Finite Group Databasebeta

← #27385448

Subgroups — #27385448

Order 13568.

2234 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133242C211yesmaximal, characteristic
2C22×#13133179C211yesmaximal, characteristic, complemented
3C2×#16810629C211yesmaximal, characteristic, complemented
4#20277206C211yesmaximal, characteristic, complemented
5#20275060C214yesmaximal, complemented
6#20275046C214yesmaximal, complemented
7#20274434C211yesmaximal, characteristic, complemented
8#20274408C211yesmaximal, characteristic, complemented
9C2×C53×#417C211yesmaximal, characteristic, complemented
10C23×D212C2211yescharacteristic
11C23×C4×D53—21no—
12C2×#13133242—24no—
13C2×#13133242—24no—
14C2×#13133242C418yescomplemented
15C22×#11719924C2212yes—
16C22×#11719922C2212yes—
17C22×#11719917C2211yescharacteristic
18C2×#13133179—24no—
19C2×#13133179—22no—
20C2×#13133179—24no—
21C2×#13133179C2214yes—
22C2×#13133176C2211yescharacteristic
23C22×#11719921—21no—
24C22×C4×Dic53C2211yescharacteristic
25#16810629—24no—
26C2×#13133139C2214yes—
27#16810530C2212yescomplemented
28#16810526C2212yescomplemented
29#16810503—21no—
30#16810503—21no—
31C4×C22⋊Dic53—22no—
32#16810385C2214yescomplemented
33#16810377C2214yescomplemented
34C22×#41×C53C2211yescharacteristic
35C22×M16×C53C2211yescharacteristic
36C22×C4⋊C4×C53C2211yescharacteristic
37C53×#417—24no—
38C2×#103×C53C2212yescomplemented
39C2×#102×C53C2212yescomplemented
40C24×D53—21no—
41C22×D212—21no—
42C22×D212—22no—
43C22×D212—22no—
44C22×D212—22no—
45C22×D212C2×C414yes—
46C22×D212D414yes—
47C22×D212D414yes—
48C22×D212C2312yes—
49C23×Dic53D411yescharacteristic
50C23×Dic53D411yescharacteristic
51C22×C4×D53—28no—
52C22×C4×D53—24no—
53C22×C4×D53—48no—
54#13133242—232no—
55C2×#11719924—24no—
56C2×#11719924—24no—
57C2×#11719924C2×C418yescomplemented
58C2×#11719922—24no—
59C2×#11719922—24no—
60C2×#11719922C2×C418yescomplemented
61C2×#11719917—22no—
62C2×#11719917—22no—
63C2×#11719917C2×C414yescomplemented
64C22×#88322C2312yes—
65#13133179—216no—
66#13133179—28no—
67#13133179—28no—
68#13133176—22no—
69#13133176—22no—
70C2×C4⋊Dic53C2312yes—
71C2×C22⋊Dic53—21no—
72C2×C22⋊Dic53—22no—
73C2×C22⋊Dic53—22no—
74C2×#11719921—28no—
75C2×#11719921—24no—
76C2×C4×Dic53—22no—
77C2×C4×Dic53—22no—
78C2×C4×Dic53—21no—
79C2×C4×Dic53D412yes—
80C2×C4×Dic53C2312yes—
81#13133139—28no—
82#13133124—22no—
83#13133124—22no—
84C2×#41×C53—22no—
85C2×#41×C53—22no—
86C2×#41×C53—22no—
87C2×#41×C53—22no—
88C2×#41×C53—22no—
89C2×#41×C53C2×C414yescomplemented
90C22×Q8×C53C2311yescharacteristic
91C22×D4×C53—21no—
92C22×D4×C53C2311yescharacteristic
93C23×C4×C53—21no—
94C23×C4×C53—21no—
95C23×C4×C53C2311yescharacteristic
96C2×M16×C53—22no—
97C2×M16×C53—22no—
98C2×M16×C53—22no—
99C22×C8×C53C2312yes—
100C2×C4⋊C4×C53—22no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53D414yes—
