Finite Group Databasebeta

← #27418410

Subgroups — #27418410

Order 13568.

1198 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811180C211yesmaximal, characteristic, complemented
2C2×#16810913C211yesmaximal, characteristic, complemented
3C2×#16810828C211yesmaximal, characteristic, complemented
4#20275911C212yesmaximal, complemented
5#20275718C212yesmaximal
6#20275663C211yesmaximal, characteristic, complemented
7#20275638C211yesmaximal, characteristic, complemented
8#20275073C212yesmaximal, complemented
9#20274982C212yesmaximal, complemented
10#20274695C211yesmaximal, characteristic, complemented
11C4×#121×C53C211yesmaximal, characteristic
12C22×#11719923C2212yes—
13#16811180—24no—
14#16811180—24no—
15C2×#13133217C2211yescharacteristic
16C2×#13133217C2211yescharacteristic
17C2×#13133209—21no—
18C2×#13133196—21no—
19C2×#13133196C2212yes—
20C2×C4×Dic106—21no—
21C2×C4×Dic106C2212yes—
22C2×#13133176—21no—
23#16810913—24no—
24#16810828—24no—
25C2×#13133148C2212yes—
26C2×C4⋊#88322C2211yescharacteristic
27C2×C4⋊#88322C2211yescharacteristic
28C2×#13133140C2212yes—
29C2×#13133140C2212yes—
30C2×#13133140C2212yes—
31C2×#13133138C2212yes—
32C2×C22⋊#88322C2212yes—
33#16810385C2212yes—
34#16810385C2212yes—
35#16810378C2212yes—
36C2×#121×C53C2211yescharacteristic
37C2×C4×D4×C53—21no—
38C2×C4×D4×C53C2211yescharacteristic
39C22×C42×C53C2211yescharacteristic
40C53×#390C2211yescharacteristic
41C53×#389C2211yescharacteristic
42C53×#388—21no—
43C53×#386—21no—
44C53×#380C2211yescharacteristic
45C4×C4⋊C4×C53C2211yescharacteristic
46C4×C22⋊C4×C53—21no—
47C22×Dic106—21no—
48C22×Dic106C2312yes—
49C2×#11719923—24no—
50C2×#11719923—24no—
51C2×#11719923—42no—
52C22×#88322C2312yes—
53C22×#88322C2312yes—
54#13133217—22no—
55#13133217—22no—
56#13133217D414yescomplemented
57#13133217D414yescomplemented
58#13133209—28no—
59#13133196—28no—
60#13133196—24no—
61#13133196—24no—
62C4×Dic106—28no—
63C4×Dic106—24no—
64C4×Dic106—24no—
65#13133176—28no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53C2312yes—
70C2×#11719919—22no—
71C2×#11719919—22no—
72C2×C22⋊Dic53—22no—
73C2×C4×Dic53—22no—
74#13133148—24no—
75C4⋊#88322—22no—
76C4⋊#88322—22no—
77#13133140—24no—
78#13133140—24no—
79#13133140—42no—
80#13133138—24no—
81C22⋊#88322—42no—
82C22×D4×C53—21no—
83C22×D4×C53C2311yescharacteristic
84C23×C4×C53—21no—
85C23×C4×C53D411yescharacteristic
86C23×C4×C53D411yescharacteristic
87C23×C4×C53C2311yescharacteristic
88#121×C53—22no—
89#121×C53—22no—
90C4×D4×C53—22no—
91C4×D4×C53—22no—
92C4×D4×C53—44no—
93C2×C4⋊C4×C53—21no—
94C2×C4⋊C4×C53C2311yescharacteristic
95C2×C4⋊C4×C53C2311yescharacteristic
96C2×C4⋊C4×C53C2311yescharacteristic
97C2×C22⋊C4×C53—21no—
98C2×C22⋊C4×C53—21no—
99C2×C22⋊C4×C53—21no—
100C2×C22⋊C4×C53—21no—
101C2×C22⋊C4×C53—21no—
102C2×C42×C53—22no—
103C2×C42×C53—22no—
104C2×C42×C53—21no—
105C2×C42×C53D411yescharacteristic
106C2×C42×C53D411yescharacteristic
107C2×C42×C53C2311yescharacteristic
108C2×C42×C53C2311yescharacteristic
109#95×C53—21no—
110#95×C53—21no—
111#95×C53—21no—
112C24×C53—21no—
113C24×C53C2×D411yescharacteristic
114C2×D4×C53—24no—
115C2×D4×C53—21no—
116C2×D4×C53—21no—
117C2×D4×C53—21no—
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—42no—
