Finite Group Databasebeta

← #27290522

Subgroups — #27290522

Order 13568.

966 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810639C211yesmaximal, characteristic
2C2×#16810622C211yesmaximal, characteristic
3C2×#16810596C211yesmaximal, characteristic
4#20277072C211yesmaximal, characteristic, complemented
5C2×#16810395C211yesmaximal, characteristic, complemented
6C2×#16810379C211yesmaximal, characteristic, complemented
7C53×#1736C211yesmaximal, characteristic, complemented
8C22×#11719945C2211yescharacteristic
9C22×#11719917C2211yescharacteristic
10C2×#13133176C2211yescharacteristic
11C2×#13133176C2211yescharacteristic
12#16810639—22no—
13#16810639C414yescomplemented
14C2×#13133143—22no—
15#16810622—22no—
16#16810622—22no—
17C2×#13133140C412yescomplemented
18C2×#13133137C412yescomplemented
19#16810596—22no—
20#16810596C414yescomplemented
21C2×#13133131—22no—
22C4⋊C4×Dic53—22no—
23C4⋊C4×Dic53—22no—
24C4×C22⋊Dic53—22no—
25#16810395—22no—
26#16810379—22no—
27C22×M16×C53C2211yescharacteristic
28C2×#117×C53C2211yescharacteristic
29C22×C4⋊C4×C53C2211yescharacteristic
30C53×#340—22no—
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×#11719945C2×C412yes—
42C2×#11719945C2×C412yes—
43C22×Dic106C2×C411yescharacteristic
44C22×D212C2×C411yescharacteristic
45C22×C22⋊D53—21no—
46C22×C4×D53—21no—
47C2×#11719917—21no—
48C2×#11719917—21no—
49C2×#11719917—21no—
50C2×#11719917—21no—
51C2×#11719917—21no—
52C2×#11719917—21no—
53C22×#88322C2×C412yes—
54#13133176—21no—
55#13133176—21no—
56#13133176—21no—
57#13133176—21no—
58#13133176C2×C412yes—
59#13133176C2×C412yes—
60#13133176D412yes—
61#13133176D412yes—
62C2×C4⋊Dic53—21no—
63C2×C4⋊Dic53—21no—
64C2×C22⋊Dic53—21no—
65C2×C22⋊Dic53D412yes—
66C2×C4×Dic53—24no—
67C2×C4×Dic53—21no—
68C2×C4×Dic53D412yes—
69#13133143—28no—
70#13133140—24no—
71#13133137—24no—
72#13133131—28no—
73#13133124—24no—
74C23×C4×C53—21no—
75C23×C4×C53C2311yescharacteristic
76C2×M16×C53—21no—
77C2×M16×C53—21no—
78C2×M16×C53—21no—
79C2×M16×C53—21no—
80C2×M16×C53—21no—
81C2×M16×C53C2×C412yes—
82C22×C8×C53—21no—
83#117×C53—21no—
84#117×C53—21no—
85#117×C53—21no—
86#117×C53C2×C412yes—
87C2×C4⋊C4×C53—21no—
88C2×C4⋊C4×C53—21no—
89C2×C4⋊C4×C53—21no—
90C2×C4⋊C4×C53—21no—
91C2×C4⋊C4×C53C2×C412yes—
92C2×C4⋊C4×C53D412yes—
93C2×C4⋊C4×C53D412yes—
94C2×C22⋊C4×C53—21no—
95C2×C42×C53—21no—
96C24×C53C2×D411yescharacteristic
97C23×D53—21no—
98C22×C4×C53—22no—
99C22×C4×C53—22no—
100C22×C4×C53—22no—
101C22×C4×C53—41no—
102C22×C4×C53—82no—
103C22×C4×C53C22×C411yescharacteristic
104C22×C4×C53C22×C411yescharacteristic
105C22×C4×C53C22×C411yescharacteristic
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53#4111yescharacteristic
112C22×C4×C53#4111yescharacteristic
113C22×C4×C53#4111yescharacteristic
114C22×C4×C53#4111yescharacteristic
115C22×C4×C53#4111yescharacteristic
116C22×C4×C53#4111yescharacteristic
117C2×C22⋊D53—22no—
118C2×C22⋊D53—22no—
119C2×C22⋊D53—41no—
120C2×C22⋊D53—41no—
