Finite Group Databasebeta

← #27402310

Subgroups — #27402310

Order 13568.

978 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810699C211yesmaximal, characteristic
2C2×#16810665C211yesmaximal, characteristic, complemented
3C2×#16810662C211yesmaximal, characteristic, complemented
4#20277646C211yesmaximal, characteristic
5#20277637C211yesmaximal, characteristic
6#20275486C211yesmaximal, characteristic, complemented
7#20275482C211yesmaximal, characteristic, complemented
8#20274915C211yesmaximal, characteristic, complemented
9#20274912C211yesmaximal, characteristic, complemented
10#20274904C211yesmaximal, characteristic, complemented
11#20274902C211yesmaximal, characteristic, complemented
12#20274876C211yesmaximal, characteristic, complemented
13#20274749C211yesmaximal, characteristic, complemented
14#107×Dic53C211yesmaximal, characteristic, complemented
15C53×#1809C211yesmaximal, characteristic, complemented
16C22×#11719912C2211yescharacteristic
17C2×C4×C22⋊D53C2211yescharacteristic
18C2×#13133176C2211yescharacteristic
19C2×#13133175C2211yescharacteristic
20C22×C4⋊Dic53C2211yescharacteristic
21C2×C4⋊C4×D53C2211yescharacteristic
22#16810699C418yescomplemented
23#16810665—24no—
24#16810662—24no—
25C2×#13133151C2211yescharacteristic
26C2×#13133151C2211yescharacteristic
27C2×#13133150C2211yescharacteristic
28C2×#13133150C2211yescharacteristic
29C2×#13133150C2211yescharacteristic
30C2×#13133149C2211yescharacteristic
31C2×#13133149C2211yescharacteristic
32C2×#13133148C2211yescharacteristic
33C2×#13133146C2211yescharacteristic
34C2×C22⋊#88322C2211yescharacteristic
35C2×#13133129C2211yescharacteristic
36C2×C8×Dic53C2211yescharacteristic
37#16810515—21no—
38#16810511—21no—
39#16810510—21no—
40#16810505—21no—
41#16810500—21no—
42#16810499—21no—
43#16810488C2211yescharacteristic
44#16810488C2211yescharacteristic
45#16810480C2211yescharacteristic
46#16810480C2211yescharacteristic
47C4⋊C4×Dic53C2211yescharacteristic
48C4⋊C4×Dic53C2211yescharacteristic
49C22⋊C4×Dic53—21no—
50#16810470—21no—
51#16810385C2211yescharacteristic
52#16810385C2211yescharacteristic
53#16810385C2211yescharacteristic
54#16810385C2211yescharacteristic
55C22×M16×C53C2211yescharacteristic
56C2×#107×C53C2211yescharacteristic
57C2×#106×C53C2211yescharacteristic
58C53×#380C2211yescharacteristic
59C53×#380C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C53×#340C2211yescharacteristic
62C22×C22⋊D53D411yescharacteristic
63C23×Dic53—21no—
64C22×C4×D53—21no—
65C22×C4×D53C2311yescharacteristic
66C2×#11719912—22no—
67C2×#11719912—22no—
68C2×#11719912C2×C414yescomplemented
69C22×#88322C2311yescharacteristic
70C22×#88322C2311yescharacteristic
71C4×C22⋊D53C2×C414yes—
72C4×C22⋊D53Q814yes—
73#13133176—22no—
74#13133176—41no—
75#13133175—22no—
76#13133175—41no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53—22no—
79C2×C4⋊Dic53—41no—
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×C4⋊Dic53C2311yescharacteristic
84C2×#11719919—21no—
85C2×#11719919D411yescharacteristic
86C4⋊C4×D53—22no—
87C4⋊C4×D53—41no—
88C2×C22⋊Dic53—21no—
