Finite Group Databasebeta

← #27385072

Subgroups — #27385072

Order 13568.

1550 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11719945C211yesmaximal, characteristic
2C2×#16810665C211yesmaximal, characteristic, complemented
3C2×#16810625C211yesmaximal, characteristic
4C2×#16810625C211yesmaximal, characteristic
5C2×#16810521C211yesmaximal, characteristic, complemented
6#20274986C212yesmaximal, complemented
7#20274979C212yesmaximal, complemented
8#20274972C212yesmaximal, complemented
9#20274965C212yesmaximal, complemented
10#20274435C211yesmaximal, characteristic, complemented
11C53×#1728C211yesmaximal, characteristic, complemented
12C22×#11719945C2211yescharacteristic
13C4×#11719945—24no—
14C4×#11719945—24no—
15C2×C4×Dic106C2211yescharacteristic
16C2×C4×Dic106C2211yescharacteristic
17C2×C4×D212C2211yescharacteristic
18C2×C4×D212C2211yescharacteristic
19C2×C4×C22⋊D53—21no—
20C2×C4×C22⋊D53—21no—
21C2×#13133176C2211yescharacteristic
22C2×#13133174—21no—
23C22×C4⋊Dic53C2211yescharacteristic
24C22×C4⋊Dic53C2211yescharacteristic
25C22×#11719919—21no—
26C2×C42×D53—21no—
27#16810665—24no—
28C2×#13133151C2212yes—
29C2×#13133149C2212yes—
30#16810625—24no—
31#16810625C418yescomplemented
32C2×#13133142C2212yes—
33C2×#13133142C2212yes—
34C2×#13133139C2212yes—
35C2×#13133139C2212yes—
36#16810521—21no—
37#16810521—21no—
38#16810521C2212yes—
39#16810521C2212yes—
40C2×#13133124—21no—
41#16810386C2212yes—
42#16810386C2212yes—
43C22×M16×C53C2211yescharacteristic
44C22×M16×C53C2211yescharacteristic
45C22×C42×C53C2211yescharacteristic
46C53×#336C2212yes—
47C53×#336C2212yes—
48C2×#11719945—22no—
49C2×#11719945—22no—
50C2×#11719945—22no—
51C2×#11719945C2×C414yes—
52C2×#11719945D414yes—
53C2×#11719945Q814yes—
54C22×Dic106C2311yescharacteristic
55C22×D212C2311yescharacteristic
56C22×C22⋊D53—21no—
57C23×Dic53—21no—
58C23×Dic53—21no—
59C22×C4×D53—21no—
60C22×C4×D53—21no—
61C22×C4×D53—21no—
62C4×Dic106—22no—
63C4×Dic106—22no—
64C4×Dic106—22no—
65C4×Dic106—22no—
66C4×D212—22no—
67C4×D212—22no—
68C4×D212—22no—
69C4×D212—22no—
70C4×C22⋊D53—28no—
71C4×C22⋊D53—28no—
72#13133176—22no—
73#13133176D414yes—
74#13133174—28no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53—22no—
78C2×C4⋊Dic53—22no—
79C2×C4⋊Dic53—22no—
80C2×C4⋊Dic53C2×C414yescomplemented
81C2×C4⋊Dic53C2312yes—
82C2×C4⋊Dic53C2312yes—
83C2×C4⋊Dic53C2311yescharacteristic
84C2×C4⋊Dic53C2311yescharacteristic
85C2×#11719919—21no—
86C2×#11719919—21no—
87C2×#11719919—22no—
88C2×#11719919—22no—
89C2×#11719919—44no—
90C2×C22⋊Dic53—21no—
91C2×#11719921—21no—
92C2×#11719921—21no—
93C2×C4×Dic53—21no—
94C2×C4×Dic53—21no—
95C2×C4×Dic53—21no—
96C42×D53—28no—
97#13133151—24no—
98#13133149—24no—
99#13133142—24no—
100#13133142C2×C418yescomplemented
101#13133139—24no—
102#13133139C2×C418yescomplemented
103#13133124—22no—
104#13133124—22no—
105C23×C4×C53D411yescharacteristic
106C23×C4×C53D411yescharacteristic
107C23×C4×C53C2311yescharacteristic
108C2×M16×C53—22no—
109C2×M16×C53—22no—
110C2×M16×C53—22no—
