Finite Group Databasebeta

← #27400725

Subgroups — #27400725

Order 13568.

1250 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811288C211yesmaximal, characteristic
2C2×#16810751C211yesmaximal, characteristic, complemented
3#20277667C211yesmaximal, characteristic, complemented
4#20275686C211yesmaximal, characteristic, complemented
5#20275674C211yesmaximal, characteristic, complemented
6#20275665C211yesmaximal, characteristic, complemented
7#20275664C211yesmaximal, characteristic, complemented
8#20275661C211yesmaximal, characteristic, complemented
9#20274724C211yesmaximal, characteristic
10#20274722C211yesmaximal, characteristic
11#20274714C211yesmaximal, characteristic
12#20274541C211yesmaximal, characteristic
13#20274539C211yesmaximal, characteristic
14#20274529C211yesmaximal, characteristic
15C53×#1993C211yesmaximal, characteristic, complemented
16C22×Q8×D53—21no—
17#16811288—24no—
18#16811288C418yescomplemented
19C2×#13133216C2211yescharacteristic
20C2×#13133216C2211yescharacteristic
21C2×#13133195—21no—
22C2×#13133195—21no—
23C2×#13133193—21no—
24C2×C4×Dic106C2211yescharacteristic
25C2×C4×D212C2211yescharacteristic
26C2×#13133179C2211yescharacteristic
27C2×#13133179C2211yescharacteristic
28#16810751—24no—
29C2×C4⋊#88322C2211yescharacteristic
30C2×#13133146C2211yescharacteristic
31C2×#13133146C2211yescharacteristic
32C2×#13133139C2211yescharacteristic
33C2×#13133139C2211yescharacteristic
34C2×#13133137C2211yescharacteristic
35C2×#13133137C2211yescharacteristic
36C2×#13133127C2211yescharacteristic
37C2×C8×Dic53C2211yescharacteristic
38#16810527—21no—
39#16810508—21no—
40#16810505—21no—
41#16810492C2211yescharacteristic
42#16810488C2211yescharacteristic
43C4⋊C4×Dic53C2211yescharacteristic
44C4⋊C4×Dic53C2211yescharacteristic
45C4×C4⋊Dic53C2211yescharacteristic
46#16810474—21no—
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53#16810385C2211yescharacteristic
54#16810385C2211yescharacteristic
55C2×C4⋊Q8×C53C2211yescharacteristic
56C2×#97×C53C2211yescharacteristic
57C53×#382C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C53×#340C2211yescharacteristic
62C22×Dic106—21no—
63C22×Dic106—21no—
64C22×Dic106D411yescharacteristic
65C2×Q8×D53—28no—
66C2×Q8×D53—44no—
67C2×Q8×D53—44no—
68C22×D212C2311yescharacteristic
69C22×C4×D53—21no—
70C22×C4×D53—21no—
71C22×C4×D53—21no—
72C22×#88322C2311yescharacteristic
73C22×#88322C2311yescharacteristic
74#13133216—22no—
75#13133216—22no—
76#13133216C2×C414yes—
77#13133216C2×C414yes—
78#13133195—28no—
79#13133195—28no—
80#13133193—28no—
81C4×Dic106C2×C414yes—
82C4×Dic106D414yes—
83C4×D212—22no—
84C4×D212C2×C414yes—
85#13133179—22no—
86#13133179—22no—
87#13133179C2×C414yes—
88#13133179C2×C414yes—
89C2×C4⋊Dic53D411yescharacteristic
90C2×C4⋊Dic53C2311yescharacteristic
91C2×C4⋊Dic53C2311yescharacteristic
92C2×C4⋊Dic53C2311yescharacteristic
93C2×#11719919—21no—
94C2×#11719919—21no—
95C2×#11719919—21no—
96C2×#11719919—21no—
