Finite Group Databasebeta

← #27290605

Subgroups — #27290605

Order 13568.

1578 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133141C212yesmaximal, complemented
2C22×C22⋊#88322C211yesmaximal, characteristic
3C2×#16810483C211yesmaximal, characteristic
4C2×#16810385C212yesmaximal, complemented
5C22×C22⋊Q8×C53C211yesmaximal, characteristic
6C23×#88322C2212yes—
7C22×C4⋊Dic53C2212yes—
8C22×C22⋊Dic53—21no—
9C2×#13133141—24no—
10C2×#13133141—24no—
11C2×#13133141—24no—
12C2×C22⋊#88322—24no—
13C2×C22⋊#88322—22no—
14#16810483—24no—
15C2×#13133124—21no—
16#16810385—24no—
17C23×Q8×C53C2211yescharacteristic
18C24×C4×C53C2211yescharacteristic
19C2×C22⋊Q8×C53—24no—
20C2×C22⋊Q8×C53—22no—
21C2×C22⋊Q8×C53—24no—
22C2×C22⋊Q8×C53C414yescomplemented
23C22×C4⋊C4×C53—21no—
24C22×C4⋊C4×C53C2211yescharacteristic
25C22×C22⋊C4×C53—21no—
26C23×Dic53—22no—
27C22×#88322—22no—
28C22×#88322—22no—
29C22×#88322—22no—
30C22×#88322—22no—
31C22×#88322—22no—
32C22×#88322—22no—
33C22×#88322D414yes—
34C2×C4⋊Dic53—22no—
35C2×C4⋊Dic53—22no—
36C2×C4⋊Dic53—22no—
37C2×C4⋊Dic53—22no—
38C2×C4⋊Dic53—22no—
39C2×C4⋊Dic53D414yes—
40C2×C22⋊Dic53—24no—
41C2×C22⋊Dic53—42no—
42C2×C22⋊Dic53—42no—
43#13133141—216no—
44C22⋊#88322—28no—
45#13133124—24no—
46C25×C53D411yescharacteristic
47C22×Q8×C53—21no—
48C22×Q8×C53—22no—
49C22×Q8×C53—22no—
50C22×Q8×C53—22no—
51C22×Q8×C53—22no—
52C22×Q8×C53—22no—
53C22×Q8×C53C2×C412yes—
54C22×Q8×C53D414yes—
55C23×C4×C53—21no—
56C23×C4×C53—21no—
57C23×C4×C53—22no—
58C23×C4×C53—22no—
59C23×C4×C53—22no—
60C23×C4×C53—22no—
61C23×C4×C53—22no—
62C23×C4×C53—22no—
63C23×C4×C53C2×C412yes—
64C23×C4×C53D411yescharacteristic
65C23×C4×C53D412yes—
66C23×C4×C53C2311yescharacteristic
67C22⋊Q8×C53—216no—
68C22⋊Q8×C53—28no—
69C22⋊Q8×C53—28no—
70C2×C4⋊C4×C53—28no—
71C2×C4⋊C4×C53—24no—
72C2×C4⋊C4×C53—21no—
73C2×C4⋊C4×C53—22no—
74C2×C4⋊C4×C53C2×C412yes—
75C2×C4⋊C4×C53D414yes—
76C2×C22⋊C4×C53—28no—
77C2×C22⋊C4×C53—24no—
78C24×C53—22no—
79C24×C53—21no—
80C24×C53—22no—
81C24×C53—21no—
82C24×C53—22no—
83C24×C53—21no—
84C24×C53—21no—
85C24×C53—22no—
86C24×C53C22⋊C412yes—
87C24×C53SD1612yes—
88C24×C53Q1612yes—
89C24×C53C2×D411yescharacteristic
90C2×Q8×C53—22no—
91C2×Q8×C53—24no—
92C2×Q8×C53—24no—
93C2×Q8×C53—24no—
94C2×Q8×C53—24no—
95C2×Q8×C53—24no—
96C2×Q8×C53—24no—
97C2×Q8×C53—24no—
98C2×Q8×C53—41no—
99C2×Q8×C53—42no—
100C2×Q8×C53—42no—
101C2×Q8×C53—42no—
102C2×Q8×C53—41no—
103C2×Q8×C53—42no—
104C2×Q8×C53—42no—
105C2×Q8×C53C22⋊C414yescomplemented
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—24no—
109C22×C4×C53—24no—
110C22×C4×C53—22no—
111C22×C4×C53—21no—
112C22×C4×C53—22no—
