Finite Group Databasebeta

← #27383989

Subgroups — #27383989

Order 13568.

1426 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#117×D53C211yesmaximal, characteristic
2C2×#16810678C211yesmaximal, characteristic, complemented
3C2×#16810674C211yesmaximal, characteristic, complemented
4#20274782C212yesmaximal, complemented
5#20274698C212yesmaximal, complemented
6#20274697C212yesmaximal, complemented
7#20274624C212yesmaximal
8#20274484C211yesmaximal, characteristic, complemented
9#20274482C211yesmaximal, characteristic, complemented
10#20274481C211yesmaximal, characteristic, complemented
11C53×#1750C211yesmaximal, characteristic, complemented
12C23×C4×D53C2211yescharacteristic
13C2×#13133176C2211yescharacteristic
14C2×#13133175C411yescharacteristic
15C2×#13133175C411yescharacteristic
16C2×#13133174C2211yescharacteristic
17C2×#13133174C2211yescharacteristic
18C2×#13133173C411yescharacteristic
19C2×#13133173C411yescharacteristic
20#117×D53—24no—
21#117×D53—24no—
22C2×C4⋊C4×D53C411yescharacteristic
23C2×C4⋊C4×D53C411yescharacteristic
24C2×C22⋊C4×D53C411yescharacteristic
25C2×C22⋊C4×D53C411yescharacteristic
26C2×C42×D53C2211yescharacteristic
27C2×C42×D53C2211yescharacteristic
28#16810678—24no—
29#16810674—24no—
30C2×#13133146C2212yescomplemented
31C2×#13133146C2212yescomplemented
32C2×C22⋊#88322C2212yes—
33C2×#13133129C2212yes—
34C2×#13133129C2212yes—
35C2×#13133129C2212yes—
36C2×#13133129C2212yes—
37C2×#13133128C2211yescharacteristic, complemented
38C2×C4×#88322C2211yescharacteristic, complemented
39#16810378C2212yescomplemented
40#16810377C2212yescomplemented
41#16810377C2212yescomplemented
42C2×#117×C53C2211yescharacteristic
43C2×C22⋊C8×C53C2212yes—
44C2×#97×C53C2211yescharacteristic, complemented
45C2×C4×C8×C53C2211yescharacteristic, complemented
46C53×#336C2212yescomplemented
47C24×D53C2×C411yescharacteristic
48C23×Dic53C2×C411yescharacteristic
49C22×C4×D53—24no—
50C22×C4×D53—22no—
51C22×C4×D53—22no—
52C22×C4×D53—24no—
53C22×C4×D53C2×C411yescharacteristic
54C22×C4×D53C2×C411yescharacteristic
55C22×C4×D53C2×C411yescharacteristic
56C22×C4×D53C2×C411yescharacteristic
57C22×C4×D53C2×C411yescharacteristic
58C22×C4×D53C2×C411yescharacteristic
59C22×C4×D53C2311yescharacteristic
60C22×C4×D53C2311yescharacteristic
61C22×#88322C2312yes—
62C22×#88322C2312yes—
63#13133176—22no—
64#13133176—22no—
65#13133175—24no—
66#13133175—24no—
67#13133174—22no—
68#13133174—22no—
69#13133174—22no—
70#13133174—22no—
71#13133173—24no—
72#13133173—24no—
73C2×C4⋊Dic53C2×C411yescharacteristic
74C2×C4⋊Dic53C2×C411yescharacteristic
75C2×#11719919C2×C411yescharacteristic
76C2×#11719919C2×C411yescharacteristic
77C2×#11719919C2×C411yescharacteristic
78C2×#11719919C2×C411yescharacteristic
79C4⋊C4×D53—24no—
80C4⋊C4×D53—24no—
81C2×C22⋊Dic53C2×C411yescharacteristic
82C2×C22⋊Dic53C2×C411yescharacteristic
83C2×#11719921C2×C411yescharacteristic
84C2×#11719921C2×C411yescharacteristic
85C2×#11719921C2×C411yescharacteristic
86C2×#11719921C2×C411yescharacteristic
87C22⋊C4×D53—24no—
88C22⋊C4×D53—24no—
