Finite Group Databasebeta

← #28314535

Subgroups — #28314535

Order 13568.

1214 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20291392C211yesmaximal, characteristic
2#20291392C211yesmaximal, characteristic
3#20290058C214yesmaximal, complemented
4#20290044C214yesmaximal, complemented
5#20290019C212yesmaximal, complemented
6#20290017C212yesmaximal, complemented
7C2×C4×Q8×D53C211yesmaximal, characteristic
8#16811645—21no—
9#16811645C2212yes—
10#16811644—21no—
11#16811644—21no—
12#16811642—21no—
13#16811642C2212yes—
14#16811639C2212yes—
15#16811639C2212yes—
16#16811627—21no—
17#16811627C2212yes—
18#16811594—21no—
19#16811594—21no—
20C4×#11719936—21no—
21C4×#11719936C2212yes—
22#16811473C2218yescomplemented
23Q8⋊#88324C2218yescomplemented
24C22×Q8×D53C411yescharacteristic
25C2×Q8×Dic53C2211yescharacteristic
26C2×C4×Dic106—21no—
27C2×C4×Dic106C411yescharacteristic
28C2×#13133182—21no—
29C2×#13133182C2211yescharacteristic
30C4×Q8×D53—21no—
31C4×Q8×D53—21no—
32C4×Q8×D53—21no—
33C4×Q8×D53—21no—
34C4×Q8×D53C412yes—
35C4×Q8×D53C412yes—
36C4×Q8×D53C2212yes—
37C4×Q8×D53C2212yes—
38C2×C4⋊C4×D53—21no—
39C2×C4⋊C4×D53C411yescharacteristic
40C2×C42×D53—21no—
41C2×C42×D53C2211yescharacteristic
42C2×C4×Q8×C53C411yescharacteristic
43C2×#11719936—22no—
44C2×#11719936—22no—
45C4⋊#88324—22no—
46C4⋊#88324—22no—
47C4⋊#88324—22no—
48C4⋊#88324C2314yes—
49#13133274—22no—
50#13133274—22no—
51C22⋊#88324—22no—
52C22⋊#88324—22no—
53#13133272—22no—
54#13133272—22no—
55C4×#88324—22no—
56C4×#88324C2314yes—
57C22×Dic106—21no—
58C22×Dic106C2×C411yescharacteristic
59C2×Q8×D53—24no—
60C2×Q8×D53—21no—
61C2×Q8×D53—21no—
62C2×Q8×D53—24no—
63C2×Q8×D53C2×C412yes—
64C2×Q8×D53C2×C412yes—
65C22×C4×D53—21no—
66C22×C4×D53—21no—
67C22×C4×D53C2×C411yescharacteristic
68C22×C4×D53D411yescharacteristic
69C22×C4×D53D411yescharacteristic
70Q8×Dic53—21no—
71Q8×Dic53—21no—
72Q8×Dic53C2×C412yes—
73Q8×Dic53C2312yes—
74C4×Dic106—21no—
75C4×Dic106—21no—
76C4×Dic106—22no—
77C4×Dic106—22no—
78C4×Dic106—22no—
79C4×Dic106—22no—
80C4×Dic106C2×C412yes—
81C4×Dic106C2×C412yes—
82#13133182—22no—
83#13133182—22no—
84#13133182—21no—
85#13133182—21no—
86#13133182—22no—
87#13133182—22no—
88#13133182C2×C412yes—
89#13133182C2312yes—
90C2×C4⋊Dic53—21no—
91C2×C4⋊Dic53C2×C411yescharacteristic
92C2×#11719919—21no—
93C2×#11719919—21no—
94C2×#11719919—21no—
95C4⋊C4×D53—22no—
96C4⋊C4×D53—22no—
97C4⋊C4×D53—21no—
98C4⋊C4×D53—21no—
99C4⋊C4×D53—22no—
100C4⋊C4×D53—22no—
101C4⋊C4×D53C2×C412yes—
102C4⋊C4×D53C2×C412yes—
103C2×C4×Dic53—21no—
104C2×C4×Dic53—21no—
105C2×C4×Dic53—21no—
106C2×C4×Dic53D411yescharacteristic
107C2×C4×Dic53D411yescharacteristic
108C2×C4×Dic53C2311yescharacteristic
109C42×D53—22no—
110C42×D53—22no—
111C42×D53—22no—
112C42×D53—22no—
113C42×D53C2×C411yescharacteristic
114C42×D53C2×C411yescharacteristic
115C42×D53D411yescharacteristic
116C42×D53D411yescharacteristic
