Finite Group Databasebeta

← #28306480

Subgroups — #28306480

Order 13568.

1822 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D53⋊(C22⋊C8)C211yesmaximal, characteristic
2C2×#16811565C211yesmaximal, characteristic
3C2×#16811496C214yesmaximal, complemented
4C22×C4⋊C4×D53C211yesmaximal, characteristic
5C22×D53⋊C8C2212yes—
6C22×C4⋊#6157C2212yes—
7C22×C22⋊#6157—21no—
8C2×#13133274—21no—
9D53⋊(C22⋊C8)—22no—
10D53⋊(C22⋊C8)C414yescomplemented
11C2×C22⋊#88324—21no—
12#16811565—22no—
13#16811565C414yescomplemented
14C2×(C2×#6157)⋊C4—21no—
15#16811496—28no—
16C23×C4×D53—21no—
17C23×C4×D53C2211yescharacteristic
18C22×C4⋊Dic53C411yescharacteristic, complemented
19C22×#11719919—21no—
20C2×C4⋊C4×D53—21no—
21C2×C4⋊C4×D53—21no—
22C2×C4⋊C4×D53—28no—
23C2×C4⋊C4×D53C412yescomplemented
24C2×C4⋊C4×D53C2212yes—
25C22×C4⋊C4×C53C411yescharacteristic, complemented
26C23×#6157—22no—
27C2×D53⋊C8—22no—
28C2×D53⋊C8—24no—
29C2×D53⋊C8—24no—
30C2×D53⋊C8C2×C414yescomplemented
31C22×#88324—22no—
32C2×C4⋊#6157—24no—
33C2×C4⋊#6157—24no—
34C2×C4⋊#6157—22no—
35C2×C4⋊#6157C2×C414yescomplemented
36C2×C22⋊#6157—24no—
37C2×C22⋊#6157—42no—
38C2×C22⋊#6157—42no—
39#13133274—24no—
40C22⋊#88324—24no—
41(C2×#6157)⋊C4—24no—
42C24×D53D411yescharacteristic
43C23×Dic53—21no—
44C23×Dic53D411yescharacteristic
45C22×C4×D53—24no—
46C22×C4×D53—24no—
47C22×C4×D53—28no—
48C22×C4×D53—22no—
49C22×C4×D53—22no—
50C22×C4×D53—44no—
51C22×C4×D53—42no—
52C22×C4×D53—42no—
53C22×C4×D53C2×C411yescharacteristic
54C22×C4×D53C2×C412yes—
55C22×C4×D53C2×C412yes—
56C22×C4×D53D411yescharacteristic
57C22×C4×D53D411yescharacteristic
58C22×C4×D53D412yes—
59C22×C4×D53Q812yes—
60C22×C4×D53C2311yescharacteristic
61C2×C4⋊Dic53—24no—
62C2×C4⋊Dic53—21no—
63C2×C4⋊Dic53C2×C412yes—
64C2×#11719919—28no—
65C2×#11719919—22no—
66C2×#11719919—22no—
67C4⋊C4×D53—28no—
68C4⋊C4×D53—28no—
69C4⋊C4×D53—24no—
70C4⋊C4×D53—24no—
71C4⋊C4×D53—24no—
72C4⋊C4×D53D418yes—
73C23×C4×C53—21no—
74C23×C4×C53C2×C411yescharacteristic
75C2×C4⋊C4×C53—24no—
76C2×C4⋊C4×C53—21no—
77C2×C4⋊C4×C53C2×C412yes—
78C24×C53C22⋊C411yescharacteristic
79C23×D53—22no—
80C23×D53—22no—
81C23×D53—22no—
82C23×D53—22no—
83C23×D53C22⋊C411yescharacteristic
84C23×D53C22⋊C412yes—
85C23×D53C4⋊C412yes—
86C23×D53D812yes—
87C23×D53SD1612yes—
88C23×D53SD1612yes—
89C23×D53Q1612yes—
90C23×D53C2×D411yescharacteristic
91C22×#6157—24no—
92C22×#6157—42no—
93C22×#6157—42no—
94C22×#6157—42no—
95C22×#6157—42no—
96C22×#6157—42no—
97C22×#6157—42no—
98C22×C4×C53—22no—
99C22×C4×C53—22no—
100C22×C4×C53—24no—
101C22×C4×C53—22no—
102C22×C4×C53—42no—
103C22×C4×C53—42no—
104C22×C4×C53C4212yes—
105C22×C4×C53C22⋊C411yescharacteristic
106C22×C4×C53C4⋊C412yes—
107C22×C4×C53C22×C411yescharacteristic
108C22×Dic53—22no—
109C22×Dic53—22no—
110C22×Dic53—24no—
111C22×Dic53—22no—
112C22×Dic53—42no—
