Finite Group Databasebeta

← #27469460

Subgroups — #27469460

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×#16810942C211yesmaximal, characteristic, complemented
2C2×#16810948C211yesmaximal, characteristic, complemented
3C2×#16810876C211yesmaximal, characteristic
4C2×#16810665C211yesmaximal, characteristic, complemented
5#20279186C211yesmaximal, characteristic, complemented
6#20279177C211yesmaximal, characteristic, complemented
7#20276247C211yesmaximal, characteristic, complemented
8#20276247C211yesmaximal, characteristic, complemented
9#20276242C211yesmaximal, characteristic, complemented
10#20276235C211yesmaximal, characteristic, complemented
11#20275591C211yesmaximal, characteristic, complemented
12#20275581C211yesmaximal, characteristic, complemented
13#20274883C211yesmaximal, characteristic, complemented
14#20274872C211yesmaximal, characteristic, complemented
15C53×#1753C211yesmaximal, characteristic, complemented
16C22×D424C2211yescharacteristic
17C2×#13133201—22no—
18C2×#13133191C2211yescharacteristic
19C2×#13133191C2211yescharacteristic
20C2×#13133176C2211yescharacteristic
21C22×C4⋊Dic53C2211yescharacteristic
22#16810948—24no—
23#16810942—24no—
24#16810876—24no—
25#16810665—24no—
26C2×#13133151C2211yescharacteristic
27C2×#13133151C2211yescharacteristic
28C2×#13133151C2211yescharacteristic
29C2×#13133149C2211yescharacteristic
30C2×#13133149C2211yescharacteristic
31C2×#13133149C2211yescharacteristic
32C2×#13133149C2211yescharacteristic
33C2×#13133149C2211yescharacteristic
34C2×#13133149C2211yescharacteristic
35C2×#13133139C2211yescharacteristic
36C2×#13133139C2211yescharacteristic
37C2×#13133139C2211yescharacteristic
38C2×#13133139C2211yescharacteristic
39C2×#13133139C2211yescharacteristic
40C2×#13133139C2211yescharacteristic
41#16810560—22no—
42#16810539—22no—
43#16810525C2211yescharacteristic
44#16810525C2211yescharacteristic
45#16810525C2211yescharacteristic
46#16810525C2211yescharacteristic
47#16810515—22no—
48#16810493C2211yescharacteristic
49#16810493C2211yescharacteristic
50#16810493C2211yescharacteristic
51#16810489C2211yescharacteristic
52C22×M16×C53C2211yescharacteristic
53C2×C22⋊C8×C53C2211yescharacteristic
54C2×#97×C53C2211yescharacteristic
55C2×C4×C8×C53C2211yescharacteristic
56C4×C22⋊C4×C53C2211yescharacteristic
57C53×#336C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×D212C2311yescharacteristic
60C22×D212C2311yescharacteristic
61C22×C22⋊D53—21no—
62C22×C22⋊D53—21no—
63C23×Dic53—21no—
64C2×D424—22no—
65C2×D424—22no—
66C2×D424—22no—
67#13133201—48no—
68#13133191—22no—
69#13133191—22no—
70#13133191—22no—
71#13133191—22no—
72#13133176—22no—
73#13133176—22no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—22no—
76C2×C4⋊Dic53—22no—
77C2×C4⋊Dic53—22no—
78C2×C4⋊Dic53—22no—
79C2×C4⋊Dic53—22no—
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×C4⋊Dic53C2311yescharacteristic
84C2×C4⋊Dic53C2311yescharacteristic
85C2×C4⋊Dic53C2311yescharacteristic
86C2×C22⋊Dic53—21no—
87C2×#11719921—21no—
88C2×#11719921—22no—
89C2×#11719921—22no—
90C2×#11719921—22no—
91C2×#11719921—21no—
92C2×#11719921—22no—
93C2×C4×Dic53—21no—
94#13133151—22no—
95#13133151—22no—
96#13133151—22no—
97#13133149—22no—
98#13133149—22no—
