Finite Group Databasebeta

← #27401400

Subgroups — #27401400

Order 13568.

1010 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810624C211yesmaximal, characteristic
2#20277586C211yesmaximal, characteristic, complemented
3#20277556C211yesmaximal, characteristic
4#20275409C211yesmaximal, characteristic, complemented
5#20275395C211yesmaximal, characteristic, complemented
6#20275345C211yesmaximal, characteristic, complemented
7#20275338C211yesmaximal, characteristic, complemented
8#20274981C211yesmaximal, characteristic, complemented
9#20274967C211yesmaximal, characteristic, complemented
10#20274806C211yesmaximal, characteristic, complemented
11#20274746C211yesmaximal, characteristic, complemented
12#20274547C211yesmaximal, characteristic, complemented
13#103×Dic53C211yesmaximal, characteristic, complemented
14#20274501C211yesmaximal, characteristic, complemented
15C53×#1788C211yesmaximal, characteristic
16C22×#11719917C2211yescharacteristic
17C2×Q8×Dic53C2211yescharacteristic
18C2×Q8×Dic53C2211yescharacteristic
19C2×#13133173—21no—
20C22×C4⋊Dic53C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133148C2211yescharacteristic
23#16810624C418yescomplemented
24C2×#13133141C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28C2×#13133127C2211yescharacteristic
29C2×C8×Dic53C2211yescharacteristic
30#16810526—21no—
31#16810526—21no—
32#16810526C2211yescharacteristic
33#16810526C2211yescharacteristic
34#16810519C2211yescharacteristic
35#16810519C2211yescharacteristic
36#16810506—21no—
37#16810506—21no—
38#16810484—21no—
39#16810483—21no—
40#16810483C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42C4⋊C4×Dic53C2211yescharacteristic
43C4×C22⋊Dic53—21no—
44C4×C22⋊Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C22×#41×C53C2211yescharacteristic
54C2×#103×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C2×#102×C53C2211yescharacteristic
57C2×#102×C53C2211yescharacteristic
58C2×C22⋊C8×C53C2211yescharacteristic
59C53×#380C2211yescharacteristic
60C23×Dic53—21no—
61C2×#11719917—41no—
62C2×#11719917—41no—
63C2×#11719917C2×C414yes—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66Q8×Dic53—22no—
67Q8×Dic53—22no—
68Q8×Dic53—41no—
69Q8×Dic53—41no—
70#13133173—44no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53—22no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53C2×C414yes—
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53—21no—
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53D411yescharacteristic
88C2×C22⋊Dic53D411yescharacteristic
89C2×C4×Dic53—21no—
90C2×C4×Dic53—21no—
91C2×C4×Dic53—21no—
92C2×C4×Dic53—21no—
93C2×C4×Dic53C2311yescharacteristic
94C2×C4×Dic53C2311yescharacteristic
95#13133150—41no—
96#13133148—41no—
97#13133141C2×C414yes—
98#13133141C2×C414yes—
99#13133138C2×C414yes—
100#13133138C2×C414yes—
101#13133127—41no—
102C8×Dic53—41no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124D411yescharacteristic
109#13133124D411yescharacteristic
110C2×#41×C53—22no—
111C2×#41×C53—22no—
112C2×#41×C53—22no—
