Finite Group Databasebeta

← #27401875

Subgroups — #27401875

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
1C2×#16810662C211yesmaximal, characteristic
2#20277599C211yesmaximal, characteristic
3#20277583C211yesmaximal, characteristic
4#20275444C211yesmaximal, characteristic, complemented
5#20275304C211yesmaximal, characteristic, complemented
6#20275303C211yesmaximal, characteristic, complemented
7#20275290C211yesmaximal, characteristic, complemented
8#20275289C211yesmaximal, characteristic, complemented
9#20274808C211yesmaximal, characteristic
10#20274804C211yesmaximal, characteristic
11#20274749C211yesmaximal, characteristic
12#20274612C211yesmaximal, characteristic, complemented
13#107×Dic53C211yesmaximal, characteristic, complemented
14#20274500C211yesmaximal, characteristic
15C53×#1944C211yesmaximal, characteristic, complemented
16C22×#11719912C2211yescharacteristic
17C2×#13133175—21no—
18C2×#13133175C2211yescharacteristic
19C2×C4⋊C4×D53C2211yescharacteristic
20#16810662C418yescomplemented
21C2×#13133151C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133129C2211yescharacteristic
26C2×C8×Dic53C2211yescharacteristic
27#16810522C2211yescharacteristic
28#16810522C2211yescharacteristic
29#16810515—21no—
30#16810513—21no—
31#16810510—21no—
32#16810504—21no—
33#16810488—21no—
34#16810485—21no—
35#16810480C2211yescharacteristic
36#16810480C2211yescharacteristic
37C4⋊C4×Dic53C2211yescharacteristic
38C4⋊C4×Dic53C2211yescharacteristic
39C4×C4⋊Dic53C2211yescharacteristic
40C4×C4⋊Dic53C2211yescharacteristic
41C4×#11719921C2211yescharacteristic
42#16810468—21no—
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810378C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×#107×C53C2211yescharacteristic
54C2×#106×C53C2211yescharacteristic
55C2×#97×C53C2211yescharacteristic
56C53×#389C2211yescharacteristic
57C53×#389C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C22×C4×D53—21no—
61C22×C4×D53C2311yescharacteristic
62C2×#11719912—22no—
63C2×#11719912—22no—
64C2×#11719912C2×C414yes—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133175—22no—
68#13133175—44no—
69#13133175C2×C414yes—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C4⋊C4×D53—22no—
78C4⋊C4×D53C2×C414yes—
79C2×#11719921—21no—
80C2×#11719921—21no—
81C2×#11719921—21no—
82C2×#11719921D411yescharacteristic
83C2×#11719921D411yescharacteristic
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53C2311yescharacteristic
90#13133151C2×C414yes—
91#13133150C2×C414yes—
92#13133149—22no—
93#13133148C2×C414yes—
94#13133129—22no—
95C8×Dic53—22no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124D411yescharacteristic
105#13133124D411yescharacteristic
106C22×C8×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C2×C4⋊C4×C53—21no—
109C2×C4⋊C4×C53—21no—
110C2×C4⋊C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C42×C53C2311yescharacteristic
115#107×C53—22no—
116#106×C53C2×C414yes—
117#97×C53—22no—
118#95×C53—21no—
119#95×C53—21no—
120C23×D53C2×D411yescharacteristic
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2411yescharacteristic
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53—21no—
131C22×Dic53—21no—
132C22×Dic53C2×D411yescharacteristic
133C22×Dic53C2×D411yescharacteristic
134C22×Dic53C2×D411yescharacteristic
135C2×C4×D53—21no—
136C2×C4×D53—21no—
137C2×C4×D53—21no—
138C2×C4×D53—21no—
139C2×C4×D53—24no—
140C2×C4×D53—42no—
141C2×C4×D53—42no—
142C2×C4×D53C22×C412yes—
143C2×C4×D53#4112yes—
144C2×C8×C53—21no—
145C2×C8×C53—21no—
146C2×C8×C53—21no—
147C2×C8×C53C22×C412yes—
148C2×C8×C53#4112yes—
149C2×C8×C53#4112yes—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—42no—
157C4⋊C4×C53—81no—
158C4⋊C4×C53C22×C412yes—
159C4⋊C4×C53C22×C412yes—
160C42×C53—21no—
161C42×C53—21no—
162#11719921—24no—
163#11719921—22no—
164#11719921—41no—
165#11719921—42no—
166#11719921—81no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—21no—
172C4⋊Dic53—21no—
173C4⋊Dic53—42no—
174C4⋊Dic53—81no—
175C4⋊Dic53C22×C412yes—
176C4⋊Dic53C22×C412yes—
177#11719919—24no—
178C4×Dic53—21no—
179C4×Dic53—21no—
180C4×Dic53—24no—