97C2×#11719919D411yescharacteristic
98C2×#11719919D411yescharacteristic
99C2×#11719921—21no—
100C2×#11719921—21no—
101C2×#11719921—21no—
102C2×C4×Dic53—21no—
103C2×C4×Dic53—21no—
104C2×C4×Dic53—21no—
105C2×C4×Dic53C2311yescharacteristic
106C2×C4×Dic53C2311yescharacteristic
107C4⋊#88322—22no—
108#13133146—22no—
109#13133146—22no—
110#13133139—22no—
111#13133139Q814yescomplemented
112#13133137—22no—
113#13133137—22no—
114#13133127—22no—
115C8×Dic53—22no—
116#13133124—21no—
117#13133124—21no—
118#13133124—21no—
119#13133124—21no—
120#13133124—21no—
121C22×Q8×C53—21no—
122C22×C8×C53C2311yescharacteristic
123C22×C8×C53C2311yescharacteristic
124C4⋊Q8×C53—22no—
125C4⋊Q8×C53C2×C414yes—
126C2×C4⋊C4×C53—21no—
127C2×C4⋊C4×C53C2311yescharacteristic
128C2×C4⋊C4×C53C2311yescharacteristic
129C2×C4⋊C4×C53C2311yescharacteristic
130C2×C4⋊C4×C53C2311yescharacteristic
131C2×C42×C53C2311yescharacteristic
132#97×C53—22no—
133#95×C53—21no—
134#95×C53—21no—
135C23×D53—21no—
136C2×Q8×C53—24no—
137C2×Q8×C53—42no—
138C2×Q8×C53—42no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2411yescharacteristic
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53C2×D411yescharacteristic
150C22×Dic53C2×D411yescharacteristic
151C22×Dic53C2×D411yescharacteristic
152C22×Dic53C2×D411yescharacteristic
153Q8×D53—432no—
154C2×D212—21no—
155C2×D212—21no—
156C2×D212—21no—
157C2×D212C22×C412yes—
158C2×D212C2×Q812yes—
159C2×D212#4112yes—
160C2×C4×D53—24no—
161C2×C4×D53—24no—
162C2×C4×D53—24no—
163C2×C4×D53—42no—
164C2×C4×D53—42no—
165C2×C4×D53—42no—
166C2×C4×D53—42no—
167C2×C4×D53—42no—
168C2×C4×D53—42no—
169C2×Dic106—22no—
170C2×Dic106—22no—
171C2×Dic106—24no—
172C2×Dic106—24no—
173C2×Dic106—42no—
174C2×Dic106—42no—
175C2×Dic106—42no—
176C2×Dic106—42no—
177C2×Dic106C22⋊C414yes—
178C2×C8×C53—21no—
179C2×C8×C53—21no—
180C2×C8×C53—21no—
181C2×C8×C53—21no—
182C2×C8×C53—21no—
183C2×C8×C53C2×Q812yescomplemented
184C4⋊C4×C53—24no—
185C4⋊C4×C53—21no—
186C4⋊C4×C53—21no—
187C4⋊C4×C53—21no—
188C4⋊C4×C53—21no—
189C4⋊C4×C53—21no—
190C4⋊C4×C53—21no—
191C4⋊C4×C53C22×C412yes—
192C4⋊C4×C53C22×C412yes—
193C42×C53C22×C412yes—
194C42×C53C2×D412yes—
195#11719921—24no—
196#11719921—24no—
197#11719921—24no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53C22⋊C414yes—
203C4⋊Dic53C22×C412yes—
204C4⋊Dic53C2×Q812yescomplemented
205#11719919—24no—
206#11719919—24no—
207#11719919—42no—
208#11719919—81no—
209#11719919C22⋊C414yes—
210#11719919C22⋊C414yes—
211C4×Dic53—42no—
212C4×Dic53—81no—
213C4×Dic53—81no—
214C4×Dic53C22×C412yes—
215C4×Dic53C22×C412yes—
216C4×Dic53C2×D412yes—
217C4×Dic53C2×D412yes—
218C2×#88322—21no—
219C2×#88322—21no—
220C2×#88322—21no—
221C2×#88322—21no—
222C2×#88322—21no—
223C2×#88322—21no—
224C23×C53C22×D411yescharacteristic
225C22×D53—22no—
226C22×D53—22no—
227C22×D53—22no—
228C22×D53—41no—
229C22×D53—41no—
230C22×D53—41no—
231C22×D53—41no—
232Q8×C53—48no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—41no—
