Finite Group Databasebeta

← #27383298

Subgroups — #27383298

Order 13568.

3838 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
1C23×Q8×D53C211yesmaximal, characteristic
2C2×#16810633C211yesmaximal, characteristic, complemented
3C2×#16810632C211yesmaximal, characteristic, complemented
4C2×#16810631C211yesmaximal, characteristic, complemented
5C2×#16810586C211yesmaximal, characteristic
6#20277094C211yesmaximal, characteristic
7#20274648C218yesmaximal, complemented
8C2×C53×#419C211yesmaximal, characteristic, complemented
9C23×Dic106—21no—
10C23×Dic106C2211yescharacteristic
11C22×Q8×D53—24no—
12C22×Q8×D53—24no—
13C22×Q8×D53—24no—
14C22×Q8×D53—22no—
15C22×Q8×D53—24no—
16C22×Q8×D53—24no—
17C22×Q8×D53C418yescomplemented
18C22×Q8×D53C2214yes—
19C23×C4×D53—21no—
20C23×C4×D53C2211yescharacteristic
21C22×#11719917C2211yescharacteristic
22C2×#13133176C2211yescharacteristic
23#16810633—24no—
24#16810632—24no—
25#16810631—24no—
26C2×#13133142C2214yescomplemented
27C2×#13133141C2214yescomplemented
28C2×#13133140C2214yescomplemented
29#16810586—24no—
30C2×#13133129C2214yes—
31#16810484—21no—
32#16810480C2214yes—
33#16810478—21no—
34C4×#11719921—22no—
35C23×Q8×C53C2211yescharacteristic
36C22×M16×C53C2211yescharacteristic
37C2×#117×C53C2211yescharacteristic
38C53×#419—24no—
39C2×#103×C53C2214yescomplemented
40C24×D53D411yescharacteristic
41C22×Dic106—28no—
42C22×Dic106—28no—
43C22×Dic106—21no—
44C22×Dic106—22no—
45C22×Dic106—22no—
46C22×Dic106—22no—
47C22×Dic106—28no—
48C22×Dic106—24no—
49C22×Dic106C2×C414yes—
50C22×Dic106D414yes—
51C22×Dic106D414yes—
52C22×Dic106C2312yes—
53C2×Q8×D53—216no—
54C2×Q8×D53—216no—
55C2×Q8×D53—232no—
56C2×Q8×D53—232no—
57C2×Q8×D53—232no—
58C2×Q8×D53—28no—
59C2×Q8×D53—28no—
60C2×Q8×D53—216no—
61C2×Q8×D53—28no—
62C2×Q8×D53—232no—
63C2×Q8×D53—28no—
64C2×Q8×D53—28no—
65C2×Q8×D53C2×C4116yes—
66C23×Dic53—21no—
67C23×Dic53D411yescharacteristic
68C22×C4×D53—28no—
69C22×C4×D53—28no—
70C22×C4×D53—28no—
71C22×C4×D53—24no—
72C22×C4×D53—22no—
73C22×C4×D53—22no—
74C22×C4×D53—22no—
75C22×C4×D53—22no—
76C22×C4×D53C2×C414yes—
77C22×C4×D53D412yes—
78C22×C4×D53D414yes—
79C22×C4×D53C2312yes—
80C2×#11719917—22no—
81C2×#11719917—22no—
82C2×#11719917—22no—
83C22×#88322C2312yes—
84#13133176—22no—
85#13133176—22no—
86C2×C4⋊Dic53C2312yes—
87C2×C22⋊Dic53—21no—
88C2×#11719921—24no—
89C2×C4×Dic53—21no—
90#13133142—28no—
91#13133141—28no—
92#13133140—28no—
93#13133129—28no—
94#13133124—24no—
95C22×Q8×C53—22no—
96C22×Q8×C53—22no—
97C22×Q8×C53—22no—
98C22×Q8×C53—21no—
99C22×Q8×C53C2×C414yes—
100C22×Q8×C53D414yes—
101C22×Q8×C53D414yes—
102C22×Q8×C53C2312yes—
103C23×C4×C53—21no—
104C23×C4×C53C2311yescharacteristic
105C2×M16×C53—22no—
106C2×M16×C53—22no—
107C2×M16×C53—22no—
