Finite Group Databasebeta

← #27402199

Subgroups — #27402199

Order 13568.

1010 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×#16810688C211yesmaximal, characteristic
2C2×#16810660C211yesmaximal, characteristic, complemented
3C2×#106×D53C211yesmaximal, characteristic, complemented
4#20277647C211yesmaximal, characteristic
5#20277641C211yesmaximal, characteristic
6#20275485C211yesmaximal, characteristic, complemented
7#20274915C211yesmaximal, characteristic, complemented
8#20274913C211yesmaximal, characteristic, complemented
9#20274904C211yesmaximal, characteristic, complemented
10#20274901C211yesmaximal, characteristic, complemented
11#20274877C211yesmaximal, characteristic, complemented
12#20274739C211yesmaximal, characteristic, complemented
13#20274739C211yesmaximal, characteristic, complemented
14C4×#13133148C211yesmaximal, characteristic, complemented
15C53×#1949C211yesmaximal, characteristic, complemented
16C22×C8×D53C2211yescharacteristic
17C22×C8×D53C2211yescharacteristic
18C2×C4×D212C2211yescharacteristic
19C2×#13133175C2211yescharacteristic
20C2×#13133175C2211yescharacteristic
21C2×C4⋊C4×D53C2211yescharacteristic
22C2×C4⋊C4×D53C2211yescharacteristic
23#16810688C418yescomplemented
24#16810660—24no—
25C2×#13133151C2211yescharacteristic
26C2×#13133150C2211yescharacteristic
27C2×#13133150C2211yescharacteristic
28C2×#13133149C2211yescharacteristic
29C2×#13133148C2211yescharacteristic
30C2×#13133148C2211yescharacteristic
31C2×#13133148C2211yescharacteristic
32C2×#13133148C2211yescharacteristic
33#106×D53—24no—
34C2×#13133129C2211yescharacteristic
35C2×#13133129C2211yescharacteristic
36C2×C4×#88322C2211yescharacteristic
37#16810514—21no—
38#16810512—21no—
39#16810508—21no—
40#16810506—21no—
41#16810501—21no—
42#16810501—21no—
43#16810489C2211yescharacteristic
44#16810480C2211yescharacteristic
45#16810480C2211yescharacteristic
46#16810480C2211yescharacteristic
47#16810480C2211yescharacteristic
48C4×C4⋊Dic53C2211yescharacteristic
49C4×#11719921—21no—
50#16810470—21no—
51#16810386C2211yescharacteristic
52#16810386C2211yescharacteristic
53#16810385C2211yescharacteristic
54#16810385C2211yescharacteristic
55C2×#107×C53C2211yescharacteristic
56C2×#106×C53C2211yescharacteristic
57C2×C4⋊C8×C53C2211yescharacteristic
58C53×#382C2211yescharacteristic
59C4×C4⋊C4×C53C2211yescharacteristic
60C53×#340C2211yescharacteristic
61C53×#340C2211yescharacteristic
62C22×D212D411yescharacteristic
63C22×C4×D53—21no—
64C22×C4×D53—21no—
65C22×C4×D53C2311yescharacteristic
66C22×C4×D53C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69C2×C8×D53—22no—
70C2×C8×D53—22no—
71C2×C8×D53—22no—
72C2×C8×D53—22no—
73C2×C8×D53C2×C414yes—
74C2×C8×D53C2×C414yescomplemented
75C4×D212C2×C414yes—
76C4×D212Q814yes—
77#13133175—22no—
78#13133175—22no—
79#13133175—41no—
80#13133175—41no—
81C2×C4⋊Dic53D411yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×C4⋊Dic53C2311yescharacteristic
84C2×C4⋊Dic53C2311yescharacteristic
85C2×C4⋊Dic53C2311yescharacteristic
86C2×#11719919—21no—
87C2×#11719919—21no—
88C4⋊C4×D53—22no—
89C4⋊C4×D53—22no—
90C4⋊C4×D53—41no—
91C4⋊C4×D53—41no—
92C2×#11719921—21no—
93C2×#11719921—21no—
94C2×#11719921—21no—
95C2×#11719921—21no—
96C2×#11719921—21no—
97C2×#11719921—21no—
98C2×#11719921D411yescharacteristic
99C2×#11719921D411yescharacteristic
100C2×C4×Dic53—21no—
101C2×C4×Dic53—21no—
102C2×C4×Dic53—21no—
103#13133151—22no—
104#13133150—41no—
105#13133150—41no—
106#13133149—22no—
107#13133148—22no—
108#13133148—22no—
109#13133148—22no—
110#13133148D414yescomplemented
