Finite Group Databasebeta

← #27390655

Subgroups — #27390655

Order 13568.

1014 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810862C211yesmaximal, characteristic
2C2×#16810727C211yesmaximal, characteristic, complemented
3C22×#13133127C211yesmaximal, characteristic
4C22⋊Q8×Dic53C211yesmaximal, characteristic
5#20277243C211yesmaximal, characteristic
6#20275583C211yesmaximal, characteristic, complemented
7#20275452C211yesmaximal, characteristic, complemented
8#20275406C211yesmaximal, characteristic, complemented
9#20275402C211yesmaximal, characteristic, complemented
10#20275138C211yesmaximal, characteristic
11#20274738C211yesmaximal, characteristic
12#20274611C211yesmaximal, characteristic, complemented
13#20274547C211yesmaximal, characteristic, complemented
14#20274544C211yesmaximal, characteristic, complemented
15C53×#1892C211yesmaximal, characteristic
16C22×#11719925C2211yescharacteristic
17C23×#88322C2211yescharacteristic
18C2×#13133196C2211yescharacteristic
19C2×Q8×Dic53C2211yescharacteristic
20C2×#13133182C2211yescharacteristic
21C22×C4×Dic53C2211yescharacteristic
22#16810862C418yescomplemented
23#16810727—24no—
24C2×#13133150C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×#13133142C2211yescharacteristic
28C2×#13133142C2211yescharacteristic
29C2×#13133141C2211yescharacteristic
30C2×#13133140C2211yescharacteristic
31C2×#13133127—24no—
32C2×#13133127C2211yescharacteristic
33C2×#13133127C2211yescharacteristic
34C2×#13133127C2211yescharacteristic
35C2×#13133127C2211yescharacteristic
36#16810530C2211yescharacteristic
37#16810528C2211yescharacteristic
38#16810522C2211yescharacteristic
39#16810519C2211yescharacteristic
40#16810509—21no—
41#16810505—21no—
42#16810498—21no—
43#16810495—21no—
44#16810483C2211yescharacteristic
45C4⋊C4×Dic53—21no—
46C4⋊C4×Dic53C2211yescharacteristic
47C4⋊C4×Dic53C2211yescharacteristic
48#16810474—21no—
49C22⋊C4×Dic53—21no—
50C22⋊C4×Dic53—21no—
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53#16810385C2211yescharacteristic
54#16810385C2211yescharacteristic
55C23×C8×C53C2211yescharacteristic
56C2×C22⋊Q8×C53C2211yescharacteristic
57C2×#103×C53C2211yescharacteristic
58C2×#103×C53C2211yescharacteristic
59C53×#380C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C53×#340C2211yescharacteristic
62C22×Dic106C2311yescharacteristic
63C23×Dic53D411yescharacteristic
64C23×Dic53D411yescharacteristic
65C2×#11719925—22no—
66C2×#11719925—22no—
67C2×#11719925C2×C414yescomplemented
68C22×#88322—22no—
69C22×#88322—22no—
70C22×#88322C2×C414yes—
71C22×#88322C2311yescharacteristic
72C22×#88322C2311yescharacteristic
73#13133196—22no—
74#13133196C2×C414yes—
75Q8×Dic53—22no—
76Q8×Dic53—22no—
77#13133182—22no—
78#13133182—22no—
79C2×C4⋊Dic53—21no—
80C2×C4⋊Dic53—21no—
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×C4⋊Dic53C2311yescharacteristic
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×#11719919—21no—
87C2×C22⋊Dic53—21no—
88C2×C22⋊Dic53—21no—
