Finite Group Databasebeta

← #27291927

Subgroups — #27291927

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×#16810625C211yesmaximal, characteristic
2C2×#16810625C211yesmaximal, characteristic
3C2×#16810589C211yesmaximal, characteristic
4C2×#16810391C211yesmaximal, characteristic, complemented
5C2×#16810391C211yesmaximal, characteristic, complemented
6C2×#16810386C211yesmaximal, characteristic, complemented
7C2×C53×#341C211yesmaximal, characteristic, complemented
8C22×#11719945C2211yescharacteristic
9C22×#11719917C2211yescharacteristic
10C22×C4⋊Dic53C2211yescharacteristic
11C22×C4⋊Dic53C2211yescharacteristic
12#16810625—22no—
13#16810625—22no—
14#16810625—22no—
15#16810625C414yescomplemented
16C2×#13133142—21no—
17C2×#13133142—21no—
18C2×#13133139—21no—
19C2×#13133139—21no—
20#16810589—22no—
21#16810589—22no—
22C2×#13133129—22no—
23#16810391—22no—
24#16810391—22no—
25#16810386—22no—
26C22×M16×C53C2211yescharacteristic
27C22×M16×C53C2211yescharacteristic
28C23×C8×C53C2211yescharacteristic
29C53×#341—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—
45C23×Dic53—21no—
46C23×Dic53—21no—
47C22×C4×D53—21no—
48C2×#11719917—21no—
49C2×#11719917—21no—
50C2×#11719917—21no—
51C2×#11719917—21no—
52C2×#11719917—21no—
53C2×#11719917D412yes—
54C22×#88322—21no—
55C2×C4⋊Dic53—21no—
56C2×C4⋊Dic53—21no—
57C2×C4⋊Dic53—21no—
58C2×C4⋊Dic53—21no—
59C2×C4⋊Dic53—21no—
60C2×C4⋊Dic53—21no—
61C2×C4⋊Dic53—21no—
62C2×C4⋊Dic53—21no—
63C2×C4⋊Dic53—21no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53C2×C412yes—
66C2×C4⋊Dic53D412yes—
67#13133142—24no—
68#13133142—24no—
69#13133139—24no—
70#13133139—24no—
71#13133129—28no—
72C23×C4×C53C2311yescharacteristic
73C2×M16×C53—21no—
74C2×M16×C53—21no—
75C2×M16×C53—21no—
76C2×M16×C53—21no—
77C2×M16×C53—21no—
78C2×M16×C53—21no—
79C2×M16×C53—21no—
80C2×M16×C53—21no—
81C2×M16×C53—21no—
82C2×M16×C53—21no—
83C2×M16×C53C2×C412yes—
84C2×M16×C53D412yes—
85C22×C8×C53—21no—
86C22×C8×C53—21no—
87C22×C8×C53—21no—
88C22×C8×C53—21no—
89C22×C8×C53—21no—
90C22×C8×C53—21no—
91C22×C8×C53—21no—
92C22×C8×C53—21no—
93C22×C8×C53D412yes—
94C24×C53#4111yescharacteristic
95C23×D53—21no—
96C22×C4×C53C22×C411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53#4111yescharacteristic
104C22×C4×C53#4111yescharacteristic
105C22×C4×C53#4111yescharacteristic
106C22×C4×C53#4111yescharacteristic
107C22×C4×C53#4111yescharacteristic
108C22×C4×C53#4111yescharacteristic
109C22×C4×C53#4111yescharacteristic
110C2×C22⋊D53—22no—
111C2×C22⋊D53—22no—
112C2×C22⋊D53—41no—
113C2×C22⋊D53—41no—
114C2×C22⋊D53—41no—
115C2×C22⋊D53—41no—
116C22×Dic53—21no—
117C22×Dic53—22no—
118C22×Dic53—22no—
119C22×Dic53—41no—
120C22×Dic53—41no—
121C22×Dic53—41no—
122C22×Dic53—41no—
123C22×Dic53—81no—
124C22×Dic53—81no—
125C22×Dic53—81no—
126C22×Dic53—81no—
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—
161M16×C53—22no—
162M16×C53—22no—
163M16×C53—22no—
164M16×C53—22no—
165M16×C53—41no—
166M16×C53—41no—
167M16×C53—41no—
168M16×C53—41no—
169C2×C8×C53—22no—
170C2×C8×C53—22no—
171C2×C8×C53—22no—
172C2×C8×C53—22no—
173C2×C8×C53—22no—
174C2×C8×C53—41no—
175C2×C8×C53—41no—
176C2×C8×C53—41no—
177C2×C8×C53—41no—
178C2×C8×C53—41no—
179C2×C8×C53—41no—
180C2×C8×C53—41no—
181C2×C8×C53—41no—
182C4⋊Dic53—22no—
183C4⋊Dic53—22no—
184C4⋊Dic53—22no—
185C4⋊Dic53—22no—
186C4⋊Dic53—41no—
187C4⋊Dic53—41no—
188C4⋊Dic53—41no—
189C4⋊Dic53—41no—
190#11719917—22no—
191#11719917—22no—
192#11719917—41no—
193#11719917—41no—
194C2×#88322—22no—
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×C53C2×C22⋊C411yescharacteristic
216C2×C4×C53#11711yescharacteristic