89C2×C22⋊Dic53D411yescharacteristic
90C2×#11719921—21no—
91C2×#11719921—21no—
92C2×#11719921—21no—
93C2×#11719921D411yescharacteristic
94C2×C4×Dic53—21no—
95C2×C4×Dic53—21no—
96C2×C4×Dic53—21no—
97C2×C4×Dic53—21no—
98C2×C4×Dic53C2311yescharacteristic
99#13133151—22no—
100#13133151D414yescomplemented
101#13133150—22no—
102#13133150—41no—
103#13133150—41no—
104#13133149—22no—
105#13133149—22no—
106#13133148—22no—
107#13133146C2×C414yescomplemented
108C22⋊#88322C2×C414yescomplemented
109#13133129C2×C414yescomplemented
110C8×Dic53C2×C414yescomplemented
111#13133124—21no—
112#13133124—21no—
113#13133124—21no—
114#13133124—21no—
115#13133124—21no—
116#13133124—21no—
117#13133124—21no—
118#13133124—21no—
119C23×C4×C53C2311yescharacteristic
120C2×M16×C53—22no—
121C2×M16×C53—22no—
122C2×M16×C53C2×C414yescomplemented
123C22×C8×C53C2311yescharacteristic
124C22×C8×C53C2311yescharacteristic
125C2×C4⋊C4×C53C2311yescharacteristic
126C2×C4⋊C4×C53C2311yescharacteristic
127C2×C4⋊C4×C53C2311yescharacteristic
128C2×C4⋊C4×C53C2311yescharacteristic
129C2×C22⋊C4×C53—21no—
130C2×C22⋊C4×C53—21no—
131#107×C53—22no—
132#106×C53—22no—
133#95×C53—21no—
134#95×C53—21no—
135C24×C53C2×D411yescharacteristic
136C23×D53C2×D411yescharacteristic
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—21no—
145C22×C4×C53C22×C412yes—
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53C2×Q812yes—
148C22×C4×C53C2411yescharacteristic
149C2×C22⋊D53—41no—
150C2×C22⋊D53—41no—
151C2×C22⋊D53C4⋊C414yes—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—42no—
157C22×Dic53—42no—
158C22×Dic53—81no—
159C22×Dic53C2×D411yescharacteristic
160C22×Dic53C2×D411yescharacteristic
161C22×Dic53C2×D411yescharacteristic
162C2×C4×D53—21no—
163C2×C4×D53—21no—
164C2×C4×D53—21no—
165C2×C4×D53—21no—
166C2×C4×D53—42no—
167C2×C4×D53—42no—
168C2×C4×D53—81no—
169C2×C4×D53C22×C412yes—
170C2×C4×D53C2×Q812yes—
171M16×C53—28no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53C22×C412yes—
175C2×C8×C53C22×C412yes—
176C2×C8×C53C2×D412yes—
177C2×C8×C53#4112yes—
178C4⋊C4×C53—21no—
179C4⋊C4×C53—21no—
180C4⋊C4×C53—21no—
181C4⋊C4×C53—21no—
182C4⋊C4×C53—21no—
183C4⋊C4×C53—21no—
184C4⋊C4×C53—21no—
185C4⋊C4×C53—21no—
186C22⋊C4×C53—42no—
187C22⋊C4×C53—81no—
188C22⋊Dic53—42no—
189C22⋊Dic53C4⋊C414yes—
190#11719921—42no—
191#11719921—42no—
192#11719921—81no—
193#11719921C4⋊C414yes—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—42no—
201C4⋊Dic53C2×D412yes—
202C4⋊Dic53C2×D412yes—
203#11719919—42no—
204#11719919C4⋊C414yes—
205C4×Dic53—42no—
206C4×Dic53—42no—
207C4×Dic53—81no—
208C4×Dic53—81no—
209C4×Dic53C22×C412yes—
210C4×Dic53C2×Q812yes—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C2×#88322C22×C412yes—
216C2×#88322C22×C412yes—
217#11719912—28no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—21no—
222C23×C53—21no—
223C23×C53—21no—
224C23×C53C2×C4⋊C412yes—
225C23×C53C22×D411yescharacteristic
226C22×D53—21no—
227C22×D53—21no—
228C22×D53—21no—
229C22×D53—21no—
230C22×D53—21no—
231C22×D53—21no—
232C22×D53C2×C4⋊C412yes—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