181C4×Dic53—42no—
182C4×Dic53—42no—
183C4×Dic53—42no—
184C4×Dic53—42no—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322—21no—
189C2×#88322—21no—
190C2×#88322C22×C412yes—
191#11719912—28no—
192C23×C53C22×D411yescharacteristic
193C22×D53—21no—
194C22×D53—21no—
195C22×D53—21no—
196C22×D53—21no—
197C22×D53—21no—
198C22×D53—21no—
199C22×D53C4×D412yes—
200C2×C4×C53—22no—
201C2×C4×C53—22no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—41no—
209C2×C4×C53—41no—
210C2×C4×C53—41no—
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×C53C23×C411yescharacteristic
219C2×C4×C53C2×#4111yescharacteristic
220C2×C4×C53C2×#4111yescharacteristic
221C2×C4×C53C2×#4111yescharacteristic
222C2×C4×C532−1+411yescharacteristic
223C2×C4×C532−1+411yescharacteristic
224C2×Dic53—21no—
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—21no—
228C2×Dic53—22no—
229C2×Dic53—22no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236C2×Dic53—41no—
237C2×Dic53—41no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53C4×D412yes—
244C2×Dic53AGL(1,ℤ/8)12yes—
245C4×D53—22no—
246C4×D53—22no—
247C4×D53—22no—
248C4×D53—22no—
249C4×D53—48no—
250C8×C53—41no—
251C8×C53#11714yescomplemented
252#88322—22no—
253#88322—41no—
254#22624—531nomaximal, Sylow
255C22×C53C2×C4×D411yescharacteristic
256C22×C53#54711yescharacteristic
257C22×C53#54711yescharacteristic
258C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
259C22×C53#57511yescharacteristic
260C22×C53#57711yescharacteristic
261C22×C53#57811yescharacteristic
262C2×D53—22no—
263C2×D53—22no—
264C2×D53—22no—
265C2×D53—41no—
266C2×D53—41no—
267C2×D53—41no—
268C2×D53—41no—
269C4×C53—41no—
270C4×C53—41no—
271C4×C53—42no—
272C4×C53—42no—
273C4×C53—81no—
274C4×C53—81no—
275C4×C53C2×#11711yescharacteristic
276C4×C53#51911yescharacteristic
277C4×C53#55211yescharacteristic
278C4×C53#55211yescharacteristic
279Dic53—22no—
280Dic53—22no—
281Dic53—41no—
282Dic53—42no—
283Dic53—42no—
284Dic53—81no—
285Dic53—81no—
286C2×#428—531no—
287#2346—531no—
288#2344—531no—
289#1944D5311yescharacteristic, complemented
290#1941—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×#107—531no—
300#1753—531no—
301C2×C53#234411yescharacteristic
302C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
303C2×C53#294711yescharacteristic
304C2×C53#334311yescharacteristic
305C2×C53#334811yescharacteristic
306C2×C53#335211yescharacteristic
307C2×C53#335411yescharacteristic
308D53—42no—
309C22×M16—531no—
310C2×#117—531no—
311C2×#117—1061no—
312C22×C4⋊C4—531no—
313#428—538no—
314C2×#107—531no—
315C2×#107—531no—
316C2×#107C2×D5311yescharacteristic
317C2×#106—531no—
318C2×#106—531no—
319C2×#106C2×D5311yescharacteristic
320C2×C22⋊C8—531no—
321C2×#97C2×D5311yescharacteristic
322C2×C4×C8—531no—
323#389—531no—
324#389—531no—
325#389C2×D5311yescharacteristic
326#389C2×D5311yescharacteristic
327#388—1061no—
328#388—1061no—
329#387—1061no—
330#387—1061no—
331#382—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×C22⋊C4—531no—
340#376—1061no—
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#340C2×D5311yescharacteristic
350#340C2×D5311yescharacteristic
351#336—531no—
352#336—531no—
353C53#2262411yescharacteristic, complemented, Sylow
354C23×C4—531no—
355C23×C4—1061no—
356C2×M16—534no—
357C2×M16—1062no—
358C2×M16—1062no—
359C22×C8—531no—
360C22×C8—531no—
361C22×C8C22×D5311yescharacteristic
362C22×C8C22×D5311yescharacteristic
363#117—534no—
364#117—1062no—
365#117—2124no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—531no—
369C2×C4⋊C4—531no—
370C2×C4⋊C4—534no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—531no—
373C2×C4⋊C4—1062no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4—1061no—
377C2×C4⋊C4C22×D5311yescharacteristic
378C2×C4⋊C4C22×D5311yescharacteristic
379C2×C4⋊C4C22×D5311yescharacteristic
380C2×C4⋊C4C22×D5311yescharacteristic
381C2×C22⋊C4—531no—
382C2×C22⋊C4—531no—
383C2×C22⋊C4—1061no—
384C2×C22⋊C4—1061no—
385C2×C22⋊C4—1061no—
386C2×C42—531no—
387C2×C42—1061no—
388C2×C42—1061no—
389C2×C42—1061no—
390C2×C42—1061no—
391C2×C42—1061no—
392C2×C42C22×D5311yescharacteristic
393#107—22no—