108C22×C8×C53C2312yes—
109#117×C53—22no—
110#117×C53—22no—
111C2×C4⋊C4×C53C2312yes—
112C2×C22⋊C4×C53—21no—
113C2×C42×C53—21no—
114#103×C53—28no—
115C24×C53C2×D411yescharacteristic
116C23×D53—22no—
117C23×D53—22no—
118C23×D53—28no—
119C23×D53C22⋊C414yes—
120C23×D53C2×D412yes—
121C2×Q8×C53—24no—
122C2×Q8×C53—28no—
123C2×Q8×C53—24no—
124C2×Q8×C53—24no—
125C2×Q8×C53—28no—
126C2×Q8×C53—28no—
127C2×Q8×C53—22no—
128C2×Q8×C53—22no—
129C2×Q8×C53—22no—
130C2×Q8×C53C22⋊C4116yescomplemented
131C2×Q8×C53C22×C414yes—
132C2×Q8×C53C2×D414yes—
133C2×Q8×C53C2×D414yes—
134C22×C4×C53—24no—
135C22×C4×C53—24no—
136C22×C4×C53—24no—
137C22×C4×C53—22no—
138C22×C4×C53—22no—
139C22×C4×C53C22×C412yes—
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D412yes—
142C22×C4×C53C2×D412yes—
143C22×C4×C53C2×D412yes—
144C22×C4×C53C2×D412yes—
145C22×C4×C53C2×D412yes—
146C22×C4×C53C2411yescharacteristic
147C22×Dic53—24no—
148C22×Dic53—24no—
149C22×Dic53—24no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53C22⋊C414yes—
155C22×Dic53C2×D412yes—
156Q8×D53—232no—
157Q8×D53—232no—
158Q8×D53—2128no—
159Q8×D53—232no—
160Q8×D53—232no—
161C2×C4×D53—216no—
162C2×C4×D53—216no—
163C2×C4×D53—264no—
164C2×C4×D53—216no—
165C2×C4×D53—28no—
166C2×C4×D53—28no—
167C2×C4×D53—28no—
168C2×C4×D53—28no—
169C2×C4×D53—24no—
170C2×C4×D53—24no—
171C2×C4×D53—22no—
172C2×C4×D53—22no—
173C2×C4×D53—22no—
174C2×C4×D53—22no—
175C2×C4×D53C22⋊C418yes—
176C2×C4×D53C22×C414yes—
177C2×C4×D53C2×D414yes—
178C2×Dic106—216no—
179C2×Dic106—216no—
180C2×Dic106—264no—
181C2×Dic106—28no—
182C2×Dic106—24no—
183C2×Dic106—28no—
184C2×Dic106—28no—
185C2×Dic106—24no—
186C2×Dic106—24no—
187C2×Dic106—22no—
188C2×Dic106—22no—
189C2×Dic106—22no—
190C2×Dic106—216no—
191C2×Dic106C22⋊C4116yescomplemented
192C2×Dic106C22×C414yes—
193C2×Dic106C2×D414yes—
194C2×Dic106C2×D414yes—
195M16×C53—44no—
196C2×C8×C53—22no—
197C2×C8×C53—22no—
198C2×C8×C53—22no—
199C4⋊C4×C53—22no—
200C4⋊C4×C53—22no—
201C22⋊C4×C53—42no—
202C42×C53—42no—
203C22⋊Dic53—42no—
204#11719921—48no—
205C4⋊Dic53—22no—
206C4⋊Dic53—22no—
207C4×Dic53—42no—
208#11719917—44no—
209C2×#88322—22no—
210C2×#88322—22no—
211C2×#88322—22no—
212C23×C53—24no—
213C23×C53C2×C22⋊C412yes—
214C23×C53#12012yes—
215C23×C53#12012yes—
216C23×C53C22×D411yescharacteristic
217C22×D53—216no—
218C22×D53—216no—
219C22×D53—22no—
220C22×D53—22no—
221C22×D53—24no—
222C22×D53—28no—
223C22×D53—216no—
224C22×D53C2×C22⋊C414yes—
225C22×D53#13718yes—
226Q8×C53—24no—
227Q8×C53—24no—
228Q8×C53—24no—
229Q8×C53—216no—
230Q8×C53C2×C22⋊C418yes—
231C2×C4×C53—24no—
232C2×C4×C53—24no—
233C2×C4×C53—24no—
234C2×C4×C53—216no—
235C2×C4×C53—42no—
236C2×C4×C53—44no—
237C2×C4×C53C2×C22⋊C414yes—
238C2×C4×C53C2×C22⋊C412yes—
239C2×C4×C53C2×C22⋊C414yes—
