Finite Group Databasebeta

← #27429383

Subgroups — #27429383

Order 13568.

716 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C4⋊Dic53C211yesmaximal, characteristic, complemented
2#20276278C211yesmaximal, characteristic, complemented
3#20276273C211yesmaximal, characteristic, complemented
4#20276118C211yesmaximal, characteristic
5#20276118C211yesmaximal, characteristic
6#20275656C211yesmaximal, characteristic, complemented
7#20275481C211yesmaximal, characteristic, complemented
8#20275339C211yesmaximal, characteristic
9#20275332C211yesmaximal, characteristic
10#20275304C211yesmaximal, characteristic, complemented
11#20275297C211yesmaximal, characteristic, complemented
12#20274848C211yesmaximal, characteristic, complemented
13#20274715C211yesmaximal, characteristic
14#20274540C211yesmaximal, characteristic
15C53×#2580C211yesmaximal, characteristic
16C2×Q8×Dic53—21no—
17C2×#13133150C2211yescharacteristic
18C2×#13133148C2211yescharacteristic
19C2×C4⋊#88322C2211yescharacteristic
20C2×#13133138C2211yescharacteristic
21C2×#13133138C2211yescharacteristic
22C2×#13133138C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24C2×#13133128C2211yescharacteristic
25#16810522—21no—
26#16810522—21no—
27#16810522—21no—
28#16810493C2211yescharacteristic
29#16810493C2211yescharacteristic
30#16810493C2211yescharacteristic
31C4×C4⋊Dic53C2211yescharacteristic
32C4×C4⋊Dic53C2211yescharacteristic
33C4×C4⋊Dic53C2211yescharacteristic
34#16810385C2211yescharacteristic
35#16810385C2211yescharacteristic
36#16810385C2211yescharacteristic
37#16810385C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810378C2211yescharacteristic
45#16810378C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×C4×Q8×C53C2211yescharacteristic
49C2×C4×D4×C53C2211yescharacteristic
50C53×#399—21no—
51C53×#393—21no—
52C53×#390C2211yescharacteristic
53C53×#390C2211yescharacteristic
54C53×#389C2211yescharacteristic
55C53×#389C2211yescharacteristic
56C53×#385—21no—
57C53×#385—21no—
58C53×#382C2211yescharacteristic
59C22×#88322C2311yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63Q8×Dic53—44no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4×Dic53—21no—
75C2×C4×Dic53—21no—
76C2×C4×Dic53—21no—
77#13133150—22no—
78#13133148—22no—
79C4⋊#88322—22no—
80#13133138—22no—
81#13133138—22no—
82#13133138—22no—
83#13133138—22no—
84#13133128—22no—
85#13133124—21no—
86#13133124—21no—
87#13133124—21no—
88C22×Q8×C53D411yescharacteristic
89C22×D4×C53C2311yescharacteristic
90C23×C4×C53—21no—
91C4×Q8×C53—22no—
92C4×Q8×C53—22no—
93C4×D4×C53—22no—
94C4×D4×C53—22no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53D411yescharacteristic
98C2×C4⋊C4×C53D411yescharacteristic
99C2×C4⋊C4×C53D411yescharacteristic
100C2×C4⋊C4×C53C2311yescharacteristic
101C2×C4⋊C4×C53C2311yescharacteristic
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C22⋊C4×C53—21no—
104C2×C22⋊C4×C53—21no—
105C2×C22⋊C4×C53—21no—
106C2×C22⋊C4×C53—21no—
107C2×C22⋊C4×C53—21no—
108C2×C42×C53C2311yescharacteristic
109C2×C42×C53C2311yescharacteristic
110C2×C42×C53C2311yescharacteristic
111#95×C53—21no—
112#95×C53—21no—
113#95×C53—21no—
114#95×C53—21no—
115C24×C53—21no—
116C2×Q8×C53—22no—
117C2×Q8×C53—22no—
118C2×Q8×C53—22no—
119C2×D4×C53—21no—
120C2×D4×C53—21no—
121C2×D4×C53—21no—
122C2×D4×C53—21no—
123C2×D4×C53—21no—
124C2×D4×C53#4112yes—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—42no—
129C22×C4×C53—81no—
130C22×C4×C53—81no—
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2411yescharacteristic
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C4⋊C4×C53—22no—
143C4⋊C4×C53—22no—
144C4⋊C4×C53—22no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—81no—
152C4⋊C4×C53—81no—
153C22⋊C4×C53—42no—
154C22⋊C4×C53—42no—
155C22⋊C4×C53—42no—
156C22⋊C4×C53—81no—
157C22⋊C4×C53—81no—
158C42×C53—21no—
159C42×C53—21no—
160C42×C53—21no—
161C42×C53—21no—
162C42×C53#4112yes—
163C42×C53#4112yes—
