Finite Group Databasebeta

← #27411715

Subgroups — #27411715

Order 13568.

1422 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810666C211yesmaximal, characteristic, complemented
2C2×#16810640C211yesmaximal, characteristic, complemented
3C2×#16810624C211yesmaximal, characteristic, complemented
4C2×#16810607C211yesmaximal, characteristic, complemented
5#20277307C211yesmaximal, characteristic, complemented
6#20275929C214yesmaximal, complemented
7#20275922C214yesmaximal, complemented
8#20274424C211yesmaximal, characteristic, complemented
9C53×#3449C211yesmaximal, characteristic
10C22×#11719917C2211yescharacteristic
11C22×#11719917C2211yescharacteristic
12C2×#13133176C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14#16810666—22no—
15#16810666C2214yescomplemented
16C2×#13133150—21no—
17C2×#13133148—21no—
18#16810640—24no—
19C2×#13133144C2214yes—
20#16810624—22no—
21#16810624C2214yes—
22C2×#13133141—21no—
23C2×#13133138—21no—
24#16810607—24no—
25C2×#13133136C2212yescomplemented
26C2×#13133134C2212yescomplemented
27#16810530C2212yescomplemented
28#16810526C2212yescomplemented
29#16810503—22no—
30#16810483—22no—
31#16810382C2214yescomplemented
32C22×#41×C53C2211yescharacteristic
33C22×#41×C53C2211yescharacteristic
34C53×#537—21no—
35C53×#537—21no—
36C53×#536—21no—
37C53×#536C2212yes—
38C53×#535—21no—
39C53×#535C2212yes—
40C53×#534—22no—
41C53×#532—22no—
42C53×#525—22no—
43C53×#525—22no—
44C2×#117×C53C2211yescharacteristic
45C23×Dic53—21no—
46C2×#11719917—21no—
47C2×#11719917—21no—
48C2×#11719917—22no—
49C2×#11719917—22no—
50C2×#11719917—22no—
51C2×#11719917D414yescomplemented
52C2×#11719917C2312yes—
53C2×#11719917C2312yes—
54C22×#88322—21no—
55C22×#88322—21no—
56#13133176—22no—
57#13133176—22no—
58C2×C4⋊Dic53—21no—
59C2×C4⋊Dic53—21no—
60C2×C4⋊Dic53—21no—
61C2×C4⋊Dic53—21no—
62C2×C4⋊Dic53—21no—
63C2×C4⋊Dic53—21no—
64C2×C4⋊Dic53C2312yes—
65C2×C22⋊Dic53—21no—
66C2×C22⋊Dic53—24no—
67C2×C4×Dic53C2312yes—
68#13133150—24no—
69#13133148—24no—
70#13133144—28no—
71#13133141—24no—
72#13133138—24no—
73#13133136—24no—
74#13133134—24no—
75#13133124—24no—
76C2×#41×C53—21no—
77C2×#41×C53—21no—
78C2×#41×C53—21no—
79C2×#41×C53—21no—
80C2×#41×C53—21no—
81C2×#41×C53—21no—
82C2×#41×C53—22no—
83C2×#41×C53—22no—
84C2×#41×C53—22no—
85C2×#41×C53—22no—
86C2×#41×C53—22no—
87C2×#41×C53—22no—
88C2×#41×C53—22no—
89C2×#41×C53—22no—
90C2×#41×C53D412yes—
91C2×#41×C53C2312yes—
92C22×Q8×C53D411yescharacteristic
93C22×Q8×C53C2311yescharacteristic
94C22×D4×C53—21no—
95C22×D4×C53D412yes—
96C22×D4×C53D411yescharacteristic
97C22×D4×C53C2311yescharacteristic
