Finite Group Databasebeta

← #27435976

Subgroups — #27435976

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×#16810747C211yesmaximal, characteristic, complemented
2#20280104C211yesmaximal, characteristic, complemented
3#20280069C211yesmaximal, characteristic
4#20276214C211yesmaximal, characteristic, complemented
5#20276209C211yesmaximal, characteristic, complemented
6#20276101C211yesmaximal, characteristic
7#20276087C211yesmaximal, characteristic
8#20275681C211yesmaximal, characteristic, complemented
9#20275485C211yesmaximal, characteristic, complemented
10#20275446C211yesmaximal, characteristic, complemented
11#20275441C211yesmaximal, characteristic, complemented
12#20275395C211yesmaximal, characteristic
13#20275395C211yesmaximal, characteristic
14C4×#13133138C211yesmaximal, characteristic
15C2×C53×#478C211yesmaximal, characteristic
16C2×C4×Dic106C2211yescharacteristic
17#16810747—24no—
18C2×#13133151C2211yescharacteristic
19C2×#13133149C2211yescharacteristic
20C2×#13133146C2211yescharacteristic
21C2×#13133146C2211yescharacteristic
22C2×#13133138C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24C2×C4×#88322C2211yescharacteristic
25C2×C8×Dic53C2211yescharacteristic
26C2×C8×Dic53C2211yescharacteristic
27#16810559—21no—
28#16810553—21no—
29#16810543—21no—
30#16810535—21no—
31#16810526C2211yescharacteristic
32#16810526C2211yescharacteristic
33#16810519C2211yescharacteristic
34#16810519C2211yescharacteristic
35#16810519C2211yescharacteristic
36#16810519C2211yescharacteristic
37#16810519C2211yescharacteristic
38#16810519C2211yescharacteristic
39#16810518—21no—
40#16810518—21no—
41#16810496—21no—
42#16810496—21no—
43#16810489C2211yescharacteristic
44C4×C4⋊Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53C2×#125×C53C2211yescharacteristic
54C2×C4×D4×C53C2211yescharacteristic
55C53×#478—24no—
56C2×#107×C53C2211yescharacteristic
57C2×#106×C53C2211yescharacteristic
58C2×C4⋊C8×C53C2211yescharacteristic
59C2×#102×C53C2211yescharacteristic
60C2×#102×C53C2211yescharacteristic
61C22×Dic106D411yescharacteristic
62C22×#88322C2311yescharacteristic
63C22×#88322C2311yescharacteristic
64C4×Dic106—22no—
65C4×Dic106—22no—
66C2×C4⋊Dic53D411yescharacteristic
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×#11719919—21no—
72C2×#11719919—21no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C2×#11719919—21no—
76C2×#11719919—21no—
77C2×#11719919D411yescharacteristic
78C2×#11719919D411yescharacteristic
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53—21no—
82C2×C22⋊Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53C2311yescharacteristic
85C2×C4×Dic53C2311yescharacteristic
86#13133151—22no—
87#13133149—22no—
88#13133146—22no—
89#13133146—22no—
90#13133138—22no—
91#13133138Q814yescomplemented
92C4×#88322—22no—
93C8×Dic53—22no—
94C8×Dic53—22no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104C22×D4×C53C2311yescharacteristic
105C23×C4×C53—21no—
106C22×C8×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108#125×C53—22no—
109#125×C53—22no—
110C4×D4×C53—22no—
111C4×D4×C53—22no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C22⋊C4×C53—21no—
118C2×C42×C53C2311yescharacteristic
119#107×C53—22no—
120#106×C53—22no—
121C4⋊C8×C53D414yes—
122#102×C53—22no—
123#102×C53—22no—
124C24×C53—21no—
125C2×D4×C53—21no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53C2×Q812yes—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—42no—
135C22×C4×C53—81no—
136C22×C4×C53—81no—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2411yescharacteristic
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—21no—
144C22×Dic53C2×D411yescharacteristic
145C22×Dic53C2×D411yescharacteristic
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C2×Dic106—22no—
149C2×Dic106—22no—
150C2×Dic106—22no—
151C2×C8×C53—21no—
152C2×C8×C53—21no—
153C2×C8×C53—21no—
154C2×C8×C53—21no—
155C2×C8×C53C2×D412yes—
156C2×C8×C53C2×D412yes—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—42no—
164C4⋊C4×C53—42no—
165C22⋊C4×C53—42no—
166C42×C53C2×D412yes—
167C42×C53#4112yes—
168C22⋊Dic53—42no—
169C22⋊Dic53—42no—
170C22⋊Dic53—81no—
