Finite Group Databasebeta

← #27385326

Subgroups — #27385326

Order 13568.

1954 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×#16811135C211yesmaximal, characteristic
2C22×#13133175C211yesmaximal, characteristic, complemented
3C2×#16810625C211yesmaximal, characteristic, complemented
4C2×#16810622C211yesmaximal, characteristic, complemented
5C2×#16810586C211yesmaximal, characteristic, complemented
6#20275036C214yesmaximal, complemented
7#20275026C214yesmaximal, complemented
8#20274436C211yesmaximal, characteristic, complemented
9C53×#1736C211yesmaximal, characteristic, complemented
10C22×#11719945C2211yescharacteristic
11C23×C4×D53C2211yescharacteristic
12C22×#11719917C2211yescharacteristic
13#16811135—24no—
14#16811135C418yescomplemented
15C2×#13133201—21no—
16C2×#13133197C2212yes—
17C2×#13133192C2212yes—
18C2×#13133187—21no—
19C2×C4×D212—22no—
20C2×#13133175—22no—
21C2×#13133175—24no—
22C2×#13133175—24no—
23C2×#13133175C2214yes—
24C22×C4⋊Dic53C2211yescharacteristic
25C22×#11719921—21no—
26C22×C4×Dic53—21no—
27#16810625—24no—
28#16810622—24no—
29C2×#13133142C2212yescomplemented
30C2×#13133140C2212yescomplemented
31C2×#13133139C2212yescomplemented
32C2×#13133137C2212yescomplemented
33#16810586—24no—
34C2×#13133129C2214yes—
35#16810385C2214yescomplemented
36C22×M16×C53C2211yescharacteristic
37C2×#117×C53C2211yescharacteristic
38C22×C4⋊C4×C53C2211yescharacteristic
39C53×#340C2214yescomplemented
40C24×D53D411yescharacteristic
41C2×#11719945—22no—
42C2×#11719945—22no—
43C2×#11719945—22no—
44C2×#11719945—22no—
45C2×#11719945—22no—
46C2×#11719945C2×C414yes—
47C22×Dic106C2311yescharacteristic
48C22×D212—22no—
49C22×D212C2311yescharacteristic
50C22×C22⋊D53—21no—
51C23×Dic53—21no—
52C23×Dic53D411yescharacteristic
53C22×C4×D53—22no—
54C22×C4×D53—22no—
55C22×C4×D53—22no—
56C22×C4×D53—22no—
57C22×C4×D53—21no—
58C22×C4×D53C2×C414yes—
59C22×C4×D53D412yes—
60C22×C4×D53D414yes—
61C22×C4×D53C2312yes—
62C2×#11719917—22no—
63C2×#11719917—22no—
64C2×#11719917—22no—
65C22×#88322C2312yes—
66#13133201—28no—
67#13133197—24no—
68#13133197C2×C418yescomplemented
69#13133192—24no—
70#13133192C2×C418yescomplemented
71#13133187—28no—
72C4×D212—216no—
73#13133175—216no—
74#13133175—28no—
75#13133175—28no—
76C2×C4⋊Dic53—22no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53—22no—
79C2×C4⋊Dic53D414yescomplemented
80C2×C4⋊Dic53C2312yes—
81C2×#11719921—28no—
82C2×#11719921—24no—
83C2×#11719921—22no—
84C2×#11719921—22no—
85C2×C4×Dic53—28no—
86C2×C4×Dic53—24no—
87#13133142—24no—
88#13133140—24no—
89#13133139—24no—
90#13133137—24no—
91#13133129—28no—
92C23×C4×C53—21no—
93C23×C4×C53C2311yescharacteristic
94C2×M16×C53—22no—
95C2×M16×C53—22no—
96C2×M16×C53—22no—
97C22×C8×C53C2312yes—
98#117×C53—22no—
99#117×C53C2×C414yes—
100C2×C4⋊C4×C53—22no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53D414yescomplemented