217C2×C4×C53#11711yescharacteristic
218C2×C4×C53C4×D411yescharacteristic
219C2×C4×C53C4×D411yescharacteristic
220C2×C4×C53C4×D411yescharacteristic
221C2×C4×C53#12011yescharacteristic
222C2×C4×C53#12011yescharacteristic
223C2×C4×C53#12111yescharacteristic
224C2×C4×C53#12111yescharacteristic
225C2×C4×C53#12111yescharacteristic
226C2×C4×C53#12111yescharacteristic
227C2×C4×C53#12111yescharacteristic
228C2×C4×C53#12111yescharacteristic
229C2×C4×C53#12111yescharacteristic
230C2×C4×C53#12311yescharacteristic
231C2×C4×C53#12311yescharacteristic
232C2×C4×C53#12311yescharacteristic
233C2×C4×C53#12311yescharacteristic
234C2×C4×C53#12311yescharacteristic
235C2×C4×C53#12311yescharacteristic
236C2×C4×C53#12311yescharacteristic
237C2×C4×C53#12311yescharacteristic
238C2×C4×C53#12411yescharacteristic
239C2×C4×C53#12411yescharacteristic
240C2×C4×C53#12411yescharacteristic
241C2×C4×C53#12611yescharacteristic
242C2×C4×C53#12611yescharacteristic
243C22⋊D53—42no—
244C22⋊D53—42no—
245C22⋊D53—42no—
246C22⋊D53—42no—
247C2×Dic53—22no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—82no—
253C2×Dic53—82no—
254C2×Dic53—81no—
255C2×Dic53—81no—
256C2×Dic53—81no—
257C2×Dic53—81no—
258C2×Dic53—82no—
259C2×Dic53—82no—
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—
273C8×C53—42no—
274C8×C53—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#38111yescharacteristic
308C4×C53#38611yescharacteristic
309C4×C53#38811yescharacteristic
310C4×C53#38811yescharacteristic
311C4×C53#39411yescharacteristic
312C4×C53#39411yescharacteristic
313C4×C53#39411yescharacteristic
314C4×C53#39711yescharacteristic
315Dic53—42no—
316Dic53—82no—
317Dic53—82no—
318C2×#417—531no—
319C2×#417—531no—
320C2×#408—531no—
321C2×#343—531no—
322C2×#343—531no—
323C2×#341D5311yescharacteristic, complemented
324C2×#340—531no—
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×M16C2×D5311yescharacteristic
340C22×M16C2×D5311yescharacteristic
341C23×C8C2×D5311yescharacteristic
342C22×C4⋊C4—531no—
343C22×C4⋊C4—531no—
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—22no—
358#340—1062no—
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—1061no—
376C23×C4—1061no—
377C23×C4—1061no—
378C23×C4C22×D5311yescharacteristic
379C2×M16—21no—
380C2×M16—21no—
381C2×M16—21no—
382C2×M16—21no—
383C2×M16—21no—
384C2×M16—21no—
385C2×M16—21no—
386C2×M16—21no—
387C2×M16—21no—
388C2×M16—21no—
389C2×M16—532no—
390C2×M16—1061no—
391C2×M16—1061no—
392C2×M16—1061no—
393C2×M16—1061no—
394C2×M16—1061no—
395C2×M16C4×D5312yes—
396C2×M16C22⋊D5312yes—
397C22×C8—21no—
398C22×C8—21no—
399C22×C8—21no—
400C22×C8—21no—
401C22×C8—21no—
402C22×C8—21no—
403C22×C8—21no—
404C22×C8—21no—
405C22×C8—1061no—
406C22×C8D21212yes—
407C2×C4⋊C4—532no—
408C2×C4⋊C4—532no—
409C2×C4⋊C4—1061no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—1061no—
419#103—1064no—
420#103—1064no—
421#102—1064no—
422#102—1064no—
423C22⋊C8—1068no—
424C24—1061no—
425C24#1171994911yescharacteristic
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—1061no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—1062no—
465C22×C4—4241no—
466C22×C4—4241no—
467C22×C4—4241no—
468C22×C4—4241no—
469C22×C4—2121no—
470C22×C4—2121no—
471C22×C4—2121no—
472C22×C4—2121no—
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#1171994711yescharacteristic
489C22×C4#1171994911yescharacteristic
490C22×C4C2×C22⋊D5311yescharacteristic
491M16—22no—
492M16—22no—
493M16—22no—
494M16—22no—
495M16—41no—
496M16—41no—
497M16—41no—
498M16—41no—
499M16—1062no—
500M16—1062no—
501M16—2121no—
502M16—2121no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—41no—
509C2×C8—41no—
