Finite Group Databasebeta

← #27391736

Subgroups — #27391736

Order 13568.

988 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810911C211yesmaximal, characteristic
2#20277286C211yesmaximal, characteristic, complemented
3C4×D4×Dic53C211yesmaximal, characteristic
4#20275661C211yesmaximal, characteristic, complemented
5#20275470C211yesmaximal, characteristic, complemented
6#20275294C211yesmaximal, characteristic, complemented
7#20275290C211yesmaximal, characteristic, complemented
8#20274962C211yesmaximal, characteristic
9#20274713C211yesmaximal, characteristic
10#20274614C211yesmaximal, characteristic, complemented
11#20274611C211yesmaximal, characteristic, complemented
12#20274567C211yesmaximal, characteristic, complemented
13#20274542C211yesmaximal, characteristic
14#20274542C211yesmaximal, characteristic
15C53×#1862C211yesmaximal, characteristic
16C2×#13133216C2211yescharacteristic
17C2×#13133182—21no—
18C2×D4×Dic53C2211yescharacteristic
19C2×D4×Dic53C2211yescharacteristic
20C22×C4×Dic53—21no—
21#16810911C418yescomplemented
22C2×#13133150C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133146C2211yescharacteristic
26C2×#13133146C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×#13133138C2211yescharacteristic
29C2×#13133127C2211yescharacteristic
30C2×#13133127C2211yescharacteristic
31#16810527—21no—
32#16810521C2211yescharacteristic
33#16810518—21no—
34#16810492C2211yescharacteristic
35C4⋊C4×Dic53C2211yescharacteristic
36C4⋊C4×Dic53C2211yescharacteristic
37C4⋊C4×Dic53C2211yescharacteristic
38C4⋊C4×Dic53C2211yescharacteristic
39C4×C4⋊Dic53C2211yescharacteristic
40#16810474—21no—
41C22⋊C4×Dic53—21no—
42C22⋊C4×Dic53—21no—
43C4×C22⋊Dic53—21no—
44C42×Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810378C2211yescharacteristic
53C2×C4×D4×C53C2211yescharacteristic
54C2×#107×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#336C2211yescharacteristic
60C22×Dic106—21no—
61C23×Dic53—21no—
62C23×Dic53—21no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65#13133216—22no—
66#13133216C2×C414yescomplemented
67#13133182—44no—
68D4×Dic53—22no—
69D4×Dic53—22no—
70D4×Dic53C2×C414yes—
71D4×Dic53Q814yescomplemented
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—44no—
88C2×C4×Dic53D411yescharacteristic
89C2×C4×Dic53D411yescharacteristic
90C2×C4×Dic53D411yescharacteristic
91C2×C4×Dic53D411yescharacteristic
92C2×C4×Dic53C2311yescharacteristic
93C2×C4×Dic53C2311yescharacteristic
94#13133150—22no—
95#13133149C2×C414yescomplemented
96#13133148C2×C414yescomplemented
97#13133146—22no—
98#13133146C2×C414yescomplemented
99#13133138—22no—
100#13133138C2×C414yes—
101#13133127—22no—
102#13133127—22no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109C22×D4×C53C2311yescharacteristic
110C23×C4×C53—21no—
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C4×D4×C53—22no—
114C4×D4×C53—22no—
115C2×C4⋊C4×C53—21no—
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120C2×C42×C53C2311yescharacteristic
121#107×C53—22no—
122#102×C53—22no—
123#102×C53C2×C414yes—
124#95×C53—21no—
125C24×C53—21no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53C22×C412yes—
130C2×D4×C53C2×Q812yes—
131C2×D4×C53#4112yes—
132C22×C4×C53—24no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—42no—
