Finite Group Databasebeta

← #27447115

Subgroups — #27447115

Order 13568.

864 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810905C211yesmaximal, characteristic, complemented
2C2×C4×#11719923C211yesmaximal, characteristic
3C4⋊C4×Dic106C211yesmaximal, characteristic, complemented
4#20276248C211yesmaximal, characteristic
5#20276191C211yesmaximal, characteristic, complemented
6#20276089C211yesmaximal, characteristic
7#20276045C211yesmaximal, characteristic, complemented
8#20275440C211yesmaximal, characteristic, complemented
9#20275274C211yesmaximal, characteristic, complemented
10#20275073C211yesmaximal, characteristic
11#20275073C211yesmaximal, characteristic
12#20274982C211yesmaximal, characteristic, complemented
13#20274982C211yesmaximal, characteristic, complemented
14#20274976C211yesmaximal, characteristic
15C53×#2452C211yesmaximal, characteristic
16C22×#11719923C2211yescharacteristic
17C2×C4×Dic106C2211yescharacteristic
18C2×C4×Dic106C2211yescharacteristic
19C2×C4×Dic106C2211yescharacteristic
20#16810905—24no—
21C4×#11719923—24no—
22C2×#13133148C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×C4⋊#88322C2211yescharacteristic
28C2×#13133140C2211yescharacteristic
29C2×#13133140C2211yescharacteristic
30C2×#13133140C2211yescharacteristic
31C2×#13133138C2211yescharacteristic
32C2×#13133138C2211yescharacteristic
33C2×#13133138C2211yescharacteristic
34C2×C4×#88322C2211yescharacteristic
35#16810521C2211yescharacteristic
36#16810521C2211yescharacteristic
37#16810520—21no—
38#16810520—21no—
39#16810493C2211yescharacteristic
40#16810492—21no—
41C4⋊C4×Dic53—21no—
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48#16810378C2211yescharacteristic
49#16810378C2211yescharacteristic
50C2×C4⋊Q8×C53C2211yescharacteristic
51C2×C22⋊Q8×C53—21no—
52C2×C4×D4×C53C2211yescharacteristic
53C53×#396—21no—
54C53×#393—21no—
55C53×#390C2211yescharacteristic
56C53×#390C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#389C2211yescharacteristic
59C53×#385—21no—
60C4×C4⋊C4×C53C2211yescharacteristic
61C22×Dic106C2311yescharacteristic
62C2×#11719923—22no—
63C2×#11719923—22no—
64C2×#11719923Q814yescomplemented
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68C22×#88322C2311yescharacteristic
69C4×Dic106—22no—
70C4×Dic106—22no—
71C4×Dic106—22no—
72C4×Dic106—22no—
73C4×Dic106—41no—
74C4×Dic106—41no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88#13133148—22no—
89#13133148—22no—
90#13133148—22no—
91#13133148—22no—
92#13133148—41no—
93C4⋊#88322—22no—
94#13133140—22no—
95#13133140—22no—
96#13133140D414yes—
97#13133138—22no—
98#13133138—22no—
99#13133138—41no—
100C4×#88322—22no—
101#13133124—21no—
102#13133124—21no—
103C22×Q8×C53—21no—
104C22×D4×C53C2311yescharacteristic
105C23×C4×C53—21no—
106C4⋊Q8×C53—22no—
107C4⋊Q8×C53—41no—
108C22⋊Q8×C53—44no—
109C4×D4×C53—22no—
110C4×D4×C53—41no—
111C2×C4⋊C4×C53—21no—
112C2×C4⋊C4×C53—21no—
113C2×C4⋊C4×C53—21no—
114C2×C4⋊C4×C53D411yescharacteristic
115C2×C4⋊C4×C53D411yescharacteristic
116C2×C4⋊C4×C53C2311yescharacteristic
117C2×C4⋊C4×C53C2311yescharacteristic
118C2×C4⋊C4×C53C2311yescharacteristic
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53—21no—
121C2×C22⋊C4×C53—21no—
122C2×C22⋊C4×C53—21no—
123C2×C22⋊C4×C53—21no—
124C2×C42×C53C2311yescharacteristic