510C2×C8—41no—
511C2×C8—41no—
512C2×C8—41no—
513C2×C8—41no—
514C2×C8—41no—
515C2×C8—41no—
516C2×C8—1062no—
517C2×C8—2121no—
518C2×C8—2121no—
519C4⋊C4—1062no—
520C4⋊C4—1062no—
521C4⋊C4—1062no—
522C4⋊C4—1062no—
523C4⋊C4—2121no—
524C4⋊C4—2121no—
525C4⋊C4—2121no—
526C4⋊C4—2121no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—1062no—
532C23—2121no—
533C23—2121no—
534C23—2121no—
535C23—2121no—
536C23—2121no—
537C23—2121no—
538C23#1313317911yescharacteristic
539C23#1313319211yescharacteristic
540C23#1313320211yescharacteristic
541C23#1313320211yescharacteristic
542C23#1313320611yescharacteristic
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—1062no—
558C2×C4—4242no—
559C2×C4—4242no—
560C2×C4—4241no—
561C2×C4—4241no—
562C2×C4—4241no—
563C2×C4—4241no—
564C2×C4—4242no—
565C2×C4—4242no—
566C2×C4—2122no—
567C2×C4—2122no—
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×C4#1313317711yescharacteristic
578C2×C4C4×C22⋊D5311yescharacteristic
579C2×C4C4×D21211yescharacteristic
580C2×C4#1313318511yescharacteristic
581C2×C4#1313318711yescharacteristic
582C2×C4#1313318811yescharacteristic
583C2×C4#1313318911yescharacteristic
584C2×C4#1313319011yescharacteristic
585C2×C4#1313319011yescharacteristic
586C2×C4#1313319011yescharacteristic
587C2×C4#1313319111yescharacteristic
588C2×C4#1313319211yescharacteristic
589C2×C4#1313319911yescharacteristic
590C2×C4#1313319911yescharacteristic
591C2×C4#1313319911yescharacteristic
592C2×C4#1313320011yescharacteristic
593C2×C4#1313320111yescharacteristic
594C2×C4#1313320211yescharacteristic
595C2×C4#1313320211yescharacteristic
596C2×C4#1313320211yescharacteristic
597C2×C4#1313320311yescharacteristic
598C2×C4#1313320311yescharacteristic
599C2×C4#1313320511yescharacteristic
600C2×C4#1313321111yescharacteristic
601C2×C4#1313321211yescharacteristic
602C8—42no—
603C8—42no—
604C8—42no—
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#1681050111yescharacteristic
626C22#1681051211yescharacteristic
627C22#1681051311yescharacteristic
628C22#1681054011yescharacteristic
629C22#1681054911yescharacteristic
630C22#1681055011yescharacteristic
631C22#1681055211yescharacteristic
632C22#1681088711yescharacteristic
633C22#1681088711yescharacteristic
634C22#1681089411yescharacteristic
635C22#1681089911yescharacteristic
636C4—4242no—
637C4—4242no—
638C4—2122no—
639C4#1681048511yescharacteristic
640C4#1681049911yescharacteristic
641C4#1681051411yescharacteristic
642C4#1681051511yescharacteristic
643C4#1681053811yescharacteristic
644C4#1681053911yescharacteristic
645C4#1681054111yescharacteristic
646C4#1681055211yescharacteristic
647C2—21no—
648C2—21no—
649C2—21no—
650C2—21no—
651C2—4242no—
652C2#2027296311yescharacteristic
653C2#2027522211yescharacteristic
654C2#2027522911yescharacteristic
655C2#2027600911yescharacteristic
656C2#2027601611yescharacteristic
657C2#2027615711yescharacteristic
658C2#2027616211yescharacteristic
subgroupG/H
centerC23#13133192
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#341C2
socleC23×C53#121
prime radicalC22×#11719945C22
cosocleC23×C4×C53C23
hypercenterC2×#341D53
hypocenterC53#11192
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#341

Complements

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

NH ≅ G/Ntype
4. C2×#16810391C2semidirect
5. C2×#16810391C2semidirect
6. C2×#16810386C2semidirect
7. C2×C53×#341C2semidirect
15. #16810625C4semidirect
323. C2×#341D53semidirect
359. C53#11192semidirect

Maximal subgroups

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

subgroupindex
C2×#168106252
C2×#168105892
C2×#168103912
C2×#168103862
C2×C53×#3412
#1119253