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×C53—41no—
249C2×C4×C53C2×C22⋊C412yes—
250C2×C4×C53C2×C22⋊C412yes—
251C2×C4×C53C23×C411yescharacteristic
252C2×C4×C53C22×D411yescharacteristic
253C2×C4×C53C22×Q811yescharacteristic
254C2×C4×C53C2×#4111yescharacteristic
255C2×C4×C532−1+411yescharacteristic
256C2×C4×C532−1+411yescharacteristic
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—22no—
264C2×Dic53—22no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53C2×C22⋊C412yes—
273C2×Dic53C2×C22⋊C412yes—
274C2×Dic53C2×C22⋊C412yes—
275C2×Dic53C2×C22⋊C412yes—
276C2×Dic53AGL(1,ℤ/8)12yes—
277C2×Dic53AGL(1,ℤ/8)12yes—
278D212—22no—
279D212—22no—
280D212—22no—
281D212C4×Q814yes—
282C4×D53—48no—
283C4×D53—48no—
284C4×D53—48no—
285Dic106—28no—
286Dic106—48no—
287Dic106—48no—
288C8×C53—22no—
289C8×C53—41no—
290#88322—41no—
291#88322—41no—
292#22523—531nomaximal, Sylow
293C22×C53C22×C22⋊C411yescharacteristic
294C22×C53#54811yescharacteristic
295C22×C53D4×Q811yescharacteristic
296C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
297C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
298C22×C53#57711yescharacteristic
299C22×C53#57811yescharacteristic
300C2×D53—24no—
301C2×D53—42no—
302C2×D53—42no—
303C2×D53—42no—
304C2×D53—42no—
305C2×D53—42no—
306C2×D53—42no—
307C4×C53—22no—
308C4×C53—22no—
309C4×C53—42no—
310C4×C53—42no—
311C4×C53—81no—
312C4×C53—81no—
313C4×C53C2×C4×Q811yescharacteristic
314C4×C53#51911yescharacteristic
315C4×C53#54811yescharacteristic
316C4×C53D4×Q811yescharacteristic
317Dic53—22no—
318Dic53—22no—
319Dic53—22no—
320Dic53—42no—
321Dic53—42no—
322Dic53—81no—
323Dic53#41814yes—
324C2×D4×Q8—531no—
325C2×#446—531no—
326#2355—531no—
327#1993—531no—
328#1993D5311yescharacteristic, complemented
329#1990—531no—
330#1988—531no—
331#1987—531no—
332#1987—531no—
333#1804—531no—
334#1804—531no—
335#1802—531no—
336#1762—531no—
337#1762—531no—
338C4×#102—531no—
339C2×C53Q8×C22⋊C411yescharacteristic
340C2×C53C2×#41811yescharacteristic
341C2×C53#289811yescharacteristic
342C2×C53#336711yescharacteristic
343C2×C53#337311yescharacteristic
344C2×C53#338411yescharacteristic
345C2×C53#338611yescharacteristic
346D53—44no—
347C23×Q8—1061no—
348D4×Q8—538no—
349D4×Q8—1064no—
350C2×C4⋊Q8—531no—
351C2×C4⋊Q8—531no—
352C2×C4⋊Q8C2×D5311yescharacteristic
353C2×C22⋊Q8—1061no—
354C2×C22⋊Q8—1061no—
355C2×C22⋊Q8—1061no—
356C2×C4×Q8—531no—
357C2×C4×D4—531no—
358C2×C4×D4—531no—
359C2×C4×D4—531no—
360#446—1064no—
361C2×C4⋊C8—531no—
362C2×C4⋊C8—531no—
363C2×C4⋊C8—531no—
364C2×#102—531no—
365C2×#102—531no—
366C2×#102—531no—
367C2×#102—531no—
368C2×#97—531no—
369C2×#97C2×D5311yescharacteristic
370C2×C4×C8—531no—
371#391—1061no—
372#387—1061no—
373#387—1061no—
374#384—531no—
375#382—531no—
376#382C2×D5311yescharacteristic
377C4×C4⋊C4—531no—
378C4×C4⋊C4—531no—
379C4×C4⋊C4—531no—
380C4×C4⋊C4—1061no—
381#340—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#340—531no—