89C2×C4×Dic53C2×C411yescharacteristic
90C2×C4×Dic53C2×C411yescharacteristic
91C2×C4×Dic53C2×C411yescharacteristic
92C2×C4×Dic53C2×C411yescharacteristic
93C2×C4×Dic53C2311yescharacteristic
94C2×C4×Dic53C2311yescharacteristic
95C42×D53—22no—
96C42×D53—22no—
97C42×D53D414yes—
98C42×D53D414yes—
99#13133146—24no—
100#13133146—24no—
101C22⋊#88322—24no—
102#13133129—24no—
103#13133129—24no—
104#13133129—24no—
105#13133129—24no—
106#13133128—22no—
107C4×#88322—22no—
108C23×C4×C53C2311yescharacteristic
109C22×C8×C53C2312yes—
110C22×C8×C53C2312yes—
111#117×C53—22no—
112#117×C53—22no—
113C2×C4⋊C4×C53C2×C411yescharacteristic
114C2×C4⋊C4×C53C2×C411yescharacteristic
115C2×C22⋊C4×C53C2×C411yescharacteristic
116C2×C22⋊C4×C53C2×C411yescharacteristic
117C2×C42×C53C2311yescharacteristic
118C2×C42×C53C2311yescharacteristic
119C22⋊C8×C53—24no—
120#97×C53—22no—
121C4×C8×C53—22no—
122C24×C53C22×C411yescharacteristic
123C23×D53—22no—
124C23×D53—22no—
125C23×D53—22no—
126C23×D53—24no—
127C23×D53—42no—
128C23×D53C22×C411yescharacteristic
129C23×D53C22×C411yescharacteristic
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—22no—
133C22×C4×C53C22×C411yescharacteristic
134C22×C4×C53C22×C411yescharacteristic
135C22×C4×C53C22×C411yescharacteristic
136C22×C4×C53C22×C411yescharacteristic
137C22×C4×C53C22×C411yescharacteristic
138C22×C4×C53#4114yes—
139C22×C4×C53C2411yescharacteristic
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53C22×C411yescharacteristic
144C22×Dic53C22×C411yescharacteristic
145C22×Dic53C22×C411yescharacteristic
146C22×Dic53C22×C411yescharacteristic
147C22×Dic53C22×C411yescharacteristic
148C22×Dic53C22×C411yescharacteristic
149C2×C4×D53—216no—
150C2×C4×D53—216no—
151C2×C4×D53—22no—
152C2×C4×D53—22no—
153C2×C4×D53—24no—
154C2×C4×D53—22no—
155C2×C4×D53—22no—
156C2×C4×D53—21no—
157C2×C4×D53—21no—
158C2×C4×D53—22no—
159C2×C4×D53—22no—
160C2×C4×D53—22no—
161C2×C4×D53—22no—
162C2×C4×D53—22no—
163C2×C4×D53—22no—
164C2×C4×D53—22no—
165C2×C4×D53—24no—
166C2×C4×D53—42no—
167C2×C4×D53—42no—
168C2×C4×D53C22⋊C414yes—
169C2×C4×D53C22⋊C414yes—
170C2×C4×D53C2×D412yes—
171C2×C4×D53C2×D412yes—
172C2×C4×D53#4114yes—
173C2×C8×C53—22no—
174C2×C8×C53—22no—
175C2×C8×C53—22no—
176C2×C8×C53—22no—
177C2×C8×C53—22no—
178C2×C8×C53—22no—
179C4⋊C4×C53—22no—
180C4⋊C4×C53—22no—
181C22⋊C4×C53—22no—
182C22⋊C4×C53—22no—
183C42×C53—21no—
184C42×C53—21no—
185C42×C53C2×D412yes—
186C42×C53C2×D412yes—
187C22⋊Dic53—22no—
188C22⋊Dic53—22no—
189#11719921—22no—
190#11719921—22no—
191#11719921—22no—
192#11719921—22no—
193C4⋊Dic53—22no—
194C4⋊Dic53—22no—
195#11719919—22no—
196#11719919—22no—
197#11719919—22no—
198#11719919—22no—
199C4×Dic53—21no—
200C4×Dic53—21no—
201C4×Dic53—22no—
202C4×Dic53—22no—
203C4×Dic53C22⋊C414yes—
204C4×Dic53C22⋊C414yes—
205C4×Dic53C2×D412yes—
206C4×Dic53C2×D412yes—
207C2×#88322—22no—
208C2×#88322—22no—
209C2×#88322—22no—
210C2×#88322—22no—
211C2×#88322—22no—
212C2×#88322—22no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—22no—
216C23×C53—22no—
217C23×C53#11712yes—
