Finite Group Databasebeta

← #27469462

Subgroups — #27469462

Order 13568.

986 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810939C211yesmaximal, characteristic, complemented
2C2×#16810944C211yesmaximal, characteristic, complemented
3C2×#16810665C211yesmaximal, characteristic, complemented
4#20279187C211yesmaximal, characteristic, complemented
5#20279178C211yesmaximal, characteristic, complemented
6#20276246C211yesmaximal, characteristic, complemented
7#20276246C211yesmaximal, characteristic, complemented
8#20276241C211yesmaximal, characteristic, complemented
9#20276234C211yesmaximal, characteristic, complemented
10#20275592C211yesmaximal, characteristic, complemented
11#20275582C211yesmaximal, characteristic, complemented
12#20274880C211yesmaximal, characteristic, complemented
13#20274870C211yesmaximal, characteristic, complemented
14#20274496C211yesmaximal, characteristic, complemented
15C2×C53×#469C211yesmaximal, characteristic
16C2×#13133198—22no—
17C2×#13133176C2211yescharacteristic
18C22×C4⋊Dic53C2211yescharacteristic
19#16810944—24no—
20#16810939—24no—
21#16810665—24no—
22C2×#13133151C2211yescharacteristic
23C2×#13133151C2211yescharacteristic
24C2×#13133150C2211yescharacteristic
25C2×#13133149C2211yescharacteristic
26C2×#13133149C2211yescharacteristic
27C2×#13133148C2211yescharacteristic
28C2×#13133148C2211yescharacteristic
29C2×#13133148C2211yescharacteristic
30C2×#13133138C2211yescharacteristic
31C2×#13133138C2211yescharacteristic
32C2×#13133138C2211yescharacteristic
33C2×#13133138C2211yescharacteristic
34C2×C22⋊#88322C2211yescharacteristic
35C2×#13133127C2211yescharacteristic
36C2×C8×Dic53C2211yescharacteristic
37#16810559—22no—
38#16810537—22no—
39#16810526C2211yescharacteristic
40#16810526C2211yescharacteristic
41#16810526C2211yescharacteristic
42#16810526C2211yescharacteristic
43#16810511—22no—
44#16810492C2211yescharacteristic
45#16810492C2211yescharacteristic
46#16810492C2211yescharacteristic
47#16810488C2211yescharacteristic
48C4×C22⋊Dic53C2211yescharacteristic
49#16810377C2211yescharacteristic
50#16810377C2211yescharacteristic
51C22×D8×C53C2211yescharacteristic
52C22×M16×C53C2211yescharacteristic
53C2×#121×C53C2211yescharacteristic
54C2×#121×C53C2211yescharacteristic
55C53×#469—24no—
56C2×#106×C53C2211yescharacteristic
57C2×#102×C53C2211yescharacteristic
58C2×#102×C53C2211yescharacteristic
59C23×Dic53—21no—
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62#13133198—48no—
63#13133176—22no—
64#13133176—22no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—22no—
67C2×C4⋊Dic53—22no—
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×#11719919—22no—
73C2×#11719919—22no—
74C2×#11719919—22no—
75C2×C22⋊Dic53—21no—
76C2×C22⋊Dic53—22no—
77C2×C22⋊Dic53—22no—
78C2×C22⋊Dic53—22no—
79C2×C22⋊Dic53—22no—
80C2×C22⋊Dic53D412yes—
81C2×C4×Dic53—21no—
82C2×C4×Dic53C2311yescharacteristic
83C2×C4×Dic53C2311yescharacteristic
84#13133151—22no—
85#13133151—22no—
86#13133150—22no—
87#13133149—22no—
88#13133149—22no—
89#13133148—22no—
90#13133148—22no—
91#13133148—22no—
92#13133138—22no—
93#13133138—22no—
94#13133138—22no—
95#13133138—22no—
96C22⋊#88322D414yes—
97#13133127—22no—
98C8×Dic53—22no—
99#13133124—22no—
100#13133124D412yes—
101C22×D4×C53—21no—
102C22×D4×C53—21no—
103C22×D4×C53C2311yescharacteristic
104C22×D4×C53C2311yescharacteristic