111C2×M16×C53—22no—
112C2×M16×C53—22no—
113C2×M16×C53C2×C414yescomplemented
114C22×C8×C53C2312yes—
115C22×C8×C53C2312yes—
116C2×C42×C53—22no—
117C2×C42×C53—22no—
118C2×C42×C53D411yescharacteristic
119C2×C42×C53D411yescharacteristic
120C2×C42×C53C2311yescharacteristic
121C2×C42×C53C2311yescharacteristic
122C24×C53C2×D411yescharacteristic
123C23×D53—21no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—22no—
128C22×C4×C53—41no—
129C22×C4×C53—41no—
130C22×C4×C53C22×C412yes—
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D412yes—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D412yes—
138C22×C4×C53C2×Q812yes—
139C22×C4×C53#4112yes—
140C22×C4×C53#4112yes—
141C22×C4×C53C2411yescharacteristic
142C2×C22⋊D53—24no—
143C2×C22⋊D53—24no—
144C2×C22⋊D53—24no—
145C22×Dic53—21no—
146C22×Dic53—24no—
147C22×Dic53—22no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—22no—
151C22×Dic53—42no—
152C22×Dic53—42no—
153C22×Dic53—42no—
154C22×Dic53—42no—
155C22×Dic53—42no—
156#11719945—28no—
157#11719945—28no—
158#11719945—28no—
159#11719945C4⋊C4116yescomplemented
160C2×D212—21no—
161C2×D212—21no—
162C2×D212—21no—
163C2×D212C22×C412yes—
164C2×D212C2×D412yes—
165C2×D212C2×Q812yes—
166C2×C4×D53—24no—
167C2×C4×D53—24no—
168C2×C4×D53—24no—
169C2×C4×D53—24no—
170C2×C4×D53—24no—
171C2×C4×D53—42no—
172C2×C4×D53—42no—
173C2×C4×D53—42no—
174C2×C4×D53—42no—
175C2×Dic106—21no—
176C2×Dic106—21no—
177C2×Dic106—21no—
178C2×Dic106C22×C412yes—
179C2×Dic106C2×D412yes—
180C2×Dic106C2×Q812yes—
181M16×C53—28no—
182M16×C53—44no—
183C2×C8×C53—22no—
184C2×C8×C53—22no—
185C2×C8×C53—22no—
186C2×C8×C53—22no—
187C2×C8×C53—22no—
188C2×C8×C53C22×C414yes—
189C42×C53—22no—
190C42×C53—22no—
191C42×C53—21no—
192C42×C53—21no—
193C42×C53—21no—
194C42×C53—21no—
195C22⋊Dic53—24no—
196#11719921—24no—
197#11719921—24no—
198C4⋊Dic53—24no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—22no—
202C4⋊Dic53—22no—
203C4⋊Dic53—22no—
204C4⋊Dic53—42no—
205C4⋊Dic53C22×C414yes—
206C4⋊Dic53C2×D412yes—
207C4⋊Dic53C2×D412yes—
208#11719919—24no—
209#11719919—24no—
210#11719919—44no—
211#11719919—44no—
212C4×Dic53—24no—
213C4×Dic53—24no—
214C4×Dic53—24no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53C4×D412yes—
220C23×C53#12012yes—
221C23×C53C22⋊Q812yes—
222C23×C53C22×D411yescharacteristic
223C22×D53—22no—
224C22×D53—22no—
225C22×D53—22no—
226C22×D53—41no—
227C22×D53—41no—
228C22×D53—41no—
229C22×D53—41no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—44no—
241C2×C4×C53—44no—
242C2×C4×C53C2×C4⋊C414yes—
243C2×C4×C53C4×D414yes—
244C2×C4×C53C4×D412yes—
245C2×C4×C53#12012yes—
246C2×C4×C53C22⋊Q814yes—
247C2×C4×C53C22⋊Q812yes—
248C2×C4×C53#12314yes—
249C2×C4×C53AGL(1,ℤ/8)12yes—
250C2×C4×C53#13712yes—
251C2×C4×C53C23×C411yescharacteristic
252C2×C4×C53C22×D411yescharacteristic
253C2×C4×C53C22×D411yescharacteristic
254C2×C4×C53C22×Q811yescharacteristic
255C2×C4×C53C2×#4111yescharacteristic
256C2×C4×C53C2×#4111yescharacteristic
257C22⋊D53—216no—
258C2×Dic53—22no—
259C2×Dic53—22no—
