Finite Group Databasebeta

← #27442381

Subgroups — #27442381

Order 13568.

744 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279741C211yesmaximal, characteristic, complemented
2#20277686C211yesmaximal, characteristic
3#20276102C211yesmaximal, characteristic
4#20276088C211yesmaximal, characteristic
5#20275852C211yesmaximal, characteristic
6#20275852C211yesmaximal, characteristic
7#20275666C211yesmaximal, characteristic, complemented
8#20275662C211yesmaximal, characteristic, complemented
9#20275485C211yesmaximal, characteristic, complemented
10#20275308C211yesmaximal, characteristic, complemented
11#20275294C211yesmaximal, characteristic, complemented
12#20274849C211yesmaximal, characteristic, complemented
13#20274716C211yesmaximal, characteristic
14#20274541C211yesmaximal, characteristic
15C53×#1940C211yesmaximal, characteristic, complemented
16C2×#13133217C2211yescharacteristic
17C2×#13133217C2211yescharacteristic
18C2×#13133216—21no—
19C2×C4×D212C2211yescharacteristic
20C2×#13133151C2211yescharacteristic
21C2×#13133149C2211yescharacteristic
22C2×#13133139C2211yescharacteristic
23C2×#13133139C2211yescharacteristic
24C2×#13133139C2211yescharacteristic
25C2×#13133139C2211yescharacteristic
26#16810544—21no—
27#16810544—21no—
28#16810525C2211yescharacteristic
29#16810525C2211yescharacteristic
30#16810522—21no—
31#16810521C2211yescharacteristic
32#16810521C2211yescharacteristic
33#16810513—21no—
34#16810513—21no—
35#16810493C2211yescharacteristic
36#16810489C2211yescharacteristic
37#16810485—21no—
38C4×C4⋊Dic53C2211yescharacteristic
39C4×C4⋊Dic53C2211yescharacteristic
40C4×C4⋊Dic53C2211yescharacteristic
41#16810470—21no—
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52C2×C4⋊C8×C53C2211yescharacteristic
53C2×C4⋊C8×C53C2211yescharacteristic
54C2×C4⋊C8×C53C2211yescharacteristic
55C2×#97×C53C2211yescharacteristic
56C4×C4⋊C4×C53C2211yescharacteristic
57C53×#336C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C22×Dic106—21no—
60C22×Dic106—21no—
61C22×D212C2311yescharacteristic
62C22×C4×D53—21no—
63#13133217—22no—
64#13133217—22no—
65#13133217—41no—
66#13133217—41no—
67#13133216—44no—
68C4×D212—22no—
69C4×D212Q814yescomplemented
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—21no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719921—21no—
84C2×#11719921—21no—
85C2×#11719921—21no—
86C2×#11719921—21no—
87C2×#11719921—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53—21no—
90C2×C4×Dic53—21no—
91#13133151—22no—
92#13133149—22no—
93#13133139—22no—
94#13133139—22no—
95#13133139—41no—
96#13133139—41no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104C22×C8×C53C2311yescharacteristic
105C22×C8×C53C2311yescharacteristic
106C22×C8×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C2×C4⋊C4×C53D411yescharacteristic
109C2×C4⋊C4×C53D411yescharacteristic
110C2×C42×C53C2311yescharacteristic
111C2×C42×C53C2311yescharacteristic
112C2×C42×C53C2311yescharacteristic
113C4⋊C8×C53—22no—
114C4⋊C8×C53—41no—
115C4⋊C8×C53—41no—
116#97×C53—22no—
117#95×C53D411yescharacteristic
118#95×C53D411yescharacteristic
119C23×D53—21no—
