Finite Group Databasebeta

← #27465788

Subgroups — #27465788

Order 13568.

922 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810941C211yesmaximal, characteristic
2C2×#16810936C211yesmaximal, characteristic
3C4×#13133191C211yesmaximal, characteristic
4#20276197C212yesmaximal, complemented
5#20276129C212yesmaximal, complemented
6#20275481C212yesmaximal, complemented
7#20274805C211yesmaximal, characteristic, complemented
8#20274801C211yesmaximal, characteristic, complemented
9#20274722C212yesmaximal, complemented
10#20274695C211yesmaximal, characteristic, complemented
11C2×C53×#382C211yesmaximal, characteristic, complemented
12C2×#13133191C2211yescharacteristic
13C2×C4×D212C2211yescharacteristic
14C2×C4×C22⋊D53—21no—
15#16810941—24no—
16#16810936—24no—
17C2×#13133150C2212yes—
18C2×#13133148C2212yes—
19C2×C4⋊#88322C2212yes—
20C2×#13133137C2212yes—
21C2×#13133137C2212yes—
22C2×C22⋊#88322C2211yescharacteristic
23C2×C22⋊#88322C2211yescharacteristic
24#16810525C2211yescharacteristic
25#16810521C2211yescharacteristic
26#16810514—21no—
27#16810502—21no—
28#16810483C2211yescharacteristic
29C4×C4⋊Dic53C2211yescharacteristic
30C4×#11719921—21no—
31#16810385C2212yes—
32#16810385C2212yes—
33#16810385C2212yes—
34#16810385C2212yes—
35#16810378C2212yes—
36C22×C4⋊C4×C53C2211yescharacteristic
37C22×C4⋊C4×C53C2211yescharacteristic
38C22×C42×C53C2211yescharacteristic
39C53×#382—21no—
40C53×#382—21no—
41C53×#382C2212yes—
42C53×#382C2212yes—
43C2×#95×C53—21no—
44C2×#95×C53—21no—
45C22×D212C2311yescharacteristic
46C22×C22⋊D53—21no—
47C22×C4×D53—21no—
48C22×#88322C2312yes—
49C22×#88322C2312yes—
50#13133191—22no—
51#13133191D414yes—
52C4×D212—24no—
53C4×C22⋊D53—44no—
54C2×C4⋊Dic53C2311yescharacteristic
55C2×C4⋊Dic53C2311yescharacteristic
56C2×C4⋊Dic53C2311yescharacteristic
57C2×#11719919—21no—
58C2×C22⋊Dic53—21no—
59C2×#11719921—21no—
60C2×#11719921—21no—
61C2×#11719921—21no—
62C2×#11719921—21no—
63C2×C4×Dic53—21no—
64#13133150—24no—
65#13133148—24no—
66C4⋊#88322—24no—
67#13133137—24no—
68#13133137—24no—
69C22⋊#88322—22no—
70C22⋊#88322—22no—
71#13133124—21no—
72#13133124—21no—
73#13133124—21no—
74C23×C4×C53—21no—
75C23×C4×C53—21no—
76C23×C4×C53D411yescharacteristic
77C23×C4×C53D411yescharacteristic
78C23×C4×C53C2311yescharacteristic
79C2×C4⋊C4×C53—21no—
80C2×C4⋊C4×C53—21no—
81C2×C4⋊C4×C53—22no—
82C2×C4⋊C4×C53—22no—
83C2×C4⋊C4×C53—22no—
84C2×C4⋊C4×C53—22no—
85C2×C4⋊C4×C53C2312yes—
86C2×C4⋊C4×C53C2312yes—
87C2×C42×C53—24no—
88C2×C42×C53D411yescharacteristic
89C2×C42×C53D411yescharacteristic
90C2×C42×C53C2311yescharacteristic
91C2×C42×C53C2311yescharacteristic
92#95×C53—22no—
93#95×C53—22no—
94#95×C53—22no—
95#95×C53—22no—
96C24×C53C2×D411yescharacteristic
97C23×D53—21no—
98C22×C4×C53—22no—
99C22×C4×C53—22no—
100C22×C4×C53—22no—
101C22×C4×C53—22no—
102C22×C4×C53—22no—