260C2×Dic53—22no—
261C2×Dic53—24no—
262C2×Dic53—22no—
263C2×Dic53—22no—
264C2×Dic53—48no—
265C2×Dic53—41no—
266C2×Dic53—42no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—42no—
271C2×Dic53—42no—
272C2×Dic53—42no—
273C2×Dic53—42no—
274C2×Dic53—84no—
275D212—22no—
276D212—22no—
277D212—22no—
278D212C2×C4⋊C414yescomplemented
279C4×D53—216no—
280C4×D53—48no—
281C4×D53—48no—
282Dic106—22no—
283Dic106—22no—
284Dic106—22no—
285Dic106C2×C4⋊C414yescomplemented
286C8×C53—24no—
287C8×C53—42no—
288#20875—531nomaximal, Sylow
289C22×C53—22no—
290C22×C53—22no—
291C22×C53—22no—
292C22×C53—22no—
293C22×C53—22no—
294C22×C53—22no—
295C22×C53#38514yes—
296C22×C53C2×C4×D411yescharacteristic
297C22×C53C2×#12011yescharacteristic
298C22×C53C2×C22⋊Q811yescharacteristic
299C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C53C2×#13711yescharacteristic
301C22×C53#57511yescharacteristic
302C22×C53#57511yescharacteristic
303C2×D53—24no—
304C2×D53—42no—
305C2×D53—42no—
306C2×D53—42no—
307C2×D53—42no—
308C2×D53—42no—
309C2×D53—42no—
310C4×C53—22no—
311C4×C53—22no—
312C4×C53—41no—
313C4×C53—41no—
314C4×C53#38514yes—
315C4×C53C22×C4⋊C411yescharacteristic
316C4×C53C2×C4×D411yescharacteristic
317C4×C53C2×C22⋊Q811yescharacteristic
318C4×C53C2×#12311yescharacteristic
319Dic53—24no—
320Dic53—44no—
321Dic53—42no—
322Dic53—42no—
323Dic53—82no—
324C2×C4×#41—531no—
325C2×#428—531no—
326C2×#417—531no—
327C2×#417—531no—
328C2×#389—531no—
329#1865—532no—
330#1864—532no—
331#1863—532no—
332#1862—532no—
333#1736—531no—
334#1728D5311yescharacteristic, complemented
335C2×C53—24no—
336C2×C53C2×#38511yescharacteristic
337C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
338C2×C53C4×#13711yescharacteristic
339C2×C53#300611yescharacteristic
340C2×C53#300711yescharacteristic
341C2×C53#307911yescharacteristic
342C2×C53#308011yescharacteristic
343D53—44no—
344C22×#41—531no—
345C22×M16C2×D5311yescharacteristic
346C22×M16C2×D5311yescharacteristic
347C4×#41—1064no—
348C4×#41—1064no—
349C2×C4×Q8—531no—
350C2×C4×Q8—531no—
351C2×C4×D4—531no—
352C2×C4×D4—531no—
353C2×C4×D4—1061no—
354C2×C4×D4—1061no—
355C2×#117—531no—
356C2×#117—1061no—
357C22×C4⋊C4—531no—
358C22×C4⋊C4—531no—
359C22×C4⋊C4—1061no—
360C22×C42—1061no—
361C22×C42C2×D5311yescharacteristic
362#428—1064no—
363C2×#107—532no—
364C2×#106—532no—
365#417—538no—
366#417—1064no—
367C2×#103—532no—
368C2×#103—532no—
369C2×#102—532no—
370C2×#102—532no—
371#389—532no—
372#389—532no—
373#389—1061no—
374#389—1061no—
375C2×#95—1061no—
376#340—532no—
377#340—532no—
378#336C2×D5312yes—
379#336C2×D5312yes—
380C53#2087511yescharacteristic, complemented, Sylow
381C2×#41—534no—
382C2×#41—534no—
383C2×#41—534no—
384C2×#41—1062no—
385C2×#41—1062no—
386C2×#41—1062no—
387C22×Q8—531no—
388C22×D4—531no—
389C22×D4—1061no—
390C23×C4—1061no—
391C23×C4—1061no—
392C23×C4—1061no—
393C23×C4—1061no—
394C23×C4—1061no—
395C23×C4D21211yescharacteristic
396C23×C4D21211yescharacteristic
397C23×C4C22×D5311yescharacteristic
398C2×M16—22no—
399C2×M16—22no—
