Finite Group Databasebeta

← #27384188

Subgroups — #27384188

Order 13568.

1222 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811100C211yesmaximal, characteristic
2C2×#16810681C211yesmaximal, characteristic
3#20277322C211yesmaximal, characteristic
4#20274810C211yesmaximal, characteristic
5#20274729C211yesmaximal, characteristic, complemented
6#20274729C211yesmaximal, characteristic, complemented
7#20274719C211yesmaximal, characteristic, complemented
8#20274719C211yesmaximal, characteristic, complemented
9#20274651C211yesmaximal, characteristic
10#20274649C211yesmaximal, characteristic
11#20274544C211yesmaximal, characteristic, complemented
12#20274544C211yesmaximal, characteristic, complemented
13#20274534C211yesmaximal, characteristic, complemented
14#20274534C211yesmaximal, characteristic, complemented
15C53×#1825C211yesmaximal, characteristic
16C23×Dic106C2211yescharacteristic
17#16811100—24no—
18#16811100C418yescomplemented
19C2×C4×Dic106—21no—
20C2×C4×Dic106—21no—
21C2×#13133182C2211yescharacteristic
22C2×#13133182C2211yescharacteristic
23C2×#13133182C2211yescharacteristic
24C2×#13133182C2211yescharacteristic
25C2×#13133176C2211yescharacteristic
26C2×#13133173—21no—
27C2×#13133173—21no—
28#16810681—24no—
29C2×#13133146C2211yescharacteristic
30C2×#13133146C2211yescharacteristic
31C2×#13133142C2211yescharacteristic
32C2×#13133142C2211yescharacteristic
33C2×#13133142C2211yescharacteristic
34C2×#13133142C2211yescharacteristic
35C2×#13133140C2211yescharacteristic
36C2×#13133140C2211yescharacteristic
37C2×#13133140C2211yescharacteristic
38C2×#13133140C2211yescharacteristic
39C2×C22⋊#88322C2211yescharacteristic
40C2×#13133127C2211yescharacteristic
41C2×C8×Dic53C2211yescharacteristic
42#16810517—21no—
43#16810511—21no—
44#16810508—21no—
45#16810506—21no—
46#16810492C2211yescharacteristic
47#16810488C2211yescharacteristic
48#16810483C2211yescharacteristic
49C4⋊C4×Dic53C2211yescharacteristic
50C4⋊C4×Dic53C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53#16810385C2211yescharacteristic
54#16810385C2211yescharacteristic
55C2×#117×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C53×#380C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C53×#340C2211yescharacteristic
62C22×Dic106—21no—
63C22×Dic106—22no—
64C22×Dic106—22no—
65C22×Dic106—22no—
66C22×Dic106—22no—
67C22×Dic106—22no—
68C22×Dic106C2×C414yes—
69C22×Dic106C2311yescharacteristic
70C22×Dic106C2311yescharacteristic
71C23×Dic53—21no—
72C22×#88322C2311yescharacteristic
73C22×#88322C2311yescharacteristic
74C4×Dic106—28no—
75C4×Dic106—28no—
76#13133182—22no—
77#13133182—22no—
78#13133182—22no—
79#13133182—22no—
80#13133182C2×C414yes—
81#13133182C2×C414yes—
82#13133182C2×C414yes—
83#13133182C2×C414yes—
84#13133176C2×C414yes—
85#13133176D414yes—
86#13133173—28no—
87#13133173—28no—
88C2×C4⋊Dic53D411yescharacteristic
89C2×C4⋊Dic53D411yescharacteristic
90C2×C4⋊Dic53C2311yescharacteristic
91C2×C4⋊Dic53C2311yescharacteristic
92C2×#11719919—21no—
93C2×#11719919—21no—