240C2×C4×C53#12014yes—
241C2×C4×C53#12012yes—
242C2×C4×C53#12014yes—
243C2×C4×C53#12012yes—
244C2×C4×C53C23×C411yescharacteristic
245C2×C4×C53C22×D411yescharacteristic
246C2×C4×C53C22×D411yescharacteristic
247C2×C4×C53C22×D411yescharacteristic
248C2×C4×C53C22×D411yescharacteristic
249C2×C4×C53C22×D411yescharacteristic
250C2×Dic53—24no—
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—24no—
254C2×Dic53—24no—
255C2×Dic53—28no—
256C2×Dic53—216no—
257C2×Dic53—42no—
258C2×Dic53—44no—
259C2×Dic53C2×C22⋊C414yes—
260C4×D53—264no—
261C4×D53—24no—
262C4×D53—24no—
263C4×D53—24no—
264C4×D53—24no—
265C4×D53—216no—
266Dic106—264no—
267Dic106—24no—
268Dic106—24no—
269Dic106—24no—
270Dic106—216no—
271Dic106C2×C22⋊C418yes—
272C8×C53—42no—
273#88322—42no—
274#20599—531nomaximal, Sylow
275C22×C53—24no—
276C22×C53—24no—
277C22×C53—24no—
278C22×C53#37914yes—
279C22×C53C22×C22⋊C411yescharacteristic
280C22×C53C2×#12011yescharacteristic
281C22×C53C2×#12011yescharacteristic
282C22×C53C2×#13714yes—
283C2×D53—232no—
284C2×D53—24no—
285C2×D53—28no—
286C2×D53—28no—
287C2×D53—24no—
288C2×D53#41918yescomplemented
289C4×C53—28no—
290C4×C53—82no—
291C4×C53#37914yes—
292C4×C53C22×C22⋊C411yescharacteristic
293C4×C53C22×C22⋊C411yescharacteristic
294C4×C53C2×#12011yescharacteristic
295C4×C53C2×#12011yescharacteristic
296Dic53—24no—
297Dic53—28no—
298Dic53—82no—
299C24×Q8—531no—
300C2×#419—531no—
301C2×#419—531no—
302C2×#419—531no—
303C2×#419D5311yescharacteristic, complemented
304C2×#407—531no—
305#2280—531no—
306#1787—538no—
307C2×C53—24no—
308C2×C53C2×#37911yescharacteristic
309C2×C53C2×#41912yes—
310C2×C53#300412yes—
311C2×C53#300412yes—
312D53—28no—
313C23×Q8—538no—
314C23×Q8—534no—
315C23×Q8—531no—
316C23×Q8—1061no—
317C23×Q8—1062no—
318C23×Q8—1064no—
319C23×Q8—1064no—
320C23×Q8—1064no—
321C23×Q8—1064no—
322C23×Q8—1064no—
323C23×Q8C2×D5311yescharacteristic
324C24×C4—531no—
325C24×C4—1061no—
326C22×M16—531no—
327C22×M16C2×D5311yescharacteristic
328C2×#117—531no—
329C2×#117C2×D5311yescharacteristic
330#419—24no—
331#419—1064no—
332#419—1064no—
333#419—1064no—
334C2×#103—534no—
335C2×#103—534no—
336C2×#103—534no—
337C2×#103C2×D5314yescomplemented
338#407—1064no—
339C2×C22⋊C8—534no—
340#381—1061no—
341#380—534no—
342#379—1061no—
343C4×C22⋊C4—1062no—
344C53#2059911yescharacteristic, complemented, Sylow
345C25—531no—
346C22×Q8—22no—
347C22×Q8—22no—
348C22×Q8—22no—
349C22×Q8—21no—
350C22×Q8—5316no—
351C22×Q8—534no—
352C22×Q8—534no—
353C22×Q8—534no—
354C22×Q8—532no—
355C22×Q8—1068no—
356C22×Q8—1064no—
357C22×Q8—1068no—
358C22×Q8—1068no—
359C22×Q8—10616no—
360C22×Q8—1068no—
361C22×Q8—10632no—
362C22×Q8—1068no—
363C22×Q8—1068no—
364C22×Q8—10616no—
365C22×Q8—10632no—
366C22×Q8—10616no—
367C22×Q8—10632no—