120C22×C4×C53C2×D411yescharacteristic
121C22×C4×C53C2×D411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2411yescharacteristic
127C22×Dic53—21no—
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53—21no—
131C22×Dic53—21no—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C2×D212—21no—
135C2×D212—21no—
136C2×D212—21no—
137C2×D212—21no—
138C2×D212—21no—
139C2×D212C2×Q812yes—
140C2×C4×D53—24no—
141C2×C4×D53—81no—
142C2×C4×D53—81no—
143C2×Dic106—42no—
144C2×Dic106—42no—
145C2×Dic106—42no—
146C2×Dic106—42no—
147C2×Dic106—81no—
148C2×Dic106—81no—
149C2×C8×C53—21no—
150C2×C8×C53—21no—
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53—21no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C4⋊C4×C53—22no—
162C4⋊C4×C53—22no—
163C42×C53—21no—
164C42×C53—21no—
165C42×C53—21no—
166C42×C53—21no—
167C42×C53C2×Q812yes—
168C42×C53#4112yes—
169#11719921—24no—
170#11719921—42no—
171#11719921—42no—
172#11719921—81no—
173#11719921—81no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—42no—
186C4⋊Dic53—42no—
187C4⋊Dic53—42no—
188C4⋊Dic53—81no—
189C4⋊Dic53C2×Q812yes—
190C4⋊Dic53#4112yes—
191C4⋊Dic53#4112yes—
192#11719919—42no—
193#11719919—42no—
194C4×Dic53—42no—
195C4×Dic53—81no—
196C4×Dic53—81no—
197C23×C53C22×D411yescharacteristic
198C22×D53—22no—
199C22×D53—41no—
200C22×D53—41no—
201C22×D53—41no—
202C22×D53—41no—
203C22×D53—41no—
204C22×D53—41no—
205C2×C4×C53—21no—
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×C53C22⋊Q812yes—
220C2×C4×C53C22⋊Q812yes—
221C2×C4×C53AGL(1,ℤ/8)12yes—
222C2×C4×C53#13712yes—
223C2×C4×C53C22×Q811yescharacteristic
224C2×C4×C53C2×#4111yescharacteristic
225C2×C4×C53C2×#4111yescharacteristic
226C2×C4×C53C2×#4111yescharacteristic
227C2×C4×C53H(5,𝔽2)11yescharacteristic
228C2×C4×C532−1+411yescharacteristic
229C2×Dic53—22no—
230C2×Dic53—22no—
231C2×Dic53—22no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
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—
250D212—22no—
251D212—22no—
252D212—41no—
253D212—41no—
254C4×D53—84no—
255Dic106—84no—
256Dic106—84no—
257C8×C53—41no—
258C8×C53—41no—
259C8×C53—41no—
260C8×C53—41no—
261#26036—531nomaximal, Sylow
262C22×C53C2×C22⋊Q811yescharacteristic
263C22×C53#54611yescharacteristic
264C22×C53#54811yescharacteristic
265C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C53C2×#13711yescharacteristic
267C22×C53#57611yescharacteristic
268C22×C53#57711yescharacteristic
269C2×D53—42no—
270C2×D53—42no—
271C2×D53—42no—
272C2×D53—81no—
273C2×D53—81no—
274C2×D53—81no—
275C2×D53—81no—
276C4×C53—22no—
277C4×C53—22no—
278C4×C53—22no—
279C4×C53—22no—
280C4×C53—41no—
281C4×C53—41no—
282C4×C53#53311yescharacteristic
283C4×C53#54411yescharacteristic
284C4×C53#55511yescharacteristic
285C4×C53#55611yescharacteristic
286Dic53—42no—
287Dic53—42no—
288Dic53—42no—
289Dic53—81no—
290Dic53—81no—
291Dic53—81no—
292Dic53—81no—
293#2551—531no—
294#2357—531no—
295#2067—531no—
296#2065—531no—
297#2023—531no—
298#2023—531no—
299#1988—531no—
300#1987—531no—
301#1949—531no—
302#1940D5311yescharacteristic, complemented
303#1921—531no—
304#1919—531no—
305#1836—531no—
306#1802—531no—
307#1762—531no—