89C2×C4×Dic53—22no—
90C2×C4×Dic53—22no—
91C2×C4×Dic53—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53—21no—
94C2×C4×Dic53—21no—
95C2×C4×Dic53D411yescharacteristic
96C2×C4×Dic53D411yescharacteristic
97C2×C4×Dic53C2311yescharacteristic
98C2×C4×Dic53C2311yescharacteristic
99#13133150C2×C414yescomplemented
100#13133148—22no—
101#13133148—22no—
102#13133142—22no—
103#13133142—22no—
104#13133141C2×C414yescomplemented
105#13133140C2×C414yescomplemented
106#13133127—22no—
107#13133127—22no—
108#13133127—22no—
109#13133127D414yes—
110#13133124—21no—
111#13133124—21no—
112#13133124—21no—
113#13133124—21no—
114#13133124—21no—
115#13133124—21no—
116#13133124—21no—
117#13133124—21no—
118#13133124—21no—
119C22×Q8×C53C2311yescharacteristic
120C23×C4×C53C2311yescharacteristic
121C22×C8×C53—22no—
122C22×C8×C53—22no—
123C22×C8×C53—22no—
124C22×C8×C53C2311yescharacteristic
125C22×C8×C53C2311yescharacteristic
126C22⋊Q8×C53—22no—
127C22⋊Q8×C53C2×C414yes—
128C2×C4⋊C4×C53—21no—
129C2×C4⋊C4×C53C2311yescharacteristic
130C2×C4⋊C4×C53C2311yescharacteristic
131C2×C4⋊C4×C53C2311yescharacteristic
132C2×C22⋊C4×C53—21no—
133C2×C22⋊C4×C53—21no—
134#103×C53—22no—
135#103×C53—22no—
136#95×C53—21no—
137C24×C53C2×D411yescharacteristic
138C2×Q8×C53—21no—
139C2×Q8×C53—21no—
140C2×Q8×C53—21no—
141C2×Q8×C53—21no—
142C2×Q8×C53—21no—
143C2×Q8×C53C22×C412yes—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53C22×C412yes—
153C22×C4×C53C2×D411yescharacteristic
154C22×C4×C53#4112yes—
155C22×C4×C53C2411yescharacteristic
156C22×Dic53—22no—
157C22×Dic53—22no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—22no—
161C22×Dic53—22no—
162C22×Dic53—21no—
163C22×Dic53—21no—
164C22×Dic53—41no—
165C22×Dic53—41no—
166C22×Dic53C2×D411yescharacteristic
167C22×Dic53C2×D411yescharacteristic
168C22×Dic53C2×D411yescharacteristic
169C22×Dic53C2×D411yescharacteristic
170C2×Dic106—21no—
171C2×Dic106—21no—
172C2×Dic106—21no—
173C2×Dic106—21no—
174C2×Dic106—21no—
175C2×Dic106C22×C412yes—
176C2×C8×C53—28no—
177C2×C8×C53—21no—
178C2×C8×C53—21no—
179C2×C8×C53—21no—
180C2×C8×C53—21no—
181C2×C8×C53C2×D412yes—
182C2×C8×C53#4112yes—
183C4⋊C4×C53—24no—
184C4⋊C4×C53—21no—
185C4⋊C4×C53—21no—
186C4⋊C4×C53—21no—
187C4⋊C4×C53—21no—
188C4⋊C4×C53—21no—
189C4⋊C4×C53C22×C412yes—
190C22⋊C4×C53—24no—
191C22⋊C4×C53—42no—
192C22⋊Dic53—24no—
193C22⋊Dic53—42no—
194#11719925—28no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—42no—
201C4⋊Dic53—42no—
202C4⋊Dic53C22×C412yes—
203#11719919—24no—
204#11719919—42no—
205#11719919—42no—
206C4×Dic53—21no—
207C4×Dic53—21no—
208C4×Dic53—22no—
209C4×Dic53—22no—
210C4×Dic53—42no—
211C4×Dic53—42no—
212C4×Dic53—42no—
213C4×Dic53—42no—
214C4×Dic53C2×D412yes—
215C4×Dic53#4112yes—
216C2×#88322—21no—
217C2×#88322—21no—
218C2×#88322—28no—
219C2×#88322C22×C412yes—
220C2×#88322C22×C412yes—
221C2×#88322C2×D412yes—
222C2×#88322#4112yes—
223C23×C53—21no—
224C23×C53—21no—
225C23×C53—21no—
226C23×C53—21no—
227C23×C53C4×D412yes—
