Finite Group Databasebeta

← #27402435

Subgroups — #27402435

Order 13568.

1010 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810726C211yesmaximal, characteristic
2C2×#16810721C211yesmaximal, characteristic
3C2×#16810708C211yesmaximal, characteristic
4#20277636C211yesmaximal, characteristic, complemented
5#20277615C211yesmaximal, characteristic
6#20275452C211yesmaximal, characteristic, complemented
7#20275447C211yesmaximal, characteristic, complemented
8#20274981C211yesmaximal, characteristic, complemented
9#20274977C211yesmaximal, characteristic
10#20274967C211yesmaximal, characteristic, complemented
11#20274963C211yesmaximal, characteristic
12#20274881C211yesmaximal, characteristic, complemented
13#20274726C211yesmaximal, characteristic
14#20274592C211yesmaximal, characteristic, complemented
15C53×#1803C211yesmaximal, characteristic, complemented
16C22×#11719924C2211yescharacteristic
17C2×Q8×Dic53C2211yescharacteristic
18C2×Q8×Dic53C2211yescharacteristic
19C2×C4×D212C2211yescharacteristic
20C2×#13133179C2211yescharacteristic
21C2×#13133179C2211yescharacteristic
22C2×C4⋊C4×D53—21no—
23#16810726—24no—
24#16810721C418yescomplemented
25#16810708—24no—
26C2×#13133150C2211yescharacteristic
27C2×#13133150C2211yescharacteristic
28C2×#13133150C2211yescharacteristic
29C2×#13133148C2211yescharacteristic
30C2×#13133148C2211yescharacteristic
31C2×#13133141C2211yescharacteristic
32C2×#13133139C2211yescharacteristic
33C2×#13133139C2211yescharacteristic
34C2×#13133137C2211yescharacteristic
35C2×#13133128C2211yescharacteristic
36C2×#13133127C2211yescharacteristic
37C2×C8×Dic53C2211yescharacteristic
38#16810522—21no—
39#16810522—21no—
40#16810522C2211yescharacteristic
41#16810522C2211yescharacteristic
42#16810519C2211yescharacteristic
43#16810519C2211yescharacteristic
44#16810494—21no—
45#16810494—21no—
46#16810489—21no—
47#16810489C2211yescharacteristic
48#16810480—21no—
49C4×C4⋊Dic53—21no—
50C4×C4⋊Dic53C2211yescharacteristic
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53#16810377C2211yescharacteristic
54#16810377C2211yescharacteristic
55C2×C4×Q8×C53C2211yescharacteristic
56C2×C4⋊C8×C53C2211yescharacteristic
57C2×#103×C53C2211yescharacteristic
58C2×#103×C53C2211yescharacteristic
59C53×#384C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C53×#340C2211yescharacteristic
62C22×D212C2311yescharacteristic
63C22×C4×D53—21no—
64C22×C4×D53—21no—
65C22×C4×D53—21no—
66C2×#11719924C2×C414yes—
67C2×#11719924D414yes—
68C2×#11719924Q814yes—
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71Q8×Dic53—22no—
72Q8×Dic53—22no—
73Q8×Dic53—41no—
74Q8×Dic53—41no—
75C4×D212—22no—
76C4×D212C2×C414yes—
77#13133179—22no—
78#13133179—22no—
79#13133179—41no—
80#13133179—41no—
81C2×C4⋊Dic53—21no—
82C2×C4⋊Dic53—21no—
83C2×C4⋊Dic53—21no—
84C2×C4⋊Dic53—21no—
85C2×C4⋊Dic53D411yescharacteristic
86C2×C4⋊Dic53D411yescharacteristic
87C2×C4⋊Dic53C2311yescharacteristic
88C2×C4⋊Dic53C2311yescharacteristic