164C4⋊Dic53—21no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—42no—
172C4⋊Dic53—42no—
173C4⋊Dic53—42no—
174C4⋊Dic53—81no—
175C4⋊Dic53—81no—
176C4⋊Dic53—81no—
177C4⋊Dic53#4112yes—
178C4×Dic53—42no—
179C4×Dic53—42no—
180C4×Dic53—42no—
181C2×#88322—21no—
182C2×#88322—21no—
183C2×#88322—21no—
184C2×#88322—21no—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C23×C53—22no—
194C23×C53—41no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53—41no—
198C23×C53—41no—
199C23×C53—41no—
200C23×C53C22×D411yescharacteristic
201Q8×C53—28no—
202D4×C53—22no—
203D4×C53—22no—
204D4×C53—41no—
205D4×C53—41no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—21no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—84no—
230C2×C4×C53#12312yes—
231C2×C4×C53#12312yes—
232C2×C4×C53AGL(1,ℤ/8)12yes—
233C2×C4×C53#13712yes—
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C53C2×#4111yescharacteristic
238C2×C4×C532−1+411yescharacteristic
239C2×C4×C532−1+411yescharacteristic
240C2×Dic53—22no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252#88322—41no—
253#88322—41no—
254#88322—41no—
255#88322—41no—
256#25002—531nomaximal, Sylow
257C22×C53—42no—
258C22×C53—42no—
259C22×C53—42no—
260C22×C53—81no—
261C22×C53—81no—
262C22×C53—81no—
263C22×C53—81no—
264C22×C53C2×#12311yescharacteristic
265C22×C53#54811yescharacteristic
266C22×C53#54811yescharacteristic
267C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
268C22×C53C2×#13711yescharacteristic
269C22×C53#57611yescharacteristic
270C22×C53#57711yescharacteristic
271C4×C53—22no—
272C4×C53—22no—
273C4×C53—22no—
274C4×C53—22no—
275C4×C53—41no—
276C4×C53—41no—
277C4×C53—81no—
278C4×C53—81no—
279C4×C53—81no—
280C4×C53#52911yescharacteristic
281C4×C53#54111yescharacteristic
282C4×C53#55211yescharacteristic
283C4×C53#55611yescharacteristic
284Dic53—42no—
285Dic53—81no—
286Dic53—81no—
287Dic53—81no—
288#2580D5311yescharacteristic
289Q8×C4⋊C4—531no—
290#2100—531no—
291#2099—531no—
292#2071—531no—
293#2071—531no—
294#1985—531no—
295#1949—531no—
296#1926—531no—
297#1925—531no—
298#1921—531no—
299#1920—531no—
300#1836—531no—
301#1802—531no—
302#1762—531no—
303C2×C53—82no—
304C2×C53#242111yescharacteristic
305C2×C53#309211yescharacteristic
306C2×C53#309611yescharacteristic
307C2×C53#336611yescharacteristic
308C2×C53#337211yescharacteristic
309C2×C53#337311yescharacteristic
310C2×C53#337511yescharacteristic
311C2×C4×Q8—1061no—
312C2×C4×Q8C2×D5311yescharacteristic
313C2×C4×D4C2×D5311yescharacteristic
314C2×#107—531no—
315C2×#106—531no—
316C2×C4⋊C8—531no—
317C2×#102—531no—
318C2×#102—531no—
319C2×#102—531no—
320C2×#102—531no—
321C2×#97—531no—
322#399—21no—
323#393—21no—
324#390C2×D5311yescharacteristic
325#390C2×D5311yescharacteristic
326#389—1061no—
327#389—1061no—
328#389—1061no—
329#389C2×D5311yescharacteristic
330#389C2×D5311yescharacteristic
331#385—21no—
332#385—21no—
333#384—531no—
334#384—531no—
335#384—531no—
336#382C2×D5311yescharacteristic
337C4×C4⋊C4—531no—
338C4×C4⋊C4—531no—
339C4×C4⋊C4—531no—
340#340—531no—
341#340—531no—
342#340—531no—
343#340—531no—
344#340—531no—
345#340—531no—
346#340—531no—
347#340—531no—
348#340—531no—
349#340—531no—
350#336—531no—
351#336—531no—
352#336—531no—
353#336—531no—
354C53#2500211yescharacteristic, complemented, Sylow
355C22×Q8C22⋊D5311yescharacteristic
356C22×D4C22×D5311yescharacteristic
357C23×C4—21no—
358C22×C8—531no—
359C22×C8—531no—
360C22×C8—531no—
361C22×C8—531no—
362C4×Q8—22no—
363C4×Q8—22no—
364C4×Q8—2124no—
365C4×D4—22no—
366C4×D4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—531no—
370C2×C4⋊C4—531no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—531no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4—1061no—
377C2×C4⋊C4—1061no—
378C2×C4⋊C4—1061no—
379C2×C4⋊C4C22⋊D5311yescharacteristic
380C2×C4⋊C4C22⋊D5311yescharacteristic
381C2×C4⋊C4C22⋊D5311yescharacteristic
382C2×C4⋊C4C22×D5311yescharacteristic
383C2×C4⋊C4C22×D5311yescharacteristic
