Finite Group Databasebeta

← #27435683

Subgroups — #27435683

Order 13568.

834 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811020C211yesmaximal, characteristic, complemented
2C2×#16810911C211yesmaximal, characteristic
3#20279666C211yesmaximal, characteristic
4#20277801C211yesmaximal, characteristic, complemented
5#20276195C211yesmaximal, characteristic, complemented
6#20276191C211yesmaximal, characteristic, complemented
7#20276087C211yesmaximal, characteristic
8#20275665C211yesmaximal, characteristic, complemented
9#20275479C211yesmaximal, characteristic, complemented
10#20275475C211yesmaximal, characteristic, complemented
11#20275430C211yesmaximal, characteristic, complemented
12#20275345C211yesmaximal, characteristic
13#20275339C211yesmaximal, characteristic
14#102×Dic53C211yesmaximal, characteristic
15C53×#2065C211yesmaximal, characteristic
16C2×#13133216C2211yescharacteristic
17C2×#13133216C2211yescharacteristic
18C2×#13133194—21no—
19C2×#13133194—21no—
20C2×D4×Dic53C2211yescharacteristic
21#16811020—24no—
22#16810911—24no—
23C2×#13133149C2211yescharacteristic
24C2×#13133149C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×#13133148C2211yescharacteristic
28C2×#13133146C2211yescharacteristic
29C2×#13133146C2211yescharacteristic
30C2×#13133138C2211yescharacteristic
31C2×#13133138C2211yescharacteristic
32C2×C8×Dic53C2211yescharacteristic
33#16810559—21no—
34#16810559—21no—
35#16810528—21no—
36#16810527—21no—
37#16810526C2211yescharacteristic
38#16810519C2211yescharacteristic
39#16810518—21no—
40#16810493C2211yescharacteristic
41#16810492C2211yescharacteristic
42C4⋊C4×Dic53C2211yescharacteristic
43C4⋊C4×Dic53C2211yescharacteristic
44C4×#11719919C2211yescharacteristic
45C4×C4⋊Dic53C2211yescharacteristic
46#16810474—21no—
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810378C2211yescharacteristic
53#16810377C2211yescharacteristic
54C2×#106×C53C2211yescharacteristic
55C2×C4⋊C8×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C2×#102×C53C2211yescharacteristic
58C53×#390C2211yescharacteristic
59C53×#389C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C22×Dic106—21no—
62C22×Dic106—21no—
63C23×Dic53—21no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133216—22no—
67#13133216—22no—
68#13133216—22no—
69#13133216D414yes—
70#13133194—44no—
71#13133194—44no—
72D4×Dic53—22no—
73D4×Dic53—22no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919D411yescharacteristic
86C2×#11719919D411yescharacteristic
87C2×C22⋊Dic53—21no—
88C2×C22⋊Dic53—21no—
89C2×C22⋊Dic53—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53C2311yescharacteristic
94C2×C4×Dic53C2311yescharacteristic
95#13133149—22no—
96#13133149—22no—
97#13133148—22no—
98#13133148—22no—
99#13133148—22no—
100#13133146—22no—
101#13133146—22no—
102#13133138—22no—
103#13133138Q814yescomplemented
104C8×Dic53—22no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124D411yescharacteristic
110#13133124D411yescharacteristic
111C22×D4×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114C2×C4⋊C4×C53—21no—
115C2×C4⋊C4×C53C2311yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C22⋊C4×C53—21no—
119C2×C22⋊C4×C53—21no—
120C2×C42×C53C2311yescharacteristic
121#106×C53—22no—
122C4⋊C8×C53—22no—
123#102×C53—22no—
124#102×C53—22no—
