Finite Group Databasebeta

← #27412785

Subgroups — #27412785

Order 13568.

1178 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810665C211yesmaximal, characteristic, complemented
2C2×#16810646C211yesmaximal, characteristic, complemented
3C2×#16810624C211yesmaximal, characteristic
4#20277206C211yesmaximal, characteristic
5C2×#16810519C211yesmaximal, characteristic, complemented
6#20276108C212yesmaximal, complemented
7#20276101C212yesmaximal, complemented
8#20276094C212yesmaximal, complemented
9#20276087C212yesmaximal, complemented
10#20274434C211yesmaximal, characteristic, complemented
11C2×C53×#417C211yesmaximal, characteristic
12C22×#11719917C2211yescharacteristic
13C2×#13133176C2211yescharacteristic
14C22×C4⋊Dic53C2211yescharacteristic
15C22×#11719919—21no—
16C22×C4×Dic53C2211yescharacteristic
17#16810665—24no—
18C2×#13133151C2212yes—
19C2×#13133149C2212yes—
20#16810646—24no—
21C2×#13133146C2212yes—
22C2×#13133146C2212yes—
23#16810624—24no—
24C2×#13133141C2212yes—
25C2×#13133138C2212yes—
26#16810530C2211yescharacteristic
27#16810530C2211yescharacteristic
28#16810526C2211yescharacteristic
29#16810526C2211yescharacteristic
30#16810519—21no—
31#16810519—21no—
32#16810519C2212yes—
33#16810519C2212yes—
34#16810503—21no—
35#16810503—21no—
36C4×C22⋊Dic53—21no—
37C4×C22⋊Dic53—21no—
38C2×#13133124—21no—
39#16810385C2212yes—
40#16810385C2212yes—
41C22×#41×C53C2211yescharacteristic
42C22×M16×C53C2211yescharacteristic
43C22×C4⋊C4×C53C2211yescharacteristic
44C53×#417—24no—
45C2×#103×C53C2212yes—
46C2×#102×C53C2212yes—
47C23×Dic53—21no—
48C23×Dic53D411yescharacteristic
49C23×Dic53D411yescharacteristic
50C2×#11719917—22no—
51C2×#11719917—22no—
52C2×#11719917—22no—
53C22×#88322C2312yes—
54#13133176—22no—
55#13133176—22no—
56C2×C4⋊Dic53—21no—
57C2×C4⋊Dic53—22no—
58C2×C4⋊Dic53D414yes—
59C2×C4⋊Dic53C2311yescharacteristic
60C2×C4⋊Dic53C2311yescharacteristic
61C2×C4⋊Dic53C2312yes—
62C2×#11719919—22no—
63C2×#11719919—22no—
64C2×#11719919—44no—
65C2×C22⋊Dic53—21no—
66C2×C22⋊Dic53—21no—
67C2×C22⋊Dic53—21no—
68C2×C22⋊Dic53—21no—
69C2×C22⋊Dic53—21no—
70C2×C4×Dic53—22no—
71C2×C4×Dic53—21no—
72C2×C4×Dic53—22no—
73C2×C4×Dic53D411yescharacteristic
74C2×C4×Dic53D411yescharacteristic
75C2×C4×Dic53C2311yescharacteristic
76C2×C4×Dic53C2311yescharacteristic
77#13133151—24no—
78#13133149—24no—
79#13133146—24no—
80#13133146—24no—
81#13133141—24no—
82#13133138—24no—
83#13133124—21no—
84#13133124—21no—
85#13133124—21no—
86#13133124—21no—
87#13133124—22no—
88#13133124—22no—
89C2×#41×C53—22no—
90C2×#41×C53—22no—
91C2×#41×C53—22no—
92C2×#41×C53—22no—
93C2×#41×C53—22no—
94C2×#41×C53Q814yes—
95C22×Q8×C53C2311yescharacteristic
96C22×D4×C53—21no—
97C22×D4×C53C2311yescharacteristic
98C23×C4×C53—21no—
99C23×C4×C53—21no—
100C23×C4×C53C2311yescharacteristic
101C2×M16×C53—22no—
102C2×M16×C53—22no—
103C2×M16×C53D414yes—
104C22×C8×C53C2312yes—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—22no—
107C2×C4⋊C4×C53—22no—
108C2×C4⋊C4×C53C2312yes—
109#103×C53—24no—
110#102×C53—24no—
111C24×C53—21no—
112C24×C53C2×D411yescharacteristic
113#41×C53—28no—
114#41×C53—28no—
115#41×C53—44no—
116#41×C53—44no—
117C2×Q8×C53—21no—
118C2×Q8×C53—21no—
119C2×Q8×C53—21no—
120C2×Q8×C53—21no—
121C2×Q8×C53—21no—
122C2×Q8×C53C2×Q812yes—
123C2×D4×C53—24no—
124C2×D4×C53—21no—
