Finite Group Databasebeta

← #27313100

Subgroups — #27313100

Order 13568.

1918 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133206C211yesmaximal, characteristic
2C2×#16810530C211yesmaximal, characteristic, complemented
3#20272961C212yesmaximal, complemented
4#20272957C212yesmaximal, complemented
5#20272955C212yesmaximal
6#20272953C212yesmaximal, complemented
7#20272951C212yesmaximal, complemented
8#20272939C212yesmaximal, complemented
9C2×C53×#391C211yesmaximal, characteristic, complemented
10C23×D212C2211yescharacteristic
11C2×#13133206—24no—
12C2×#13133206—24no—
13C2×#13133206C418yescomplemented
14C22×C4⋊Dic53C2211yescharacteristic
15C22×#11719921C2212yes—
16C22×#11719921C2212yes—
17C22×#11719921C2212yes—
18C22×#11719921C2212yes—
19C22×#11719921C2212yes—
20C22×#11719921C2212yes—
21C22×C4×Dic53C2211yescharacteristic
22C22×C4×Dic53C2211yescharacteristic
23#16810530—24no—
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C23×Q8×C53C2211yescharacteristic
31C22×C4⋊C4×C53C2211yescharacteristic
32C22×C42×C53C2211yescharacteristic
33C53×#391—24no—
34C2×#95×C53C2212yes—
35C2×#95×C53C2212yes—
36C24×D53C2312yes—
37C22×D212—22no—
38C22×D212—22no—
39C22×D212—41no—
40C22×D212—41no—
41C22×D212—41no—
42C22×D212—41no—
43C22×D212C2×C414yes—
44C23×Dic53C2312yes—
45C23×Dic53C2312yes—
46C23×Dic53C2312yes—
47#13133206—232no—
48C2×C4⋊Dic53—22no—
49C2×C4⋊Dic53—41no—
50C2×C4⋊Dic53—41no—
51C2×#11719921—24no—
52C2×#11719921—24no—
53C2×#11719921—24no—
54C2×#11719921—24no—
55C2×#11719921—24no—
56C2×#11719921—24no—
57C2×#11719921—24no—
58C2×#11719921—24no—
59C2×#11719921—42no—
60C2×#11719921—42no—
61C2×#11719921—42no—
62C2×#11719921—42no—
63C2×#11719921—42no—
64C2×#11719921—42no—
65C2×#11719921—42no—
66C2×#11719921—42no—
67C2×#11719921C2×C418yes—
68C2×#11719921C2×C418yes—
69C2×C4×Dic53—22no—
70C2×C4×Dic53—22no—
71C2×C4×Dic53—22no—
72C2×C4×Dic53—41no—
73C2×C4×Dic53—41no—
74C2×C4×Dic53C2×C414yes—
75#13133124—24no—
76#13133124—24no—
77#13133124—24no—
78#13133124—24no—
79#13133124—42no—
80#13133124—42no—
81C22×Q8×C53—41no—
82C22×Q8×C53—41no—
83C22×Q8×C53—41no—
84C22×Q8×C53—41no—
85C22×Q8×C53C2×C414yes—
86C22×Q8×C53D414yes—
87C22×Q8×C53D414yes—
88C23×C4×C53C2312yes—
89C23×C4×C53C2312yes—
90C23×C4×C53C2311yescharacteristic
91C23×C4×C53C2311yescharacteristic
92C23×C4×C53C2311yescharacteristic
93C2×C4⋊C4×C53—22no—
94C2×C4⋊C4×C53—41no—
95C2×C4⋊C4×C53—41no—
96C2×C42×C53—22no—
97C2×C42×C53—41no—
98C2×C42×C53—41no—
99#95×C53—24no—
100#95×C53—24no—
101C24×C53C2411yescharacteristic
102C23×D53—22no—
103C23×D53—22no—
104C23×D53—22no—
105C23×D53—22no—
106C23×D53—22no—
107C23×D53—22no—
108C23×D53—22no—
109C23×D53—22no—
110C23×D53—22no—
111C23×D53—22no—
112C23×D53—22no—
113C23×D53—22no—
114C23×D53—22no—
115C23×D53—22no—
116C23×D53C22×C414yes—
117C2×Q8×C53—44no—
118C2×Q8×C53—44no—
119C2×Q8×C53—44no—
120C2×Q8×C53—44no—
121C2×Q8×C53—44no—
122C2×Q8×C53—44no—