117C42×D53D411yescharacteristic
118C42×D53D411yescharacteristic
119C42×D53C2311yescharacteristic
120C42×D53C2311yescharacteristic
121C22×Q8×C53C2×C411yescharacteristic
122C4×Q8×C53—21no—
123C4×Q8×C53—21no—
124C4×Q8×C53C2×C412yes—
125C4×Q8×C53C2×C412yes—
126C2×C4⋊C4×C53—21no—
127C2×C4⋊C4×C53C2×C411yescharacteristic
128C2×C42×C53—21no—
129C2×C42×C53C2×C411yescharacteristic
130C23×D53C22⋊C411yescharacteristic
131C2×Q8×C53—21no—
132C2×Q8×C53—22no—
133C2×Q8×C53M1614yescomplemented
134C2×Q8×C53C22×C412yes—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53C22⋊C411yescharacteristic
138C22×C4×C53C22⋊C411yescharacteristic
139C22×C4×C53C22×C411yescharacteristic
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53C22⋊C411yescharacteristic
143C22×Dic53C2×D411yescharacteristic
144C22×Dic53C2×D411yescharacteristic
145Q8×D53—28no—
146Q8×D53—28no—
147Q8×D53—24no—
148Q8×D53—24no—
149Q8×D53—24no—
150Q8×D53—24no—
151C2×C4×D53—28no—
152C2×C4×D53—22no—
153C2×C4×D53—22no—
154C2×C4×D53—22no—
155C2×C4×D53—22no—
156C2×C4×D53—24no—
157C2×C4×D53—22no—
158C2×C4×D53—22no—
159C2×C4×D53—42no—
160C2×C4×D53—44no—
161C2×C4×D53C22⋊C411yescharacteristic
162C2×C4×D53C22⋊C411yescharacteristic
163C2×C4×D53C22⋊C411yescharacteristic
164C2×C4×D53C22⋊C411yescharacteristic
165C2×C4×D53C22⋊C411yescharacteristic
166C2×C4×D53C22⋊C411yescharacteristic
167C2×C4×D53C22×C411yescharacteristic
168C2×C4×D53C22×C411yescharacteristic
169C2×C4×D53C2×D411yescharacteristic
170C2×C4×D53C2×D411yescharacteristic
171C2×C4×D53C2×D411yescharacteristic
172C2×C4×D53C2×D411yescharacteristic
173C2×Dic106—28no—
174C2×Dic106—21no—
175C2×Dic106—22no—
176C2×Dic106—22no—
177C2×Dic106—22no—
178C2×Dic106M1614yescomplemented
179C2×Dic106C22×C412yes—
180C2×#88324—24no—
181C2×#88324—24no—
182#11719936—44no—
183#11719936—44no—
184C4⋊C4×C53—22no—
185C4⋊C4×C53—22no—
186C4⋊C4×C53—21no—
187C4⋊C4×C53C22×C412yes—
188C42×C53—22no—
189C42×C53—22no—
190C42×C53C22⋊C411yescharacteristic
191C42×C53C22⋊C411yescharacteristic
192C42×C53C22×C411yescharacteristic
193C42×C53C22×C411yescharacteristic
194C4⋊Dic53—21no—
195C4⋊Dic53—22no—
196C4⋊Dic53—22no—
197C4⋊Dic53C22×C412yes—
198#11719919—22no—
199#11719919—22no—
200#11719919—22no—
201#11719919—22no—
202#11719919—22no—
203#11719919—22no—
204C4×Dic53—22no—
205C4×Dic53—22no—
206C4×Dic53—22no—
207C4×Dic53—22no—
208C4×Dic53—22no—
209C4×Dic53—22no—
210C4×Dic53C22⋊C411yescharacteristic
211C4×Dic53C22⋊C411yescharacteristic
212C4×Dic53C22⋊C411yescharacteristic
213C4×Dic53C22⋊C411yescharacteristic
214C4×Dic53C22×C411yescharacteristic
215C4×Dic53C2×D411yescharacteristic
216C4×Dic53C2×D411yescharacteristic
217C4×Dic53C2×D411yescharacteristic
218C4×Dic53C2×D411yescharacteristic
219C4×Dic53C2×D411yescharacteristic
220C4×Dic53C2×D411yescharacteristic
221C4×Dic53C2411yescharacteristic
222C23×C53C2×C22⋊C411yescharacteristic
223C22×D53—22no—
224C22×D53—22no—
