Finite Group Databasebeta

← #27447013

Subgroups — #27447013

Order 13568.

896 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810926C211yesmaximal, characteristic, complemented
2#20278422C211yesmaximal, characteristic, complemented
3#20277731C211yesmaximal, characteristic
4#20276245C211yesmaximal, characteristic, complemented
5#20276214C211yesmaximal, characteristic, complemented
6#20276098C211yesmaximal, characteristic, complemented
7#20276051C211yesmaximal, characteristic, complemented
8#20275475C211yesmaximal, characteristic, complemented
9#20275288C211yesmaximal, characteristic, complemented
10#20275153C211yesmaximal, characteristic
11#20275037C211yesmaximal, characteristic
12#20274973C211yesmaximal, characteristic, complemented
13#20274908C211yesmaximal, characteristic, complemented
14#20274898C211yesmaximal, characteristic, complemented
15C2×C53×#428C211yesmaximal, characteristic, complemented
16C2×#13133208C2211yescharacteristic
17C2×#13133191C2211yescharacteristic
18C2×#13133179C2211yescharacteristic
19C2×C4×C22⋊D53—21no—
20#16810926—24no—
21C2×#13133148C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23C2×#13133146C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133139C2211yescharacteristic
26C2×#13133139C2211yescharacteristic
27C2×#13133137C2211yescharacteristic
28C2×#13133137C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30#16810559—21no—
31#16810546—21no—
32#16810524C2211yescharacteristic
33#16810522C2211yescharacteristic
34#16810521—21no—
35#16810519C2211yescharacteristic
36#16810519C2211yescharacteristic
37#16810513—21no—
38#16810507—21no—
39#16810502—21no—
40#16810494—21no—
41#16810492C2211yescharacteristic
42#16810483C2211yescharacteristic
43C4⋊C4×Dic53C2211yescharacteristic
44C4×C22⋊Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C22×M16×C53C2211yescharacteristic
54C2×#117×C53C2211yescharacteristic
55C22×C4⋊C4×C53C2211yescharacteristic
56C53×#428—24no—
57C2×#107×C53C2211yescharacteristic
58C2×#107×C53C2211yescharacteristic
59C2×#106×C53C2211yescharacteristic
60C2×#106×C53C2211yescharacteristic
61C22×D212C2311yescharacteristic
62C22×C22⋊D53—21no—
63C22×C4×D53—21no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133208—22no—
67#13133208—41no—
68#13133191—22no—
69#13133191Q814yes—
70#13133179—22no—
71#13133179—41no—
72C4×C22⋊D53—44no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×#11719919—21no—
79C2×#11719919—21no—
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53D411yescharacteristic
85C2×C22⋊Dic53D411yescharacteristic
86C2×#11719921—21no—
87C2×#11719921—21no—
88C2×#11719921—21no—
89C2×#11719921—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53C2311yescharacteristic
92C2×C4×Dic53C2311yescharacteristic
93#13133148—22no—
94#13133148—41no—
95#13133146—22no—
96#13133146—22no—
97#13133139—22no—
98#13133139—22no—
99#13133137—22no—
100#13133137—41no—
101C22⋊#88322—22no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124D411yescharacteristic
108#13133124D411yescharacteristic
109C23×C4×C53—21no—
110C23×C4×C53C2311yescharacteristic
111C2×M16×C53—22no—
112C2×M16×C53—22no—
113C2×M16×C53—22no—
114C22×C8×C53C2311yescharacteristic
115C22×C8×C53C2311yescharacteristic
116#117×C53—22no—
117#117×C53—41no—
118C2×C4⋊C4×C53—21no—
119C2×C4⋊C4×C53—22no—
120C2×C4⋊C4×C53—41no—
121C2×C4⋊C4×C53C2311yescharacteristic
122C2×C4⋊C4×C53C2311yescharacteristic
123C2×C4⋊C4×C53C2311yescharacteristic
124C2×C4⋊C4×C53C2311yescharacteristic
125C2×C22⋊C4×C53—21no—
126C2×C42×C53—21no—