137C22×C4×C53—42no—
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2411yescharacteristic
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—24no—
145C22×Dic53—24no—
146C22×Dic53—21no—
147C22×Dic53—42no—
148C22×Dic53—42no—
149C22×Dic53—42no—
150C22×Dic53—42no—
151C22×Dic53C2×D411yescharacteristic
152C22×Dic53C2×D411yescharacteristic
153C22×Dic53C2×D411yescharacteristic
154C22×Dic53C2×D411yescharacteristic
155C2×Dic106—24no—
156C2×Dic106—42no—
157C2×Dic106—42no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53C22×C412yes—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—42no—
169C4⋊C4×C53C22×C412yes—
170C4⋊C4×C53#4112yes—
171C22⋊C4×C53—24no—
172C42×C53—21no—
173C42×C53—21no—
174C22⋊Dic53—24no—
175C22⋊Dic53—24no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53C22×C412yes—
182C4⋊Dic53C22×C412yes—
183C4⋊Dic53C2×Q812yes—
184#11719919—24no—
185#11719919—42no—
186#11719919—42no—
187#11719919—42no—
188C4×Dic53—22no—
189C4×Dic53—22no—
190C4×Dic53—22no—
191C4×Dic53—41no—
192C4×Dic53—42no—
193C4×Dic53—42no—
194C4×Dic53—42no—
195C4×Dic53—42no—
196C4×Dic53—42no—
197C4×Dic53C22×C412yes—
198C4×Dic53C2×Q812yes—
199C4×Dic53#4112yes—
200C4×Dic53#4112yes—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322C22×C412yes—
207C23×C53—22no—
208C23×C53—22no—
209C23×C53—22no—
210C23×C53—41no—
211C23×C53—41no—
212C23×C53—41no—
213C23×C53—41no—
214C23×C53C22×D411yescharacteristic
215D4×C53—22no—
216D4×C53—22no—
217D4×C53—22no—
218D4×C53C4×Q814yescomplemented
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—22no—
222C2×C4×C53—22no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—48no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53AGL(1,ℤ/8)12yes—
235C2×C4×C53C23×C411yescharacteristic
236C2×C4×C53C22×Q811yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C53C2×#4111yescharacteristic
239C2×C4×C53C2×#4111yescharacteristic
240C2×C4×C53C2×#4111yescharacteristic
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—21no—
247C2×Dic53—22no—
248C2×Dic53—22no—
249C2×Dic53—22no—
250C2×Dic53—48no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—48no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53C4×D412yes—
262C2×Dic53C4×D412yes—
263C2×Dic53C22⋊Q812yes—
264C2×Dic53C22⋊Q812yes—
265C2×Dic53AGL(1,ℤ/8)12yes—
266C2×Dic53#13712yes—
267Dic106—48no—
268C8×C53—22no—
269C8×C53—41no—
270#88322—22no—
271#88322—41no—
272#21614—531nomaximal, Sylow
273C22×C53—24no—
274C22×C53—42no—
275C22×C53—42no—
276C22×C53—42no—
277C22×C53—42no—
278C22×C53—42no—
279C22×C53—42no—
280C22×C53C2×C4×D411yescharacteristic
281C22×C53C2×C22⋊Q811yescharacteristic
282C22×C53#52511yescharacteristic
283C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
284C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
285C22×C53C2×#13711yescharacteristic
286C22×C53#57511yescharacteristic
287C4×C53—22no—
288C4×C53—42no—
289C4×C53—41no—
290C4×C53—42no—
291C4×C53—81no—
292C4×C53C2×C4×Q811yescharacteristic
293C4×C53C4×#4111yescharacteristic
294C4×C53#52911yescharacteristic
295C4×C53#53311yescharacteristic
296Dic53—22no—
297Dic53—22no—
298Dic53—22no—
299Dic53—42no—
300Dic53—42no—
301Dic53—42no—
302Dic53—81no—
303Dic53#44014yes—
304C2×#476—531no—
305C4×C4⋊Q8—531no—
306C42×D4—531no—
307#1987—531no—
308#1947—531no—