103C2×C4⋊C4×C53C2312yes—
104C2×C4⋊C4×C53C2312yes—
105C2×C22⋊C4×C53—21no—
106C2×C42×C53—21no—
107C24×C53C2×D411yescharacteristic
108C23×D53—22no—
109C23×D53—22no—
110C23×D53—21no—
111C23×D53—42no—
112C23×D53—42no—
113C23×D53C22⋊C414yes—
114C23×D53C2×D412yes—
115C22×C4×C53—24no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—44no—
119C22×C4×C53C22×C412yes—
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2×D412yes—
122C22×C4×C53C2×D412yes—
123C22×C4×C53C2×D412yes—
124C22×C4×C53#4112yes—
125C22×C4×C53#4112yes—
126C22×C4×C53C2411yescharacteristic
127C2×C22⋊D53—24no—
128C2×C22⋊D53—42no—
129C2×C22⋊D53—42no—
130C22×Dic53—22no—
131C22×Dic53—22no—
132C22×Dic53—24no—
133C22×Dic53—22no—
134C22×Dic53—21no—
135C22×Dic53—44no—
136C22×Dic53C22⋊C414yes—
137C22×Dic53C2×D412yes—
138#11719945—28no—
139#11719945—28no—
140#11719945—44no—
141#11719945—44no—
142C2×D212—21no—
143C2×D212—21no—
144C2×D212—21no—
145C2×D212—21no—
146C2×D212—28no—
147C2×D212—21no—
148C2×D212—44no—
149C2×D212—44no—
150C2×D212C22×C412yes—
151C2×C4×D53—28no—
152C2×C4×D53—28no—
153C2×C4×D53—24no—
154C2×C4×D53—28no—
155C2×C4×D53—28no—
156C2×C4×D53—24no—
157C2×C4×D53—22no—
158C2×C4×D53—22no—
159C2×C4×D53—22no—
160C2×C4×D53—22no—
161C2×C4×D53—24no—
162C2×C4×D53—42no—
163C2×C4×D53—42no—
164C2×C4×D53C22⋊C418yes—
165C2×C4×D53C22×C414yes—
166C2×C4×D53C2×D414yes—
167C2×Dic106—21no—
168C2×Dic106—21no—
169C2×Dic106—21no—
170C2×Dic106—21no—
171C2×Dic106—21no—
172C2×Dic106C22×C412yes—
173M16×C53—44no—
174C2×C8×C53—22no—
175C2×C8×C53—22no—
176C2×C8×C53—22no—
177C4⋊C4×C53—24no—
178C4⋊C4×C53—22no—
179C4⋊C4×C53—22no—
180C4⋊C4×C53C22×C414yes—
181C4⋊C4×C53C2×D414yes—
182C22⋊C4×C53—24no—
183C42×C53—24no—
184#11719921—216no—
185#11719921—28no—
186#11719921—28no—
187C4⋊Dic53—24no—
188C4⋊Dic53—22no—
189C4⋊Dic53—28no—
190C4⋊Dic53C2×D414yes—
191C4×Dic53—216no—
192#11719917—44no—
193C2×#88322—22no—
194C2×#88322—22no—
195C2×#88322—22no—
196C23×C53—22no—
197C23×C53—22no—
198C23×C53C2×C22⋊C412yes—
199C23×C53#12012yes—
200C23×C53#12312yes—
201C23×C53C22×D411yescharacteristic
202C22×D53—22no—
203C22×D53—22no—
204C22×D53—22no—
205C22×D53—22no—
206C22×D53—22no—
207C22×D53—22no—
208C22×D53—28no—
209C22×D53—22no—
210C22×D53—48no—
211C22×D53—48no—
212C22×D53—44no—
213C22×D53—44no—
214C22×D53—41no—
215C22×D53—41no—
216C22×D53—42no—
217C22×D53—42no—
218C22×D53C2×C22⋊C414yes—
219C2×C4×C53—24no—
220C2×C4×C53—24no—
221C2×C4×C53—48no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—44no—
225C2×C4×C53C2×C22⋊C412yes—
226C2×C4×C53C4×D414yes—
227C2×C4×C53C4×D414yes—
228C2×C4×C53#12012yes—
229C2×C4×C53#12114yes—
230C2×C4×C53#12114yes—
231C2×C4×C53#12312yes—
232C2×C4×C53C23×C411yescharacteristic
233C2×C4×C53C22×D411yescharacteristic
234C2×C4×C53C22×D411yescharacteristic