94C2×#11719919—21no—
95C2×#11719919—21no—
96C2×#11719919—21no—
97C2×#11719919—21no—
98C2×C22⋊Dic53—21no—
99C2×C22⋊Dic53D411yescharacteristic
100C2×C22⋊Dic53D411yescharacteristic
101C2×C4×Dic53—21no—
102C2×C4×Dic53—21no—
103C2×C4×Dic53—21no—
104C2×C4×Dic53—21no—
105C2×C4×Dic53—21no—
106C2×C4×Dic53—21no—
107C2×C4×Dic53C2311yescharacteristic
108C2×C4×Dic53C2311yescharacteristic
109#13133146—22no—
110#13133146—22no—
111#13133142—22no—
112#13133142—22no—
113#13133142—22no—
114#13133142—22no—
115#13133140—22no—
116#13133140—22no—
117#13133140—22no—
118#13133140—22no—
119C22⋊#88322—22no—
120#13133127—22no—
121C8×Dic53—22no—
122#13133124—21no—
123#13133124—21no—
124#13133124—21no—
125#13133124—21no—
126#13133124—21no—
127C23×C4×C53C2311yescharacteristic
128C22×C8×C53C2311yescharacteristic
129C22×C8×C53C2311yescharacteristic
130#117×C53—22no—
131#117×C53C2×C414yes—
132C2×C4⋊C4×C53C2311yescharacteristic
133C2×C4⋊C4×C53C2311yescharacteristic
134C2×C4⋊C4×C53C2311yescharacteristic
135C2×C4⋊C4×C53C2311yescharacteristic
136C2×C22⋊C4×C53—21no—
137C2×C22⋊C4×C53—21no—
138C2×C42×C53—21no—
139C22⋊C8×C53—22no—
140#95×C53—21no—
141C24×C53C2×D411yescharacteristic
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53C22×C412yes—
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D412yes—
153C22×C4×C53C2411yescharacteristic
154C22×Dic53—24no—
155C22×Dic53—24no—
156C22×Dic53—24no—
157C22×Dic53—21no—
158C22×Dic53—21no—
159C22×Dic53—21no—
160C22×Dic53—21no—
161C22×Dic53C2×D411yescharacteristic
162C22×Dic53C2×D411yescharacteristic
163C22×Dic53C2×D411yescharacteristic
164C22×Dic53C2×D411yescharacteristic
165C2×Dic106—24no—
166C2×Dic106—24no—
167C2×Dic106—28no—
168C2×Dic106—28no—
169C2×Dic106—24no—
170C2×Dic106—21no—
171C2×Dic106—21no—
172C2×Dic106—21no—
173C2×Dic106—21no—
174C2×Dic106—21no—
175C2×Dic106—21no—
176C2×Dic106—28no—
177C2×Dic106—28no—
178C2×Dic106C22×C412yes—
179C2×Dic106C22×C412yes—
180C2×Dic106#4112yes—
181C2×Dic106#4112yes—
182C2×Dic106#4112yes—
183C2×Dic106#4112yes—
184C2×C8×C53—21no—
185C2×C8×C53—21no—
186C2×C8×C53—21no—
187C2×C8×C53—21no—
188C2×C8×C53—21no—
189C2×C8×C53—21no—
190C4⋊C4×C53—21no—
191C4⋊C4×C53—21no—
192C4⋊C4×C53—21no—
193C4⋊C4×C53—21no—
194C4⋊C4×C53—21no—
195C4⋊C4×C53—21no—
196C4⋊C4×C53C22×C412yes—
197C4⋊C4×C53C22×C412yes—
198C22⋊C4×C53—24no—
199C22⋊C4×C53—81no—
200C42×C53—24no—
201C22⋊Dic53—81no—
202C22⋊Dic53C22⋊C414yes—
203C22⋊Dic53C22⋊C414yes—
204C4⋊Dic53—21no—
205C4⋊Dic53—21no—
206C4⋊Dic53—21no—
207C4⋊Dic53—21no—
208C4⋊Dic53C22⋊C414yes—
209C4⋊Dic53C22⋊C414yes—
210#11719919—24no—
211#11719919—24no—
212#11719919—24no—
213#11719919—24no—
214#11719919—81no—
215#11719919—81no—
216C4×Dic53—24no—
217C4×Dic53—24no—
218C4×Dic53—24no—
219C4×Dic53—24no—
220C4×Dic53—81no—
221C4×Dic53—81no—
222C4×Dic53C22×C412yes—
223C4×Dic53C22×C412yes—
224C4×Dic53C2×D412yes—
225C4×Dic53C2×D412yes—
226C2×#88322—21no—
227C2×#88322—21no—
228C2×#88322—21no—
229C2×#88322—21no—
230C2×#88322—21no—
231C2×#88322—21no—
232C23×C53—21no—