309#1919—531no—
310#1919—531no—
311#1862—531no—
312#1862D5311yescharacteristic
313#1802—531no—
314#1776—531no—
315#1776—531no—
316#1767—531no—
317#1762—531no—
318#1762—531no—
319C2×C53—44no—
320C2×C53C4×C22⋊Q811yescharacteristic
321C2×C53C2×#44011yescharacteristic
322C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
323C2×C53#307711yescharacteristic
324C2×C53#307911yescharacteristic
325C2×C53#310911yescharacteristic
326C2×C53#311611yescharacteristic
327C2×C4⋊Q8—531no—
328C2×C4×Q8—1061no—
329C2×C4×D4—531no—
330C2×C4×D4—531no—
331C2×C4×D4C2×D5311yescharacteristic
332C22×C42—1061no—
333#476—538no—
334C2×#107—531no—
335C2×#107C2×D5311yescharacteristic
336C2×#106—531no—
337C2×#106—531no—
338C2×C4⋊C8—531no—
339C2×C4⋊C8—531no—
340C2×#102—531no—
341C2×#102—531no—
342C2×#102C2×D5311yescharacteristic
343C2×#102C2×D5311yescharacteristic
344C2×#97—531no—
345C2×#97—531no—
346#391—1061no—
347#389—531no—
348#389—1061no—
349#389C2×D5311yescharacteristic
350#384—531no—
351C4×C4⋊C4—531no—
352C4×C4⋊C4—531no—
353C4×C4⋊C4—531no—
354C4×C4⋊C4—531no—
355C4×C4⋊C4—531no—
356C4×C4⋊C4—1061no—
357C4×C22⋊C4—1061no—
358C4×C22⋊C4—1061no—
359C4×C22⋊C4—1061no—
360C43—531no—
361#340—531no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340—531no—
368#340C2×D5311yescharacteristic
369#336—531no—
370#336C2×D5311yescharacteristic
371C53#2161411yescharacteristic, complemented, Sylow
372C22×Q8—1061no—
373C22×D4C22×D5311yescharacteristic
374C23×C4—21no—
375C23×C4—1061no—
376C23×C4—1061no—
377C22×C8—531no—
378C22×C8—531no—
379C22×C8C22×D5311yescharacteristic
380C22×C8C22×D5311yescharacteristic
381C4⋊Q8—534no—
382C4⋊Q8—1062no—
383C4×Q8—2124no—
384C4×D4—22no—
385C4×D4—22no—
386C4×D4—534no—
387C4×D4—534no—
388C4×D4—1062no—
389C4×D4—1062no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4C22×D5311yescharacteristic
400C2×C4⋊C4C22×D5311yescharacteristic
401C2×C4⋊C4C22×D5311yescharacteristic
402C2×C22⋊C4—21no—
403C2×C22⋊C4—1061no—
404C2×C22⋊C4—1061no—
405C2×C42—531no—
406C2×C42—531no—
407C2×C42—531no—
408C2×C42—531no—
409C2×C42—531no—
410C2×C42—531no—
411C2×C42—1061no—
412C2×C42—1061no—
413C2×C42—1061no—
414C2×C42—1061no—
415C2×C42—1061no—
416C2×C42—2124no—
417C2×C42C22×D5311yescharacteristic
418#107—22no—
419#107—1062no—
420#106—534no—
421#106—534no—
422C4⋊C8—534no—
423C4⋊C8—1062no—
424#102—22no—
425#102—534no—
426#102—1062no—
427#102C4×D5314yes—
428#97—1062no—
429#97—1062no—
430#95—21no—
431#95—1061no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436#95—1061no—
437C24—21no—
438C2×Q8—1064no—
439C2×Q8—2122no—
440C2×Q8—2122no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4C2×C4×D5312yes—
445C2×D4#1171994712yes—
446C2×D4Q8×D5312yes—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—531no—
454C22×C4—531no—
455C22×C4—531no—
456C22×C4—531no—
457C22×C4—1064no—
458C22×C4—1064no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—2122no—
464C22×C4—2122no—
465C22×C4—2122no—
466C22×C4—2122no—
467C22×C4D4×D5311yescharacteristic
468C22×C4D4×D5311yescharacteristic
469C22×C4C23×D5311yescharacteristic
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—532no—
476C2×C8—1061no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8C2×C4×D5312yes—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—42no—
487C4⋊C4—532no—