228C23×C53#13712yes—
229C23×C53#13712yes—
230C23×C53C22×D411yescharacteristic
231Q8×C53—22no—
232Q8×C53—22no—
233Q8×C53—41no—
234Q8×C53—41no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
242C2×C4×C53—22no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53C4×D412yes—
254C2×C4×C53C23×C411yescharacteristic
255C2×C4×C53C22×D411yescharacteristic
256C2×C4×C53C2×#4111yescharacteristic
257C2×C4×C53C2×#4111yescharacteristic
258C2×C4×C53C2×#4111yescharacteristic
259C2×C4×C53C2×#4111yescharacteristic
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—22no—
265C2×Dic53—22no—
266C2×Dic53—21no—
267C2×Dic53—21no—
268C2×Dic53—21no—
269C2×Dic53—21no—
270C2×Dic53—21no—
271C2×Dic53—21no—
272C2×Dic53—44no—
273C2×Dic53—44no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53—41no—
284C2×Dic53#12112yes—
285C2×Dic53#12112yes—
286Dic106—22no—
287Dic106—22no—
288Dic106—41no—
289Dic106—41no—
290C8×C53—22no—
291C8×C53—22no—
292#88322—22no—
293#88322C4×D414yes—
294#21527—531nomaximal, Sylow
295C22×C53—22no—
296C22×C53—22no—
297C22×C53—22no—
298C22×C53—22no—
299C22×C53—22no—
300C22×C53—22no—
301C22×C53#44114yescomplemented
302C22×C53C2×C4×D411yescharacteristic
303C22×C53C2×#12111yescharacteristic
304C22×C53#53211yescharacteristic
305C22×C53C2×#13711yescharacteristic
306C22×C53C2×#13711yescharacteristic
307C22×C53#57511yescharacteristic
308C22×C53#57511yescharacteristic
309C4×C53—22no—
310C4×C53—42no—
311C4×C53—42no—
312C4×C53—81no—
313C4×C53—81no—
314C4×C53C2×C4×D411yescharacteristic
315C4×C53C4×#4111yescharacteristic
316C4×C53#52511yescharacteristic
317C4×C53#52911yescharacteristic
318Dic53—22no—
319Dic53—22no—
320Dic53—42no—
321Dic53—41no—
322Dic53—41no—
323Dic53—42no—
324Dic53—81no—
325Dic53—81no—
326C2×#467—531no—
327C2×#441—531no—
328C22×#97—531no—
329C4×C22⋊Q8—531no—
330C4×C22⋊Q8—531no—
331#1971—531no—
332#1943—531no—
333#1935—531no—
334#1935—531no—
335#1892—531no—
336#1892D5311yescharacteristic
337#1807—531no—
338#1776—531no—
339#1763—531no—
340#1763—531no—
341C2×C53—24no—
342C2×C53C4×#12111yescharacteristic
343C2×C53C2×#44111yescharacteristic
344C2×C53#294311yescharacteristic
345C2×C53#305611yescharacteristic
346C2×C53#305611yescharacteristic
347C2×C53#318111yescharacteristic
348C2×C53#318111yescharacteristic
349C22×Q16—531no—
350C23×C8—531no—
351C23×C8C2×D5311yescharacteristic
352C2×C22⋊Q8—531no—
353C2×C22⋊Q8C2×D5311yescharacteristic
354C2×C4×Q8—531no—
355C2×C4×Q8—531no—
356C22×C42—531no—
357#467—538no—
358#441—1064no—
359C2×#107—531no—
360C2×#106—531no—
361C2×#106—531no—
362C2×#103—531no—
363C2×#103—531no—
364C2×#103—531no—
365C2×#103—531no—
366C2×#103C2×D5311yescharacteristic
367C2×#103C2×D5311yescharacteristic
368C2×#97—531no—
369C2×#97—531no—
370C2×#97—531no—
371C2×#97—531no—
372C2×#97—1064no—
373#391—531no—
374#391—531no—
375#389—531no—
376#389—531no—
377#387—1061no—
378#387—1061no—
379#385—1061no—
380#385—1061no—
381#380—531no—
382#380C2×D5311yescharacteristic
383C4×C4⋊C4—531no—
