Finite Group Databasebeta

← #27291924

Subgroups — #27291924

Order 13568.

866 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810622C211yesmaximal, characteristic
2C2×#16810622C211yesmaximal, characteristic
3C2×#16810589C211yesmaximal, characteristic
4C2×#16810392C211yesmaximal, characteristic, complemented
5C2×#16810392C211yesmaximal, characteristic, complemented
6C2×#16810388C211yesmaximal, characteristic, complemented
7C2×C53×#340C211yesmaximal, characteristic, complemented
8C22×#11719945C2211yescharacteristic
9C22×#11719917C2211yescharacteristic
10C22×#11719917C2211yescharacteristic
11C22×#11719917C2211yescharacteristic
12#16810622—22no—
13#16810622—22no—
14#16810622—22no—
15#16810622C414yescomplemented
16C2×#13133140—21no—
17C2×#13133140—21no—
18C2×#13133137—21no—
19C2×#13133137—21no—
20#16810589—22no—
21#16810589—22no—
22C2×#13133129—22no—
23#16810392—22no—
24#16810392—22no—
25#16810388—22no—
26C23×C8×C53C2211yescharacteristic
27C22×C4⋊C4×C53C2211yescharacteristic
28C22×C4⋊C4×C53C2211yescharacteristic
29C53×#340—22no—
30C2×#11719945—21no—
31C2×#11719945—21no—
32C2×#11719945—21no—
33C2×#11719945—21no—
34C2×#11719945—21no—
35C2×#11719945—21no—
36C2×#11719945—21no—
37C2×#11719945—21no—
38C2×#11719945—21no—
39C2×#11719945—21no—
40C2×#11719945C2×C412yes—
41C2×#11719945D412yes—
42C22×Dic106D411yescharacteristic
43C22×D212D411yescharacteristic
44C22×C22⋊D53—21no—
45C22×C4×D53—21no—
46C2×#11719917—21no—
47C2×#11719917—21no—
48C2×#11719917—21no—
49C2×#11719917—21no—
50C2×#11719917—21no—
51C2×#11719917—21no—
52C2×#11719917—21no—
53C2×#11719917—21no—
54C2×#11719917—21no—
55C2×#11719917—21no—
56C2×#11719917—21no—
57C2×#11719917—21no—
58C2×#11719917—21no—
59C2×#11719917—21no—
60C2×#11719917—21no—
61C2×#11719917C2×C412yes—
62C2×#11719917D412yes—
63C2×#11719917D412yes—
64C22×#88322—21no—
65C22×#88322—21no—
66C22×#88322—21no—
67#13133140—24no—
68#13133140—24no—
69#13133137—24no—
70#13133137—24no—
71#13133129—28no—
72C23×C4×C53—21no—
73C23×C4×C53—21no—
74C23×C4×C53C2311yescharacteristic
75C22×C8×C53—21no—
76C22×C8×C53—21no—
77C22×C8×C53—21no—
78C22×C8×C53—21no—
79C22×C8×C53—21no—
80C22×C8×C53—21no—
81C22×C8×C53D412yes—
82C2×C4⋊C4×C53—21no—
83C2×C4⋊C4×C53—21no—
84C2×C4⋊C4×C53—21no—
85C2×C4⋊C4×C53—21no—
86C2×C4⋊C4×C53—21no—
87C2×C4⋊C4×C53—21no—
88C2×C4⋊C4×C53—21no—
89C2×C4⋊C4×C53—21no—
90C2×C4⋊C4×C53—21no—
91C2×C4⋊C4×C53—21no—
92C2×C4⋊C4×C53C2×C412yes—
93C2×C4⋊C4×C53D412yes—
94C24×C53#4111yescharacteristic
95C23×D53—21no—
96C22×C4×C53—22no—
97C22×C4×C53—22no—
98C22×C4×C53—41no—
99C22×C4×C53—41no—
100C22×C4×C53—41no—
101C22×C4×C53—41no—
102C22×C4×C53—81no—
103C22×C4×C53—81no—
104C22×C4×C53—81no—
105C22×C4×C53—81no—
106C22×C4×C53C22×C411yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53#4111yescharacteristic
114C22×C4×C53#4111yescharacteristic
115C22×C4×C53#4111yescharacteristic
116C22×C4×C53#4111yescharacteristic
117C22×C4×C53#4111yescharacteristic
118C22×C4×C53#4111yescharacteristic
119C22×C4×C53#4111yescharacteristic