89C2×C4⋊Dic53C2311yescharacteristic
90C2×#11719919—21no—
91C2×#11719919—21no—
92C4⋊C4×D53—44no—
93C2×#11719921—21no—
94C2×#11719921—21no—
95C2×#11719921—21no—
96C2×C4×Dic53—21no—
97C2×C4×Dic53—21no—
98C2×C4×Dic53C2311yescharacteristic
99C2×C4×Dic53C2311yescharacteristic
100#13133150—22no—
101#13133150—41no—
102#13133150C2×C414yescomplemented
103#13133148—22no—
104#13133148—41no—
105#13133141C2×C414yescomplemented
106#13133139—22no—
107#13133139—22no—
108#13133137C2×C414yes—
109#13133128C2×C414yescomplemented
110#13133127—22no—
111C8×Dic53—22no—
112#13133124—21no—
113#13133124—21no—
114#13133124—21no—
115#13133124—21no—
116#13133124—21no—
117#13133124—21no—
118#13133124D411yescharacteristic
119#13133124D411yescharacteristic
120C22×Q8×C53C2311yescharacteristic
121C22×C8×C53C2311yescharacteristic
122C22×C8×C53C2311yescharacteristic
123C4×Q8×C53—22no—
124C4×Q8×C53C2×C414yescomplemented
125C2×C4⋊C4×C53—21no—
126C2×C4⋊C4×C53—21no—
127C2×C4⋊C4×C53—21no—
128C2×C4⋊C4×C53C2311yescharacteristic
129C2×C4⋊C4×C53C2311yescharacteristic
130C2×C4⋊C4×C53C2311yescharacteristic
131C2×C42×C53—21no—
132C2×C42×C53C2311yescharacteristic
133C4⋊C8×C53—22no—
134#103×C53—22no—
135#103×C53—22no—
136C23×D53—21no—
137C2×Q8×C53—21no—
138C2×Q8×C53—21no—
139C2×Q8×C53—21no—
140C2×Q8×C53C22×C412yes—
141C2×Q8×C53C2×D412yes—
142C2×Q8×C53C2×Q812yes—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2411yescharacteristic
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53C2×D411yescharacteristic
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53C2×D411yescharacteristic
156C22×Dic53C2×D411yescharacteristic
157C2×D212—21no—
158C2×D212—21no—
159C2×D212—21no—
160C2×D212C22×C412yes—
161C2×D212C2×D412yes—
162C2×D212C2×Q812yes—
163C2×C4×D53—24no—
164C2×C4×D53—42no—
165C2×C4×D53—42no—
166C2×C4×D53—42no—
167C2×C4×D53—42no—
168C2×C4×D53—42no—
169C2×C4×D53—42no—
170C2×C4×D53—81no—
171C2×C4×D53—81no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
175C2×C8×C53—21no—
176C2×C8×C53—21no—
177C2×C8×C53—21no—
178C4⋊C4×C53—24no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53—21no—
184C4⋊C4×C53—42no—
185C4⋊C4×C53—81no—
186C4⋊C4×C53C22×C412yes—
187C42×C53—24no—
188C42×C53C22×C412yes—
189C42×C53#4112yes—
190#11719924C4⋊C4116yescomplemented
191#11719921—24no—
192#11719921—42no—
193#11719921—42no—
194C4⋊Dic53—21no—
195C4⋊Dic53—22no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—41no—
201C4⋊Dic53—42no—
202C4⋊Dic53—42no—
203C4⋊Dic53—42no—
204C4⋊Dic53—42no—
205C4⋊Dic53C22×C412yes—
206#11719919—42no—
207#11719919—42no—
208C4×Dic53—21no—
209C4×Dic53—21no—
210C4×Dic53—21no—
211C4×Dic53—21no—
212C4×Dic53—42no—
213C4×Dic53—42no—
214C2×#88322—21no—
215C2×#88322—21no—
216C2×#88322C22×C412yes—
217C2×#88322C22×C412yes—
218C2×#88322C2×D412yes—
219C2×#88322C2×Q812yes—
220C23×C53C22×D411yescharacteristic
221C22×D53—22no—
222C22×D53—22no—
223C22×D53—22no—
224C22×D53—41no—
225C22×D53—41no—
226C22×D53—41no—
227C22×D53—41no—
228Q8×C53—22no—