121C2×C22⋊D53—41no—
122C2×C22⋊D53—41no—
123C22×Dic53—21no—
124C22×Dic53—22no—
125C22×Dic53—22no—
126#11719945—22no—
127#11719945—22no—
128#11719945—22no—
129#11719945—22no—
130#11719945—22no—
131#11719945—22no—
132#11719945—22no—
133#11719945—22no—
134#11719945—41no—
135#11719945—41no—
136#11719945—41no—
137#11719945—41no—
138#11719945—41no—
139#11719945—41no—
140#11719945—41no—
141#11719945—41no—
142C2×D212—21no—
143C2×D212—21no—
144C2×D212—21no—
145C2×D212—21no—
146C2×D212—21no—
147C2×D212C4212yescomplemented
148C2×C4×D53—22no—
149C2×C4×D53—22no—
150C2×C4×D53—41no—
151C2×C4×D53—41no—
152C2×C4×D53—41no—
153C2×C4×D53—41no—
154C2×Dic106—21no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106C4212yescomplemented
160M16×C53—22no—
161M16×C53—22no—
162M16×C53—41no—
163M16×C53—41no—
164C2×C8×C53—22no—
165C2×C8×C53—41no—
166C2×C8×C53—41no—
167C4⋊C4×C53—22no—
168C4⋊C4×C53—22no—
169C4⋊C4×C53—22no—
170C4⋊C4×C53—22no—
171C4⋊C4×C53—41no—
172C4⋊C4×C53C22⋊C414yes—
173C22⋊C4×C53—22no—
174C22⋊C4×C53—41no—
175C42×C53—22no—
176C42×C53—41no—
177C22⋊Dic53—22no—
178C22⋊Dic53—22no—
179C22⋊Dic53—22no—
180C22⋊Dic53C22⋊C414yes—
181C4⋊Dic53—22no—
182C4⋊Dic53—22no—
183C4⋊Dic53—22no—
184C4⋊Dic53—22no—
185C4×Dic53—22no—
186C4×Dic53—22no—
187C4×Dic53—22no—
188C4×Dic53—84no—
189C4×Dic53C22⋊C414yes—
190#11719917—41no—
191#11719917—41no—
192#11719917—41no—
193#11719917—41no—
194C2×#88322—22no—
195C2×#88322—22no—
196C2×#88322—22no—
197C23×C53—41no—
198C23×C53—41no—
199C23×C53C2×C22⋊C411yescharacteristic
200C23×C53C4×D411yescharacteristic
201C23×C53C4×D411yescharacteristic
202C23×C53#12011yescharacteristic
203C23×C53#12111yescharacteristic
204C23×C53#12111yescharacteristic
205C23×C53#12311yescharacteristic
206C22×D53—22no—
207C22×D53—41no—
208C22×D53—41no—
209C22×D53—81no—
210C22×D53—81no—
211C2×C4×C53—24no—
212C2×C4×C53—44no—
213C2×C4×C53—42no—
214C2×C4×C53—82no—
215C2×C4×C53—82no—
216C2×C4×C53—81no—
217C2×C4×C53—81no—
218C2×C4×C53C2×C4211yescharacteristic
219C2×C4×C53C2×C22⋊C411yescharacteristic
220C2×C4×C53C2×C22⋊C411yescharacteristic
221C2×C4×C53#11711yescharacteristic
222C2×C4×C53#11711yescharacteristic
223C2×C4×C53#11711yescharacteristic
224C2×C4×C53C4×D411yescharacteristic
225C2×C4×C53C4×D411yescharacteristic
226C2×C4×C53C4×D411yescharacteristic
227C2×C4×C53C4×D411yescharacteristic
228C2×C4×C53C4×D411yescharacteristic
229C2×C4×C53C4×D411yescharacteristic
230C2×C4×C53C4×D411yescharacteristic
231C2×C4×C53C4×D411yescharacteristic
232C2×C4×C53C4×D411yescharacteristic
233C2×C4×C53C4×D411yescharacteristic
234C2×C4×C53#12111yescharacteristic
235C2×C4×C53#12111yescharacteristic
236C2×C4×C53#12111yescharacteristic
237C2×C4×C53#12111yescharacteristic
238C2×C4×C53#12311yescharacteristic
239C2×C4×C53#12311yescharacteristic
240C2×C4×C53#12411yescharacteristic
241C2×C4×C53#12411yescharacteristic
242C2×C4×C53#12411yescharacteristic
243C2×C4×C53#12611yescharacteristic
244C2×C4×C53#12611yescharacteristic
245C2×C4×C53#12711yescharacteristic