125#95×C53—21no—
126C24×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53C2×Q812yes—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2411yescharacteristic
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—42no—
144C22×Dic53—42no—
145C22×Dic53—42no—
146C22×Dic53C2×D411yescharacteristic
147C22×Dic53C2×D411yescharacteristic
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C2×Dic106—24no—
151C2×Dic106—42no—
152C2×Dic106—42no—
153C2×Dic106—42no—
154C2×Dic106—81no—
155C2×Dic106—81no—
156C2×C8×C53—21no—
157C2×C8×C53—21no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
161C2×C8×C53#4112yes—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—81no—
168C4⋊C4×C53C2×D412yes—
169C22⋊C4×C53—42no—
170C22⋊C4×C53—42no—
171C42×C53—21no—
172C42×C53—21no—
173C22⋊Dic53—42no—
174C22⋊Dic53—42no—
175C22⋊Dic53—42no—
176C4⋊Dic53—21no—
177C4⋊Dic53—21no—
178C4⋊Dic53—21no—
179C4⋊Dic53—21no—
180C4⋊Dic53—21no—
181C4⋊Dic53—21no—
182C4⋊Dic53—42no—
183C4⋊Dic53—42no—
184C4⋊Dic53C2×D412yes—
185C4⋊Dic53C2×Q812yes—
186#11719919—24no—
187#11719919—22no—
188#11719919—42no—
189#11719919—42no—
190#11719919—42no—
191#11719919—81no—
192#11719919SD1614yes—
193C4×Dic53—21no—
194C4×Dic53—21no—
195C4×Dic53—42no—
196C4×Dic53—81no—
197C4×Dic53—81no—
198C4×Dic53C2×D412yes—
199C4×Dic53#4112yes—
200C2×#88322—21no—
201C2×#88322—21no—
202C2×#88322—21no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322C2×Q812yes—
206C23×C53—22no—
207C23×C53—41no—
208C23×C53—41no—
209C23×C53—41no—
210C23×C53—41no—
211C23×C53—41no—
212C23×C53—41no—
213C23×C53C22×D411yescharacteristic
214D4×C53—22no—
215D4×C53—22no—
216D4×C53—41no—
217D4×C53—41no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—22no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—41no—
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×C53C2×SD1612yes—
233C2×C4×C53C22×D411yescharacteristic
234C2×C4×C53C22×Q811yescharacteristic
235C2×C4×C53C2×#4111yescharacteristic
236C2×C4×C53C2×#4111yescharacteristic
237C2×C4×C532−1+411yescharacteristic
238C2×C4×C532−1+411yescharacteristic
239C2×Dic53—21no—
240C2×Dic53—21no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—22no—
244C2×Dic53—21no—
245C2×Dic53—22no—
246C2×Dic53—21no—
247C2×Dic53—21no—
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—41no—
257C2×Dic53—41no—
258C2×Dic53—84no—
259C2×Dic53#12112yes—
260C2×Dic53#12112yes—
261C2×Dic53C2×SD1612yes—
262C2×Dic53C2×SD1612yes—
263C2×Dic53AGL(1,ℤ/8)12yes—
264Dic106—84no—
265Dic106—84no—
266C8×C53—22no—
267C8×C53—41no—
268#88322—22no—
269#88322—41no—
270#25451—531nomaximal, Sylow
271C22×C53—42no—
272C22×C53—42no—
273C22×C53—42no—
274C22×C53—81no—
275C22×C53—81no—
276C22×C53—81no—
277C22×C53—81no—
278C22×C53C2×#12111yescharacteristic
279C22×C53#54711yescharacteristic
280C22×C53D4×Q811yescharacteristic
281C22×C53C22×SD1611yescharacteristic
282C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
283C22×C53#57711yescharacteristic
284C22×C53#57711yescharacteristic
285C4×C53—22no—
286C4×C53—41no—
287C4×C53—42no—
288C4×C53—81no—
289C4×C53—81no—
290C4×C53#53311yescharacteristic
291C4×C53#54111yescharacteristic
292C4×C53#54811yescharacteristic
293C4×C53D4×Q811yescharacteristic
294Dic53—22no—
295Dic53—22no—
296Dic53—41no—
297Dic53—42no—
298Dic53—42no—
299Dic53—81no—
300Dic53—81no—
