Finite Group Databasebeta

← #27401326

Subgroups — #27401326

Order 13568.

1010 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810625C211yesmaximal, characteristic
2#20277590C211yesmaximal, characteristic, complemented
3#20277546C211yesmaximal, characteristic
4#20275405C211yesmaximal, characteristic, complemented
5#20275391C211yesmaximal, characteristic, complemented
6#20275357C211yesmaximal, characteristic, complemented
7#20275350C211yesmaximal, characteristic, complemented
8#20274979C211yesmaximal, characteristic, complemented
9#20274965C211yesmaximal, characteristic, complemented
10#20274802C211yesmaximal, characteristic, complemented
11#20274748C211yesmaximal, characteristic, complemented
12#20274656C211yesmaximal, characteristic
13#20274541C211yesmaximal, characteristic, complemented
14C4×#13133139C211yesmaximal, characteristic, complemented
15C53×#1753C211yesmaximal, characteristic, complemented
16C22×#11719945C2211yescharacteristic
17C2×C4×D212C2211yescharacteristic
18C2×C4×D212C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20C2×#13133151C2211yescharacteristic
21C2×#13133149C2211yescharacteristic
22#16810625C418yescomplemented
23C2×#13133142C2211yescharacteristic
24C2×#13133142C2211yescharacteristic
25C2×#13133142C2211yescharacteristic
26C2×#13133142C2211yescharacteristic
27C2×#13133139C2211yescharacteristic
28C2×#13133139C2211yescharacteristic
29C2×#13133139C2211yescharacteristic
30C2×#13133139C2211yescharacteristic
31#16810529C2211yescharacteristic
32#16810529C2211yescharacteristic
33#16810521C2211yescharacteristic
34#16810521C2211yescharacteristic
35#16810506—21no—
36#16810506—21no—
37#16810499—21no—
38#16810499—21no—
39#16810485—21no—
40#16810483C2211yescharacteristic
41#16810483C2211yescharacteristic
42#16810481—21no—
43C4×C4⋊Dic53C2211yescharacteristic
44C4×C4⋊Dic53C2211yescharacteristic
45C22⋊C4×Dic53—21no—
46#95×D53—21no—
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53C22×M16×C53C2211yescharacteristic
54C2×C22⋊C8×C53C2211yescharacteristic
55C2×#97×C53C2211yescharacteristic
56C2×C4×C8×C53C2211yescharacteristic
57C4×C22⋊C4×C53C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C2×#11719945—22no—
61C2×#11719945—22no—
62C2×#11719945—22no—
63C2×#11719945—41no—
64C2×#11719945—41no—
65C2×#11719945C2×C414yes—
66C22×Dic106C2311yescharacteristic
67C22×D212C2311yescharacteristic
68C22×C22⋊D53—21no—
69C23×Dic53—21no—
70C22×C4×D53—21no—
71C22×C4×D53—21no—
72C22×C4×D53—21no—
73C4×D212—22no—
74C4×D212—22no—
75C4×D212—41no—
76C4×D212—41no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53—22no—
79C2×C4⋊Dic53C2×C414yescomplemented
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×C4⋊Dic53C2311yescharacteristic
84C2×C4⋊Dic53C2311yescharacteristic
85C2×C4⋊Dic53C2311yescharacteristic
86C2×#11719919—21no—
87C2×#11719919—21no—
88C2×C22⋊Dic53—21no—
89C2×C22⋊Dic53—21no—
90C2×#11719921—21no—
91C2×#11719921—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53—21no—
94#13133151—41no—
95#13133149—41no—
96#13133142—41no—
97#13133142—41no—
