Finite Group Databasebeta

← #27291016

Subgroups — #27291016

Order 13568.

1454 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133132C212yesmaximal
2C2×#16810483C211yesmaximal, characteristic, complemented
3C2×#16810390C212yesmaximal, complemented
4#20272938C211yesmaximal, characteristic, complemented
5C2×C53×#386C211yesmaximal, characteristic
6C22×C4⋊Dic53—21no—
7C22×C22⋊Dic53C2212yes—
8C22×C22⋊Dic53C2212yes—
9C2×#13133132—24no—
10C2×#13133132—24no—
11C2×#13133132—24no—
12#16810483—24no—
13C2×#13133124—21no—
14C2×#13133124—22no—
15#16810390—24no—
16C23×D4×C53C2211yescharacteristic
17C24×C4×C53C2211yescharacteristic
18C22×C22⋊C4×C53C412yescomplemented
19C22×C22⋊C4×C53C2211yescharacteristic
20C53×#386—21no—
21C53×#386—22no—
22C53×#386—21no—
23C2×#95×C53C412yescomplemented
24C23×Dic53—22no—
25C23×Dic53—22no—
26C2×C4⋊Dic53—24no—
27C2×C4⋊Dic53—44no—
28C2×C22⋊Dic53—22no—
29C2×C22⋊Dic53—22no—
30C2×C22⋊Dic53—22no—
31C2×C22⋊Dic53—22no—
32C2×C22⋊Dic53—22no—
33C2×C22⋊Dic53—22no—
34C2×C22⋊Dic53—22no—
35C2×C22⋊Dic53—22no—
36C2×C22⋊Dic53—22no—
37C2×C22⋊Dic53—22no—
38C2×C22⋊Dic53D414yes—
39C2×C22⋊Dic53D414yes—
40#13133132—216no—
41#13133124—24no—
42#13133124—44no—
43C25×C53C2×C411yescharacteristic
44C25×C53C2311yescharacteristic
45C22×D4×C53—22no—
46C22×D4×C53—22no—
47C22×D4×C53—21no—
48C22×D4×C53—21no—
49C22×D4×C53—21no—
50C22×D4×C53—21no—
51C22×D4×C53—22no—
52C22×D4×C53—22no—
53C22×D4×C53D412yes—
54C22×D4×C53Q812yes—
55C23×C4×C53—22no—
56C23×C4×C53—22no—
57C23×C4×C53—44no—
58C23×C4×C53C2×C411yescharacteristic
59C23×C4×C53C2×C412yes—
60C23×C4×C53C2×C412yes—
61C23×C4×C53D411yescharacteristic
62C23×C4×C53D412yes—
63C23×C4×C53D411yescharacteristic
64C2×C22⋊C4×C53—22no—
65C2×C22⋊C4×C53—21no—
66C2×C22⋊C4×C53—21no—
67C2×C22⋊C4×C53—22no—
68C2×C22⋊C4×C53—22no—
69C2×C22⋊C4×C53—22no—
70C2×C22⋊C4×C53—44no—
71#95×C53—22no—
72#95×C53—22no—
73C24×C53—21no—
74C24×C53—21no—
75C24×C53—22no—
76C24×C53—22no—
77C24×C53—42no—
78C24×C53—42no—
79C24×C53—42no—
80C24×C53—42no—
81C24×C53C4⋊C412yescomplemented
82C24×C53C22×C411yescharacteristic
83C24×C53C2×D411yescharacteristic
84C24×C53C2×D411yescharacteristic
85C24×C53C2×D411yescharacteristic
86C24×C53C2×D412yes—
87C24×C53C2×D412yes—
88C24×C53C2×Q811yescharacteristic
89C24×C53#4111yescharacteristic
90C24×C53#4111yescharacteristic
91C24×C53#4112yes—
92C24×C53#4112yes—
93C2×D4×C53—24no—
94C2×D4×C53—22no—
95C2×D4×C53—22no—
96C2×D4×C53—22no—
97C2×D4×C53—22no—
98C2×D4×C53—24no—
99C2×D4×C53—24no—
100C2×D4×C53—24no—
101C2×D4×C53—41no—
102C2×D4×C53—42no—
103C2×D4×C53—42no—
104C2×D4×C53—42no—
105C2×D4×C53—41no—
106C2×D4×C53—42no—
107C2×D4×C53—42no—
108C2×D4×C53—41no—
109C2×D4×C53—42no—
110C2×D4×C53—41no—
111C22×C4×C53—21no—
112C22×C4×C53—21no—
113C22×C4×C53—24no—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53—24no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—21no—
120C22×C4×C53—24no—
121C22×C4×C53—21no—
122C22×C4×C53—24no—
123C22×C4×C53—24no—
124C22×C4×C53—22no—
125C22×C4×C53—24no—
126C22×C4×C53—42no—
127C22×C4×C53—42no—
128C22×C4×C53—42no—
129C22×C4×C53—48no—
130C22×C4×C53—48no—
131C22×C4×C53—42no—
132C22×C4×C53—42no—
133C22×C4×C53C22⋊C412yes—
134C22×C4×C53C22⋊C412yes—
135C22×C4×C53C4⋊C412yescomplemented