103C22×C4×C53—22no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—41no—
107C22×C4×C53—41no—
108C22×C4×C53—42no—
109C22×C4×C53—42no—
110C22×C4×C53—42no—
111C22×C4×C53—42no—
112C22×C4×C53—42no—
113C22×C4×C53—42no—
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×D412yes—
118C22×C4×C53C2×D411yescharacteristic
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53#4112yes—
121C22×C4×C53C2411yescharacteristic
122C2×C22⋊D53—24no—
123C2×C22⋊D53—42no—
124C2×C22⋊D53—42no—
125C22×Dic53—21no—
126C22×Dic53—21no—
127C22×Dic53—21no—
128C2×D212—22no—
129C2×D212—22no—
130C2×D212—21no—
131C2×D212C2×D412yes—
132C2×C4×D53—42no—
133C2×C4×D53—42no—
134C2×C4×D53—42no—
135C4⋊C4×C53—22no—
136C4⋊C4×C53—22no—
137C4⋊C4×C53—22no—
138C4⋊C4×C53—22no—
139C4⋊C4×C53—42no—
140C4⋊C4×C53—42no—
141C42×C53—22no—
142C42×C53—22no—
143C42×C53—22no—
144C42×C53#4114yes—
145C22⋊Dic53—42no—
146#11719921—24no—
147#11719921—42no—
148#11719921—42no—
149#11719921—42no—
150C4⋊Dic53—21no—
151C4⋊Dic53—22no—
152C4⋊Dic53—22no—
153C4⋊Dic53C2×D412yes—
154#11719919—42no—
155C4×Dic53—42no—
156C2×#88322—22no—
157C2×#88322—22no—
158C2×#88322—22no—
159C2×#88322—22no—
160C2×#88322—22no—
161C2×#88322—22no—
162C23×C53—21no—
163C23×C53—21no—
164C23×C53—21no—
165C23×C53—21no—
166C23×C53#12112yes—
167C23×C53#13512yes—
168C23×C53AGL(1,ℤ/8)12yes—
169C23×C53C22×D411yescharacteristic
170C22×D53—22no—
171C22×D53—41no—
172C22×D53—41no—
173C22×D53—41no—
174C22×D53—41no—
175C22×D53—41no—
176C22×D53—41no—
177C2×C4×C53—21no—
178C2×C4×C53—21no—
179C2×C4×C53—21no—
180C2×C4×C53—21no—
181C2×C4×C53—21no—
182C2×C4×C53—21no—
183C2×C4×C53—24no—
184C2×C4×C53—24no—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—21no—
188C2×C4×C53—21no—
189C2×C4×C53—21no—
190C2×C4×C53—21no—
191C2×C4×C53—44no—
192C2×C4×C53—44no—
193C2×C4×C53—42no—
194C2×C4×C53—42no—
195C2×C4×C53—42no—
196C2×C4×C53—42no—
197C2×C4×C53—42no—
198C2×C4×C53—42no—
199C2×C4×C53—84no—
200C2×C4×C53—84no—
201C2×C4×C53#12112yes—
202C2×C4×C53#12312yes—
203C2×C4×C53#12312yes—
204C2×C4×C53C22×D411yescharacteristic
205C2×C4×C53C2×#4111yescharacteristic
206C2×C4×C53C2×#4112yes—
207C2×C4×C532−1+412yes—
208C22⋊D53—48no—
209C2×Dic53—22no—
210C2×Dic53—41no—
211C2×Dic53—41no—
212C2×Dic53—41no—
213C2×Dic53—41no—
214C2×Dic53—41no—
215C2×Dic53—41no—
216C2×Dic53—41no—
217C2×Dic53—41no—
218D212—24no—
219D212—42no—
220C4×D53—84no—
221#88322—42no—
222#88322—42no—
223#27651—531nomaximal, Sylow
224C22×C53—22no—
225C22×C53—22no—
226C22×C53—22no—
227C22×C53—22no—
228C22×C53—22no—
229C22×C53—22no—
230C22×C53#46314yes—
231C22×C53C2×#12111yescharacteristic
232C22×C53C2×#12311yescharacteristic
233C22×C53#53611yescharacteristic
234C22×C53C2×#13511yescharacteristic
235C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
236C22×C53#57511yescharacteristic
237C22×C53#57511yescharacteristic
238C2×D53—42no—
239C2×D53—42no—
240C2×D53—42no—
241C2×D53—81no—