400C2×M16—22no—
401C2×M16—22no—
402C2×M16—22no—
403C2×M16C4×D5314yescomplemented
404C22×C8C22×D5312yes—
405C22×C8C22×D5312yes—
406C4×Q8—1062no—
407C4×Q8—1062no—
408C4×Q8—1062no—
409C4×Q8—1062no—
410C4×D4—1062no—
411C4×D4—1062no—
412C4×D4—1062no—
413C4×D4—1062no—
414C4×D4—1068no—
415C4×D4—1068no—
416#117—534no—
417#117—1068no—
418#117—1062no—
419C2×C4⋊C4—534no—
420C2×C4⋊C4—532no—
421C2×C4⋊C4—531no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4—532no—
424C2×C4⋊C4—1061no—
425C2×C4⋊C4—1061no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—1062no—
428C2×C4⋊C4—1062no—
429C2×C4⋊C4—1062no—
430C2×C4⋊C4—1061no—
431C2×C4⋊C4—1062no—
432C2×C4⋊C4—1062no—
433C2×C4⋊C4—2124no—
434C2×C22⋊C4—1061no—
435C2×C22⋊C4—1061no—
436C2×C22⋊C4—1061no—
437C2×C42—22no—
438C2×C42—22no—
439C2×C42—1068no—
440C2×C42—1061no—
441C2×C42—1061no—
442C2×C42—1061no—
443C2×C42D21211yescharacteristic
444C2×C42D21211yescharacteristic
445C2×C42C22×D5311yescharacteristic
446C2×C42C22×D5311yescharacteristic
447#107—1064no—
448#106—1064no—
449#103—538no—
450#103—1064no—
451#102—538no—
452#102—1064no—
453#95—1062no—
454#95—1062no—
455C24—1061no—
456C24C2×D21211yescharacteristic
457#41—5316no—
458#41—1068no—
459#41—1068no—
460#41—1068no—
461C2×Q8—532no—
462C2×Q8—532no—
463C2×Q8—532no—
464C2×Q8—1061no—
465C2×Q8—1061no—
466C2×Q8—1061no—
467C2×D4—532no—
468C2×D4—532no—
469C2×D4—532no—
470C2×D4—1064no—
471C2×D4—1064no—
472C2×D4—1064no—
473C2×D4—1061no—
474C2×D4—1061no—
475C2×D4—1061no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—1064no—
483C22×C4—1064no—
484C22×C4—1061no—
485C22×C4—1064no—
486C22×C4—1064no—
487C22×C4—1064no—
488C22×C4—1064no—
489C22×C4—1062no—
490C22×C4—1061no—
491C22×C4—1061no—
492C22×C4—1062no—
493C22×C4—2122no—
494C22×C4—2122no—
495C22×C4—2122no—
496C22×C4—2122no—
497C22×C4—2122no—
498C22×C4—2122no—
499C22×C4—2122no—
500C22×C4—2122no—
501C22×C4—2122no—
502C22×C4C2×C4×D5312yes—
503C22×C4C2×D21211yescharacteristic
504C22×C4C2×D21211yescharacteristic
505C22×C4C2×D21211yescharacteristic
506C22×C4C2×D21211yescharacteristic
507C22×C4C2×D21211yescharacteristic
508C22×C4#1171994512yes—
509C22×C4D4×D5312yes—
510C22×C4D4×D5312yes—
511C22×C4#1171994712yes—
512C22×C4Q8×D5312yes—
513C22×C4C23×D5311yescharacteristic
514M16—28no—
515M16—44no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8C2×C4×D5314yes—
522C4⋊C4—534no—
523C4⋊C4—532no—
524C4⋊C4—532no—
525C4⋊C4—1064no—
526C4⋊C4—1064no—
527C4⋊C4—1064no—
528C4⋊C4—1061no—
529C4⋊C4—1061no—
530C4⋊C4—1062no—
531C4⋊C4—1062no—
532C4⋊C4—1062no—
533C4⋊C4—2124no—
534C4⋊C4—2122no—
535C4⋊C4—2124no—
536C22⋊C4—1064no—
537C22⋊C4—1064no—
538C22⋊C4—1064no—
539C42—22no—
540C42—22no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—1064no—
546C42—1064no—
547C42—1064no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—1062no—
553C23—1062no—
554C23—1062no—
555C23—2121no—
556C23—2121no—
557C23—2121no—
558C23—2121no—
559C23C4×D21212yes—
560C23#1313318712yes—
561C23#1313319712yes—
562C23C22×D21211yescharacteristic
563Q8—534no—