98#13133142C2×C414yescomplemented
99#13133142C2×C414yescomplemented
100#13133139—22no—
101#13133139—22no—
102#13133139C2×C414yes—
103#13133139C2×C414yes—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112#13133124—21no—
113#13133124—21no—
114C23×C4×C53C2311yescharacteristic
115C2×M16×C53—41no—
116C2×M16×C53—41no—
117C2×M16×C53C2×C414yescomplemented
118C22×C8×C53C2311yescharacteristic
119C22×C8×C53C2311yescharacteristic
120C22×C8×C53C2311yescharacteristic
121C22×C8×C53C2311yescharacteristic
122C2×C22⋊C4×C53D411yescharacteristic
123C2×C22⋊C4×C53D411yescharacteristic
124C2×C42×C53C2311yescharacteristic
125C2×C42×C53C2311yescharacteristic
126C22⋊C8×C53—22no—
127#97×C53—41no—
128C4×C8×C53—41no—
129#95×C53D411yescharacteristic
130#95×C53D411yescharacteristic
131C24×C53C2×D411yescharacteristic
132C23×D53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53C22×C412yes—
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53#4112yes—
144C22×C4×C53C2411yescharacteristic
145C2×C22⋊D53—24no—
146C2×C22⋊D53—42no—
147C2×C22⋊D53—42no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—24no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—81no—
157C22×Dic53—81no—
158#11719945—28no—
159#11719945—44no—
160#11719945—44no—
161#11719945—44no—
162C2×D212—21no—
163C2×D212—21no—
164C2×D212—21no—
165C2×D212C22×C412yes—
166C2×D212#4112yes—
167C2×D212#4112yes—
168C2×C4×D53—24no—
169C2×C4×D53—42no—
170C2×C4×D53—42no—
171C2×C4×D53—42no—
172C2×C4×D53—42no—
173C2×C4×D53—42no—
174C2×C4×D53—42no—
175C2×C4×D53—81no—
176C2×C4×D53—81no—
177C2×Dic106—21no—
178C2×Dic106—21no—
179C2×Dic106—21no—
180C2×Dic106—21no—
181C2×Dic106—21no—
182C2×Dic106C22×C412yes—
183M16×C53—44no—
184C2×C8×C53—21no—
185C2×C8×C53—21no—
186C2×C8×C53—21no—
187C2×C8×C53—21no—
188C2×C8×C53—21no—
189C2×C8×C53—21no—
190C2×C8×C53—21no—
191C2×C8×C53—21no—
192C2×C8×C53—21no—
193C2×C8×C53—21no—
194C2×C8×C53C22×C412yes—
195C2×C8×C53C22×C412yes—
196C22⋊C4×C53—22no—
197C22⋊C4×C53—41no—
198C42×C53—21no—
199C42×C53—21no—
200C42×C53—21no—
201C42×C53—21no—
202C22⋊Dic53—42no—
203C22⋊Dic53—81no—
204#11719921—42no—
205#11719921—42no—
206C4⋊Dic53—24no—
207C4⋊Dic53—21no—
208C4⋊Dic53—21no—
209C4⋊Dic53—21no—
210C4⋊Dic53—21no—
211C4⋊Dic53—21no—
212C4⋊Dic53—21no—
213C4⋊Dic53—21no—
214C4⋊Dic53—21no—
215C4⋊Dic53—21no—
216C4⋊Dic53—21no—
217C4⋊Dic53C22×C412yes—
218C4⋊Dic53C22×C412yes—
219#11719919—81no—
220#11719919—81no—
221C4×Dic53—81no—
222C4×Dic53—81no—
223C23×C53—21no—
224C23×C53—21no—
225C23×C53—21no—
226C23×C53—21no—
227C23×C53—21no—
228C23×C53—21no—
229C23×C53C4×D412yes—
230C23×C53C22×D411yescharacteristic
231C22×D53—22no—
232C22×D53—22no—
233C22×D53—22no—
234C22×D53—41no—
235C22×D53—41no—
236C22×D53—41no—
237C22×D53—41no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—22no—
243C2×C4×C53—22no—
244C2×C4×C53—22no—
245C2×C4×C53—22no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—21no—
249C2×C4×C53—21no—
250C2×C4×C53—21no—
251C2×C4×C53—21no—
252C2×C4×C53—21no—
253C2×C4×C53—21no—
254C2×C4×C53—21no—
255C2×C4×C53C4×D412yes—
256C2×C4×C53#13712yes—