98C23×C4×C53—21no—
99C23×C4×C53D412yes—
100C23×C4×C53C2311yescharacteristic
101C4⋊Q8×C53—22no—
102#127×C53—22no—
103#124×C53—22no—
104#124×C53—22no—
105#123×C53—24no—
106#123×C53—24no—
107C22⋊Q8×C53—22no—
108C22⋊Q8×C53—22no—
109C22⋊Q8×C53—22no—
110C22⋊Q8×C53—22no—
111#121×C53—22no—
112#121×C53—22no—
113#121×C53—24no—
114#121×C53—24no—
115#121×C53—22no—
116#121×C53—22no—
117#120×C53—24no—
118#120×C53—24no—
119C4×D4×C53—24no—
120C4×D4×C53—24no—
121#117×C53—21no—
122#117×C53—21no—
123#117×C53D412yes—
124#117×C53C2312yes—
125C2×C4⋊C4×C53D411yescharacteristic
126C2×C4⋊C4×C53D411yescharacteristic
127C2×C22⋊C4×C53—21no—
128C2×C42×C53—21no—
129C24×C53—21no—
130C24×C53C2×D412yes—
131C24×C53C2×D411yescharacteristic
132#41×C53—24no—
133#41×C53—24no—
134#41×C53—22no—
135#41×C53—22no—
136#41×C53—22no—
137#41×C53—22no—
138#41×C53—42no—
139#41×C53—42no—
140#41×C53—42no—
141#41×C53—42no—
142#41×C53—42no—
143#41×C53—42no—
144#41×C53—42no—
145#41×C53—42no—
146#41×C53—42no—
147#41×C53—42no—
148#41×C53—42no—
149#41×C53—42no—
150C2×Q8×C53—21no—
151C2×Q8×C53—21no—
152C2×Q8×C53—21no—
153C2×Q8×C53—21no—
154C2×Q8×C53—21no—
155C2×Q8×C53—21no—
156C2×Q8×C53—21no—
157C2×Q8×C53—22no—
158C2×Q8×C53—22no—
159C2×Q8×C53C2×D412yes—
160C2×D4×C53—22no—
161C2×D4×C53—22no—
162C2×D4×C53—22no—
163C2×D4×C53—24no—
164C2×D4×C53—24no—
165C2×D4×C53—22no—
166C2×D4×C53—24no—
167C2×D4×C53—24no—
168C2×D4×C53—21no—
169C2×D4×C53—22no—
170C2×D4×C53—22no—
171C2×D4×C53—21no—
172C2×D4×C53—21no—
173C2×D4×C53—21no—
174C2×D4×C53—21no—
175C2×D4×C53—21no—
176C2×D4×C53—21no—
177C2×D4×C53—42no—
178C2×D4×C53—42no—
179C2×D4×C53C2×D412yes—
180C22×C4×C53—21no—
181C22×C4×C53—21no—
182C22×C4×C53—21no—
183C22×C4×C53—22no—
184C22×C4×C53—22no—
185C22×C4×C53—22no—
186C22×C4×C53—24no—
187C22×C4×C53—24no—
188C22×C4×C53—22no—
189C22×C4×C53—42no—
190C22×C4×C53—42no—
191C22×C4×C53C2×D412yes—
192C22×C4×C53C2×D411yescharacteristic
193C22×C4×C53C2×D411yescharacteristic
194C22×C4×C53C2×D411yescharacteristic
195C22×C4×C53C2×D411yescharacteristic
196C22×C4×C53C2×D411yescharacteristic
197C22×C4×C53C2×D412yes—
198C22×C4×C53#4112yes—
199C22×C4×C53#4112yes—
200C22×C4×C53#4112yes—
201C22×C4×C53C2411yescharacteristic
202C22×Dic53—22no—
203C22×Dic53—22no—
204C22×Dic53—41no—
205C22×Dic53—41no—
206C22×Dic53—82no—
207C4⋊C4×C53—24no—
208C4⋊C4×C53—24no—
209C4⋊C4×C53—21no—
210C4⋊C4×C53—21no—
211C4⋊C4×C53C2×D412yes—
212C4⋊C4×C53C2×D412yes—
213C22⋊C4×C53—24no—
214C22⋊C4×C53—24no—
215C22⋊C4×C53—24no—
216C22⋊C4×C53—24no—
217C22⋊C4×C53—22no—
218C22⋊C4×C53—22no—
219C42×C53—22no—
220C42×C53—22no—
221C22⋊Dic53—42no—
222C22⋊Dic53—84no—