225C22×D53—22no—
226C22×D53C2×C22⋊C411yescharacteristic
227C22×D53C2×C22⋊C411yescharacteristic
228Q8×C53—22no—
229Q8×C53—24no—
230Q8×C53C2×M1614yes—
231C2×C4×C53—22no—
232C2×C4×C53—24no—
233C2×C4×C53—22no—
234C2×C4×C53—22no—
235C2×C4×C53—41no—
236C2×C4×C53—42no—
237C2×C4×C53C2×C22⋊C411yescharacteristic
238C2×C4×C53C2×C22⋊C411yescharacteristic
239C2×C4×C53C2×C22⋊C411yescharacteristic
240C2×C4×C53C2×C22⋊C411yescharacteristic
241C2×C4×C53C2×C22⋊C411yescharacteristic
242C2×C4×C53C2×M1612yes—
243C2×C4×C53C2×M1612yes—
244C2×C4×C53C23×C411yescharacteristic
245C2×Dic53—22no—
246C2×Dic53—22no—
247C2×Dic53—24no—
248C2×Dic53—22no—
249C2×Dic53—22no—
250C2×Dic53—41no—
251C2×Dic53—42no—
252C2×Dic53C2×C22⋊C411yescharacteristic
253C2×Dic53C2×C22⋊C411yescharacteristic
254C2×Dic53C2×C22⋊C411yescharacteristic
255C2×Dic53C2×C22⋊C411yescharacteristic
256C2×Dic53C22×D411yescharacteristic
257C2×Dic53C22×D411yescharacteristic
258C4×D53—28no—
259C4×D53—28no—
260C4×D53—22no—
261C4×D53—22no—
262C4×D53—22no—
263C4×D53—21no—
264C4×D53—21no—
265C4×D53—21no—
266C4×D53—21no—
267C4×D53—22no—
268C4×D53—22no—
269C4×D53—22no—
270C4×D53—22no—
271C4×D53—41no—
272C4×D53—41no—
273C4×D53—44no—
274C4×D53—44no—
275C4×D53#13714yes—
276Dic106—28no—
277Dic106—28no—
278Dic106—22no—
279Dic106—24no—
280Dic106C2×M1614yes—
281#88324—44no—
282#88324—44no—
283#17449—531nomaximal, Sylow
284C22×C53—21no—
285C22×C53—21no—
286C22×C53#40712yes—
287C22×C53C22×C22⋊C411yescharacteristic
288C2×D53—28no—
289C2×D53—21no—
290C2×D53—21no—
291C2×D53—21no—
292C2×D53—21no—
293C2×D53#41912yes—
294C2×D53#42112yes—
295C4×C53—24no—
296C4×C53—21no—
297C4×C53—21no—
298C4×C53—21no—
299C4×C53—42no—
300C4×C53#40712yes—
301C4×C53#41912yes—
302C4×C53C22×M1611yescharacteristic
303C4×C53C22×M1611yescharacteristic
304Dic53—21no—
305Dic53—21no—
306Dic53—24no—
307Dic53—21no—
308Dic53—21no—
309Dic53—21no—
310Dic53—42no—
311Dic53C2×#13712yes—
312C22×C4×Q8—531no—
313#2920—531no—
314#2920—531no—
315#1491—534no—
316#1487—534no—
317#1479—532no—
318#1479—532no—
319C2×C53—22no—
320C2×C53C2×#40711yescharacteristic
321C2×C53C2×#41911yescharacteristic
322C2×C53C2×#42111yescharacteristic
323D53—22no—
324D53—22no—
325C23×Q8—531no—
326C2×C4×Q8—531no—
327C2×C4×Q8—532no—
328C2×C4×Q8—532no—
329C2×C4×Q8—532no—
330C2×C4×Q8—532no—
331C2×C4×Q8—531no—
332C2×C4×Q8—531no—
333C2×C4×Q8—1061no—
334C2×C4×Q8—1061no—
335C2×C4×Q8—1061no—
336C2×C4×Q8—1061no—
337C2×C4×Q8—1061no—
338C2×C4×Q8—1061no—
339C2×C4×Q8#615711yescharacteristic
340C22×C4⋊C4—531no—
341C22×C4⋊C4—1061no—
342C22×C42—531no—
343C22×C42—1061no—
344#432—532no—
345#432—1061no—
346#432—1061no—
347#432—1061no—
348#431—532no—
349#431—532no—
350#431—532no—
351#431—1061no—
352#423—532no—
353#423—1061no—
354#407—1061no—
355#407—1061no—
356C4×M16—532no—
357C4×M16—1061no—
358Q8⋊C8—538no—
359Q8⋊C8—538no—
360C53#1744911yescharacteristic, complemented, Sylow