113C2×#41×C53—41no—
114C2×#41×C53—41no—
115C2×#41×C53C2×C414yes—
116C22×Q8×C53C2311yescharacteristic
117C22×D4×C53—21no—
118C22×D4×C53C2311yescharacteristic
119C23×C4×C53—21no—
120C23×C4×C53C2311yescharacteristic
121C22×C8×C53C2311yescharacteristic
122C22×C8×C53C2311yescharacteristic
123C2×C4⋊C4×C53C2311yescharacteristic
124C2×C4⋊C4×C53C2311yescharacteristic
125C2×C22⋊C4×C53—21no—
126#103×C53—22no—
127#103×C53—22no—
128#102×C53—41no—
129#102×C53—41no—
130C22⋊C8×C53—22no—
131#95×C53—21no—
132C24×C53—21no—
133C24×C53C2×D411yescharacteristic
134#41×C53—28no—
135#41×C53—44no—
136#41×C53—44no—
137#41×C53—44no—
138C2×Q8×C53—21no—
139C2×Q8×C53—21no—
140C2×Q8×C53—21no—
141C2×Q8×C53C22×C412yes—
142C2×Q8×C53#4112yes—
143C2×Q8×C53#4112yes—
144C2×D4×C53—24no—
145C2×D4×C53—21no—
146C2×D4×C53—21no—
147C2×D4×C53—21no—
148C2×D4×C53—21no—
149C2×D4×C53—21no—
150C2×D4×C53—42no—
151C2×D4×C53—42no—
152C2×D4×C53C22×C412yes—
153C22×C4×C53—21no—
154C22×C4×C53—21no—
155C22×C4×C53—21no—
156C22×C4×C53—21no—
157C22×C4×C53—21no—
158C22×C4×C53—24no—
159C22×C4×C53—21no—
160C22×C4×C53—21no—
161C22×C4×C53—42no—
162C22×C4×C53—42no—
163C22×C4×C53C22×C412yes—
164C22×C4×C53C2×D411yescharacteristic
165C22×C4×C53#4112yes—
166C22×C4×C53C2411yescharacteristic
167C22×Dic53—24no—
168C22×Dic53—21no—
169C22×Dic53—21no—
170C22×Dic53—21no—
171C22×Dic53—21no—
172C22×Dic53—42no—
173C22×Dic53—42no—
174C22×Dic53C2×D411yescharacteristic
175C22×Dic53C2×D411yescharacteristic
176C22×Dic53C2×D411yescharacteristic
177C22×Dic53C2×D411yescharacteristic
178C2×C8×C53—21no—
179C2×C8×C53—21no—
180C2×C8×C53—21no—
181C2×C8×C53—21no—
182C2×C8×C53—21no—
183C2×C8×C53—21no—
184C4⋊C4×C53—21no—
185C4⋊C4×C53—21no—
186C4⋊C4×C53—21no—
187C4⋊C4×C53—21no—
188C22⋊C4×C53—42no—
189C22⋊Dic53—22no—
190C22⋊Dic53—41no—
191C22⋊Dic53—42no—
192C22⋊Dic53—42no—
193C22⋊Dic53—81no—
194C22⋊Dic53—81no—
195C22⋊Dic53—81no—
196C4⋊Dic53—24no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—21no—
203C4⋊Dic53—42no—
204C4⋊Dic53—42no—
205C4⋊Dic53C22×C412yes—
206C4⋊Dic53C22×C412yes—
207#11719919—42no—
208#11719919—42no—
209C4×Dic53—21no—
210C4×Dic53—21no—
211C4×Dic53—21no—
212C4×Dic53—21no—
213C4×Dic53—42no—
214C4×Dic53—42no—
215C4×Dic53—42no—
216C4×Dic53—42no—
217#11719917—44no—
218C2×#88322—21no—
219C2×#88322—21no—
220C2×#88322—21no—
221C2×#88322—21no—
222C2×#88322C22×C412yes—
223C2×#88322C22×C412yes—
224C23×C53—21no—
225C23×C53—21no—
226C23×C53—21no—
227C23×C53—21no—
228C23×C53—21no—
229C23×C53—21no—
230C23×C53—22no—
231C23×C53—41no—
232C23×C53—41no—
233C23×C53—41no—
234C23×C53—41no—
235C23×C53—41no—
236C23×C53—41no—
237C23×C53C4×D412yes—
238C23×C53C22×D411yescharacteristic
239Q8×C53—22no—
240Q8×C53—22no—
241Q8×C53—22no—
242Q8×C53#11714yescomplemented
243D4×C53—22no—
244D4×C53—22no—
245D4×C53—48no—
246D4×C53—41no—
247D4×C53—41no—
248C2×C4×C53—22no—
249C2×C4×C53—21no—
250C2×C4×C53—21no—
251C2×C4×C53—22no—
252C2×C4×C53—22no—
253C2×C4×C53—22no—
254C2×C4×C53—22no—
255C2×C4×C53—22no—
256C2×C4×C53—22no—
257C2×C4×C53—48no—
