Finite Group Databasebeta

← #27435986

Subgroups — #27435986

Order 13568.

774 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810920C211yesmaximal, characteristic
2C2×Q8×#88322C211yesmaximal, characteristic, complemented
3#20280073C211yesmaximal, characteristic
4#20276216C211yesmaximal, characteristic, complemented
5#20276211C211yesmaximal, characteristic, complemented
6#20276100C211yesmaximal, characteristic
7#20276086C211yesmaximal, characteristic
8#20275681C211yesmaximal, characteristic, complemented
9#20275483C211yesmaximal, characteristic, complemented
10#20275445C211yesmaximal, characteristic, complemented
11#20275440C211yesmaximal, characteristic, complemented
12#20275397C211yesmaximal, characteristic
13#20275397C211yesmaximal, characteristic
14C4×#13133137C211yesmaximal, characteristic
15C53×#2584C211yesmaximal, characteristic, complemented
16C2×C4×D212C2211yescharacteristic
17#16810920—24no—
18Q8×#88322—24no—
19C2×#13133150C2211yescharacteristic
20C2×#13133150C2211yescharacteristic
21C2×#13133148C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×C4⋊#88322C2211yescharacteristic
24C2×C4⋊#88322C2211yescharacteristic
25C2×C4⋊#88322C2211yescharacteristic
26C2×#13133137C2211yescharacteristic
27C2×#13133137C2211yescharacteristic
28C2×#13133137C2211yescharacteristic
29C2×#13133137C2211yescharacteristic
30C2×C4×#88322C2211yescharacteristic
31C2×C4×#88322C2211yescharacteristic
32C2×C4×#88322C2211yescharacteristic
33#16810558—21no—
34#16810534—21no—
35#16810525C2211yescharacteristic
36#16810525C2211yescharacteristic
37#16810521C2211yescharacteristic
38#16810521C2211yescharacteristic
39#16810494—21no—
40#16810494—21no—
41#16810489C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50C2×#125×C53C2211yescharacteristic
51C2×C4×Q8×C53C2211yescharacteristic
52C53×#398—21no—
53C53×#395—21no—
54C53×#389—21no—
55C53×#389—21no—
56C53×#389C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#389C2211yescharacteristic
59C53×#389C2211yescharacteristic
60C4×C4⋊C4×C53C2211yescharacteristic
61C22×D212C2311yescharacteristic
62C22×C4×D53—21no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C4×D212—22no—
68C4×D212—22no—
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C4⋊Dic53C2311yescharacteristic
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719921—21no—
75C2×#11719921—21no—
76C2×#11719921—21no—
77C2×#11719921—21no—
78C2×#11719921—21no—
79#13133150—22no—
80#13133150—22no—
81#13133148—22no—
82#13133148—22no—
83C4⋊#88322—22no—
84C4⋊#88322—22no—
85C4⋊#88322D414yes—
86#13133137—22no—
87#13133137—22no—
88#13133137—22no—
89#13133137Q814yescomplemented
90C4×#88322—22no—
91C4×#88322—22no—
92C4×#88322—22no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97C22×Q8×C53D411yescharacteristic
98#125×C53—22no—
99#125×C53—22no—
100C4×Q8×C53—22no—
101C4×Q8×C53—22no—
102C2×C4⋊C4×C53—21no—
103C2×C4⋊C4×C53—21no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53D411yescharacteristic
109C2×C4⋊C4×C53D411yescharacteristic
110C2×C4⋊C4×C53D411yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C42×C53—21no—
116C2×C42×C53C2311yescharacteristic
117C2×C42×C53C2311yescharacteristic
118C2×C42×C53C2311yescharacteristic
119#95×C53—21no—
120#95×C53—21no—
121#95×C53—21no—
122#95×C53—21no—
123#95×C53—21no—
124C23×D53—21no—
125C2×Q8×C53—22no—
126C2×Q8×C53—22no—
127C2×Q8×C53—22no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2411yescharacteristic
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C2×D212—21no—
143C2×D212—21no—
144C2×D212—21no—
145C2×D212—21no—
146C2×D212—21no—
147C2×D212C2×Q812yes—
148C2×C4×D53—42no—
149C2×C4×D53—81no—
150C2×C4×D53—81no—
151C4⋊C4×C53—22no—
152C4⋊C4×C53—22no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—42no—
161C4⋊C4×C53—42no—
162C4⋊C4×C53—42no—
163C4⋊C4×C53—42no—
164C4⋊C4×C53—81no—
165C4⋊C4×C53—81no—
166C4⋊C4×C53C2×Q812yes—
167C4⋊C4×C53#4112yes—
168C42×C53—21no—
169C42×C53—21no—
170C42×C53—21no—
171C42×C53—21no—
172C42×C53—81no—
173C42×C53C2×D412yes—
174C42×C53#4112yes—
175#11719921—42no—
176#11719921—42no—
177#11719921—42no—
178#11719921—81no—