308C2×C53#243911yescharacteristic
309C2×C53#307911yescharacteristic
310C2×C53#308111yescharacteristic
311C2×C53#331811yescharacteristic
312C2×C53#332011yescharacteristic
313C2×C53#337311yescharacteristic
314C2×C53#337511yescharacteristic
315D53—82no—
316C2×C4⋊Q8—531no—
317C2×C4⋊Q8—531no—
318C2×C4⋊Q8—1061no—
319C2×C4×D4—531no—
320C2×#107—531no—
321C2×#106—531no—
322C2×C4⋊C8C2×D5311yescharacteristic
323C2×C4⋊C8C2×D5311yescharacteristic
324C2×C4⋊C8C2×D5311yescharacteristic
325C2×#102—531no—
326C2×#102—531no—
327C2×#102—531no—
328C2×#102—531no—
329C2×#97C2×D5311yescharacteristic
330#395—1061no—
331#395—1061no—
332#390—531no—
333#390—531no—
334#389—531no—
335#389—531no—
336#389—1061no—
337#388—1061no—
338#388—1061no—
339#384—531no—
340#382—531no—
341#381—1061no—
342C4×C4⋊C4—531no—
343C4×C4⋊C4—531no—
344C4×C4⋊C4—531no—
345C4×C4⋊C4C2×D5311yescharacteristic
346C4×C22⋊C4—1061no—
347#340—531no—
348#340—531no—
349#340—531no—
350#340—531no—
351#340—531no—
352#340—531no—
353#340—531no—
354#340—531no—
355#340—531no—
356#340—531no—
357#336C2×D5311yescharacteristic
358#336C2×D5311yescharacteristic
359C53#2603611yescharacteristic, complemented, Sylow
360C22×Q8—1061no—
361C22×Q8—1061no—
362C22×D4—531no—
363C23×C4—1061no—
364C22×C8C22×D5311yescharacteristic
365C22×C8C22×D5311yescharacteristic
366C22×C8C22×D5311yescharacteristic
367C22×C8C22×D5311yescharacteristic
368C4⋊Q8—1062no—
369C4⋊Q8—1062no—
370C4⋊Q8—2121no—
371C4⋊Q8—2121no—
372C4⋊Q8—2124no—
373C4×D4—534no—
374C4×D4—1062no—
375C2×C4⋊C4—531no—
376C2×C4⋊C4—531no—
377C2×C4⋊C4—531no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4—1061no—
385C2×C4⋊C4—1061no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4D21211yescharacteristic
389C2×C4⋊C4D21211yescharacteristic
390C2×C22⋊C4—1061no—
391C2×C22⋊C4—1061no—
392C2×C22⋊C4—1061no—
393C2×C22⋊C4—1061no—
394C2×C22⋊C4—1061no—
395C2×C42—1061no—
396C2×C42—1061no—
397C2×C42—1061no—
398C2×C42C22×D5311yescharacteristic
399C2×C42C22×D5311yescharacteristic
400C2×C42C22×D5311yescharacteristic
401#107—1062no—
402#106—1062no—
403C4⋊C8—22no—
404C4⋊C8—41no—
405C4⋊C8—41no—
406#102—1062no—
407#102—1062no—
408#102—2121no—
409#102—2121no—
410#97—22no—
411#95—1061no—
412#95—1061no—
413#95—1061no—
414#95—1061no—
415#95—1061no—
416#95—1061no—
417#95—1061no—
418#95D21211yescharacteristic
419#95D21211yescharacteristic
420C24—1061no—
421C2×Q8—4241no—
422C2×Q8—4241no—
423C2×Q8—2122no—
424C2×Q8—2122no—
425C2×Q8—2122no—
426C2×Q8—2122no—
427C2×D4—532no—
428C2×D4—1061no—
429C2×D4—1061no—
430C2×D4—1061no—
431C2×D4—1061no—
432C2×D4—1061no—
433C22×C4—1061no—
434C22×C4—1061no—
435C22×C4—1061no—
436C22×C4—1064no—
437C22×C4—1061no—
438C22×C4—1061no—
439C22×C4—1061no—
440C22×C4—1061no—
441C22×C4—4241no—
442C22×C4—4241no—
443C22×C4C2×D21211yescharacteristic
444C22×C4C2×D21211yescharacteristic
445C22×C4C2×D21211yescharacteristic
446C22×C4C2×D21211yescharacteristic
447C22×C4C2×D21211yescharacteristic
448C22×C4C2×D21211yescharacteristic
449C22×C4C23×D5311yescharacteristic
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—532no—
465C4⋊C4—532no—
466C4⋊C4—532no—
467C4⋊C4—1061no—
468C4⋊C4—1061no—