258C2×C4×C53—41no—
259C2×C4×C53—41no—
260C2×C4×C53—41no—
261C2×C4×C53—41no—
262C2×C4×C53—41no—
263C2×C4×C53—41no—
264C2×C4×C53C4×D412yes—
265C2×C4×C53C23×C411yescharacteristic
266C2×C4×C53C2×#4111yescharacteristic
267C2×C4×C53C2×#4111yescharacteristic
268C2×C4×C53C2×#4111yescharacteristic
269C2×C4×C53H(5,𝔽2)11yescharacteristic
270C2×C4×C53H(5,𝔽2)11yescharacteristic
271C2×Dic53—21no—
272C2×Dic53—21no—
273C2×Dic53—22no—
274C2×Dic53—22no—
275C2×Dic53—21no—
276C2×Dic53—21no—
277C2×Dic53—21no—
278C2×Dic53—21no—
279C2×Dic53—21no—
280C2×Dic53—21no—
281C2×Dic53—21no—
282C2×Dic53—21no—
283C2×Dic53—21no—
284C2×Dic53—48no—
285C2×Dic53—41no—
286C2×Dic53—41no—
287C2×Dic53—41no—
288C2×Dic53—41no—
289C2×Dic53—41no—
290C2×Dic53—41no—
291C2×Dic53—41no—
292C2×Dic53—41no—
293C2×Dic53—41no—
294C2×Dic53—41no—
295C2×Dic53AGL(1,ℤ/8)12yes—
296C8×C53—41no—
297C8×C53—41no—
298#88322—22no—
299#88322—22no—
300#22583—531nomaximal, Sylow
301C22×C53—22no—
302C22×C53—22no—
303C22×C53—22no—
304C22×C53—41no—
305C22×C53—41no—
306C22×C53—41no—
307C22×C53—41no—
308C22×C53—42no—
309C22×C53—42no—
310C22×C53—42no—
311C22×C53—81no—
312C22×C53—81no—
313C22×C53—81no—
314C22×C53—81no—
315C22×C53C2×C4×D411yescharacteristic
316C22×C53#54611yescharacteristic
317C22×C53#54611yescharacteristic
318C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
319C22×C53#57511yescharacteristic
320C22×C53#57711yescharacteristic
321C22×C53#57811yescharacteristic
322C4×C53—22no—
323C4×C53—24no—
324C4×C53—81no—
325C4×C53—81no—
326C4×C53C2×#11711yescharacteristic
327C4×C53#51811yescharacteristic
328C4×C53#55111yescharacteristic
329C4×C53#55111yescharacteristic
330Dic53—22no—
331Dic53—41no—
332Dic53—42no—
333Dic53—42no—
334Dic53—41no—
335Dic53—41no—
336Dic53—81no—
337Dic53—81no—
338C2×#417—531no—
339#2344—531no—
340#2341—531no—
341#1936—531no—
342#1934—531no—
343#1926—531no—
344#1925—531no—
345#1865—531no—
346#1863—531no—
347#1824—531no—
348#1809—531no—
349#1788D5311yescharacteristic
350#1763—531no—
351C4×#103—531no—
352#1753—531no—
353C2×C53—42no—
354C2×C53—82no—
355C2×C53#234211yescharacteristic
356C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
357C2×C53#294711yescharacteristic
358C2×C53#330811yescharacteristic
359C2×C53#331711yescharacteristic
360C2×C53#332711yescharacteristic
361C2×C53#333211yescharacteristic
362C22×#41C2×D5311yescharacteristic
363C22×M16—531no—
364C2×C4×Q8—531no—
365C2×C4×Q8—531no—
366C2×#117—1061no—
367C22×C4⋊C4—531no—
368C2×#107—531no—
369C2×#106—531no—
370#417—538no—
371C2×#103—531no—
372C2×#103—531no—
373C2×#103C2×D5311yescharacteristic
374C2×#103C2×D5311yescharacteristic
375C2×#102—531no—
376C2×#102—531no—
377C2×#102C2×D5311yescharacteristic
378C2×#102C2×D5311yescharacteristic
379C2×C22⋊C8C2×D5311yescharacteristic
380C2×#97—531no—
381C2×C4×C8—531no—
382#390—531no—
383#390—531no—
384#390—1061no—
385#390—1061no—
386#389—531no—
387#389—531no—
388#387—1061no—
389#387—1061no—
390#381—1061no—
391#380—531no—
392#380—1061no—
393#380C2×D5311yescharacteristic
394C4×C4⋊C4—531no—