113C22×C4×C53—21no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—24no—
117C22×C4×C53—24no—
118C22×C4×C53—24no—
119C22×C4×C53—22no—
120C22×C4×C53—24no—
121C22×C4×C53—22no—
122C22×C4×C53—41no—
123C22×C4×C53—42no—
124C22×C4×C53—41no—
125C22×C4×C53—42no—
126C22×C4×C53—41no—
127C22×C4×C53—41no—
128C22×C4×C53—42no—
129C22×C4×C53—42no—
130C22×C4×C53—42no—
131C22×C4×C53—42no—
132C22×C4×C53—42no—
133C22×C4×C53—42no—
134C22×C4×C53—42no—
135C22×C4×C53—42no—
136C22×C4×C53C22⋊C412yes—
137C22×C4×C53C22×C411yescharacteristic
138C22×C4×C53C2×D412yes—
139C22×C4×C53C2×D412yes—
140C22×C4×C53C2×D412yes—
141C22×C4×C53C2×D412yes—
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53#4112yes—
144C22×C4×C53#4112yes—
145C22×Dic53—24no—
146C22×Dic53—42no—
147C22×Dic53—42no—
148C22×Dic53—42no—
149C22×Dic53—42no—
150C22×Dic53—42no—
151C22×Dic53—42no—
152C4⋊C4×C53—216no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—24no—
155C4⋊C4×C53C22⋊C414yescomplemented
156C22⋊C4×C53—216no—
157C22⋊Dic53—48no—
158C4⋊Dic53—24no—
159C4⋊Dic53—24no—
160C4⋊Dic53—42no—
161C4⋊Dic53—42no—
162C2×#88322—24no—
163C2×#88322—24no—
164C2×#88322—24no—
165C2×#88322—42no—
166C2×#88322—42no—
167C2×#88322—42no—
168C2×#88322—42no—
169C23×C53—22no—
170C23×C53—22no—
171C23×C53—22no—
172C23×C53—22no—
173C23×C53—22no—
174C23×C53—22no—
175C23×C53—24no—
176C23×C53—22no—
177C23×C53—22no—
178C23×C53—24no—
179C23×C53—24no—
180C23×C53—24no—
181C23×C53—22no—
182C23×C53—22no—
183C23×C53—42no—
184C23×C53—42no—
185C23×C53—42no—
186C23×C53—42no—
187C23×C53—42no—
188C23×C53—42no—
189C23×C53—42no—
190C23×C53—42no—
191C23×C53#10314yescomplemented
192C23×C53C2×C22⋊C411yescharacteristic
193C23×C53#12011yescharacteristic
194C23×C53#12012yes—
195C23×C53#12112yes—
196C23×C53#12311yescharacteristic
197C23×C53C2×SD1611yescharacteristic
198C23×C53C2×Q1611yescharacteristic
199C23×C53#13512yes—
200C23×C53AGL(1,ℤ/8)11yescharacteristic
201C23×C53#13711yescharacteristic
202C23×C53#13712yes—
203Q8×C53—28no—
204Q8×C53—44no—
205Q8×C53—44no—
206Q8×C53—44no—
207C2×C4×C53—22no—
208C2×C4×C53—24no—
209C2×C4×C53—24no—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—42no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—44no—
216C2×C4×C53—42no—
217C2×C4×C53—44no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—44no—
221C2×C4×C53—44no—
222C2×C4×C53—44no—
223C2×C4×C53—44no—
224C2×C4×C53—44no—
225C2×C4×C53—44no—
226C2×C4×C53C2×C22⋊C411yescharacteristic
227C2×C4×C53C2×C22⋊C411yescharacteristic
228C2×C4×C53C4×D412yes—
229C2×C4×C53C4×D412yes—
230C2×C4×C53#12012yes—
231C2×C4×C53#12011yescharacteristic
232C2×C4×C53#12012yes—