105C23×C4×C53C2311yescharacteristic
106C2×D8×C53—22no—
107C2×D8×C53—22no—
108C2×D8×C53—22no—
109C2×M16×C53—22no—
110C2×M16×C53—22no—
111C2×M16×C53D414yes—
112C22×C8×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114#121×C53—22no—
115#121×C53—22no—
116#121×C53—22no—
117#121×C53—22no—
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C22⋊C4×C53—21no—
121C2×C22⋊C4×C53—21no—
122#106×C53—22no—
123#102×C53—22no—
124#102×C53—22no—
125C24×C53—21no—
126C24×C53—21no—
127C24×C53C2×D411yescharacteristic
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53—21no—
136C2×D4×C53—21no—
137C2×D4×C53—21no—
138C2×D4×C53—21no—
139C2×D4×C53—21no—
140C2×D4×C53—44no—
141C2×D4×C53—42no—
142C2×D4×C53—42no—
143C2×D4×C53—82no—
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×C53C2×D411yescharacteristic
151C22×C4×C53C2×D412yes—
152C22×C4×C53C2×D412yes—
153C22×C4×C53C2411yescharacteristic
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—21no—
158C22×Dic53—44no—
159C22×Dic53—42no—
160C22×Dic53C2×D412yes—
161C22×Dic53C2×D412yes—
162D8×C53—44no—
163M16×C53—28no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53C2×D412yes—
167C2×C8×C53C2×D412yes—
168C2×C8×C53#4112yes—
169C2×C8×C53#4112yes—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C22⋊C4×C53—42no—
175C22⋊C4×C53—42no—
176C22⋊Dic53—24no—
177C22⋊Dic53—44no—
178C22⋊Dic53—42no—
179C22⋊Dic53—44no—
180C22⋊Dic53—44no—
181C22⋊Dic53—82no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—42no—
191#11719919—44no—
192#11719919—44no—
193#11719919—82no—
194C4×Dic53—21no—
195C4×Dic53—21no—
196C4×Dic53—21no—
197C4×Dic53—21no—
198C4×Dic53—42no—
199C2×#88322—21no—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322C2×D412yes—
204C2×#88322C2×D412yes—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—22no—
208C23×C53—22no—
209C23×C53—41no—
210C23×C53—41no—
211C23×C53—41no—
212C23×C53—41no—
213C23×C53—41no—
214C23×C53—41no—
215C23×C53—42no—
216C23×C53—42no—
217C23×C53—42no—
218C23×C53—42no—
219C23×C53#12712yes—
220C23×C53C22×D411yescharacteristic
221D4×C53—41no—
222D4×C53—41no—
223D4×C53—41no—
224D4×C53—41no—
225D4×C53—41no—
226D4×C53—41no—
227D4×C53—41no—
228D4×C53—41no—
229D4×C53—84no—
230D4×C53—84no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—42no—
240C2×C4×C53—42no—
241C2×C4×C53#12712yes—
242C2×C4×C53C22×D411yescharacteristic
243C2×C4×C53C22×D411yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C2×C4×C532−1+411yescharacteristic
247C2×C4×C532−1+411yescharacteristic
248C2×Dic53—22no—
249C2×Dic53—22no—
250C2×Dic53—22no—
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—42no—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C2×Dic53—42no—
261C2×Dic53—84no—
262C2×Dic53AGL(1,ℤ/8)14yes—
263C8×C53—22no—
264C8×C53#12314yescomplemented
265#88322—22no—
266#88322—22no—
267#27941—531nomaximal, Sylow
268C22×C53—22no—
269C22×C53—22no—
270C22×C53—22no—
271C22×C53—42no—
272C22×C53—42no—
273C22×C53—82no—
274C22×C53—82no—
275C22×C53—82no—
276C22×C53—82no—
277C22×C53—81no—
278C22×C53—81no—
279C22×C53—81no—
280C22×C53—81no—
281C22×C53—81no—
282C22×C53—81no—
283C22×C53C2×#12711yescharacteristic
284C22×C53#54711yescharacteristic
285C22×C53#54711yescharacteristic
286C22×C53C2×AGL(1,ℤ/8)12yes—
287C22×C53#57712yes—
288C4×C53—22no—
289C4×C53—81no—
290C4×C53—81no—
291C4×C53C2×#12311yescharacteristic