123C2×Q8×C53C22⋊C4116yescomplemented
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—22no—
137C22×C4×C53—22no—
138C22×C4×C53—22no—
139C22×C4×C53—22no—
140C22×C4×C53—22no—
141C22×C4×C53—22no—
142C22×C4×C53—22no—
143C22×C4×C53—22no—
144C22×C4×C53—22no—
145C22×C4×C53—22no—
146C22×C4×C53—22no—
147C22×C4×C53—22no—
148C22×C4×C53—22no—
149C22×C4×C53C22×C412yes—
150C22×C4×C53C22×C412yes—
151C22×C4×C53C22×C412yes—
152C22×C4×C53C2×D412yes—
153C22×C4×C53C2×D412yes—
154C22×C4×C53C2×D412yes—
155C22×C4×C53C2×D412yes—
156C22×C4×C53C2×D412yes—
157C22×C4×C53C2×D412yes—
158C22×C4×C53#4114yes—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53—22no—
162C22×Dic53—22no—
163C22×Dic53—22no—
164C22×Dic53—22no—
165C22×Dic53—22no—
166C22×Dic53—22no—
167C22×Dic53—22no—
168C22×Dic53—22no—
169C22×Dic53—22no—
170C22×Dic53—22no—
171C22×Dic53—22no—
172C22×Dic53—22no—
173C22×Dic53—22no—
174C22×Dic53—22no—
175C22×Dic53—22no—
176C22×Dic53—22no—
177C22×Dic53—22no—
178C22×Dic53C22×C414yes—
179C22×Dic53#4114yes—
180C2×D212—28no—
181C2×D212—44no—
182C2×D212—44no—
183C2×D212—44no—
184C2×D212—44no—
185C2×D212—44no—
186C2×D212—44no—
187C4⋊C4×C53—44no—
188C42×C53—44no—
189#11719921—216no—
190#11719921—216no—
191#11719921—48no—
192#11719921—48no—
193#11719921—48no—
194#11719921—48no—
195C4⋊Dic53—44no—
196C4×Dic53—28no—
197C4×Dic53—44no—
198C23×C53C23×C411yescharacteristic
199C23×C53C22×D411yescharacteristic
200C23×C53C22×D411yescharacteristic
201C23×C53C2×#4112yes—
202C23×C53C2×#4112yes—
203C23×C53H(5,𝔽2)11yescharacteristic
204C23×C53H(5,𝔽2)11yescharacteristic
205C23×C53H(5,𝔽2)11yescharacteristic
206C23×C53H(5,𝔽2)11yescharacteristic
207C23×C53H(5,𝔽2)12yes—
208C23×C532−1+412yes—
209C22×D53—24no—
210C22×D53—24no—
211C22×D53—24no—
212C22×D53—24no—
213C22×D53—24no—
214C22×D53—24no—
215C22×D53—24no—
216C22×D53—42no—
217C22×D53—42no—
218C22×D53—42no—
219C22×D53—42no—
220C22×D53—42no—
221C22×D53—42no—
222C22×D53—42no—
223C22×D53—42no—
224C22×D53—42no—
225C22×D53—42no—
226C22×D53—42no—
227C22×D53—42no—
228C22×D53—42no—
229C22×D53—42no—
230C22×D53—42no—
231C22×D53—42no—
232C22×D53—42no—
233C22×D53—42no—
234C22×D53—42no—
235C22×D53—42no—
236C22×D53—42no—
237C22×D53—42no—
238C22×D53—42no—
239C22×D53—42no—
240C22×D53—42no—
241C22×D53—42no—
242C22×D53—42no—
243C22×D53—42no—
244Q8×C53—416no—
245C2×C4×C53—22no—
246C2×C4×C53—22no—
247C2×C4×C53—22no—
248C2×C4×C53—22no—
249C2×C4×C53—22no—
250C2×C4×C53—22no—
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—22no—
258C2×C4×C53—22no—
259C2×C4×C53—22no—
260C2×C4×C53—22no—
261C2×C4×C53—22no—
262C2×C4×C53—22no—
263C2×C4×C53—24no—
264C2×C4×C53—24no—
265C2×C4×C53—24no—
266C2×C4×C53—42no—
267C2×C4×C53—42no—
268C2×C4×C53—42no—
269C2×C4×C53—42no—
270C2×C4×C53—42no—
271C2×C4×C53—42no—
272C2×C4×C53—42no—
273C2×C4×C53—42no—
274C2×C4×C53—42no—
275C2×C4×C53—42no—
276C2×C4×C53—42no—
277C2×C4×C53C2×C22⋊C414yes—
278C2×C4×C53C2×C22⋊C414yes—