113C22×Dic53—42no—
114C22×Dic53C22⋊C411yescharacteristic
115C22×Dic53C22⋊C412yes—
116C22×Dic53C4⋊C412yes—
117C22×Dic53C2×D411yescharacteristic
118C2×C4×D53—22no—
119C2×C4×D53—22no—
120C2×C4×D53—22no—
121C2×C4×D53—22no—
122C2×C4×D53—22no—
123C2×C4×D53—22no—
124C2×C4×D53—28no—
125C2×C4×D53—28no—
126C2×C4×D53—28no—
127C2×C4×D53—28no—
128C2×C4×D53—28no—
129C2×C4×D53—28no—
130C2×C4×D53—44no—
131C2×C4×D53—416no—
132C2×C4×D53—416no—
133C2×C4×D53—44no—
134C2×C4×D53—42no—
135C2×C4×D53—42no—
136C2×C4×D53—44no—
137C2×C4×D53C4212yes—
138C2×C4×D53C22⋊C412yes—
139C2×C4×D53C22⋊C412yes—
140C2×C4×D53C4⋊C412yes—
141C2×C4×D53C4⋊C412yes—
142C2×C4×D53C4⋊C412yes—
143C2×C4×D53C22×C411yescharacteristic
144C2×C4×D53C22×C411yescharacteristic
145C2×C4×D53C2×D414yes—
146C2×C4×D53C2×D411yescharacteristic
147C2×C4×D53C2×Q811yescharacteristic
148C2×C4×D53#4114yes—
149C22⋊#6157—48no—
150C2×#88324—24no—
151C2×#88324—42no—
152C2×#88324—42no—
153C4⋊#6157—28no—
154C4⋊#6157—44no—
155D53⋊C8—28no—
156D53⋊C8—44no—
157C4⋊C4×C53—24no—
158C4⋊C4×C53—22no—
159C4⋊C4×C53C22⋊C414yescomplemented
160C4⋊Dic53—24no—
161C4⋊Dic53—22no—
162C4⋊Dic53C22⋊C414yescomplemented
163#11719919—28no—
164#11719919—28no—
165C23×C53—21no—
166C23×C53—21no—
167C23×C53—21no—
168C23×C53—21no—
169C23×C53#9512yes—
170C23×C53#10212yes—
171C23×C53#10312yes—
172C23×C53C2×C22⋊C411yescharacteristic
173C22×D53—21no—
174C22×D53—21no—
175C22×D53—21no—
176C22×D53—21no—
177C22×D53—24no—
178C22×D53—24no—
179C22×D53—24no—
180C22×D53—24no—
181C22×D53—24no—
182C22×D53—24no—
183C22×D53—24no—
184C22×D53—24no—
185C22×D53—24no—
186C22×D53—24no—
187C22×D53—24no—
188C22×D53—24no—
189C22×D53#9512yes—
190C22×D53#10212yes—
191C22×D53#10214yes—
192C22×D53#10312yes—
193C22×D53#10314yes—
194C22×D53#10614yes—
195C22×D53#10714yes—
196C22×D53C2×C22⋊C411yescharacteristic
197C22×D53C2×C4⋊C411yescharacteristic
198C22×D53#12012yes—
199C22×D53#12312yes—
200C22×D53C2×D811yescharacteristic
201C22×D53C2×SD1611yescharacteristic
202C22×D53C2×SD1611yescharacteristic
203C22×D53C2×Q1611yescharacteristic
204C22×D53AGL(1,ℤ/8)12yes—
205C22×D53#13712yes—
206C2×#6157—44no—
207C2×#6157—44no—
208C2×#6157—44no—
209C2×#6157—82no—
210C2×#6157—82no—
211C2×#6157—82no—
212C2×#6157—82no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—24no—
216C2×C4×C53—24no—
217C2×C4×C53—24no—
218C2×C4×C53—48no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53#9512yes—
222C2×C4×C53C2×C4211yescharacteristic
223C2×C4×C53C2×C22⋊C412yes—
224C2×C4×C53C2×C4⋊C411yescharacteristic
225C2×C4×C53#11712yes—
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—24no—
229C2×Dic53—24no—
230C2×Dic53—24no—
231C2×Dic53—48no—
232C2×Dic53—42no—
233C2×Dic53—42no—
234C2×Dic53#9512yes—
235C2×Dic53C2×C22⋊C411yescharacteristic
236C2×Dic53C2×C4⋊C411yescharacteristic
237C2×Dic53#12012yes—