179#11719921—81no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186#11719919—81no—
187#11719919—81no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C2×#88322—21no—
193C2×#88322—21no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322C2×D412yes—
198C2×#88322C2×D412yes—
199C2×#88322C2×Q812yes—
200C23×C53C22×D411yescharacteristic
201C22×D53—22no—
202C22×D53—41no—
203C22×D53—41no—
204C22×D53—41no—
205C22×D53—41no—
206C22×D53—41no—
207C22×D53—41no—
208Q8×C53—28no—
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—22no—
220C2×C4×C53—22no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53#12112yes—
236C2×C4×C53#12112yes—
237C2×C4×C53#13512yes—
238C2×C4×C53#13512yes—
239C2×C4×C53C22×D411yescharacteristic
240C2×C4×C53C22×Q811yescharacteristic
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C532−1+411yescharacteristic
244C2×C4×C532−1+411yescharacteristic
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254D212—22no—
255D212—22no—
256D212—41no—
257D212—41no—
258C4×D53—84no—
259#88322—22no—
260#88322—22no—
261#88322—22no—
262#88322—41no—
263#25473—531nomaximal, Sylow
264C22×C53C2×#12111yescharacteristic
265C22×C53#54811yescharacteristic
266C22×C53D4×Q811yescharacteristic
267C22×C53C2×#13511yescharacteristic
268C22×C53C2×#13511yescharacteristic
269C22×C53#57611yescharacteristic
270C22×C53#57711yescharacteristic
271C2×D53—42no—
272C2×D53—42no—
273C2×D53—42no—
274C2×D53—81no—
275C2×D53—81no—
276C2×D53—81no—
277C2×D53—81no—
278C4×C53—22no—
279C4×C53—22no—
280C4×C53—22no—
281C4×C53—22no—
282C4×C53—42no—
283C4×C53—42no—
284C4×C53—41no—
285C4×C53—41no—
286C4×C53—81no—
287C4×C53—81no—
288C4×C53#53311yescharacteristic
289C4×C53#54111yescharacteristic
290C4×C53#54711yescharacteristic
291C4×C53D4×Q811yescharacteristic
292Dic53—81no—
293Dic53—81no—
294Dic53—81no—
295C2×#478—531no—
296C2×C8×Q8—531no—
297#2584D5311yescharacteristic, complemented
298#2580—531no—
299#2088—531no—
300#2087—531no—
301#2067—531no—
302#2065—531no—
303#1991—531no—
304#1949—531no—
305#1942—531no—
306#1941—531no—
307#1934—531no—
308#1934—531no—
309C4×#102—531no—
310C2×C53#242911yescharacteristic
311C2×C53#305011yescharacteristic
312C2×C53#306211yescharacteristic
313C2×C53#336111yescharacteristic
314C2×C53#336511yescharacteristic
315C2×C53#338011yescharacteristic
316C2×C53#338111yescharacteristic
317D53—82no—
318C2×#125C2×D5311yescharacteristic
319C2×C4×Q8C2×D5311yescharacteristic
320C2×C4×D4—531no—
321#478—1064no—
322C8×Q8—1064no—
323C2×#107—531no—
324C2×#107—531no—
325C2×#106—531no—
326C2×#106—531no—
327C2×C4⋊C8—531no—
328C2×C4⋊C8—531no—
329C2×C4⋊C8—531no—
330C2×#102—531no—
331C2×#102—531no—
332C2×#102—531no—
333C2×#102—531no—
334C2×C4×C8—531no—
335C2×C4×C8—531no—
336C2×C4×C8—531no—
337#399—1061no—
338#398—21no—
339#395—21no—
340#393—1061no—
341#390—531no—
342#390—531no—
343#389—21no—
344#389—21no—
345#389—531no—
346#389—531no—
347#389C2×D5311yescharacteristic
348#389C2×D5311yescharacteristic
349#389C2×D5311yescharacteristic
350#389C2×D5311yescharacteristic
351#385—1061no—
352#385—1061no—
353#382—531no—
354C4×C4⋊C4C2×D5311yescharacteristic
355#340—531no—
356#340—531no—
357#340—531no—
358#340—531no—
359#340—531no—
360#340—531no—
361#340—531no—
362#340—531no—
363C53#2547311yescharacteristic, complemented, Sylow
364C22×Q8C22⋊D5311yescharacteristic
365C22×D4—531no—
366C23×C4—1061no—
367C22×C8—531no—
368C22×C8—531no—
369C22×C8—531no—
370C22×C8—531no—
371#125—22no—
372#125—22no—
373C4×Q8—22no—
374C4×Q8—22no—
375C4×D4—1062no—
376C4×D4—1062no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—531no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—531no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4C22⋊D5311yescharacteristic
389C2×C4⋊C4C22⋊D5311yescharacteristic
390C2×C4⋊C4C22⋊D5311yescharacteristic
391C2×C4⋊C4C22×D5311yescharacteristic
392C2×C4⋊C4C22×D5311yescharacteristic
393C2×C4⋊C4C22×D5311yescharacteristic
394C2×C4⋊C4C22×D5311yescharacteristic
395C2×C22⋊C4—1061no—
396C2×C22⋊C4—1061no—
397C2×C22⋊C4—1061no—
398C2×C22⋊C4—1061no—
399C2×C22⋊C4—1061no—
400C2×C42—21no—
401C2×C42C22×D5311yescharacteristic
402C2×C42C22×D5311yescharacteristic