120C2×C22⋊D53—22no—
121C2×C22⋊D53—22no—
122C2×C22⋊D53—41no—
123C2×C22⋊D53—41no—
124C2×C22⋊D53—41no—
125C2×C22⋊D53—41no—
126C22×Dic53—21no—
127#11719945—22no—
128#11719945—22no—
129#11719945—22no—
130#11719945—22no—
131#11719945—22no—
132#11719945—22no—
133#11719945—22no—
134#11719945—22no—
135#11719945—41no—
136#11719945—41no—
137#11719945—41no—
138#11719945—41no—
139#11719945—41no—
140#11719945—41no—
141#11719945—41no—
142#11719945—41no—
143C2×D212—21no—
144C2×D212—21no—
145C2×D212—21no—
146C2×D212—21no—
147C2×D212—21no—
148C2×D212C22⋊C412yes—
149C2×C4×D53—22no—
150C2×C4×D53—22no—
151C2×C4×D53—41no—
152C2×C4×D53—41no—
153C2×C4×D53—41no—
154C2×C4×D53—41no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106C22⋊C412yes—
161C2×C8×C53—22no—
162C2×C8×C53—22no—
163C2×C8×C53—22no—
164C2×C8×C53—41no—
165C2×C8×C53—41no—
166C2×C8×C53—41no—
167C2×C8×C53—41no—
168C4⋊C4×C53—22no—
169C4⋊C4×C53—22no—
170C4⋊C4×C53—22no—
171C4⋊C4×C53—22no—
172C4⋊C4×C53—41no—
173C4⋊C4×C53—41no—
174C4⋊C4×C53—41no—
175C4⋊C4×C53—41no—
176#11719917—22no—
177#11719917—22no—
178#11719917—22no—
179#11719917—22no—
180#11719917—22no—
181#11719917—22no—
182#11719917—41no—
183#11719917—41no—
184#11719917—41no—
185#11719917—41no—
186#11719917—41no—
187#11719917—41no—
188C2×#88322—22no—
189C2×#88322—22no—
190C2×#88322—22no—
191C2×#88322—41no—
192C2×#88322—41no—
193C2×#88322—41no—
194C2×#88322—41no—
195C2×#88322—41no—
196C2×#88322—41no—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53—21no—
200C23×C53—21no—
201C23×C53C4×D411yescharacteristic
202C23×C53#12111yescharacteristic
203C23×C53#12311yescharacteristic
204C23×C53#12311yescharacteristic
205C23×C53#12411yescharacteristic
206C23×C53#12611yescharacteristic
207C23×C53#12611yescharacteristic
208C22×D53—22no—
209C22×D53—41no—
210C22×D53—41no—
211C22×D53—41no—
212C22×D53—41no—
213C22×D53—41no—
214C22×D53—41no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53—82no—
218C2×C4×C53—82no—
219C2×C4×C53—81no—
220C2×C4×C53—81no—
221C2×C4×C53—81no—
222C2×C4×C53—81no—
223C2×C4×C53—82no—
224C2×C4×C53—82no—
225C2×C4×C53C2×C22⋊C411yescharacteristic
226C2×C4×C53#11711yescharacteristic
227C2×C4×C53#11711yescharacteristic
228C2×C4×C53C4×D411yescharacteristic
229C2×C4×C53C4×D411yescharacteristic
230C2×C4×C53C4×D411yescharacteristic
231C2×C4×C53#12011yescharacteristic
232C2×C4×C53#12011yescharacteristic
233C2×C4×C53#12111yescharacteristic
234C2×C4×C53#12111yescharacteristic
235C2×C4×C53#12111yescharacteristic
236C2×C4×C53#12111yescharacteristic
237C2×C4×C53#12111yescharacteristic
238C2×C4×C53#12111yescharacteristic
239C2×C4×C53#12111yescharacteristic
240C2×C4×C53#12311yescharacteristic
241C2×C4×C53#12311yescharacteristic
242C2×C4×C53#12311yescharacteristic
243C2×C4×C53#12311yescharacteristic
244C2×C4×C53#12311yescharacteristic
245C2×C4×C53#12311yescharacteristic
246C2×C4×C53#12311yescharacteristic
247C2×C4×C53#12311yescharacteristic
248C2×C4×C53#12411yescharacteristic
249C2×C4×C53#12411yescharacteristic
250C2×C4×C53#12411yescharacteristic