246C22⋊D53—42no—
247C22⋊D53—42no—
248C22⋊D53—42no—
249C22⋊D53—42no—
250C2×Dic53—22no—
251C2×Dic53—24no—
252C2×Dic53—24no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256C2×Dic53—42no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259D212—22no—
260D212—22no—
261D212—41no—
262D212—41no—
263C4×D53—42no—
264C4×D53—42no—
265C4×D53—42no—
266C4×D53—42no—
267Dic106—22no—
268Dic106—22no—
269Dic106—41no—
270Dic106—41no—
271C8×C53—42no—
272#88322—42no—
273#10954—531nomaximal, Sylow
274C22×C53—21no—
275C22×C53—21no—
276C22×C53—21no—
277C22×C53—21no—
278C22×C53—21no—
279C22×C53—21no—
280C22×C53—41no—
281C22×C53—41no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53C4×C22⋊C411yescharacteristic
285C22×C53#38611yescharacteristic
286C22×C53#38611yescharacteristic
287C22×C53#38611yescharacteristic
288C22×C53#38811yescharacteristic
289C22×C53#38811yescharacteristic
290C22×C53#39011yescharacteristic
291C2×D53—42no—
292C2×D53—82no—
293C2×D53—81no—
294C2×D53—81no—
295C2×D53—82no—
296C4×C53—44no—
297C4×C53—82no—
298C4×C53C4×C22⋊C411yescharacteristic
299C4×C53C4×C22⋊C411yescharacteristic
300C4×C53#38811yescharacteristic
301C4×C53#38811yescharacteristic
302C4×C53#38811yescharacteristic
303C4×C53#38811yescharacteristic
304C4×C53#39011yescharacteristic
305C4×C53#39011yescharacteristic
306Dic53—44no—
307Dic53—44no—
308Dic53—42no—
309C2×#421—531no—
310C2×#417—531no—
311C2×#410—531no—
312#2274—531no—
313C2×#344—531no—
314#1736D5311yescharacteristic, complemented
315C2×H(3,ℤ/4)—531no—
316C2×C53—21no—
317C2×C53—21no—
318C2×C53—21no—
319C2×C53—21no—
320C2×C53—21no—
321C2×C53—21no—
322C2×C53#143711yescharacteristic
323C2×C53#190411yescharacteristic
324C2×C53#190511yescharacteristic
325D53—82no—
326C22×#41—531no—
327C22×M16—531no—
328C22×M16C2×D5311yescharacteristic
329C2×#117—531no—
330C2×#117—531no—
331C2×#117C2×D5311yescharacteristic
332C22×C4⋊C4C2×D5311yescharacteristic
333#421—534no—
334#421—1062no—
335C2×#104—1062no—
336#417—1062no—
337#417—1062no—
338C2×#103—532no—
339C2×#102—532no—
340#410—534no—
341#410—1062no—
342C2×#99—1062no—
343C4×C4⋊C4—1062no—
344C4×C4⋊C4—1062no—
345C4×C22⋊C4—1062no—
346#344—1062no—
347#340—22no—
348H(3,ℤ/4)—1062no—
349C53#1095411yescharacteristic, complemented, Sylow
350C2×#41—532no—
351C2×#41—532no—
352C2×#41—1061no—
353C2×#41—1061no—
354C2×#41—1061no—
355C2×#41—1061no—
356C2×#41—1061no—
357C2×#41—1061no—
358C2×#41—1061no—
359C2×#41—1061no—
360C2×#41—1061no—
361C2×#41—1061no—
362C22×Q8—531no—
363C22×D4—531no—
364C22×D4—1061no—
365C23×C4—21no—
366C23×C4—1061no—
367C23×C4C22×D5311yescharacteristic
368C2×M16—21no—
369C2×M16—21no—
370C2×M16—21no—
371C2×M16—21no—
372C2×M16—21no—
373C2×M16—1061no—
374C2×M16—1061no—
375C2×M16—1061no—
376C2×M16—1061no—
377C2×M16—1061no—
378C2×M16—1061no—
379C2×M16C4×D5312yes—
380C22×C8—21no—
381C22×C8—532no—
382#117—21no—
383#117—21no—
384#117—21no—
385#117—532no—
386#117—532no—
387#117—532no—
388#117—532no—
389#117—1061no—