136C22×Dic53—24no—
137C22×Dic53—24no—
138C22×Dic53—44no—
139C22×Dic53—44no—
140C22×Dic53—44no—
141C22×Dic53—44no—
142C22×Dic53—42no—
143C22×Dic53—42no—
144C22×Dic53—42no—
145C22×Dic53—42no—
146C22⋊C4×C53—44no—
147C22⋊C4×C53—42no—
148C22⋊C4×C53—41no—
149C22⋊C4×C53—41no—
150C22⋊C4×C53—44no—
151C22⋊Dic53—24no—
152C22⋊Dic53—24no—
153C22⋊Dic53—24no—
154C22⋊Dic53—24no—
155C22⋊Dic53—42no—
156C22⋊Dic53—42no—
157C22⋊Dic53—42no—
158C22⋊Dic53—42no—
159C4⋊Dic53—48no—
160C23×C53—22no—
161C23×C53—22no—
162C23×C53—22no—
163C23×C53—22no—
164C23×C53—22no—
165C23×C53—22no—
166C23×C53—22no—
167C23×C53—24no—
168C23×C53—42no—
169C23×C53—42no—
170C23×C53—42no—
171C23×C53—42no—
172C23×C53—44no—
173C23×C53—42no—
174C23×C53—42no—
175C23×C53—42no—
176C23×C53—42no—
177C23×C53—44no—
178C23×C53—44no—
179C23×C53—44no—
180C23×C53—41no—
181C23×C53—42no—
182C23×C53—41no—
183C23×C53—42no—
184C23×C53—42no—
185C23×C53—82no—
186C23×C53—82no—
187C23×C53—84no—
188C23×C53C2×C22⋊C411yescharacteristic
189C23×C53C2×C4⋊C411yescharacteristic
190C23×C53#11711yescharacteristic
191C23×C53C4×D412yes—
192C23×C53C4×D412yes—
193C23×C53#12011yescharacteristic
194C23×C53#12012yes—
195C23×C53#12112yes—
196C23×C53#12112yes—
197C23×C53#12112yes—
198C23×C53#12111yescharacteristic
199C23×C53#12111yescharacteristic
200C23×C53C22⋊Q812yes—
201C23×C53C22⋊Q812yes—
202C23×C53C22⋊Q811yescharacteristic
203C23×C53C22⋊Q811yescharacteristic
204C23×C53#12311yescharacteristic
205C23×C53#12312yes—
206C23×C53#12312yes—
207C23×C53#12412yes—
208C23×C53#12411yescharacteristic
209C23×C53#12612yes—
210C23×C53#12711yescharacteristic
211D4×C53—42no—
212D4×C53—42no—
213D4×C53—44no—
214D4×C53—44no—
215D4×C53—42no—
216D4×C53—42no—
217C2×C4×C53—24no—
218C2×C4×C53—24no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—24no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—48no—
227C2×C4×C53—48no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53—41no—
231C2×C4×C53—42no—
232C2×C4×C53—41no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53—84no—
237C2×C4×C53#9914yescomplemented
238C2×Dic53—44no—
239C2×Dic53—48no—
240C2×Dic53—44no—
241C2×Dic53—44no—
242C2×Dic53—44no—
243C2×Dic53—84no—
244C2×Dic53—84no—
245C2×Dic53—82no—
246C2×Dic53—82no—
247C2×Dic53—84no—
248#11043—531nomaximal, Sylow
249C22×C53—22no—
250C22×C53—22no—
251C22×C53—22no—
252C22×C53—22no—
253C22×C53—22no—
254C22×C53—22no—
255C22×C53—21no—
256C22×C53—22no—
257C22×C53—21no—
258C22×C53—22no—
259C22×C53—21no—
260C22×C53—22no—
261C22×C53—22no—
262C22×C53—21no—
263C22×C53—24no—
264C22×C53—22no—
265C22×C53—24no—
266C22×C53—22no—
267C22×C53—42no—
268C22×C53—42no—
269C22×C53—42no—
270C22×C53—42no—
271C22×C53—42no—
272C22×C53—42no—
273C22×C53—42no—
274C22×C53—42no—
275C22×C53—44no—
276C22×C53—42no—
277C22×C53—84no—
278C22×C53—84no—
279C22×C53—81no—
280C22×C53—81no—
281C22×C53—82no—
282C22×C53#38011yescharacteristic
283C22×C53#38512yes—
284C22×C53#38611yescharacteristic
285C22×C53#38812yes—
286C22×C53#39011yescharacteristic
287C22×C53#39211yescharacteristic
288C22×C53#39311yescharacteristic
289C22×C53#39412yes—
290C22×C53#39612yes—
291C22×C53#39711yescharacteristic
292C22×C53#39911yescharacteristic
293C22×C53C2×#9912yes—
294C22×C53#41012yes—
