Finite Group Databasebeta

← #27291664

Subgroups — #27291664

Order 13568.

894 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810605C211yesmaximal, characteristic
2C2×#16810596C211yesmaximal, characteristic
3#20277205C211yesmaximal, characteristic
4C2×#16810383C211yesmaximal, characteristic, complemented
5C2×#16810380C211yesmaximal, characteristic, complemented
6C2×#16810379C211yesmaximal, characteristic, complemented
7C2×C53×#339C211yesmaximal, characteristic, complemented
8C22×#11719945C2211yescharacteristic
9C22×#11719917C2211yescharacteristic
10C2×#13133176C2211yescharacteristic
11C2×#13133176C2211yescharacteristic
12#16810605—22no—
13#16810605—22no—
14C2×#13133135—21no—
15C2×#13133133—21no—
16#16810596—22no—
17#16810596—22no—
18C2×#13133131—22no—
19#16810529—21no—
20#16810525—21no—
21#16810502—22no—
22C4×#11719921C414yescomplemented
23#16810383—22no—
24#16810380—22no—
25#16810379—22no—
26C22×M16×C53C2211yescharacteristic
27C2×#117×C53C2211yescharacteristic
28C22×C42×C53C2211yescharacteristic
29C53×#339—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×#11719945—21no—
41C2×#11719945—21no—
42C22×Dic106D411yescharacteristic
43C22×D212D411yescharacteristic
44C22×C22⋊D53D412yes—
45C22×C4×D53C2×C412yes—
46C2×#11719917—21no—
47C2×#11719917—21no—
48C2×#11719917—21no—
49C2×#11719917—21no—
50C2×#11719917—21no—
51C2×#11719917D412yes—
52C22×#88322—21no—
53#13133176—21no—
54#13133176—21no—
55#13133176—21no—
56#13133176—21no—
57#13133176—21no—
58#13133176—21no—
59#13133176—21no—
60#13133176D412yes—
61C2×C4⋊Dic53—21no—
62C2×C4⋊Dic53—21no—
63C2×C22⋊Dic53—21no—
64C2×C22⋊Dic53—21no—
65C2×#11719921—24no—
66C2×C4×Dic53—21no—
67C2×C4×Dic53C2×C412yes—
68#13133135—24no—
69#13133133—24no—
70#13133131—28no—
71#13133124—24no—
72C23×C4×C53—21no—
73C23×C4×C53C2311yescharacteristic
74C2×M16×C53—21no—
75C2×M16×C53—21no—
76C2×M16×C53—21no—
77C2×M16×C53—21no—
78C2×M16×C53—21no—
79C2×M16×C53D412yes—
80C22×C8×C53—21no—
81#117×C53—21no—
82#117×C53—21no—
83#117×C53—21no—
84#117×C53D412yes—
85C2×C4⋊C4×C53—21no—
86C2×C22⋊C4×C53—21no—
87C2×C42×C53—21no—
88C2×C42×C53—21no—
89C2×C42×C53—21no—
90C2×C42×C53—21no—
91C2×C42×C53—21no—
92C2×C42×C53—21no—
93C2×C42×C53C2×C412yes—
94C24×C53C2×D411yescharacteristic
95C23×D53C22⋊C412yes—
96C22×C4×C53—22no—
97C22×C4×C53—44no—
98C22×C4×C53—41no—
99C22×C4×C53—41no—
100C22×C4×C53—41no—
101C22×C4×C53C22×C411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53#4111yescharacteristic
108C22×C4×C53#4111yescharacteristic
109C22×C4×C53#4111yescharacteristic
110C22×C4×C53#4111yescharacteristic
111C22×C4×C53#4111yescharacteristic
112C22×C4×C53#4111yescharacteristic
113C22×C4×C53#4111yescharacteristic
114C22×C4×C53#4111yescharacteristic
115C2×C22⋊D53—22no—
116C2×C22⋊D53—22no—
117C2×C22⋊D53—22no—
118C2×C22⋊D53—22no—
119C2×C22⋊D53—22no—
120C2×C22⋊D53—22no—
121C22×Dic53—22no—
122C22×Dic53—41no—
123C22×Dic53C22⋊C412yes—
124#11719945—41no—
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—
140C2×D212—21no—
141C2×D212—21no—
142C2×D212—21no—
143C2×D212—21no—
144C2×D212—21no—
145C2×D212—21no—
146C2×C4×D53—22no—
147C2×C4×D53—22no—
148C2×C4×D53—22no—
149C2×C4×D53—22no—
150C2×C4×D53—22no—
151C2×C4×D53—22no—
152C2×Dic106—21no—
153C2×Dic106—21no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158M16×C53—22no—
159M16×C53—22no—
160M16×C53—41no—
161M16×C53—41no—
162C2×C8×C53—22no—
163C2×C8×C53—41no—
164C2×C8×C53—41no—
165C4⋊C4×C53—22no—
166C4⋊C4×C53—41no—