103C2×C4⋊C4×C53C2312yes—
104#103×C53—24no—
105#102×C53—24no—
106C24×C53—21no—
107C24×C53C2×D411yescharacteristic
108C23×D53—24no—
109C23×D53—24no—
110C23×D53—24no—
111C23×D53—22no—
112C23×D53—44no—
113C23×D53—44no—
114#41×C53—28no—
115#41×C53—28no—
116#41×C53—44no—
117#41×C53—44no—
118C2×Q8×C53—21no—
119C2×Q8×C53—21no—
120C2×Q8×C53—21no—
121C2×Q8×C53—21no—
122C2×Q8×C53—21no—
123C2×Q8×C53C22×C412yescomplemented
124C2×D4×C53—24no—
125C2×D4×C53—21no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—42no—
131C2×D4×C53—42no—
132C2×D4×C53C22×C412yescomplemented
133C22×C4×C53—24no—
134C22×C4×C53—22no—
135C22×C4×C53—21no—
136C22×C4×C53—24no—
137C22×C4×C53—44no—
138C22×C4×C53—42no—
139C22×C4×C53—42no—
140C22×C4×C53C22×C412yes—
141C22×C4×C53C2×D412yes—
142C22×C4×C53C2×D412yes—
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D412yes—
145C22×C4×C53#4112yes—
146C22×C4×C53#4112yes—
147C22×C4×C53C2411yescharacteristic
148C22×Dic53—22no—
149C22×Dic53—24no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—42no—
153C22×Dic53C2×D412yes—
154C22×Dic53C2×D412yes—
155C2×D212—28no—
156C2×D212—24no—
157C2×D212—24no—
158C2×D212—28no—
159C2×D212—24no—
160C2×D212—28no—
161C2×D212—22no—
162C2×D212—22no—
163C2×D212—22no—
164C2×D212C22⋊C4116yescomplemented
165C2×D212C22×C414yes—
166C2×D212C2×D414yes—
167C2×D212C2×D414yes—
168C2×C4×D53—216no—
169C2×C4×D53—432no—
170C2×C4×D53—48no—
171C2×C4×D53—48no—
172M16×C53—44no—
173C2×C8×C53—22no—
174C2×C8×C53—22no—
175C2×C8×C53—22no—
176C4⋊C4×C53—24no—
177C4⋊C4×C53—22no—
178C4⋊C4×C53C2×D414yes—
179C22⋊Dic53—44no—
180C22⋊Dic53—42no—
181C22⋊Dic53—44no—
182#11719924—216no—
183#11719922—216no—
184#11719921—216no—
185C4⋊Dic53—22no—
186C4⋊Dic53—22no—
187C4×Dic53—24no—
188C4×Dic53—22no—
189C4×Dic53—22no—
190C4×Dic53—42no—
191#11719917—28no—
192C2×#88322—22no—
193C2×#88322—22no—
194C2×#88322C22×C414yescomplemented
195C23×C53—22no—
196C23×C53—22no—
197C23×C53—22no—
198C23×C53—41no—
199C23×C53—41no—
200C23×C53—42no—
201C23×C53—42no—
202C23×C53C4×D412yes—
203C23×C53#12012yes—
204C23×C53#12112yes—
205C23×C53C22×D411yescharacteristic
206C22×D53—24no—
207C22×D53—24no—
208C22×D53—24no—
209C22×D53—216no—
210C22×D53—48no—
211C22×D53—416no—
212C22×D53—48no—
213C22×D53—416no—
214C22×D53—44no—
215C22×D53—44no—
216Q8×C53—22no—
217Q8×C53—22no—
218Q8×C53—41no—
219Q8×C53—41no—
220D4×C53—22no—
221D4×C53—22no—
222D4×C53—48no—
223D4×C53—41no—
224D4×C53—41no—
225C2×C4×C53—22no—
226C2×C4×C53—24no—
227C2×C4×C53—48no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—44no—
231C2×C4×C53—48no—
232C2×C4×C53C2×C22⋊C414yes—
233C2×C4×C53C4×D412yes—
234C2×C4×C53C4×D414yes—
235C2×C4×C53#12012yes—
236C2×C4×C53#12114yes—
237C2×C4×C53#12112yes—
238C2×C4×C53#12314yes—
239C2×C4×C53C23×C411yescharacteristic
240C2×C4×C53C22×D411yescharacteristic
