Finite Group Databasebeta

← #27440144

Subgroups — #27440144

Order 13568.

1046 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811333C211yesmaximal, characteristic
2C2×#16811022C211yesmaximal, characteristic, complemented
3C2×#16810920C211yesmaximal, characteristic
4#20276053C211yesmaximal, characteristic, complemented
5#20276053C211yesmaximal, characteristic, complemented
6#20276048C211yesmaximal, characteristic, complemented
7#20276043C211yesmaximal, characteristic, complemented
8#20275672C211yesmaximal, characteristic, complemented
9#20275652C211yesmaximal, characteristic, complemented
10#20274975C211yesmaximal, characteristic
11#20274975C211yesmaximal, characteristic
12#20274961C211yesmaximal, characteristic
13#20274961C211yesmaximal, characteristic
14C4×#13133137C211yesmaximal, characteristic
15C53×#2584C211yesmaximal, characteristic, complemented
16#16811333—24no—
17#16811333—24no—
18C2×#13133217C2211yescharacteristic
19C2×#13133209C2211yescharacteristic
20C2×#13133209C2211yescharacteristic
21C2×#13133195—21no—
22C2×#13133195—21no—
23C2×#13133193—21no—
24C2×C4×D212C2211yescharacteristic
25C2×C4×D212C2211yescharacteristic
26C2×C4×D212C2211yescharacteristic
27C2×C4⋊C4×D53—21no—
28#16811022—24no—
29#16810920—24no—
30C2×#13133150C2211yescharacteristic
31C2×#13133150C2211yescharacteristic
32C2×#13133150C2211yescharacteristic
33C2×#13133148C2211yescharacteristic
34C2×#13133148C2211yescharacteristic
35C2×#13133148C2211yescharacteristic
36C2×C4⋊#88322C2211yescharacteristic
37C2×#13133137C2211yescharacteristic
38C2×#13133137C2211yescharacteristic
39C2×#13133137C2211yescharacteristic
40C2×#13133137C2211yescharacteristic
41C2×C4×#88322C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50#16810378C2211yescharacteristic
51#16810378C2211yescharacteristic
52C2×#125×C53C2211yescharacteristic
53C2×C4×Q8×C53C2211yescharacteristic
54C53×#398—21no—
55C53×#395—21no—
56C53×#389—21no—
57C53×#389—21no—
58C53×#389C2211yescharacteristic
59C53×#389C2211yescharacteristic
60C53×#389C2211yescharacteristic
61C53×#389C2211yescharacteristic
62C4×C4⋊C4×C53C2211yescharacteristic
63C22×Dic106—21no—
64C22×D212C2311yescharacteristic
65C22×C4×D53—21no—
66C22×C4×D53—21no—
67C22×C4×D53—21no—
68C22×#88322C2311yescharacteristic
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71C22×#88322C2311yescharacteristic
72#13133217—22no—
73#13133217—22no—
74#13133209—22no—
75#13133209—22no—
76#13133209—22no—
77#13133209D414yes—
78#13133195—28no—
79#13133195—28no—
80#13133193—28no—
81C4×D212—22no—
82C4×D212—22no—
83C4×D212—22no—
84C4×D212—22no—
85C4×D212—22no—
86C4×D212D414yes—
87C2×C4⋊Dic53—21no—
88C2×C4⋊Dic53C2311yescharacteristic
89C2×C4⋊Dic53C2311yescharacteristic
90C2×C4⋊Dic53C2311yescharacteristic
91C2×#11719919—21no—
92C2×#11719919—21no—
93C2×#11719919—21no—
94C2×#11719919—21no—
95C4⋊C4×D53—28no—
96C2×#11719921—21no—
97C2×#11719921—21no—
98C2×#11719921—21no—
99#13133150—22no—
100#13133150—22no—
101#13133150—22no—
102#13133148—22no—
103#13133148—22no—
104#13133148—22no—
105C4⋊#88322—22no—
106#13133137—22no—
107#13133137—22no—
108#13133137Q814yescomplemented
109#13133137Q814yescomplemented
110C4×#88322—22no—
111C22×Q8×C53D411yescharacteristic
112#125×C53—22no—
113#125×C53—22no—
114C4×Q8×C53—22no—
115C4×Q8×C53—22no—
116C2×C4⋊C4×C53—21no—
117C2×C4⋊C4×C53—21no—
118C2×C4⋊C4×C53—21no—
119C2×C4⋊C4×C53—21no—
120C2×C4⋊C4×C53—21no—
121C2×C4⋊C4×C53—21no—
122C2×C4⋊C4×C53D411yescharacteristic