127#107×C53—22no—
128#107×C53D414yescomplemented
129#106×C53—22no—
130#106×C53—22no—
131C24×C53C2×D411yescharacteristic
132C23×D53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—42no—
142C22×C4×C53—42no—
143C22×C4×C53—81no—
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×Q812yes—
146C22×C4×C53#4112yes—
147C22×C4×C53C2411yescharacteristic
148C2×C22⋊D53—24no—
149C2×C22⋊D53—42no—
150C2×C22⋊D53—42no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53C2×D411yescharacteristic
155C22×Dic53C2×D411yescharacteristic
156C22×Dic53C2×D411yescharacteristic
157C22×Dic53C2×D411yescharacteristic
158C2×D212—21no—
159C2×D212—21no—
160C2×D212—21no—
161C2×D212—21no—
162C2×D212—21no—
163C2×D212C2×Q812yes—
164C2×C4×D53—42no—
165C2×C4×D53—42no—
166C2×C4×D53—81no—
167M16×C53—28no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C2×C8×C53—21no—
172C2×C8×C53C2×D412yes—
173C2×C8×C53#4112yes—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—42no—
181C4⋊C4×C53C2×D412yes—
182C4⋊C4×C53C2×D412yes—
183C22⋊C4×C53—42no—
184C42×C53—42no—
185C22⋊Dic53—22no—
186C22⋊Dic53—41no—
187C22⋊Dic53—81no—
188#11719921—24no—
189#11719921—42no—
190#11719921—42no—
191#11719921—81no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—42no—
198C4⋊Dic53—42no—
199C4⋊Dic53C2×Q812yes—
200#11719919—42no—
201#11719919—42no—
202#11719919—42no—
203#11719919—81no—
204#11719919—81no—
205C4×Dic53—21no—
206C4×Dic53—21no—
207C4×Dic53—21no—
208C4×Dic53—21no—
209C4×Dic53—42no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C2×#88322—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—21no—
222C23×C53C22⋊Q812yes—
223C23×C53C22×D411yescharacteristic
224C22×D53—22no—
225C22×D53—41no—
226C22×D53—41no—
227C22×D53—41no—
228C22×D53—41no—
229C22×D53—41no—
230C22×D53—41no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—22no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53—41no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—84no—
250C2×C4×C53C22⋊Q812yes—
251C2×C4×C53C22×D411yescharacteristic
252C2×C4×C53C22×Q811yescharacteristic
253C2×C4×C53C2×#4111yescharacteristic
254C2×C4×C53C2×#4111yescharacteristic
255C2×C4×C53H(5,𝔽2)11yescharacteristic
256C2×C4×C532−1+411yescharacteristic
257C22⋊D53—48no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—22no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—21no—
267C2×Dic53—21no—
268C2×Dic53—21no—
269C2×Dic53—21no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53AGL(1,ℤ/8)12yes—
279D212—22no—
280D212—22no—
281D212—41no—
282D212—41no—
283C4×D53—84no—
284C8×C53—22no—
285C8×C53—22no—
286#88322—41no—
287#88322—41no—
288#26366—531nomaximal, Sylow
289C22×C53—22no—
290C22×C53—22no—
291C22×C53—22no—
292C22×C53—41no—
293C22×C53—41no—
294C22×C53—41no—
295C22×C53—41no—
296C22×C53C2×C22⋊Q811yescharacteristic
297C22×C53D4211yescharacteristic
298C22×C53#54711yescharacteristic
299C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C53#57511yescharacteristic
301C22×C53#57711yescharacteristic
302C22×C53#57711yescharacteristic
303C2×D53—42no—
304C2×D53—42no—
305C2×D53—42no—
306C2×D53—81no—
307C2×D53—81no—
308C2×D53—81no—
309C2×D53—81no—
310C4×C53—22no—
311C4×C53—42no—
312C4×C53—42no—
313C4×C53—81no—
314C4×C53—81no—
315C4×C53#52511yescharacteristic
316C4×C53#53911yescharacteristic
317C4×C53D4×Q811yescharacteristic
318C4×C53#55411yescharacteristic
319Dic53—22no—
320Dic53—42no—
321Dic53—41no—
322Dic53—41no—
323Dic53—41no—
324Dic53—81no—
325Dic53—81no—
326C2×#478—531no—
327C2×#428D5311yescharacteristic, complemented