242C2×D53—81no—
243C2×D53—81no—
244C2×D53—81no—
245C4×C53—22no—
246C4×C53—22no—
247C4×C53—22no—
248C4×C53—41no—
249C4×C53—41no—
250C4×C53—82no—
251C4×C53—82no—
252C4×C53#54712yes—
253C4×C53#55212yes—
254Dic53—42no—
255Dic53—81no—
256Dic53—81no—
257C2×#481—531no—
258C2×#480—531no—
259C4×#121—531no—
260C2×#382D5311yescharacteristic, complemented
261#2085—532no—
262#2073—532no—
263#1949—532no—
264#1824—531no—
265#1823—531no—
266#1804—532no—
267#1797—531no—
268C2×C53—24no—
269C2×C53#248611yescharacteristic
270C2×C53C2×#46311yescharacteristic
271C2×C53#304011yescharacteristic
272C2×C53#309111yescharacteristic
273C2×C53#309411yescharacteristic
274C2×C53#323211yescharacteristic
275C2×C53#323411yescharacteristic
276D53—82no—
277C2×#121—531no—
278C2×C4×D4—531no—
279C2×C4×D4—1061no—
280C22×C4⋊C4C2×D5311yescharacteristic
281C22×C4⋊C4C2×D5311yescharacteristic
282C22×C42C2×D5311yescharacteristic
283#481—1064no—
284#480—1064no—
285C2×#107—532no—
286C2×#106—532no—
287C2×C4⋊C8—532no—
288C2×#102—532no—
289C2×#102—532no—
290C2×C22⋊C8—531no—
291C2×C22⋊C8—531no—
292#390—531no—
293#389—531no—
294#388—1061no—
295#386—1061no—
296#382—21no—
297#382—21no—
298#382C2×D5312yes—
299#382C2×D5312yes—
300#380—531no—
301C4×C4⋊C4—531no—
302C4×C22⋊C4—1061no—
303C2×#95—21no—
304C2×#95—21no—
305#340—532no—
306#340—532no—
307#340—532no—
308#340—532no—
309#336—532no—
310C53#2765111yescharacteristic, complemented, Sylow
311C22×D4—531no—
312C22×D4—1061no—
313C23×C4—21no—
314C23×C4—21no—
315C23×C4—1061no—
316C23×C4C22⋊D5311yescharacteristic
317C23×C4C22⋊D5311yescharacteristic
318C23×C4C22×D5311yescharacteristic
319C22×C8—532no—
320C22×C8—532no—
321#121—534no—
322#121—1062no—
323C4×D4—1064no—
324C4×D4—2124no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—531no—
332C2×C4⋊C4—531no—
333C2×C4⋊C4—531no—
334C2×C4⋊C4—1061no—
335C2×C4⋊C4C22×D5312yes—
336C2×C4⋊C4C22×D5312yes—
337C2×C22⋊C4—1061no—
338C2×C22⋊C4—1061no—
339C2×C22⋊C4—1061no—
340C2×C22⋊C4—1061no—
341C2×C22⋊C4—1061no—
342C2×C42—24no—
343C2×C42—1061no—
344C2×C42C22⋊D5311yescharacteristic
345C2×C42C22⋊D5311yescharacteristic
346C2×C42C22×D5311yescharacteristic
347C2×C42C22×D5311yescharacteristic
348#107—1064no—
349#106—1064no—
350C4⋊C8—1064no—
351#102—1064no—
352#102—1064no—
353C22⋊C8—1062no—
354C22⋊C8—1062no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—1061no—
360#95—1061no—
361#95—1061no—
362C24—1061no—
363C24C2×C22⋊D5311yescharacteristic
364C2×D4—532no—
365C2×D4—1064no—
366C2×D4—1061no—
367C2×D4—1062no—
368C2×D4—1062no—
369C2×D4—2122no—
370C2×D4—2122no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—1061no—
388C22×C4—1061no—
389C22×C4—1061no—
390C22×C4—2122no—
391C22×C4—2122no—
392C22×C4—2122no—
393C22×C4D4×D5312yes—
394C22×C4#1171994712yes—
395C22×C4C2×C22⋊D5311yescharacteristic
396C22×C4C2×C22⋊D5311yescharacteristic
397C22×C4C2×C22⋊D5311yescharacteristic