123C2×C4⋊C4×C53D411yescharacteristic
124C2×C4⋊C4×C53D411yescharacteristic
125C2×C4⋊C4×C53C2311yescharacteristic
126C2×C4⋊C4×C53C2311yescharacteristic
127C2×C4⋊C4×C53C2311yescharacteristic
128C2×C4⋊C4×C53C2311yescharacteristic
129C2×C42×C53—21no—
130C2×C42×C53C2311yescharacteristic
131C2×C42×C53C2311yescharacteristic
132C2×C42×C53C2311yescharacteristic
133#95×C53—21no—
134#95×C53—21no—
135#95×C53—21no—
136#95×C53—21no—
137#95×C53—21no—
138C23×D53—21no—
139C2×Q8×C53—22no—
140C2×Q8×C53—22no—
141C2×Q8×C53—41no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2411yescharacteristic
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C2×D212—21no—
157C2×D212—21no—
158C2×D212—21no—
159C2×D212C2×D412yes—
160C2×D212C2×Q812yes—
161C2×D212C2×Q812yes—
162C2×C4×D53—24no—
163C2×C4×D53—24no—
164C2×C4×D53—24no—
165C2×C4×D53—42no—
166C2×C4×D53—42no—
167C2×C4×D53—81no—
168C2×C4×D53—81no—
169C2×C4×D53—81no—
170C2×C4×D53—81no—
171C2×Dic106—24no—
172C2×Dic106—81no—
173C2×Dic106—81no—
174C4⋊C4×C53—22no—
175C4⋊C4×C53—22no—
176C4⋊C4×C53—22no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53—42no—
184C4⋊C4×C53—42no—
185C4⋊C4×C53—42no—
186C4⋊C4×C53—42no—
187C4⋊C4×C53—81no—
188C4⋊C4×C53—81no—
189C4⋊C4×C53C2×Q812yes—
190C4⋊C4×C53C2×Q812yes—
191C42×C53—21no—
192C42×C53—21no—
193C42×C53—21no—
194C42×C53—21no—
195C42×C53—81no—
196C42×C53C2×D412yes—
197C42×C53C2×D412yes—
198#11719921—24no—
199#11719921—24no—
200#11719921—24no—
201C4⋊Dic53—24no—
202C4⋊Dic53—21no—
203C4⋊Dic53—21no—
204C4⋊Dic53—21no—
205C4⋊Dic53C2×D412yes—
206C4⋊Dic53C2×D412yes—
207C4⋊Dic53C2×D412yes—
208#11719919—24no—
209#11719919—24no—
210#11719919—24no—
211#11719919—24no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C2×#88322—21no—
216C2×#88322—21no—
217C2×#88322—21no—
218C2×#88322—21no—
219C2×#88322—21no—
220C2×#88322—21no—
221C2×#88322—21no—
222C2×#88322C2×Q812yes—
223C2×#88322C2×Q812yes—
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—44no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
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—21no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—21no—
249C2×C4×C53—21no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—41no—
257C2×C4×C53—41no—
258C2×C4×C53—41no—
259C2×C4×C53—41no—
260C2×C4×C53#12012yes—
261C2×C4×C53#12012yes—
262C2×C4×C53#13512yes—
263C2×C4×C53C22×D411yescharacteristic
264C2×C4×C53C22×D411yescharacteristic
265C2×C4×C53C22×Q811yescharacteristic
266C2×C4×C53C22×Q811yescharacteristic
267C2×C4×C532−1+411yescharacteristic
268C2×C4×C532−1+411yescharacteristic
269C2×Dic53—22no—
270C2×Dic53—22no—
271C2×Dic53—22no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278D212—22no—
279D212—22no—
280D212—22no—
281D212C4⋊Q814yes—
282C4×D53—48no—
283C4×D53—84no—
284C4×D53—84no—
285Dic106—84no—
286#88322—22no—
287#88322—22no—
288#88322—41no—
289#88322—41no—
290#25843—531nomaximal, Sylow
291C22×C53C2×#12011yescharacteristic
292C22×C53D4×Q811yescharacteristic
293C22×C53D4×Q811yescharacteristic
294C22×C53C2×#13511yescharacteristic
295C22×C53#57511yescharacteristic
296C22×C53#57611yescharacteristic
297C22×C53#57711yescharacteristic
298C2×D53—24no—
299C2×D53—42no—
300C2×D53—42no—
301C2×D53—42no—
302C2×D53—42no—
303C2×D53—42no—
304C2×D53—42no—
305C4×C53—22no—
306C4×C53—22no—
307C4×C53—22no—
308C4×C53—42no—
309C4×C53—42no—
310C4×C53—41no—
311C4×C53—41no—
312C4×C53—41no—
313C4×C53—81no—
314C4×C53—81no—
315C4×C53C2×C4⋊Q811yescharacteristic