121C2×D4×C53—42no—
122C2×D4×C53#4112yes—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—21no—
126C22×C4×C53—22no—
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—41no—
136C22×C4×C53—41no—
137C22×C4×C53—42no—
138C22×C4×C53—42no—
139C22×C4×C53—42no—
140C22×C4×C53SD1614yescomplemented
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2411yescharacteristic
147C22×Dic53—22no—
148C22×Dic53—22no—
149C2×Dic106—22no—
150C2×Dic106—22no—
151C2×Dic106—22no—
152C2×Dic106—22no—
153C2×Dic106—24no—
154C2×Dic106—22no—
155C2×Dic106—42no—
156C2×Dic106—42no—
157C2×Dic106C2×D414yes—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—42no—
164C4⋊C4×C53#4112yes—
165C22⋊C4×C53—24no—
166C22⋊C4×C53—42no—
167C22⋊C4×C53—42no—
168C22⋊C4×C53—42no—
169C22⋊C4×C53—42no—
170C42×C53—22no—
171C42×C53—22no—
172C42×C53—21no—
173C42×C53—21no—
174C42×C53—42no—
175C42×C53C2×D412yes—
176C42×C53C2×D412yes—
177C22⋊Dic53—28no—
178#11719923—48no—
179C4⋊Dic53—24no—
180C4⋊Dic53—24no—
181C4⋊Dic53—24no—
182C4⋊Dic53—22no—
183C4⋊Dic53C2×D414yes—
184#11719919—28no—
185#11719919—28no—
186C4×Dic53—28no—
187C2×#88322—22no—
188C2×#88322—22no—
189C2×#88322—22no—
190C2×#88322—22no—
191C2×#88322—22no—
192C2×#88322—22no—
193C23×C53—21no—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53—21no—
199C23×C53—22no—
200C23×C53—41no—
201C23×C53—41no—
202C23×C53—41no—
203C23×C53—41no—
204C23×C53—41no—
205C23×C53—41no—
206C23×C53C2×SD1612yes—
207C23×C53C22×D411yescharacteristic
208D4×C53—22no—
209D4×C53—22no—
210D4×C53—48no—
211D4×C53—41no—
212D4×C53—41no—
213C2×C4×C53—21no—
214C2×C4×C53—21no—
215C2×C4×C53—21no—
216C2×C4×C53—21no—
217C2×C4×C53—28no—
218C2×C4×C53—21no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—22no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—44no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—84no—
239C2×C4×C53#12012yes—
240C2×C4×C53#12012yes—
241C2×C4×C53C2×SD1612yes—
242C2×C4×C53C2×SD1612yes—
243C2×C4×C53#13712yes—
244C2×C4×C53C22×D411yescharacteristic
245C2×C4×C53C22×D411yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53H(5,𝔽2)11yescharacteristic
249C2×C4×C532−1+411yescharacteristic
250C2×Dic53—24no—
251C2×Dic53—24no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53—42no—
256Dic106—24no—
257Dic106—24no—
258Dic106—48no—
259Dic106—42no—
260Dic106—42no—
261#88322—42no—
262#88322—42no—
263#24151—531nomaximal, Sylow
264C22×C53—22no—
265C22×C53—22no—
266C22×C53—22no—
267C22×C53—41no—
268C22×C53—41no—
269C22×C53—41no—
270C22×C53—41no—
271C22×C53—42no—
272C22×C53—42no—
273C22×C53—42no—
274C22×C53—81no—
275C22×C53—81no—
276C22×C53—81no—
277C22×C53—81no—
278C22×C53C2×#12011yescharacteristic
279C22×C53#52511yescharacteristic
280C22×C53#53711yescharacteristic
281C22×C53C22×SD1611yescharacteristic
282C22×C53C2×#13711yescharacteristic
283C22×C53#57511yescharacteristic
284C22×C53#57511yescharacteristic
285C4×C53—22no—
286C4×C53—22no—
287C4×C53—42no—
288C4×C53—41no—
289C4×C53—41no—
290C4×C53—81no—
291C4×C53—81no—
292C4×C53#44814yes—
293C4×C53#54611yescharacteristic
294C4×C53#54711yescharacteristic
295C4×C53#54811yescharacteristic
296C4×C53#55011yescharacteristic
297Dic53—44no—
298Dic53—44no—
299C2×#533—531no—
300C2×#476—531no—