251C2×C4×C53#12611yescharacteristic
252C2×C4×C53#12611yescharacteristic
253C22⋊D53—42no—
254C22⋊D53—42no—
255C22⋊D53—42no—
256C22⋊D53—42no—
257C2×Dic53—22no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260D212—22no—
261D212—22no—
262D212—41no—
263D212—41no—
264C4×D53—42no—
265C4×D53—42no—
266C4×D53—42no—
267C4×D53—42no—
268Dic106—22no—
269Dic106—22no—
270Dic106—41no—
271Dic106—41no—
272C8×C53—42no—
273#88322—42no—
274#88322—42no—
275#88322—42no—
276#11192—531nomaximal, Sylow
277C22×C53—21no—
278C22×C53—21no—
279C22×C53—21no—
280C22×C53—21no—
281C22×C53—21no—
282C22×C53—21no—
283C22×C53—21no—
284C22×C53—21no—
285C22×C53—21no—
286C22×C53—21no—
287C22×C53—21no—
288C22×C53—21no—
289C22×C53#38611yescharacteristic
290C22×C53#38811yescharacteristic
291C22×C53#38811yescharacteristic
292C22×C53#39411yescharacteristic
293C22×C53#39711yescharacteristic
294C22×C53#39711yescharacteristic
295C22×C53#39711yescharacteristic
296C22×C53#47111yescharacteristic
297C22×C53#47111yescharacteristic
298C22×C53#47211yescharacteristic
299C22×C53#47311yescharacteristic
300C2×D53—42no—
301C2×D53—42no—
302C2×D53—42no—
303C2×D53—81no—
304C2×D53—81no—
305C2×D53—81no—
306C2×D53—81no—
307C4×C53—82no—
308C4×C53—82no—
309C4×C53#38111yescharacteristic
310C4×C53#38611yescharacteristic
311C4×C53#38811yescharacteristic
312C4×C53#38811yescharacteristic
313C4×C53#39411yescharacteristic
314C4×C53#39411yescharacteristic
315C4×C53#39411yescharacteristic
316C4×C53#39711yescharacteristic
317Dic53—42no—
318C2×#417—531no—
319C2×#417—531no—
320C2×#408—531no—
321C2×#343—531no—
322C2×#343—531no—
323C2×#341—531no—
324C2×#340D5311yescharacteristic, complemented
325C2×C53—21no—
326C2×C53—21no—
327C2×C53—21no—
328C2×C53—21no—
329C2×C53#144011yescharacteristic
330C2×C53#190711yescharacteristic
331C2×C53#190811yescharacteristic
332C2×C53#205111yescharacteristic
333C2×C53#205211yescharacteristic
334C2×C53#207811yescharacteristic
335C2×C53#207911yescharacteristic
336D53—82no—
337C22×#41—531no—
338C22×M16—531no—
339C22×M16—531no—
340C22×M16—531no—
341C23×C8C2×D5311yescharacteristic
342C22×C4⋊C4C2×D5311yescharacteristic
343C22×C4⋊C4C2×D5311yescharacteristic
344#417—534no—
345#417—1062no—
346#417—1062no—
347#417—1062no—
348C2×#103—1061no—
349C2×#103—1061no—
350C2×#102—1061no—
351C2×#102—1061no—
352#408—1062no—
353#408—1062no—
354C2×C22⋊C8—1062no—
355#343—1062no—
356#343—1062no—
357#341—1062no—
358#340—22no—
359C53#1119211yescharacteristic, complemented, Sylow
360C2×#41—532no—
361C2×#41—532no—
362C2×#41—1061no—
363C2×#41—1061no—
364C2×#41—1061no—
365C2×#41—1061no—
366C2×#41—1061no—
367C2×#41—1061no—
368C2×#41—1061no—
369C2×#41—1061no—
370C2×#41—1061no—
371C2×#41—1061no—
372C22×Q8—531no—
373C22×D4—531no—
374C22×D4—1061no—
375C23×C4—21no—
376C23×C4—21no—
377C23×C4—1061no—
378C23×C4C22×D5311yescharacteristic
379C2×M16—532no—
380C2×M16—532no—
381C2×M16—532no—
382C2×M16—1061no—
383C2×M16—1061no—
384C2×M16—1061no—
385C2×M16—1061no—
386C2×M16—1061no—
387C2×M16—1061no—
388C2×M16—1061no—