167C22⋊C4×C53—22no—
168C22⋊C4×C53—41no—
169C42×C53—22no—
170C42×C53—22no—
171C42×C53—22no—
172C42×C53—41no—
173C42×C53—41no—
174C42×C53—41no—
175C22⋊Dic53—22no—
176C22⋊Dic53—41no—
177C22⋊Dic53—41no—
178C22⋊Dic53—41no—
179#11719921—48no—
180C4⋊Dic53—22no—
181C4⋊Dic53—41no—
182C4⋊Dic53—41no—
183C4⋊Dic53—41no—
184C4×Dic53—22no—
185C4×Dic53—22no—
186C4×Dic53—22no—
187C4×Dic53—41no—
188#11719917—22no—
189#11719917—22no—
190#11719917—41no—
191#11719917—41no—
192C2×#88322—22no—
193C2×#88322—41no—
194C2×#88322—41no—
195C23×C53—42no—
196C23×C53C2×C22⋊C411yescharacteristic
197C23×C53#12011yescharacteristic
198C23×C53#12111yescharacteristic
199C23×C53#12111yescharacteristic
200C23×C53#12311yescharacteristic
201C23×C53#12311yescharacteristic
202C23×C53#12311yescharacteristic
203C22×D53—22no—
204C22×D53—22no—
205C22×D53—22no—
206C22×D53—44no—
207C2×C4×C53—24no—
208C2×C4×C53—42no—
209C2×C4×C53—44no—
210C2×C4×C53—44no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—82no—
214C2×C4×C53#11711yescharacteristic
215C2×C4×C53#11711yescharacteristic
216C2×C4×C53C4×D411yescharacteristic
217C2×C4×C53C4×D411yescharacteristic
218C2×C4×C53C4×D411yescharacteristic
219C2×C4×C53C4×D411yescharacteristic
220C2×C4×C53#12011yescharacteristic
221C2×C4×C53#12111yescharacteristic
222C2×C4×C53#12111yescharacteristic
223C2×C4×C53#12111yescharacteristic
224C2×C4×C53#12111yescharacteristic
225C2×C4×C53#12111yescharacteristic
226C2×C4×C53#12111yescharacteristic
227C2×C4×C53#12311yescharacteristic
228C2×C4×C53#12311yescharacteristic
229C2×C4×C53#12311yescharacteristic
230C2×C4×C53#12311yescharacteristic
231C2×C4×C53#12311yescharacteristic
232C2×C4×C53#12311yescharacteristic
233C2×C4×C53#12311yescharacteristic
234C2×C4×C53#12411yescharacteristic
235C2×C4×C53#12411yescharacteristic
236C2×C4×C53#12411yescharacteristic
237C2×C4×C53#12411yescharacteristic
238C2×C4×C53#12611yescharacteristic
239C2×C4×C53#12611yescharacteristic
240C2×C4×C53#12611yescharacteristic
241C2×C4×C53#12611yescharacteristic
242C22⋊D53—42no—
243C22⋊D53—42no—
244C22⋊D53—42no—
245C22⋊D53—42no—
246C2×Dic53—22no—
247C2×Dic53—22no—
248C2×Dic53—22no—
249C2×Dic53—44no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—82no—
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—42no—
266#88322—42no—
267#11148—531nomaximal, Sylow
268C22×C53—22no—
269C22×C53—22no—
270C22×C53—22no—
271C22×C53—41no—
272C22×C53—41no—
273C22×C53—41no—
274C22×C53—41no—
275C22×C53#38111yescharacteristic
276C22×C53#38611yescharacteristic
277C22×C53#38611yescharacteristic
278C22×C53#39411yescharacteristic
279C22×C53#39411yescharacteristic
280C22×C53#39711yescharacteristic
281C22×C53#39711yescharacteristic
282C2×D53—44no—
283C2×D53—42no—
284C2×D53—82no—
285C2×D53—82no—
286C4×C53—44no—
287C4×C53—82no—
288C4×C53#38811yescharacteristic
289C4×C53#38811yescharacteristic
290C4×C53#38811yescharacteristic
291C4×C53#38811yescharacteristic
292C4×C53#39411yescharacteristic
293C4×C53#39411yescharacteristic
294C4×C53#39711yescharacteristic
295C4×C53#39711yescharacteristic
296Dic53—42no—
297Dic53—82no—
298Dic53—82no—
299C2×#413—531no—
300C2×#410—531no—
301#2297—531no—
302C2×#339—531no—
303C2×#339D5311yescharacteristic, complemented
304C2×H(3,ℤ/4)—531no—
305C2×H(3,ℤ/4)—531no—
306C2×C53—21no—
307C2×C53—21no—
308C2×C53—21no—
309C2×C53—21no—
310C2×C53—21no—
311C2×C53—21no—
312C2×C53#144011yescharacteristic
313C2×C53#182712yes—
314D53—82no—
315C22×#41—531no—
316C22×M16—531no—
