Finite Group Databasebeta

← #27401839

Subgroups — #27401839

Order 13568.

810 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277608C211yesmaximal, characteristic
2#20277584C211yesmaximal, characteristic
3#20275454C211yesmaximal, characteristic, complemented
4#20275449C211yesmaximal, characteristic, complemented
5#20275309C211yesmaximal, characteristic, complemented
6#20275302C211yesmaximal, characteristic, complemented
7#20275295C211yesmaximal, characteristic, complemented
8#20275288C211yesmaximal, characteristic, complemented
9#20274810C211yesmaximal, characteristic
10#20274805C211yesmaximal, characteristic
11#20274748C211yesmaximal, characteristic
12#20274615C211yesmaximal, characteristic, complemented
13#106×Dic53C211yesmaximal, characteristic, complemented
14#20274496C211yesmaximal, characteristic
15C2×C53×#428C211yesmaximal, characteristic
16C2×#13133176—21no—
17C2×#13133151C2211yescharacteristic
18C2×#13133149C2211yescharacteristic
19C2×C22⋊#88322C2211yescharacteristic
20C2×#13133127C2211yescharacteristic
21C2×C8×Dic53C2211yescharacteristic
22#16810522C2211yescharacteristic
23#16810522C2211yescharacteristic
24#16810522C2211yescharacteristic
25#16810522C2211yescharacteristic
26#16810516—21no—
27#16810516—21no—
28#16810509—21no—
29#16810509—21no—
30#16810491—21no—
31#16810486—21no—
32#16810483C2211yescharacteristic
33#16810483C2211yescharacteristic
34C4⋊C4×Dic53C2211yescharacteristic
35C4⋊C4×Dic53C2211yescharacteristic
36C4⋊C4×Dic53C2211yescharacteristic
37C4⋊C4×Dic53C2211yescharacteristic
38C4×C4⋊Dic53—21no—
39C4×C22⋊Dic53C2211yescharacteristic
40#16810386C2211yescharacteristic
41#16810386C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C22×M16×C53C2211yescharacteristic
53C2×#117×C53C2211yescharacteristic
54C22×C4⋊C4×C53C2211yescharacteristic
55C53×#428C418yescomplemented
56C2×#107×C53C2211yescharacteristic
57C2×#107×C53C2211yescharacteristic
58C2×#106×C53C2211yescharacteristic
59C2×#106×C53C2211yescharacteristic
60C22×#88322C2311yescharacteristic
61C22×#88322C2311yescharacteristic
62#13133176—44no—
63C2×C4⋊Dic53—21no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2311yescharacteristic
69C2×C4⋊Dic53C2311yescharacteristic
70C2×C4⋊Dic53C2311yescharacteristic
71C2×C22⋊Dic53—21no—
72C2×C22⋊Dic53—21no—
73C2×C22⋊Dic53D411yescharacteristic
74C2×C22⋊Dic53D411yescharacteristic
75C2×C4×Dic53—21no—
76C2×C4×Dic53—21no—
77C2×C4×Dic53—21no—
78C2×C4×Dic53—21no—
79C2×C4×Dic53C2311yescharacteristic
80C2×C4×Dic53C2311yescharacteristic
81#13133151—22no—
82#13133149—22no—
83C22⋊#88322—22no—
84#13133127—22no—
85C8×Dic53—22no—
86#13133124—21no—
87#13133124—21no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95#13133124—21no—
96#13133124D411yescharacteristic
97#13133124D411yescharacteristic
98C23×C4×C53—21no—
99C23×C4×C53C2311yescharacteristic
100C2×M16×C53—22no—
101C2×M16×C53—22no—
102C2×M16×C53C2×C414yes—
103C22×C8×C53C2311yescharacteristic
104C22×C8×C53C2311yescharacteristic
105#117×C53—22no—
106#117×C53C2×C414yes—
107C2×C4⋊C4×C53—21no—
108C2×C4⋊C4×C53—22no—
109C2×C4⋊C4×C53C2×C414yes—
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C22⋊C4×C53—21no—
115C2×C42×C53—21no—