223C4⋊Dic53—22no—
224C4⋊Dic53—22no—
225C4⋊Dic53—41no—
226C4⋊Dic53—41no—
227C4⋊Dic53—42no—
228C4×Dic53—22no—
229C4×Dic53—22no—
230#11719917—24no—
231#11719917—22no—
232#11719917—22no—
233#11719917—22no—
234#11719917—42no—
235#11719917C2×D414yes—
236C2×#88322—24no—
237C2×#88322—22no—
238C2×#88322—22no—
239C2×#88322—42no—
240C23×C53—22no—
241C23×C53—22no—
242C23×C53—22no—
243C23×C53—22no—
244C23×C53—41no—
245C23×C53—41no—
246C23×C53—42no—
247C23×C53—42no—
248C23×C53—42no—
249C23×C53—81no—
250C23×C53—81no—
251C23×C53#12011yescharacteristic
252C23×C53#12011yescharacteristic
253C23×C53#12112yes—
254C23×C53#12312yes—
255C23×C53C22×D411yescharacteristic
256Q8×C53—22no—
257Q8×C53—22no—
258Q8×C53—41no—
259Q8×C53—41no—
260Q8×C53—41no—
261Q8×C53—41no—
262Q8×C53—42no—
263D4×C53—22no—
264D4×C53—22no—
265D4×C53—42no—
266D4×C53—42no—
267D4×C53—44no—
268D4×C53—44no—
269D4×C53—44no—
270D4×C53—41no—
271D4×C53—41no—
272D4×C53—41no—
273D4×C53—41no—
274D4×C53—42no—
275D4×C53—84no—
276D4×C53—84no—
277C2×C4×C53—22no—
278C2×C4×C53—22no—
279C2×C4×C53—22no—
280C2×C4×C53—22no—
281C2×C4×C53—22no—
282C2×C4×C53—22no—
283C2×C4×C53—42no—
284C2×C4×C53—42no—
285C2×C4×C53—42no—
286C2×C4×C53—44no—
287C2×C4×C53—44no—
288C2×C4×C53—44no—
289C2×C4×C53—81no—
290C2×C4×C53—81no—
291C2×C4×C53#12011yescharacteristic
292C2×C4×C53#12011yescharacteristic
293C2×C4×C53#12112yes—
294C2×C4×C53#12112yes—
295C2×C4×C53#12112yes—
296C2×C4×C53#12112yes—
297C2×C4×C53#12112yes—
298C2×C4×C53#12312yes—
299C2×C4×C53#12312yes—
300C2×C4×C53#12312yes—
301C2×C4×C53#12412yes—
302C2×C4×C53#12412yes—
303C2×C4×C53C22×D411yescharacteristic
304C2×C4×C53C22×D411yescharacteristic
305C2×C4×C53C22×D411yescharacteristic
306C2×C4×C53C2×#4111yescharacteristic
307C2×C4×C53C2×#4111yescharacteristic
308C2×C4×C53C2×#4111yescharacteristic
309C2×Dic53—42no—
310C2×Dic53—44no—
311C2×Dic53—42no—
312C2×Dic53—42no—
313C2×Dic53—42no—
314C2×Dic53—82no—
315C2×Dic53—82no—
316#88322—24no—
317#88322—42no—
318#23495—531nomaximal, Sylow
319C22×C53—21no—
320C22×C53—21no—
321C22×C53—21no—
322C22×C53—21no—
323C22×C53—21no—
324C22×C53—21no—
325C22×C53—42no—
326C22×C53—42no—
327C22×C53—44no—
328C22×C53—42no—
329C22×C53—42no—
330C22×C53—42no—
331C22×C53—42no—
332C22×C53—42no—
333C22×C53—42no—
334C22×C53—82no—
335C22×C53—84no—
336C22×C53#39412yes—
337C22×C53#39412yes—
338C22×C53C2×#12011yescharacteristic
339C22×C53C2×#12111yescharacteristic
340C22×C53C2×#12311yescharacteristic
341C4×C53—42no—
342C4×C53—42no—
343C4×C53—42no—
344C4×C53—42no—
345C4×C53#39412yes—
346C4×C53#39412yes—
347C4×C53C2×#12111yescharacteristic