229Q8×C53—22no—
230Q8×C53—22no—
231Q8×C53C2×C4⋊C414yescomplemented
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—22no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53C4×D412yes—
249C2×C4×C53C4×D412yes—
250C2×C4×C53C23×C411yescharacteristic
251C2×C4×C53C22×D411yescharacteristic
252C2×C4×C53C22×Q811yescharacteristic
253C2×C4×C53C2×#4111yescharacteristic
254C2×C4×C53H(5,𝔽2)11yescharacteristic
255C2×C4×C532−1+411yescharacteristic
256C2×Dic53—21no—
257C2×Dic53—21no—
258C2×Dic53—22no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—21no—
267C2×Dic53—21no—
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×Dic53AGL(1,ℤ/8)12yes—
277D212—22no—
278D212—22no—
279D212—22no—
280D212C2×C4⋊C414yes—
281C4×D53—48no—
282C4×D53—84no—
283C4×D53—84no—
284C8×C53—41no—
285C8×C53—41no—
286#88322—22no—
287#88322C2×C4⋊C414yescomplemented
288#22677—531nomaximal, Sylow
289C22×C53C2×C4×D411yescharacteristic
290C22×C53D4211yescharacteristic
291C22×C53D4×Q811yescharacteristic
292C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C53#57511yescharacteristic
294C22×C53#57711yescharacteristic
295C22×C53#57811yescharacteristic
296C2×D53—24no—
297C2×D53—42no—
298C2×D53—42no—
299C2×D53—42no—
300C2×D53—42no—
301C2×D53—42no—
302C2×D53—42no—
303C4×C53—24no—
304C4×C53—22no—
305C4×C53—22no—
306C4×C53—42no—
307C4×C53—81no—
308C4×C53—81no—
309C4×C53C22×C4⋊C411yescharacteristic
310C4×C53#52011yescharacteristic
311C4×C53#54711yescharacteristic
312C4×C53#55411yescharacteristic
313Dic53—22no—
314Dic53—42no—
315Dic53—41no—
316Dic53—41no—
317Dic53—41no—
318Dic53—81no—
319Dic53—81no—
320C2×#440—531no—
321C2×#440—531no—
322C2×C4×SD16—531no—
323#2351—531no—
324D4×C4⋊C4—531no—
325#1943—531no—
326#1942—531no—
327#1865—531no—
328#1864—531no—
329#1863—531no—
330#1862—531no—
331#1847—531no—
332#1804—531no—
333#1803D5311yescharacteristic, complemented
334#1771—531no—
335C2×C53D4×C4⋊C411yescharacteristic
336C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
337C2×C53#294711yescharacteristic
338C2×C53#326511yescharacteristic
339C2×C53#326911yescharacteristic
340C2×C53#338611yescharacteristic
341C2×C53#338811yescharacteristic
342D53—44no—
343C22×SD16—531no—
344C2×C4×Q8—531no—
345C2×C4×Q8—531no—
346C2×C4×Q8C2×D5311yescharacteristic
347C2×C4×D4—531no—
348C2×C4×D4—531no—
349C2×C4×D4—531no—
350C22×C4⋊C4—1061no—
351#440—538no—
352#440—1064no—
353C4×SD16—1064no—
354C2×#107—531no—
355C2×#107—531no—
356C2×#107—531no—
357C2×#106—531no—
358C2×#106—531no—
359C2×C4⋊C8C2×D5311yescharacteristic
360C2×#103—531no—
361C2×#103C2×D5311yescharacteristic
362C2×#103C2×D5311yescharacteristic
363C2×#102—531no—
364C2×#102—531no—
365C2×#102—531no—
366C2×#97—531no—
367C2×#97—531no—
368C2×C4×C8—531no—
369#389—531no—
370#389—531no—
371#389—531no—
372#389—531no—
373#389—1061no—
374#389—1061no—
375#385—1061no—
376#385—1061no—
377#384C2×D5311yescharacteristic
378#382—531no—
379#382—1061no—
380#380—1061no—
381C4×C4⋊C4—531no—
382C4×C4⋊C4—1061no—