301Dic53#46014yes—
302C2×#498—531no—
303C2×#476—531no—
304#2542—531no—
305#2369—531no—
306#2084—531no—
307#2084—531no—
308#2065—531no—
309#2065D5311yescharacteristic
310#1988—531no—
311#1948—531no—
312#1948—531no—
313#1939—531no—
314#1926—531no—
315#1926—531no—
316C4×#102—531no—
317C2×C53—82no—
318C2×C53#242911yescharacteristic
319C2×C53C2×#46011yescharacteristic
320C2×C53#304211yescharacteristic
321C2×C53#333511yescharacteristic
322C2×C53#335411yescharacteristic
323C2×C53Q8×SD1611yescharacteristic
324C2×C53#338411yescharacteristic
325C2×C4⋊Q8—531no—
326C2×C4⋊Q8—531no—
327C2×C22⋊Q8—1061no—
328C2×C22⋊Q8—1061no—
329C2×C4×D4—531no—
330#498—1064no—
331#476—1064no—
332C2×#106—531no—
333C2×#106—531no—
334C2×#106—531no—
335C2×#106—531no—
336C2×#106—531no—
337C2×#106C2×D5311yescharacteristic
338C2×C4⋊C8—531no—
339C2×C4⋊C8—531no—
340C2×C4⋊C8C2×D5311yescharacteristic
341C2×#102—531no—
342C2×#102—531no—
343C2×#102C2×D5311yescharacteristic
344C2×#102C2×D5311yescharacteristic
345C2×C4×C8—531no—
346#399—1061no—
347#399—1061no—
348#391—1061no—
349#391—1061no—
350#390—531no—
351#390C2×D5311yescharacteristic
352#389—531no—
353#389—1061no—
354#389C2×D5311yescharacteristic
355#384—531no—
356#384—531no—
357C4×C4⋊C4—531no—
358C4×C4⋊C4—531no—
359C4×C4⋊C4—531no—
360C4×C4⋊C4—531no—
361C4×C4⋊C4—1061no—
362#340—531no—
363#340—531no—
364#340—531no—
365#340—531no—
366#340—531no—
367#340C2×D5311yescharacteristic
368#336—531no—
369#336—531no—
370C53#2545111yescharacteristic, complemented, Sylow
371C22×Q8—1061no—
372C22×Q8—1061no—
373C22×D4C22×D5311yescharacteristic
374C23×C4—1061no—
375C22×C8—531no—
376C22×C8—531no—
377C22×C8C22×D5311yescharacteristic
378C22×C8C22×D5311yescharacteristic
379C4⋊Q8—534no—
380C4⋊Q8—1062no—
381C4⋊Q8—1062no—
382C4⋊Q8—1062no—
383C22⋊Q8—2124no—
384C22⋊Q8—2124no—
385C4×D4—1062no—
386C4×D4—1062no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—531no—
394C2×C4⋊C4—1061no—
395C2×C4⋊C4—1061no—
396C2×C4⋊C4—1061no—
397C2×C4⋊C4—1061no—
398C2×C4⋊C4—1061no—
399C2×C4⋊C4—1061no—
400C2×C4⋊C4—1061no—
401C2×C4⋊C4C22×D5311yescharacteristic
402C2×C4⋊C4C22×D5311yescharacteristic
403C2×C4⋊C4C22×D5311yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1061no—
407C2×C22⋊C4—1061no—
408C2×C22⋊C4—1061no—
409C2×C42—531no—
410C2×C42—531no—
411C2×C42—1061no—
412C2×C42—1061no—
413C2×C42—1061no—
414C2×C42C22×D5311yescharacteristic
415#106—22no—
416#106—1062no—
417#106—1062no—
418#106—1062no—
419#106—1062no—
420#106—1062no—
421C4⋊C8—22no—
422C4⋊C8—1062no—
423C4⋊C8—1062no—
424#102—22no—
425#102—22no—
426#102—534no—
427#102—1062no—
428C4×C8—1062no—
429#95—21no—
430#95—531no—
431#95—531no—
432#95—1061no—
433#95—1061no—
434#95—1061no—
435#95—1061no—
436C24—21no—
437C2×Q8—1064no—
438C2×Q8—4241no—
439C2×Q8—4241no—
440C2×Q8—2122no—
441C2×Q8—2122no—
442C2×Q8—2122no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4Q8×D5312yes—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—531no—
453C22×C4—531no—
454C22×C4—531no—
455C22×C4—531no—
456C22×C4—1061no—
457C22×C4—1061no—
458C22×C4—1061no—
459C22×C4—1061no—
460C22×C4—2122no—
461C22×C4—2122no—
462C22×C4—2122no—
463C22×C4D4×D5311yescharacteristic
464C22×C4D4×D5311yescharacteristic
465C22×C4C23×D5311yescharacteristic
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—532no—