348C4×C53C2×#12111yescharacteristic
349C4×C53C2×#12311yescharacteristic
350C4×C53C2×#12411yescharacteristic
351Dic53—82no—
352Dic53—82no—
353#3449D5311yescharacteristic
354C2×#428—531no—
355C2×#421—531no—
356C2×#417—531no—
357C2×#413—531no—
358#2311—531no—
359#2038—534no—
360#2037—534no—
361#1733—531no—
362C2×C53—21no—
363C2×C53—21no—
364C2×C53—22no—
365C2×C53—22no—
366C2×C53—82no—
367C2×C53—82no—
368C2×C53C2×#39411yescharacteristic
369C2×C53#302111yescharacteristic
370C2×C53#302211yescharacteristic
371C22×#41C2×D5311yescharacteristic
372C22×#41C2×D5311yescharacteristic
373C22×M16—531no—
374C22×M16—531no—
375#537—21no—
376#537—21no—
377#536—21no—
378#536C2×D5312yes—
379#535—21no—
380#535C2×D5312yes—
381#534—22no—
382#532—22no—
383#525—22no—
384#525—22no—
385C2×#117—531no—
386C2×#117C2×D5311yescharacteristic
387C22×C4⋊C4—531no—
388#428—534no—
389#428—1062no—
390C2×#107—1061no—
391C2×#106—1061no—
392#421—1064no—
393C2×#104—534no—
394#417—534no—
395#417—1062no—
396C2×#103—1061no—
397C2×#102—1061no—
398#413—1064no—
399C2×#101—532no—
400C2×#100—532no—
401#391—532no—
402#390—532no—
403#386—1062no—
404#380—1062no—
405#339—534no—
406C53#2349511yescharacteristic, complemented, Sylow
407C2×#41—22no—
408C2×#41—22no—
409C2×#41—22no—
410C2×#41—22no—
411C2×#41—21no—
412C2×#41—21no—
413C2×#41—21no—
414C2×#41—21no—
415C2×#41—21no—
416C2×#41—21no—
417C2×#41—22no—
418C2×#41—22no—
419C2×#41—22no—
420C2×#41—22no—
421C2×#41C22⋊D5312yes—
422C2×#41C22×D5312yes—
423C22×Q8C22⋊D5311yescharacteristic
424C22×Q8C22×D5311yescharacteristic
425C22×D4—21no—
426C22×D4C22⋊D5312yes—
427C22×D4C22⋊D5311yescharacteristic
428C22×D4C22×D5311yescharacteristic
429C23×C4—21no—
430C23×C4—1061no—
431C23×C4C22⋊D5312yes—
432C23×C4C22×D5311yescharacteristic
433C2×M16—534no—
434C2×M16—532no—
435C2×M16—532no—
436C2×M16—1062no—
437C2×M16—1061no—
438C2×M16—1061no—
439C2×M16—1062no—
440C2×M16—1062no—
441C22×C8—1061no—
442C22×C8—1061no—
443C4⋊Q8—22no—
444#127—22no—
445#124—22no—
446#124—22no—
447#123—24no—
448#123—24no—
449C22⋊Q8—22no—
450C22⋊Q8—22no—
451C22⋊Q8—22no—
452C22⋊Q8—22no—
453#121—22no—
454#121—22no—
455#121—24no—
456#121—24no—
457#121—22no—
458#121—22no—
459#120—24no—
460#120—24no—
461C4×D4—24no—
462C4×D4—24no—
463#117—21no—
464#117—21no—
465#117—1062no—
466#117—1062no—
467#117C22⋊D5312yes—
468#117C22×D5312yes—
469C2×C4⋊C4—532no—
470C2×C4⋊C4—1061no—
471C2×C4⋊C4—1061no—
472C2×C4⋊C4—1061no—
473C2×C4⋊C4—1061no—
474C2×C4⋊C4—1061no—
475C2×C4⋊C4—1061no—
476C2×C4⋊C4C22⋊D5311yescharacteristic
477C2×C4⋊C4C22⋊D5311yescharacteristic
478C2×C22⋊C4—21no—
479C2×C22⋊C4—1064no—
480C2×C22⋊C4—1061no—