388#340—531no—
389#340C2×D5311yescharacteristic
390#340C2×D5311yescharacteristic
391#340C2×D5311yescharacteristic
392#340C2×D5311yescharacteristic
393C53#2252311yescharacteristic, complemented, Sylow
394C22×Q8—21no—
395C22×Q8—531no—
396C22×Q8—1061no—
397C22×Q8—1061no—
398C22×Q8—1068no—
399C22×Q8—2124no—
400C22×Q8—2124no—
401C22×D4—531no—
402C23×C4—1061no—
403C23×C4—1061no—
404C23×C4—1061no—
405C22×C8—531no—
406C22×C8—531no—
407C22×C8C22×D5311yescharacteristic
408C22×C8C22×D5311yescharacteristic
409C4⋊Q8—22no—
410C4⋊Q8—534no—
411C4⋊Q8—534no—
412C4⋊Q8—1062no—
413C4⋊Q8—1062no—
414C4⋊Q8C4×D5314yes—
415C22⋊Q8—1068no—
416C22⋊Q8—1068no—
417C22⋊Q8—1068no—
418C4×Q8—534no—
419C4×Q8—534no—
420C4×D4—534no—
421C4×D4—534no—
422C4×D4—534no—
423C4×D4—1062no—
424C4×D4—1062no—
425C4×D4—1062no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—531no—
428C2×C4⋊C4—531no—
429C2×C4⋊C4—531no—
430C2×C4⋊C4—531no—
431C2×C4⋊C4—531no—
432C2×C4⋊C4—531no—
433C2×C4⋊C4—1061no—
434C2×C4⋊C4—1061no—
435C2×C4⋊C4—1061no—
436C2×C4⋊C4—1061no—
437C2×C4⋊C4C22×D5311yescharacteristic
438C2×C4⋊C4C22×D5311yescharacteristic
439C2×C4⋊C4C22×D5311yescharacteristic
440C2×C4⋊C4C22×D5311yescharacteristic
441C2×C22⋊C4—1061no—
442C2×C22⋊C4—1061no—
443C2×C22⋊C4—1061no—
444C2×C42—531no—
445C2×C42—531no—
446C2×C42—1061no—
447C2×C42—1061no—
448C2×C42—1061no—
449C2×C42C22×D5311yescharacteristic
450C4⋊C8—1062no—
451C4⋊C8—1062no—
452C4⋊C8—1062no—
453#102—534no—
454#102—1062no—
455#102—1062no—
456#102—1062no—
457#97—22no—
458#97—1062no—
459C4×C8—1062no—
460#95—21no—
461#95—21no—
462#95—1061no—
463#95—1061no—
464#95—1061no—
465#95—1061no—
466#95—1061no—
467C24—1061no—
468C2×Q8—24no—
469C2×Q8—42no—
470C2×Q8—42no—
471C2×Q8—534no—
472C2×Q8—1062no—
473C2×Q8—1062no—
474C2×Q8—1064no—
475C2×Q8—1064no—
476C2×Q8—2122no—
477C2×Q8—2122no—
478C2×Q8—2122no—
479C2×Q8—2122no—
480C2×Q8—21232no—
481C2×D4—532no—
482C2×D4—532no—
483C2×D4—532no—
484C2×D4—1061no—
485C2×D4—1061no—
486C2×D4—1061no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—531no—
492C22×C4—531no—
493C22×C4—531no—
494C22×C4—531no—
495C22×C4—1064no—
496C22×C4—1064no—
497C22×C4—1061no—
498C22×C4—1061no—
499C22×C4—1064no—
500C22×C4—1061no—
501C22×C4—2122no—
502C22×C4—2122no—
503C22×C4—2122no—
504C22×C4—2122no—
505C22×C4—2122no—
506C22×C4—2122no—
507C22×C4D4×D5311yescharacteristic
508C22×C4D4×D5311yescharacteristic
509C22×C4C23×D5311yescharacteristic
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—1061no—
516C2×C8—1061no—
517C2×C8—1061no—
518C2×C8—1061no—
519C2×C8—1061no—
520C2×C8—1061no—
521C2×C8Q8×D5312yescomplemented
522C4⋊C4—24no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—534no—
530C4⋊C4—534no—
531C4⋊C4—534no—
532C4⋊C4—532no—
533C4⋊C4—532no—
534C4⋊C4—1064no—
535C4⋊C4—1064no—
536C4⋊C4—1061no—
537C4⋊C4—1061no—
538C4⋊C4—1061no—
539C4⋊C4—1061no—
540C4⋊C4—4241no—
541C4⋊C4—2122no—
542C4⋊C4C2×C4×D5312yes—
543C4⋊C4C2×C4×D5312yes—
544C22⋊C4—1064no—