241C2×C4×C53C22×D411yescharacteristic
242C2×C4×C53C22×D411yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×Dic53—28no—
246C2×Dic53—22no—
247C2×Dic53—22no—
248C2×Dic53—22no—
249C2×Dic53—22no—
250C2×Dic53—22no—
251C2×Dic53—44no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53AGL(1,ℤ/8)14yescomplemented
256D212—24no—
257D212—24no—
258D212—24no—
259D212—216no—
260D212C2×C22⋊C418yes—
261C4×D53—432no—
262C8×C53—42no—
263#88322—24no—
264#20925—531nomaximal, Sylow
265C22×C53—24no—
266C22×C53—22no—
267C22×C53—22no—
268C22×C53—24no—
269C22×C53—44no—
270C22×C53—42no—
271C22×C53—81no—
272C22×C53—81no—
273C22×C53—82no—
274C22×C53#38614yes—
275C22×C53C2×C4×D411yescharacteristic
276C22×C53C2×#12011yescharacteristic
277C22×C53C2×#12111yescharacteristic
278C22×C53C2×AGL(1,ℤ/8)12yes—
279C22×C53#57512yes—
280C2×D53—28no—
281C2×D53—432no—
282C2×D53—48no—
283C2×D53—44no—
284C2×D53—44no—
285C2×D53—48no—
286C4×C53—44no—
287C4×C53—42no—
288C4×C53#38614yes—
289C4×C53C22×C22⋊C411yescharacteristic
290C4×C53C2×C4×D411yescharacteristic
291C4×C53C2×#12111yescharacteristic
292C4×C53C2×#12311yescharacteristic
293Dic53—24no—
294Dic53—42no—
295Dic53—82no—
296C22×AGL(1,ℤ/8)—531no—
297C22×C4×D4—531no—
298C2×#417D5311yescharacteristic, complemented
299C2×#418—531no—
300#2297—531no—
301#1879—534no—
302#1877—534no—
303#1736—531no—
304#1728—531no—
305C2×C53—24no—
306C2×C53—82no—
307C2×C53C2×#38611yescharacteristic
308C2×C53C4×AGL(1,ℤ/8)12yes—
309C2×C53#300311yescharacteristic
310C2×C53#300511yescharacteristic
311C2×C53#303612yes—
312D53—48no—
313C22×#41C2×D5311yescharacteristic
314C23×D4—531no—
315C24×C4—1061no—
316C2×AGL(1,ℤ/8)—538no—
317C2×AGL(1,ℤ/8)—1064no—
318C2×AGL(1,ℤ/8)—1064no—
319C22×SD16—532no—
320C22×D8—532no—
321C22×M16—531no—
322C22×M16C2×D5311yescharacteristic
323C2×C4×D4—534no—
324C2×C4×D4—1062no—
325C2×C4×D4—1064no—
326C2×C4×D4—1064no—
327C2×#117—531no—
328C22×C4⋊C4C2×D5311yescharacteristic
329C22×C22⋊C4—1061no—
330C22×C42—531no—
331#418—1064no—
332#417—24no—
333C2×#103C2×D5312yescomplemented
334C2×#102—534no—
335C2×#102C2×D5312yescomplemented
336#391—532no—
337#390—532no—
338#386—1061no—
339#386—1061no—
340C4×C22⋊C4—1062no—
341#340—534no—
342#336—534no—
343C53#2092511yescharacteristic, complemented, Sylow
344C25—1061no—
345C2×#41—22no—
346C2×#41—22no—
347C2×#41—22no—
348C2×#41—22no—
349C2×#41—22no—
350C2×#41C4×D5314yescomplemented
351C22×Q8C22×D5311yescharacteristic
352C22×D4—21no—
353C22×D4—534no—
354C22×D4—534no—
355C22×D4—534no—
356C22×D4—532no—
357C22×D4—1062no—
358C22×D4—1061no—
359C22×D4—1062no—
360C22×D4—1062no—
361C22×D4C22×D5311yescharacteristic
362C23×C4—21no—
363C23×C4—21no—
364C23×C4—531no—
365C23×C4—531no—
366C23×C4—1064no—
367C23×C4—1068no—
368C23×C4—2128no—
369C23×C4C22×D5311yescharacteristic
370AGL(1,ℤ/8)—10632no—