125C2×D4×C53—21no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—42no—
130C2×D4×C53—42no—
131C2×D4×C53C2×Q812yes—
132C22×C4×C53—21no—
133C22×C4×C53—24no—
134C22×C4×C53—22no—
135C22×C4×C53—42no—
136C22×C4×C53—42no—
137C22×C4×C53—42no—
138C22×C4×C53—42no—
139C22×C4×C53—42no—
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D412yes—
142C22×C4×C53C2×D412yes—
143C22×C4×C53C2×Q812yes—
144C22×C4×C53#4112yes—
145C22×C4×C53#4112yes—
146C22×C4×C53#4112yes—
147C22×C4×C53C2411yescharacteristic
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—24no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—41no—
157C22×Dic53—41no—
158C22×Dic53—42no—
159C22×Dic53—42no—
160C22×Dic53C2×D411yescharacteristic
161C22×Dic53C2×D411yescharacteristic
162C22×Dic53C2×D411yescharacteristic
163C22×Dic53C2×D411yescharacteristic
164M16×C53—28no—
165C2×C8×C53—22no—
166C2×C8×C53—22no—
167C2×C8×C53C2×D414yes—
168C4⋊C4×C53—22no—
169C4⋊C4×C53—22no—
170C4⋊C4×C53—42no—
171C22⋊Dic53—42no—
172C22⋊Dic53—42no—
173C22⋊Dic53—42no—
174C22⋊Dic53—42no—
175C22⋊Dic53—42no—
176C4⋊Dic53—24no—
177C4⋊Dic53—22no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53C2×D414yes—
183#11719919—44no—
184#11719919—44no—
185C4×Dic53—22no—
186C4×Dic53—22no—
187C4×Dic53—21no—
188C4×Dic53—21no—
189C4×Dic53—21no—
190C4×Dic53—21no—
191C4×Dic53—42no—
192#11719917—44no—
193C2×#88322—22no—
194C2×#88322—22no—
195C2×#88322—22no—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53—21no—
200C23×C53—22no—
201C23×C53—41no—
202C23×C53—41no—
203C23×C53—41no—
204C23×C53—41no—
205C23×C53—41no—
206C23×C53—41no—
207C23×C53#12112yes—
208C23×C53#12112yes—
209C23×C53C22⋊Q812yes—
210C23×C53C22×D411yescharacteristic
211Q8×C53—22no—
212Q8×C53—22no—
213Q8×C53—41no—
214Q8×C53—41no—
215D4×C53—22no—
216D4×C53—22no—
217D4×C53—48no—
218D4×C53—41no—
219D4×C53—41no—
220C2×C4×C53—22no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—48no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—84no—
228C2×C4×C53#12112yes—
229C2×C4×C53#12114yes—
230C2×C4×C53#12112yes—
231C2×C4×C53C22⋊Q814yes—
232C2×C4×C53C22⋊Q812yes—
233C2×C4×C53C22⋊Q814yes—
234C2×C4×C53#12614yes—
235C2×C4×C53C22×D411yescharacteristic
236C2×C4×C53C22×D411yescharacteristic
237C2×C4×C53C22×Q811yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—24no—
246C2×Dic53—21no—
247C2×Dic53—21no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—44no—
253C2×Dic53—44no—
254C2×Dic53—48no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53AGL(1,ℤ/8)12yes—
264C2×Dic53#13712yes—
265C8×C53—24no—
266#88322—42no—
267#23617—531nomaximal, Sylow
268C22×C53—22no—
269C22×C53—22no—
270C22×C53—22no—
271C22×C53—22no—
272C22×C53—22no—
273C22×C53—22no—
274C22×C53—42no—
275C22×C53—42no—
276C22×C53—42no—
277C22×C53—81no—
278C22×C53—81no—
279C22×C53—81no—
280C22×C53—81no—
281C22×C53#39614yes—
282C22×C53C2×#12111yescharacteristic
283C22×C53C2×#12111yescharacteristic
284C22×C53C2×C22⋊Q811yescharacteristic
285C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
286C22×C53C2×#13711yescharacteristic
287C22×C53#57511yescharacteristic
288C22×C53#57511yescharacteristic
289C4×C53—42no—
290C4×C53—82no—
291C4×C53#39614yes—
292C4×C53C2×#12111yescharacteristic
293C4×C53C2×C22⋊Q811yescharacteristic
294C4×C53C2×C22⋊Q811yescharacteristic
295C4×C53C2×#12611yescharacteristic
296Dic53—22no—