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—41no—
250C2×C4×C53—41no—
251C2×C4×C53C2×C4⋊C412yes—
252C2×C4×C53C23×C411yescharacteristic
253C2×C4×C53C22×D411yescharacteristic
254C2×C4×C53C22×Q811yescharacteristic
255C2×C4×C53C2×#4111yescharacteristic
256C2×C4×C53H(5,𝔽2)11yescharacteristic
257C2×C4×C532−1+411yescharacteristic
258C22⋊D53—44no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—21no—
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—22no—
266C2×Dic53—22no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—84no—
278C2×Dic53C2×C4⋊C412yes—
279C2×Dic53C2×C4⋊C412yes—
280C2×Dic53C2×C4⋊C412yes—
281C4×D53—22no—
282C4×D53—22no—
283C4×D53—22no—
284C4×D53—22no—
285C4×D53—84no—
286C8×C53—22no—
287C8×C53C4×D414yes—
288#88322—22no—
289#88322—22no—
290#22664—531nomaximal, Sylow
291C22×C53—22no—
292C22×C53—22no—
293C22×C53—22no—
294C22×C53—41no—
295C22×C53—41no—
296C22×C53—41no—
297C22×C53—41no—
298C22×C53C22×C4⋊C411yescharacteristic
299C22×C53D4211yescharacteristic
300C22×C53#54711yescharacteristic
301C22×C53#57511yescharacteristic
302C22×C53#57511yescharacteristic
303C22×C53#57611yescharacteristic
304C22×C53#57811yescharacteristic
305C2×D53—22no—
306C2×D53—22no—
307C2×D53—22no—
308C2×D53—41no—
309C2×D53—41no—
310C2×D53—41no—
311C2×D53—41no—
312C4×C53—22no—
313C4×C53—81no—
314C4×C53—81no—
315C4×C53—81no—
316C4×C53—81no—
317C4×C53C2×C4×D411yescharacteristic
318C4×C53#52011yescharacteristic
319C4×C53D4×Q811yescharacteristic
320C4×C53#55411yescharacteristic
321Dic53—22no—
322Dic53—22no—
323Dic53—22no—
324Dic53—42no—
325Dic53—42no—
326Dic53—81no—
327Dic53—81no—
328C2×#436—531no—
329C2×#428—531no—
330C2×#428—531no—
331#2352—531no—
332#2352—531no—
333#1949—531no—
334#1949—531no—
335#1854—531no—
336#1854—531no—
337#1852—531no—
338#1852—531no—
339#1846—531no—
340#1809—531no—
341#1809D5311yescharacteristic, complemented
342C4×#107—531no—
343C2×C53—42no—
344C2×C53D4×C4⋊C411yescharacteristic
345C2×C53#291111yescharacteristic
346C2×C53#291311yescharacteristic
347C2×C53#327311yescharacteristic
348C2×C53#327711yescharacteristic
349C2×C53#335211yescharacteristic
350C2×C53#335511yescharacteristic
351D53—42no—
352C22×M16—531no—
353C22×M16C2×D5311yescharacteristic
354C2×C4×D4—531no—
355C2×#117—531no—
356C2×#117—531no—
357C22×C4⋊C4—531no—
358C22×C4⋊C4—531no—
359#436—538no—
360#428—1064no—
361#428—1064no—
362C2×#107—531no—
363C2×#107—531no—
364C2×#107—531no—
365C2×#107—531no—
366C2×#107—531no—
367C2×#107C2×D5311yescharacteristic
368C2×#106—531no—
369C2×#106—531no—
370C2×#106—531no—
371C2×#106C2×D5311yescharacteristic
372C2×C4⋊C8—531no—
373C2×C22⋊C8—531no—
374C2×C22⋊C8—531no—
375C2×C4×C8—531no—
376#388—1061no—
377#388—1061no—
378#387—1061no—
379#387—1061no—
380#386—1061no—
381#386—1061no—
382#382—531no—
383#382—531no—
384#380—531no—
385#380—531no—
386#380C2×D5311yescharacteristic
387#380C2×D5311yescharacteristic
388C4×C4⋊C4—531no—
389C4×C4⋊C4—531no—
390C4×C22⋊C4—1061no—
391C4×C22⋊C4—1061no—
392#340—531no—
393#340—531no—
394#340—531no—
395#340—531no—