390#117—1061no—
391#117—1061no—
392#117—1061no—
393#117C4×D5312yes—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4C4×D5312yes—
401C2×C4⋊C4D21212yes—
402C2×C4⋊C4C22⋊D5312yes—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—532no—
405C2×C22⋊C4—1061no—
406C2×C42—21no—
407C2×C42—532no—
408C2×C42—1061no—
409C2×C42—1064no—
410#104—1068no—
411#103—1064no—
412#102—1064no—
413#99—1068no—
414#95—1064no—
415C24—1061no—
416C24D4×D5311yescharacteristic
417#41—1062no—
418#41—1062no—
419#41—1062no—
420#41—1062no—
421#41—1062no—
422#41—1062no—
423#41—1062no—
424#41—1062no—
425#41—2121no—
426#41—2121no—
427#41—2121no—
428#41—2121no—
429#41—2121no—
430#41—2121no—
431#41—2121no—
432#41—2121no—
433C2×Q8—532no—
434C2×Q8—1061no—
435C2×Q8—1061no—
436C2×Q8—1061no—
437C2×Q8—1061no—
438C2×Q8—1061no—
439C2×D4—532no—
440C2×D4—1062no—
441C2×D4—1062no—
442C2×D4—1061no—
443C2×D4—1061no—
444C2×D4—1061no—
445C2×D4—1061no—
446C2×D4—1061no—
447C2×D4—2121no—
448C2×D4—2121no—
449C2×D4—2121no—
450C2×D4—2121no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—41no—
455C22×C4—82no—
456C22×C4—1061no—
457C22×C4—1062no—
458C22×C4—1062no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—2121no—
462C22×C4—2121no—
463C22×C4—2121no—
464C22×C4—2121no—
465C22×C4C2×C4×D5311yescharacteristic
466C22×C4C2×C4×D5311yescharacteristic
467C22×C4C2×C4×D5311yescharacteristic
468C22×C4C2×D21211yescharacteristic
469C22×C4#1171994511yescharacteristic
470C22×C4#1171994511yescharacteristic
471C22×C4D4×D5311yescharacteristic
472C22×C4D4×D5311yescharacteristic
473C22×C4D4×D5311yescharacteristic
474C22×C4#1171994711yescharacteristic
475C22×C4#1171994711yescharacteristic
476C22×C4#1171994911yescharacteristic
477C22×C4#1171994911yescharacteristic
478C22×C4C2×C22⋊D5311yescharacteristic
479M16—22no—
480M16—22no—
481M16—41no—
482M16—41no—
483M16—2121no—
484M16—2121no—
485M16—2121no—
486M16—2121no—
487C2×C8—22no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8—1062no—
491C2×C8—1062no—
492C2×C8—1062no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—41no—
498C4⋊C4—1062no—
499C4⋊C4—1062no—
500C4⋊C4—1062no—
501C4⋊C4—1062no—
502C4⋊C4#1171992114yes—
503C22⋊C4—22no—
504C22⋊C4—41no—
505C22⋊C4—534no—
506C22⋊C4—1062no—
507C22⋊C4—1062no—
508C22⋊C4—1062no—
509C42—22no—
510C42—41no—
511C42—534no—
512C42—1062no—
513C42—1062no—
514C42—1062no—
515C42—4244no—
516C23—41no—
517C23—41no—
518C23—1062no—
519C23—4241no—
520C23—4241no—
521C23—2121no—
522C23—2121no—
523C23C22⋊C4×D5311yescharacteristic
524C23#1313317711yescharacteristic
525C23#1313317911yescharacteristic
526C23#1313318711yescharacteristic
527C23#1313318911yescharacteristic
528C23#1313319011yescharacteristic
529C23#1313320211yescharacteristic
530Q8—1062no—
531Q8—1062no—
532Q8—2121no—
533Q8—2121no—
534D4—1062no—
535D4—1062no—
536D4—2122no—
537D4—2122no—
538D4—2122no—
539D4—2122no—
540D4—2121no—
541D4—2121no—
542C2×C4—24no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—1064no—
