2×#95×C53C2214yes—
34C24×D53C2311yescharacteristic
35C23×Dic53C2312yes—
36C23×Dic53C2312yes—
37C23×Dic53C2312yes—
38C23×Dic53C2311yescharacteristic
39C22×C4×D53—22no—
40C22×C4×D53—22no—
41C22×C4×D53—22no—
42C22×C4×D53—22no—
43C22×C4×D53—24no—
44C22×C4×D53—24no—
45C22×C4×D53—24no—
46C22×C4×D53—28no—
47C22×C4×D53—44no—
48C22×C4×D53C2×C414yes—
49C22×C4×D53D414yes—
50C22×C4×D53Q814yes—
51C2×C4⋊Dic53—22no—
52C2×C4⋊Dic53—22no—
53C2×C4⋊Dic53—22no—
54C2×C4⋊Dic53—22no—
55C2×C4⋊Dic53—22no—
56C2×C4⋊Dic53D414yes—
57C2×#11719919—24no—
58C2×#11719919—24no—
59C2×#11719919—24no—
60C4⋊C4×D53—232no—
61C2×#11719921—24no—
62C2×#11719921—24no—
63C2×#11719921—24no—
64C2×#11719921—28no—
65C2×#11719921C2×C418yes—
66C2×C4×Dic53—22no—
67C2×C4×Dic53—22no—
68C2×C4×Dic53C2×C414yescomplemented
69#13133124—28no—
70#13133124—24no—
71#13133124—28no—
72#13133124—44no—
73#13133124C2×C418yescomplemented
74C23×C4×C53C2312yes—
75C23×C4×C53C2312yes—
76C23×C4×C53C2312yes—
77C23×C4×C53C2311yescharacteristic
78C2×C4⋊C4×C53—22no—
79C2×C4⋊C4×C53—22no—
80C2×C4⋊C4×C53—22no—
81C2×C4⋊C4×C53—22no—
82C2×C4⋊C4×C53—22no—
83C2×C4⋊C4×C53D414yes—
84C2×C42×C53—22no—
85C2×C42×C53—22no—
86C2×C42×C53C2×C414yescomplemented
87#95×C53—44no—
88C24×C53C2411yescharacteristic
89C23×D53—21no—
90C23×D53—21no—
91C23×D53—21no—
92C23×D53—21no—
93C23×D53—22no—
94C23×D53—22no—
95C23×D53C22×C412yes—
96C23×D53C2×D412yes—
97C23×D53C2×Q812yes—
98C23×D53#4114yes—
99C23×D53#4114yes—
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—24no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—24no—
114C22×C4×C53—24no—
115C22×C4×C53C22×C414yes—
116C22×C4×C53C22×C412yes—
117C22×C4×C53C2×D414yes—
118C22×C4×C53C2×D412yes—
119C22×C4×C53C2×D412yes—
120C22×C4×C53C2×D412yes—
121C22×C4×C53C2×Q812yes—
122C22×C4×C53#4112yes—
123C22×C4×C53#4112yes—
124C22×Dic53—21no—
125C22×Dic53—21no—
126C22×Dic53—21no—
127C22×Dic53—21no—
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—24no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—24no—
142C22×Dic53—24no—
143C22×Dic53C22×C414yes—
144C22×Dic53C22×C412yes—
145C22×Dic53C2×D414yes—
146C22×Dic53C2×D412yes—
147C22×Dic53C2×Q812yes—
148C2×C4×D53—28no—
149C2×C4×D53—28no—
150C2×C4×D53—28no—
151C2×C4×D53—28no—
152C2×C4×D53—28no—
153C2×C4×D53—28no—
154C2×C4×D53—216no—
155C2×C4×D53—416no—
156C2×C4×D53—416no—
157C2×C4×D53—48no—
158C2×C4×D53—48no—
159C2×C4×D53C4⋊C4116yes—
160C4⋊C4×C53—28no—
161C4⋊C4×C53—44no—
162C42×C53—28no—
163#11719921—216no—
164#11719921—48no—
165C4⋊Dic53—28no—
166C4⋊Dic53—44no—
167#11719919—216no—
168C4×Dic53—28no—
169C23×C53C23×C411yescharacteristic
170C23×C53C22×D411yescharacteristic
171C23×C53C22×D411yescharacteristic
172C23×C53C22×D411yescharacteristic
173C23×C53C22×Q811yescharacteristic
174C23×C53C2×#4112yes—
175C23×C53C2×#4112yes—
176C23×C53C2×#4111yescharacteristic
177C23×C53C2×#4111yescharacteristic
178C23×C53H(5,𝔽2)12yes—
179C23×C532−1+412yes—
180C22×D53—22no—
181C22×D53—22no—
182C22×D53—24no—