295C4×C53—24no—
296C4×C53—42no—
297C4×C53—42no—
298C4×C53—82no—
299Dic53—84no—
300Dic53—84no—
301C22×#99—532no—
302C2×#386D5311yescharacteristic
303C2×#380—531no—
304C2×#342—532no—
305C22⋊#95—531no—
306C2×C53—21no—
307C2×C53—22no—
308C2×C53—22no—
309C2×C53—22no—
310C2×C53—21no—
311C2×C53—22no—
312C2×C53—21no—
313C2×C53—21no—
314C2×C53—82no—
315C2×C53#143911yescharacteristic
316C2×C53#179912yes—
317C2×C53#189811yescharacteristic
318C2×C53#189911yescharacteristic
319C2×C53#196412yes—
320C23×D4C2×D5311yescharacteristic
321C24×C4C2×D5311yescharacteristic
322C22×C4⋊C4—1061no—
323C22×C22⋊C4—532no—
324C22×C22⋊C4—532no—
325C22×C22⋊C4Dic5312yescomplemented
326C22×C22⋊C4C2×D5311yescharacteristic
327C2×#99—1064no—
328C2×#99—1064no—
329C2×#99—1064no—
330#386—21no—
331#386—22no—
332#386—21no—
333#380—1064no—
334C2×#95—1061no—
335C2×#95—1062no—
336C2×#95Dic5312yescomplemented
337#342—1064no—
338C53#1104311yescharacteristic, complemented, Sylow
339C25C2×Dic5311yescharacteristic
340C25C22×D5311yescharacteristic
341C22×D4—21no—
342C22×D4—21no—
343C22×D4—22no—
344C22×D4—22no—
345C22×D4—22no—
346C22×D4—22no—
347C22×D4—21no—
348C22×D4—21no—
349C22×D4Dic10612yes—
350C22×D4D21212yes—
351C23×C4—22no—
352C23×C4—22no—
353C23×C4—44no—
354C23×C4—1062no—
355C23×C4—1062no—
356C23×C4C2×Dic5311yescharacteristic
357C23×C4C2×Dic5312yes—
358C23×C4C2×Dic5312yes—
359C23×C4C22⋊D5311yescharacteristic
360C23×C4C22⋊D5312yes—
361C23×C4C22⋊D5311yescharacteristic
362C2×C4⋊C4—1064no—
363C2×C4⋊C4—2124no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—44no—
371C2×C22⋊C4—534no—
372C2×C22⋊C4—534no—
373C2×C22⋊C4—1062no—
374C2×C22⋊C4—1062no—
375C2×C22⋊C4—1062no—
376C2×C22⋊C4—1062no—
377C2×C22⋊C4—1062no—
378C2×C22⋊C4—1062no—
379C2×C22⋊C4—1062no—
380C2×C22⋊C4—1062no—
381C2×C22⋊C4—1062no—
382C2×C22⋊C4—1062no—
383#99—10616no—
384#95—22no—
385#95—22no—
386#95—1064no—
387#95—2124no—
388C24—21no—
389C24—21no—
390C24—22no—
391C24—22no—
392C24—42no—
393C24—42no—
394C24—42no—
395C24—42no—
396C24C4⋊Dic5312yescomplemented
397C24C2×Dic10611yescharacteristic
398C24C2×D21211yescharacteristic
399C24#1171994511yescharacteristic
400C24#1171994511yescharacteristic
401C24D4×D5312yes—
402C24D4×D5312yes—
403C24#1171994712yes—
404C24#1171994712yes—
405C24C22×Dic5311yescharacteristic
406C24C2×C22⋊D5311yescharacteristic
407C24C2×C22⋊D5311yescharacteristic
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—24no—
416C2×D4—41no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—41no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—41no—
424C2×D4—42no—
425C2×D4—41no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—1064no—
449C22×C4—1064no—
450C22×C4—2124no—
451C22×C4—2124no—
452C22×C4—2124no—
453C22×C4—2124no—
454C22×C4—2122no—
455C22×C4—2122no—
456C22×C4—2122no—
457C22×C4—2122no—
458C22×C4C4⋊Dic5312yescomplemented
459C22×C4C22⋊Dic5312yes—
460C22×C4C22⋊Dic5312yes—
461C4⋊C4—2128no—
462C22⋊C4—44no—
463C22⋊C4—42no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—44no—
467C22⋊C4—1064no—
468C22⋊C4—1064no—
469C22⋊C4—1064no—
470C22⋊C4—1064no—
471C22⋊C4—2122no—
472C22⋊C4—2122no—
473C22⋊C4—2122no—
474C22⋊C4—2122no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—44no—
490C23—42no—