368C22×Q8—1068no—
369C22×Q8—1068no—
370C22×Q8—10632no—
371C22×Q8—1062no—
372C22×Q8—1062no—
373C22×Q8—1061no—
374C22×Q8—1062no—
375C22×Q8C4×D5314yes—
376C22×Q8D21214yes—
377C22×Q8C22⋊D5314yes—
378C22×Q8C22×D5312yes—
379C23×C4—21no—
380C23×C4—531no—
381C23×C4—534no—
382C23×C4—532no—
383C23×C4—534no—
384C23×C4—532no—
385C23×C4—1068no—
386C23×C4—1064no—
387C23×C4—1068no—
388C23×C4—1068no—
389C23×C4—1061no—
390C23×C4—1062no—
391C23×C4—1062no—
392C23×C4—1062no—
393C23×C4—1062no—
394C23×C4C22×D5311yescharacteristic
395C2×M16—22no—
396C2×M16—22no—
397C2×M16—22no—
398C2×M16—1062no—
399C2×M16—1062no—
400C2×M16—1062no—
401C22×C8—532no—
402C22×C8C22×D5312yes—
403#117—22no—
404#117—22no—
405#117—1062no—
406#117—1062no—
407C2×C4⋊C4—532no—
408C2×C4⋊C4C22×D5312yes—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—1064no—
411C2×C22⋊C4—1061no—
412C2×C42—21no—
413C2×C42—1061no—
414#103—28no—
415#103—1068no—
416#103—1068no—
417#103—1068no—
418C22⋊C8—1068no—
419#95—1064no—
420C24—532no—
421C24—534no—
422C24—1068no—
423C24—1062no—
424C24—1062no—
425C24D4×D5311yescharacteristic
426C2×Q8—24no—
427C2×Q8—28no—
428C2×Q8—24no—
429C2×Q8—28no—
430C2×Q8—24no—
431C2×Q8—28no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—5316no—
436C2×Q8—534no—
437C2×Q8—534no—
438C2×Q8—534no—
439C2×Q8—10616no—
440C2×Q8—10616no—
441C2×Q8—10616no—
442C2×Q8—10664no—
443C2×Q8—10632no—
444C2×Q8—10632no—
445C2×Q8—10632no—
446C2×Q8—106128no—
447C2×Q8—10632no—
448C2×Q8—1064no—
449C2×Q8—1068no—
450C2×Q8—1064no—
451C2×Q8—1068no—
452C2×Q8—1064no—
453C2×Q8—1068no—
454C2×Q8—1062no—
455C2×Q8—1062no—
456C2×Q8—1062no—
457C2×Q8#11719921116yescomplemented
458C2×Q8C2×C4×D5314yes—
459C2×Q8C2×D21214yes—
460C2×Q8C2×C22⋊D5314yes—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—534no—
467C22×C4—532no—
468C22×C4—538no—
469C22×C4—534no—
470C22×C4—534no—
471C22×C4—10616no—
472C22×C4—10616no—
473C22×C4—10616no—
474C22×C4—10664no—
475C22×C4—1064no—
476C22×C4—1062no—
477C22×C4—1064no—
478C22×C4—1062no—
479C22×C4—1064no—
480C22×C4—1062no—
481C22×C4—1068no—
482C22×C4—1068no—
483C22×C4—1064no—
484C22×C4—1068no—
485C22×C4—1064no—
486C22×C4—1068no—
487C22×C4—1062no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4C2×C4×D5312yes—
493C22×C4C2×D21212yes—
494C22×C4D4×D5311yescharacteristic
495C22×C4D4×D5312yes—
496C22×C4D4×D5312yes—
497C22×C4D4×D5312yes—
498C22×C4C2×C22⋊D5312yes—
499C22×C4C23×D5311yescharacteristic
500M16—44no—
501M16—2124no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—1062no—
506C2×C8—1062no—
507C2×C8—1062no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—1062no—
511C4⋊C4—1062no—
512C22⋊C4—42no—
513C22⋊C4—2128no—
514C22⋊C4—2122no—
515C42—42no—
516C42—2122no—
517C23—24no—
518C23—534no—
519C23—538no—