183C22×D53—22no—
184C22×D53—22no—
185C22×D53—22no—
186C22×D53—22no—
187C22×D53—22no—
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×D53—24no—
198C22×D53C2×C4⋊C414yes—
199C22×D53#11718yes—
200C22×D53#12414yes—
201C22×D53#12414yes—
202C22×D53#12518yes—
203C2×C4×C53—24no—
204C2×C4×C53—24no—
205C2×C4×C53—24no—
206C2×C4×C53—24no—
207C2×C4×C53—28no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53—44no—
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—42no—
218C2×C4×C53—44no—
219C2×C4×C53—44no—
220C2×C4×C53C2×C4⋊C414yes—
221C2×C4×C53C4×D414yes—
222C2×C4×C53C4×D414yes—
223C2×C4×C53#12014yes—
224C2×C4×C53C22⋊Q814yes—
225C2×C4×C53C22⋊Q814yes—
226C2×C4×C53#12314yes—
227C2×Dic53—22no—
228C2×Dic53—22no—
229C2×Dic53—22no—
230C2×Dic53—22no—
231C2×Dic53—22no—
232C2×Dic53—22no—
233C2×Dic53—24no—
234C2×Dic53—24no—
235C2×Dic53—24no—
236C2×Dic53—24no—
237C2×Dic53—28no—
238C2×Dic53—42no—
239C2×Dic53—42no—
240C2×Dic53—44no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248C2×Dic53—44no—
249C2×Dic53—44no—
250C2×Dic53C2×C4⋊C414yes—
251C4×D53—232no—
252C4×D53—432no—
253#13881—531nomaximal, Sylow
254C22×C53C22×C4⋊C411yescharacteristic
255C22×C53C2×#11712yes—
256C22×C53C2×C4×D411yescharacteristic
257C22×C53C2×C4×D411yescharacteristic
258C22×C53#52012yes—
259C22×C53C2×#12011yescharacteristic
260C22×C53C2×C22⋊Q811yescharacteristic
261C22×C53C2×C22⋊Q811yescharacteristic
262C22×C53C2×#12311yescharacteristic
263C22×C53C2×#12411yescharacteristic
264C22×C53C2×#12411yescharacteristic
265C22×C53C2×#12512yes—
266C22×C53#53511yescharacteristic
267C22×C53#53611yescharacteristic
268C22×C53#54412yes—
269C22×C53#54612yes—
270C22×C53#54612yes—
271C22×C53#54812yes—
272C22×C53#54812yes—
273C22×C53#55212yes—
274C22×C53#55212yes—
275C22×C53#55312yes—
276C22×C53#55312yes—
277C2×D53—24no—
278C2×D53—28no—
279C2×D53—28no—
280C2×D53—24no—
281C2×D53—24no—
282C2×D53—28no—
283C2×D53—28no—
284C2×D53—24no—
285C2×D53—24no—
286C2×D53—24no—
287C2×D53#38218yescomplemented
288C4×C53—44no—
289C4×C53—44no—
290C4×C53—82no—
291C4×C53#38518yes—
292Dic53—24no—
293Dic53—44no—
294Dic53—44no—
295Dic53—82no—
296C23×C4⋊C4—531no—
297C2×#382—531no—
298C2×#382—531no—
299C2×#380—531no—
300C2×C4×C22⋊C4—531no—
301#1442—534no—
302C4⋊#95—531no—
303C4⋊#95D5311yescharacteristic, complemented
304C22⋊#95—534no—
305C2×C53C2×#38211yescharacteristic
306C2×C53C2×#38511yescharacteristic
307C2×C53#231711yescharacteristic
308C2×C53#235212yes—
309C2×C53#235212yes—
310C2×C53#243511yescharacteristic
311C2×C53#243611yescharacteristic
312C2×C53#244212yes—
313C2×C53#244212yes—
314C2×C53#250311yescharacteristic
315C2×C53#250411yescharacteristic
316D53—28no—
317C24×C4—531no—
318C24×C4—532no—
319C22×C4⋊C4—531no—
320C22×C4⋊C4—532no—
321C22×C4⋊C4—531no—
322C22×C4⋊C4—1064no—
323C22×C4⋊C4—1064no—
324C22×C4⋊C4—1064no—
325C22×C4⋊C4C2×D5311yescharacteristic
326C22×C4⋊C4C2×D5311yescharacteristic
327C22×C22⋊C4—532no—
328C22×C22⋊C4—532no—