279C2×C4×C53C2×C22⋊C414yes—
280C2×Dic53—24no—
281C2×Dic53—24no—
282C2×Dic53—24no—
283C2×Dic53—24no—
284C2×Dic53—24no—
285C2×Dic53—24no—
286C2×Dic53—42no—
287C2×Dic53—42no—
288C2×Dic53—42no—
289C2×Dic53—42no—
290C2×Dic53—42no—
291C2×Dic53—42no—
292C2×Dic53—42no—
293C2×Dic53—42no—
294C2×Dic53—42no—
295C2×Dic53—42no—
296C2×Dic53—42no—
297C2×Dic53—42no—
298C2×Dic53—42no—
299C2×Dic53—42no—
300C2×Dic53—42no—
301D212—416no—
302#14492—531nomaximal, Sylow
303C22×C53C22×C22⋊C411yescharacteristic
304C22×C53C4×#4112yes—
305C22×C53#51811yescharacteristic
306C22×C53#51811yescharacteristic
307C22×C53#52012yes—
308C22×C53#53411yescharacteristic
309C22×C53#53411yescharacteristic
310C22×C53#53411yescharacteristic
311C22×C53#53411yescharacteristic
312C22×C53#53812yes—
313C22×C53#53812yes—
314C22×C53#54212yes—
315C22×C53#54212yes—
316C22×C53#54812yes—
317C22×C53#54812yes—
318C22×C53#55012yes—
319C22×C53#55012yes—
320C22×C53#55112yes—
321C22×C53#55511yescharacteristic
322C22×C53#55511yescharacteristic
323C22×C53H(3,𝔽4)11yescharacteristic
324C22×C53#56212yes—
325C22×C53#56311yescharacteristic
326C2×D53—44no—
327C2×D53—44no—
328C2×D53—44no—
329C2×D53—44no—
330C2×D53—44no—
331C2×D53—44no—
332C2×D53—44no—
333C2×D53—82no—
334C2×D53—82no—
335C2×D53—82no—
336C2×D53—82no—
337C2×D53—82no—
338C2×D53—82no—
339C2×D53—82no—
340C2×D53—82no—
341C4×C53—24no—
342C4×C53—24no—
343C4×C53—24no—
344C4×C53—44no—
345C4×C53—82no—
346Dic53—44no—
347Dic53—44no—
348Dic53—82no—
349C22×#124—531no—
350C2×#391—531no—
351C2×#391D5311yescharacteristic, complemented
352#1439—532no—
353#1439—532no—
354#1438—532no—
355#1437—532no—
356#1437—532no—
357#1434—532no—
358C2×C53#230811yescharacteristic
359C2×C53#233712yes—
360C2×C53#234112yes—
361C2×C53#237211yescharacteristic
362C2×C53#237911yescharacteristic
363C2×C53#273012yes—
364C2×C53#273012yes—
365C2×C53#273111yescharacteristic
366C2×C53#273111yescharacteristic
367C2×C53#273311yescharacteristic
368C2×C53#273311yescharacteristic
369D53—84no—
370C23×Q8C2×D5311yescharacteristic
371C23×D4—531no—
372C2×#124—538no—
373C2×#124—1064no—
374C2×#124—1064no—
375C22×C4⋊C4—531no—
376C22×C4⋊C4C2×D5311yescharacteristic
377C22×C22⋊C4—532no—
378C22×C22⋊C4—532no—
379C22×C22⋊C4—532no—
380C22×C22⋊C4—532no—
381C22×C22⋊C4—532no—
382C22×C22⋊C4—532no—
383C22×C42—531no—
384C22×C42—531no—
385C22×C42C2×D5311yescharacteristic
386#391—24no—
387#391—1064no—
388C2×#95—532no—
389C2×#95—532no—
390C2×#95—532no—
391C2×#95—532no—
392C2×#95—532no—
393C2×#95—532no—
394C2×#95C2×D5312yes—
395C2×#95C2×D5312yes—
396C53#1449211yescharacteristic, complemented, Sylow
397C25—532no—
398C22×Q8—41no—
399C22×Q8—41no—
400C22×Q8—41no—
401C22×Q8—41no—
402C22×Q8C4×D5314yes—
403C22×Q8D21214yes—
404C22×Q8C22⋊D5314yes—
405C22×D4—534no—
406C22×D4—1062no—
407C22×D4—1062no—
408C22×D4—2121no—
409C22×D4—2121no—
410C22×D4—2121no—
411C22×D4—2121no—
412C23×C4—532no—
413C23×C4—532no—
414C23×C4—532no—
415C23×C4C22×D5312yes—