233C2×C4×C53#12112yes—
234C2×C4×C53#12112yes—
235C2×C4×C53#12112yes—
236C2×C4×C53#12112yes—
237C2×C4×C53#12112yes—
238C2×C4×C53#12312yes—
239C2×C4×C53#12311yescharacteristic
240C2×C4×C53#12412yes—
241C2×C4×C53#12712yes—
242C2×Dic53—44no—
243C2×Dic53—44no—
244C2×Dic53—44no—
245C2×Dic53—82no—
246C2×Dic53—82no—
247C2×Dic53—82no—
248C2×Dic53—82no—
249#88322—44no—
250#10970—531nomaximal, Sylow
251C22×C53—24no—
252C22×C53—24no—
253C22×C53—24no—
254C22×C53—42no—
255C22×C53—42no—
256C22×C53—42no—
257C22×C53—44no—
258C22×C53—44no—
259C22×C53—44no—
260C22×C53—44no—
261C22×C53—42no—
262C22×C53#37911yescharacteristic
263C22×C53#38611yescharacteristic
264C22×C53#38611yescharacteristic
265C22×C53#39212yes—
266C22×C53#39412yes—
267C22×C53C2×#10311yescharacteristic
268C22×C53#41611yescharacteristic
269C22×C53#41711yescharacteristic
270C22×C53#41911yescharacteristic
271C22×C53#44812yes—
272C22×C53#44912yes—
273C22×C53#45011yescharacteristic
274C22×C53#45111yescharacteristic
275C22×C53#45112yes—
276C22×C53#45212yes—
277C22×C53#45212yes—
278C22×C53#46512yes—
279C22×C53#46712yes—
280C22×C53#46812yes—
281C22×C53#47012yes—
282C22×C53#48311yescharacteristic
283C22×C53#48411yescharacteristic
284C22×C53#48512yes—
285C4×C53—42no—
286C4×C53—44no—
287C4×C53—44no—
288C4×C53#38612yes—
289C4×C53#39012yes—
290C4×C53#39212yes—
291C4×C53#39412yes—
292Dic53—84no—
293C22×C22⋊Q8D5311yescharacteristic
294C22×#103—532no—
295C22×C22⋊C8—531no—
296C2×#380—531no—
297C2×#340—532no—
298C2×C53—44no—
299C2×C53#143811yescharacteristic
300C2×C53#178711yescharacteristic
301C2×C53#178811yescharacteristic
302C2×C53#187211yescharacteristic
303C2×C53#187311yescharacteristic
304C2×C53#189211yescharacteristic
305C2×C53#189411yescharacteristic
306C2×C53#200012yes—
307C2×C53#200112yes—
308C2×C53#203512yes—
309C2×C53#203612yes—
310C23×Q8C2×D5311yescharacteristic
311C24×C4C2×D5311yescharacteristic
312C23×C8—532no—
313C2×C22⋊Q8—24no—
314C2×C22⋊Q8—22no—
315C2×C22⋊Q8—24no—
316C2×C22⋊Q8Dic5314yescomplemented
317C22×C4⋊C4—21no—
318C22×C4⋊C4—532no—
319C22×C4⋊C4C2×D5311yescharacteristic
320C22×C22⋊C4—21no—
321C22×C22⋊C4—1061no—
322C2×#103—1064no—
323C2×#103—1064no—
324C2×#103—1064no—
325C2×C22⋊C8—1064no—
326C2×C22⋊C8—1062no—
327#380—1064no—
328C2×#95—1061no—
329#340—1064no—
330C53#1097011yescharacteristic, complemented, Sylow
331C25C22⋊D5311yescharacteristic
332C22×Q8—21no—
333C22×Q8—22no—
334C22×Q8—22no—
335C22×Q8—22no—
336C22×Q8—22no—
337C22×Q8—22no—
338C22×Q8C2×Dic5312yes—
339C22×Q8C22⋊D5314yes—
340C23×C4—21no—
341C23×C4—21no—
342C23×C4—22no—
343C23×C4—22no—
344C23×C4—22no—
345C23×C4—22no—
346C23×C4—22no—
347C23×C4—22no—