329C22×C42—531no—
330C22×C42C2×D5311yescharacteristic
331#382—1064no—
332#382—1064no—
333#380—1064no—
334C4×C22⋊C4—538no—
335C2×#95—532no—
336C2×#95—534no—
337C2×#95—534no—
338C2×#95—532no—
339C2×#95—534no—
340C2×#95C2×D5314yes—
341C53#1388111yescharacteristic, complemented, Sylow
342C25—531no—
343C23×C4—531no—
344C23×C4—534no—
345C23×C4—534no—
346C23×C4—534no—
347C23×C4—532no—
348C23×C4—532no—
349C23×C4—532no—
350C23×C4—1064no—
351C23×C4—1064no—
352C23×C4—1068no—
353C23×C4—1062no—
354C23×C4—1062no—
355C23×C4—1062no—
356C23×C4—1062no—
357C23×C4—1064no—
358C23×C4—2124no—
359C23×C4C22×D5312yes—
360C23×C4C22×D5312yes—
361C23×C4C22×D5312yes—
362C23×C4C22×D5311yescharacteristic
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—534no—
369C2×C4⋊C4—10632no—
370C2×C4⋊C4—1064no—
371C2×C4⋊C4—1064no—
372C2×C4⋊C4—1064no—
373C2×C4⋊C4—1062no—
374C2×C4⋊C4—1062no—
375C2×C4⋊C4—1062no—
376C2×C4⋊C4—1062no—
377C2×C4⋊C4—1062no—
378C2×C4⋊C4D21214yes—
379C2×C22⋊C4—538no—
380C2×C22⋊C4—1064no—
381C2×C22⋊C4—1064no—
382C2×C22⋊C4—1068no—
383C2×C22⋊C4—1064no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—534no—
387C2×C42—1062no—
388C2×C42—1062no—
389C2×C42C4×D5314yescomplemented
390#95—44no—
391#95—538no—
392#95—1064no—
393#95—1068no—
394#95—1068no—
395#95—2124no—
396C24—532no—
397C24—532no—
398C24—532no—
399C24—534no—
400C24—534no—
401C24—1061no—
402C24—1061no—
403C24—1061no—
404C24—1061no—
405C24—1062no—
406C24—1062no—
407C24C23×D5311yescharacteristic
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—532no—
424C22×C4—532no—
425C22×C4—532no—
426C22×C4—5316no—
427C22×C4—534no—
428C22×C4—534no—
429C22×C4—10616no—
430C22×C4—1061no—
431C22×C4—1061no—
432C22×C4—1061no—
433C22×C4—1061no—
434C22×C4—1068no—
435C22×C4—1068no—
436C22×C4—1068no—
437C22×C4—1068no—
438C22×C4—1068no—
439C22×C4—1068no—
440C22×C4—1062no—
441C22×C4—1062no—
442C22×C4—1062no—
443C22×C4—1062no—
444C22×C4—1062no—
445C22×C4—1064no—
446C22×C4—1062no—
447C22×C4—1062no—
448C22×C4—1064no—
449C22×C4—1062no—
450C22×C4—1062no—
451C22×C4—1062no—
452C22×C4—1062no—
453C22×C4—1064no—
454C22×C4—1064no—
455C22×C4—21216no—
456C22×C4—21216no—
457C22×C4—2128no—
458C22×C4—2128no—
459C22×C4C2×C4×D5314yes—
460C22×C4C2×C4×D5312yes—
461C22×C4C2×D21214yes—
462C22×C4C2×D21212yes—
463C22×C4#1171994512yes—
464C22×C4D4×D5312yes—
465C22×C4D4×D5312yes—
466C22×C4#1171994712yes—
467C22×C4Q8×D5312yes—
468C4⋊C4—28no—
469C4⋊C4—44no—
470C4⋊C4—10616no—
471C4⋊C4—1068no—
472C4⋊C4—2124no—
473C22⋊C4—10616no—
474C22⋊C4—2128no—
475C42—28no—
476C42—1068no—
477C23—534no—
478C23—538no—
479C23—534no—
480C23—534no—
481C23—538no—
482C23—1064no—
483C23—1064no—
484C23—1064no—
485C23—1064no—
486C23—1064no—
487C23—1064no—
488C23—1064no—
489C23—1064no—
490C23—1062no—
491C23—1062no—
492C23—1064no—
493C23—1062no—
494C23—1062no—
495C23—1062no—
496C23—1062no—
497C23—1062no—
498C23—1062no—
499C23—1064no—
500C23C22×C4×D5311yescharacteristic
501C23C2×D4×D5311yescharacteristic