171C22⋊Dic53—81no—
172C4⋊Dic53—22no—
173C4⋊Dic53—21no—
174C4⋊Dic53—21no—
175C4⋊Dic53—21no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53C2×Q812yes—
180C4⋊Dic53#4112yes—
181#11719919—22no—
182#11719919—22no—
183#11719919—42no—
184#11719919—42no—
185#11719919—81no—
186#11719919—81no—
187#11719919—81no—
188#11719919—81no—
189C4×Dic53—21no—
190C4×Dic53—21no—
191C4×Dic53—21no—
192C4×Dic53—21no—
193C4×Dic53—81no—
194C2×#88322—21no—
195C2×#88322—21no—
196C2×#88322—21no—
197C2×#88322—21no—
198C2×#88322—21no—
199C2×#88322C2×Q812yes—
200C23×C53—22no—
201C23×C53—41no—
202C23×C53—41no—
203C23×C53—41no—
204C23×C53—41no—
205C23×C53—41no—
206C23×C53—41no—
207C23×C53C22×D411yescharacteristic
208D4×C53—22no—
209D4×C53—22no—
210D4×C53—41no—
211D4×C53—41no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—84no—
226C2×C4×C53#12112yes—
227C2×C4×C53#12112yes—
228C2×C4×C53C22×D411yescharacteristic
229C2×C4×C53C22×Q811yescharacteristic
230C2×C4×C53C2×#4111yescharacteristic
231C2×C4×C53C2×#4111yescharacteristic
232C2×C4×C532−1+411yescharacteristic
233C2×C4×C532−1+411yescharacteristic
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—21no—
239C2×Dic53—21no—
240C2×Dic53—22no—
241C2×Dic53—22no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—21no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53#13512yes—
257C2×Dic53#13512yes—
258Dic106—28no—
259C8×C53—22no—
260C8×C53—22no—
261#88322—22no—
262#88322—41no—
263#25473—531nomaximal, Sylow
264C22×C53—42no—
265C22×C53—42no—
266C22×C53—42no—
267C22×C53—81no—
268C22×C53—81no—
269C22×C53—81no—
270C22×C53—81no—
271C22×C53C2×#12111yescharacteristic
272C22×C53#54811yescharacteristic
273C22×C53D4×Q811yescharacteristic
274C22×C53C2×#13511yescharacteristic
275C22×C53C2×#13511yescharacteristic
276C22×C53#57611yescharacteristic
277C22×C53#57711yescharacteristic
278C4×C53—22no—
279C4×C53—22no—
280C4×C53—81no—
281C4×C53—81no—
282C4×C53—81no—
283C4×C53#53311yescharacteristic
284C4×C53#54111yescharacteristic
285C4×C53#54711yescharacteristic
286C4×C53D4×Q811yescharacteristic
287Dic53—22no—
288Dic53—22no—
289Dic53—42no—
290Dic53—42no—
291Dic53—41no—
292Dic53—41no—
293Dic53—81no—
294Dic53—81no—
295C2×#478D5311yescharacteristic
296C2×C8×Q8—531no—
297#2584—531no—
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—82no—
311C2×C53#242911yescharacteristic
312C2×C53#305011yescharacteristic
313C2×C53#306211yescharacteristic
314C2×C53#336111yescharacteristic
315C2×C53#336511yescharacteristic
316C2×C53#338011yescharacteristic
317C2×C53#338111yescharacteristic
318C2×#125C2×D5311yescharacteristic
319C2×C4×Q8—531no—
320C2×C4×D4C2×D5311yescharacteristic
321#478—24no—
322C8×Q8—1064no—
323C2×#107—531no—
324C2×#107C2×D5311yescharacteristic
325C2×#106—531no—
326C2×#106C2×D5311yescharacteristic
327C2×C4⋊C8—531no—
328C2×C4⋊C8—531no—
329C2×C4⋊C8C2×D5311yescharacteristic
330C2×#102—531no—
331C2×#102—531no—
332C2×#102C2×D5311yescharacteristic
333C2×#102C2×D5311yescharacteristic
334C2×C4×C8—531no—
335C2×C4×C8—531no—
336C2×C4×C8—531no—
337#399—1061no—
338#398—1061no—
339#395—1061no—
340#393—1061no—
341#390—531no—
342#390—531no—
343#389—531no—
344#389—531no—
345#389—531no—
346#389—531no—
347#389—531no—
348#389—531no—
349#389—1061no—
350#389—1061no—
351#385—1061no—
352#385—1061no—
353#382—531no—
354C4×C4⋊C4—531no—
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×Q8—531no—
365C22×D4C22×D5311yescharacteristic
366C23×C4—21no—
367C22×C8—531no—
368C22×C8—531no—
369C22×C8C22×D5311yescharacteristic
370C22×C8C22×D5311yescharacteristic
371#125—22no—
372#125—22no—
373C4×Q8—1062no—
374C4×Q8—1062no—
375C4×D4—22no—
376C4×D4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4—531no—
382C2×C4⋊C4—531no—
383C2×C4⋊C4—531no—
384C2×C4⋊C4—531no—
385C2×C4⋊C4—531no—
386C2×C4⋊C4—1061no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4—1061no—
389C2×C4⋊C4—1061no—
390C2×C4⋊C4—1061no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4C22×D5311yescharacteristic
393C2×C4⋊C4C22×D5311yescharacteristic
394C2×C4⋊C4C22×D5311yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—1061no—
397C2×C22⋊C4—1061no—
398C2×C22⋊C4—1061no—