116#107×C53C2×C414yes—
117#107×C53C2×C414yes—
118#106×C53C2×C414yes—
119#106×C53C2×C414yes—
120C24×C53C2×D411yescharacteristic
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—24no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—42no—
131C22×C4×C53—42no—
132C22×C4×C53C22×C412yes—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53#4112yes—
135C22×C4×C53C2411yescharacteristic
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—21no—
139C22×Dic53—21no—
140C22×Dic53C2×D411yescharacteristic
141C22×Dic53C2×D411yescharacteristic
142C22×Dic53C2×D411yescharacteristic
143C22×Dic53C2×D411yescharacteristic
144M16×C53—28no—
145C2×C8×C53—21no—
146C2×C8×C53—21no—
147C2×C8×C53C22×C412yes—
148C2×C8×C53C22×C412yes—
149C2×C8×C53#4112yes—
150C2×C8×C53#4112yes—
151C4⋊C4×C53—24no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53C22×C412yes—
157C4⋊C4×C53C22×C412yes—
158C4⋊C4×C53C22×C412yes—
159C4⋊C4×C53C22×C412yes—
160C22⋊C4×C53—24no—
161C42×C53—24no—
162C22⋊Dic53—22no—
163C22⋊Dic53—41no—
164C22⋊Dic53—42no—
165C22⋊Dic53—81no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53—42no—
175C4⋊Dic53—42no—
176C4⋊Dic53—42no—
177C4⋊Dic53—42no—
178C4×Dic53—21no—
179C4×Dic53—21no—
180C4×Dic53—21no—
181C4×Dic53—21no—
182C4×Dic53—42no—
183C4×Dic53—42no—
184C4×Dic53—81no—
185C4×Dic53—81no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322—21no—
191C2×#88322—21no—
192C23×C53—21no—
193C23×C53—21no—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53C4×D412yes—
199C23×C53C22×D411yescharacteristic
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—22no—
203C2×C4×C53—22no—
204C2×C4×C53—22no—
205C2×C4×C53—22no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—22no—
210C2×C4×C53—48no—
211C2×C4×C53—41no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53C4×D412yes—
220C2×C4×C53C23×C411yescharacteristic
221C2×C4×C53C2×#4111yescharacteristic
222C2×C4×C53C2×#4111yescharacteristic
223C2×C4×C53C2×#4111yescharacteristic
224C2×C4×C532−1+411yescharacteristic
225C2×C4×C532−1+411yescharacteristic
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—21no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53AGL(1,ℤ/8)12yes—
250C8×C53—22no—
251C8×C53#11714yescomplemented
252#88322—41no—
253#88322—41no—
254#22622—531nomaximal, Sylow
255C22×C53—22no—
256C22×C53—22no—
257C22×C53—22no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53—41no—
261C22×C53—41no—
262C22×C53C2×C4×D411yescharacteristic
263C22×C53#54711yescharacteristic
264C22×C53#54711yescharacteristic
265C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C53#57511yescharacteristic
267C22×C53#57711yescharacteristic
268C22×C53#57811yescharacteristic
269C4×C53—22no—
270C4×C53—42no—
271C4×C53—42no—
272C4×C53—42no—
273C4×C53—42no—
274C4×C53C2×#11711yescharacteristic
275C4×C53#51911yescharacteristic
276C4×C53#55211yescharacteristic
277C4×C53#55211yescharacteristic
278Dic53—22no—
279Dic53—41no—
280Dic53—41no—
281Dic53—41no—
282Dic53—81no—
283Dic53—81no—
284Dic53—81no—
285Dic53—81no—
286C2×#428D5311yescharacteristic
287#2347—531no—
288#2344—531no—
289#1943—531no—
290#1942—531no—
291#1921—531no—
292#1920—531no—
293#1919—531no—