233C23×C53—21no—
234C23×C53—21no—
235C23×C53—21no—
236C23×C53—21no—
237C23×C53—21no—
238C23×C53C2×C22⋊C412yes—
239C23×C53C22×D411yescharacteristic
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—22no—
247C2×C4×C53—22no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
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×C53C2×C22⋊C412yes—
259C2×C4×C53C23×C411yescharacteristic
260C2×C4×C53C22×D411yescharacteristic
261C2×C4×C53C2×#4111yescharacteristic
262C2×C4×C53C2×#4111yescharacteristic
263C2×C4×C53C2×#4111yescharacteristic
264C2×C4×C53C2×#4111yescharacteristic
265C2×Dic53—216no—
266C2×Dic53—22no—
267C2×Dic53—22no—
268C2×Dic53—22no—
269C2×Dic53—22no—
270C2×Dic53—22no—
271C2×Dic53—22no—
272C2×Dic53—21no—
273C2×Dic53—21no—
274C2×Dic53—21no—
275C2×Dic53—21no—
276C2×Dic53—21no—
277C2×Dic53—21no—
278C2×Dic53—21no—
279C2×Dic53—21no—
280C2×Dic53—41no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53—41no—
284C2×Dic53—41no—
285C2×Dic53—41no—
286C2×Dic53C2×C22⋊C412yes—
287C2×Dic53C2×C22⋊C412yes—
288C2×Dic53C2×C22⋊C412yes—
289C2×Dic53C2×C22⋊C412yes—
290Dic106—216no—
291Dic106—22no—
292Dic106—22no—
293Dic106—22no—
294Dic106—22no—
295Dic106—22no—
296Dic106—22no—
297Dic106#11714yes—
298Dic106#11714yes—
299C8×C53—41no—
300C8×C53—41no—
301#88322—41no—
302#88322—41no—
303#20745—531nomaximal, Sylow
304C22×C53—22no—
305C22×C53—22no—
306C22×C53—22no—
307C22×C53—41no—
308C22×C53—41no—
309C22×C53—41no—
310C22×C53—41no—
311C22×C53C22×C22⋊C411yescharacteristic
312C22×C53#52511yescharacteristic
313C22×C53#52511yescharacteristic
314C22×C53#57711yescharacteristic
315C22×C53#57711yescharacteristic
316C22×C53#57811yescharacteristic
317C22×C53#57811yescharacteristic
318C4×C53—22no—
319C4×C53—42no—
320C4×C53—42no—
321C4×C53—81no—
322C4×C53—81no—
323C4×C53C2×#11711yescharacteristic
324C4×C53C2×#11711yescharacteristic
325C4×C53#52511yescharacteristic
326C4×C53#52511yescharacteristic
327Dic53—22no—
328Dic53—22no—
329Dic53—22no—
330Dic53—22no—
331Dic53—24no—
332Dic53—24no—
333Dic53—81no—
334Dic53—81no—
335C2×#519—531no—
336C2×#433—531no—
337#2314—531no—
338#1825—531no—
339#1825D5311yescharacteristic
340#1805—531no—
341#1805—531no—
342#1803—531no—
343#1803—531no—
344#1787—531no—
345#1787—531no—
346#1763—531no—
347#1763—531no—
348C4×#103—531no—
349C4×#103—531no—
350C2×C53—42no—
351C2×C53#228011yescharacteristic
352C2×C53#289811yescharacteristic
353C2×C53#289811yescharacteristic
354C2×C53#312111yescharacteristic
355C2×C53#312111yescharacteristic
356C2×C53#312611yescharacteristic
357C2×C53#312711yescharacteristic
358C23×Q8—531no—
359#519—538no—
360#519—1064no—
361C2×C4×Q8—531no—
362C2×C4×Q8—531no—
363C2×C4×Q8—531no—
364C2×C4×Q8—531no—
365C2×C4×Q8—1061no—
366C2×C4×Q8—1061no—
367C2×#117—531no—
368C2×#117—1061no—
369C2×#117—1061no—
370C2×#117C2×D5311yescharacteristic
371#433—1064no—
372C2×C4⋊C8—531no—
373C2×C4⋊C8—531no—
374C2×#103—531no—
375C2×#103—531no—
376C2×#103—531no—
377C2×#103—531no—
378C2×#103—531no—
379C2×#103—531no—
380C2×#103—531no—