395C4×C4⋊C4—531no—
396C4×C22⋊C4—531no—
397C4×C22⋊C4—1061no—
398#340—531no—
399#340—531no—
400#340—531no—
401#340—531no—
402#340—531no—
403#340—531no—
404#336—531no—
405#336—531no—
406C53#2258311yescharacteristic, complemented, Sylow
407C2×#41—22no—
408C2×#41—22no—
409C2×#41—22no—
410C2×#41—41no—
411C2×#41—41no—
412C2×#41C4×D5314yes—
413C22×Q8C22×D5311yescharacteristic
414C22×D4—21no—
415C22×D4C22×D5311yescharacteristic
416C23×C4—21no—
417C23×C4—1061no—
418C23×C4C22×D5311yescharacteristic
419C2×M16—534no—
420C2×M16—2121no—
421C2×M16—2121no—
422C22×C8—531no—
423C22×C8—531no—
424C22×C8C22×D5311yescharacteristic
425C22×C8C22×D5311yescharacteristic
426C4×Q8—1062no—
427C4×Q8—1062no—
428C4×Q8—2121no—
429C4×Q8—2121no—
430#117—2124no—
431C2×C4⋊C4—534no—
432C2×C4⋊C4—531no—
433C2×C4⋊C4—531no—
434C2×C4⋊C4—531no—
435C2×C4⋊C4—531no—
436C2×C4⋊C4—1061no—
437C2×C4⋊C4—1062no—
438C2×C4⋊C4—1061no—
439C2×C4⋊C4—1061no—
440C2×C4⋊C4—1061no—
441C2×C4⋊C4—1061no—
442C2×C4⋊C4C22×D5311yescharacteristic
443C2×C4⋊C4C22×D5311yescharacteristic
444C2×C22⋊C4—21no—
445C2×C22⋊C4—531no—
446C2×C22⋊C4—531no—
447C2×C22⋊C4—1061no—
448C2×C22⋊C4—1061no—
449C2×C22⋊C4—1061no—
450C2×C22⋊C4—1061no—
451C2×C22⋊C4—1061no—
452C2×C42—531no—
453C2×C42—531no—
454C2×C42—1061no—
455C2×C42—1061no—
456C2×C42—1061no—
457C2×C42—1061no—
458#107—2121no—
459#106—2121no—
460#103—22no—
461#103—22no—
462#103—534no—
463#103—534no—
464#102—41no—
465#102—41no—
466#102—534no—
467#102—534no—
468C22⋊C8—22no—
469#97—2121no—
470C4×C8—2121no—
471#95—21no—
472#95—531no—
473#95—531no—
474#95—1061no—
475#95—1061no—
476#95—1061no—
477#95—1061no—
478#95—1061no—
479C24—21no—
480C24D4×D5311yescharacteristic
481#41—28no—
482#41—44no—
483#41—44no—
484#41—44no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8C2×C4×D5312yes—
489C2×Q8#1171994712yes—
490C2×Q8#1171994712yes—
491C2×D4—24no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—42no—
498C2×D4—42no—
499C2×D4C2×C4×D5312yes—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—24no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—531no—
511C22×C4—531no—
512C22×C4—531no—
513C22×C4—531no—
514C22×C4—1064no—
515C22×C4—1061no—
516C22×C4—1061no—
517C22×C4—1061no—
518C22×C4—1061no—
519C22×C4—2122no—
520C22×C4—2122no—
521C22×C4C2×C4×D5312yes—
522C22×C4D4×D5311yescharacteristic
523C22×C4#1171994712yes—
524C22×C4C23×D5311yescharacteristic
525M16—2124no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—532no—
533C2×C8—532no—
534C2×C8—1061no—
535C2×C8—1061no—
536C2×C8—1061no—
537C2×C8—1061no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—532no—
543C4⋊C4—532no—
544C4⋊C4—1064no—
545C4⋊C4—1061no—
546C4⋊C4—1061no—
547C4⋊C4—1061no—
548C4⋊C4—1061no—
549C4⋊C4—1061no—
550C4⋊C4—1061no—
551C4⋊C4—2122no—
552C4⋊C4—2122no—
553C4⋊C4—2122no—
554C4⋊C4—2122no—
555C22⋊C4—42no—
556C22⋊C4—1062no—
557C22⋊C4—4241no—
558C22⋊C4—4241no—
559C22⋊C4—4241no—
560C22⋊C4—2122no—
561C22⋊C4—2121no—
562C22⋊C4—2122no—
563C42—1061no—
564C42—1061no—
565C42—1061no—
566C42—1061no—
567C42—2122no—