383#340—531no—
384#340—531no—
385#340C2×D5311yescharacteristic
386#340C2×D5311yescharacteristic
387#336—531no—
388#336—531no—
389C53#2267711yescharacteristic, complemented, Sylow
390C22×Q8C22×D5311yescharacteristic
391C22×D4—531no—
392C23×C4—1061no—
393C23×C4—1061no—
394C23×C4—1061no—
395C2×SD16—534no—
396C2×SD16—534no—
397C2×SD16—534no—
398C22×C8—531no—
399C22×C8—531no—
400C22×C8C22×D5311yescharacteristic
401C22×C8C22×D5311yescharacteristic
402C4×Q8—22no—
403C4×Q8—1062no—
404C4×Q8—1062no—
405C4×Q8—2121no—
406C4×Q8—2121no—
407C4×Q8C4×D5314yescomplemented
408C4×D4—534no—
409C4×D4—1062no—
410C4×D4—1062no—
411C4×D4—1062no—
412C4×D4—2121no—
413C4×D4—2121no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—531no—
418C2×C4⋊C4—531no—
419C2×C4⋊C4—531no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—531no—
422C2×C4⋊C4—1061no—
423C2×C4⋊C4—1061no—
424C2×C4⋊C4—1061no—
425C2×C4⋊C4—1061no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4—2124no—
429C2×C4⋊C4C22×D5311yescharacteristic
430C2×C4⋊C4C22×D5311yescharacteristic
431C2×C4⋊C4C22×D5311yescharacteristic
432C2×C22⋊C4—1061no—
433C2×C22⋊C4—1061no—
434C2×C22⋊C4—1061no—
435C2×C42—21no—
436C2×C42—531no—
437C2×C42—531no—
438C2×C42—1061no—
439C2×C42—1061no—
440C2×C42C22×D5311yescharacteristic
441#107—534no—
442#107—1062no—
443#107—2121no—
444#106—1062no—
445#106—2121no—
446C4⋊C8—22no—
447#103—22no—
448#103—22no—
449#103—534no—
450#102—534no—
451#102—1062no—
452#102—1062no—
453#97—534no—
454#97—1062no—
455C4×C8—1062no—
456#95—531no—
457#95—531no—
458#95—1061no—
459#95—1061no—
460#95—1061no—
461#95—1061no—
462#95—1061no—
463#95—1061no—
464C24—1061no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8C2×C4×D5312yes—
469C2×Q8D4×D5312yes—
470C2×Q8Q8×D5312yes—
471C2×D4—532no—
472C2×D4—532no—
473C2×D4—532no—
474C2×D4—1061no—
475C2×D4—1061no—
476C2×D4—1061no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—531no—
482C22×C4—531no—
483C22×C4—531no—
484C22×C4—531no—
485C22×C4—1061no—
486C22×C4—1064no—
487C22×C4—1061no—
488C22×C4—1061no—
489C22×C4—4241no—
490C22×C4—4241no—
491C22×C4—2122no—
492C22×C4—2122no—
493C22×C4—2122no—
494C22×C4—2122no—
495C22×C4—2122no—
496C22×C4—2122no—
497C22×C4D4×D5311yescharacteristic
498C22×C4D4×D5311yescharacteristic
499C22×C4C23×D5311yescharacteristic
500SD16—5316no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—532no—
508C2×C8—532no—
509C2×C8—532no—
510C2×C8—532no—
511C2×C8—1061no—
512C2×C8—1061no—
513C4⋊C4—24no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—42no—
520C4⋊C4—81no—
521C4⋊C4—532no—
522C4⋊C4—1062no—
523C4⋊C4—1061no—
524C4⋊C4—1061no—
525C4⋊C4—1061no—
526C4⋊C4—1061no—
527C4⋊C4—1061no—
528C4⋊C4—2121no—
529C4⋊C4—2122no—
530C4⋊C4—2122no—
531C4⋊C4—2122no—
532C4⋊C4—2122no—
533C4⋊C4—2122no—
534C4⋊C4—2122no—
535C4⋊C4C2×C4×D5312yes—
536C22⋊C4—1064no—
537C22⋊C4—2122no—
538C22⋊C4—2122no—
539C42—24no—
540C42—1061no—
541C42—1061no—
542C42—1061no—
543C42—1061no—
544C42—2122no—
545C42—2122no—