294#1918—531no—
295#1825—531no—
296#1824—531no—
297#1809—531no—
298#1776—531no—
299C4×#106—531no—
300#1753—531no—
301C2×C53—42no—
302C2×C53#234411yescharacteristic
303C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
304C2×C53#294711yescharacteristic
305C2×C53#334511yescharacteristic
306C2×C53#334911yescharacteristic
307C2×C53#335111yescharacteristic
308C2×C53#335311yescharacteristic
309C22×M16C2×D5311yescharacteristic
310C2×#117—1061no—
311C2×#117C2×D5311yescharacteristic
312C22×C4⋊C4C2×D5311yescharacteristic
313#428Dic5318yescomplemented
314C2×#107—531no—
315C2×#107C2×D5311yescharacteristic
316C2×#107C2×D5311yescharacteristic
317C2×#106—531no—
318C2×#106C2×D5311yescharacteristic
319C2×#106C2×D5311yescharacteristic
320C2×C22⋊C8—531no—
321C2×#97—531no—
322C2×C4×C8—531no—
323#389—531no—
324#389—531no—
325#389—531no—
326#389—531no—
327#388—1061no—
328#388—1061no—
329#387—1061no—
330#387—1061no—
331#383—1061no—
332#381—1061no—
333#380—531no—
334#380—531no—
335C4×C4⋊C4—531no—
336C4×C4⋊C4—531no—
337C4×C4⋊C4—531no—
338C4×C4⋊C4—531no—
339C4×C4⋊C4—1061no—
340C4×C22⋊C4—531no—
341#340—531no—
342#340—531no—
343#340—531no—
344#340—531no—
345#340—531no—
346#340—531no—
347#340—531no—
348#340—531no—
349#340—531no—
350#340—531no—
351#336—531no—
352#336—531no—
353C53#2262211yescharacteristic, complemented, Sylow
354C23×C4—21no—
355C23×C4C22×D5311yescharacteristic
356C2×M16—22no—
357C2×M16—22no—
358C2×M16C2×Dic5314yes—
359C22×C8—531no—
360C22×C8—531no—
361C22×C8C22×D5311yescharacteristic
362C22×C8C22×D5311yescharacteristic
363#117—22no—
364#117—2124no—
365#117C2×Dic5314yes—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—531no—
369C2×C4⋊C4—531no—
370C2×C4⋊C4—531no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—1061no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4C2×Dic5314yes—
377C2×C4⋊C4C22×D5311yescharacteristic
378C2×C4⋊C4C22×D5311yescharacteristic
379C2×C4⋊C4C22×D5311yescharacteristic
380C2×C4⋊C4C22×D5311yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—531no—
383C2×C22⋊C4—531no—
384C2×C22⋊C4—1061no—
385C2×C22⋊C4—1061no—
386C2×C42—21no—
387C2×C42—531no—
388C2×C42—531no—
389C2×C42—1061no—
390C2×C42—1061no—
391C2×C42—1061no—
392C2×C42—1061no—
393#107—1062no—
394#107C2×Dic5314yes—
395#107C2×Dic5314yes—
396#106—1062no—
397#106C2×Dic5314yes—
398#106C2×Dic5314yes—
399C22⋊C8—1062no—
400#97—1062no—
401C4×C8—1062no—
402#95—531no—
403#95—531no—
404#95—1061no—
405#95—1061no—
406#95—1061no—
407#95—1061no—
408#95—1061no—
409#95—1061no—
410#95—1061no—
411#95—1061no—
412#95—1061no—
413#95—1061no—
414C24D4×D5311yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—531no—
427C22×C4—531no—
428C22×C4—531no—
429C22×C4—531no—
430C22×C4—1061no—
431C22×C4—1061no—
432C22×C4—1061no—
433C22×C4—1061no—
434C22×C4D4×D5311yescharacteristic
435C22×C4#1171994712yes—
436C22×C4C22×Dic5312yes—
437C22×C4C23×D5311yescharacteristic
438M16—28no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—1061no—
442C2×C8—1061no—
443C2×C8—1061no—
444C2×C8—1061no—
445C2×C8—1061no—
446C2×C8—1061no—
447C2×C8#1171994512yes—
448C2×C8#1171994512yes—
449C2×C8C22×Dic5312yes—
450C2×C8C22×Dic5312yes—