371C2×SD16—538no—
372C2×SD16—1064no—
373C2×SD16—1064no—
374C2×D8—538no—
375C2×D8—1064no—
376C2×D8—1064no—
377C2×M16—22no—
378C2×M16—22no—
379C2×M16—22no—
380C2×M16—534no—
381C2×M16—1062no—
382C2×M16—1062no—
383C22×C8—532no—
384C22×C8C22×D5312yes—
385C4×D4—1068no—
386C4×D4—10616no—
387C4×D4—1068no—
388#117—1062no—
389#117—1062no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—532no—
393C2×C4⋊C4C22⋊D5314yes—
394C2×C4⋊C4C22×D5312yes—
395C2×C22⋊C4—1064no—
396C2×C22⋊C4—1068no—
397C2×C22⋊C4—1062no—
398C2×C22⋊C4—1062no—
399C2×C22⋊C4—1061no—
400C2×C42—532no—
401C2×C42—532no—
402C2×C42—1062no—
403C2×C42—1062no—
404C2×C42—1061no—
405#103—24no—
406#102—24no—
407#102—1068no—
408#95—1062no—
409#95—1062no—
410C24—21no—
411C24—1064no—
412C24—1064no—
413C24—1064no—
414C24—1062no—
415C24—2124no—
416C24—2124no—
417C24D4×D5311yescharacteristic
418#41—28no—
419#41—28no—
420#41—44no—
421#41—44no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8C2×C4×D5312yescomplemented
428C2×D4—24no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—5316no—
437C2×D4—534no—
438C2×D4—534no—
439C2×D4—534no—
440C2×D4—1064no—
441C2×D4—1068no—
442C2×D4—1064no—
443C2×D4—1068no—
444C2×D4—1064no—
445C2×D4—1068no—
446C2×D4—1062no—
447C2×D4—1062no—
448C2×D4—1062no—
449C2×D4C2×C4×D5312yescomplemented
450C22×C4—24no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—44no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—532no—
458C22×C4—532no—
459C22×C4—10616no—
460C22×C4—1064no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—2128no—
465C22×C4—21232no—
466C22×C4—2128no—
467C22×C4—2122no—
468C22×C4C2×C4×D5312yes—
469C22×C4#1171994512yes—
470C22×C4D4×D5312yes—
471C22×C4D4×D5312yes—
472C22×C4D4×D5311yescharacteristic
473C22×C4#1171994712yes—
474C22×C4C2×C22⋊D5312yes—
475C22×C4C23×D5311yescharacteristic
476SD16—10616no—
477D8—10616no—
478M16—44no—
479M16—1068no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—534no—
484C2×C8—1062no—
485C2×C8—1062no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—1062no—
489C4⋊C4—1062no—
490C4⋊C4C2×C22⋊D5314yes—
491C22⋊C4—10616no—
492C22⋊C4—2124no—
493C22⋊C4—2124no—
494C22⋊C4—2122no—
495C42—1064no—
496C42—1062no—
497C42—1062no—
498C42—2122no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—10616no—
507C23—1064no—
508C23—1064no—
509C23—1064no—
510C23—2128no—
511C23—21216no—
512C23—2128no—
513C23—21216no—
514C23—2124no—
515C23—2124no—
516C23#1313317712yes—
517C23#1313318512yes—
518C23#1313319012yes—
519C23C2×D4×D5311yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—41no—
523Q8—41no—
524D4—22no—
525D4—22no—
526D4—48no—
527D4—41no—
528D4—41no—
529D4—538no—
530D4—10616no—
531D4—1064no—
532D4—1064no—
533D4—1064no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—48no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—44no—