235C2×C4×C53C22×D411yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C22⋊D53—48no—
239C2×Dic53—22no—
240C2×Dic53—22no—
241C2×Dic53—28no—
242C2×Dic53—22no—
243C2×Dic53—24no—
244C2×Dic53—48no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—44no—
248C2×Dic53C2×C22⋊C414yes—
249D212—22no—
250D212—22no—
251D212—416no—
252D212—41no—
253D212—41no—
254C4×D53—24no—
255C4×D53—24no—
256C4×D53—24no—
257C4×D53—24no—
258C4×D53—216no—
259C4×D53—48no—
260Dic106—22no—
261Dic106—22no—
262Dic106—41no—
263Dic106—41no—
264C8×C53—42no—
265#88322—42no—
266#20910—531nomaximal, Sylow
267C22×C53—24no—
268C22×C53—24no—
269C22×C53—22no—
270C22×C53—22no—
271C22×C53#38614yes—
272C22×C53C22×C22⋊C411yescharacteristic
273C22×C53C2×#12011yescharacteristic
274C22×C53C2×#12311yescharacteristic
275C22×C53#57512yes—
276C22×C53#57512yes—
277C2×D53—24no—
278C2×D53—24no—
279C2×D53—24no—
280C2×D53—416no—
281C2×D53—42no—
282C2×D53—42no—
283C2×D53—42no—
284C2×D53—42no—
285C2×D53—44no—
286C2×D53—42no—
287C2×D53—82no—
288C2×D53—81no—
289C2×D53—81no—
290C4×C53—44no—
291C4×C53—44no—
292C4×C53#38614yes—
293C4×C53C2×C4×D411yescharacteristic
294C4×C53C2×C4×D411yescharacteristic
295C4×C53C2×#12111yescharacteristic
296C4×C53C2×#12111yescharacteristic
297Dic53—24no—
298Dic53—44no—
299Dic53—42no—
300C2×#525—531no—
301C22×#117—531no—
302C2×#417—531no—
303C2×#417—531no—
304C2×#407—531no—
305#1875—534no—
306#1873—534no—
307#1736—531no—
308#1736D5311yescharacteristic, complemented
309C2×C53—24no—
310C2×C53C2×#38611yescharacteristic
311C2×C53#289612yes—
312C2×C53#300511yescharacteristic
313C2×C53#300511yescharacteristic
314C2×C53#309312yes—
315D53—44no—
316D53—82no—
317C22×#41—531no—
318C24×C4—531no—
319C22×M16—531no—
320C22×M16C2×D5311yescharacteristic
321#525—538no—
322#525—1064no—
323C2×#123—1061no—
324C2×C22⋊Q8—532no—
325C2×#121—532no—
326C2×#120—1061no—
327C2×C4×D4—1062no—
328C2×#117—534no—
329C2×#117—1064no—
330C2×#117—1062no—
331C2×#117—1064no—
332C2×#117C2×D5311yescharacteristic
333C22×C4⋊C4—531no—
334C22×C4⋊C4C2×D5311yescharacteristic
335C22×C22⋊C4—1061no—
336C22×C42—1061no—
337#417—1064no—
338#417—1064no—
339C2×#103—532no—
340C2×#103—532no—
341C2×#102—532no—
342C2×#102—532no—
343#407—1064no—
344C2×C22⋊C8—534no—
345#340—534no—
346#340C2×D5314yescomplemented
347C53#2091011yescharacteristic, complemented, Sylow
348C25—531no—
349C2×#41—534no—
350C2×#41—1062no—
351C2×#41—1062no—
352C2×#41—1062no—
353C2×#41—1062no—
354C2×#41—1062no—
355C22×Q8—531no—
356C22×D4—531no—
357C22×D4—1062no—
358C22×D4—1061no—
359C23×C4—21no—
360C23×C4—531no—
361C23×C4—534no—
362C23×C4—532no—
363C23×C4—534no—
364C23×C4—532no—
365C23×C4—1062no—
366C23×C4—1062no—
367C23×C4—1062no—
368C23×C4—1062no—
369C23×C4—1061no—
370C23×C4—1061no—
371C23×C4C22×D5311yescharacteristic
372C2×M16—22no—
373C2×M16—22no—
374C2×M16—22no—
375C2×M16—1062no—