550C2×C4—1064no—
551C2×C4—1062no—
552C2×C4—2122no—
553C2×C4—2122no—
554C2×C4—2122no—
555C2×C4—2122no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2122no—
559C2×C4—2122no—
560C2×C4—2122no—
561C2×C4—2122no—
562C2×C4C42×D5311yescharacteristic
563C2×C4C22⋊C4×D5311yescharacteristic
564C2×C4C2×#1171992111yescharacteristic
565C2×C4#1313317411yescharacteristic
566C2×C4#1313317511yescharacteristic
567C2×C4#1313317511yescharacteristic
568C2×C4#1313317711yescharacteristic
569C2×C4#1313317711yescharacteristic
570C2×C4#1313317711yescharacteristic
571C2×C4C4×C22⋊D5311yescharacteristic
572C2×C4C4×C22⋊D5311yescharacteristic
573C2×C4#1313317911yescharacteristic
574C2×C4#1313317911yescharacteristic
575C2×C4#1313317911yescharacteristic
576C2×C4C4×D21211yescharacteristic
577C2×C4C4×D21211yescharacteristic
578C2×C4#1313318911yescharacteristic
579C2×C4#1313319011yescharacteristic
580C2×C4#1313319211yescharacteristic
581C2×C4#1313319211yescharacteristic
582C2×C4#1313320111yescharacteristic
583C2×C4#1313320211yescharacteristic
584C2×C4#1313320311yescharacteristic
585C2×C4#1313320311yescharacteristic
586C2×C4#1313320611yescharacteristic
587C2×C4#1313321211yescharacteristic
588C2×C4#1313321211yescharacteristic
589C2×C4#1313321311yescharacteristic
590C8—42no—
591C8—2122no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—4242no—
603C22—4242no—
604C22—4241no—
605C22—4241no—
606C22—2122no—
607C22#1681047011yescharacteristic
608C22#1681050011yescharacteristic
609C22#1681050111yescharacteristic
610C22#1681050111yescharacteristic
611C22#1681051211yescharacteristic
612C22#1681051311yescharacteristic
613C22#1681052311yescharacteristic
614C4—44no—
615C4—82no—
616C4—2124no—
617C4—2124no—
618C4—2122no—
619C4#1681047011yescharacteristic
620C4C4×#1171992111yescharacteristic
621C4#1681051211yescharacteristic
622C4#1681051311yescharacteristic
623C4#1681051511yescharacteristic
624C4#1681051511yescharacteristic
625C4#1681052311yescharacteristic
626C4#1681052411yescharacteristic
627C2—21no—
628C2—21no—
629C2—21no—
630C2—21no—
631C2—21no—
632C2—21no—
633C2—4242no—
634C2#2027295111yescharacteristic
635C2#2027520011yescharacteristic
636C2#2027520711yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC2×C4×C53C2×C42
FrattiniC23×C4C22×D53
FittingC53×#1736C2
socleC22×C53#386
prime radicalC22×#11719945C22
cosocleC23×C4×C53C23
hypercenter#1736D53
hypocenterC53#10954
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
4. #20277072C2semidirect
5. C2×#16810395C2semidirect
6. C2×#16810379C2semidirect
7. C53×#1736C2semidirect
13. #16810639C4semidirect
17. C2×#13133140C4semidirect
18. C2×#13133137C4semidirect
20. #16810596C4semidirect
147. C2×D212C42semidirect
159. C2×Dic106C42semidirect
314. #1736D53semidirect
349. C53#10954semidirect

Maximal subgroups

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

subgroupindex
C2×#168106392
C2×#168106222
C2×#168105962
#202770722
C2×#168103952
C2×#168103792
C53×#17362
#1095453