111#13133129C2×C414yes—
112#13133129C2×C414yescomplemented
113C4×#88322C2×C414yescomplemented
114#13133124—21no—
115#13133124—21no—
116#13133124—21no—
117#13133124—21no—
118#13133124—21no—
119#13133124—21no—
120#13133124—21no—
121C22×C8×C53C2311yescharacteristic
122C22×C8×C53C2311yescharacteristic
123C2×C4⋊C4×C53C2311yescharacteristic
124C2×C4⋊C4×C53C2311yescharacteristic
125C2×C4⋊C4×C53C2311yescharacteristic
126C2×C4⋊C4×C53C2311yescharacteristic
127C2×C42×C53—21no—
128C2×C42×C53C2311yescharacteristic
129#107×C53—22no—
130#106×C53—22no—
131C4⋊C8×C53C2×C414yescomplemented
132#95×C53—21no—
133#95×C53—21no—
134#95×C53—21no—
135C23×D53C2×D411yescharacteristic
136C23×D53C2×D411yescharacteristic
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2411yescharacteristic
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53C2×D411yescharacteristic
149C22×Dic53C2×D411yescharacteristic
150C2×D212—22no—
151C2×D212—22no—
152C2×D212C4⋊C414yes—
153C2×C4×D53—21no—
154C2×C4×D53—21no—
155C2×C4×D53—21no—
156C2×C4×D53—21no—
157C2×C4×D53—21no—
158C2×C4×D53—21no—
159C2×C4×D53—21no—
160C2×C4×D53—21no—
161C2×C4×D53—42no—
162C2×C4×D53—42no—
163C2×C4×D53—42no—
164C2×C4×D53—42no—
165C2×C4×D53—81no—
166C2×C4×D53—81no—
167C2×C4×D53C22×C412yes—
168C2×C4×D53C22×C412yes—
169C2×C4×D53C2×Q812yes—
170C2×C4×D53C2×Q812yes—
171C2×C8×C53—21no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C2×C8×C53—21no—
175C2×C8×C53C22×C412yes—
176C2×C8×C53C22×C412yes—
177C4⋊C4×C53—21no—
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×C53C2×D412yes—
185C42×C53—81no—
186C42×C53C22×C412yes—
187C42×C53C2×Q812yes—
188#11719921—42no—
189#11719921—42no—
190#11719921—42no—
191#11719921—42no—
192#11719921—81no—
193#11719921—81no—
194#11719921C4⋊C414yes—
195#11719921C4⋊C414yes—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—21no—
203C4⋊Dic53C4⋊C414yes—
204C4⋊Dic53C2×D412yes—
205#11719919—42no—
206#11719919—42no—
207C4×Dic53—42no—
208C4×Dic53—42no—
209C4×Dic53—81no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322C22×C412yes—
213C2×#88322C22×C412yes—
214C2×#88322C2×D412yes—
215C2×#88322#4112yes—
216C8×D53—28no—
217C8×D53—28no—
218C23×C53C22×D411yescharacteristic
219C22×D53—21no—
220C22×D53—21no—
221C22×D53—21no—
222C22×D53—21no—
223C22×D53—21no—
224C22×D53—21no—
225C22×D53—21no—
226C22×D53—21no—
227C22×D53—21no—
228C22×D53—21no—
229C22×D53C2×C4⋊C412yes—
230C22×D53C2×C4⋊C412yes—
231C22×D53#13512yes—
232C22×D53#13512yes—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—22no—
238C2×C4×C53—41no—
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×C53C2×C4⋊C412yes—
250C2×C4×C53C2×C4⋊C412yes—
251C2×C4×C53C23×C411yescharacteristic
252C2×C4×C53C22×D411yescharacteristic
253C2×C4×C53C22×Q811yescharacteristic
254C2×C4×C53C2×#4111yescharacteristic
255C2×C4×C53H(5,𝔽2)11yescharacteristic
256C2×C4×C532−1+411yescharacteristic
257C2×Dic53—21no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—22no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53C2×C4⋊C412yes—
274C2×Dic53C2×C4⋊C412yes—
275D212—28no—
276C4×D53—22no—
277C4×D53—22no—
278C4×D53—22no—
279C4×D53—22no—
280C4×D53—22no—
281C4×D53—22no—
282C4×D53—22no—
283C4×D53—22no—
284C4×D53—84no—
285C4×D53—84no—
286C8×C53—22no—
287C8×C53—22no—
288#88322—22no—
289#88322C4×D414yes—
290#22654—531nomaximal, Sylow
291C22×C53C22×C4⋊C411yescharacteristic
292C22×C53D4211yescharacteristic
293C22×C53#54711yescharacteristic
294C22×C53C2×#13511yescharacteristic
295C22×C53C2×#13511yescharacteristic
296C22×C53#57711yescharacteristic