238C2×Dic53#12312yes—
239C4×D53—24no—
240C4×D53—24no—
241C4×D53—24no—
242C4×D53—24no—
243C4×D53—24no—
244C4×D53—24no—
245C4×D53—416no—
246C4×D53—48no—
247C4×D53—48no—
248C4×D53C4×D414yes—
249C4×D53C4×D414yes—
250C4×D53#12114yes—
251C4×D53C22⋊Q814yes—
252#88324—44no—
253#10730—531nomaximal, Sylow
254C22×C53—22no—
255C22×C53—22no—
256C22×C53—22no—
257C22×C53—22no—
258C22×C53—22no—
259C22×C53—22no—
260C22×C53#34014yescomplemented
261C22×C53C2×#9511yescharacteristic
262C22×C53#37911yescharacteristic
263C22×C53#38111yescharacteristic
264C22×C53C2×#10211yescharacteristic
265C22×C53C2×#10311yescharacteristic
266C22×C53#41811yescharacteristic
267C22×C53#41911yescharacteristic
268C2×D53—22no—
269C2×D53—22no—
270C2×D53—22no—
271C2×D53—22no—
272C2×D53—22no—
273C2×D53—22no—
274C2×D53—28no—
275C2×D53—28no—
276C2×D53—28no—
277C2×D53—28no—
278C2×D53#34014yescomplemented
279C2×D53#38512yes—
280C2×D53#38612yes—
281C2×D53C2×#10211yescharacteristic
282C2×D53C2×#10311yescharacteristic
283C2×D53#41712yes—
284C2×D53C2×#10611yescharacteristic
285C2×D53C2×#10711yescharacteristic
286C2×D53#42812yes—
287C2×D53#44712yes—
288C2×D53#44812yes—
289C2×D53#45012yes—
290C2×D53#45112yes—
291C2×D53#48012yes—
292C2×D53#48112yes—
293C2×D53#48312yes—
294C2×D53#48412yes—
295#6157—84no—
296C4×C53—22no—
297C4×C53—44no—
298C4×C53C4×C22⋊C412yes—
299C4×C53#38012yes—
300Dic53—22no—
301Dic53—44no—
302Dic53#38512yes—
303Dic53#38612yes—
304C23×C4⋊C4—531no—
305C22×C22⋊C8—531no—
306C2×#380—531no—
307C2×#340—534no—
308C2×C53—24no—
309C2×C53C22⋊#9511yescharacteristic
310C2×C53C2×#34011yescharacteristic
311C2×C53#173611yescharacteristic
312C2×C53#178611yescharacteristic
313C2×C53#178711yescharacteristic
314C2×C53#182311yescharacteristic
315C2×C53#182411yescharacteristic
316D53—24no—
317D53#185212yes—
318D53#185312yes—
319D53#187212yes—
320D53#187412yes—
321C24×C4—531no—
322C24×C4—1061no—
323C23×C8—532no—
324C22×C4⋊C4—532no—
325C22×C4⋊C4—532no—
326C22×C4⋊C4—531no—
327C22×C4⋊C4—532no—
328C22×C4⋊C4—1068no—
329C22×C4⋊C4—1061no—
330C22×C4⋊C4—1061no—
331C22×C4⋊C4—1061no—
332C22×C4⋊C4#615711yescharacteristic, complemented
333C22×C22⋊C4—1061no—
334C2×C22⋊C8—534no—
335C2×C22⋊C8—1061no—
336C2×C22⋊C8—1062no—
337C2×C22⋊C8—1061no—
338#380—534no—
339#380—1062no—
340C2×#95—1061no—
341#340—1068no—
342C53#1073011yescharacteristic, complemented, Sylow
343C25—531no—
344C23×C4—21no—
345C23×C4—531no—
346C23×C4—531no—
347C23×C4—532no—
348C23×C4—532no—
349C23×C4—532no—
350C23×C4—532no—
351C23×C4—531no—
352C23×C4—531no—
353C23×C4—531no—
354C23×C4—1064no—
355C23×C4—1064no—
356C23×C4—1062no—
357C23×C4—1062no—
358C23×C4—1068no—
359C23×C4—1061no—
360C23×C4—1062no—
361C23×C4—2122no—
362C23×C4—2122no—
363C23×C4—2124no—
364C23×C4C2×#615711yescharacteristic
365C22×C8—534no—
366C22×C8—1064no—
367C22×C8—1062no—
368C22×C8—1062no—