292C4×C53#53611yescharacteristic
293C4×C53#54711yescharacteristic
294C4×C53#54711yescharacteristic
295Dic53—24no—
296Dic53—42no—
297Dic53—81no—
298Dic53—81no—
299Dic53—81no—
300Dic53—81no—
301C2×#481—531no—
302C2×#482—531no—
303C2×#469D5311yescharacteristic
304C2×#428—531no—
305#2501—531no—
306#2500—531no—
307#2094—531no—
308#2094—531no—
309#2093—531no—
310#2092—531no—
311#1972—531no—
312#1970—531no—
313#1847—531no—
314#1844—531no—
315#1753—531no—
316C2×C53—42no—
317C2×C53—161no—
318C2×C53—161no—
319C2×C53#250111yescharacteristic
320C2×C53#315111yescharacteristic
321C2×C53#315411yescharacteristic
322C2×C53#334512yes—
323C2×C53#335412yes—
324C22×D8C2×D5311yescharacteristic
325C22×M16C2×D5311yescharacteristic
326C2×#123—1062no—
327C2×#121C2×D5311yescharacteristic
328C2×#121C2×D5311yescharacteristic
329C2×#117—531no—
330C22×C4⋊C4—531no—
331#482—1064no—
332#481—1064no—
333#469—24no—
334#428—1064no—
335C2×#107—531no—
336C2×#107—531no—
337C2×#107—531no—
338C2×#106—531no—
339C2×#106—531no—
340C2×#106—531no—
341C2×#106—531no—
342C2×#106—531no—
343C2×#106C2×D5311yescharacteristic
344C2×#102—531no—
345C2×#102—531no—
346C2×#102—531no—
347C2×#102—531no—
348C2×#102C2×D5311yescharacteristic
349C2×#102C2×D5311yescharacteristic
350C2×C22⋊C8—531no—
351C2×#97—531no—
352C2×C4×C8—531no—
353#399—1062no—
354#394—1062no—
355#390—531no—
356#390—531no—
357#390—531no—
358#390—531no—
359#388—1062no—
360#384—531no—
361#384—531no—
362#384—531no—
363#382—531no—
364C4×C22⋊C4—531no—
365#336—531no—
366#336—531no—
367C53#2794111yescharacteristic, complemented, Sylow
368C22×D4—21no—
369C22×D4—21no—
370C22×D4C22×D5311yescharacteristic
371C22×D4C22×D5311yescharacteristic
372C23×C4—1061no—
373C23×C4C22×D5311yescharacteristic
374C2×D8—22no—
375C2×D8—22no—
376C2×D8—22no—
377C2×M16—22no—
378C2×M16—22no—
379C2×M16C22⋊D5314yes—
380C22×C8—531no—
381C22×C8—531no—
382C22×C8C22×D5311yescharacteristic
383C22×C8C22×D5311yescharacteristic
384#123—2128no—
385#121—22no—
386#121—22no—
387#121—22no—
388#121—22no—
389#117—1062no—
390#117—1062no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—531no—
395C2×C4⋊C4—1062no—
396C2×C4⋊C4—1062no—
397C2×C4⋊C4—1062no—
398C2×C4⋊C4—1062no—
399C2×C4⋊C4—1062no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4C22×D5311yescharacteristic
402C2×C4⋊C4C22×D5311yescharacteristic
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—532no—
406C2×C22⋊C4—1062no—
407C2×C22⋊C4—1062no—
408C2×C22⋊C4—1062no—
409C2×C22⋊C4—1062no—
410C2×C22⋊C4—1061no—
411C2×C42—531no—
412C2×C42—531no—
413C2×C42—1061no—
414#107—1062no—
415#107—1062no—
416#107—1062no—
417#106—22no—
418#106—1062no—
419#106—1062no—
420#106—1062no—
421#106—1062no—
422#106—1062no—
423#102—22no—
424#102—22no—
425#102—1062no—
426#102—1062no—
427#102—1062no—
428#102—1062no—
429C22⋊C8—534no—
430#97—1062no—
431C4×C8—1062no—
432#95—532no—
433#95—1062no—
434C24—21no—
435C24—21no—
436C24D4×D5311yescharacteristic
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—44no—
450C2×D4—42no—
451C2×D4—42no—
452C2×D4—82no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—532no—
460C22×C4—532no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—2124no—
466C22×C4—2122no—
467C22×C4D4×D5311yescharacteristic
468C22×C4D4×D5312yes—
469C22×C4C2×C22⋊D5312yes—
470C22×C4C23×D5311yescharacteristic
471D8—44no—
