Finite Group Databasebeta

← #27385186

Subgroups — #27385186

Order 13568.

1714 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811096C211yesmaximal, characteristic
2C22×#13133131C211yesmaximal, characteristic
3#20277329C211yesmaximal, characteristic, complemented
4#20277327C211yesmaximal, characteristic, complemented
5#20277290C211yesmaximal, characteristic
6#20274997C218yesmaximal, complemented
7C2×#16810389C211yesmaximal, characteristic, complemented
8C2×C53×#342C211yesmaximal, characteristic, complemented
9C23×C22⋊D53C2211yescharacteristic
10#16811096—22no—
11#16811096—22no—
12#16811096—22no—
13#16811096C2214yes—
14C2×C4×C22⋊D53—22no—
15C2×#13133177—22no—
16C2×#13133173—21no—
17C22×C22⋊Dic53C2211yescharacteristic
18C22×C22⋊Dic53C2211yescharacteristic
19C22×C22⋊Dic53C2211yescharacteristic
20C2×C22⋊C4×D53—21no—
21C2×#13133131C418yescomplemented
22C2×#13133131C2214yes—
23#16810502—22no—
24#16810500C2214yes—
25#16810498—22no—
26#16810498C2214yescomplemented
27#16810496—22no—
28#16810495C2214yescomplemented
29#16810484—21no—
30#16810483—22no—
31#16810481—22no—
32#16810478—21no—
33#16810389C2214yescomplemented
34C22×C22⋊C4×C53C2211yescharacteristic
35C22×C22⋊C4×C53C2211yescharacteristic
36C22×C22⋊C4×C53C2211yescharacteristic
37C53×#342C2214yescomplemented
38C24×D53D411yescharacteristic
39C22×C22⋊D53—21no—
40C22×C22⋊D53—21no—
41C22×C22⋊D53—21no—
42C22×C22⋊D53—21no—
43C22×C22⋊D53—22no—
44C22×C22⋊D53—22no—
45C22×C22⋊D53C2×C414yes—
46C22×C22⋊D53D412yes—
47C22×C22⋊D53Q814yes—
48C22×C22⋊D53C2312yes—
49C23×Dic53—21no—
50C23×Dic53D411yescharacteristic
51C23×Dic53D412yes—
52C23×Dic53C2312yes—
53C22×C4×D53—22no—
54C4×C22⋊D53—28no—
55C4×C22⋊D53—28no—
56#13133177—28no—
57#13133177—28no—
58#13133173—24no—
59#13133173—24no—
60C2×C4⋊Dic53—24no—
61C2×#11719919—22no—
62C2×#11719919—24no—
63C2×C22⋊Dic53—21no—
64C2×C22⋊Dic53—21no—
65C2×C22⋊Dic53—21no—
66C2×C22⋊Dic53—21no—
67C2×C22⋊Dic53—21no—
68C2×C22⋊Dic53—21no—
69C2×C22⋊Dic53—21no—
70C2×C22⋊Dic53—21no—
71C2×C22⋊Dic53—21no—
72C2×C22⋊Dic53—21no—
73C2×C22⋊Dic53—21no—
74C2×C22⋊Dic53—22no—
75C2×C22⋊Dic53C2×C414yescomplemented
76C2×C22⋊Dic53D412yes—
77C2×C22⋊Dic53C2312yes—
78C2×C22⋊Dic53C2312yes—
79C2×#11719921—22no—
80C2×#11719921—24no—
81C22⋊C4×D53—24no—
82C22⋊C4×D53—24no—
83C2×C4×Dic53—22no—
84#13133131C2×C4116yescomplemented
85#13133124—24no—
86#13133124—24no—
87#13133124—24no—
88C25×C53C2311yescharacteristic
89C23×C4×C53—21no—
90C23×C4×C53—21no—
91C23×C4×C53D412yes—
92C2×C22⋊C4×C53—21no—
93C2×C22⋊C4×C53—21no—
94C2×C22⋊C4×C53—21no—
95C2×C22⋊C4×C53—21no—
96C2×C22⋊C4×C53—21no—
97C2×C22⋊C4×C53—21no—
98C2×C22⋊C4×C53—22no—
99C2×C22⋊C4×C53—22no—
100C2×C22⋊C4×C53—22no—
101C2×C22⋊C4×C53C2×C414yescomplemented
102C2×C22⋊C4×C53D412yes—
103C2×C22⋊C4×C53C2312yes—
104C2×C22⋊C4×C53C2312yes—
105C2×C22⋊C4×C53C2312yes—
106C24×C53—82no—
107C24×C53C22×C412yes—
108C24×C53C2×D411yescharacteristic
109C24×C53C2×D411yescharacteristic