381C2×#103—531no—
382C2×C22⋊C8—531no—
383C2×C22⋊C8C2×D5311yescharacteristic
384C2×#97—531no—
385C2×C4×C8—531no—
386#388—1061no—
387#388—1061no—
388#387—1061no—
389#387—1061no—
390#384—531no—
391#382—531no—
392#380—531no—
393#380C2×D5311yescharacteristic
394C4×C4⋊C4—531no—
395C4×C4⋊C4—531no—
396#340—531no—
397#340—531no—
398#340—531no—
399#340—531no—
400#340C2×D5311yescharacteristic
401#340C2×D5311yescharacteristic
402#340C2×D5311yescharacteristic
403#340C2×D5311yescharacteristic
404C53#2074511yescharacteristic, complemented, Sylow
405C22×Q8—534no—
406C22×Q8—531no—
407C22×Q8—531no—
408C22×Q8—1062no—
409C22×Q8—1062no—
410C22×Q8—1062no—
411C22×Q8—1062no—
412C22×Q8—1061no—
413C22×Q8—1062no—
414C23×C4—1061no—
415C23×C4C22×D5311yescharacteristic
416C22×C8—531no—
417C22×C8—531no—
418C22×C8C22×D5311yescharacteristic
419C22×C8C22×D5311yescharacteristic
420C4×Q8—534no—
421C4×Q8—534no—
422C4×Q8—534no—
423C4×Q8—534no—
424C4×Q8—1068no—
425C4×Q8—1068no—
426C4×Q8—1062no—
427C4×Q8—1062no—
428C4×Q8—1062no—
429C4×Q8—1062no—
430#117—22no—
431#117—534no—
432#117—534no—
433#117—1068no—
434#117—1068no—
435#117C4×D5314yes—
436C2×C4⋊C4—531no—
437C2×C4⋊C4—531no—
438C2×C4⋊C4—531no—
439C2×C4⋊C4—531no—
440C2×C4⋊C4—1061no—
441C2×C4⋊C4—1061no—
442C2×C4⋊C4—1061no—
443C2×C4⋊C4—1061no—
444C2×C4⋊C4—1061no—
445C2×C4⋊C4—1061no—
446C2×C4⋊C4C22×D5311yescharacteristic
447C2×C4⋊C4C22×D5311yescharacteristic
448C2×C4⋊C4C22×D5311yescharacteristic
449C2×C4⋊C4C22×D5311yescharacteristic
450C2×C22⋊C4—21no—
451C2×C22⋊C4—21no—
452C2×C22⋊C4—531no—
453C2×C22⋊C4—531no—
454C2×C22⋊C4—1061no—
455C2×C42—21no—
456C2×C42—531no—
457C2×C42—531no—
458C2×C42—1061no—
459C2×C42—1061no—
460C2×C42—1061no—
461C2×C42—1061no—
462C2×C42—1061no—
463C2×C42—1061no—
464C4⋊C8—1062no—
465C4⋊C8—1062no—
466#103—1062no—
467#103—1062no—
468#103—1062no—
469#103—1062no—
470#103—1062no—
471#103—1062no—
472#103—1062no—
473#103—1062no—
474C22⋊C8—22no—
475C22⋊C8—1062no—
476#97—1062no—
477C4×C8—1062no—
478#95—21no—
479#95—1061no—
480#95—1061no—
481#95—1061no—
482#95—1061no—
483#95—1061no—
484C24D4×D5311yescharacteristic
485C2×Q8—532no—
486C2×Q8—532no—
487C2×Q8—532no—
488C2×Q8—532no—
489C2×Q8—532no—
490C2×Q8—532no—
491C2×Q8—1068no—
492C2×Q8—1068no—
493C2×Q8—1064no—
494C2×Q8—1064no—
495C2×Q8—1068no—
496C2×Q8—1068no—
497C2×Q8—1064no—
498C2×Q8—1061no—
499C2×Q8—1061no—
500C2×Q8—1061no—
501C2×Q8—1061no—
502C2×Q8—1061no—
503C2×Q8—1061no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—531no—
513C22×C4—531no—
514C22×C4—531no—
515C22×C4—531no—
516C22×C4—1064no—
517C22×C4—1064no—
518C22×C4—1064no—
519C22×C4—1061no—
520C22×C4—1061no—
521C22×C4—1061no—
522C22×C4—1061no—
523C22×C4C2×C4×D5312yes—
524C22×C4D4×D5311yescharacteristic
525C22×C4D4×D5312yes—
526C22×C4C23×D5311yescharacteristic
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—1061no—
534C2×C8—1061no—
535C2×C8—1061no—
536C2×C8—1061no—
537C2×C8—1061no—
538C2×C8—1061no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—534no—
546C4⋊C4—534no—
547C4⋊C4—1064no—