469C4⋊C4—1061no—
470C4⋊C4—1061no—
471C4⋊C4—1061no—
472C4⋊C4—1061no—
473C4⋊C4—1061no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1061no—
477C4⋊C4—1061no—
478C4⋊C4—4241no—
479C4⋊C4—2122no—
480C4⋊C4—2122no—
481C4⋊C4—2122no—
482C4⋊C4—2122no—
483C4⋊C4—2122no—
484C22⋊C4—1064no—
485C22⋊C4—4241no—
486C22⋊C4—4241no—
487C22⋊C4—2122no—
488C22⋊C4—2122no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—4241no—
494C42—4241no—
495C42—2122no—
496C42#1171994712yes—
497C42Q8×D5312yes—
498C23—1062no—
499C23—2121no—
500C23—2121no—
501C23—2121no—
502C23—2121no—
503C23—2121no—
504C23—2121no—
505C23C22×D21211yescharacteristic
506Q8—4244no—
507Q8—4244no—
508D4—1062no—
509D4—1062no—
510D4—2121no—
511D4—2121no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—1062no—
527C2×C4—1062no—
528C2×C4—1062no—
529C2×C4—4244no—
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—2121no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4—2121no—
542C2×C4—2121no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4—2121no—
548C2×C4#1313319712yes—
549C2×C4#1313319712yes—
550C2×C4#1313324512yes—
551C2×C4#1313325012yes—
552C2×C4C2×Q8×D5311yescharacteristic
553C2×C4#41×D5311yescharacteristic
554C2×C4#41×D5311yescharacteristic
555C2×C4C2×#1171994711yescharacteristic
556C2×C4#1313326211yescharacteristic
557C2×C4#1313326511yescharacteristic
558C8—41no—
559C8—41no—
560C8—41no—
561C8—41no—
562C22—4241no—
563C22—4241no—
564C22—4241no—
565C22—4241no—
566C22—2122no—
567C22—2122no—
568C22—2122no—
569C22C2×#1313319711yescharacteristic
570C22#1681126811yescharacteristic
571C22#1681128511yescharacteristic
572C22C2×#1313324511yescharacteristic
573C22C2×#1313325011yescharacteristic
574C22#1681142811yescharacteristic
575C22#1681143511yescharacteristic
576C4—22no—
577C4—22no—
578C4—22no—
579C4—22no—
580C4—41no—
581C4—41no—
582C4—4241no—
583C4—4241no—
584C4—4241no—
585C4—4241no—
586C4—2122no—
587C4—2122no—
588C4—2122no—
589C4#1681117911yescharacteristic
590C4#1681125011yescharacteristic
591C4#1681133711yescharacteristic
592C4#1681134411yescharacteristic
593C2—4242no—
594C2#2027840411yescharacteristic
595C2#2028453111yescharacteristic
596C2#2028454711yescharacteristic
597C2#2028685811yescharacteristic
598C2#2028688811yescharacteristic
599C2#2028750911yescharacteristic
600C2#2028753111yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1940C2
socleC23×C53C22×D4
prime radicalC22×D212C23
hypercenter#1940D53
hypocenterC53#26036
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1940

Complements

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

NH ≅ G/Ntype
1. #20279741C2semidirect
7. #20275666C2semidirect
8. #20275662C2semidirect
9. #20275485C2semidirect
10. #20275308C2semidirect
11. #20275294C2semidirect
12. #20274849C2semidirect
15. C53×#1940C2semidirect
69. C4×D212Q8semidirect
302. #1940D53semidirect
359. C53#26036semidirect

Maximal subgroups

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

subgroupindex
#202797412
#202776862
#202761022
#202760882
#202758522
#202756662
#202756622
#202754852
#202753082
#202752942
#202748492
#202747162
#202745412
C53×#19402
#2603653