110C24×C53C2×D411yescharacteristic
111C24×C53C2×D411yescharacteristic
112C24×C53C2×D411yescharacteristic
113C24×C53C2×D411yescharacteristic
114C24×C53C2×Q812yes—
115C24×C53#4112yes—
116C24×C53#4112yes—
117C24×C53C2411yescharacteristic
118C23×D53—21no—
119C23×D53—21no—
120C23×D53—22no—
121C23×D53—82no—
122C23×D53C4⋊C414yescomplemented
123C23×D53C2×D412yes—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—24no—
127C22×C4×C53—22no—
128C22×C4×C53—22no—
129C22×C4×C53—42no—
130C22×C4×C53—82no—
131C22×C4×C53—82no—
132C22×C4×C53—82no—
133C22×C4×C53C2×D414yes—
134C2×C22⋊D53—24no—
135C2×C22⋊D53—24no—
136C2×C22⋊D53—24no—
137C2×C22⋊D53—24no—
138C2×C22⋊D53—22no—
139C2×C22⋊D53—22no—
140C2×C22⋊D53—24no—
141C2×C22⋊D53—24no—
142C2×C22⋊D53—24no—
143C2×C22⋊D53—24no—
144C2×C22⋊D53—24no—
145C2×C22⋊D53—22no—
146C2×C22⋊D53—22no—
147C2×C22⋊D53—22no—
148C2×C22⋊D53—22no—
149C2×C22⋊D53C4⋊C418yescomplemented
150C2×C22⋊D53C22×C414yes—
151C2×C22⋊D53C2×Q814yes—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—24no—
158C22×Dic53—22no—
159C22×Dic53—22no—
160C22×Dic53—22no—
161C22×Dic53—22no—
162C22×Dic53—82no—
163C22×Dic53—82no—
164C22×Dic53—82no—
165C22×Dic53C4⋊C414yescomplemented
166C22×Dic53C2×D412yes—
167C22×Dic53C2×D414yes—
168C2×C4×D53—28no—
169C2×C4×D53—84no—
170C22⋊C4×C53—24no—
171C22⋊C4×C53—22no—
172C22⋊C4×C53—22no—
173C22⋊C4×C53—22no—
174C22⋊C4×C53—22no—
175C22⋊C4×C53—22no—
176C22⋊C4×C53—22no—
177C22⋊C4×C53—22no—
178C22⋊C4×C53—42no—
179C22⋊C4×C53C22×C414yescomplemented
180C22⋊Dic53—24no—
181C22⋊Dic53—22no—
182C22⋊Dic53—22no—
183C22⋊Dic53—22no—
184C22⋊Dic53—22no—
185C22⋊Dic53—22no—
186C22⋊Dic53—41no—
187C22⋊Dic53—41no—
188C22⋊Dic53—41no—
189C22⋊Dic53—41no—
190C22⋊Dic53C22×C414yescomplemented
191#11719921—28no—
192#11719921—84no—
193C4⋊Dic53—84no—
194#11719919—28no—
195#11719919—84no—
196C4×Dic53—28no—
197C23×C53—42no—
198C23×C53—42no—
199C23×C53—42no—
200C23×C53—42no—
201C23×C53—42no—
202C23×C53—42no—
203C23×C53—42no—
204C23×C53—82no—
205C23×C53—82no—
206C23×C53—82no—
207C23×C53—82no—
208C23×C53C2×C4⋊C412yes—
209C23×C53C2×C4⋊C412yes—
210C23×C53C4×D412yes—
211C23×C53C4×D412yes—
212C23×C53C4×D412yes—
213C23×C53C4×D412yes—
214C23×C53#12011yescharacteristic
215C23×C53#12011yescharacteristic
216C23×C53#12011yescharacteristic
217C23×C53#12011yescharacteristic
218C23×C53C22⋊Q812yes—
219C23×C53C22⋊Q812yes—
220C23×C53C22⋊Q812yes—
221C23×C53C22⋊Q812yes—
222C23×C53#12312yes—
223C23×C53#12312yes—
224C23×C53C23×C411yescharacteristic
225C23×C53C22×D411yescharacteristic
226C23×C53C22×D411yescharacteristic
227C23×C53C22×D411yescharacteristic
228C23×C53C22×Q811yescharacteristic
229C23×C53C2×#4111yescharacteristic
230C23×C53C2×#4111yescharacteristic
231C22×D53—22no—
232C22×D53—24no—
233C22×D53—22no—
234C22×D53—24no—
235C22×D53—44no—
236C22×D53—44no—
237C22×D53—84no—
238C22×D53—82no—
239C22×D53—84no—
240C22×D53C2×C4⋊C414yes—
241C2×C4×C53—24no—
242C2×C4×C53—24no—