520C23—10616no—
521C23—10616no—
522C23—10616no—
523C23—1062no—
524C23—1062no—
525C23—1064no—
526C23—1068no—
527C23C22⋊C4×D5312yes—
528C23#1313318512yes—
529C23#1313318712yes—
530C23C2×D4×D5311yescharacteristic
531Q8—24no—
532Q8—24no—
533Q8—24no—
534Q8—216no—
535Q8—538no—
536Q8—10664no—
537Q8—1064no—
538Q8—1064no—
539Q8—1064no—
540Q8—10616no—
541Q8C2×#1171992118yes—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—216no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—534no—
549C2×C4—10664no—
550C2×C4—1064no—
551C2×C4—1062no—
552C2×C4—1064no—
553C2×C4—1062no—
554C2×C4—1064no—
555C2×C4—1068no—
556C2×C4—10616no—
557C2×C4—1064no—
558C2×C4—1064no—
559C2×C4—1064no—
560C2×C4—1064no—
561C2×C4—10616no—
562C2×C4—2122no—
563C2×C4—2124no—
564C2×C4C22⋊C4×D5314yes—
565C2×C4C22⋊C4×D5312yes—
566C2×C4C2×#1171992114yes—
567C2×C4#1313318514yes—
568C2×C4#1313318512yes—
569C2×C4#1313318714yes—
570C2×C4#1313318712yes—
571C2×C4C22×C4×D5311yescharacteristic
572C2×C4C2×D4×D5311yescharacteristic
573C2×C4C2×D4×D5311yescharacteristic
574C2×C4C2×D4×D5311yescharacteristic
575C2×C4C22×C22⋊D5311yescharacteristic
576C2×C4C22×D21211yescharacteristic
577C8—42no—
578C8—2122no—
579C22—24no—
580C22—24no—
581C22—24no—
582C22—538no—
583C22—10632no—
584C22—1068no—
585C22—1068no—
586C22—1064no—
587C22—1064no—
588C22#1681047814yes—
589C22C2×C22⋊C4×D5311yescharacteristic
590C22C2×#1313318511yescharacteristic
591C22C2×#1313318711yescharacteristic
592C22#137×D5314yes—
593C4—28no—
594C4—82no—
595C4—1064no—
596C4—1068no—
597C4—4242no—
598C4#1681047814yes—
599C4C2×C22⋊C4×D5311yescharacteristic
600C4C22×#1171992111yescharacteristic
601C4C2×#1313318511yescharacteristic
602C4C2×#1313318711yescharacteristic
603C2—24no—
604C2—1068no—
605C2C2×#1681047811yescharacteristic
606C2#419×D5312yes—
607C2#2028394712yes—
608C2#2028395112yes—
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#419C2
socleC23×C53C22×D4
prime radicalC24×D53D4
cosocleC22×C4×C53C24
hypercenterC2×#419D53
hypocenterC53#20599
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
2. C2×#16810633C2semidirect
3. C2×#16810632C2semidirect
4. C2×#16810631C2semidirect
7. #20274648C2semidirect
8. C2×C53×#419C2semidirect
17. C22×Q8×D53C4semidirect
26. C2×#13133142C22semidirect
27. C2×#13133141C22semidirect
28. C2×#13133140C22semidirect
39. C2×#103×C53C22semidirect
130. C2×Q8×C53C22⋊C4semidirect
191. C2×Dic106C22⋊C4semidirect
288. C2×D53#419semidirect
303. C2×#419D53semidirect
337. C2×#103C2×D53semidirect
344. C53#20599semidirect
457. C2×Q8#11719921semidirect

Maximal subgroups

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

subgroupindex
C23×Q8×D532
C2×#168106332
C2×#168106322
C2×#168106312
C2×#168105862
#202770942
#202746482
C2×C53×#4192
#2059953