451C4⋊C4—24no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—1061no—
457C4⋊C4—1061no—
458C4⋊C4—1061no—
459C4⋊C4—1061no—
460C4⋊C4—1061no—
461C4⋊C4—1061no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—2122no—
465C4⋊C4—2122no—
466C4⋊C4—2122no—
467C4⋊C4—2122no—
468C4⋊C4C22×Dic5312yes—
469C4⋊C4C22×Dic5312yes—
470C4⋊C4C22×Dic5312yes—
471C4⋊C4C22×Dic5312yes—
472C22⋊C4—24no—
473C22⋊C4—1062no—
474C22⋊C4—4241no—
475C22⋊C4—2121no—
476C22⋊C4—2122no—
477C42—24no—
478C42—1061no—
479C42—1061no—
480C42—1061no—
481C42—1061no—
482C42—4241no—
483C42—4241no—
484C42—2122no—
485C42—2122no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23D4×Dic5312yes—
493C23C2×D4×D5311yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—48no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—532no—
514C2×C4—1061no—
515C2×C4—1061no—
516C2×C4—1061no—
517C2×C4—1061no—
518C2×C4—1061no—
519C2×C4—1061no—
520C2×C4—1061no—
521C2×C4—1061no—
522C2×C4—1061no—
523C2×C4—1061no—
524C2×C4—1061no—
525C2×C4—2121no—
526C2×C4—2121no—
527C2×C4—2121no—
528C2×C4—2121no—
529C2×C4—2121no—
530C2×C4—2121no—
531C2×C4—2121no—
532C2×C4—2121no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4D4×Dic5312yes—
538C2×C4C23×Dic5311yescharacteristic
539C2×C4C2×#1171994711yescharacteristic
540C2×C4C2×#1171994511yescharacteristic
541C2×C4C2×#1171994511yescharacteristic
542C2×C4#1313326511yescharacteristic
543C2×C4#1313326511yescharacteristic
544C8—22no—
545C8—2121no—
546C8—2121no—
547C8#1313317614yescomplemented
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×D4×Dic5311yescharacteristic
556C22#1681127411yescharacteristic
557C22#1681127411yescharacteristic
558C22AGL(1,ℤ/8)×D5311yescharacteristic
559C22#1681141911yescharacteristic
560C22#1681143211yescharacteristic
561C22#1681143911yescharacteristic
562C4—22no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—1062no—
568C4—4241no—
569C4—4241no—
570C4—4241no—
571C4—4241no—
572C4—2121no—
573C4—2121no—
574C4—2121no—
575C4C2×#1313317611yescharacteristic
576C4#1681110111yescharacteristic
577C4#1681130811yescharacteristic
578C4#1681130811yescharacteristic
579C2—42no—
580C2#2027758411yescharacteristic
581C2AGL(1,ℤ/8)×Dic5311yescharacteristic
582C2#2028355911yescharacteristic
583C2#2028718611yescharacteristic
584C2#2028724611yescharacteristic
585C2#2028727111yescharacteristic
586C2#2028729511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#428C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
cosocleC22×C4×C53C24
hypercenterC2×#428D53
hypocenterC53#22622
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
3. #20275454C2semidirect
4. #20275449C2semidirect
5. #20275309C2semidirect
6. #20275302C2semidirect
7. #20275295C2semidirect
8. #20275288C2semidirect
12. #20274615C2semidirect
13. #106×Dic53C2semidirect
55. C53×#428C4semidirect
251. C8×C53#117semidirect
313. #428Dic53semidirect
353. C53#22622semidirect
547. C8#13133176semidirect

Maximal subgroups

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

subgroupindex
#202776082
#202775842
#202754542
#202754492
#202753092
#202753022
#202752952
#202752882
#202748102
#202748052
#202747482
#202746152
#106×Dic532
#202744962
C2×C53×#4282
#2262253