301C2×#461—531no—
302C4×#121D5311yescharacteristic
303#2035—532no—
304#2000—532no—
305#1987—531no—
306#1981—531no—
307#1880—532no—
308#1865—532no—
309#1797—531no—
310C2×C53—42no—
311C2×C53—82no—
312C2×C53#240711yescharacteristic
313C2×C53C2×#44811yescharacteristic
314C2×C53#300511yescharacteristic
315C2×C53#310111yescharacteristic
316C2×C53#311611yescharacteristic
317C2×C53#324611yescharacteristic
318C2×C53#325211yescharacteristic
319C22×SD16—532no—
320#533—1064no—
321#533—1064no—
322C2×C4⋊Q8—531no—
323C2×C4⋊Q8—531no—
324C2×#125—1061no—
325C2×C22⋊Q8—532no—
326C2×C22⋊Q8—1061no—
327C2×#121C2×D5311yescharacteristic
328C2×C4×Q8—532no—
329C2×C4×Q8—1061no—
330C2×C4×D4—21no—
331C2×C4×D4C2×D5311yescharacteristic
332C2×#117—1061no—
333C22×C42C2×D5311yescharacteristic
334#476—1064no—
335#461—1064no—
336C2×#106—532no—
337C2×C4⋊C8—531no—
338C2×C4⋊C8—531no—
339C2×#103—532no—
340C2×#103—532no—
341C2×#103—532no—
342C2×#102—532no—
343C2×C22⋊C8—532no—
344#390C2×D5311yescharacteristic
345#389C2×D5311yescharacteristic
346#388—21no—
347#386—21no—
348#380C2×D5311yescharacteristic
349C4×C4⋊C4C2×D5311yescharacteristic
350C4×C22⋊C4—21no—
351#340—532no—
352#340—532no—
353#336—532no—
354C53#2415111yescharacteristic, complemented, Sylow
355C22×Q8—532no—
356C22×Q8—1061no—
357C22×D4—21no—
358C22×D4C22×D5311yescharacteristic
359C23×C4—21no—
360C23×C4C22⋊D5311yescharacteristic
361C23×C4C22⋊D5311yescharacteristic
362C23×C4C22×D5311yescharacteristic
363C2×SD16—1064no—
364C2×SD16—1064no—
365C2×SD16—2122no—
366C22×C8—532no—
367C22×C8—532no—
368C4⋊Q8—534no—
369C4⋊Q8—534no—
370C4⋊Q8—1062no—
371C4⋊Q8—1062no—
372#125—1068no—
373C22⋊Q8—1064no—
374C22⋊Q8—1064no—
375C22⋊Q8—1068no—
376#121—22no—
377#121—22no—
378C4×Q8—1064no—
379C4×Q8—1064no—
380C4×Q8—1068no—
381C4×D4—22no—
382C4×D4—22no—
383C4×D4—44no—
384#117—1068no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—532no—
387C2×C4⋊C4—1061no—
388C2×C4⋊C4—1061no—
389C2×C4⋊C4—1062no—
390C2×C4⋊C4—1062no—
391C2×C4⋊C4—1061no—
392C2×C4⋊C4C22×D5311yescharacteristic
393C2×C4⋊C4C22×D5311yescharacteristic
394C2×C4⋊C4C22×D5311yescharacteristic
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—1062no—
401C2×C42—22no—
402C2×C42—22no—
403C2×C42—21no—
404C2×C42—1062no—
405C2×C42C22⋊D5311yescharacteristic
406C2×C42C22⋊D5311yescharacteristic
407C2×C42C22×D5311yescharacteristic
408C2×C42C22×D5311yescharacteristic
409#106—1064no—
410C4⋊C8—1062no—
411C4⋊C8—1062no—
412#103—1064no—
413#103—1064no—
414#103—2122no—
415#102—1064no—
416C22⋊C8—2122no—
417#95—21no—
418#95—21no—
419#95—21no—
420C24—21no—
421C24C2×C22⋊D5311yescharacteristic
422C2×Q8—534no—
423C2×Q8—1064no—
424C2×Q8—1062no—
425C2×Q8—1062no—
426C2×Q8—1062no—
427C2×Q8—1062no—
428C2×Q8—1062no—
429C2×Q8—2122no—
430C2×Q8—2122no—
431C2×D4—24no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4#1171994512yes—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—1062no—
458C22×C4—1062no—
459C22×C4#1171992314yescomplemented
460C22×C4C2×C22⋊D5311yescharacteristic
461C22×C4C2×C22⋊D5311yescharacteristic
462C22×C4C2×C22⋊D5311yescharacteristic
463C22×C4C2×C22⋊D5311yescharacteristic
464C22×C4C2×C22⋊D5311yescharacteristic
465C22×C4C23×D5311yescharacteristic
466SD16—2128no—
467C2×C8—1062no—