99#13133149—22no—
100#13133149—22no—
101#13133149—22no—
102#13133149—22no—
103#13133139—22no—
104#13133139—22no—
105#13133139—22no—
106#13133139—22no—
107#13133139—22no—
108#13133139—22no—
109#13133124—22no—
110C23×C4×C53C2311yescharacteristic
111C2×M16×C53—22no—
112C2×M16×C53—22no—
113C2×M16×C53D414yes—
114C22×C8×C53C2311yescharacteristic
115C22×C8×C53C2311yescharacteristic
116C22×C8×C53C2311yescharacteristic
117C22×C8×C53C2311yescharacteristic
118C2×C22⋊C4×C53D412yes—
119C2×C42×C53C2311yescharacteristic
120C2×C42×C53C2311yescharacteristic
121C22⋊C8×C53D414yes—
122#97×C53—22no—
123C4×C8×C53—22no—
124#95×C53D412yes—
125C24×C53C2×D411yescharacteristic
126C23×D53—21no—
127C23×D53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53C2×D412yes—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D412yes—
135C22×C4×C53C2×D412yes—
136C22×C4×C53C2×D412yes—
137C22×C4×C53C2411yescharacteristic
138C2×C22⋊D53—44no—
139C2×C22⋊D53—42no—
140C2×C22⋊D53—42no—
141C2×C22⋊D53—82no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—44no—
149C22×Dic53—42no—
150C2×D212—21no—
151C2×D212—21no—
152C2×D212—21no—
153C2×D212—21no—
154C2×D212—21no—
155C2×D212—21no—
156C2×D212—21no—
157C2×D212—21no—
158C2×D212—21no—
159C2×D212—21no—
160C2×D212—21no—
161C2×D212—21no—
162M16×C53—28no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53C2×D412yes—
170C2×C8×C53C2×D412yes—
171C2×C8×C53C2×D412yes—
172C2×C8×C53C2×D412yes—
173C2×C8×C53#4112yes—
174C2×C8×C53#4112yes—
175C22⋊C4×C53—24no—
176C42×C53—21no—
177C42×C53—21no—
178C42×C53—21no—
179C42×C53—21no—
180C22⋊Dic53—42no—
181#11719921—44no—
182#11719921—42no—
183#11719921—42no—
184#11719921—44no—
185#11719921—44no—
186#11719921—82no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—44no—
200C4⋊Dic53—42no—
201C4⋊Dic53—44no—
202C4⋊Dic53—82no—
203C4×Dic53—42no—
204D424—44no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—22no—
208C23×C53—22no—
209C23×C53#12712yes—
210C23×C53C22×D411yescharacteristic
211C22×D53—41no—
212C22×D53—41no—
213C22×D53—41no—
214C22×D53—41no—
215C22×D53—41no—
216C22×D53—41no—
217C22×D53—42no—
218C22×D53—42no—
219C22×D53—42no—
220C22×D53—42no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—22no—
224C2×C4×C53—22no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53#12712yes—
233C2×C4×C53AGL(1,ℤ/8)14yes—
234C2×C4×C53C22×D411yescharacteristic
235C2×C4×C53C22×D411yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C532−1+411yescharacteristic
239C2×C4×C532−1+411yescharacteristic
240C22⋊D53—84no—
241C22⋊D53—84no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—84no—
255D212—41no—
256D212—41no—
257D212—41no—
258D212—41no—
259D212—41no—
260D212—41no—
261D212—41no—
262D212—41no—
263C8×C53—22no—
264C8×C53—22no—
265C8×C53—22no—
266C8×C53#12314yescomplemented
267#27941—531nomaximal, Sylow
268C22×C53—22no—
269C22×C53—22no—
270C22×C53—22no—
271C22×C53—42no—
272C22×C53—42no—
273C22×C53C2×#12711yescharacteristic
274C22×C53#54711yescharacteristic
275C22×C53#54711yescharacteristic
276C22×C53C2×AGL(1,ℤ/8)12yes—
277C22×C53#57712yes—
278C2×D53—82no—
279C2×D53—82no—
280C2×D53—82no—
281C2×D53—82no—
282C2×D53—81no—
283C2×D53—81no—
284C2×D53—81no—
285C2×D53—81no—
286C2×D53—81no—
287C2×D53—81no—
288C4×C53—24no—