257C2×C4×C53C23×C411yescharacteristic
258C2×C4×C53C2×#4111yescharacteristic
259C2×C4×C53C2×#4111yescharacteristic
260C2×C4×C53C2×#4111yescharacteristic
261C2×C4×C53H(5,𝔽2)11yescharacteristic
262C2×C4×C53H(5,𝔽2)11yescharacteristic
263C22⋊D53—48no—
264C2×Dic53—22no—
265C2×Dic53—22no—
266C2×Dic53—22no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
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—41no—
285C2×Dic53—84no—
286D212—22no—
287D212—22no—
288D212—22no—
289D212#11714yes—
290C4×D53—48no—
291C4×D53—84no—
292C4×D53—84no—
293Dic106—22no—
294Dic106—22no—
295Dic106—41no—
296Dic106—41no—
297C8×C53—22no—
298C8×C53—22no—
299C8×C53—41no—
300C8×C53—41no—
301#22578—531nomaximal, Sylow
302C22×C53—22no—
303C22×C53—22no—
304C22×C53—22no—
305C22×C53—41no—
306C22×C53—41no—
307C22×C53—41no—
308C22×C53—41no—
309C22×C53C2×C4×D411yescharacteristic
310C22×C53#54611yescharacteristic
311C22×C53#54611yescharacteristic
312C22×C53C2×#13711yescharacteristic
313C22×C53#57511yescharacteristic
314C22×C53#57611yescharacteristic
315C22×C53#57711yescharacteristic
316C2×D53—24no—
317C2×D53—42no—
318C2×D53—42no—
319C2×D53—42no—
320C2×D53—42no—
321C2×D53—42no—
322C2×D53—42no—
323C4×C53—22no—
324C4×C53—22no—
325C4×C53—41no—
326C4×C53—41no—
327C4×C53—41no—
328C4×C53C2×#11711yescharacteristic
329C4×C53#51811yescharacteristic
330C4×C53#55111yescharacteristic
331C4×C53#55111yescharacteristic
332Dic53—42no—
333Dic53—42no—
334Dic53—42no—
335Dic53—81no—
336Dic53—81no—
337Dic53—81no—
338Dic53—81no—
339C2×#417—531no—
340#2345—531no—
341#2339—531no—
342#1935—531no—
343#1933—531no—
344#1928—531no—
345#1927—531no—
346#1864—531no—
347#1862—531no—
348#1823—531no—
349#1809—531no—
350#1788—531no—
351#1762—531no—
352C4×#102—531no—
353#1753D5311yescharacteristic, complemented
354C2×C53—42no—
355C2×C53#234211yescharacteristic
356C2×C53C4×#13711yescharacteristic
357C2×C53#294611yescharacteristic
358C2×C53#330711yescharacteristic
359C2×C53#331411yescharacteristic
360C2×C53#332211yescharacteristic
361C2×C53#333111yescharacteristic
362D53—44no—
363C22×#41—531no—
364C22×M16C2×D5311yescharacteristic
365C2×C4×D4—531no—
366C2×C4×D4—531no—
367C22×C4⋊C4—531no—
368C2×#107—531no—
369C2×#106—531no—
370#417—538no—
371C2×#103—531no—
372C2×#103—531no—
373C2×#103—531no—
374C2×#103—531no—
375C2×#102—531no—
376C2×#102—531no—
377C2×#102—531no—
378C2×#102—531no—
379C2×C22⋊C8C2×D5311yescharacteristic
380C2×#97C2×D5311yescharacteristic
381C2×C4×C8C2×D5311yescharacteristic
382#391—531no—
383#391—531no—
384#389—531no—
385#389—531no—
386#387—1061no—
387#387—1061no—
388#386—1061no—
389#386—1061no—
390#381—1061no—
391#380—531no—
392#380—531no—
393#380—1061no—
394C4×C4⋊C4—531no—
395C4×C4⋊C4—531no—
396C4×C22⋊C4—1061no—
397C4×C22⋊C4C2×D5311yescharacteristic
398C2×#95—1061no—
399#340—531no—
400#340—531no—
401#340—531no—
402#340—531no—
403#340—531no—
404#340—531no—
405#336C2×D5311yescharacteristic
406#336C2×D5311yescharacteristic
407C53#2257811yescharacteristic, complemented, Sylow
408C2×#41—534no—
409C2×#41—1062no—
410C2×#41—1062no—
411C2×#41—1062no—