394#107—534no—
395#107—534no—
396#106—534no—
397#106—1062no—
398#106C4×D5314yes—
399C22⋊C8—1062no—
400#97—22no—
401C4×C8—1062no—
402#95—21no—
403#95—21no—
404#95—531no—
405#95—531no—
406#95—1061no—
407#95—1061no—
408#95—1061no—
409#95—1061no—
410#95—1061no—
411#95—1061no—
412#95—1061no—
413#95—1061no—
414C24—531no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—531no—
420C22×C4—531no—
421C22×C4—532no—
422C22×C4—532no—
423C22×C4—531no—
424C22×C4—1064no—
425C22×C4—1061no—
426C22×C4—1061no—
427C22×C4—1061no—
428C22×C4—1061no—
429C22×C4—1061no—
430C22×C4—1061no—
431C22×C4—1061no—
432C22×C4—1061no—
433C22×C4—2122no—
434C22×C4—2122no—
435C22×C4D4×D5311yescharacteristic
436C22×C4D4×D5311yescharacteristic
437C22×C4C23×D5311yescharacteristic
438M16—1068no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—532no—
443C2×C8—1061no—
444C2×C8—1061no—
445C2×C8—1061no—
446C2×C8—1061no—
447C2×C8—1061no—
448C2×C8C2×C4×D5312yes—
449C2×C8#1171994712yes—
450C2×C8#1171994912yes—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—42no—
458C4⋊C4—81no—
459C4⋊C4—532no—
460C4⋊C4—532no—
461C4⋊C4—1064no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—1061no—
467C4⋊C4—1061no—
468C4⋊C4—4241no—
469C4⋊C4—2122no—
470C4⋊C4C2×C4×D5312yes—
471C4⋊C4C2×C4×D5312yes—
472C22⋊C4—1062no—
473C22⋊C4—1064no—
474C22⋊C4—4241no—
475C22⋊C4—2121no—
476C22⋊C4—2122no—
477C42—21no—
478C42—21no—
479C42—1064no—
480C42—1061no—
481C42—1061no—
482C42—2122no—
483C42—2122no—
484C42—2122no—
485C42—2122no—
486C23—532no—
487C23—1061no—
488C23—1061no—
489C23—1061no—
490C23—1061no—
491C23—1061no—
492C23—1061no—
493C23C2×D4×D5311yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—532no—
513C2×C4—532no—
514C2×C4—1061no—
515C2×C4—1061no—
516C2×C4—1061no—
517C2×C4—1061no—
518C2×C4—1062no—
519C2×C4—1062no—
520C2×C4—1062no—
521C2×C4—1062no—
522C2×C4—1061no—
523C2×C4—1062no—
524C2×C4—1062no—
525C2×C4—1061no—
526C2×C4—1061no—
527C2×C4—2128no—
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×C4—2121no—
538C2×C4C22×C4×D5311yescharacteristic
539C2×C4#41×D5311yescharacteristic
540C2×C4C2×#1171994711yescharacteristic
541C2×C4C2×#1171994911yescharacteristic
542C2×C4#1313326411yescharacteristic
543C2×C4#1313326511yescharacteristic
544C8—41no—
545C8—1062no—
546C8—2121no—
547C8#1313317514yescomplemented
548C22—1062no—
549C22—1062no—
550C22—1062no—
551C22—2121no—
552C22—2121no—
553C22—2121no—
554C22—2121no—
555C22C4×D4×D5311yescharacteristic
556C22#1681127311yescharacteristic
557C22#1681127611yescharacteristic
558C22AGL(1,ℤ/8)×D5311yescharacteristic
559C22#1681142211yescharacteristic
560C22#1681143311yescharacteristic
561C22#1681143911yescharacteristic
562C4—41no—
563C4—41no—
564C4—42no—
565C4—42no—
566C4—81no—
567C4—81no—
568C4—1062no—
569C4—1062no—
570C4—4241no—
571C4—4241no—
572C4—2122no—
573C4—2122no—
574C4—2121no—
575C4C2×#1313317511yescharacteristic
576C4#1681109911yescharacteristic
577C4#1681130711yescharacteristic
578C4#1681131111yescharacteristic
579C2—2122no—
580C2#2027758311yescharacteristic
581C2#2028352011yescharacteristic
582C2#2028355511yescharacteristic
583C2#2028715411yescharacteristic
584C2#2028722911yescharacteristic
585C2#2028728511yescharacteristic
586C2#2028730511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1944C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1944D53
hypocenterC53#22624
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1944

Complements

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

NH ≅ G/Ntype
4. #20275444C2semidirect
5. #20275304C2semidirect
6. #20275303C2semidirect
7. #20275290C2semidirect
8. #20275289C2semidirect
12. #20274612C2semidirect
13. #107×Dic53C2semidirect
15. C53×#1944C2semidirect
20. #16810662C4semidirect
251. C8×C53#117semidirect
289. #1944D53semidirect
353. C53#22624semidirect
547. C8#13133175semidirect

Maximal subgroups

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

subgroupindex
C2×#168106622
#202775992
#202775832
#202754442
#202753042
#202753032
#202752902
#202752892
#202748082
#202748042
#202747492
#202746122
#107×Dic532
#202745002
C53×#19442
#2262453