398C22×C4C2×C22⋊D5311yescharacteristic
399C22×C4C2×C22⋊D5311yescharacteristic
400C22×C4C23×D5311yescharacteristic
401C2×C8—1062no—
402C2×C8—1062no—
403C2×C8—1062no—
404C2×C8—1062no—
405C2×C8—1062no—
406C2×C8—1062no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—532no—
414C4⋊C4—1061no—
415C4⋊C4—1062no—
416C4⋊C4—1062no—
417C4⋊C4—2122no—
418C22⋊C4—1064no—
419C22⋊C4—2122no—
420C22⋊C4—2122no—
421C22⋊C4—2122no—
422C22⋊C4—2122no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—2122no—
427C42#1171994514yes—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—1062no—
433C23—2121no—
434C23—2121no—
435C23—2121no—
436C23—2121no—
437C23—2121no—
438C23—2121no—
439C23#1313318812yes—
440C23#1313323712yes—
441C23#1313324212yes—
442C23C22×C22⋊D5311yescharacteristic
443D4—1064no—
444D4—2128no—
445D4—2122no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—84no—
469C2×C4—84no—
470C2×C4—1062no—
471C2×C4—4244no—
472C2×C4—2121no—
473C2×C4—2121no—
474C2×C4—2121no—
475C2×C4—2121no—
476C2×C4—2121no—
477C2×C4—2121no—
478C2×C4—2121no—
479C2×C4—2121no—
480C2×C4#1313318812yes—
481C2×C4#1313320012yes—
482C2×C4#1313320012yes—
483C2×C4C2×D4×D5311yescharacteristic
484C2×C4C2×#1171994711yescharacteristic
485C2×C4C2×#1171994512yes—
486C2×C4#1313326512yes—
487C8—2122no—
488C8—2122no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—4241no—
496C22—4241no—
497C22—4241no—
498C22—4241no—
499C22—2122no—
500C22—2122no—
501C22—2122no—
502C22#1681083514yes—
503C22C2×#1313318811yescharacteristic
504C22C2×#1313320011yescharacteristic
505C22#1681119711yescharacteristic
506C22C2×#1313323711yescharacteristic
507C22C2×#1313324211yescharacteristic
508C22#1681142111yescharacteristic
509C22#1681142111yescharacteristic
510C4—22no—
511C4—22no—
512C4—22no—
513C4—41no—
514C4—41no—
515C4—82no—
516C4—82no—
517C4—4241no—
518C4—4241no—
519C4—2122no—
520C4#1681127412yes—
521C4#1681130812yes—
522C2—24no—
523C2—4242no—
524C2#2027901711yescharacteristic
525C2C2×#1681083511yescharacteristic
526C2#2028420511yescharacteristic
527C2#2028462811yescharacteristic
528C2#2028465511yescharacteristic
529C2#2028588311yescharacteristic
530C2#2028590511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#382C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenterC2×#382D53
hypocenterC53#27651
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
4. #20276197C2semidirect
5. #20276129C2semidirect
6. #20275481C2semidirect
7. #20274805C2semidirect
8. #20274801C2semidirect
9. #20274722C2semidirect
10. #20274695C2semidirect
11. C2×C53×#382C2semidirect
260. C2×#382D53semidirect
310. C53#27651semidirect

Maximal subgroups

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

subgroupindex
C2×#168109412
C2×#168109362
C4×#131331912
#202761972
#202761292
#202754812
#202748052
#202748012
#202747222
#202746952
C2×C53×#3822
#2765153