316C4×C53#53611yescharacteristic
317C4×C53D4×Q811yescharacteristic
318C4×C53D4×Q811yescharacteristic
319Dic53—42no—
320Dic53—42no—
321Dic53—42no—
322C2×#554—531no—
323C2×#499—531no—
324C2×#478—531no—
325#2584D5311yescharacteristic, complemented
326#2059—531no—
327#2059—531no—
328#2058—531no—
329#2057—531no—
330#1989—531no—
331#1984—531no—
332#1864—531no—
333#1864—531no—
334#1862—531no—
335#1862—531no—
336C4×#102—531no—
337C2×C53#243411yescharacteristic
338C2×C53#300211yescharacteristic
339C2×C53#301211yescharacteristic
340C2×C53#338011yescharacteristic
341C2×C53#338111yescharacteristic
342C2×C53#338811yescharacteristic
343C2×C53#338911yescharacteristic
344D53—44no—
345#554—1064no—
346#554—1064no—
347C2×C4⋊Q8—531no—
348C2×#125—531no—
349C2×#125—531no—
350C2×#125C2×D5311yescharacteristic
351C2×C22⋊Q8—1061no—
352C2×C22⋊Q8—1061no—
353C2×C22⋊Q8—1061no—
354C2×C4×Q8C2×D5311yescharacteristic
355C2×C4×D4—531no—
356C2×C4×D4—531no—
357C2×C4×D4—531no—
358C22×C4⋊C4—1061no—
359#499—1064no—
360#478—1064no—
361C2×#107—531no—
362C2×#107—531no—
363C2×#107—531no—
364C2×#106—531no—
365C2×#106—531no—
366C2×#106—531no—
367C2×C4⋊C8—531no—
368C2×#102—531no—
369C2×#102—531no—
370C2×#102—531no—
371C2×#102—531no—
372C2×C4×C8—531no—
373#398—21no—
374#395—21no—
375#389—21no—
376#389—21no—
377#389C2×D5311yescharacteristic
378#389C2×D5311yescharacteristic
379#389C2×D5311yescharacteristic
380#389C2×D5311yescharacteristic
381C4×C4⋊C4C2×D5311yescharacteristic
382#340—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#340—531no—
388#336—531no—
389#336—531no—
390#336—531no—
391#336—531no—
392C53#2584311yescharacteristic, complemented, Sylow
393C22×Q8—1061no—
394C22×Q8C22⋊D5311yescharacteristic
395C22×D4—531no—
396C23×C4—1061no—
397C23×C4—1061no—
398C23×C4—1061no—
399C22×C8—531no—
400C22×C8—531no—
401C22×C8—531no—
402C22×C8—531no—
403C4⋊Q8—1062no—
404C4⋊Q8—1062no—
405#125—22no—
406#125—22no—
407#125—534no—
408#125—1062no—
409#125—1062no—
410#125—1062no—
411C22⋊Q8—1068no—
412C22⋊Q8—1068no—
413C22⋊Q8—1068no—
414C4×Q8—22no—
415C4×Q8—22no—
416C4×D4—534no—
417C4×D4—1062no—
418C4×D4—1062no—
419C4×D4—1062no—
420C4×D4—1062no—
421C4×D4—1062no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—531no—
429C2×C4⋊C4—531no—
430C2×C4⋊C4—531no—
431C2×C4⋊C4—1068no—
432C2×C4⋊C4—1061no—
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×C4⋊C4C22×D5311yescharacteristic
442C2×C4⋊C4C22×D5311yescharacteristic
443C2×C4⋊C4C22×D5311yescharacteristic
444C2×C22⋊C4—1061no—
445C2×C22⋊C4—1061no—
446C2×C22⋊C4—1061no—
447C2×C42—21no—
448C2×C42C22×D5311yescharacteristic
449C2×C42C22×D5311yescharacteristic
450C2×C42C22×D5311yescharacteristic
451#107—1062no—
452#107—1062no—
453#107—1062no—
454#106—1062no—
455#106—1062no—
456#106—1062no—
457C4⋊C8—1062no—
458#102—534no—
459#102—534no—
460#102—1062no—
461#102—1062no—
462C4×C8—1062no—
463#95—21no—
464#95—21no—
465#95—21no—
466#95—21no—
467#95—21no—
468C24—1061no—
469C2×Q8—22no—
470C2×Q8—22no—
471C2×Q8—41no—
472C2×Q8—1064no—
473C2×Q8—4241no—
474C2×Q8—4241no—
475C2×D4—532no—
476C2×D4—532no—
477C2×D4—532no—
478C2×D4—1061no—
479C2×D4—1061no—
480C2×D4—1061no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—1064no—
486C22×C4—1064no—
487C22×C4—1061no—
488C22×C4—1061no—
489C22×C4—1061no—
490C22×C4—1064no—
491C22×C4—4241no—
492C22×C4—4241no—
493C22×C4—4241no—
494C22×C4—4241no—
495C22×C4—2122no—
496C22×C4—2122no—
497C22×C4C2×C22⋊D5311yescharacteristic
498C22×C4C2×C22⋊D5311yescharacteristic
