Finite Group Databasebeta

← #27290564

Subgroups — #27290564

Order 13568.

926 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810605C211yesmaximal, characteristic
2C2×#16810596C211yesmaximal, characteristic
3C2×#16810589C211yesmaximal, characteristic
4C2×#16810395C211yesmaximal, characteristic, complemented
5C2×#16810393C211yesmaximal, characteristic, complemented
6#20274411C211yesmaximal, characteristic, complemented
7C2×C53×#405C211yesmaximal, characteristic, complemented
8C22×#11719945C2211yescharacteristic
9C22×#11719917C2211yescharacteristic
10C22×#11719917C2211yescharacteristic
11C2×#13133176C2211yescharacteristic
12#16810605—22no—
13#16810605—22no—
14C2×#13133135C412yescomplemented
15C2×#13133133C412yescomplemented
16#16810596—22no—
17#16810596—22no—
18C2×#13133131C414yes—
19#16810589—22no—
20#16810589—21no—
21#16810589—21no—
22C2×#13133129C414yescomplemented
23#16810395—22no—
24#16810393—22no—
25#16810377—22no—
26C22×M16×C53C2211yescharacteristic
27C23×C8×C53C2211yescharacteristic
28C2×#117×C53C2211yescharacteristic
29C53×#405—22no—
30C53×#405—22no—
31C2×#97×C53—21no—
32C2×C4×C8×C53—21no—
33C2×#11719945—21no—
34C2×#11719945—21no—
35C2×#11719945—21no—
36C2×#11719945—21no—
37C2×#11719945—21no—
38C2×#11719945—21no—
39C2×#11719945—21no—
40C2×#11719945—21no—
41C2×#11719945—21no—
42C2×#11719945—21no—
43C2×#11719945—21no—
44C2×#11719945—21no—
45C22×Dic106C2×C411yescharacteristic
46C22×D212C2×C411yescharacteristic
47C22×C22⋊D53C2×C412yes—
48C22×C4×D53C2×C412yes—
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×#11719917C2×C412yes—
61C22×#88322—21no—
62C22×#88322C2×C412yes—
63#13133176—21no—
64#13133176—21no—
65#13133176—21no—
66#13133176—21no—
67C2×C4⋊Dic53—21no—
68C2×C22⋊Dic53C2×C412yes—
69C2×C4×Dic53—21no—
70#13133135—24no—
71#13133133—24no—
72#13133131—28no—
73#13133129—24no—
74#13133129—24no—
75C23×C4×C53C2311yescharacteristic
76C2×M16×C53—21no—
77C2×M16×C53—21no—
78C2×M16×C53—21no—
79C2×M16×C53C2×C412yes—
80C2×M16×C53D412yes—
81C2×M16×C53D412yes—
82C22×C8×C53—21no—
83C22×C8×C53—21no—
84C22×C8×C53—21no—
85C22×C8×C53—21no—
86C22×C8×C53—21no—
87C22×C8×C53C2×C412yes—
88C22×C8×C53D412yes—
89C22×C8×C53D411yescharacteristic
90C22×C8×C53D411yescharacteristic
91#117×C53—21no—
92#117×C53—21no—
93#117×C53D412yes—
94#117×C53D412yes—
95C2×C4⋊C4×C53—21no—
96C2×C22⋊C4×C53C2×C412yes—
97C2×C42×C53—21no—
98#97×C53—24no—
99C4×C8×C53—24no—
100C24×C53C22×C411yescharacteristic
101C23×D53C4212yes—
102C22×C4×C53—22no—
103C22×C4×C53C22×C411yescharacteristic
104C22×C4×C53C22×C411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53#4111yescharacteristic
112C22×C4×C53#4111yescharacteristic
113C22×C4×C53#4111yescharacteristic
114C22×C4×C53#4111yescharacteristic
115C22×C4×C53#4111yescharacteristic
116C22×C4×C53#4111yescharacteristic
117C2×C22⋊D53—22no—
118C2×C22⋊D53—22no—