396#340C2×D5311yescharacteristic
397#340C2×D5311yescharacteristic
398C53#2266411yescharacteristic, complemented, Sylow
399C22×D4—531no—
400C23×C4—531no—
401C23×C4—1061no—
402C23×C4—1061no—
403C23×C4C22×D5311yescharacteristic
404C2×M16—22no—
405C2×M16—22no—
406C2×M16—534no—
407C2×M16—1062no—
408C2×M16—1062no—
409C2×M16C4×D5314yescomplemented
410C22×C8—531no—
411C22×C8—531no—
412C22×C8C22×D5311yescharacteristic
413C22×C8C22×D5311yescharacteristic
414C4×D4—534no—
415C4×D4—534no—
416#117—1062no—
417#117—1062no—
418#117—2121no—
419#117—2121no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—531no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4—531no—
424C2×C4⋊C4—531no—
425C2×C4⋊C4—1062no—
426C2×C4⋊C4—1062no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4—1061no—
429C2×C4⋊C4—2121no—
430C2×C4⋊C4—2121no—
431C2×C4⋊C4C22×D5311yescharacteristic
432C2×C4⋊C4C22×D5311yescharacteristic
433C2×C4⋊C4C22×D5311yescharacteristic
434C2×C4⋊C4C22×D5311yescharacteristic
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—531no—
438C2×C22⋊C4—531no—
439C2×C22⋊C4—1061no—
440C2×C22⋊C4—1061no—
441C2×C22⋊C4—1061no—
442C2×C22⋊C4—1061no—
443C2×C42—531no—
444C2×C42—1061no—
445C2×C42—1061no—
446C2×C42—1061no—
447C2×C42—1061no—
448#107—22no—
449#107—534no—
450#107—1062no—
451#107—1062no—
452#107—2121no—
453#107—2121no—
454#106—22no—
455#106—1062no—
456#106—1062no—
457#106—1062no—
458C4⋊C8—534no—
459C22⋊C8—534no—
460C22⋊C8—534no—
461C4×C8—534no—
462#95—21no—
463#95—21no—
464#95—1061no—
465#95—1061no—
466#95—1061no—
467#95—1061no—
468#95—1061no—
469#95—1061no—
470#95—1061no—
471#95—1061no—
472C24—531no—
473C24D4×D5311yescharacteristic
474C2×D4—534no—
475C2×D4—2121no—
476C2×D4—2121no—
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—531no—
486C22×C4—531no—
487C22×C4—531no—
488C22×C4—532no—
489C22×C4—532no—
490C22×C4—1061no—
491C22×C4—1061no—
492C22×C4—1061no—
493C22×C4—1061no—
494C22×C4—1061no—
495C22×C4—1061no—
496C22×C4—1061no—
497C22×C4—1061no—
498C22×C4—4241no—
499C22×C4—4241no—
500C22×C4—2122no—
501C22×C4—2122no—
502C22×C4—2122no—
503C22×C4—2122no—
504C22×C4C2×C4×D5312yes—
505C22×C4D4×D5311yescharacteristic
506C22×C4Q8×D5312yes—
507C22×C4C23×D5311yescharacteristic
508M16—28no—
509M16—1068no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—532no—
513C2×C8—532no—
514C2×C8—1061no—
515C2×C8—1061no—
516C2×C8—1061no—
517C2×C8—1061no—
518C2×C8C2×C4×D5312yes—
519C2×C8C2×C4×D5312yes—
520C2×C8D4×D5312yes—
521C2×C8#1171994712yes—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—534no—
531C4⋊C4—532no—
532C4⋊C4—532no—
533C4⋊C4—1061no—
534C4⋊C4—1061no—
535C4⋊C4—1061no—
536C4⋊C4—1061no—
537C4⋊C4—1061no—
538C4⋊C4—1061no—
539C4⋊C4—2122no—
540C4⋊C4—2122no—
541C22⋊C4—42no—
542C22⋊C4—81no—
543C22⋊C4—534no—
544C22⋊C4—534no—
545C22⋊C4—4241no—
546C22⋊C4—2122no—
547C22⋊C4—2122no—
548C22⋊C4—2122no—
549C42—532no—
550C42—532no—
551C42—4241no—
552C42—4241no—
553C42—2122no—
554C42—2122no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—532no—
562C23—1061no—
563C23—1061no—
564C23—1061no—
565C23—1061no—
566C23—1061no—