297Dic53—22no—
298Dic53—44no—
299Dic53—41no—
300Dic53—41no—
301Dic53—81no—
302Dic53—81no—
303C2×#428—531no—
304C2×#423—531no—
305C2×#417—531no—
306C2×#417D5311yescharacteristic
307#2297—531no—
308C2×#389—531no—
309#2068—532no—
310#2067—532no—
311#2066—532no—
312#2065—532no—
313#1736—531no—
314C2×C53—24no—
315C2×C53—82no—
316C2×C53C2×#39611yescharacteristic
317C2×C53#303611yescharacteristic
318C2×C53#303711yescharacteristic
319C2×C53#305511yescharacteristic
320C2×C53#305611yescharacteristic
321C2×C53#307911yescharacteristic
322C2×C53#308011yescharacteristic
323C22×#41C2×D5311yescharacteristic
324C22×M16—531no—
325C22×M16C2×D5311yescharacteristic
326C2×#117—531no—
327C22×C4⋊C4—531no—
328C22×C4⋊C4—1061no—
329C22×C4⋊C4C2×D5311yescharacteristic
330C22×C42—531no—
331#428—1064no—
332C2×#107—532no—
333C2×#106—532no—
334#423—1064no—
335C2×C4⋊C8—532no—
336C2×C4⋊C8—532no—
337#417—24no—
338#417—1064no—
339C2×#103—532no—
340C2×#103C2×D5312yes—
341C2×#102—532no—
342C2×#102C2×D5312yes—
343#391—531no—
344#391—531no—
345#390—531no—
346#390—531no—
347#389—532no—
348#389—532no—
349#389—1061no—
350#389—1061no—
351#386—1061no—
352#386—1061no—
353C4×C22⋊C4—1061no—
354C4×C22⋊C4—1061no—
355C2×#95—1061no—
356#340—532no—
357#340—532no—
358C53#2361711yescharacteristic, complemented, Sylow
359C2×#41—22no—
360C2×#41—22no—
361C2×#41—22no—
362C2×#41—22no—
363C2×#41—22no—
364C2×#41Dic10614yes—
365C22×Q8C22×D5311yescharacteristic
366C22×D4—21no—
367C22×D4C22×D5311yescharacteristic
368C23×C4—21no—
369C23×C4—21no—
370C23×C4—531no—
371C23×C4—531no—
372C23×C4—1061no—
373C23×C4C22×D5311yescharacteristic
374C2×M16—22no—
375C2×M16—22no—
376C2×M16—1062no—
377C2×M16—1062no—
378C2×M16—1062no—
379C2×M16C22⋊D5314yes—
380C22×C8—532no—
381C22×C8C22×D5312yes—
382#117—1062no—
383#117—1062no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—534no—
388C2×C4⋊C4—532no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—1062no—
392C2×C4⋊C4—1061no—
393C2×C4⋊C4—1062no—
394C2×C4⋊C4—1062no—
395C2×C4⋊C4—2124no—
396C2×C4⋊C4C22×D5312yes—
397C2×C22⋊C4—1061no—
398C2×C22⋊C4—1061no—
399C2×C22⋊C4—1061no—
400C2×C22⋊C4—1061no—
401C2×C22⋊C4—1061no—
402C2×C42—531no—
403C2×C42—531no—
404C2×C42—531no—
405C2×C42—531no—
406C2×C42—1062no—
407C2×C42—1062no—
408C2×C42—1061no—
409#107—1064no—
410#106—1064no—
411C4⋊C8—1064no—
412C4⋊C8—1064no—
413#103—24no—
414#103—1064no—
415#102—24no—
416#102—1064no—
417#95—1062no—
418#95—1062no—
419#95—1061no—
420#95—1061no—
421#95—1061no—
422#95—1061no—
423C24—21no—
424C24D4×D5311yescharacteristic
425#41—28no—
426#41—28no—
427#41—44no—
428#41—44no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8C2×Dic10612yes—
435C2×D4—24no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4C2×Dic10612yes—
444C22×C4—21no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—531no—
453C22×C4—531no—
454C22×C4—531no—
455C22×C4—531no—
456C22×C4—1062no—
457C22×C4—1062no—
458C22×C4—1062no—
459C22×C4—1062no—
460C22×C4—1064no—
461C22×C4—1062no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—2121no—
465C22×C4—2121no—
466C22×C4—2122no—
467C22×C4—2122no—
468C22×C4C2×Dic10612yes—
469C22×C4#1171994512yes—
470C22×C4D4×D5311yescharacteristic
471C22×C4D4×D5312yes—
472C22×C4#1171994712yes—
473C22×C4#1171994712yes—
474C22×C4C2×C22⋊D5312yes—
475C22×C4C23×D5311yescharacteristic