119C2×C22⋊D53—22no—
120C2×C22⋊D53—22no—
121C2×C22⋊D53—22no—
122C2×C22⋊D53—22no—
123C22×Dic53—22no—
124C22×Dic53C4212yes—
125#11719945—41no—
126#11719945—41no—
127#11719945—41no—
128#11719945—41no—
129#11719945—41no—
130#11719945—41no—
131#11719945—41no—
132#11719945—41no—
133#11719945—41no—
134#11719945—41no—
135#11719945—41no—
136#11719945—41no—
137#11719945—41no—
138#11719945—41no—
139#11719945—41no—
140#11719945—41no—
141C2×D212—21no—
142C2×D212—21no—
143C2×D212—21no—
144C2×D212—21no—
145C2×D212—21no—
146C2×D212—21no—
147C2×C4×D53—22no—
148C2×C4×D53—22no—
149C2×C4×D53—22no—
150C2×C4×D53—22no—
151C2×C4×D53—22no—
152C2×C4×D53—22no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159M16×C53—22no—
160M16×C53—22no—
161M16×C53—22no—
162M16×C53C22⋊C414yes—
163C2×C8×C53—22no—
164C2×C8×C53—22no—
165C2×C8×C53—22no—
166C2×C8×C53—21no—
167C2×C8×C53—22no—
168C2×C8×C53—21no—
169C2×C8×C53—22no—
170C2×C8×C53—22no—
171C2×C8×C53—22no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53C22⋊C412yes—
175C2×C8×C53C22⋊C412yes—
176C4⋊C4×C53—22no—
177C4⋊C4×C53—22no—
178C22⋊C4×C53—22no—
179C22⋊C4×C53C22⋊C414yes—
180C42×C53—22no—
181C42×C53—22no—
182C22⋊Dic53—22no—
183C22⋊Dic53—22no—
184C4⋊Dic53—41no—
185C4⋊Dic53—41no—
186C4×Dic53—41no—
187C4×Dic53—41no—
188#11719917—22no—
189#11719917—22no—
190#11719917—41no—
191#11719917—41no—
192#11719917—41no—
193#11719917—41no—
194#11719917—41no—
195#11719917—41no—
196C2×#88322—22no—
197C2×#88322—22no—
198C2×#88322—22no—
199C2×#88322—22no—
200C2×#88322—41no—
201C2×#88322—41no—
202C23×C53—42no—
203C23×C53C2×C4211yescharacteristic
204C23×C53C2×C22⋊C411yescharacteristic
205C23×C53#11711yescharacteristic
206C23×C53C4×D411yescharacteristic
207C23×C53C4×D411yescharacteristic
208C23×C53C4×D411yescharacteristic
209C23×C53C4×D411yescharacteristic
210C22×D53—22no—
211C22×D53—22no—
212C22×D53—22no—
213C22×D53—44no—
214C2×C4×C53—24no—
215C2×C4×C53—44no—
216C2×C4×C53C2×C22⋊C411yescharacteristic
217C2×C4×C53C2×C22⋊C411yescharacteristic
218C2×C4×C53#11711yescharacteristic
219C2×C4×C53#11711yescharacteristic
220C2×C4×C53C4×D411yescharacteristic
221C2×C4×C53C4×D411yescharacteristic
222C2×C4×C53C4×D411yescharacteristic
223C2×C4×C53C4×D411yescharacteristic
224C2×C4×C53C4×D411yescharacteristic
225C2×C4×C53C4×D411yescharacteristic
226C2×C4×C53C4×D411yescharacteristic
227C2×C4×C53C4×D411yescharacteristic
228C2×C4×C53#12011yescharacteristic
229C2×C4×C53#12111yescharacteristic
230C2×C4×C53#12111yescharacteristic
231C2×C4×C53#12111yescharacteristic
232C2×C4×C53#12111yescharacteristic
233C2×C4×C53#12111yescharacteristic
234C2×C4×C53#12111yescharacteristic
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
243C2×C4×C53#12711yescharacteristic
244C22⋊D53—42no—
245C22⋊D53—42no—
246C22⋊D53—42no—
247C22⋊D53—42no—
248C2×Dic53—22no—
249C2×Dic53—22no—
250C2×Dic53—22no—
251C2×Dic53—44no—