472C2×C8—1061no—
473C2×C8—1061no—
474C2×C8—1061no—
475C2×C8—1061no—
476C2×C8—1061no—
477C2×C8#1171994712yes—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—81no—
484C4⋊C4—534no—
485C4⋊C4—532no—
486C4⋊C4—532no—
487C4⋊C4—1062no—
488C4⋊C4—1064no—
489C4⋊C4—1061no—
490C4⋊C4—1061no—
491C4⋊C4—1061no—
492C4⋊C4—1061no—
493C4⋊C4—1061no—
494C4⋊C4—1061no—
495C4⋊C4—4241no—
496C4⋊C4—2122no—
497C4⋊C4—2122no—
498C4⋊C4—2122no—
499C4⋊C4—2122no—
500C4⋊C4—2122no—
501C4⋊C4D4×D5312yes—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—2122no—
505C22⋊C4—2122no—
506C22⋊C4—2122no—
507C42—21no—
508C42—21no—
509C42—532no—
510C42—532no—
511C42—1061no—
512C42—1061no—
513C42—4241no—
514C42—4241no—
515C42—2122no—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23C2×D4×D5311yescharacteristic
524Q8—4244no—
525Q8—4244no—
526D4—22no—
527D4—22no—
528D4—41no—
529D4—41no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—532no—
545C2×C4—532no—
546C2×C4—532no—
547C2×C4—532no—
548C2×C4—532no—
549C2×C4—1061no—
550C2×C4—1061no—
551C2×C4—1061no—
552C2×C4—1061no—
553C2×C4—1061no—
554C2×C4—1062no—
555C2×C4—1062no—
556C2×C4—1061no—
557C2×C4—1061no—
558C2×C4—4244no—
559C2×C4—2121no—
560C2×C4—2121no—
561C2×C4—2121no—
562C2×C4—2121no—
563C2×C4—2121no—
564C2×C4—2121no—
565C2×C4—2121no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4SD16×D5312yes—
570C2×C4C2×D4×D5311yescharacteristic
571C2×C4C2×Q8×D5311yescharacteristic
572C2×C4#41×D5311yescharacteristic
573C2×C4C2×#1171994711yescharacteristic
574C2×C4#1313326411yescharacteristic
575C2×C4#1313326411yescharacteristic
576C8—22no—
577C8—41no—
578C8—1062no—
579C8—2121no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22#121×D5311yescharacteristic
588C22#1681127311yescharacteristic
589C22#1681128611yescharacteristic
590C22C2×SD16×D5311yescharacteristic
591C22AGL(1,ℤ/8)×D5311yescharacteristic
592C22#1681143011yescharacteristic
593C22#1681143311yescharacteristic
594C4—22no—
595C4—42no—
596C4—41no—
597C4—81no—
598C4—81no—
599C4—534no—
600C4—1062no—
601C4—1062no—
602C4—4241no—
603C4—4241no—
604C4—2122no—
605C4—2121no—
606C4—2122no—
607C4#1681117511yescharacteristic
608C4#1681123011yescharacteristic
609C4#1681127911yescharacteristic
610C4#1681128611yescharacteristic
611C2—82no—
612C2#2027830711yescharacteristic
613C2#460×D5311yescharacteristic
614C2#2028422511yescharacteristic
615C2#2028706711yescharacteristic
616C2#2028730511yescharacteristic
617C2#2028757011yescharacteristic
618C2#2028761711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2065C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2065D53
hypocenterC53#25451
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2065

Complements

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

NH ≅ G/Ntype
1. C2×#16811020C2semidirect
4. #20277801C2semidirect
5. #20276195C2semidirect
6. #20276191C2semidirect
8. #20275665C2semidirect
9. #20275479C2semidirect
10. #20275475C2semidirect
11. #20275430C2semidirect
103. #13133138Q8semidirect
370. C53#25451semidirect

Maximal subgroups

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

subgroupindex
C2×#168110202
C2×#168109112
#202796662
#202778012
#202761952
#202761912
#202760872
#202756652
#202754792
#202754752
#202754302
#202753452
#202753392
#102×Dic532
C53×#20652
#2545153