491C23—44no—
492C23—42no—
493C23—44no—
494C23—44no—
495C23—41no—
496C23—42no—
497C23—41no—
498C23—42no—
499C23—42no—
500C23—84no—
501C23—82no—
502C23—82no—
503C23C2×C22⋊Dic5311yescharacteristic
504C23C2×C4⋊Dic5311yescharacteristic
505C23#1313317611yescharacteristic
506C23D4×Dic5312yes—
507C23D4×Dic5312yes—
508C23#1313318511yescharacteristic
509C23#1313318712yes—
510C23#1313318812yes—
511C23#1313318912yes—
512C23#1313319012yes—
513C23#1313319111yescharacteristic
514C23#1313319111yescharacteristic
515C23#1313319412yes—
516C23#1313319412yes—
517C23#1313319611yescharacteristic
518C23#1313319611yescharacteristic
519C23#1313319811yescharacteristic
520C23#1313319912yes—
521C23#1313320112yes—
522C23#1313320312yes—
523C23#1313320411yescharacteristic
524C23#1313321012yes—
525C23#1313321311yescharacteristic
526D4—42no—
527D4—42no—
528D4—44no—
529D4—44no—
530D4—42no—
531D4—42no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—48no—
542C2×C4—48no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—84no—
552C2×C4—4244no—
553C2×C4—4244no—
554C2×C4—4244no—
555C2×C4—4242no—
556C2×C4—4242no—
557C2×C4—2124no—
558C2×C4—2128no—
559C2×C4—2124no—
560C2×C4—2124no—
561C2×C4—2124no—
562C2×C4#1313313214yescomplemented
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—24no—
570C22—22no—
571C22—24no—
572C22—22no—
573C22—21no—
574C22—21no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—21no—
579C22—22no—
580C22—21no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—44no—
590C22—42no—
591C22—84no—
592C22—84no—
593C22—82no—
594C22—81no—
595C22—81no—
596C22#1681048311yescharacteristic
597C22#1681049812yes—
598C22#1681050311yescharacteristic
599C22#1681051612yes—
600C22#1681052611yescharacteristic
601C22#1681053211yescharacteristic
602C22#1681053511yescharacteristic
603C22#1681053912yes—
604C22#1681054612yes—
605C22#1681054811yescharacteristic
606C22#1681055911yescharacteristic
607C22C2×#1313313212yes—
608C22#1681059712yes—
609C4—24no—
610C4—42no—
611C4—42no—
612C4—82no—
613C4—4244no—
614C4—4244no—
615C2—21no—
616C2—22no—
617C2—22no—
618C2—22no—
619C2—21no—
620C2—22no—
621C2—21no—
622C2—21no—
623C2—82no—
624C2#2027296011yescharacteristic
625C2#2027470312yes—
626C2#2027516911yescharacteristic
627C2#2027517411yescharacteristic
628C2#2027555212yes—
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC24×C53C22×C4
FrattiniC25C22×D53
FittingC2×C53×#386C2
socleC23×C53C2×C22⋊C4
prime radicalC23×D4×C53C22
cosocleC25×C53C23
hypercenterC2×#386D53
hypocenterC53#11043
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#386

Complements

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

NH ≅ G/Ntype
2. C2×#16810483C2semidirect
3. C2×#16810390C2semidirect
4. #20272938C2semidirect
18. C22×C22⋊C4×C53C4semidirect
23. C2×#95×C53C4semidirect
81. C24×C53C4⋊C4semidirect
135. C22×C4×C53C4⋊C4semidirect
237. C2×C4×C53#99semidirect
325. C22×C22⋊C4Dic53semidirect
336. C2×#95Dic53semidirect
338. C53#11043semidirect
396. C24C4⋊Dic53semidirect
458. C22×C4C4⋊Dic53semidirect
562. C2×C4#13133132semidirect

Maximal subgroups

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

subgroupindex
C22×#131331322
C2×#168104832
C2×#168103902
#202729382
C2×C53×#3862
#1104353