252C2×Dic53—42no—
253D212—41no—
254D212—41no—
255D212—41no—
256D212—41no—
257C4×D53—42no—
258C4×D53—42no—
259C4×D53—42no—
260C4×D53—42no—
261Dic106—41no—
262Dic106—41no—
263Dic106—41no—
264Dic106—41no—
265C8×C53—22no—
266C8×C53—22no—
267C8×C53—24no—
268#88322—42no—
269#88322—42no—
270#10961—531nomaximal, Sylow
271C22×C53—22no—
272C22×C53—22no—
273C22×C53—22no—
274C22×C53—41no—
275C22×C53—41no—
276C22×C53—41no—
277C22×C53—41no—
278C22×C53C4×C22⋊C411yescharacteristic
279C22×C53C4×C22⋊C411yescharacteristic
280C22×C53C4×C22⋊C411yescharacteristic
281C22×C53#38611yescharacteristic
282C22×C53#38811yescharacteristic
283C22×C53#38811yescharacteristic
284C22×C53#39011yescharacteristic
285C2×D53—42no—
286C2×D53—44no—
287C2×D53—82no—
288C2×D53—82no—
289C4×C53—44no—
290C4×C53#38611yescharacteristic
291C4×C53#38611yescharacteristic
292C4×C53#38811yescharacteristic
293C4×C53#38811yescharacteristic
294C4×C53#38811yescharacteristic
295C4×C53#38811yescharacteristic
296C4×C53#39011yescharacteristic
297C4×C53#39011yescharacteristic
298Dic53—42no—
299Dic53—82no—
300C2×#413—531no—
301C2×#410—531no—
302C2×#408—531no—
303C2×#405D5311yescharacteristic, complemented
304C2×#344—531no—
305C2×#344—531no—
306#1729—531no—
307C2×C53—21no—
308C2×C53—21no—
309C2×C53—21no—
310C2×C53—21no—
311C2×C53—21no—
312C2×C53—21no—
313C2×C53#143711yescharacteristic
314C2×C53#175712yes—
315D53—82no—
316C22×#41—531no—
317C22×M16—531no—
318C22×M16—531no—
319C22×M16C2×D5311yescharacteristic
320C23×C8C2×D5311yescharacteristic
321C2×#117—531no—
322C2×#117C2×D5311yescharacteristic
323#413—1062no—
324#413—1062no—
325C2×#101—532no—
326C2×#100—532no—
327#410—1062no—
328#410—1062no—
329C2×#99—534no—
330#408—1062no—
331#408—1061no—
332#408—1061no—
333C2×C22⋊C8—534no—
334#405—22no—
335#405—22no—
336C2×#97—21no—
337C2×C4×C8—21no—
338#344—1062no—
339#344—1062no—
340#336—1062no—
341C53#1096111yescharacteristic, complemented, Sylow
342C2×#41—1061no—
343C2×#41—1061no—
344C2×#41—1061no—
345C2×#41—1061no—
346C2×#41—1061no—
347C2×#41—1061no—
348C2×#41—1061no—
349C2×#41—1061no—
350C2×#41—1061no—
351C2×#41—1061no—
352C2×#41—1061no—
353C2×#41—1061no—
354C22×Q8—531no—
355C22×D4—532no—
356C22×D4—531no—
357C23×C4—532no—
358C23×C4C22×D5311yescharacteristic
359C2×M16—21no—
360C2×M16—21no—
361C2×M16—21no—
362C2×M16—532no—
363C2×M16—1061no—
364C2×M16—1061no—
365C2×M16—1061no—
366C2×M16—1061no—
367C2×M16—1061no—
368C2×M16—1061no—
369C2×M16—1061no—
370C2×M16—1061no—
371C2×M16—1061no—
372C2×M16—1061no—
373C2×M16—1061no—
374C2×M16C4×D5312yes—
375C2×M16D21212yes—
376C2×M16C22⋊D5312yes—
377C22×C8—21no—
378C22×C8—21no—
379C22×C8—21no—
380C22×C8—21no—
381C22×C8—21no—
382C22×C8—532no—
383C22×C8—1061no—
384C22×C8C4×D5312yes—
385C22×C8D21212yes—
386C22×C8C22⋊D5311yescharacteristic
387C22×C8C22⋊D5311yescharacteristic
388#117—21no—
389#117—21no—
390#117—1061no—