348C23×C4—1062no—
349C23×C4C2×Dic5312yes—
350C23×C4C22⋊D5312yes—
351C23×C4C22⋊D5311yescharacteristic
352C23×C4C22×D5311yescharacteristic
353C22×C8—534no—
354C22×C8—1062no—
355C22×C8—1062no—
356C22×C8—1062no—
357C22×C8—1062no—
358C22×C8—1062no—
359C22×C8—1062no—
360C22⋊Q8—216no—
361C22⋊Q8—28no—
362C22⋊Q8—28no—
363C2×C4⋊C4—28no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—24no—
367C2×C4⋊C4—534no—
368C2×C4⋊C4—1062no—
369C2×C4⋊C4—1062no—
370C2×C4⋊C4—1062no—
371C2×C4⋊C4—1062no—
372C2×C4⋊C4—1062no—
373C2×C4⋊C4C2×Dic5312yes—
374C2×C4⋊C4C22⋊D5314yes—
375C2×C22⋊C4—28no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—1064no—
378C2×C22⋊C4—2122no—
379C2×C22⋊C4—2122no—
380#103—10616no—
381C22⋊C8—1068no—
382#95—1064no—
383C24—21no—
384C24—22no—
385C24—22no—
386C24—21no—
387C24—22no—
388C24—21no—
389C24—21no—
390C24—22no—
391C24#1171992412yes—
392C24#1171992512yes—
393C24C22⋊Dic5312yes—
394C24C2×C22⋊D5311yescharacteristic
395C2×Q8—22no—
396C2×Q8—24no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—24no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—41no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—41no—
408C2×Q8—42no—
409C2×Q8—42no—
410C2×Q8C22⋊Dic5314yescomplemented
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—41no—
428C22×C4—42no—
429C22×C4—41no—
430C22×C4—42no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—1064no—
442C22×C4—2122no—
443C22×C4—2122no—
444C22×C4—2122no—
445C22×C4—2122no—
446C22×C4—2122no—
447C22×C4—2122no—
448C22×C4C22⋊Dic5312yes—
449C22×C4D4×D5312yes—
450C22×C4D4×D5312yes—
451C22×C4#1171994712yes—
452C22×C4#1171994712yes—
453C22×C4C22×Dic5311yescharacteristic
454C22×C4C2×C22⋊D5312yes—
455C22×C4C2×C22⋊D5312yes—
456C22×C4C2×C22⋊D5311yescharacteristic
457C2×C8—1064no—
458C2×C8—1064no—
459C2×C8—1064no—
460C2×C8—2122no—
461C2×C8—2122no—
462C2×C8—2122no—
463C2×C8—2122no—
464C4⋊C4—216no—
465C4⋊C4—22no—
466C4⋊C4—24no—
467C4⋊C4—1064no—
468C4⋊C4—1064no—
469C4⋊C4—2122no—
470C4⋊C4—2122no—
471C4⋊C4C22⋊Dic5314yescomplemented
472C22⋊C4—216no—
473C22⋊C4—2128no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23#1313314114yescomplemented
497C23C2×C22⋊Dic5311yescharacteristic
498C23#1313318511yescharacteristic
499C23#1313318612yes—
500C23#1313318912yes—
501C23#1313319811yescharacteristic
502C23C2×#1171992411yescharacteristic
503C23C2×#1171992511yescharacteristic
504C23#1313323712yes—
505C23#1313324211yescharacteristic
506C23#1313324711yescharacteristic
507C23#1313324812yes—
508Q8—28no—
509Q8—44no—
510Q8—44no—
511Q8—44no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—4242no—
532C2×C4—4242no—
533C2×C4—4242no—