389C2×M16—1061no—
390C2×M16—1061no—
391C2×M16—1061no—
392C2×M16—1061no—
393C2×M16—1061no—
394C2×M16—1061no—
395C2×M16—1061no—
396C2×M16—1061no—
397C22×C8—21no—
398C22×C8—21no—
399C22×C8—21no—
400C22×C8—21no—
401C22×C8—21no—
402C22×C8—21no—
403C22×C8—1061no—
404C22×C8—1061no—
405C22×C8—1061no—
406C22×C8C22⋊D5312yes—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4C4×D5312yes—
418C2×C4⋊C4D21212yes—
419#103—1064no—
420#103—1064no—
421#102—1064no—
422#102—1064no—
423C22⋊C8—1068no—
424C24—1061no—
425C24#1171994711yescharacteristic
426#41—1062no—
427#41—1062no—
428#41—1062no—
429#41—1062no—
430#41—1062no—
431#41—1062no—
432#41—1062no—
433#41—1062no—
434#41—2121no—
435#41—2121no—
436#41—2121no—
437#41—2121no—
438#41—2121no—
439#41—2121no—
440#41—2121no—
441#41—2121no—
442C2×Q8—532no—
443C2×Q8—1061no—
444C2×Q8—1061no—
445C2×Q8—1061no—
446C2×Q8—1061no—
447C2×Q8—1061no—
448C2×D4—532no—
449C2×D4—1062no—
450C2×D4—1062no—
451C2×D4—1061no—
452C2×D4—1061no—
453C2×D4—1061no—
454C2×D4—1061no—
455C2×D4—1061no—
456C2×D4—2121no—
457C2×D4—2121no—
458C2×D4—2121no—
459C2×D4—2121no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—81no—
467C22×C4—81no—
468C22×C4—81no—
469C22×C4—81no—
470C22×C4—1061no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—2121no—
474C22×C4—2121no—
475C22×C4—2121no—
476C22×C4—2121no—
477C22×C4C2×C4×D5311yescharacteristic
478C22×C4C2×D21211yescharacteristic
479C22×C4#1171994511yescharacteristic
480C22×C4#1171994511yescharacteristic
481C22×C4#1171994511yescharacteristic
482C22×C4#1171994511yescharacteristic
483C22×C4D4×D5311yescharacteristic
484C22×C4D4×D5311yescharacteristic
485C22×C4D4×D5311yescharacteristic
486C22×C4D4×D5311yescharacteristic
487C22×C4#1171994711yescharacteristic
488C22×C4#1171994911yescharacteristic
489C22×C4#1171994911yescharacteristic
490C22×C4C2×C22⋊D5311yescharacteristic
491M16—1062no—
492M16—1062no—
493M16—1062no—
494M16—1062no—
495M16—1062no—
496M16—1062no—
497M16—2121no—
498M16—2121no—
499M16—2121no—
500M16—2121no—
501M16—2121no—
502M16—2121no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—41no—
507C2×C8—41no—
508C2×C8—41no—
509C2×C8—41no—
510C2×C8—1062no—
511C2×C8—1062no—
512C2×C8—1062no—
513C2×C8—2121no—
514C2×C8—2121no—
515C2×C8—2121no—
516C2×C8—2121no—
517C2×C8—2121no—
518C2×C8—2121no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—41no—
524C4⋊C4—41no—
525C4⋊C4—41no—
526C4⋊C4—41no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—1062no—
532C23—2121no—
533C23—2121no—
534C23—2121no—
535C23—2121no—
536C23—2121no—
537C23—2121no—
538C23#1313317711yescharacteristic
539C23#1313318811yescharacteristic
540C23#1313319911yescharacteristic
541C23#1313320111yescharacteristic
542C23#1313320311yescharacteristic
543C23#1313321211yescharacteristic
544C23#1313321211yescharacteristic
545Q8—1062no—
546Q8—1062no—
547Q8—2121no—
548Q8—2121no—
549D4—1062no—
550D4—1062no—
551D4—2122no—
552D4—2122no—
553D4—2122no—
554D4—2122no—
555D4—2121no—