412C2×#41—2121no—
413C2×#41—2121no—
414C22×Q8—531no—
415C22×D4—531no—
416C22×D4—1061no—
417C23×C4—1061no—
418C23×C4—1061no—
419C23×C4—1061no—
420C23×C4—1061no—
421C23×C4C22×D5311yescharacteristic
422C2×M16—41no—
423C2×M16—41no—
424C2×M16C4×D5314yescomplemented
425C22×C8C22×D5311yescharacteristic
426C22×C8C22×D5311yescharacteristic
427C22×C8C22×D5311yescharacteristic
428C22×C8C22×D5311yescharacteristic
429C4×D4—1062no—
430C4×D4—1062no—
431C4×D4—2121no—
432C4×D4—2121no—
433C2×C4⋊C4—534no—
434C2×C4⋊C4—531no—
435C2×C4⋊C4—531no—
436C2×C4⋊C4—531no—
437C2×C4⋊C4—531no—
438C2×C4⋊C4—531no—
439C2×C4⋊C4—531no—
440C2×C4⋊C4—1061no—
441C2×C4⋊C4—1062no—
442C2×C4⋊C4—1061no—
443C2×C4⋊C4—1061no—
444C2×C22⋊C4—1061no—
445C2×C22⋊C4—1061no—
446C2×C22⋊C4—1061no—
447C2×C22⋊C4—1061no—
448C2×C22⋊C4D21211yescharacteristic
449C2×C22⋊C4D21211yescharacteristic
450C2×C42—1061no—
451C2×C42—1061no—
452C2×C42C22×D5311yescharacteristic
453C2×C42C22×D5311yescharacteristic
454#107—2121no—
455#106—2121no—
456#103—534no—
457#103—534no—
458#103—2121no—
459#103—2121no—
460#102—534no—
461#102—534no—
462#102—1062no—
463#102—1062no—
464C22⋊C8—22no—
465#97—41no—
466C4×C8—41no—
467#95—1061no—
468#95—1061no—
469#95—1061no—
470#95—1061no—
471#95—1061no—
472#95—1061no—
473#95—1061no—
474#95—1061no—
475#95—1061no—
476#95—1061no—
477#95D21211yescharacteristic
478#95D21211yescharacteristic
479C24—1061no—
480C24C2×D21211yescharacteristic
481#41—1068no—
482#41—2124no—
483#41—2124no—
484#41—2124no—
485C2×Q8—532no—
486C2×Q8—1061no—
487C2×Q8—1061no—
488C2×Q8—1061no—
489C2×Q8—1061no—
490C2×Q8—1061no—
491C2×D4—532no—
492C2×D4—532no—
493C2×D4—532no—
494C2×D4—1064no—
495C2×D4—1061no—
496C2×D4—1061no—
497C2×D4—1061no—
498C2×D4—2122no—
499C2×D4—2122no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—1064no—
505C22×C4—1061no—
506C22×C4—1061no—
507C22×C4—1061no—
508C22×C4—1064no—
509C22×C4—1061no—
510C22×C4—1061no—
511C22×C4—1061no—
512C22×C4—1061no—
513C22×C4—4241no—
514C22×C4—4241no—
515C22×C4—4241no—
516C22×C4—4241no—
517C22×C4—2122no—
518C22×C4—2122no—
519C22×C4—2122no—
520C22×C4—2122no—
521C22×C4—2122no—
522C22×C4—2122no—
523C22×C4C2×C4×D5312yes—
524C22×C4C2×D21211yescharacteristic
525C22×C4C2×D21211yescharacteristic
526C22×C4C2×D21211yescharacteristic
527C22×C4C2×D21211yescharacteristic
528C22×C4C2×D21211yescharacteristic
529C22×C4#1171994512yes—
530C22×C4C23×D5311yescharacteristic
531M16—44no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8C2×C4×D5312yes—
543C2×C8C2×C4×D5312yes—
544C4⋊C4—532no—
545C4⋊C4—532no—
546C4⋊C4—1064no—
547C4⋊C4—1061no—
548C4⋊C4—1061no—
549C4⋊C4—1061no—
550C4⋊C4—1061no—
551C4⋊C4—1061no—
552C4⋊C4—1061no—
553C4⋊C4—1061no—
554C4⋊C4—1061no—
555C4⋊C4—1061no—
556C4⋊C4—1061no—
557C4⋊C4—4241no—
558C4⋊C4—4241no—
559C22⋊C4—22no—
560C22⋊C4—41no—
561C22⋊C4—4241no—
562C22⋊C4—2122no—
563C22⋊C4—2122no—
564C22⋊C4—2122no—
565C42—21no—
566C42—21no—
567C42—21no—
568C42—21no—
569C42—4241no—
570C42—4241no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—1062no—
578C23—1062no—
579C23—1062no—
580C23—2121no—