534C2×C4—4242no—
535C2×C4—2124no—
536C2×C4—2124no—
537C2×C4—2124no—
538C2×C4C2×C22⋊Dic5311yescharacteristic
539C2×C4C2×C22⋊Dic5311yescharacteristic
540C2×C4D4×Dic5312yes—
541C2×C4D4×Dic5312yes—
542C2×C4#1313318512yes—
543C2×C4#1313318511yescharacteristic
544C2×C4#1313318612yes—
545C2×C4#1313318812yes—
546C2×C4#1313318812yes—
547C2×C4#1313318912yes—
548C2×C4#1313318912yes—
549C2×C4#1313318912yes—
550C2×C4#1313319812yes—
551C2×C4#1313319811yescharacteristic
552C2×C4#1313320412yes—
553C2×C4#1313321312yes—
554C8—2124no—
555C22—24no—
556C22—24no—
557C22—24no—
558C22—44no—
559C22—44no—
560C22—42no—
561C22—42no—
562C22—44no—
563C22—44no—
564C22—42no—
565C22—42no—
566C22#1681047911yescharacteristic
567C22#1681050311yescharacteristic
568C22#1681050311yescharacteristic
569C22#1681053112yes—
570C22#1681053712yes—
571C22C2×#1313314111yescharacteristic
572C22#1681061911yescharacteristic
573C22#1681062411yescharacteristic
574C22#1681063211yescharacteristic
575C22#1681076212yes—
576C22#1681076812yes—
577C22#1681077311yescharacteristic
578C22#1681077811yescharacteristic
579C22#1681077912yes—
580C22#1681078312yes—
581C22#1681078512yes—
582C22#1681085412yes—
583C22#1681086212yes—
584C22#1681087112yes—
585C22#1681087912yes—
586C22#1681095111yescharacteristic
587C22#1681095611yescharacteristic
588C22#1681096112yes—
589C4—42no—
590C4—44no—
591C4—44no—
592C4—4244no—
593C4#1681050312yes—
594C4#1681052612yes—
595C4#1681053112yes—
596C4#1681053712yes—
597C2—44no—
598C2#2027295611yescharacteristic
599C2#2027465011yescharacteristic
600C2#2027465511yescharacteristic
601C2#2027502311yescharacteristic
602C2#2027502811yescharacteristic
603C2#2027513911yescharacteristic
604C2#2027514911yescharacteristic
605C2#2027571612yes—
606C2#2027572312yes—
607C2#2027590812yes—
608C2#2027591512yes—
subgroupG/H
centerC24C2×C22⋊D53
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC22×C22⋊Q8×C53C2
socleC24×C53C2×D4
prime radicalC25×C53D4
cosocleC23×C4×C53C23
hypercenterC22×C22⋊Q8D53
hypocenterC53#10970
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C24 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#13133141C2semidirect
4. C2×#16810385C2semidirect
22. C2×C22⋊Q8×C53C4semidirect
105. C2×Q8×C53C22⋊C4semidirect
155. C4⋊C4×C53C22⋊C4semidirect
191. C23×C53#103semidirect
316. C2×C22⋊Q8Dic53semidirect
330. C53#10970semidirect
410. C2×Q8C22⋊Dic53semidirect
471. C4⋊C4C22⋊Dic53semidirect
496. C23#13133141semidirect

Maximal subgroups

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

subgroupindex
C22×#131331412
C22×C22⋊#883222
C2×#168104832
C2×#168103852
C22×C22⋊Q8×C532
#1097053