125C2×C42×C53C2311yescharacteristic
126C2×C42×C53C2311yescharacteristic
127#95×C53—21no—
128#95×C53—21no—
129#95×C53D411yescharacteristic
130#95×C53D411yescharacteristic
131C24×C53—21no—
132C2×Q8×C53—42no—
133C2×Q8×C53—42no—
134C2×Q8×C53—81no—
135C2×D4×C53—21no—
136C2×D4×C53—21no—
137C2×D4×C53—21no—
138C2×D4×C53—21no—
139C2×D4×C53—21no—
140C2×D4×C53C2×Q812yes—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—42no—
145C22×C4×C53—42no—
146C22×C4×C53—81no—
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C22×C4×C53C2411yescharacteristic
154C22×Dic53—21no—
155C22×Dic53—21no—
156C22×Dic53—21no—
157C22×Dic53—21no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106—21no—
161C2×Dic106C2×D412yes—
162C2×Dic106C2×Q812yes—
163C2×Dic106#4112yes—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—41no—
170C4⋊C4×C53—42no—
171C4⋊C4×C53—42no—
172C4⋊C4×C53—42no—
173C4⋊C4×C53SD1614yescomplemented
174C4⋊C4×C53C2×D412yes—
175C22⋊C4×C53—42no—
176C22⋊C4×C53—42no—
177C22⋊C4×C53—42no—
178C22⋊C4×C53—42no—
179C22⋊C4×C53—81no—
180C42×C53—21no—
181C42×C53—21no—
182C42×C53—21no—
183C42×C53—21no—
184C42×C53—21no—
185C42×C53—21no—
186#11719923—28no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—42no—
193C4⋊Dic53—81no—
194C4⋊Dic53#4112yes—
195#11719919—24no—
196#11719919—42no—
197#11719919—42no—
198#11719919—81no—
199#11719919—81no—
200C4×Dic53—24no—
201C4×Dic53—42no—
202C4×Dic53—42no—
203C2×#88322—21no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322C2×D412yes—
214C2×#88322C2×Q812yes—
215C23×C53—22no—
216C23×C53—41no—
217C23×C53—41no—
218C23×C53—41no—
219C23×C53—41no—
220C23×C53—41no—
221C23×C53—41no—
222C23×C53C22×D411yescharacteristic
223Q8×C53—84no—
224D4×C53—22no—
225D4×C53—22no—
226D4×C53—41no—
227D4×C53—41no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—22no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—21no—
239C2×C4×C53—21no—
240C2×C4×C53—21no—
241C2×C4×C53—21no—
242C2×C4×C53—21no—
243C2×C4×C53—21no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—84no—
253C2×C4×C53C2×SD1612yes—
254C2×C4×C53C2×SD1612yes—
255C2×C4×C53C2×SD1612yes—
256C2×C4×C53C22×D411yescharacteristic
257C2×C4×C53C22×Q811yescharacteristic
258C2×C4×C53C2×#4111yescharacteristic
259C2×C4×C53C2×#4111yescharacteristic
260C2×C4×C53H(5,𝔽2)11yescharacteristic
261C2×C4×C532−1+411yescharacteristic
262C2×Dic53—22no—
263C2×Dic53—22no—
264C2×Dic53—22no—
265C2×Dic53—22no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274Dic106—22no—
275Dic106—22no—
276Dic106—22no—
277Dic106C22⋊Q814yescomplemented
278#88322—22no—
279#88322—22no—
280#88322—41no—
281#88322—41no—
282#26373—531nomaximal, Sylow
283C22×C53—42no—
284C22×C53—42no—
285C22×C53—42no—
286C22×C53—81no—
287C22×C53—81no—
288C22×C53—81no—
289C22×C53—81no—
290C22×C53#52511yescharacteristic
291C22×C53D4211yescharacteristic
292C22×C53D4×Q811yescharacteristic
293C22×C53C22×SD1611yescharacteristic
294C22×C53#57511yescharacteristic
295C22×C53#57711yescharacteristic
296C22×C53#57711yescharacteristic
297C4×C53—22no—
298C4×C53—22no—
299C4×C53—22no—
300C4×C53—41no—
301C4×C53—41no—
302C4×C53—41no—
303C4×C53—42no—
304C4×C53—81no—
305C4×C53—81no—
306C4×C53C2×C22⋊Q811yescharacteristic
307C4×C53#53911yescharacteristic
308C4×C53#54711yescharacteristic
309C4×C53#55411yescharacteristic
310Dic53—24no—
311Dic53—42no—