384C4×C4⋊C4—531no—
385C4×C4⋊C4—1061no—
386C4×C4⋊C4—1061no—
387C4×C22⋊C4—1061no—
388C4×C22⋊C4—1061no—
389#340—531no—
390#340—531no—
391#340—531no—
392#340—531no—
393#340C2×D5311yescharacteristic
394#340C2×D5311yescharacteristic
395C53#2152711yescharacteristic, complemented, Sylow
396C22×Q8—531no—
397C22×Q8C22×D5311yescharacteristic
398C23×C4—531no—
399C23×C4—531no—
400C23×C4C22×D5311yescharacteristic
401C2×Q16—534no—
402C2×Q16—1062no—
403C2×Q16—1062no—
404C22×C8—22no—
405C22×C8—22no—
406C22×C8—22no—
407C22×C8—531no—
408C22×C8—534no—
409C22×C8—531no—
410C22×C8—1062no—
411C22×C8—1062no—
412C22×C8C22×D5311yescharacteristic
413C22×C8C22×D5311yescharacteristic
414C22⋊Q8—22no—
415C22⋊Q8—534no—
416C22⋊Q8—1062no—
417C22⋊Q8C4×D5314yes—
418C4×Q8—1062no—
419C4×Q8—1062no—
420C4×Q8—1062no—
421C4×Q8—1062no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—531no—
424C2×C4⋊C4—531no—
425C2×C4⋊C4—531no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4—1061no—
429C2×C4⋊C4—1061no—
430C2×C4⋊C4—1061no—
431C2×C4⋊C4C22×D5311yescharacteristic
432C2×C4⋊C4C22×D5311yescharacteristic
433C2×C4⋊C4C22×D5311yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—1061no—
437C2×C22⋊C4—1061no—
438C2×C42—531no—
439C2×C42—531no—
440C2×C42—531no—
441C2×C42—531no—
442C2×C42—1062no—
443C2×C42—1062no—
444C2×C42—1061no—
445C2×C42—1061no—
446C2×C42—1061no—
447C2×C42—1061no—
448#107—534no—
449#106—1062no—
450#106—1062no—
451#103—22no—
452#103—22no—
453#103—534no—
454#103—534no—
455#103—1062no—
456#103—1062no—
457#97—534no—
458#97—1062no—
459#97—1062no—
460#97—1062no—
461#95—21no—
462#95—1061no—
463#95—1061no—
464#95—1061no—
465#95—1061no—
466#95—1061no—
467#95—1061no—
468#95—1061no—
469#95—1061no—
470#95—1061no—
471C24D4×D5311yescharacteristic
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—532no—
478C2×Q8—1061no—
479C2×Q8—1061no—
480C2×Q8—1061no—
481C2×Q8—1061no—
482C2×Q8—1061no—
483C2×Q8C2×C4×D5312yes—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—531no—
493C22×C4—531no—
494C22×C4—531no—
495C22×C4—531no—
496C22×C4—1062no—
497C22×C4—1062no—
498C22×C4—1062no—
499C22×C4—1062no—
500C22×C4—1061no—
501C22×C4—1061no—
502C22×C4—1061no—
503C22×C4—1061no—
504C22×C4—2121no—
505C22×C4—2121no—
506C22×C4C2×C4×D5312yes—
507C22×C4D4×D5311yescharacteristic
508C22×C4#1171994712yes—
509C22×C4C23×D5311yescharacteristic
510Q16—1068no—
511C2×C8—28no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—532no—
517C2×C8—532no—
518C2×C8—532no—
519C2×C8—532no—
520C2×C8—1068no—
521C2×C8—1061no—
522C2×C8—1061no—
523C2×C8D4×D5312yes—
524C2×C8#1171994712yes—
525C4⋊C4—24no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—532no—
532C4⋊C4—1064no—
533C4⋊C4—1061no—
534C4⋊C4—1061no—
535C4⋊C4—1061no—
536C4⋊C4—1061no—
537C4⋊C4—1061no—
538C4⋊C4—2122no—
539C4⋊C4—2122no—
540C4⋊C4—2122no—
541C4⋊C4—2122no—
542C4⋊C4C2×C4×D5312yes—
543C22⋊C4—24no—
544C22⋊C4—42no—
545C22⋊C4—1064no—
546C22⋊C4—2122no—
547C42—532no—
548C42—532no—
549C42—1062no—
550C42—1062no—