391#117—1061no—
392#117—1061no—
393#117—1061no—
394#117D21212yes—
395#117C22⋊D5312yes—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—1061no—
398C2×C22⋊C4—532no—
399C2×C22⋊C4C4×D5312yes—
400C2×C42—21no—
401C2×C42—1061no—
402#101—1064no—
403#100—1064no—
404#99—1068no—
405C22⋊C8—1064no—
406C22⋊C8—1064no—
407#97—24no—
408C4×C8—24no—
409C24—532no—
410C24C2×C4×D5311yescharacteristic
411#41—2121no—
412#41—2121no—
413#41—2121no—
414#41—2121no—
415#41—2121no—
416#41—2121no—
417#41—2121no—
418#41—2121no—
419#41—2121no—
420#41—2121no—
421#41—2121no—
422#41—2121no—
423#41—2121no—
424#41—2121no—
425#41—2121no—
426#41—2121no—
427C2×Q8—1061no—
428C2×Q8—1061no—
429C2×Q8—1061no—
430C2×Q8—1061no—
431C2×Q8—1061no—
432C2×Q8—1061no—
433C2×D4—1062no—
434C2×D4—1062no—
435C2×D4—1062no—
436C2×D4—1062no—
437C2×D4—1062no—
438C2×D4—1062no—
439C2×D4—1061no—
440C2×D4—1061no—
441C2×D4—1061no—
442C2×D4—1061no—
443C2×D4—1061no—
444C2×D4—1061no—
445C22×C4—22no—
446C22×C4—532no—
447C22×C4—1062no—
448C22×C4—1062no—
449C22×C4—1062no—
450C22×C4—1062no—
451C22×C4—1062no—
452C22×C4—1062no—
453C22×C4—1062no—
454C22×C4C2×C4×D5311yescharacteristic
455C22×C4C2×C4×D5311yescharacteristic
456C22×C4C2×D21211yescharacteristic
457C22×C4C2×D21211yescharacteristic
458C22×C4C2×D21211yescharacteristic
459C22×C4#1171994511yescharacteristic
460C22×C4#1171994511yescharacteristic
461C22×C4#1171994511yescharacteristic
462C22×C4#1171994511yescharacteristic
463C22×C4#1171994511yescharacteristic
464C22×C4#1171994511yescharacteristic
465C22×C4C2×C22⋊D5311yescharacteristic
466C22×C4C2×C22⋊D5311yescharacteristic
467C22×C4C2×C22⋊D5311yescharacteristic
468M16—22no—
469M16—22no—
470M16—22no—
471M16—1062no—
472M16—1062no—
473M16—2121no—
474M16—2121no—
475M16—2121no—
476M16—2121no—
477M16—2121no—
478M16—2121no—
479M16#1171992114yes—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—21no—
484C2×C8—22no—
485C2×C8—21no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—1062no—
492C2×C8—1062no—
493C2×C8—1062no—
494C2×C8—1062no—
495C2×C8—2121no—
496C2×C8—2121no—
497C2×C8#1171992112yes—
498C2×C8#1171992112yes—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—2121no—
502C4⋊C4—2121no—
503C22⋊C4—22no—
504C22⋊C4—1062no—
505C22⋊C4—1062no—
506C22⋊C4#1171992114yes—
507C42—22no—
508C42—22no—
509C42—2121no—
510C42—2121no—
511C23—42no—
512C23—1062no—
513C23—1062no—
514C23—1062no—
515C23—2124no—
516C23C42×D5311yescharacteristic
517C23C2×#1171992111yescharacteristic
518C23#1313317411yescharacteristic
519C23C4×C22⋊D5311yescharacteristic
520C23C4×C22⋊D5311yescharacteristic
521C23C4×D21211yescharacteristic
522C23C4×D21211yescharacteristic
523Q8—2121no—
524Q8—2121no—
525Q8—2121no—
526Q8—2121no—
527D4—2122no—
528D4—2122no—
529D4—2122no—
530D4—2122no—
531D4—2121no—
532D4—2121no—
533D4—2121no—
534D4—2121no—
535C2×C4—24no—
536C2×C4—44no—
537C2×C4—1062no—