476M16—28no—
477M16—2124no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—1062no—
481C2×C8—1062no—
482C2×C8—1062no—
483C2×C8C2×C22⋊D5314yes—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—42no—
487C4⋊C4—534no—
488C4⋊C4—1064no—
489C4⋊C4—1062no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—2124no—
495C4⋊C4—2124no—
496C22⋊C4—2122no—
497C22⋊C4—2122no—
498C22⋊C4—2122no—
499C22⋊C4—2122no—
500C22⋊C4—2122no—
501C42—1062no—
502C42—1062no—
503C42—1061no—
504C42—1061no—
505C42—1061no—
506C42—1061no—
507C42—2122no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—22no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23#1313318812yes—
520C23#1313319012yes—
521C23#1313319412yes—
522C23C2×D4×D5311yescharacteristic
523Q8—22no—
524Q8—22no—
525Q8—41no—
526Q8—41no—
527D4—22no—
528D4—22no—
529D4—48no—
530D4—41no—
531D4—41no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—48no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—84no—
540C2×C4—532no—
541C2×C4—532no—
542C2×C4—1061no—
543C2×C4—1061no—
544C2×C4—1061no—
545C2×C4—1061no—
546C2×C4—1061no—
547C2×C4—1061no—
548C2×C4—1064no—
549C2×C4—1061no—
550C2×C4—1061no—
551C2×C4—1061no—
552C2×C4—1061no—
553C2×C4—2124no—
554C2×C4—2124no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2128no—
562C2×C4—2122no—
563C2×C4—2122no—
564C2×C4#1313318812yes—
565C2×C4#1313318914yes—
566C2×C4#1313319012yes—
567C2×C4#1313319412yes—
568C2×C4#1313319414yes—
569C2×C4#1313319614yes—
570C2×C4#1313321014yes—
571C2×C4C2×D4×D5311yescharacteristic
572C2×C4C22×C22⋊D5311yescharacteristic
573C2×C4C22×Dic10611yescharacteristic
574C2×C4C2×#1171994711yescharacteristic
575C2×C4C2×#1171994711yescharacteristic
576C2×C4C2×#1171994511yescharacteristic
577C8—24no—
578C8—2122no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22#1681054614yes—
593C22C2×#1313318811yescharacteristic
594C22C2×#1313319011yescharacteristic
595C22C2×#1313319411yescharacteristic
596C22AGL(1,ℤ/8)×D5311yescharacteristic
597C22#137×D5311yescharacteristic
598C22#1681141911yescharacteristic
599C22#1681142211yescharacteristic
600C4—42no—
601C4—82no—
602C4—1062no—
603C4—1062no—
604C4—4241no—
605C4—4241no—
606C4—2124no—
607C4—2121no—
608C4—2121no—
609C4#1681054614yes—
610C4C2×#1313318911yescharacteristic
611C4C2×#1313319411yescharacteristic
612C4C2×#1313319611yescharacteristic
613C4C2×#1313321011yescharacteristic
614C2—24no—
615C2—82no—
616C2C2×#1681054611yescharacteristic
617C2#2028417811yescharacteristic
618C2#2028418611yescharacteristic
619C2#2028432811yescharacteristic
620C2#2028433911yescharacteristic
621C2#2028452611yescharacteristic
622C2#2028453511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#417C2
socleC23×C53C22×D4
prime radicalC22×#41×C53C22
hypercenterC2×#417D53
hypocenterC53#23617
LCSG ▷ C22×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
1. C2×#16810665C2semidirect
2. C2×#16810646C2semidirect
5. C2×#16810519C2semidirect
6. #20276108C2semidirect
7. #20276101C2semidirect
8. #20276094C2semidirect
9. #20276087C2semidirect
10. #20274434C2semidirect
358. C53#23617semidirect

Maximal subgroups

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

subgroupindex
C2×#168106652
C2×#168106462
C2×#168106242
#202772062
C2×#168105192
#202761082
#202761012
#202760942
#202760872
#202744342
C2×C53×#4172
#2361753