545C22⋊C4—1064no—
546C22⋊C4—1064no—
547C42—532no—
548C42—532no—
549C42—532no—
550C42—532no—
551C42—4241no—
552C42—4241no—
553C42—2122no—
554C42C2×C4×D5312yes—
555C42D4×D5312yes—
556C23—1062no—
557C23—1062no—
558C23—1062no—
559C23—2121no—
560C23—2121no—
561C23—2121no—
562C23—2121no—
563C23C2×D4×D5311yescharacteristic
564Q8—48no—
565Q8—1068no—
566Q8—2128no—
567Q8—2128no—
568D4—534no—
569D4—1062no—
570D4—1062no—
571D4—1062no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—532no—
589C2×C4—532no—
590C2×C4—532no—
591C2×C4—532no—
592C2×C4—532no—
593C2×C4—532no—
594C2×C4—1061no—
595C2×C4—1061no—
596C2×C4—1061no—
597C2×C4—1061no—
598C2×C4—1061no—
599C2×C4—1061no—
600C2×C4—1062no—
601C2×C4—1062no—
602C2×C4—2128no—
603C2×C4—2128no—
604C2×C4—2128no—
605C2×C4—2121no—
606C2×C4—2121no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4—2121no—
611C2×C4—2121no—
612C2×C4C22⋊C4×D5312yes—
613C2×C4C22⋊C4×D5312yes—
614C2×C4C22×C4×D5311yescharacteristic
615C2×C4C2×D4×D5311yescharacteristic
616C2×C4C2×Q8×D5311yescharacteristic
617C2×C4#41×D5311yescharacteristic
618C2×C4#1313326411yescharacteristic
619C2×C4#1313326511yescharacteristic
620C8—22no—
621C8—41no—
622C8—2121no—
623C8—2121no—
624C22—1064no—
625C22—2122no—
626C22—2122no—
627C22—2122no—
628C22—2122no—
629C22—2122no—
630C22—2122no—
631C22C2×C22⋊C4×D5311yescharacteristic
632C22#1681128111yescharacteristic
633C22#1681128811yescharacteristic
634C22AGL(1,ℤ/8)×D5311yescharacteristic
635C22AGL(1,ℤ/8)×D5311yescharacteristic
636C22#1681143211yescharacteristic
637C22#1681143911yescharacteristic
638C4—22no—
639C4—22no—
640C4—42no—
641C4—42no—
642C4—81no—
643C4—81no—
644C4—534no—
645C4—1062no—
646C4—1062no—
647C4—1062no—
648C4—4241no—
649C4—2122no—
650C4—2122no—
651C4C4×Q8×D5311yescharacteristic
652C4#1681109911yescharacteristic
653C4#1681128311yescharacteristic
654C4#1681128611yescharacteristic
655C2—2124no—
656C2#2027754911yescharacteristic
657C2#418×D5311yescharacteristic
658C2#2028324711yescharacteristic
659C2#2028743011yescharacteristic
660C2#2028750411yescharacteristic
661C2#2028761111yescharacteristic
662C2#2028763311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1993C2
socleC23×C53C22×D4
prime radicalC22×D212C23
cosocleC22×C4×C53C24
hypercenter#1993D53
hypocenterC53#22523
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
2. C2×#16810751C2semidirect
3. #20277667C2semidirect
4. #20275686C2semidirect
5. #20275674C2semidirect
6. #20275665C2semidirect
7. #20275664C2semidirect
8. #20275661C2semidirect
15. C53×#1993C2semidirect
18. #16811288C4semidirect
111. #13133139Q8semidirect
183. C2×C8×C53C2×Q8semidirect
204. C4⋊Dic53C2×Q8semidirect
328. #1993D53semidirect
393. C53#22523semidirect
521. C2×C8Q8×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168112882
C2×#168107512
#202776672
#202756862
#202756742
#202756652
#202756642
#202756612
#202747242
#202747222
#202747142
#202745412
#202745392
#202745292
C53×#19932
#2252353