361C22×Q8—532no—
362C22×Q8—532no—
363C22×Q8—531no—
364C22×Q8—1061no—
365C22×Q8—1061no—
366C22×Q8—1061no—
367C22×Q8—1064no—
368C22×Q8—1064no—
369C22×Q8C2×#615711yescharacteristic
370C23×C4—531no—
371C23×C4—531no—
372C23×C4—531no—
373C23×C4—1061no—
374C23×C4—1061no—
375C2×M16—1062no—
376C2×M16—1062no—
377C4×Q8—21no—
378C4×Q8—21no—
379C4×Q8—532no—
380C4×Q8—532no—
381C4×Q8—532no—
382C4×Q8—532no—
383C4×Q8—532no—
384C4×Q8—532no—
385C4×Q8—1062no—
386C4×Q8—1062no—
387C4×Q8—1062no—
388C4×Q8—1062no—
389C4×Q8—1061no—
390C4×Q8—1062no—
391C4×Q8—1062no—
392C4×Q8—1061no—
393C4×Q8—1062no—
394C4×Q8—1062no—
395C4×Q8—1061no—
396C4×Q8—1061no—
397C4×Q8—1061no—
398C4×Q8—1061no—
399C4×Q8C2×#615712yes—
400C4×Q8C2×#615712yes—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—532no—
404C2×C4⋊C4—532no—
405C2×C4⋊C4—1061no—
406C2×C4⋊C4—1061no—
407C2×C4⋊C4—1062no—
408C2×C4⋊C4—1062no—
409C2×C4⋊C4—1062no—
410C2×C4⋊C4—1062no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4C2×#615711yescharacteristic
416C2×C42—21no—
417C2×C42—531no—
418C2×C42—531no—
419C2×C42—531no—
420C2×C42—531no—
421C2×C42—531no—
422C2×C42—531no—
423C2×C42—531no—
424C2×C42—531no—
425C2×C42—531no—
426C2×C42—531no—
427C2×C42—531no—
428C2×C42—1062no—
429C2×C42—1062no—
430C2×C42—1062no—
431C2×C42—1062no—
432C2×C42—1061no—
433C2×C42—1061no—
434C2×C42—1061no—
435C2×C42C2×#615711yescharacteristic
436C4⋊C8—534no—
437C4⋊C8—1062no—
438C4⋊C8—1062no—
439C4⋊C8—1062no—
440C22⋊C8—1062no—
441C22⋊C8—1062no—
442C22⋊C8—1062no—
443C22⋊C8—1062no—
444#97—1062no—
445#97—1062no—
446C4×C8—534no—
447C4×C8—1062no—
448C24—531no—
449C2×Q8—21no—
450C2×Q8—22no—
451C2×Q8—534no—
452C2×Q8—532no—
453C2×Q8—1062no—
454C2×Q8—1062no—
455C2×Q8—1068no—
456C2×Q8—1064no—
457C2×Q8—1064no—
458C2×Q8—1068no—
459C2×Q8—1068no—
460C2×Q8—1064no—
461C2×Q8—1064no—
462C2×Q8—1061no—
463C2×Q8—1062no—
464C2×Q8#1171993614yescomplemented
465C2×Q8C22×#615712yes—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—531no—
469C22×C4—531no—
470C22×C4—531no—
471C22×C4—531no—
472C22×C4—531no—
473C22×C4—531no—
474C22×C4—531no—
475C22×C4—531no—
476C22×C4—531no—
477C22×C4—531no—
478C22×C4—531no—
479C22×C4—531no—
480C22×C4—531no—
481C22×C4—531no—
482C22×C4—531no—
483C22×C4—1064no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1068no—
487C22×C4—1062no—
488C22×C4—1062no—
489C22×C4—1061no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1061no—
493C22×C4—2122no—
494C22×C4—2124no—
495C22×C4C22⋊#615711yescharacteristic
496C22×C4C22⋊#615711yescharacteristic
497C22×C4C22×#615711yescharacteristic
498M16—2124no—
499M16—2124no—
500C2×C8—1064no—
501C2×C8—1064no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—532no—
506C4⋊C4—1062no—
507C4⋊C4—1062no—
508C4⋊C4—1062no—
509C4⋊C4—1062no—
510C4⋊C4—1062no—
511C4⋊C4—1062no—
512C4⋊C4—1061no—
513C4⋊C4—1062no—
514C4⋊C4—1062no—
515C4⋊C4C22×#615712yes—
516C42—22no—
517C42—22no—