369C22×C8—1064no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—24no—
372C2×C4⋊C4—532no—
373C2×C4⋊C4—538no—
374C2×C4⋊C4—534no—
375C2×C4⋊C4—1064no—
376C2×C4⋊C4—1064no—
377C2×C4⋊C4—1068no—
378C2×C4⋊C4—1068no—
379C2×C4⋊C4—1064no—
380C2×C4⋊C4—1062no—
381C2×C4⋊C4—1062no—
382C2×C4⋊C4—1068no—
383C2×C4⋊C4—1061no—
384C2×C4⋊C4—1064no—
385C2×C4⋊C4—1064no—
386C2×C4⋊C4—1064no—
387C2×C4⋊C4—1062no—
388C2×C4⋊C4C2×#615712yes—
389C2×C22⋊C4—1064no—
390C2×C22⋊C4—2122no—
391C2×C22⋊C4—2122no—
392C22⋊C8—1064no—
393C22⋊C8—1064no—
394#95—1064no—
395C24—531no—
396C24—532no—
397C24—532no—
398C24—532no—
399C24—532no—
400C24—532no—
401C24—532no—
402C24—531no—
403C24—1062no—
404C24—1062no—
405C24—1062no—
406C24—1062no—
407C24C22⋊#615711yescharacteristic
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—531no—
415C22×C4—531no—
416C22×C4—531no—
417C22×C4—534no—
418C22×C4—531no—
419C22×C4—531no—
420C22×C4—534no—
421C22×C4—532no—
422C22×C4—532no—
423C22×C4—531no—
424C22×C4—532no—
425C22×C4—532no—
426C22×C4—532no—
427C22×C4—532no—
428C22×C4—532no—
429C22×C4—532no—
430C22×C4—1062no—
431C22×C4—1062no—
432C22×C4—1068no—
433C22×C4—1068no—
434C22×C4—1068no—
435C22×C4—1068no—
436C22×C4—1062no—
437C22×C4—1062no—
438C22×C4—1068no—
439C22×C4—1068no—
440C22×C4—1062no—
441C22×C4—1062no—
442C22×C4—1062no—
443C22×C4—1062no—
444C22×C4—1064no—
445C22×C4—1062no—
446C22×C4—1064no—
447C22×C4—2122no—
448C22×C4—2122no—
449C22×C4—2124no—
450C22×C4—2124no—
451C22×C4—21216no—
452C22×C4—21216no—
453C22×C4—2124no—
454C22×C4—2122no—
455C22×C4—2122no—
456C22×C4—2122no—
457C22×C4—2122no—
458C22×C4—2122no—
459C22×C4—2122no—
460C22×C4—2122no—
461C22×C4—2122no—
462C22×C4C4×#615712yes—
463C22×C4C4⋊#615712yes—
464C22×C4C22⋊#615711yescharacteristic
465C22×C4C22×#615711yescharacteristic
466C2×C8—1064no—
467C2×C8—1068no—
468C2×C8—2122no—
469C2×C8—2122no—
470C2×C8—2124no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—534no—
474C4⋊C4—1068no—
475C4⋊C4—1068no—
476C4⋊C4—1064no—
477C4⋊C4—1068no—
478C4⋊C4—1062no—
479C4⋊C4—2124no—
480C4⋊C4C22⋊#615714yescomplemented
481C22⋊C4—2128no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—531no—
487C23—531no—
488C23—532no—
489C23—532no—
490C23—531no—
491C23—531no—
492C23—531no—
493C23—531no—
494C23—532no—
495C23—532no—
496C23—532no—
497C23—532no—
498C23—532no—
499C23—534no—
500C23—534no—
501C23—534no—
502C23—534no—
503C23—1064no—
504C23—1064no—
505C23—1064no—
506C23—1064no—
507C23—1064no—
508C23—1064no—
509C23—1064no—
510C23—1064no—
511C23—1064no—
512C23—1064no—
513C23—1064no—
514C23—1064no—
515C23—1061no—
516C23—1061no—
517C23—1061no—
518C23—1061no—
519C23(C2×#6157)⋊C412yes—
520C23#1313327912yes—
521C23#1313328012yes—
522C23C2×C22⋊#615711yescharacteristic
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—48no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—531no—