548C4⋊C4—1064no—
549C4⋊C4—1064no—
550C4⋊C4—1064no—
551C4⋊C4—1061no—
552C4⋊C4—1061no—
553C4⋊C4—1061no—
554C4⋊C4—1061no—
555C4⋊C4—4241no—
556C4⋊C4—4241no—
557C4⋊C4C2×C4×D5312yes—
558C4⋊C4C2×C4×D5312yes—
559C22⋊C4—24no—
560C22⋊C4—81no—
561C22⋊C4—534no—
562C22⋊C4—534no—
563C22⋊C4—4241no—
564C42—24no—
565C42—532no—
566C42—532no—
567C42—532no—
568C42—532no—
569C42—1064no—
570C42—1064no—
571C42—1064no—
572C42—1064no—
573C42—4241no—
574C42—4241no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23C22⋊C4×D5312yes—
582C23C2×D4×D5311yescharacteristic
583Q8—534no—
584Q8—534no—
585Q8—10616no—
586Q8—1062no—
587Q8—1062no—
588Q8—1062no—
589Q8—1062no—
590Q8—1062no—
591Q8—1062no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—532no—
611C2×C4—532no—
612C2×C4—532no—
613C2×C4—532no—
614C2×C4—10616no—
615C2×C4—1062no—
616C2×C4—1062no—
617C2×C4—1062no—
618C2×C4—1062no—
619C2×C4—1062no—
620C2×C4—1062no—
621C2×C4—1061no—
622C2×C4—1061no—
623C2×C4—1061no—
624C2×C4—1061no—
625C2×C4—1061no—
626C2×C4—1061no—
627C2×C4—1061no—
628C2×C4—1061no—
629C2×C4—2121no—
630C2×C4—2121no—
631C2×C4—2121no—
632C2×C4—2121no—
633C2×C4—2121no—
634C2×C4—2121no—
635C2×C4C22⋊C4×D5312yes—
636C2×C4C22×C4×D5311yescharacteristic
637C2×C4C2×D4×D5311yescharacteristic
638C2×C4#41×D5311yescharacteristic
639C2×C4#41×D5311yescharacteristic
640C2×C4#41×D5311yescharacteristic
641C2×C4#41×D5311yescharacteristic
642C8—41no—
643C8—41no—
644C8—2121no—
645C8—2121no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22C2×C22⋊C4×D5311yescharacteristic
654C22#1681113111yescharacteristic
655C22#1681113111yescharacteristic
656C22#1681143011yescharacteristic
657C22#1681143311yescharacteristic
658C22#1681143711yescharacteristic
659C22#1681143911yescharacteristic
660C4—22no—
661C4—42no—
662C4—42no—
663C4—81no—
664C4—81no—
665C4—1064no—
666C4—1064no—
667C4—1062no—
668C4—1062no—
669C4—1062no—
670C4—1062no—
671C4—4241no—
672C4—4241no—
673C4#117×D5311yescharacteristic
674C4#117×D5311yescharacteristic
675C4#1681113111yescharacteristic
676C4#1681113111yescharacteristic
677C2—42no—
678C2#2027709511yescharacteristic
679C2#2028324411yescharacteristic
680C2#2028324611yescharacteristic
681C2#2028493911yescharacteristic
682C2#2028494711yescharacteristic
683C2#2028500211yescharacteristic
684C2#2028501111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1825C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
cosocleC22×C4×C53C24
hypercenter#1825D53
hypocenterC53#20745
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
5. #20274729C2semidirect
6. #20274729C2semidirect
7. #20274719C2semidirect
8. #20274719C2semidirect
11. #20274544C2semidirect
12. #20274544C2semidirect
13. #20274534C2semidirect
14. #20274534C2semidirect
18. #16811100C4semidirect
404. C53#20745semidirect

Maximal subgroups

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

subgroupindex
C2×#168111002
C2×#168106812
#202773222
#202748102
#202747292
#202747192
#202746512
#202746492
#202745442
#202745342
C53×#18252
#2074553