581C23—2121no—
582C23—2121no—
583C23—2121no—
584C23C4×D21212yes—
585C23C22×D21211yescharacteristic
586Q8—1062no—
587Q8—1062no—
588Q8—2121no—
589Q8—2121no—
590D4—534no—
591D4—1062no—
592D4—1062no—
593D4—1062no—
594D4—2128no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—1062no—
613C2×C4—1062no—
614C2×C4—1062no—
615C2×C4—4244no—
616C2×C4—4244no—
617C2×C4—4244no—
618C2×C4—2128no—
619C2×C4—2121no—
620C2×C4—2121no—
621C2×C4—2121no—
622C2×C4—2121no—
623C2×C4—2121no—
624C2×C4—2121no—
625C2×C4—2121no—
626C2×C4—2121no—
627C2×C4—2121no—
628C2×C4—2121no—
629C2×C4—2121no—
630C2×C4—2121no—
631C2×C4—2121no—
632C2×C4—2121no—
633C2×C4—2121no—
634C2×C4—2121no—
635C2×C4—2121no—
636C2×C4—2121no—
637C2×C4C4×D21212yes—
638C2×C4#1313325012yes—
639C2×C4C22×C4×D5311yescharacteristic
640C2×C4#41×D5311yescharacteristic
641C2×C4#41×D5311yescharacteristic
642C2×C4C2×#1171994511yescharacteristic
643C2×C4#1313326211yescharacteristic
644C2×C4#1313326211yescharacteristic
645C8—22no—
646C8—22no—
647C8—41no—
648C8—41no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—41no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—1064no—
657C22—2122no—
658C22—2122no—
659C22—2122no—
660C22—2122no—
661C22—2122no—
662C22—2122no—
663C22C2×C4×D21211yescharacteristic
664C22#1681126811yescharacteristic
665C22#1681126811yescharacteristic
666C22C2×#1313325011yescharacteristic
667C22#1681142311yescharacteristic
668C22#1681142811yescharacteristic
669C22#1681143511yescharacteristic
670C4—22no—
671C4—22no—
672C4—41no—
673C4—41no—
674C4—41no—
675C4—4241no—
676C4—4241no—
677C4—4241no—
678C4—4241no—
679C4—2122no—
680C4—2122no—
681C4—2122no—
682C4#117×D5311yescharacteristic
683C4#1681109611yescharacteristic
684C4#1681130211yescharacteristic
685C4#1681130211yescharacteristic
686C2—42no—
687C2—2124no—
688C2#2027756711yescharacteristic
689C2C4×#1313325011yescharacteristic
690C2#2028354911yescharacteristic
691C2#2028669311yescharacteristic
692C2#2028679811yescharacteristic
693C2#2028691811yescharacteristic
694C2#2028703211yescharacteristic
subgroupG/H
centerC23C22×D212
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1753C2
socleC23×C53C22×D4
prime radicalC22×#11719945C22
cosocleC22×C4×C53C24
hypercenter#1753D53
hypocenterC53#22578
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
2. #20277590C2semidirect
4. #20275405C2semidirect
5. #20275391C2semidirect
6. #20275357C2semidirect
7. #20275350C2semidirect
8. #20274979C2semidirect
9. #20274965C2semidirect
10. #20274802C2semidirect
11. #20274748C2semidirect
13. #20274541C2semidirect
14. C4×#13133139C2semidirect
15. C53×#1753C2semidirect
22. #16810625C4semidirect
79. C2×C4⋊Dic53C2×C4semidirect
98. #13133142C2×C4semidirect
99. #13133142C2×C4semidirect
117. C2×M16×C53C2×C4semidirect
353. #1753D53semidirect
407. C53#22578semidirect
424. C2×M16C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168106252
#202775902
#202775462
#202754052
#202753912
#202753572
#202753502
#202749792
#202749652
#202748022
#202747482
#202746562
#202745412
C4×#131331392
C53×#17532
#2257853