556D4—2121no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—1062no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2122no—
571C2×C4—2122no—
572C2×C4—2122no—
573C2×C4—2122no—
574C2×C4C22⋊C4×D5311yescharacteristic
575C2×C4#1313317411yescharacteristic
576C2×C4#1313317511yescharacteristic
577C2×C4C4×C22⋊D5311yescharacteristic
578C2×C4#1313317911yescharacteristic
579C2×C4C4×D21211yescharacteristic
580C2×C4#1313318511yescharacteristic
581C2×C4#1313318711yescharacteristic
582C2×C4#1313318911yescharacteristic
583C2×C4#1313319011yescharacteristic
584C2×C4#1313319011yescharacteristic
585C2×C4#1313319011yescharacteristic
586C2×C4#1313319111yescharacteristic
587C2×C4#1313319211yescharacteristic
588C2×C4#1313319211yescharacteristic
589C2×C4#1313319911yescharacteristic
590C2×C4#1313319911yescharacteristic
591C2×C4#1313320011yescharacteristic
592C2×C4#1313320211yescharacteristic
593C2×C4#1313320211yescharacteristic
594C2×C4#1313320211yescharacteristic
595C2×C4#1313320211yescharacteristic
596C2×C4#1313320211yescharacteristic
597C2×C4#1313320311yescharacteristic
598C2×C4#1313320511yescharacteristic
599C2×C4#1313320611yescharacteristic
600C2×C4#1313321111yescharacteristic
601C2×C4#1313321211yescharacteristic
602C8—42no—
603C8—2122no—
604C8—2122no—
605C8—2122no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—4241no—
619C22—4241no—
620C22—4241no—
621C22—4241no—
622C22—2122no—
623C22—2122no—
624C22—2122no—
625C22#1681049911yescharacteristic
626C22#1681051211yescharacteristic
627C22#1681051511yescharacteristic
628C22#1681053911yescharacteristic
629C22#1681054911yescharacteristic
630C22#1681055011yescharacteristic
631C22#1681055211yescharacteristic
632C22#1681088311yescharacteristic
633C22#1681088311yescharacteristic
634C22#1681089011yescharacteristic
635C22#1681089511yescharacteristic
636C4—82no—
637C4—82no—
638C4—2122no—
639C4#1681048511yescharacteristic
640C4#1681050111yescharacteristic
641C4#1681051311yescharacteristic
642C4#1681051411yescharacteristic
643C4#1681053811yescharacteristic
644C4#1681054011yescharacteristic
645C4#1681054111yescharacteristic
646C4#1681055211yescharacteristic
647C2—21no—
648C2—21no—
649C2—21no—
650C2—21no—
651C2—4242no—
652C2#2027296311yescharacteristic
653C2#2027522611yescharacteristic
654C2#2027523311yescharacteristic
655C2#2027600811yescharacteristic
656C2#2027601511yescharacteristic
657C2#2027615611yescharacteristic
658C2#2027616111yescharacteristic
subgroupG/H
centerC23#13133188
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#340C2
socleC23×C53#121
prime radicalC22×#11719945C22
cosocleC23×C4×C53C23
hypercenterC2×#340D53
hypocenterC53#11192
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
4. C2×#16810392C2semidirect
5. C2×#16810392C2semidirect
6. C2×#16810388C2semidirect
7. C2×C53×#340C2semidirect
15. #16810622C4semidirect
324. C2×#340D53semidirect
359. C53#11192semidirect

Maximal subgroups

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

subgroupindex
C2×#168106222
C2×#168105892
C2×#168103922
C2×#168103882
C2×C53×#3402
#1119253