297C22×C53#57711yescharacteristic
298C2×D53—22no—
299C2×D53—22no—
300C2×D53—22no—
301C2×D53—22no—
302C2×D53—22no—
303C2×D53—22no—
304C2×D53—22no—
305C2×D53—22no—
306C2×D53—22no—
307C2×D53—41no—
308C2×D53—41no—
309C2×D53—41no—
310C2×D53—41no—
311C2×D53#42714yescomplemented
312C4×C53—22no—
313C4×C53—22no—
314C4×C53—42no—
315C4×C53—81no—
316C4×C53—81no—
317C4×C53—81no—
318C4×C53C2×C4×D411yescharacteristic
319C4×C53#52011yescharacteristic
320C4×C53D4×Q811yescharacteristic
321C4×C53#55411yescharacteristic
322Dic53—22no—
323Dic53—22no—
324Dic53—42no—
325Dic53—81no—
326Dic53—81no—
327Dic53—81no—
328C2×C8×D4—531no—
329C2×#427—531no—
330C22×#106—531no—
331#2352—531no—
332#2352—531no—
333#1949—531no—
334#1949D5311yescharacteristic, complemented
335#1854—531no—
336#1854—531no—
337#1852—531no—
338#1852—531no—
339#1846—531no—
340#1807—531no—
341#1807—531no—
342C4×#106—531no—
343C2×C53D4×C4⋊C411yescharacteristic
344C2×C53C2×#42711yescharacteristic
345C2×C53#291211yescharacteristic
346C2×C53#328211yescharacteristic
347C2×C53#328211yescharacteristic
348C2×C53#333711yescharacteristic
349C2×C53#333711yescharacteristic
350D53—24no—
351D53—42no—
352C23×C8—531no—
353C23×C8—531no—
354C2×C4×D4—531no—
355C2×#117—531no—
356C2×#117—531no—
357C22×C4⋊C4—531no—
358C22×C4⋊C4—531no—
359C8×D4—538no—
360#427—1064no—
361C2×#107—531no—
362C2×#107—531no—
363C2×#107—531no—
364C2×#107C2×D5311yescharacteristic
365C2×#106—531no—
366C2×#106—531no—
367C2×#106—531no—
368C2×#106—531no—
369C2×#106—531no—
370C2×#106—1064no—
371C2×#106C2×D5311yescharacteristic
372C2×C4⋊C8C2×D5311yescharacteristic
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#382C2×D5311yescharacteristic
384#380—531no—
385#380—531no—
386#380—531no—
387#380—531no—
388C4×C4⋊C4—531no—
389C4×C4⋊C4C2×D5311yescharacteristic
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#2265411yescharacteristic, complemented, Sylow
399C22×D4—531no—
400C23×C4—531no—
401C23×C4—531no—
402C23×C4—1061no—
403C23×C4—1061no—
404C22×C8—531no—
405C22×C8—531no—
406C22×C8—534no—
407C22×C8—534no—
408C22×C8—1062no—
409C22×C8—1062no—
410C22×C8—1062no—
411C22×C8—1062no—
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—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×C22⋊C4—1061no—
443C2×C42—21no—
444C2×C42—1061no—
445C2×C42—1061no—
446C2×C42—1061no—
447C2×C42C22×D5311yescharacteristic
448#107—22no—
449#107—1062no—
450#107—2121no—
451#107—2121no—
452#106—22no—
453#106—534no—
454#106—1062no—
455#106—1062no—
456#106—1062no—
457#106—1062no—
458C4⋊C8C4×D5314yescomplemented
459C22⋊C8—534no—
460C22⋊C8—534no—
461C4×C8—534no—
462#95—21no—
463#95—21no—
464#95—21no—
465#95—1061no—
466#95—1061no—
467#95—1061no—
468#95—1061no—
469#95—1061no—
470#95—1061no—
471#95—1061no—
472C24—531no—
473C24—531no—
474C2×D4—534no—
475C2×D4—1062no—
476C2×D4—1062no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—531no—
482C22×C4—531no—
483C22×C4—532no—
484C22×C4—532no—
485C22×C4—532no—
486C22×C4—532no—
487C22×C4—1061no—
488C22×C4—1061no—
489C22×C4—1061no—
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—1061no—
499C22×C4—4241no—
500C22×C4—4241no—
501C22×C4—2122no—
502C22×C4—2122no—
503C22×C4—2122no—
504C22×C4—2122no—
505C22×C4D4×D5311yescharacteristic
506C22×C4D4×D5311yescharacteristic
507C22×C4C23×D5311yescharacteristic
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—532no—
513C2×C8—532no—
514C2×C8—532no—
515C2×C8—532no—
516C2×C8—1068no—
517C2×C8—1068no—
518C2×C8—1061no—
519C2×C8—1061no—
520C2×C8C2×C4×D5312yes—
521C2×C8C2×C4×D5312yes—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—534no—