567C23—1061no—
568C23C4⋊C4×D5312yes—
569C23C2×D4×D5311yescharacteristic
570D4—2124no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—41no—
578C2×C4—41no—
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—532no—
590C2×C4—532no—
591C2×C4—532no—
592C2×C4—1061no—
593C2×C4—1061no—
594C2×C4—1061no—
595C2×C4—1061no—
596C2×C4—1061no—
597C2×C4—1061no—
598C2×C4—1062no—
599C2×C4—1062no—
600C2×C4—1062no—
601C2×C4—1062no—
602C2×C4—1062no—
603C2×C4—1062no—
604C2×C4—4244no—
605C2×C4—4244no—
606C2×C4—2121no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4—2121no—
611C2×C4—2121no—
612C2×C4—2121no—
613C2×C4—2121no—
614C2×C4—2121no—
615C2×C4—2121no—
616C2×C4C4⋊C4×D5312yes—
617C2×C4C22×C4×D5311yescharacteristic
618C2×C4C2×D4×D5311yescharacteristic
619C2×C4C2×Q8×D5311yescharacteristic
620C2×C4C2×#1171994711yescharacteristic
621C2×C4#1313326311yescharacteristic
622C2×C4#1313326411yescharacteristic
623C8—22no—
624C8—1062no—
625C8—1062no—
626C8#1313317714yes—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—41no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—1062no—
635C22—1062no—
636C22—1062no—
637C22—2121no—
638C22—2121no—
639C22—2121no—
640C22—2121no—
641C22C2×C4⋊C4×D5311yescharacteristic
642C22#1681125711yescharacteristic
643C22#1681127311yescharacteristic
644C22#1681141911yescharacteristic
645C22#1681142211yescharacteristic
646C22#1681142711yescharacteristic
647C22#1681143911yescharacteristic
648C4—22no—
649C4—81no—
650C4—81no—
651C4—81no—
652C4—81no—
653C4—1062no—
654C4—1062no—
655C4—1062no—
656C4—4241no—
657C4—4241no—
658C4—2122no—
659C4—2122no—
660C4C2×#1313317711yescharacteristic
661C4#1681110311yescharacteristic
662C4#1681128611yescharacteristic
663C4#1681133111yescharacteristic
664C2—42no—
665C2—2122no—
666C2#2027761111yescharacteristic
667C2#2028330211yescharacteristic
668C2#2028331411yescharacteristic
669C2#2028632311yescharacteristic
670C2#2028636911yescharacteristic
671C2#2028728511yescharacteristic
672C2#2028731611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1809C2
socleC23×C53C22×D4
prime radicalC22×C22⋊D53D4
cosocleC22×C4×C53C24
hypercenter#1809D53
hypocenterC53#22664
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
2. C2×#16810665C2semidirect
3. C2×#16810662C2semidirect
6. #20275486C2semidirect
7. #20275482C2semidirect
8. #20274915C2semidirect
9. #20274912C2semidirect
10. #20274904C2semidirect
11. #20274902C2semidirect
12. #20274876C2semidirect
13. #20274749C2semidirect
14. #107×Dic53C2semidirect
15. C53×#1809C2semidirect
22. #16810699C4semidirect
68. C2×#11719912C2×C4semidirect
100. #13133151D4semidirect
107. #13133146C2×C4semidirect
108. C22⋊#88322C2×C4semidirect
109. #13133129C2×C4semidirect
110. C8×Dic53C2×C4semidirect
122. C2×M16×C53C2×C4semidirect
341. #1809D53semidirect
398. C53#22664semidirect
409. C2×M16C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168106992
C2×#168106652
C2×#168106622
#202776462
#202776372
#202754862
#202754822
#202749152
#202749122
#202749042
#202749022
#202748762
#202747492
#107×Dic532
C53×#18092
#2266453