538C2×C4—1062no—
539C2×C4—1062no—
540C2×C4—2124no—
541C2×C4—2122no—
542C2×C4—2122no—
543C2×C4—2122no—
544C2×C4—2122no—
545C2×C4—2122no—
546C2×C4C2×#1171992111yescharacteristic
547C2×C4C2×#1171992111yescharacteristic
548C2×C4#1313317411yescharacteristic
549C2×C4#1313317411yescharacteristic
550C2×C4C4×C22⋊D5311yescharacteristic
551C2×C4C4×C22⋊D5311yescharacteristic
552C2×C4C4×C22⋊D5311yescharacteristic
553C2×C4C4×C22⋊D5311yescharacteristic
554C2×C4C4×D21211yescharacteristic
555C2×C4C4×D21211yescharacteristic
556C2×C4C4×D21211yescharacteristic
557C2×C4C4×D21211yescharacteristic
558C2×C4#1313318611yescharacteristic
559C2×C4#1313319111yescharacteristic
560C2×C4#1313319111yescharacteristic
561C2×C4#1313319111yescharacteristic
562C2×C4#1313319111yescharacteristic
563C2×C4#1313319111yescharacteristic
564C2×C4#1313319111yescharacteristic
565C2×C4#1313320011yescharacteristic
566C2×C4#1313320011yescharacteristic
567C2×C4#1313320011yescharacteristic
568C2×C4#1313320511yescharacteristic
569C2×C4#1313320511yescharacteristic
570C2×C4#1313320511yescharacteristic
571C2×C4#1313321111yescharacteristic
572C2×C4#1313321111yescharacteristic
573C2×C4#1313321411yescharacteristic
574C8—22no—
575C8—22no—
576C8—24no—
577C8—2122no—
578C8—2122no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—4242no—
587C22—4242no—
588C22—2122no—
589C22—2124no—
590C22C4×#1171992111yescharacteristic
591C22C4×#1171992111yescharacteristic
592C22C4×#1171992111yescharacteristic
593C22#1681050211yescharacteristic
594C22#1681051411yescharacteristic
595C22#1681051411yescharacteristic
596C22#1681052511yescharacteristic
597C4—44no—
598C4—4242no—
599C4—2122no—
600C4#1681050211yescharacteristic
601C4#1681050211yescharacteristic
602C4#1681051411yescharacteristic
603C4#1681051411yescharacteristic
604C4#1681051411yescharacteristic
605C4#1681051411yescharacteristic
606C4#1681052511yescharacteristic
607C4#1681052511yescharacteristic
608C2—21no—
609C2—21no—
610C2—21no—
611C2—21no—
612C2—21no—
613C2—21no—
614C2—4242no—
615C2#2027295311yescharacteristic
616C2#2027451612yes—
subgroupG/H
centerC2×C4C2×#11719921
commutatorC23×C53C2×C42
FrattiniC23×C4C22×D53
FittingC2×C53×#405C2
socleC22×C53C4×C22⋊C4
prime radicalC22×#11719945C22
cosocleC23×C4×C53C23
hypercenterC2×#405D53
hypocenterC53#10961
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C8 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
4. C2×#16810395C2semidirect
5. C2×#16810393C2semidirect
6. #20274411C2semidirect
7. C2×C53×#405C2semidirect
14. C2×#13133135C4semidirect
15. C2×#13133133C4semidirect
22. C2×#13133129C4semidirect
303. C2×#405D53semidirect
341. C53#10961semidirect

Maximal subgroups

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

subgroupindex
C2×#168106052
C2×#168105962
C2×#168105892
C2×#168103952
C2×#168103932
#202744112
C2×C53×#4052
#1096153