540C2×C4—48no—
541C2×C4—534no—
542C2×C4—1068no—
543C2×C4—1062no—
544C2×C4—1062no—
545C2×C4—1062no—
546C2×C4—1062no—
547C2×C4—1062no—
548C2×C4—21232no—
549C2×C4—2122no—
550C2×C4—2122no—
551C2×C4—2122no—
552C2×C4—2124no—
553C2×C4C22⋊C4×D5314yes—
554C2×C4#1313317712yes—
555C2×C4C4×C22⋊D5314yes—
556C2×C4#1313318512yes—
557C2×C4#1313318814yes—
558C2×C4#1313319012yes—
559C2×C4#1313319914yes—
560C2×C4C22×C4×D5311yescharacteristic
561C2×C4C2×D4×D5311yescharacteristic
562C2×C4C2×D4×D5311yescharacteristic
563C2×C4C22×C22⋊D5311yescharacteristic
564C2×C4C2×#1171994711yescharacteristic
565C2×C4C2×#1171994511yescharacteristic
566C8—42no—
567C8—1064no—
568C22—24no—
569C22—22no—
570C22—22no—
571C22—24no—
572C22—44no—
573C22—42no—
574C22—82no—
575C22—81no—
576C22—81no—
577C22—1068no—
578C22—21232no—
579C22—2128no—
580C22—2124no—
581C22—2124no—
582C22—2128no—
583C22#1681049914yes—
584C22C2×#1313317711yescharacteristic
585C22C2×#1313318511yescharacteristic
586C22C2×#1313319011yescharacteristic
587C22AGL(1,ℤ/8)×D5312yes—
588C22#1681142212yes—
589C4—44no—
590C4—42no—
591C4—1064no—
592C4—4242no—
593C4—2122no—
594C4#1681049914yes—
595C4C2×C22⋊C4×D5311yescharacteristic
596C4C2×C4×C22⋊D5311yescharacteristic
597C4C2×#1313318811yescharacteristic
598C4C2×#1313319911yescharacteristic
599C2—24no—
600C2—82no—
601C2—2128no—
602C2C2×#1681049911yescharacteristic
603C2#2028352812yes—
604C2#2028394011yescharacteristic
605C2#2028395611yescharacteristic
606C2#2028417812yes—
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#417C2
socleC23×C53C22×D4
prime radicalC22×#13133242C2
cosocleC22×C4×C53C24
hypercenterC2×#417D53
hypocenterC53#20925
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C22×#13133179C2semidirect
3. C2×#16810629C2semidirect
4. #20277206C2semidirect
5. #20275060C2semidirect
6. #20275046C2semidirect
7. #20274434C2semidirect
8. #20274408C2semidirect
9. C2×C53×#417C2semidirect
14. C2×#13133242C4semidirect
27. #16810530C22semidirect
28. #16810526C22semidirect
32. #16810385C22semidirect
33. #16810377C22semidirect
38. C2×#103×C53C22semidirect
39. C2×#102×C53C22semidirect
57. C2×#11719924C2×C4semidirect
60. C2×#11719922C2×C4semidirect
63. C2×#11719917C2×C4semidirect
89. C2×#41×C53C2×C4semidirect
123. C2×Q8×C53C22×C4semidirect
132. C2×D4×C53C22×C4semidirect
164. C2×D212C22⋊C4semidirect
194. C2×#88322C22×C4semidirect
255. C2×Dic53AGL(1,ℤ/8)semidirect
298. C2×#417D53semidirect
333. C2×#103C2×D53semidirect
335. C2×#102C2×D53semidirect
343. C53#20925semidirect
350. C2×#41C4×D53semidirect
427. C2×Q8C2×C4×D53semidirect
449. C2×D4C2×C4×D53semidirect

Maximal subgroups

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

subgroupindex
C22×#131332422
C22×#131331792
C2×#168106292
#202772062
#202750602
#202750462
#202744342
#202744082
C2×C53×#4172
#2092553