376C2×M16—1062no—
377C2×M16—1062no—
378C22×C8—532no—
379C22×C8C22×D5312yes—
380#123—1068no—
381C22⋊Q8—538no—
382C22⋊Q8—1064no—
383#121—538no—
384#121—1064no—
385#120—1068no—
386C4×D4—10616no—
387#117—22no—
388#117—1068no—
389#117—10616no—
390#117—1068no—
391#117C4×D5314yes—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—534no—
395C2×C4⋊C4—532no—
396C2×C4⋊C4—1062no—
397C2×C4⋊C4—1062no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4D21214yescomplemented
400C2×C4⋊C4C22×D5312yes—
401C2×C4⋊C4C22×D5312yes—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—1062no—
404C2×C22⋊C4—1064no—
405C2×C22⋊C4—1062no—
406C2×C22⋊C4—1068no—
407C2×C42—21no—
408C2×C42—1064no—
409C2×C42—1068no—
410#103—1064no—
411#103—1064no—
412#102—1064no—
413#102—1064no—
414C22⋊C8—1068no—
415C24—532no—
416C24—534no—
417C24—1062no—
418C24—1062no—
419C24—1061no—
420C24—2122no—
421C24—2122no—
422C24D4×D5311yescharacteristic
423#41—1068no—
424#41—1068no—
425#41—2124no—
426#41—2124no—
427C2×Q8—532no—
428C2×Q8—1061no—
429C2×Q8—1061no—
430C2×Q8—1061no—
431C2×Q8—1061no—
432C2×Q8—1061no—
433C2×D4—532no—
434C2×D4—1068no—
435C2×D4—1064no—
436C2×D4—1061no—
437C2×D4—1061no—
438C2×D4—1061no—
439C2×D4—1061no—
440C2×D4—1061no—
441C2×D4—2124no—
442C2×D4—2124no—
443C2×D4—2122no—
444C2×D4—2122no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—44no—
449C22×C4—534no—
450C22×C4—532no—
451C22×C4—538no—
452C22×C4—534no—
453C22×C4—534no—
454C22×C4—1062no—
455C22×C4—1062no—
456C22×C4—1068no—
457C22×C4—1068no—
458C22×C4—1064no—
459C22×C4—1068no—
460C22×C4—1068no—
461C22×C4—1064no—
462C22×C4—1062no—
463C22×C4—1062no—
464C22×C4—1062no—
465C22×C4—1062no—
466C22×C4—1064no—
467C22×C4—1062no—
468C22×C4—1061no—
469C22×C4—1064no—
470C22×C4—2124no—
471C22×C4—2122no—
472C22×C4—2122no—
473C22×C4C2×C4×D5312yes—
474C22×C4C2×D21212yes—
475C22×C4#1171994512yes—
476C22×C4D4×D5311yescharacteristic
477C22×C4D4×D5312yes—
478C22×C4D4×D5312yes—
479C22×C4#1171994912yes—
480C22×C4C23×D5311yescharacteristic
481M16—44no—
482M16—2124no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—1062no—
487C2×C8—1062no—
488C2×C8—1062no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—534no—
493C4⋊C4—1068no—
494C4⋊C4—1064no—
495C4⋊C4—1062no—
496C4⋊C4C2×C4×D5314yes—
497C4⋊C4C2×D21214yes—
498C22⋊C4—24no—
499C22⋊C4—1068no—
500C22⋊C4—1068no—
501C22⋊C4—10616no—
502C42—24no—
503C42—10616no—
504C23—22no—
505C23—22no—
506C23—534no—
507C23—1068no—
508C23—1062no—
509C23—1062no—
510C23—1062no—
511C23—1062no—
512C23—1062no—
513C23—1062no—
514C23—1062no—
515C23—2128no—
516C23—2124no—
517C23—2124no—
518C23—2128no—
519C23—2121no—
520C23—2121no—
521C23—2122no—
522C23—2122no—
523C23C22⋊C4×D5312yes—
524C23#1313318712yes—
525C23#1313320212yes—
526C23C2×D4×D5311yescharacteristic
527Q8—1062no—
528Q8—1062no—
529Q8—2121no—
530Q8—2121no—