551C42—1061no—
552C42—1061no—
553C42—2122no—
554C42—2122no—
555C42—2122no—
556C42—2122no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23#1313317712yes—
562C23#1313324612yes—
563C23#1313324912yes—
564C23C2×D4×D5311yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—41no—
568Q8—41no—
569Q8—1062no—
570Q8—1062no—
571Q8—2121no—
572Q8—2121no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—532no—
592C2×C4—532no—
593C2×C4—1061no—
594C2×C4—1061no—
595C2×C4—1061no—
596C2×C4—1061no—
597C2×C4—1062no—
598C2×C4—1062no—
599C2×C4—1061no—
600C2×C4—1061no—
601C2×C4—1061no—
602C2×C4—1061no—
603C2×C4—1061no—
604C2×C4—1061no—
605C2×C4—2124no—
606C2×C4—2124no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4—2121no—
611C2×C4—2121no—
612C2×C4—2121no—
613C2×C4—2121no—
614C2×C4—2121no—
615C2×C4—2121no—
616C2×C4—2121no—
617C2×C4#1313317712yes—
618C2×C4C22×C4×D5311yescharacteristic
619C2×C4C2×D4×D5311yescharacteristic
620C2×C4#41×D5311yescharacteristic
621C2×C4#41×D5311yescharacteristic
622C2×C4C2×#1171994711yescharacteristic
623C2×C4C2×#1171994711yescharacteristic
624C8—22no—
625C8—22no—
626C8—534no—
627C8—1062no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22#1681072714yescomplemented
635C22C2×#1313317711yescharacteristic
636C22#121×D5311yescharacteristic
637C22#1681116811yescharacteristic
638C22C2×#1313324611yescharacteristic
639C22C2×#1313324911yescharacteristic
640C22#1681142011yescharacteristic
641C22#1681142011yescharacteristic
642C4—22no—
643C4—42no—
644C4—42no—
645C4—81no—
646C4—81no—
647C4—1062no—
648C4—1062no—
649C4—4241no—
650C4—4241no—
651C4—2122no—
652C4—2121no—
653C4—2121no—
654C4—2122no—
655C4C4×D4×D5311yescharacteristic
656C4C4×#1171994711yescharacteristic
657C4#1681112911yescharacteristic
658C4#1681115011yescharacteristic
659C2—24no—
660C2#2027722511yescharacteristic
661C2C2×#1681072711yescharacteristic
662C2#2028351111yescharacteristic
663C2#2028433811yescharacteristic
664C2#2028434211yescharacteristic
665C2#2028541711yescharacteristic
666C2#2028541911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1892C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
cosocleC22×C4×C53C24
hypercenter#1892D53
hypocenterC53#21527
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
2. C2×#16810727C2semidirect
6. #20275583C2semidirect
7. #20275452C2semidirect
8. #20275406C2semidirect
9. #20275402C2semidirect
12. #20274611C2semidirect
13. #20274547C2semidirect
14. #20274544C2semidirect
22. #16810862C4semidirect
67. C2×#11719925C2×C4semidirect
99. #13133150C2×C4semidirect
104. #13133141C2×C4semidirect
105. #13133140C2×C4semidirect
301. C22×C53#441semidirect
395. C53#21527semidirect
634. C22#16810727semidirect

Maximal subgroups

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

subgroupindex
C2×#168108622
C2×#168107272
C22×#131331272
C22⋊Q8×Dic532
#202772432
#202755832
#202754522
#202754062
#202754022
#202751382
#202747382
#202746112
#202745472
#202745442
C53×#18922
#2152753