403C2×C42C22×D5311yescharacteristic
404#107—1062no—
405#107—1062no—
406#106—1062no—
407#106—1062no—
408C4⋊C8—534no—
409C4⋊C8—1062no—
410C4⋊C8—1062no—
411#102—534no—
412#102—1062no—
413#102—1062no—
414#102—1062no—
415C4×C8—1062no—
416C4×C8—1062no—
417C4×C8—1062no—
418#95—21no—
419#95—21no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95—1061no—
424#95—1061no—
425#95—1061no—
426#95—1061no—
427C24—1061no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×D4—532no—
432C2×D4—1061no—
433C2×D4—1061no—
434C2×D4—1061no—
435C2×D4—1061no—
436C2×D4—1061no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—1061no—
442C22×C4—1061no—
443C22×C4—1061no—
444C22×C4—4241no—
445C22×C4—4241no—
446C22×C4—2122no—
447C22×C4C2×C22⋊D5311yescharacteristic
448C22×C4C2×C22⋊D5311yescharacteristic
449C22×C4C2×C22⋊D5311yescharacteristic
450C22×C4C2×C22⋊D5311yescharacteristic
451C22×C4C2×C22⋊D5311yescharacteristic
452C22×C4C2×C22⋊D5311yescharacteristic
453C22×C4C23×D5311yescharacteristic
454C2×C8—532no—
455C2×C8—532no—
456C2×C8—532no—
457C2×C8—1061no—
458C2×C8—1061no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8—1061no—
465C2×C8—1061no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—81no—
480C4⋊C4—81no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—1061no—
486C4⋊C4—1061no—
487C4⋊C4—4241no—
488C4⋊C4—4241no—
489C4⋊C4Q8×D5312yes—
490C4⋊C4#1171994912yes—
491C22⋊C4—4241no—
492C22⋊C4—4241no—
493C22⋊C4—2122no—
494C22⋊C4—2122no—
495C22⋊C4—2122no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—81no—
501C42D4×D5312yes—
502C42#1171994712yes—
503C23—1062no—
504C23—2121no—
505C23—2121no—
506C23—2121no—
507C23—2121no—
508C23—2121no—
509C23—2121no—
510C23C22×C22⋊D5311yescharacteristic
511Q8—28no—
512D4—1062no—
513D4—1062no—
514D4—2121no—
515D4—2121no—
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—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—4244no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4—2121no—
546C2×C4—2121no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2121no—
552C2×C4#1313318912yes—
553C2×C4#1313318912yes—
554C2×C4#1313323712yes—
555C2×C4#1313323712yes—
556C2×C4C2×D4×D5311yescharacteristic
557C2×C4C2×Q8×D5311yescharacteristic
558C2×C4C2×#1171994711yescharacteristic
559C2×C4C2×#1171994911yescharacteristic
560C2×C4#1313326511yescharacteristic
561C2×C4#1313326511yescharacteristic
562C8—1062no—
563C8—1062no—
564C8—1062no—
565C8—2121no—
566C22—4241no—
567C22—4241no—
568C22—4241no—
569C22—4241no—
570C22—2122no—
571C22—2122no—
572C22—2122no—
573C22C2×#1313318911yescharacteristic
574C22#1681128411yescharacteristic
575C22Q8×C22⋊D5311yescharacteristic
576C22C2×#1313323711yescharacteristic
577C22C2×#1313323711yescharacteristic
578C22#1681142611yescharacteristic
579C22#1681143111yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—42no—
585C4—42no—
586C4—41no—
587C4—41no—
588C4—81no—
589C4—81no—
590C4—4241no—
591C4—4241no—
592C4—4241no—
593C4#1681117811yescharacteristic
594C4#1681123411yescharacteristic
595C4#1681127611yescharacteristic
596C4#1681128811yescharacteristic
597C2—4242no—
598C2#2027831711yescharacteristic
599C2#2028429311yescharacteristic
600C2#2028439711yescharacteristic
601C2#2028737511yescharacteristic
602C2#2028741111yescharacteristic
603C2#2028757811yescharacteristic
604C2#2028758611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2584C2
socleC23×C53C22×D4
prime radicalC22×D212C23
hypercenter#2584D53
hypocenterC53#25473
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2584

Complements

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

NH ≅ G/Ntype
2. C2×Q8×#88322C2semidirect
4. #20276216C2semidirect
5. #20276211C2semidirect
8. #20275681C2semidirect
9. #20275483C2semidirect
10. #20275445C2semidirect
11. #20275440C2semidirect
15. C53×#2584C2semidirect
89. #13133137Q8semidirect
297. #2584D53semidirect
363. C53#25473semidirect

Maximal subgroups

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

subgroupindex
C2×#168109202
C2×Q8×#883222
#202800732
#202762162
#202762112
#202761002
#202760862
#202756812
#202754832
#202754452
#202754402
#202753972
C4×#131331372
C53×#25842
#2547353