399C2×C22⋊C4—1061no—
400C2×C42—531no—
401C2×C42—531no—
402C2×C42—1061no—
403C2×C42C22×D5311yescharacteristic
404#107—22no—
405#107—1062no—
406#106—22no—
407#106—1062no—
408C4⋊C8—1062no—
409C4⋊C8—1062no—
410C4⋊C8C22⋊D5314yes—
411#102—22no—
412#102—22no—
413#102—534no—
414#102—1062no—
415C4×C8—1062no—
416C4×C8—1062no—
417C4×C8—1062no—
418#95—1061no—
419#95—1061no—
420#95—1061no—
421#95—1061no—
422#95—1061no—
423#95—1061no—
424#95—1061no—
425#95—1061no—
426#95—1061no—
427C24—21no—
428C2×Q8—1062no—
429C2×Q8—1062no—
430C2×Q8—1062no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4Q8×D5312yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—42no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—531no—
444C22×C4—531no—
445C22×C4—531no—
446C22×C4—531no—
447C22×C4—1061no—
448C22×C4—1061no—
449C22×C4—1061no—
450C22×C4—1061no—
451C22×C4D4×D5311yescharacteristic
452C22×C4D4×D5311yescharacteristic
453C22×C4C23×D5311yescharacteristic
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—532no—
459C2×C8—1061no—
460C2×C8—1061no—
461C2×C8—1061no—
462C2×C8—1061no—
463C2×C8—1061no—
464C2×C8C2×C22⋊D5312yes—
465C2×C8C2×C22⋊D5312yes—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—532no—
475C4⋊C4—532no—
476C4⋊C4—1062no—
477C4⋊C4—1062no—
478C4⋊C4—1062no—
479C4⋊C4—1061no—
480C4⋊C4—1061no—
481C4⋊C4—1061no—
482C4⋊C4—1061no—
483C4⋊C4—1061no—
484C4⋊C4—1061no—
485C4⋊C4—4241no—
486C4⋊C4—4241no—
487C4⋊C4—4241no—
488C4⋊C4—4241no—
489C4⋊C4—2122no—
490C4⋊C4—2122no—
491C22⋊C4—42no—
492C22⋊C4—4241no—
493C22⋊C4—4241no—
494C22⋊C4—2122no—
495C22⋊C4—2122no—
496C42—1061no—
497C42—1061no—
498C42—1061no—
499C42—1061no—
500C42—4241no—
501C42#1171994712yes—
502C42C2×C22⋊D5312yes—
503C23—22no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23C2×D4×D5311yescharacteristic
511Q8—1068no—
512D4—22no—
513D4—22no—
514D4—41no—
515D4—41no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—84no—
530C2×C4—532no—
531C2×C4—532no—
532C2×C4—1061no—
533C2×C4—1061no—
534C2×C4—1061no—
535C2×C4—1061no—
536C2×C4—1062no—
537C2×C4—1062no—
538C2×C4—1061no—
539C2×C4—1061no—
540C2×C4—1061no—
541C2×C4—1061no—
542C2×C4—1061no—
543C2×C4—1061no—
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—2121no—
553C2×C4—2121no—
554C2×C4#1313318812yes—
555C2×C4#1313318812yes—
556C2×C4C22×C22⋊D5311yescharacteristic
557C2×C4C2×Q8×D5311yescharacteristic
558C2×C4#41×D5311yescharacteristic
559C2×C4C2×#1171994711yescharacteristic
560C2×C4#1313326411yescharacteristic
561C2×C4#1313326511yescharacteristic
562C8—22no—
563C8—22no—
564C8—1062no—
565C8—2121no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22C2×#1313318811yescharacteristic
574C22#1681128311yescharacteristic
575C22#1681128611yescharacteristic
576C22#135×D5311yescharacteristic
577C22#135×D5311yescharacteristic
578C22#1681142711yescharacteristic
579C22#1681143311yescharacteristic
580C4—22no—
581C4—22no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4—1062no—
586C4—1062no—
587C4—4241no—
588C4—4241no—
589C4—2122no—
590C4—2122no—
591C4—2121no—
592C4—2121no—
593C4#1681117511yescharacteristic
594C4#1681123311yescharacteristic
595C4#1681127211yescharacteristic
596C4Q8×C22⋊D5311yescharacteristic
597C2—82no—
598C2#2027831011yescharacteristic
599C2#2028429211yescharacteristic
600C2#2028438911yescharacteristic
601C2#2028737811yescharacteristic
602C2#2028741411yescharacteristic
603C2#2028757711yescharacteristic
604C2#2028758511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#478C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenterC2×#478D53
hypocenterC53#25473
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
1. C2×#16810747C2semidirect
2. #20280104C2semidirect
4. #20276214C2semidirect
5. #20276209C2semidirect
8. #20275681C2semidirect
9. #20275485C2semidirect
10. #20275446C2semidirect
11. #20275441C2semidirect
91. #13133138Q8semidirect
363. C53#25473semidirect

Maximal subgroups

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

subgroupindex
C2×#168107472
#202801042
#202800692
#202762142
#202762092
#202761012
#202760872
#202756812
#202754852
#202754462
#202754412
#202753952
C4×#131331382
C2×C53×#4782
#2547353