416C23×C4C22×D5312yes—
417C23×C4C22×D5311yescharacteristic
418C23×C4C22×D5311yescharacteristic
419C23×C4C22×D5311yescharacteristic
420#124—10632no—
421C2×C4⋊C4—22no—
422C2×C4⋊C4—41no—
423C2×C4⋊C4—41no—
424C2×C4⋊C4—1062no—
425C2×C4⋊C4—2121no—
426C2×C4⋊C4—2121no—
427C2×C22⋊C4—538no—
428C2×C22⋊C4—538no—
429C2×C22⋊C4—1064no—
430C2×C22⋊C4—1064no—
431C2×C22⋊C4—1064no—
432C2×C22⋊C4—1064no—
433C2×C22⋊C4—1064no—
434C2×C22⋊C4—1064no—
435C2×C22⋊C4—1064no—
436C2×C22⋊C4—1064no—
437C2×C22⋊C4—2122no—
438C2×C22⋊C4—2122no—
439C2×C22⋊C4—2122no—
440C2×C22⋊C4—2122no—
441C2×C22⋊C4—2122no—
442C2×C22⋊C4—2122no—
443C2×C22⋊C4—2122no—
444C2×C22⋊C4—2122no—
445C2×C42—22no—
446C2×C42—41no—
447C2×C42—41no—
448C2×C42—534no—
449C2×C42—1062no—
450C2×C42—1062no—
451C2×C42—1062no—
452C2×C42—2121no—
453C2×C42—2121no—
454#95—24no—
455#95—24no—
456#95—1064no—
457#95—1064no—
458#95—1064no—
459#95—1064no—
460#95—2122no—
461#95—2122no—
462C24—534no—
463C24—1062no—
464C24—1062no—
465C24—1062no—
466C24—1062no—
467C24—1062no—
468C24—1062no—
469C24—1062no—
470C24—1062no—
471C24—1062no—
472C24—1062no—
473C24—1062no—
474C24—1062no—
475C24—1062no—
476C24—1062no—
477C24C23×D5311yescharacteristic
478C2×Q8—44no—
479C2×Q8—44no—
480C2×Q8—44no—
481C2×Q8—44no—
482C2×Q8—44no—
483C2×Q8—44no—
484C2×Q8#11719921116yescomplemented
485C2×D4—1068no—
486C2×D4—2124no—
487C2×D4—2124no—
488C2×D4—2124no—
489C2×D4—2124no—
490C2×D4—2124no—
491C2×D4—2124no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—22no—
513C22×C4—22no—
514C22×C4—22no—
515C22×C4—22no—
516C22×C4—22no—
517C22×C4—534no—
518C22×C4—534no—
519C22×C4—1062no—
520C22×C4—1062no—
521C22×C4—1062no—
522C22×C4—1062no—
523C22×C4—1062no—
524C22×C4—1062no—
525C22×C4—1062no—
526C22×C4—1062no—
527C22×C4—1062no—
528C22×C4—1062no—
529C22×C4—1062no—
530C22×C4—1062no—
531C22×C4—1062no—
532C22×C4—1062no—
533C22×C4—1062no—
534C22×C4—1062no—
535C22×C4—1062no—
536C22×C4—1062no—
537C22×C4—1062no—
538C22×C4C2×C4×D5312yes—
539C22×C4C2×C4×D5312yes—
540C22×C4C2×C4×D5312yes—
541C22×C4C2×D21212yes—
542C22×C4C2×D21212yes—
543C22×C4C2×D21212yes—
544C22×C4#1171994914yes—
545C22×C4C2×C22⋊D5312yes—
546C22×C4C2×C22⋊D5312yes—
547C22×C4C2×C22⋊D5312yes—
548C4⋊C4—44no—
549C4⋊C4—2124no—
550C22⋊C4—10616no—
551C22⋊C4—10616no—
552C22⋊C4—2128no—
553C22⋊C4—2128no—
554C22⋊C4—2128no—
555C22⋊C4—2128no—
556C42—44no—
557C42—1068no—
558C42—2124no—
559C23—1064no—
560C23—1064no—
561C23—1064no—
562C23—1064no—
563C23—1064no—
564C23—1064no—
565C23—1064no—
566C23—2122no—
567C23—2122no—
568C23—2122no—
569C23—2122no—
570C23—2122no—
571C23—2122no—
572C23—2122no—
573C23—2122no—
574C23—2122no—
575C23—2122no—
576C23—2122no—
577C23—2122no—
578C23—2122no—
579C23—2122no—
580C23—2122no—
581C23—2122no—
582C23—2122no—
583C23—2122no—
584C23—2122no—
585C23—2122no—
586C23—2122no—
587C23—2122no—
588C23—2122no—
589C23—2122no—
590C23—2122no—
591C23—2122no—
592C23—2122no—
593C23—2122no—