532C2×C4—531no—
533C2×C4—532no—
534C2×C4—532no—
535C2×C4—534no—
536C2×C4—534no—
537C2×C4—534no—
538C2×C4—534no—
539C2×C4—532no—
540C2×C4—1064no—
541C2×C4—1064no—
542C2×C4—1064no—
543C2×C4—1064no—
544C2×C4—1064no—
545C2×C4—1064no—
546C2×C4—1061no—
547C2×C4—1061no—
548C2×C4—1064no—
549C2×C4—1064no—
550C2×C4—1064no—
551C2×C4—4242no—
552C2×C4—4242no—
553C2×C4—4242no—
554C2×C4—4242no—
555C2×C4—2128no—
556C2×C4—2128no—
557C2×C4—21216no—
558C2×C4—2124no—
559C2×C4—2124no—
560C2×C4—2124no—
561C2×C4—2128no—
562C2×C4—2122no—
563C2×C4—2122no—
564C2×C4(C2×#6157)⋊C412yes—
565C2×C4C2×C4×#615711yescharacteristic
566C2×C4C2×C22⋊#615712yes—
567C2×C4C2×C4⋊#615711yescharacteristic
568C2×C4#1313329612yes—
569C8—2124no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—532no—
577C22—532no—
578C22—532no—
579C22—532no—
580C22—531no—
581C22—531no—
582C22—531no—
583C22—531no—
584C22—532no—
585C22—532no—
586C22—532no—
587C22—532no—
588C22—532no—
589C22—532no—
590C22—532no—
591C22—532no—
592C22—534no—
593C22—1068no—
594C22—1068no—
595C22—1068no—
596C22—1068no—
597C22—1062no—
598C22—1062no—
599C22—1062no—
600C22—1062no—
601C22—1062no—
602C22—1062no—
603C22#1681149614yescomplemented
604C22C2×(C2×#6157)⋊C411yescharacteristic
605C22#1681156411yescharacteristic
606C22#1681156611yescharacteristic
607C22C2×#1313327911yescharacteristic
608C22C2×#1313328011yescharacteristic
609C22#1681161511yescharacteristic
610C22#1681161611yescharacteristic
611C4—22no—
612C4—44no—
613C4—532no—
614C4—532no—
615C4—1062no—
616C4—4244no—
617C4—2124no—
618C4C4×C22⋊#615712yes—
619C4#1681156512yes—
620C2—24no—
621C2—532no—
622C2—532no—
623C2—532no—
624C2—532no—
625C2—1064no—
626C2#2028988911yescharacteristic
627C2C2×#1681149611yescharacteristic
628C2#2029040211yescharacteristic
629C2#2029055311yescharacteristic
630C2#2029055411yescharacteristic
631C2#2029061811yescharacteristic
632C2#2029061911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C2×C42
FrattiniC22×C4C22×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×D53D4
cosocleC22×C4×D53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#10730
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#16811496C2semidirect
10. D53⋊(C22⋊C8)C4semidirect
13. #16811565C4semidirect
18. C22×C4⋊Dic53C4semidirect
23. C2×C4⋊C4×D53C4semidirect
25. C22×C4⋊C4×C53C4semidirect
30. C2×D53⋊C8C2×C4semidirect
35. C2×C4⋊#6157C2×C4semidirect
159. C4⋊C4×C53C22⋊C4semidirect
162. C4⋊Dic53C22⋊C4semidirect
260. C22×C53#340semidirect
278. C2×D53#340semidirect
332. C22×C4⋊C4#6157semidirect
342. C53#10730semidirect
480. C4⋊C4C22⋊#6157semidirect
603. C22#16811496semidirect

Maximal subgroups

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

subgroupindex
C2×D53⋊(C22⋊C8)2
C2×#168115652
C2×#168114962
C22×C4⋊C4×D532
#1073053