312Dic53—81no—
313Dic53—81no—
314C2×#475—531no—
315C2×C4×SD16—531no—
316#2452D5311yescharacteristic
317Q8×C4⋊C4—531no—
318#2094—531no—
319#2084—531no—
320#2065—531no—
321#2058—531no—
322#1941—531no—
323#1916—531no—
324#1880—531no—
325#1880—531no—
326#1865—531no—
327#1865—531no—
328#1864—531no—
329C2×C53—82no—
330C2×C53#244911yescharacteristic
331C2×C53#310311yescharacteristic
332C2×C53#312511yescharacteristic
333C2×C53#327111yescharacteristic
334C2×C53D4×SD1611yescharacteristic
335C2×C53Q8×SD1611yescharacteristic
336C2×C53#338811yescharacteristic
337C22×SD16—531no—
338C2×C4⋊Q8C2×D5311yescharacteristic
339C2×C22⋊Q8—21no—
340C2×C4×Q8—531no—
341C2×C4×Q8—531no—
342C2×C4×Q8—531no—
343C2×C4×D4C2×D5311yescharacteristic
344#475—1064no—
345C4×SD16—1064no—
346C2×#106—531no—
347C2×#106—531no—
348C2×#106—531no—
349C2×#106—531no—
350C2×#106—531no—
351C2×C4⋊C8—531no—
352C2×#103—531no—
353C2×#103—531no—
354C2×#103—531no—
355C2×#102—531no—
356C2×#102—531no—
357C2×#102—531no—
358C2×C4×C8—531no—
359#396—21no—
360#393—21no—
361#390C2×D5311yescharacteristic
362#390C2×D5311yescharacteristic
363#389—531no—
364#389—531no—
365#389—1061no—
366#389—1061no—
367#389C2×D5311yescharacteristic
368#389C2×D5311yescharacteristic
369#385—21no—
370#384—531no—
371#384—1061no—
372C4×C4⋊C4—1061no—
373C4×C4⋊C4C2×D5311yescharacteristic
374#340—531no—
375#340—531no—
376#340—531no—
377#340—531no—
378#336—531no—
379#336—531no—
380#336—531no—
381#336—531no—
382C53#2637311yescharacteristic, complemented, Sylow
383C22×Q8—21no—
384C22×Q8—531no—
385C22×D4C22×D5311yescharacteristic
386C23×C4—21no—
387C2×SD16—534no—
388C2×SD16—1062no—
389C2×SD16—1062no—
390C22×C8—531no—
391C22×C8—531no—
392C22×C8—531no—
393C22×C8—531no—
394C4⋊Q8—22no—
395C4⋊Q8—41no—
396C22⋊Q8—44no—
397C4×Q8—1062no—
398C4×Q8—1062no—
399C4×Q8—1062no—
400C4×Q8—1062no—
401C4×Q8—2121no—
402C4×Q8—2121no—
403C4×D4—22no—
404C4×D4—41no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—531no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4C22⋊D5311yescharacteristic
419C2×C4⋊C4C22⋊D5311yescharacteristic
420C2×C4⋊C4C22×D5311yescharacteristic
421C2×C4⋊C4C22×D5311yescharacteristic
422C2×C4⋊C4C22×D5311yescharacteristic
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C42—1061no—
429C2×C42—1061no—
430C2×C42—1061no—
431C2×C42C22×D5311yescharacteristic
432C2×C42C22×D5311yescharacteristic
433C2×C42C22×D5311yescharacteristic
434#106—1062no—
435#106—1062no—
436#106—1062no—
437#106—1062no—
438#106—2121no—
439C4⋊C8—1062no—
440#103—534no—
441#103—1062no—
442#103—1062no—
443#102—1062no—
444#102—1062no—
445#102—2121no—
446C4×C8—1062no—
447#95—21no—
448#95—21no—
449#95—1061no—
450#95—1061no—
451#95C22⋊D5311yescharacteristic
452#95C22⋊D5311yescharacteristic
453C24—21no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—81no—
457C2×Q8—532no—
458C2×Q8—532no—
459C2×Q8—532no—
460C2×Q8—1061no—
461C2×Q8—1061no—
462C2×Q8—1061no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4Q8×D5312yes—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—81no—
475C22×C4—1061no—
476C22×C4—1061no—
477C22×C4—1061no—
478C22×C4—1061no—
479C22×C4C2×C22⋊D5311yescharacteristic
480C22×C4C2×C22⋊D5311yescharacteristic
481C22×C4C2×C22⋊D5311yescharacteristic
482C22×C4C2×C22⋊D5311yescharacteristic
483C22×C4C2×C22⋊D5311yescharacteristic