218C23×C53C23×C411yescharacteristic
219C22×D53—21no—
220C22×D53—21no—
221C22×D53—22no—
222C22×D53—28no—
223C22×D53—21no—
224C22×D53—21no—
225C22×D53—22no—
226C22×D53—22no—
227C22×D53—48no—
228C22×D53—48no—
229C22×D53—44no—
230C22×D53—44no—
231C22×D53C2×C22⋊C412yes—
232C22×D53#11712yes—
233C22×D53#13114yes—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—22no—
237C2×C4×C53—22no—
238C2×C4×C53—24no—
239C2×C4×C53—24no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—22no—
243C2×C4×C53—22no—
244C2×C4×C53C2×C22⋊C412yes—
245C2×C4×C53C2×C22⋊C412yes—
246C2×C4×C53#11712yes—
247C2×C4×C53C22×D411yescharacteristic
248C2×C4×C53C22×D411yescharacteristic
249C2×C4×C53C2×#4112yes—
250C2×C4×C53C2×#4112yes—
251C2×Dic53—28no—
252C2×Dic53—21no—
253C2×Dic53—21no—
254C2×Dic53—21no—
255C2×Dic53—21no—
256C2×Dic53—22no—
257C2×Dic53—22no—
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—22no—
261C2×Dic53—22no—
262C2×Dic53C2×C22⋊C412yes—
263C2×Dic53C2×C22⋊C412yes—
264C2×Dic53C2×C22⋊C412yes—
265C2×Dic53#11712yes—
266C4×D53—28no—
267C4×D53—28no—
268C4×D53—28no—
269C4×D53—24no—
270C4×D53—24no—
271C4×D53—24no—
272C4×D53—24no—
273C4×D53—44no—
274C4×D53—44no—
275C4×D53—44no—
276C8×C53—42no—
277C8×C53—42no—
278#88322—42no—
279#88322—42no—
280#20717—531nomaximal, Sylow
281C22×C53—24no—
282C22×C53—22no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53—42no—
286C22×C53C22×C22⋊C411yescharacteristic
287C22×C53C2×#11711yescharacteristic
288C22×C53C2×#11711yescharacteristic
289C22×C53C2×#13112yes—
290C22×C53#56812yes—
291C2×D53—22no—
292C2×D53—24no—
293C2×D53—24no—
294C2×D53—22no—
295C2×D53—24no—
296C2×D53—22no—
297C2×D53—416no—
298C2×D53—41no—
299C2×D53—41no—
300C2×D53—42no—
301C2×D53#40814yes—
302C4×C53—22no—
303C4×C53—22no—
304C4×C53—22no—
305C4×C53—41no—
306C4×C53—41no—
307C4×C53#52512yes—
308C4×C53#52512yes—
309Dic53—22no—
310Dic53—22no—
311Dic53—22no—
312Dic53—22no—
313Dic53—41no—
314Dic53—41no—
315C22×#117—531no—
316C2×#432—531no—
317C2×#431—531no—
318#1817—532no—
319#1798—532no—
320#1798—532no—
321#1779—532no—
322#1750—531no—
323#1750—531no—
324#1750—531no—
325#1750D5311yescharacteristic, complemented
326C2×C53—42no—
327C2×C53#228011yescharacteristic
328C2×C53C2×#40811yescharacteristic
329C2×C53#287911yescharacteristic
330C2×C53#292412yes—
331C2×C53#292412yes—
332D53—24no—
333D53—42no—
334C24×C4—531no—
335C2×#117—531no—
336C2×#117—531no—
337C2×#117—531no—
338C2×#117—531no—
339C2×#117—531no—
340C2×#117—531no—
341C2×#117—531no—
342C2×#117—1064no—
343C2×#117—1064no—
344C2×#117C2×D5311yescharacteristic
345C22×C4⋊C4—531no—
346C22×C4⋊C4—531no—
347C22×C22⋊C4—531no—
348C22×C22⋊C4—531no—
349C22×C42—531no—
350C22×C42—531no—
351#432—1064no—
352#431—1064no—
353C2×C4⋊C8—532no—
354C2×C4⋊C8—532no—
355C2×C22⋊C8—532no—
356C2×C22⋊C8—532no—
357C2×C22⋊C8—532no—
358C2×C22⋊C8—532no—
359C2×C22⋊C8—532no—
360C2×C22⋊C8C2×D5312yes—
361C2×#97—531no—
362C2×#97C2×D5311yescharacteristic, complemented
363C2×C4×C8—531no—
364C2×C4×C8C2×D5311yescharacteristic, complemented
365#336—532no—
366#336—532no—
367#336—532no—