499C22×C4C2×C22⋊D5311yescharacteristic
500C22×C4C2×C22⋊D5311yescharacteristic
501C22×C4C2×C22⋊D5311yescharacteristic
502C22×C4C2×C22⋊D5311yescharacteristic
503C22×C4C23×D5311yescharacteristic
504C2×C8—532no—
505C2×C8—532no—
506C2×C8—1061no—
507C2×C8—1061no—
508C2×C8—1061no—
509C2×C8—1061no—
510C2×C8—1061no—
511C2×C8—1061no—
512C2×C8—1061no—
513C2×C8—1061no—
514C2×C8—1061no—
515C2×C8—1061no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—81no—
530C4⋊C4—81no—
531C4⋊C4—532no—
532C4⋊C4—532no—
533C4⋊C4—532no—
534C4⋊C4—1064no—
535C4⋊C4—1064no—
536C4⋊C4—1064no—
537C4⋊C4—1064no—
538C4⋊C4—1064no—
539C4⋊C4—1061no—
540C4⋊C4—1061no—
541C4⋊C4—1061no—
542C4⋊C4Q8×D5312yes—
543C4⋊C4Q8×D5312yes—
544C22⋊C4—1064no—
545C22⋊C4—1064no—
546C22⋊C4—1064no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—81no—
552C42D4×D5312yes—
553C42D4×D5312yes—
554C23—1062no—
555C23—1062no—
556C23—1062no—
557C23—2121no—
558C23—2121no—
559C23—2121no—
560C23—2121no—
561C23C22×C22⋊D5311yescharacteristic
562Q8—44no—
563Q8—4244no—
564D4—534no—
565D4—1062no—
566D4—1062no—
567D4—1062no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—1062no—
596C2×C4—1062no—
597C2×C4—1062no—
598C2×C4—4244no—
599C2×C4—4244no—
600C2×C4—2128no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4—2121no—
604C2×C4—2121no—
605C2×C4—2121no—
606C2×C4—2121no—
607C2×C4#1313318512yes—
608C2×C4#1313318512yes—
609C2×C4#1313323712yes—
610C2×C4C2×D4×D5311yescharacteristic
611C2×C4C2×D4×D5311yescharacteristic
612C2×C4C2×Q8×D5311yescharacteristic
613C2×C4C2×Q8×D5311yescharacteristic
614C2×C4#1313326511yescharacteristic
615C2×C4#1313326511yescharacteristic
616C8—1062no—
617C8—1062no—
618C8—2121no—
619C8—2121no—
620C22—1064no—
621C22—2122no—
622C22—2122no—
623C22—2122no—
624C22—2122no—
625C22—2122no—
626C22—2122no—
627C22C2×#1313318511yescharacteristic
628C22Q8×C22⋊D5311yescharacteristic
629C22Q8×C22⋊D5311yescharacteristic
630C22C2×#1313323711yescharacteristic
631C22#1681142111yescharacteristic
632C22#1681142611yescharacteristic
633C22#1681143111yescharacteristic
634C4—22no—
635C4—22no—
636C4—22no—
637C4—41no—
638C4—42no—
639C4—41no—
640C4—41no—
641C4—42no—
642C4—81no—
643C4—81no—
644C4—2122no—
645C4—2122no—
646C4—2122no—
647C4C4⋊Q8×D5311yescharacteristic
648C4#1681119811yescharacteristic
649C4#1681128811yescharacteristic
650C4#1681128811yescharacteristic
651C2—2124no—
652C2#2027835611yescharacteristic
653C2#2028393111yescharacteristic
654C2#2028401611yescharacteristic
655C2#2028757811yescharacteristic
656C2#2028758611yescharacteristic
657C2#2028765411yescharacteristic
658C2#2028766911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2584C2
socleC23×C53C22×D4
prime radicalC22×D212C23
hypercenter#2584D53
hypocenterC53#25843
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2584

Complements

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

NH ≅ G/Ntype
2. C2×#16811022C2semidirect
4. #20276053C2semidirect
5. #20276053C2semidirect
6. #20276048C2semidirect
7. #20276043C2semidirect
8. #20275672C2semidirect
9. #20275652C2semidirect
15. C53×#2584C2semidirect
108. #13133137Q8semidirect
109. #13133137Q8semidirect
325. #2584D53semidirect
392. C53#25843semidirect

Maximal subgroups

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

subgroupindex
C2×#168113332
C2×#168110222
C2×#168109202
#202760532
#202760482
#202760432
#202756722
#202756522
#202749752
#202749612
C4×#131331372
C53×#25842
#2584353