484C22×C4C2×C22⋊D5311yescharacteristic
485C22×C4C23×D5311yescharacteristic
486SD16—1068no—
487C2×C8—532no—
488C2×C8—532no—
489C2×C8—1061no—
490C2×C8—1061no—
491C2×C8—1061no—
492C2×C8—1061no—
493C2×C8—1061no—
494C2×C8—1061no—
495C2×C8—1061no—
496C2×C8—1061no—
497C2×C8—1061no—
498C2×C8—1061no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—41no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—532no—
509C4⋊C4—1064no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C4⋊C4—1061no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—4241no—
516C4⋊C4—4241no—
517C4⋊C4—4241no—
518C4⋊C4—2122no—
519C4⋊C4—2122no—
520C4⋊C4—2122no—
521C4⋊C4#1171992314yescomplemented
522C4⋊C4D4×D5312yes—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—81no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—1064no—
535C42—2122no—
536C42—2122no—
537C23—22no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23C22×C22⋊D5311yescharacteristic
545Q8—84no—
546Q8—534no—
547Q8—1062no—
548Q8—1062no—
549Q8—1062no—
550D4—22no—
551D4—22no—
552D4—41no—
553D4—41no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—84no—
579C2×C4—1062no—
580C2×C4—1062no—
581C2×C4—1062no—
582C2×C4—1062no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4C2×#1171992312yes—
592C2×C4C2×#1171992312yes—
593C2×C4C2×#1171992312yes—
594C2×C4C2×D4×D5311yescharacteristic
595C2×C4C2×Q8×D5311yescharacteristic
596C2×C4#41×D5311yescharacteristic
597C2×C4#41×D5311yescharacteristic
598C2×C4#1313326211yescharacteristic
599C2×C4#1313326511yescharacteristic
600C8—1062no—
601C8—1062no—
602C8—2121no—
603C8—2121no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22#1681113011yescharacteristic
612C22D4×C22⋊D5311yescharacteristic
613C22Q8×C22⋊D5311yescharacteristic
614C22C22×#1171992311yescharacteristic
615C22#1681142111yescharacteristic
616C22#1681143111yescharacteristic
617C22#1681143411yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—42no—
625C4—81no—
626C4—81no—
627C4—1064no—
628C4—4241no—
629C4—4241no—
630C4—2122no—
631C4C22⋊Q8×D5311yescharacteristic
632C4#1681121911yescharacteristic
633C4#1681127511yescharacteristic
634C4#1681132711yescharacteristic
635C2—82no—
636C2#2027852111yescharacteristic
637C2#2028474211yescharacteristic
638C2#2028498611yescharacteristic
639C2#2028630111yescharacteristic
640C2D4×#1171992311yescharacteristic
641C2Q8×#1171992311yescharacteristic
642C2#2028766011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2452C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
hypercenter#2452D53
hypocenterC53#26373
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2452

Complements

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

NH ≅ G/Ntype
1. C2×#16810905C2semidirect
3. C4⋊C4×Dic106C2semidirect
5. #20276191C2semidirect
7. #20276045C2semidirect
8. #20275440C2semidirect
9. #20275274C2semidirect
12. #20274982C2semidirect
13. #20274982C2semidirect
64. C2×#11719923Q8semidirect
173. C4⋊C4×C53SD16semidirect
277. Dic106C22⋊Q8semidirect
382. C53#26373semidirect
521. C4⋊C4#11719923semidirect

Maximal subgroups

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

subgroupindex
C2×#168109052
C2×C4×#117199232
C4⋊C4×Dic1062
#202762482
#202761912
#202760892
#202760452
#202754402
#202752742
#202750732
#202749822
#202749762
C53×#24522
#2637353