243C2×C4×C53—44no—
244C2×C4×C53—44no—
245C2×C4×C53—44no—
246C2×C4×C53—44no—
247C2×C4×C53—42no—
248C2×C4×C53—82no—
249C2×C4×C53—84no—
250C2×C4×C53—84no—
251C22⋊D53—24no—
252C22⋊D53—24no—
253C22⋊D53—24no—
254C22⋊D53—24no—
255C22⋊D53—28no—
256C22⋊D53—28no—
257C2×Dic53—22no—
258C2×Dic53—22no—
259C2×Dic53—24no—
260C2×Dic53—24no—
261C2×Dic53—24no—
262C2×Dic53—24no—
263C2×Dic53—44no—
264C2×Dic53—44no—
265C2×Dic53—44no—
266C2×Dic53—44no—
267C2×Dic53—44no—
268C2×Dic53—44no—
269C2×Dic53—42no—
270C2×Dic53—82no—
271C2×Dic53—84no—
272C2×Dic53C2×C4⋊C414yes—
273C4×D53—88no—
274#20889—531nomaximal, Sylow
275C22×C53—22no—
276C22×C53—22no—
277C22×C53—22no—
278C22×C53—22no—
279C22×C53—22no—
280C22×C53—22no—
281C22×C53—22no—
282C22×C53—42no—
283C22×C53—42no—
284C22×C53—42no—
285C22×C53—42no—
286C22×C53—42no—
287C22×C53—42no—
288C22×C53—42no—
289C22×C53—42no—
290C22×C53—42no—
291C22×C53—42no—
292C22×C53—42no—
293C22×C53—42no—
294C22×C53—41no—
295C22×C53—41no—
296C22×C53—41no—
297C22×C53—41no—
298C22×C53#38512yes—
299C22×C53#38512yes—
300C22×C53#38512yes—
301C22×C53#38512yes—
302C22×C53C22×C4⋊C411yescharacteristic
303C22×C53C2×C4×D411yescharacteristic
304C22×C53C2×C4×D411yescharacteristic
305C22×C53C2×#12011yescharacteristic
306C22×C53C2×C22⋊Q811yescharacteristic
307C22×C53C2×C22⋊Q811yescharacteristic
308C22×C53C2×#12311yescharacteristic
309C2×D53—24no—
310C2×D53—48no—
311C2×D53—44no—
312C2×D53—44no—
313C2×D53—82no—
314C2×D53—84no—
315C4×C53—44no—
316C4×C53—44no—
317C4×C53—82no—
318Dic53—24no—
319Dic53—44no—
320Dic53—44no—
321Dic53—82no—
322C2×#518—531no—
323C22×#99—531no—
324#2315—531no—
325#2315—531no—
326#2308—531no—
327#1867—538no—
328C2×#342—531no—
329C2×#342D5311yescharacteristic, complemented
330C2×C53—21no—
331C2×C53—21no—
332C2×C53—21no—
333C2×C53—21no—
334C2×C53—22no—
335C2×C53—22no—
336C2×C53—21no—
337C2×C53—21no—
338C2×C53—22no—
339C2×C53—22no—
340C2×C53C2×#38511yescharacteristic
341C2×C53#302612yes—
342D53—44no—
343C23×D4—531no—
344#518—534no—
345#518—1062no—
346#518—1062no—
347#518—1062no—
348C2×C4×D4—1062no—
349C2×C4×D4—1062no—
350C2×#117—1061no—
351C22×C22⋊C4—531no—
352C22×C22⋊C4—531no—
353C22×C22⋊C4—531no—
354C22×C22⋊C4—1061no—
355C22×C22⋊C4C2×D5311yescharacteristic
356C22×C22⋊C4C2×D5311yescharacteristic
357C22×C22⋊C4C2×D5311yescharacteristic
358C2×#99—538no—
359C2×#99—534no—
360#386—534no—
361#386—1062no—
362#385—534no—
363#385—534no—
364#385—1062no—
365#385—1062no—
366#381—1061no—
367#380—1062no—
368#380—1062no—
369#379—1061no—
370#342—534no—
371#342C2×D5314yescomplemented
372C53#2088911yescharacteristic, complemented, Sylow
373C25—531no—
374C25C22×D5311yescharacteristic
375C22×D4—534no—
376C22×D4—532no—
377C22×D4—534no—
378C22×D4—532no—
379C22×D4—1061no—
380C22×D4—1061no—
381C22×D4—1062no—
382C22×D4—1061no—
383C22×D4—1061no—
384C22×D4—1062no—
385C23×C4—21no—
386C23×C4—21no—
387C23×C4—531no—
388C23×C4—532no—
389C23×C4—532no—
390C23×C4—1062no—
391C23×C4—1061no—