472M16—28no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—532no—
476C2×C8—532no—
477C2×C8—1061no—
478C2×C8—1061no—
479C2×C8—1061no—
480C2×C8—1061no—
481C2×C8#1171994712yes—
482C2×C8#1171994712yes—
483C2×C8C2×C22⋊D5312yes—
484C2×C8C2×C22⋊D5312yes—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—1061no—
496C4⋊C4—1061no—
497C4⋊C4—4242no—
498C4⋊C4—2124no—
499C4⋊C4—2124no—
500C4⋊C4—2122no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—1064no—
504C22⋊C4—4242no—
505C22⋊C4—2124no—
506C22⋊C4—2124no—
507C22⋊C4—2124no—
508C22⋊C4—2122no—
509C42—1061no—
510C42—1061no—
511C42—1061no—
512C42—1061no—
513C42—2122no—
514C23—21no—
515C23—21no—
516C23—22no—
517C23—22no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—42no—
523C23—42no—
524C23—41no—
525C23—41no—
526C23—42no—
527C23—42no—
528C23#1313321312yes—
529C23C2×D4×D5311yescharacteristic
530D4—41no—
531D4—41no—
532D4—41no—
533D4—41no—
534D4—41no—
535D4—41no—
536D4—41no—
537D4—41no—
538D4—84no—
539D4—84no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—534no—
551C2×C4—1062no—
552C2×C4—1062no—
553C2×C4—1062no—
554C2×C4—1062no—
555C2×C4—1062no—
556C2×C4—4244no—
557C2×C4—2122no—
558C2×C4—2122no—
559C2×C4—2122no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2122no—
565C2×C4#1313321312yes—
566C2×C4C2×D4×D5311yescharacteristic
567C2×C4C22×C22⋊D5311yescharacteristic
568C2×C4C2×#1171994711yescharacteristic
569C2×C4C2×#1171994711yescharacteristic
570C2×C4#1313326411yescharacteristic
571C2×C4#1313326411yescharacteristic
572C8—22no—
573C8—1062no—
574C8—1062no—
575C8#1313319814yescomplemented
576C22—22no—
577C22—22no—
578C22—22no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—82no—
584C22—82no—
585C22—81no—
586C22—81no—
587C22—82no—
588C22—82no—
589C22—81no—
590C22—81no—
591C22C2×#1313321311yescharacteristic
592C22#1681127311yescharacteristic
593C22#1681127311yescharacteristic
594C22AGL(1,ℤ/8)×D5312yes—
595C22#1681143312yes—
596C4—22no—
597C4—81no—
598C4—81no—
599C4—1064no—
600C4—4241no—
601C4—4241no—
602C4—4241no—
603C4—4241no—
604C4—2122no—
605C4C2×#1313319811yescharacteristic
606C4#1681119411yescharacteristic
607C4#1681127311yescharacteristic
608C4#1681127311yescharacteristic
609C2—42no—
610C2—161no—
611C2—161no—
612C2#2027918711yescharacteristic
613C2#2028516311yescharacteristic
614C2#2028518911yescharacteristic
615C2#2028718012yes—
616C2#2028730512yes—
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#469C2
socleC23×C53C22×D4
hypercenterC2×#469D53
hypocenterC53#27941
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#469

Complements

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

NH ≅ G/Ntype
1. C2×#16810939C2semidirect
2. C2×#16810944C2semidirect
3. C2×#16810665C2semidirect
4. #20279187C2semidirect
5. #20279178C2semidirect
6. #20276246C2semidirect
7. #20276246C2semidirect
8. #20276241C2semidirect
9. #20276234C2semidirect
10. #20275592C2semidirect
11. #20275582C2semidirect
12. #20274880C2semidirect
13. #20274870C2semidirect
14. #20274496C2semidirect
264. C8×C53#123semidirect
367. C53#27941semidirect
575. C8#13133198semidirect

Maximal subgroups

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

subgroupindex
C2×#168109392
C2×#168109442
C2×#168106652
#202791872
#202791782
#202762462
#202762412
#202762342
#202755922
#202755822
#202748802
#202748702
#202744962
C2×C53×#4692
#2794153