488C4⋊C4—532no—
489C4⋊C4—532no—
490C4⋊C4—1064no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—2122no—
497C4⋊C4—2122no—
498C4⋊C4—2122no—
499C4⋊C4C2×C4×D5312yes—
500C4⋊C4#1171994712yes—
501C22⋊C4—24no—
502C22⋊C4—1064no—
503C22⋊C4—1064no—
504C42—21no—
505C42—21no—
506C42—532no—
507C42—532no—
508C42—532no—
509C42—532no—
510C42—1062no—
511C42—1062no—
512C42—1062no—
513C42—2121no—
514C42—2122no—
515C42—2122no—
516C42—2122no—
517C42—2122no—
518C42—2122no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23C2×D4×D5311yescharacteristic
527Q8—2128no—
528D4—22no—
529D4—22no—
530D4—22no—
531D4#1313318214yescomplemented
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—48no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—532no—
548C2×C4—532no—
549C2×C4—532no—
550C2×C4—532no—
551C2×C4—532no—
552C2×C4—532no—
553C2×C4—1061no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—1061no—
557C2×C4—1061no—
558C2×C4—1061no—
559C2×C4—1062no—
560C2×C4—1062no—
561C2×C4—1062no—
562C2×C4—2128no—
563C2×C4—2128no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4#1313324112yes—
574C2×C4C22×C4×D5311yescharacteristic
575C2×C4C2×Q8×D5311yescharacteristic
576C2×C4#41×D5311yescharacteristic
577C2×C4#41×D5311yescharacteristic
578C2×C4C2×#1171994711yescharacteristic
579C2×C4C2×#1171994711yescharacteristic
580C8—22no—
581C8—41no—
582C8—1062no—
583C8—2121no—
584C22—24no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22C4×D4×D5311yescharacteristic
592C22C22⋊Q8×D5311yescharacteristic
593C22#1681112911yescharacteristic
594C22AGL(1,ℤ/8)×D5311yescharacteristic
595C22C2×#1313324111yescharacteristic
596C22#137×D5311yescharacteristic
597C22#1681142011yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—41no—
602C4—81no—
603C4—534no—
604C4—1062no—
605C4—1062no—
606C4—1062no—
607C4—4241no—
608C4—2122no—
609C4—2122no—
610C4—2122no—
611C4C2×#1313318211yescharacteristic
612C4C4×#1171994711yescharacteristic
613C4#1681115011yescharacteristic
614C4#1681117511yescharacteristic
615C2—44no—
616C2#2027724711yescharacteristic
617C2#440×D5311yescharacteristic
618C2C4×#1313324111yescharacteristic
619C2#2028451011yescharacteristic
620C2#2028452311yescharacteristic
621C2#2028480111yescharacteristic
622C2#2028487011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1862C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
cosocleC22×C4×C53C24
hypercenter#1862D53
hypocenterC53#21614
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1862

Complements

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

NH ≅ G/Ntype
2. #20277286C2semidirect
4. #20275661C2semidirect
5. #20275470C2semidirect
6. #20275294C2semidirect
7. #20275290C2semidirect
10. #20274614C2semidirect
11. #20274611C2semidirect
12. #20274567C2semidirect
21. #16810911C4semidirect
66. #13133216C2×C4semidirect
71. D4×Dic53Q8semidirect
95. #13133149C2×C4semidirect
96. #13133148C2×C4semidirect
98. #13133146C2×C4semidirect
218. D4×C53C4×Q8semidirect
371. C53#21614semidirect
531. D4#13133182semidirect

Maximal subgroups

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

subgroupindex
C2×#168109112
#202772862
C4×D4×Dic532
#202756612
#202754702
#202752942
#202752902
#202749622
#202747132
#202746142
#202746112
#202745672
#202745422
C53×#18622
#2161453