518C42—531no—
519C42—531no—
520C42—531no—
521C42—531no—
522C42—531no—
523C42—531no—
524C42—531no—
525C42—531no—
526C42—531no—
527C42—531no—
528C42—531no—
529C42—531no—
530C42—1062no—
531C42—1062no—
532C42—1062no—
533C42—1062no—
534C42—1062no—
535C42—1062no—
536C42C22⋊#615711yescharacteristic
537C42C22⋊#615711yescharacteristic
538C42C22×#615711yescharacteristic
539C42C22×#615711yescharacteristic
540C23—531no—
541C23—531no—
542C23—1062no—
543C23—1062no—
544C23—1062no—
545C23C2×C22⋊#615711yescharacteristic
546Q8—24no—
547Q8—22no—
548Q8—534no—
549Q8—1068no—
550Q8—1068no—
551Q8—1064no—
552Q8—1062no—
553Q8C2×#1171993614yes—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—531no—
561C2×C4—534no—
562C2×C4—531no—
563C2×C4—531no—
564C2×C4—531no—
565C2×C4—531no—
566C2×C4—531no—
567C2×C4—1062no—
568C2×C4—1062no—
569C2×C4—1062no—
570C2×C4—1062no—
571C2×C4—1068no—
572C2×C4—1068no—
573C2×C4—1062no—
574C2×C4—1062no—
575C2×C4—1064no—
576C2×C4—1062no—
577C2×C4—1062no—
578C2×C4—1062no—
579C2×C4—1062no—
580C2×C4—1061no—
581C2×C4—1061no—
582C2×C4—1061no—
583C2×C4—1061no—
584C2×C4—1062no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2124no—
588C2×C4—2124no—
589C2×C4—2121no—
590C2×C4—2122no—
591C2×C4C2×C22⋊#615711yescharacteristic
592C2×C4C2×C22⋊#615711yescharacteristic
593C2×C4C2×C22⋊#615711yescharacteristic
594C2×C4C2×C22⋊#615711yescharacteristic
595C2×C4C2×C22⋊#615711yescharacteristic
596C2×C4D53⋊M1612yes—
597C2×C4C2×#1171993612yes—
598C2×C4C23×#615711yescharacteristic
599C8—2124no—
600C8—2124no—
601C22—21no—
602C22—21no—
603C22—532no—
604C22—532no—
605C22—1068no—
606C22—1061no—
607C22—1061no—
608C22—1061no—
609C22—1061no—
610C22#1681159412yes—
611C22C22×C22⋊#615711yescharacteristic
612C4—21no—
613C4—21no—
614C4—21no—
615C4—24no—
616C4—42no—
617C4—532no—
618C4—1061no—
619C4—1061no—
620C4—1061no—
621C4—1061no—
622C4—1061no—
623C4—1064no—
624C4—2122no—
625C4#1681159412yes—
626C4#1681161612yes—
627C4C2×D53⋊M1611yescharacteristic
628C4C22×#1171993611yescharacteristic
629C2—22no—
630C2—1062no—
631C2—1062no—
632C2C2×#1681159411yescharacteristic
633C2C2×#1681161611yescharacteristic
634C2D53⋊#42111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6157
commutatorC2×C4×C53C23×C4
FrattiniC2×C4C23×#6157
FittingC2×C4×Q8×C53C4
socleC22×C53C22×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC4×Dic53C24
hypercenterC2×C4×Q8#6157
hypocenterC53#17449
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
3. #20290058C2semidirect
4. #20290044C2semidirect
5. #20290019C2semidirect
6. #20290017C2semidirect
22. #16811473C22semidirect
23. Q8⋊#88324C22semidirect
133. C2×Q8×C53M16semidirect
178. C2×Dic106M16semidirect
360. C53#17449semidirect
464. C2×Q8#11719936semidirect

Maximal subgroups

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

subgroupindex
#202913922
#202900582
#202900442
#202900192
#202900172
C2×C4×Q8×D532
#1744953