392C23×C4D21212yes—
393C4×D4—1068no—
394C4×D4—1068no—
395C4×D4—1068no—
396C4×D4—1068no—
397#117—1064no—
398#117—1064no—
399C2×C4⋊C4—1062no—
400C2×C4⋊C4—1064no—
401C2×C4⋊C4—1064no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—22no—
409C2×C22⋊C4—22no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—534no—
412C2×C22⋊C4—532no—
413C2×C22⋊C4—532no—
414C2×C22⋊C4—532no—
415C2×C22⋊C4—1064no—
416C2×C22⋊C4—1064no—
417C2×C22⋊C4—1062no—
418C2×C22⋊C4—1061no—
419C2×C22⋊C4—1061no—
420C2×C22⋊C4—1061no—
421C2×C22⋊C4—1062no—
422C2×C22⋊C4—1061no—
423C2×C22⋊C4—1061no—
424C2×C22⋊C4—1064no—
425C2×C22⋊C4—1061no—
426C2×C22⋊C4—1061no—
427C2×C22⋊C4—1061no—
428C2×C22⋊C4—1061no—
429C2×C22⋊C4—1061no—
430C2×C22⋊C4—1061no—
431C2×C22⋊C4C4×D5314yescomplemented
432C2×C22⋊C4D21212yes—
433C2×C22⋊C4C22×D5312yes—
434C2×C22⋊C4C22×D5312yes—
435C2×C22⋊C4C22×D5312yes—
436C2×C42—1062no—
437#99—5316no—
438#95—1064no—
439#95—1064no—
440#95—1064no—
441C24—82no—
442C24—534no—
443C24—532no—
444C24—1061no—
445C24—1062no—
446C24—1061no—
447C24—4242no—
448C24C2×C4×D5312yes—
449C24C2×D21211yescharacteristic
450C24C2×D21211yescharacteristic
451C24#1171994512yes—
452C24D4×D5311yescharacteristic
453C24D4×D5311yescharacteristic
454C24D4×D5311yescharacteristic
455C24D4×D5311yescharacteristic
456C24#1171994712yes—
457C24Q8×D5312yes—
458C24C23×D5311yescharacteristic
459C2×D4—538no—
460C2×D4—534no—
461C2×D4—534no—
462C2×D4—1064no—
463C2×D4—1064no—
464C2×D4—1064no—
465C2×D4—1064no—
466C2×D4—1062no—
467C2×D4—1062no—
468C2×D4—1064no—
469C2×D4—1064no—
470C2×D4—1064no—
471C2×D4—1064no—
472C2×D4—1064no—
473C2×D4—1062no—
474C2×D4—1062no—
475C2×D4—1062no—
476C2×D4—1062no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—42no—
483C22×C4—82no—
484C22×C4—82no—
485C22×C4—82no—
486C22×C4—534no—
487C22×C4—532no—
488C22×C4—534no—
489C22×C4—1068no—
490C22×C4—1061no—
491C22×C4—1061no—
492C22×C4—1062no—
493C22×C4—1064no—
494C22×C4—1062no—
495C22×C4—1062no—
496C22×C4—1062no—
497C22×C4—1062no—
498C22×C4—1062no—
499C22×C4—1062no—
500C22×C4—4244no—
501C22×C4—4242no—
502C22×C4—4242no—
503C22×C4—4242no—
504C22×C4C2×D21214yes—
505C4⋊C4—1068no—
506C4⋊C4—4244no—
507C4⋊C4—4244no—
508C22⋊C4—24no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—42no—
517C22⋊C4—534no—
518C22⋊C4—1068no—
519C22⋊C4—1064no—
520C22⋊C4—1062no—
521C22⋊C4—1062no—
522C22⋊C4—1062no—
523C22⋊C4—1062no—
524C22⋊C4—1062no—
525C22⋊C4—4244no—
526C22⋊C4—2121no—
527C22⋊C4—2121no—
528C22⋊C4—2121no—
529C22⋊C4—2121no—
530C22⋊C4C2×C4×D5314yescomplemented
531C42—1068no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—82no—
540C23—82no—
541C23—82no—
542C23—82no—
543C23—534no—
544C23—1064no—
545C23—1062no—
546C23—1064no—
547C23—1062no—
548C23—4242no—
549C23—4244no—
550C23—4244no—
551C23—2124no—
552C23—2124no—
553C23C4⋊C4×D5312yes—
554C23C4⋊C4×D5312yes—
555C23#1313317712yes—
556C23#1313317712yes—
557C23C4×D21212yes—
558C23C4×D21212yes—