328#2441—531no—
329#2361—531no—
330#2094—531no—
331#2088—531no—
332#2067—531no—
333#2059—531no—
334#1948—531no—
335#1918—531no—
336#1895—531no—
337#1875—531no—
338#1864—531no—
339#1853—531no—
340#1851—531no—
341C2×C53—42no—
342C2×C53#244911yescharacteristic
343C2×C53#307711yescharacteristic
344C2×C53#308411yescharacteristic
345C2×C53#327111yescharacteristic
346C2×C53#327211yescharacteristic
347C2×C53#334511yescharacteristic
348C2×C53#335111yescharacteristic
349D53—82no—
350C22×M16C2×D5311yescharacteristic
351C2×#125—531no—
352C2×#121—531no—
353C2×C4×D4—531no—
354C2×C4×D4—1061no—
355C2×#117C2×D5311yescharacteristic
356C22×C4⋊C4C2×D5311yescharacteristic
357#478—1064no—
358#428—24no—
359C2×#107C2×D5311yescharacteristic
360C2×#107C2×D5311yescharacteristic
361C2×#106—531no—
362C2×#106—531no—
363C2×#106C2×D5311yescharacteristic
364C2×#106C2×D5311yescharacteristic
365C2×C4⋊C8—531no—
366C2×C4⋊C8—531no—
367C2×#102—531no—
368C2×#102—531no—
369C2×#102—531no—
370C2×#102—531no—
371C2×C22⋊C8—531no—
372#399—1061no—
373#396—1061no—
374#390—531no—
375#389—531no—
376#389—531no—
377#389—531no—
378#389—1061no—
379#388—1061no—
380#387—1061no—
381#386—1061no—
382#385—1061no—
383#384—531no—
384#380—531no—
385C4×C4⋊C4—531no—
386C4×C22⋊C4—531no—
387#340—531no—
388#340—531no—
389#340—531no—
390#340—531no—
391#340—531no—
392#340—531no—
393#336—531no—
394#336—531no—
395C53#2636611yescharacteristic, complemented, Sylow
396C22×D4—531no—
397C22×D4—1061no—
398C23×C4—21no—
399C23×C4—1061no—
400C23×C4C22×D5311yescharacteristic
401C2×M16—22no—
402C2×M16—22no—
403C2×M16—22no—
404C22×C8—531no—
405C22×C8—531no—
406C22×C8C22×D5311yescharacteristic
407C22×C8C22×D5311yescharacteristic
408#125—1062no—
409#125—2121no—
410#121—534no—
411#121—1062no—
412C4×D4—1062no—
413C4×D4—2121no—
414C4×D4—2124no—
415#117—22no—
416#117—41no—
417C2×C4⋊C4—22no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—41no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—531no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4—1061no—
424C2×C4⋊C4—1061no—
425C2×C4⋊C4—1061no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4—1061no—
429C2×C4⋊C4—1061no—
430C2×C4⋊C4C22×D5311yescharacteristic
431C2×C4⋊C4C22×D5311yescharacteristic
432C2×C4⋊C4C22×D5311yescharacteristic
433C2×C4⋊C4C22×D5311yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—531no—
436C2×C22⋊C4—531no—
437C2×C22⋊C4—1061no—
438C2×C22⋊C4—1061no—
439C2×C22⋊C4—1061no—
440C2×C22⋊C4—1061no—
441C2×C22⋊C4—1061no—
442C2×C42—21no—
443C2×C42—531no—
444C2×C42—531no—
445C2×C42—1061no—
446#107—22no—
447#107C22⋊D5314yescomplemented
448#106—22no—
449#106—22no—
450#106—1062no—
451#106—2121no—
452C4⋊C8—1062no—
453C4⋊C8—1062no—
454#102—1062no—
455#102—1062no—
456#102—1062no—
457#102—2121no—
458C22⋊C8—1062no—
459#95—531no—
460#95—531no—
461#95—1061no—
462#95—1061no—
463#95—1061no—
464#95—1061no—
465#95—1061no—
466C24—1061no—
467C24D4×D5311yescharacteristic
468C2×D4—532no—
469C2×D4—1064no—
470C2×D4—1061no—
471C2×D4—1061no—
472C2×D4—1061no—
473C2×D4—1061no—
474C2×D4—1061no—
475C2×D4—2122no—
476C2×D4—2122no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—81no—
488C22×C4—531no—
489C22×C4—531no—
490C22×C4—531no—
491C22×C4—531no—
492C22×C4—1061no—
493C22×C4—1061no—
494C22×C4—1061no—
495C22×C4—4241no—
496C22×C4—2122no—
497C22×C4—2122no—
498C22×C4#1171994512yes—
499C22×C4D4×D5311yescharacteristic
500C22×C4Q8×D5312yes—
501C22×C4C23×D5311yescharacteristic
502M16—28no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—1061no—