481C2×C42—21no—
482C2×C42—532no—
483#107—1064no—
484#106—1064no—
485#104—1068no—
486#103—1064no—
487#102—1064no—
488#101—1064no—
489#100—1064no—
490#95—1064no—
491C24—21no—
492C24C2×C22⋊D5312yes—
493C24C2×C22⋊D5311yescharacteristic
494#41—24no—
495#41—24no—
496#41—22no—
497#41—22no—
498#41—22no—
499#41—22no—
500#41—42no—
501#41—42no—
502#41—42no—
503#41—42no—
504#41—42no—
505#41—42no—
506#41—42no—
507#41—42no—
508#41—42no—
509#41—42no—
510#41—42no—
511#41—42no—
512C2×Q8—21no—
513C2×Q8—21no—
514C2×Q8—21no—
515C2×Q8—21no—
516C2×Q8—21no—
517C2×Q8—21no—
518C2×Q8—21no—
519C2×Q8—22no—
520C2×Q8—22no—
521C2×Q8C2×C22⋊D5312yes—
522C2×D4—22no—
523C2×D4—22no—
524C2×D4—22no—
525C2×D4—24no—
526C2×D4—24no—
527C2×D4—22no—
528C2×D4—24no—
529C2×D4—24no—
530C2×D4—21no—
531C2×D4—22no—
532C2×D4—22no—
533C2×D4—21no—
534C2×D4—21no—
535C2×D4—21no—
536C2×D4—21no—
537C2×D4—21no—
538C2×D4—21no—
539C2×D4—42no—
540C2×D4—42no—
541C2×D4C2×C22⋊D5312yes—
542C22×C4—21no—
543C22×C4—21no—
544C22×C4—21no—
545C22×C4—22no—
546C22×C4—22no—
547C22×C4—22no—
548C22×C4—24no—
549C22×C4—24no—
550C22×C4—22no—
551C22×C4—42no—
552C22×C4—42no—
553C22×C4—1062no—
554C22×C4—1062no—
555C22×C4—4242no—
556C22×C4—2121no—
557C22×C4—2121no—
558C22×C4D4×D5312yes—
559C22×C4#1171994712yes—
560C22×C4#1171994712yes—
561C22×C4#1171994712yes—
562C22×C4C2×C22⋊D5312yes—
563C22×C4C2×C22⋊D5311yescharacteristic
564C22×C4C2×C22⋊D5311yescharacteristic
565C22×C4C2×C22⋊D5311yescharacteristic
566C22×C4C2×C22⋊D5311yescharacteristic
567C22×C4C2×C22⋊D5311yescharacteristic
568C22×C4C23×D5311yescharacteristic
569M16—534no—
570M16—1064no—
571M16—1062no—
572M16—1062no—
573M16—1062no—
574M16—2122no—
575C2×C8—1064no—
576C2×C8—1062no—
577C2×C8—1062no—
578C2×C8—2122no—
579C4⋊C4—24no—
580C4⋊C4—24no—
581C4⋊C4—21no—
582C4⋊C4—21no—
583C4⋊C4—1062no—
584C4⋊C4—1062no—
585C4⋊C4—2121no—
586C4⋊C4—2121no—
587C4⋊C4—2122no—
588C4⋊C4C2×C22⋊D5312yes—
589C4⋊C4C2×C22⋊D5312yes—
590C22⋊C4—24no—
591C22⋊C4—24no—
592C22⋊C4—24no—
593C22⋊C4—24no—
594C22⋊C4—22no—
595C22⋊C4—22no—
596C22⋊C4—4244no—
597C22⋊C4—2122no—
598C42—22no—
599C42—22no—
600C42—1062no—
601C42—1062no—
602C23—22no—
603C23—22no—
604C23—22no—
605C23—22no—
606C23—41no—
607C23—41no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23—81no—
612C23—81no—
613C23#1313318611yescharacteristic
614C23#1313318611yescharacteristic
615C23#1313318912yes—
616C23#1313319812yes—
617C23C22×C22⋊D5311yescharacteristic
618Q8—22no—
619Q8—22no—
620Q8—42no—
621Q8—41no—
622Q8—41no—
623Q8—41no—
624Q8—41no—
625D4—22no—
626D4—22no—
627D4—44no—
628D4—44no—
629D4—44no—
630D4—42no—
631D4—42no—