559C23#1313318711yescharacteristic
560C23#1313318711yescharacteristic
561C23#1313318711yescharacteristic
562C23#1313318711yescharacteristic
563C23#1313319512yes—
564C23#1313319512yes—
565C23#1313319712yes—
566C23#1313319712yes—
567C23#1313319912yes—
568C23#1313319912yes—
569C23C22×C4×D5311yescharacteristic
570C23C2×D4×D5311yescharacteristic
571C23C2×D4×D5311yescharacteristic
572C23C22×D21211yescharacteristic
573C23C2×Q8×D5311yescharacteristic
574C23C2×#1171994711yescharacteristic
575C23C2×#1171994511yescharacteristic
576D4—1068no—
577D4—1068no—
578D4—1064no—
579D4—1064no—
580D4—1064no—
581D4—1064no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—82no—
590C2×C4—84no—
591C2×C4—84no—
592C2×C4—534no—
593C2×C4—1064no—
594C2×C4—1064no—
595C2×C4—1062no—
596C2×C4—1062no—
597C2×C4—1064no—
598C2×C4—1064no—
599C2×C4—4248no—
600C2×C4—4242no—
601C2×C4—4244no—
602C2×C4—2124no—
603C2×C4—2124no—
604C2×C4—2124no—
605C2×C4—2122no—
606C2×C4—2124no—
607C2×C4—2124no—
608C2×C4—2124no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22—1064no—
633C22—4244no—
634C22—4242no—
635C22—2124no—
636C22—2128no—
637C22—2124no—
638C22#1681049712yes—
639C22#1681049712yes—
640C22#1681049712yes—
641C22#1681049712yes—
642C22C2×C4⋊C4×D5311yescharacteristic
643C22C2×#1313317711yescharacteristic
644C22C2×C4×D21211yescharacteristic
645C22C2×#1313318711yescharacteristic
646C22C2×#1313319511yescharacteristic
647C22C2×#1313319711yescharacteristic
648C22C2×#1313319911yescharacteristic
649C4—44no—
650C4—44no—
651C4—82no—
652C4—1064no—
653C4—4242no—
654C4—2124no—
655C4—2124no—
656C2—21no—
657C2—21no—
658C2—21no—
659C2—22no—
660C2—22no—
661C2—21no—
662C2—22no—
663C2—22no—
664C2—21no—
665C2—21no—
666C2—2124no—
667C2C2×#1681049711yescharacteristic
668C2#2028411412yes—
subgroupG/H
centerC22C2×#13133187
commutatorC23×C53C23×C4
FrattiniC24C23×D53
FittingC2×C53×#342C2
socleC22×C53C2×#120
prime radicalC23×C22⋊D53C22
cosocleC24×C53C24
hypercenterC2×#342D53
hypocenterC53#20889
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C25 ◁ C2×#342

Complements

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

NH ≅ G/Ntype
3. #20277329C2semidirect
4. #20277327C2semidirect
6. #20274997C2semidirect
7. C2×#16810389C2semidirect
8. C2×C53×#342C2semidirect
21. C2×#13133131C4semidirect
26. #16810498C22semidirect
28. #16810495C22semidirect
33. #16810389C22semidirect
37. C53×#342C22semidirect
75. C2×C22⋊Dic53C2×C4semidirect
84. #13133131C2×C4semidirect
101. C2×C22⋊C4×C53C2×C4semidirect
122. C23×D53C4⋊C4semidirect
149. C2×C22⋊D53C4⋊C4semidirect
165. C22×Dic53C4⋊C4semidirect
179. C22⋊C4×C53C22×C4semidirect
190. C22⋊Dic53C22×C4semidirect
329. C2×#342D53semidirect
371. #342C2×D53semidirect
372. C53#20889semidirect
431. C2×C22⋊C4C4×D53semidirect
530. C22⋊C4C2×C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168110962
C22×#131331312
#202773292
#202773272
#202772902
#202749972
C2×#168103892
C2×C53×#3422
#2088953