594C23C22×C4×D5311yescharacteristic
595C23C22×C22⋊D5311yescharacteristic
596C23C22×D21211yescharacteristic
597C23#41×D5312yes—
598C23C2×#1171994912yes—
599C23#1313326211yescharacteristic
600C23#1313326211yescharacteristic
601C23#1313326211yescharacteristic
602C23#1313326211yescharacteristic
603C23#1313326312yes—
604C23#1313326512yes—
605Q8—416no—
606D4—21216no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—1064no—
640C2×C4—1064no—
641C2×C4—1064no—
642C2×C4—1064no—
643C2×C4—1064no—
644C2×C4—1064no—
645C2×C4—2122no—
646C2×C4—2122no—
647C2×C4—2122no—
648C2×C4—2122no—
649C2×C4—2122no—
650C2×C4—2122no—
651C2×C4—2122no—
652C2×C4—2122no—
653C2×C4—2122no—
654C2×C4—2122no—
655C2×C4—2122no—
656C2×C4—2122no—
657C2×C4—2122no—
658C2×C4—2122no—
659C2×C4—2122no—
660C2×C4C2×#1171992114yes—
661C2×C4C2×#1171992114yes—
662C2×C4C2×#1171992114yes—
663C22—4242no—
664C22—4242no—
665C22—4242no—
666C22—4242no—
667C22—4242no—
668C22—4242no—
669C22—4242no—
670C22—4242no—
671C22—2124no—
672C22—2124no—
673C22—2124no—
674C22—2124no—
675C22—2124no—
676C22—2124no—
677C22—2124no—
678C22C22×#1171992111yescharacteristic
679C22C4×#1171994912yes—
680C22#1681109611yescharacteristic
681C22#1681109611yescharacteristic
682C22#1681110512yes—
683C22#1681118211yescharacteristic
684C22#1681118211yescharacteristic
685C22#1681118511yescharacteristic
686C22#1681118511yescharacteristic
687C22#1681120712yes—
688C22#1681121312yes—
689C22#1681124312yes—
690C22#1681124712yes—
691C22#1681128412yes—
692C22#1681128512yes—
693C22#1681129212yes—
694C22#1681129512yes—
695C22#1681130012yes—
696C22#1681133511yescharacteristic
697C22#1681133911yescharacteristic
698C22#1681136511yescharacteristic
699C22#1681136912yes—
700C22#1681137211yescharacteristic
701C4—24no—
702C4—24no—
703C4—24no—
704C4—44no—
705C4—82no—
706C4—4242no—
707C4—2124no—
708C4—2124no—
709C2—4244no—
710C2#2027729011yescharacteristic
711C2#2027752712yes—
712C2#2027755312yes—
713C2#2027782611yescharacteristic
714C2#2027788711yescharacteristic
715C2#2028170612yes—
716C2#2028171012yes—
717C2#2028172111yescharacteristic
718C2#2028172411yescharacteristic
719C2#2028173911yescharacteristic
720C2#2028174111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC2×C53×#391C2
socle, cosocleC24×C53C24
prime radicalC23×D212C22
hypercenterC2×#391D53
hypocenterC53#14492
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
2. C2×#16810530C2semidirect
3. #20272961C2semidirect
4. #20272957C2semidirect
6. #20272953C2semidirect
7. #20272951C2semidirect
8. #20272939C2semidirect
9. C2×C53×#391C2semidirect
13. C2×#13133206C4semidirect
123. C2×Q8×C53C22⋊C4semidirect
351. C2×#391D53semidirect
396. C53#14492semidirect
484. C2×Q8#11719921semidirect

Maximal subgroups

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

subgroupindex
C22×#131332062
C2×#168105302
#202729612
#202729572
#202729552
#202729532
#202729512
#202729392
C2×C53×#3912
#1449253