632D4—42no—
633D4—41no—
634D4—41no—
635D4—41no—
636D4—41no—
637D4—84no—
638D4—84no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—81no—
652C2×C4—81no—
653C2×C4—4242no—
654C2×C4—4242no—
655C2×C4—2122no—
656C2×C4—2124no—
657C2×C4—2122no—
658C2×C4—2122no—
659C2×C4—2122no—
660C2×C4#1313318611yescharacteristic
661C2×C4#1313318611yescharacteristic
662C2×C4#1313318812yes—
663C2×C4#1313318812yes—
664C2×C4#1313318912yes—
665C2×C4#1313318912yes—
666C2×C4#1313318912yes—
667C2×C4#1313319812yes—
668C2×C4#1313319812yes—
669C2×C4#1313319812yes—
670C2×C4#1313320412yes—
671C2×C4#1313320412yes—
672C2×C4C2×D4×D5311yescharacteristic
673C2×C4C22×C22⋊D5311yescharacteristic
674C2×C4C22×C22⋊D5311yescharacteristic
675C2×C4C2×#1171994711yescharacteristic
676C2×C4C2×#1171994711yescharacteristic
677C2×C4C2×#1171994711yescharacteristic
678C8—1064no—
679C8—2122no—
680C22—21no—
681C22—21no—
682C22—21no—
683C22—21no—
684C22—21no—
685C22—21no—
686C22—42no—
687C22—42no—
688C22—44no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—42no—
693C22—42no—
694C22—42no—
695C22—82no—
696C22—84no—
697C22#1681053712yes—
698C22#1681053712yes—
699C22C2×#1313318611yescharacteristic
700C22C2×#1313318911yescharacteristic
701C22C2×#1313319811yescharacteristic
702C4—42no—
703C4—42no—
704C4—42no—
705C4—42no—
706C4—4242no—
707C4—4242no—
708C4#1681053712yes—
709C4#1681053712yes—
710C4C2×#1313318811yescharacteristic
711C4C2×#1313318911yescharacteristic
712C4C2×#1313319811yescharacteristic
713C4C2×#1313320411yescharacteristic
714C2—21no—
715C2—21no—
716C2—22no—
717C2—22no—
718C2—82no—
719C2—82no—
720C2C2×#1681053711yescharacteristic
721C2#2028408711yescharacteristic
722C2#2028409411yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC53×#3449C2
socleC22×C53C2×#120
hypercenter#3449D53
hypocenterC53#23495
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3449

Complements

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

NH ≅ G/Ntype
1. C2×#16810666C2semidirect
2. C2×#16810640C2semidirect
3. C2×#16810624C2semidirect
4. C2×#16810607C2semidirect
5. #20277307C2semidirect
6. #20275929C2semidirect
7. #20275922C2semidirect
8. #20274424C2semidirect
15. #16810666C22semidirect
25. C2×#13133136C22semidirect
26. C2×#13133134C22semidirect
27. #16810530C22semidirect
28. #16810526C22semidirect
31. #16810382C22semidirect
51. C2×#11719917D4semidirect
406. C53#23495semidirect

Maximal subgroups

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

subgroupindex
C2×#168106662
C2×#168106402
C2×#168106242
C2×#168106072
#202773072
#202759292
#202759222
#202744242
C53×#34492
#2349553