289C4×C53—22no—
290C4×C53—42no—
291C4×C53C2×#12311yescharacteristic
292C4×C53#53611yescharacteristic
293C4×C53#54711yescharacteristic
294C4×C53#54711yescharacteristic
295Dic53—81no—
296Dic53—81no—
297Dic53—81no—
298Dic53—81no—
299Dic53—81no—
300Dic53—81no—
301C2×#481—531no—
302C2×#482—531no—
303C2×#469—531no—
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#1753D5311yescharacteristic, complemented
316C2×C53—42no—
317C2×C53#250111yescharacteristic
318C2×C53#315111yescharacteristic
319C2×C53#315411yescharacteristic
320C2×C53#334512yes—
321C2×C53#335412yes—
322D53—161no—
323D53—161no—
324C22×D8—531no—
325C22×M16C2×D5311yescharacteristic
326C2×#123—1062no—
327C2×#121—531no—
328C2×#121—531no—
329C2×#117—531no—
330C22×C4⋊C4—531no—
331#482—1064no—
332#481—1064no—
333#469—1064no—
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×#106—531no—
344C2×#102—531no—
345C2×#102—531no—
346C2×#102—531no—
347C2×#102—531no—
348C2×#102—531no—
349C2×#102—531no—
350C2×C22⋊C8C2×D5311yescharacteristic
351C2×#97C2×D5311yescharacteristic
352C2×C4×C8C2×D5311yescharacteristic
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⋊C4C2×D5311yescharacteristic
365#336C2×D5311yescharacteristic
366#336C2×D5311yescharacteristic
367C53#2794111yescharacteristic, complemented, Sylow
368C22×D4—531no—
369C22×D4—531no—
370C22×D4—1061no—
371C22×D4—1061no—
372C23×C4—1061no—
373C23×C4C22×D5311yescharacteristic
374C2×D8—1062no—
375C2×D8—1062no—
376C2×D8—1062no—
377C2×M16—22no—
378C2×M16—22no—
379C2×M16D21214yes—
380C22×C8C22×D5311yescharacteristic
381C22×C8C22×D5311yescharacteristic
382C22×C8C22×D5311yescharacteristic
383C22×C8C22×D5311yescharacteristic
384#123—2128no—
385#121—1062no—
386#121—1062no—
387#121—1062no—
388#121—1062no—
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—531no—
396C2×C4⋊C4—531no—
397C2×C4⋊C4—1062no—
398C2×C4⋊C4—1062no—
399C2×C4⋊C4—1062no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4—1062no—
402C2×C4⋊C4—1062no—
403C2×C22⋊C4—1062no—
404C2×C22⋊C4—1061no—
405C2×C22⋊C4—1061no—
406C2×C22⋊C4—1061no—
407C2×C22⋊C4—1062no—
408C2×C22⋊C4—1062no—
409C2×C22⋊C4—1062no—
410C2×C22⋊C4D21212yes—
411C2×C42—1061no—
412C2×C42C22×D5311yescharacteristic
413C2×C42C22×D5311yescharacteristic
414#107—1062no—
415#107—1062no—
416#107—1062no—
417#106—1062no—
418#106—1062no—
419#106—1062no—
420#106—1062no—
421#106—1062no—
422#106—1062no—
423#102—1062no—
424#102—1062no—
425#102—1062no—
426#102—1062no—
427#102—1062no—
428#102—1062no—
429C22⋊C8D21214yes—
430#97—22no—
431C4×C8—22no—
432#95—1062no—
433#95D21212yes—
434C24—1061no—
435C24—1061no—
436C24C2×D21211yescharacteristic
437C2×D4—1061no—
438C2×D4—1061no—
439C2×D4—1061no—
440C2×D4—1061no—
441C2×D4—1061no—
442C2×D4—1061no—
443C2×D4—1061no—
444C2×D4—1061no—
445C2×D4—1061no—
446C2×D4—1061no—
447C2×D4—1061no—
448C2×D4—1061no—
449C2×D4—4242no—
450C2×D4—2124no—
451C2×D4—2122no—
452C2×D4—2122no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—2124no—
464C22×C4—2122no—
465C22×C4C2×D21212yes—
466C22×C4C2×D21211yescharacteristic
467C22×C4C2×D21212yes—
468C22×C4C2×D21212yes—
469C22×C4C2×D21212yes—
470C22×C4C23×D5311yescharacteristic
471D8—2124no—
472M16—28no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8C2×D21212yes—