546C42C2×C4×D5312yes—
547C42#1171994712yes—
548C23—1062no—
549C23—1062no—
550C23—1062no—
551C23—2121no—
552C23—2121no—
553C23—2121no—
554C23—2121no—
555C23C2×D4×D5311yescharacteristic
556Q8—22no—
557Q8—22no—
558Q8—22no—
559Q8C4⋊C4×D5314yescomplemented
560D4—534no—
561D4—1062no—
562D4—1062no—
563D4—1062no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—532no—
581C2×C4—1061no—
582C2×C4—1061no—
583C2×C4—1062no—
584C2×C4—1061no—
585C2×C4—1061no—
586C2×C4—1061no—
587C2×C4—1061no—
588C2×C4—1061no—
589C2×C4—1061no—
590C2×C4—1061no—
591C2×C4—1061no—
592C2×C4—1061no—
593C2×C4—4244no—
594C2×C4—4244no—
595C2×C4—2128no—
596C2×C4—2121no—
597C2×C4—2121no—
598C2×C4—2121no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4—2121no—
604C2×C4#1313317712yes—
605C2×C4#1313317712yes—
606C2×C4C22×C4×D5311yescharacteristic
607C2×C4C2×D4×D5311yescharacteristic
608C2×C4C2×Q8×D5311yescharacteristic
609C2×C4C2×#1171994711yescharacteristic
610C2×C4#1313326311yescharacteristic
611C2×C4#1313326411yescharacteristic
612C8—41no—
613C8—41no—
614C8—534no—
615C8—1062no—
616C22—1064no—
617C22—2122no—
618C22—2122no—
619C22—2122no—
620C22—2122no—
621C22—2122no—
622C22—2122no—
623C22C2×#1313317711yescharacteristic
624C22#1681125711yescharacteristic
625C22#1681128611yescharacteristic
626C22AGL(1,ℤ/8)×D5311yescharacteristic
627C22#1681142211yescharacteristic
628C22#1681142911yescharacteristic
629C22#1681143611yescharacteristic
630C4—24no—
631C4—22no—
632C4—22no—
633C4—42no—
634C4—81no—
635C4—81no—
636C4—1062no—
637C4—4241no—
638C4—4241no—
639C4—2121no—
640C4—2121no—
641C4—2122no—
642C4—2121no—
643C4C2×C4⋊C4×D5311yescharacteristic
644C4#1681110311yescharacteristic
645C4#1681127311yescharacteristic
646C4#1681133111yescharacteristic
647C2—2124no—
648C2#2027761111yescharacteristic
649C2#2028352811yescharacteristic
650C2#2028355211yescharacteristic
651C2#2028620511yescharacteristic
652C2#2028626511yescharacteristic
653C2#2028763911yescharacteristic
654C2#2028766711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1803C2
socleC23×C53C22×D4
prime radicalC22×D212C23
cosocleC22×C4×C53C24
hypercenter#1803D53
hypocenterC53#22677
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
4. #20277636C2semidirect
6. #20275452C2semidirect
7. #20275447C2semidirect
8. #20274981C2semidirect
10. #20274967C2semidirect
12. #20274881C2semidirect
14. #20274592C2semidirect
15. C53×#1803C2semidirect
24. #16810721C4semidirect
102. #13133150C2×C4semidirect
105. #13133141C2×C4semidirect
109. #13133128C2×C4semidirect
124. C4×Q8×C53C2×C4semidirect
190. #11719924C4⋊C4semidirect
231. Q8×C53C2×C4⋊C4semidirect
287. #88322C2×C4⋊C4semidirect
333. #1803D53semidirect
389. C53#22677semidirect
407. C4×Q8C4×D53semidirect
559. Q8C4⋊C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168107262
C2×#168107212
C2×#168107082
#202776362
#202776152
#202754522
#202754472
#202749812
#202749772
#202749672
#202749632
#202748812
#202747262
#202745922
C53×#18032
#2267753