530C4⋊C4—532no—
531C4⋊C4—1061no—
532C4⋊C4—1061no—
533C4⋊C4—1061no—
534C4⋊C4—1061no—
535C4⋊C4—1061no—
536C4⋊C4—1061no—
537C4⋊C4—1061no—
538C4⋊C4—2122no—
539C4⋊C4—2122no—
540C4⋊C4D4×D5312yes—
541C22⋊C4—534no—
542C22⋊C4—534no—
543C22⋊C4—4241no—
544C22⋊C4—4241no—
545C22⋊C4—2122no—
546C22⋊C4—2122no—
547C22⋊C4—2122no—
548C22⋊C4—2122no—
549C42—81no—
550C42—4241no—
551C42—2122no—
552C42—2122no—
553C42C2×C4×D5312yes—
554C42Q8×D5312yes—
555C23—532no—
556C23—532no—
557C23—532no—
558C23—532no—
559C23—1061no—
560C23—1061no—
561C23—1061no—
562C23—1061no—
563C23—1061no—
564C23—1061no—
565C23—1061no—
566C23—1061no—
567C23—1061no—
568C23—1061no—
569C23C2×D4×D5311yescharacteristic
570D4—1068no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—41no—
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—532no—
588C2×C4—532no—
589C2×C4—1061no—
590C2×C4—1061no—
591C2×C4—1061no—
592C2×C4—1061no—
593C2×C4—1062no—
594C2×C4—1062no—
595C2×C4—1062no—
596C2×C4—1062no—
597C2×C4—1062no—
598C2×C4—1062no—
599C2×C4—1062no—
600C2×C4—1062no—
601C2×C4—1062no—
602C2×C4—4244no—
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—2121no—
613C2×C4—2121no—
614C2×C4—2121no—
615C2×C4C4⋊C4×D5312yes—
616C2×C4C4⋊C4×D5312yes—
617C2×C4C22×C4×D5311yescharacteristic
618C2×C4C2×D4×D5311yescharacteristic
619C2×C4C2×Q8×D5311yescharacteristic
620C2×C4#41×D5311yescharacteristic
621C2×C4#1313326311yescharacteristic
622C2×C4#1313326511yescharacteristic
623C8—22no—
624C8—22no—
625C8—534no—
626C8—1062no—
627C22—534no—
628C22—1062no—
629C22—1062no—
630C22—1062no—
631C22—1062no—
632C22—1062no—
633C22—1062no—
634C22—1062no—
635C22—1062no—
636C22—1062no—
637C22—2121no—
638C22—2121no—
639C22—2121no—
640C22—2121no—
641C22C2×C4⋊C4×D5311yescharacteristic
642C22#1681125711yescharacteristic
643C22#1681127511yescharacteristic
644C22#135×D5311yescharacteristic
645C22#135×D5311yescharacteristic
646C22#1681143211yescharacteristic
647C22#1681143311yescharacteristic
648C4—22no—
649C4—22no—
650C4—42no—
651C4—81no—
652C4—81no—
653C4—81no—
654C4—1062no—
655C4—1062no—
656C4—4241no—
657C4—4241no—
658C4—4241no—
659C4—2122no—
660C4C4×D4×D5311yescharacteristic
661C4#1681110511yescharacteristic
662C4#1681128811yescharacteristic
663C4#1681132911yescharacteristic
664C2—1064no—
665C2—2122no—
666C2#2027761611yescharacteristic
667C2#427×D5311yescharacteristic
668C2#2028330911yescharacteristic
669C2#2028641711yescharacteristic
670C2#2028642811yescharacteristic
671C2#2028709111yescharacteristic
672C2#2028710111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1949C2
socleC23×C53C22×D4
prime radicalC22×D212D4
cosocleC22×C4×C53C24
hypercenter#1949D53
hypocenterC53#22654
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
2. C2×#16810660C2semidirect
3. C2×#106×D53C2semidirect
6. #20275485C2semidirect
7. #20274915C2semidirect
8. #20274913C2semidirect
9. #20274904C2semidirect
10. #20274901C2semidirect
11. #20274877C2semidirect
12. #20274739C2semidirect
13. #20274739C2semidirect
14. C4×#13133148C2semidirect
15. C53×#1949C2semidirect
23. #16810688C4semidirect
74. C2×C8×D53C2×C4semidirect
110. #13133148D4semidirect
112. #13133129C2×C4semidirect
113. C4×#88322C2×C4semidirect
131. C4⋊C8×C53C2×C4semidirect
311. C2×D53#427semidirect
334. #1949D53semidirect
398. C53#22654semidirect
458. C4⋊C8C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168106882
C2×#168106602
C2×#106×D532
#202776472
#202776412
#202754852
#202749152
#202749132
#202749042
#202749012
#202748772
#202747392
C4×#131331482
C53×#19492
#2265453