317C22×M16C2×D5311yescharacteristic
318C2×#117—531no—
319C2×#117—531no—
320C2×#117C2×D5311yescharacteristic
321C22×C42C2×D5311yescharacteristic
322#413—1062no—
323#413—1062no—
324C2×#101—1061no—
325C2×#100—1061no—
326#410—1062no—
327#410—1062no—
328C2×#99—1062no—
329#391—1061no—
330#390—1061no—
331#386—1062no—
332C4×C22⋊C4—534no—
333#339—22no—
334#339—1062no—
335H(3,ℤ/4)—1062no—
336H(3,ℤ/4)—1062no—
337C53#1114811yescharacteristic, complemented, Sylow
338C2×#41—1061no—
339C2×#41—1061no—
340C2×#41—1061no—
341C2×#41—1061no—
342C2×#41—1061no—
343C2×#41—1061no—
344C2×#41—1061no—
345C2×#41—1061no—
346C2×#41—1061no—
347C2×#41—1061no—
348C2×#41—1061no—
349C2×#41—1061no—
350C22×Q8—531no—
351C22×D4—532no—
352C22×D4—531no—
353C23×C4—21no—
354C23×C4—532no—
355C23×C4C22×D5311yescharacteristic
356C2×M16—21no—
357C2×M16—21no—
358C2×M16—21no—
359C2×M16—21no—
360C2×M16—21no—
361C2×M16—532no—
362C2×M16—1061no—
363C2×M16—1061no—
364C2×M16—1061no—
365C2×M16—1061no—
366C2×M16—1061no—
367C2×M16C22⋊D5312yes—
368C22×C8—21no—
369C22×C8—1061no—
370#117—21no—
371#117—21no—
372#117—21no—
373#117—532no—
374#117—1061no—
375#117—1061no—
376#117—1061no—
377#117—1061no—
378#117—1061no—
379#117—1061no—
380#117—1061no—
381#117D21212yes—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4—1061no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—1061no—
387C2×C22⋊C4—1064no—
388C2×C22⋊C4—1061no—
389C2×C42—21no—
390C2×C42—21no—
391C2×C42—21no—
392C2×C42—21no—
393C2×C42—21no—
394C2×C42—21no—
395C2×C42—532no—
396C2×C42—1061no—
397C2×C42C4×D5312yes—
398#101—1064no—
399#100—1064no—
400#99—1068no—
401#95—1064no—
402C24—532no—
403C24D4×D5311yescharacteristic
404#41—2121no—
405#41—2121no—
406#41—2121no—
407#41—2121no—
408#41—2121no—
409#41—2121no—
410#41—2121no—
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—
420C2×Q8—1061no—
421C2×Q8—1061no—
422C2×Q8—1061no—
423C2×Q8—1061no—
424C2×Q8—1061no—
425C2×Q8—1061no—
426C2×D4—1062no—
427C2×D4—1062no—
428C2×D4—1062no—
429C2×D4—1062no—
430C2×D4—1062no—
431C2×D4—1062no—
432C2×D4—1061no—
433C2×D4—1061no—
434C2×D4—1061no—
435C2×D4—1061no—
436C2×D4—1061no—
437C2×D4—1061no—
438C22×C4—22no—
439C22×C4—44no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—532no—
444C22×C4—1062no—
445C22×C4—1062no—
446C22×C4—1062no—
447C22×C4—1062no—
448C22×C4—1062no—
449C22×C4—1062no—
450C22×C4—1062no—
451C22×C4—2121no—
452C22×C4C2×C4×D5311yescharacteristic
453C22×C4C2×D21211yescharacteristic
454C22×C4#1171994511yescharacteristic
455C22×C4#1171994511yescharacteristic
456C22×C4#1171994511yescharacteristic
457C22×C4#1171994511yescharacteristic
458C22×C4D4×D5311yescharacteristic
459C22×C4D4×D5311yescharacteristic
460C22×C4D4×D5311yescharacteristic
461C22×C4#1171994711yescharacteristic
462C22×C4#1171994711yescharacteristic
463C22×C4#1171994911yescharacteristic
464C22×C4#1171994911yescharacteristic
465C22×C4C2×C22⋊D5311yescharacteristic
466M16—22no—
467M16—22no—
468M16—41no—
469M16—41no—
470M16—1062no—
471M16—1062no—
472M16—2121no—
473M16—2121no—
474C2×C8—22no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—1062no—
478C2×C8—2121no—
479C2×C8—2121no—
480C4⋊C4—22no—
481C4⋊C4—41no—
482C4⋊C4—1062no—
483C4⋊C4—2121no—
484C4⋊C4—2121no—
485C4⋊C4—2121no—
486C22⋊C4—22no—
487C22⋊C4—41no—
488C22⋊C4—1062no—
489C22⋊C4—2128no—
490C22⋊C4—2121no—
491C22⋊C4—2121no—
492C22⋊C4—2121no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—41no—
497C42—41no—
498C42—41no—
499C42—1062no—