508C2×C8—1061no—
509C2×C8—1061no—
510C2×C8—1061no—
511C2×C8—1061no—
512C2×C8—1061no—
513C2×C8#1171994512yes—
514C2×C8C2×C22⋊D5312yes—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—42no—
522C4⋊C4—532no—
523C4⋊C4—1061no—
524C4⋊C4—1061no—
525C4⋊C4—1061no—
526C4⋊C4—1061no—
527C4⋊C4—1061no—
528C4⋊C4—4241no—
529C4⋊C4—4241no—
530C4⋊C4—2122no—
531C4⋊C4—2122no—
532C4⋊C4—2122no—
533C4⋊C4—2122no—
534C4⋊C4—2122no—
535C4⋊C4C2×C22⋊D5312yes—
536C4⋊C4C2×C22⋊D5312yes—
537C22⋊C4—42no—
538C22⋊C4—1062no—
539C22⋊C4—1064no—
540C22⋊C4—4241no—
541C22⋊C4—4241no—
542C22⋊C4—2122no—
543C22⋊C4—2122no—
544C22⋊C4—2121no—
545C42—42no—
546C42—1061no—
547C42—1061no—
548C42—1061no—
549C42—1061no—
550C42—2122no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—1062no—
558C23—2121no—
559C23—2121no—
560C23—2121no—
561C23—2121no—
562C23—2121no—
563C23—2121no—
564C23#1313319512yes—
565C23C2×D4×D5311yescharacteristic
566D4—1062no—
567D4—1062no—
568D4—2128no—
569D4—2121no—
570D4—2121no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—84no—
590C2×C4—532no—
591C2×C4—1061no—
592C2×C4—1061no—
593C2×C4—1061no—
594C2×C4—1062no—
595C2×C4—1061no—
596C2×C4—1061no—
597C2×C4—1061no—
598C2×C4—1061no—
599C2×C4—1061no—
600C2×C4—1061no—
601C2×C4—1061no—
602C2×C4—1061no—
603C2×C4—4244no—
604C2×C4—2121no—
605C2×C4—2121no—
606C2×C4—2121no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4—2121no—
611C2×C4—2121no—
612C2×C4#1313319512yes—
613C2×C4C22×C22⋊D5311yescharacteristic
614C2×C4C2×Q8×D5311yescharacteristic
615C2×C4C2×#1171994511yescharacteristic
616C2×C4C2×#1171994511yescharacteristic
617C2×C4#1313326211yescharacteristic
618C2×C4#1313326511yescharacteristic
619C8—22no—
620C8—22no—
621C8—2121no—
622C8—2121no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—4241no—
631C22—4241no—
632C22—4241no—
633C22—4241no—
634C22—2122no—
635C22—2122no—
636C22—2122no—
637C22C2×#1313319511yescharacteristic
638C22D4×C22⋊D5311yescharacteristic
639C22#1681127411yescharacteristic
640C22AGL(1,ℤ/8)×D5311yescharacteristic
641C22#1681142211yescharacteristic
642C22#1681143211yescharacteristic
643C22#1681143311yescharacteristic
644C4—22no—
645C4—42no—
646C4—42no—
647C4—81no—
648C4—81no—
649C4—1062no—
650C4—4241no—
651C4—4241no—
652C4—2121no—
653C4—2121no—
654C4—2122no—
655C4—2121no—
656C4#1681113411yescharacteristic
657C4#1681122011yescharacteristic
658C4Q8×C22⋊D5311yescharacteristic
659C4#1681132811yescharacteristic
660C2—42no—
661C2—4242no—
662C2#2027852411yescharacteristic
663C2#2028451111yescharacteristic
664C2#2028457611yescharacteristic
665C2#2028629411yescharacteristic
666C2#2028630911yescharacteristic
667C2#2028718611yescharacteristic
668C2#2028727611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#428C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenterC2×#428D53
hypocenterC53#26366
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
1. C2×#16810926C2semidirect
2. #20278422C2semidirect
4. #20276245C2semidirect
5. #20276214C2semidirect
6. #20276098C2semidirect
7. #20276051C2semidirect
8. #20275475C2semidirect
9. #20275288C2semidirect
12. #20274973C2semidirect
13. #20274908C2semidirect
14. #20274898C2semidirect
15. C2×C53×#428C2semidirect
128. #107×C53D4semidirect
327. C2×#428D53semidirect
395. C53#26366semidirect
447. #107C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109262
#202784222
#202777312
#202762452
#202762142
#202760982
#202760512
#202754752
#202752882
#202751532
#202750372
#202749732
#202749082
#202748982
C2×C53×#4282
#2636653