368#336C2×D5312yescomplemented
369C53#2071711yescharacteristic, complemented, Sylow
370C25—531no—
371C23×C4—531no—
372C23×C4—531no—
373C23×C4—531no—
374C23×C4—531no—
375C23×C4—531no—
376C23×C4—531no—
377C23×C4—531no—
378C23×C4—531no—
379C23×C4—531no—
380C23×C4—1062no—
381C23×C4—1062no—
382C23×C4—1064no—
383C23×C4—1064no—
384C23×C4C22×D5311yescharacteristic
385C22×C8—532no—
386C22×C8—532no—
387C22×C8C22×D5312yes—
388C22×C8C22×D5312yes—
389#117—22no—
390#117—22no—
391#117—1062no—
392#117—1062no—
393#117—1064no—
394#117—1064no—
395#117—1064no—
396#117—1064no—
397#117—1062no—
398#117—1062no—
399#117—1062no—
400#117—1062no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—1064no—
408C2×C4⋊C4—1064no—
409C2×C4⋊C4C4×D5311yescharacteristic
410C2×C4⋊C4C4×D5311yescharacteristic
411C2×C22⋊C4—531no—
412C2×C22⋊C4—531no—
413C2×C22⋊C4—531no—
414C2×C22⋊C4—531no—
415C2×C22⋊C4—531no—
416C2×C22⋊C4—531no—
417C2×C22⋊C4—1064no—
418C2×C22⋊C4—1064no—
419C2×C22⋊C4C4×D5311yescharacteristic
420C2×C22⋊C4C4×D5311yescharacteristic
421C2×C42—531no—
422C2×C42—531no—
423C2×C42—531no—
424C2×C42—534no—
425C2×C42—534no—
426C2×C42—531no—
427C2×C42—531no—
428C2×C42—531no—
429C2×C42—1062no—
430C2×C42—1062no—
431C2×C42C22×D5311yescharacteristic
432C2×C42C22×D5311yescharacteristic
433C4⋊C8—1064no—
434C4⋊C8—1064no—
435C22⋊C8—24no—
436C22⋊C8—1064no—
437C22⋊C8—1064no—
438C22⋊C8—1064no—
439C22⋊C8—1064no—
440C22⋊C8—1064no—
441#97—22no—
442#97—1062no—
443C4×C8—22no—
444C4×C8—1062no—
445C24—531no—
446C24—531no—
447C24—1062no—
448C24—1062no—
449C24—1064no—
450C24—1062no—
451C24—2122no—
452C24C2×C4×D5311yescharacteristic
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—531no—
457C22×C4—531no—
458C22×C4—531no—
459C22×C4—531no—
460C22×C4—534no—
461C22×C4—532no—
462C22×C4—532no—
463C22×C4—534no—
464C22×C4—534no—
465C22×C4—531no—
466C22×C4—531no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1064no—
471C22×C4—1062no—
472C22×C4—10616no—
473C22×C4—10616no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1064no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1061no—
483C22×C4—1061no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—2122no—
488C22×C4—2122no—
489C22×C4C2×C4×D5311yescharacteristic
490C22×C4C2×C4×D5311yescharacteristic
491C22×C4C2×C4×D5311yescharacteristic
492C22×C4C2×C4×D5311yescharacteristic
493C22×C4C2×C4×D5311yescharacteristic
494C22×C4#1171994514yes—
495C22×C4C23×D5311yescharacteristic
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—1062no—
503C2×C8—1062no—
504C2×C8—1062no—
505C2×C8—1062no—
506C2×C8—1062no—
507C2×C8—1062no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—1062no—
511C4⋊C4—1062no—
512C4⋊C4—1062no—
513C4⋊C4—1062no—
514C4⋊C4—1062no—
515C4⋊C4—1062no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—1062no—
519C22⋊C4—1062no—
520C22⋊C4—1062no—
521C22⋊C4—1062no—
522C22⋊C4—1062no—
523C22⋊C4—1062no—
524C42—21no—
525C42—21no—
526C42—534no—
527C42—534no—
528C42—532no—
529C42—532no—
530C42—1062no—
531C42—1062no—
532C42—1061no—
533C42—1061no—
534C42D4×D5312yes—