500C42—1062no—
501C42—1062no—
502C42—2121no—
503C23—42no—
504C23—1062no—
505C23—1062no—
506C23—1062no—
507C23—2124no—
508C23C22⋊C4×D5311yescharacteristic
509C23#1313318711yescharacteristic
510C23#1313318811yescharacteristic
511C23#1313319011yescharacteristic
512C23#1313319911yescharacteristic
513C23#1313320211yescharacteristic
514C23#1313320211yescharacteristic
515Q8—2121no—
516Q8—2121no—
517Q8—2121no—
518Q8—2121no—
519D4—2122no—
520D4—2122no—
521D4—2122no—
522D4—2122no—
523D4—2121no—
524D4—2121no—
525D4—2121no—
526D4—2121no—
527C2×C4—24no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—82no—
534C2×C4—1062no—
535C2×C4—1062no—
536C2×C4—1062no—
537C2×C4—4242no—
538C2×C4—2124no—
539C2×C4—2122no—
540C2×C4—2122no—
541C2×C4—2122no—
542C2×C4—2122no—
543C2×C4—2122no—
544C2×C4—2122no—
545C2×C4#1313317411yescharacteristic
546C2×C4#1313317511yescharacteristic
547C2×C4#1313317711yescharacteristic
548C2×C4C4×C22⋊D5311yescharacteristic
549C2×C4#1313317911yescharacteristic
550C2×C4C4×D21211yescharacteristic
551C2×C4#1313318511yescharacteristic
552C2×C4#1313318911yescharacteristic
553C2×C4#1313319011yescharacteristic
554C2×C4#1313319011yescharacteristic
555C2×C4#1313319111yescharacteristic
556C2×C4#1313319211yescharacteristic
557C2×C4#1313319211yescharacteristic
558C2×C4#1313319911yescharacteristic
559C2×C4#1313319911yescharacteristic
560C2×C4#1313320011yescharacteristic
561C2×C4#1313320111yescharacteristic
562C2×C4#1313320211yescharacteristic
563C2×C4#1313320211yescharacteristic
564C2×C4#1313320211yescharacteristic
565C2×C4#1313320311yescharacteristic
566C2×C4#1313320311yescharacteristic
567C2×C4#1313320511yescharacteristic
568C2×C4#1313320611yescharacteristic
569C2×C4#1313321111yescharacteristic
570C2×C4#1313321211yescharacteristic
571C2×C4#1313321211yescharacteristic
572C2×C4#1313321211yescharacteristic
573C8—42no—
574C8—2122no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—4242no—
583C22—4242no—
584C22—2122no—
585C22—2124no—
586C22#1681048511yescharacteristic
587C22#1681049911yescharacteristic
588C22#1681050111yescharacteristic
589C22#1681053911yescharacteristic
590C22#1681054111yescharacteristic
591C22#1681054911yescharacteristic
592C22#1681055211yescharacteristic
593C4—44no—
594C4—82no—
595C4—4242no—
596C4—4242no—
597C4—2122no—
598C4#1681051211yescharacteristic
599C4#1681051311yescharacteristic
600C4#1681051411yescharacteristic
601C4#1681051511yescharacteristic
602C4#1681053811yescharacteristic
603C4#1681054011yescharacteristic
604C4#1681055011yescharacteristic
605C4#1681055211yescharacteristic
606C2—21no—
607C2—21no—
608C2—21no—
609C2—21no—
610C2—21no—
611C2—21no—
612C2—4242no—
613C2#2027296311yescharacteristic
614C2#2027481812yes—
subgroupG/H
centerC2×C4#13133175
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#339C2
socleC22×C53#381
prime radicalC22×#11719945C22
cosocleC23×C4×C53C23
hypercenterC2×#339D53
hypocenterC53#11148
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#339

Complements

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

NH ≅ G/Ntype
4. C2×#16810383C2semidirect
5. C2×#16810380C2semidirect
6. C2×#16810379C2semidirect
7. C2×C53×#339C2semidirect
22. C4×#11719921C4semidirect
303. C2×#339D53semidirect
337. C53#11148semidirect

Maximal subgroups

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

subgroupindex
C2×#168106052
C2×#168105962
#202772052
C2×#168103832
C2×#168103802
C2×#168103792
C2×C53×#3392
#1114853