480C2×C8C2×D21212yes—
481C2×C8C2×D21212yes—
482C2×C8C2×D21212yes—
483C2×C8#1171994712yes—
484C2×C8#1171994712yes—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—1061no—
488C4⋊C4—1061no—
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—2122no—
500C4⋊C4—2124no—
501C22⋊C4—24no—
502C22⋊C4—4242no—
503C22⋊C4—2124no—
504C22⋊C4—2122no—
505C22⋊C4—2122no—
506C22⋊C4—2122no—
507C22⋊C4—2124no—
508C22⋊C4—2124no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—2122no—
514C23—21no—
515C23—21no—
516C23—22no—
517C23—22no—
518C23—2121no—
519C23—2121no—
520C23—2121no—
521C23—2121no—
522C23—2122no—
523C23—2122no—
524C23—2121no—
525C23—2121no—
526C23—2122no—
527C23—2122no—
528C23#1313321412yes—
529C23C22×D21211yescharacteristic
530D4—4244no—
531D4—4244no—
532D4—2121no—
533D4—2121no—
534D4—2121no—
535D4—2121no—
536D4—2121no—
537D4—2121no—
538D4—2121no—
539D4—2121no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—4244no—
552C2×C4—2122no—
553C2×C4—2122no—
554C2×C4—2122no—
555C2×C4—2122no—
556C2×C4—2121no—
557C2×C4—2121no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2122no—
563C2×C4—2122no—
564C2×C4#1313321412yes—
565C2×C4#1313324514yes—
566C2×C4C22×D21211yescharacteristic
567C2×C4C22×D21211yescharacteristic
568C2×C4C2×#1171994711yescharacteristic
569C2×C4C2×#1171994711yescharacteristic
570C2×C4#1313326411yescharacteristic
571C2×C4#1313326411yescharacteristic
572C8—22no—
573C8—22no—
574C8—22no—
575C8#1313320114yescomplemented
576C22—22no—
577C22—22no—
578C22—22no—
579C22—42no—
580C22—42no—
581C22—4241no—
582C22—4241no—
583C22—4242no—
584C22—4242no—
585C22—4241no—
586C22—4241no—
587C22—4242no—
588C22—4242no—
589C22—4241no—
590C22—4241no—
591C22C2×#1313321411yescharacteristic
592C22#1681127711yescharacteristic
593C22#1681127711yescharacteristic
594C22C2×#1313324512yes—
595C22#1681143512yes—
596C4—22no—
597C4—24no—
598C4—42no—
599C4—4241no—
600C4—4241no—
601C4—4241no—
602C4—4241no—
603C4—4241no—
604C4—4241no—
605C4C2×#1313320111yescharacteristic
606C4#1681119911yescharacteristic
607C4#1681127711yescharacteristic
608C4#1681127711yescharacteristic
609C2—42no—
610C2—8481no—
611C2—8481no—
612C2#2027918611yescharacteristic
613C2#2028516711yescharacteristic
614C2#2028519111yescharacteristic
615C2#2028718712yes—
616C2#2028731112yes—
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1753C2
socleC23×C53C22×D4
prime radicalC2×#16810876C2
hypercenter#1753D53
hypocenterC53#27941
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
1. C2×#16810942C2semidirect
2. C2×#16810948C2semidirect
4. C2×#16810665C2semidirect
5. #20279186C2semidirect
6. #20279177C2semidirect
7. #20276247C2semidirect
8. #20276247C2semidirect
9. #20276242C2semidirect
10. #20276235C2semidirect
11. #20275591C2semidirect
12. #20275581C2semidirect
13. #20274883C2semidirect
14. #20274872C2semidirect
15. C53×#1753C2semidirect
266. C8×C53#123semidirect
315. #1753D53semidirect
367. C53#27941semidirect
575. C8#13133201semidirect

Maximal subgroups

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

subgroupindex
C2×#168109422
C2×#168109482
C2×#168108762
C2×#168106652
#202791862
#202791772
#202762472
#202762422
#202762352
#202755912
#202755812
#202748832
#202748722
C53×#17532
#2794153