535C42D4×D5312yes—
536C23—21no—
537C23—21no—
538C23—22no—
539C23—22no—
540C23—532no—
541C23—534no—
542C23—532no—
543C23—1068no—
544C23—1061no—
545C23—1061no—
546C23—1062no—
547C23—1061no—
548C23—1061no—
549C23—1062no—
550C23—1062no—
551C23—2128no—
552C23—2124no—
553C23—2128no—
554C23—2124no—
555C23#1313317412yes—
556C23C22×C4×D5311yescharacteristic
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—532no—
568C2×C4—532no—
569C2×C4—532no—
570C2×C4—532no—
571C2×C4—1068no—
572C2×C4—1068no—
573C2×C4—1068no—
574C2×C4—1068no—
575C2×C4—1061no—
576C2×C4—1061no—
577C2×C4—1061no—
578C2×C4—1061no—
579C2×C4—1062no—
580C2×C4—1062no—
581C2×C4—1064no—
582C2×C4—1064no—
583C2×C4—1064no—
584C2×C4—1064no—
585C2×C4—1061no—
586C2×C4—1061no—
587C2×C4—1062no—
588C2×C4—1062no—
589C2×C4—2124no—
590C2×C4—2124no—
591C2×C4—2124no—
592C2×C4C22⋊C4×D5312yes—
593C2×C4C22⋊C4×D5312yes—
594C2×C4#1313317412yes—
595C2×C4C2×D4×D5311yescharacteristic
596C2×C4C2×D4×D5311yescharacteristic
597C2×C4#41×D5312yes—
598C2×C4C2×#1171994512yes—
599C8—42no—
600C8—42no—
601C8—2122no—
602C8—2122no—
603C22—22no—
604C22—24no—
605C22—41no—
606C22—41no—
607C22—42no—
608C22—534no—
609C22—1062no—
610C22—1062no—
611C22—1064no—
612C22—1064no—
613C22—1062no—
614C22—1064no—
615C22—21216no—
616C22—2121no—
617C22—2121no—
618C22—2122no—
619C22C2×C22⋊C4×D5311yescharacteristic
620C22#117×D5311yescharacteristic
621C22C2×#1313317411yescharacteristic
622C22#131×D5312yes—
623C22#1681138812yes—
624C4—22no—
625C4—22no—
626C4—22no—
627C4—41no—
628C4—41no—
629C4—1062no—
630C4—1062no—
631C4—1062no—
632C4—1062no—
633C4—2121no—
634C4—2121no—
635C4#1681113212yes—
636C4#1681113212yes—
637C2—42no—
638C2—1064no—
639C2—2122no—
640C2#2027709611yescharacteristic
641C2#408×D5311yescharacteristic
642C2#2028315711yescharacteristic
643C2#2028337012yes—
644C2#2028337112yes—
subgroupG/H
center, FrattiniC22×C4C23×D53
commutator, socleC23×C53C23×C4
FittingC53×#1750C2
prime radicalC24×D53C2×C4
cosocleC22×C4×C53C24
hypercenter#1750D53
hypocenterC53#20717
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ #1750

Complements

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

NH ≅ G/Ntype
2. C2×#16810678C2semidirect
3. C2×#16810674C2semidirect
4. #20274782C2semidirect
5. #20274698C2semidirect
6. #20274697C2semidirect
8. #20274484C2semidirect
9. #20274482C2semidirect
10. #20274481C2semidirect
11. C53×#1750C2semidirect
30. C2×#13133146C22semidirect
31. C2×#13133146C22semidirect
37. C2×#13133128C22semidirect
38. C2×C4×#88322C22semidirect
39. #16810378C22semidirect
40. #16810377C22semidirect
41. #16810377C22semidirect
44. C2×#97×C53C22semidirect
45. C2×C4×C8×C53C22semidirect
46. C53×#336C22semidirect
325. #1750D53semidirect
362. C2×#97C2×D53semidirect
364. C2×C4×C8C2×D53semidirect
368. #336C2×D53semidirect
369. C53#20717semidirect

Maximal subgroups

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

subgroupindex
C2×#117×D532
C2×#168106782
C2×#168106742
#202747822
#202746982
#202746972
#202746242
#202744842
#202744822
#202744812
C53×#17502
#2071753