564Q8—1062no—
565Q8—1062no—
566Q8—1062no—
567D4—534no—
568D4—10616no—
569D4—1062no—
570D4—1062no—
571D4—1062no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—1062no—
585C2×C4—1062no—
586C2×C4—1062no—
587C2×C4—10616no—
588C2×C4—1064no—
589C2×C4—1062no—
590C2×C4—1062no—
591C2×C4—4244no—
592C2×C4—2128no—
593C2×C4—2128no—
594C2×C4—2122no—
595C2×C4—2122no—
596C2×C4—2122no—
597C2×C4—2128no—
598C2×C4—2122no—
599C2×C4—2121no—
600C2×C4—2121no—
601C2×C4—2121no—
602C2×C4—2121no—
603C2×C4—2122no—
604C2×C4C4⋊C4×D5314yes—
605C2×C4#1313317714yes—
606C2×C4C4×D21212yes—
607C2×C4#1313318712yes—
608C2×C4#1313319514yes—
609C2×C4#1313319712yes—
610C2×C4#1313319914yes—
611C2×C4#1313324512yes—
612C2×C4#1313325012yes—
613C2×C4C22×C4×D5311yescharacteristic
614C2×C4C2×D4×D5311yescharacteristic
615C2×C4C2×D4×D5311yescharacteristic
616C2×C4C2×Q8×D5311yescharacteristic
617C2×C4C2×#1171994711yescharacteristic
618C2×C4C2×#1171994511yescharacteristic
619C8—24no—
620C8—42no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—1064no—
628C22—2122no—
629C22—2122no—
630C22—2122no—
631C22—2122no—
632C22—2122no—
633C22—2122no—
634C22#1681049714yes—
635C22C2×C4×D21211yescharacteristic
636C22C2×#1313318711yescharacteristic
637C22C2×#1313319711yescharacteristic
638C22C2×#1313324511yescharacteristic
639C22C2×#1313325011yescharacteristic
640C22#1681142311yescharacteristic
641C22#1681142311yescharacteristic
642C4—22no—
643C4—22no—
644C4—41no—
645C4—41no—
646C4—1064no—
647C4—4242no—
648C4—2122no—
649C4—2122no—
650C4—2124no—
651C4#1681049714yes—
652C4C2×C4⋊C4×D5311yescharacteristic
653C4C2×#1313317711yescharacteristic
654C4C2×#1313319511yescharacteristic
655C4C2×#1313319911yescharacteristic
656C2—24no—
657C2—2124no—
658C2C2×#1681049711yescharacteristic
659C2C4×#1313324511yescharacteristic
660C2C4×#1313325011yescharacteristic
661C2#2028396911yescharacteristic
662C2#2028397711yescharacteristic
663C2#2028453111yescharacteristic
664C2#2028453811yescharacteristic
subgroupG/H
centerC23C22×D212
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1728C2
socleC23×C53C22×D4
prime radicalC22×#11719945C22
cosocleC22×C4×C53C24
hypercenter#1728D53
hypocenterC53#20875
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1728

Complements

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

NH ≅ G/Ntype
2. C2×#16810665C2semidirect
5. C2×#16810521C2semidirect
6. #20274986C2semidirect
7. #20274979C2semidirect
8. #20274972C2semidirect
9. #20274965C2semidirect
10. #20274435C2semidirect
11. C53×#1728C2semidirect
31. #16810625C4semidirect
80. C2×C4⋊Dic53C2×C4semidirect
100. #13133142C2×C4semidirect
102. #13133139C2×C4semidirect
113. C2×M16×C53C2×C4semidirect
159. #11719945C4⋊C4semidirect
278. D212C2×C4⋊C4semidirect
285. Dic106C2×C4⋊C4semidirect
334. #1728D53semidirect
380. C53#20875semidirect
403. C2×M16C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117199452
C2×#168106652
C2×#168106252
C2×#168105212
#202749862
#202749792
#202749722
#202749652
#202744352
C53×#17282
#2087553