468C2×C8—1062no—
469C2×C8—1062no—
470C2×C8—1062no—
471C2×C8—1062no—
472C2×C8—1062no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—42no—
479C4⋊C4—534no—
480C4⋊C4—1068no—
481C4⋊C4—1068no—
482C4⋊C4—1064no—
483C4⋊C4—1064no—
484C4⋊C4—1064no—
485C4⋊C4—1062no—
486C4⋊C4#1171994512yes—
487C22⋊C4—24no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—1068no—
493C42—22no—
494C42—22no—
495C42—21no—
496C42—21no—
497C42—42no—
498C42—1068no—
499C42D4×D5312yes—
500C42D4×D5312yes—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—22no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23C2×#1171992312yes—
515C23C22×C22⋊D5311yescharacteristic
516Q8—1064no—
517Q8—1064no—
518Q8—2128no—
519Q8—2122no—
520Q8—2122no—
521D4—22no—
522D4—22no—
523D4—48no—
524D4—41no—
525D4—41no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—28no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—44no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—84no—
552C2×C4—1064no—
553C2×C4—1064no—
554C2×C4—2122no—
555C2×C4—2122no—
556C2×C4—2122no—
557C2×C4—2122no—
558C2×C4#1313318512yes—
559C2×C4#1313318512yes—
560C2×C4C2×#1171992312yes—
561C2×C4C2×#1171992312yes—
562C2×C4#1313324712yes—
563C2×C4C2×D4×D5311yescharacteristic
564C2×C4C2×D4×D5311yescharacteristic
565C2×C4C2×#1171994511yescharacteristic
566C2×C4C2×#1171994511yescharacteristic
567C2×C4#1313326311yescharacteristic
568C2×C4#1313326411yescharacteristic
569C8—2122no—
570C8—2122no—
571C22—22no—
572C22—22no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22C2×#1313318511yescharacteristic
586C22#1681113411yescharacteristic
587C22#1681120111yescharacteristic
588C22C22×#1171992311yescharacteristic
589C22C2×#1313324711yescharacteristic
590C22#1681142111yescharacteristic
591C22#1681142111yescharacteristic
592C4—22no—
593C4—22no—
594C4—41no—
595C4—41no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—2124no—
600C4—2124no—
601C4#1681076114yes—
602C4#1681126911yescharacteristic
603C4#1681127811yescharacteristic
604C4#1681128211yescharacteristic
605C4#1681129411yescharacteristic
606C2—42no—
607C2—82no—
608C2#2027806011yescharacteristic
609C2C2×#1681076111yescharacteristic
610C2#2028395511yescharacteristic
611C2#2028472311yescharacteristic
612C2#2028487711yescharacteristic
613C2#2028603111yescharacteristic
614C2#2028608811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC4×#121×C53C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenterC4×#121D53
hypocenterC53#24151
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×#121

Complements

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

NH ≅ G/Ntype
1. C2×#16811180C2semidirect
2. C2×#16810913C2semidirect
3. C2×#16810828C2semidirect
4. #20275911C2semidirect
6. #20275663C2semidirect
7. #20275638C2semidirect
8. #20275073C2semidirect
9. #20274982C2semidirect
10. #20274695C2semidirect
56. #13133217D4semidirect
57. #13133217D4semidirect
140. C22×C4×C53SD16semidirect
354. C53#24151semidirect
459. C22×C4#11719923semidirect

Maximal subgroups

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

subgroupindex
C2×#168111802
C2×#168109132
C2×#168108282
#202759112
#202757182
#202756632
#202756382
#202750732
#202749822
#202746952
C4×#121×C532
#2415153