502C23C2×D4×D5311yescharacteristic
503C23C22×D21211yescharacteristic
504C23C2×Q8×D5311yescharacteristic
505C23#41×D5312yes—
506C23#41×D5312yes—
507C23C2×#1171994711yescharacteristic
508C23C2×#1171994511yescharacteristic
509C23#1313326212yes—
510C23#1313326512yes—
511C2×C4—28no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—534no—
529C2×C4—1062no—
530C2×C4—1062no—
531C2×C4—1062no—
532C2×C4—1062no—
533C2×C4—1062no—
534C2×C4—1062no—
535C2×C4—10632no—
536C2×C4—1068no—
537C2×C4—1064no—
538C2×C4—1064no—
539C2×C4—1064no—
540C2×C4—1064no—
541C2×C4—21232no—
542C2×C4—2124no—
543C2×C4—2124no—
544C2×C4—2122no—
545C2×C4—2122no—
546C2×C4—2122no—
547C2×C4—2122no—
548C2×C4—2124no—
549C2×C4—2122no—
550C2×C4—2122no—
551C2×C4—2122no—
552C2×C4—2122no—
553C2×C4—2122no—
554C2×C4C4⋊C4×D5314yes—
555C2×C4#1313317714yes—
556C2×C4C4×D21214yes—
557C2×C4#1313318714yes—
558C2×C4#1313319514yes—
559C2×C4#1313319714yes—
560C2×C4#1313319914yes—
561C22—538no—
562C22—1064no—
563C22—1064no—
564C22—1064no—
565C22—1068no—
566C22—1064no—
567C22—1068no—
568C22—1064no—
569C22—1068no—
570C22—1064no—
571C22—1068no—
572C22C2×C4⋊C4×D5311yescharacteristic
573C22#117×D5312yes—
574C22C2×#1313317711yescharacteristic
575C22C2×C4×D21211yescharacteristic
576C22#1681110412yes—
577C22C2×#1313318711yescharacteristic
578C22C2×#1313319511yescharacteristic
579C22C2×#1313319711yescharacteristic
580C22C2×#1313319911yescharacteristic
581C22#124×D5311yescharacteristic
582C22#124×D5311yescharacteristic
583C22#125×D5312yes—
584C22#1681118611yescharacteristic
585C22#1681119811yescharacteristic
586C22#1681125012yes—
587C22#1681126812yes—
588C22#1681127012yes—
589C22#1681128312yes—
590C22#1681128512yes—
591C22#1681130712yes—
592C22#1681131412yes—
593C22#1681132012yes—
594C22#1681132312yes—
595C4—44no—
596C4—44no—
597C4—82no—
598C4—1064no—
599C4—4242no—
600C4—2124no—
601C4—2124no—
602C4#1681049718yes—
603C2—1068no—
604C2#382×D5311yescharacteristic
605C2C2×#1681049711yescharacteristic
606C2#2027733611yescharacteristic
607C2#2027763712yes—
608C2#2027764712yes—
609C2#2027836511yescharacteristic
610C2#2027837211yescharacteristic
611C2#2027843712yes—
612C2#2027844112yes—
613C2#2027919911yescharacteristic
614C2#2027920611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC53×C4⋊#95C2
socle, cosocleC24×C53C24
prime radicalC24×D53C23
hypercenterC4⋊#95D53
hypocenterC53#13881
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
2. C2×#16810489C2semidirect
3. C2×#16810488C2semidirect
6. #20272970C2semidirect
7. #20272944C2semidirect
9. C53×C4⋊#95C2semidirect
24. C4×#11719921C4semidirect
68. C2×C4×Dic53C2×C4semidirect
73. #13133124C2×C4semidirect
86. C2×C42×C53C2×C4semidirect
287. C2×D53#382semidirect
303. C4⋊#95D53semidirect
341. C53#13881semidirect
389. C2×C42C4×D53semidirect

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×#168104892
C2×#168104882
C2×#168104802
C2×C4×#117199212
#202729702
#202729442
#202729362
C53×C4⋊#952
#1388153