384C2×C4⋊C4C22×D5311yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C42—1061no—
391C2×C42—1061no—
392C2×C42—1061no—
393C2×C42C22×D5311yescharacteristic
394C2×C42C22×D5311yescharacteristic
395C2×C42C22×D5311yescharacteristic
396#107—1062no—
397#106—1062no—
398C4⋊C8—1062no—
399#102—1062no—
400#102—1062no—
401#102—1062no—
402#102—1062no—
403#97—1062no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—1061no—
409#95—1061no—
410#95—1061no—
411C24—21no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4#1171994912yes—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—42no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4—1061no—
428C22×C4—1061no—
429C22×C4—1061no—
430C22×C4—1061no—
431C22×C4C2×C22⋊D5311yescharacteristic
432C22×C4C2×C22⋊D5311yescharacteristic
433C22×C4C2×C22⋊D5311yescharacteristic
434C22×C4C2×C22⋊D5311yescharacteristic
435C22×C4C2×C22⋊D5311yescharacteristic
436C22×C4C2×C22⋊D5311yescharacteristic
437C22×C4C23×D5311yescharacteristic
438C2×C8—1061no—
439C2×C8—1061no—
440C2×C8—1061no—
441C2×C8—1061no—
442C2×C8—1061no—
443C2×C8—1061no—
444C2×C8—1061no—
445C2×C8—1061no—
446C2×C8—1061no—
447C2×C8—1061no—
448C2×C8—1061no—
449C2×C8—1061no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—81no—
460C4⋊C4—81no—
461C4⋊C4—532no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—4241no—
470C4⋊C4—4241no—
471C4⋊C4—4241no—
472C4⋊C4—2122no—
473C4⋊C4—2122no—
474C4⋊C4—2122no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—81no—
479C22⋊C4—81no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—2122no—
485C42—2122no—
486C42—2122no—
487C42#1171994712yes—
488C42#1171994712yes—
489C23—22no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23C22×C22⋊D5311yescharacteristic
497Q8—28no—
498D4—22no—
499D4—22no—
500D4—41no—
501D4—41no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—84no—
526C2×C4—1062no—
527C2×C4—2121no—
528C2×C4—2121no—
529C2×C4—2121no—
530C2×C4—2121no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4—2121no—
538C2×C4#1313319812yes—
539C2×C4#1313319812yes—
540C2×C4#1313324212yes—
541C2×C4#1313324712yes—
542C2×C4#41×D5311yescharacteristic
543C2×C4C2×#1171994711yescharacteristic
544C2×C4C2×#1171994711yescharacteristic
545C2×C4C2×#1171994911yescharacteristic
546C2×C4#1313326411yescharacteristic
547C2×C4#1313326511yescharacteristic
548C8—2121no—
549C8—2121no—
550C8—2121no—
551C8—2121no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22C2×#1313319811yescharacteristic
560C22#1681128011yescharacteristic
561C22#1681128411yescharacteristic
562C22C2×#1313324211yescharacteristic
563C22C2×#1313324711yescharacteristic
564C22#1681142611yescharacteristic
565C22#1681143411yescharacteristic
566C4—22no—
567C4—22no—
568C4—22no—
569C4—22no—
570C4—41no—
571C4—41no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—4241no—
576C4—4241no—
577C4—4241no—
578C4—2122no—
579C4#1681115911yescharacteristic
580C4#1681123311yescharacteristic
581C4#1681131511yescharacteristic
582C4#1681134411yescharacteristic
583C2—82no—
584C2#2027821411yescharacteristic
585C2#2028463511yescharacteristic
586C2#2028466711yescharacteristic
587C2#2028742211yescharacteristic
588C2#2028749611yescharacteristic
589C2#2028750311yescharacteristic
590C2#2028752511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2580C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2580D53
hypocenterC53#25002
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2580

Complements

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

NH ≅ G/Ntype
1. Q8×C4⋊Dic53C2semidirect
2. #20276278C2semidirect
3. #20276273C2semidirect
6. #20275656C2semidirect
7. #20275481C2semidirect
10. #20275304C2semidirect
11. #20275297C2semidirect
12. #20274848C2semidirect
354. C53#25002semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊Dic532
#202762782
#202762732
#202761182
#202756562
#202754812
#202753392
#202753322
#202753042
#202752972
#202748482
#202747152
#202745402
C53×#25802
#2500253