568C42—2122no—
569C42—2122no—
570C42—2122no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—22no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23#1313317712yes—
585C23C2×D4×D5311yescharacteristic
586Q8—22no—
587Q8—22no—
588Q8—22no—
589Q8#1313317314yescomplemented
590D4—22no—
591D4—22no—
592D4—48no—
593D4—41no—
594D4—41no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—48no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—532no—
612C2×C4—1061no—
613C2×C4—1061no—
614C2×C4—1062no—
615C2×C4—1062no—
616C2×C4—1061no—
617C2×C4—1061no—
618C2×C4—1061no—
619C2×C4—1061no—
620C2×C4—1061no—
621C2×C4—1061no—
622C2×C4—1061no—
623C2×C4—1061no—
624C2×C4—1061no—
625C2×C4—2121no—
626C2×C4—2121no—
627C2×C4—2121no—
628C2×C4—2121no—
629C2×C4—2121no—
630C2×C4—2121no—
631C2×C4—2128no—
632C2×C4—2121no—
633C2×C4—2121no—
634C2×C4—2121no—
635C2×C4—2121no—
636C2×C4#1313317712yes—
637C2×C4C22×C4×D5311yescharacteristic
638C2×C4C2×#1171994711yescharacteristic
639C2×C4C2×#1171994711yescharacteristic
640C2×C4C2×#1171994711yescharacteristic
641C2×C4#1313326211yescharacteristic
642C2×C4#1313326211yescharacteristic
643C8—41no—
644C8—41no—
645C8—1062no—
646C8—1062no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22C2×#1313317711yescharacteristic
662C22#1681126011yescharacteristic
663C22#1681126011yescharacteristic
664C22AGL(1,ℤ/8)×D5311yescharacteristic
665C22#1681142211yescharacteristic
666C22#1681143011yescharacteristic
667C22#1681143711yescharacteristic
668C4—22no—
669C4—24no—
670C4—81no—
671C4—81no—
672C4—1062no—
673C4—4241no—
674C4—4241no—
675C4—2121no—
676C4—2121no—
677C4—2121no—
678C4—2122no—
679C4—2122no—
680C4C2×#1313317311yescharacteristic
681C4#1681109611yescharacteristic
682C4#1681130411yescharacteristic
683C4#1681130411yescharacteristic
684C2—42no—
685C2—82no—
686C2#2027756611yescharacteristic
687C2#2028352811yescharacteristic
688C2#2028356111yescharacteristic
689C2#2028669911yescharacteristic
690C2#2028683411yescharacteristic
691C2#2028697611yescharacteristic
692C2#2028703511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1788C2
socleC23×C53C22×D4
prime radicalC22×#41×C53C22
cosocleC22×C4×C53C24
hypercenter#1788D53
hypocenterC53#22583
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1788

Complements

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

NH ≅ G/Ntype
2. #20277586C2semidirect
4. #20275409C2semidirect
5. #20275395C2semidirect
6. #20275345C2semidirect
7. #20275338C2semidirect
8. #20274981C2semidirect
9. #20274967C2semidirect
10. #20274806C2semidirect
11. #20274746C2semidirect
12. #20274547C2semidirect
13. #103×Dic53C2semidirect
14. #20274501C2semidirect
23. #16810624C4semidirect
242. Q8×C53#117semidirect
406. C53#22583semidirect
589. Q8#13133173semidirect

Maximal subgroups

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

subgroupindex
C2×#168106242
#202775862
#202775562
#202754092
#202753952
#202753452
#202753382
#202749812
#202749672
#202748062
#202747462
#202745472
#103×Dic532
#202745012
C53×#17882
#2258353