531D4—1062no—
532D4—1062no—
533D4—21216no—
534D4—2128no—
535D4—2121no—
536D4—2121no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—48no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—534no—
544C2×C4—1068no—
545C2×C4—1062no—
546C2×C4—1062no—
547C2×C4—10616no—
548C2×C4—1064no—
549C2×C4—1064no—
550C2×C4—1064no—
551C2×C4—1064no—
552C2×C4—1062no—
553C2×C4—1064no—
554C2×C4—2128no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2124no—
558C2×C4—2128no—
559C2×C4C22⋊C4×D5312yes—
560C2×C4#1313317914yes—
561C2×C4C4×D21214yes—
562C2×C4#1313318712yes—
563C2×C4#1313319014yes—
564C2×C4#1313319214yes—
565C2×C4#1313320212yes—
566C2×C4C22×C4×D5311yescharacteristic
567C2×C4C2×D4×D5311yescharacteristic
568C2×C4C2×D4×D5311yescharacteristic
569C2×C4C22×D21211yescharacteristic
570C2×C4C2×#1171994511yescharacteristic
571C2×C4C2×#1171994911yescharacteristic
572C8—42no—
573C8—2122no—
574C22—22no—
575C22—22no—
576C22—24no—
577C22—24no—
578C22—1064no—
579C22—1064no—
580C22—1064no—
581C22—4241no—
582C22—4241no—
583C22—4242no—
584C22—21216no—
585C22—2124no—
586C22—2122no—
587C22—2122no—
588C22—2122no—
589C22—2122no—
590C22—2122no—
591C22#1681050114yes—
592C22C2×C22⋊C4×D5311yescharacteristic
593C22C2×#1313318711yescharacteristic
594C22C2×#1313320211yescharacteristic
595C22#1681141912yes—
596C22#1681142212yes—
597C4—44no—
598C4—44no—
599C4—1064no—
600C4—2124no—
601C4—2122no—
602C4#1681050114yes—
603C4C2×#1313317911yescharacteristic
604C4C2×C4×D21211yescharacteristic
605C4C2×#1313319011yescharacteristic
606C4C2×#1313319211yescharacteristic
607C2—24no—
608C2—4242no—
609C2—2124no—
610C2C2×#1681050111yescharacteristic
611C2#2028323112yes—
612C2#2028395911yescharacteristic
613C2#2028396111yescharacteristic
614C2#2028464912yes—
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1736C2
socleC23×C53C22×D4
prime radicalC2×#16811135C2
cosocleC22×C4×C53C24
hypercenter#1736D53
hypocenterC53#20910
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
2. C22×#13133175C2semidirect
3. C2×#16810625C2semidirect
4. C2×#16810622C2semidirect
5. C2×#16810586C2semidirect
6. #20275036C2semidirect
7. #20275026C2semidirect
8. #20274436C2semidirect
9. C53×#1736C2semidirect
14. #16811135C4semidirect
29. C2×#13133142C22semidirect
30. C2×#13133140C22semidirect
31. C2×#13133139C22semidirect
32. C2×#13133137C22semidirect
35. #16810385C22semidirect
39. C53×#340C22semidirect
68. #13133197C2×C4semidirect
70. #13133192C2×C4semidirect
79. C2×C4⋊Dic53D4semidirect
102. C2×C4⋊C4×C53D4semidirect
308. #1736D53semidirect
346. #340C2×D53semidirect
347. C53#20910semidirect
399. C2×C4⋊C4D212semidirect

Maximal subgroups

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

subgroupindex
C2×#168111352
C22×#131331752
C2×#168106252
C2×#168106222
C2×#168105862
#202750362
#202750262
#202744362
C53×#17362
#2091053