Finite Group Databasebeta

← #27425403

Subgroups — #27425403

Order 13568.

828 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810679C211yesmaximal, characteristic
2#20283062C211yesmaximal, characteristic
3C4×#13133193C211yesmaximal, characteristic
4#20276149C212yesmaximal
5#20275917C212yesmaximal
6#20275668C211yesmaximal, characteristic, complemented
7#20275474C211yesmaximal, characteristic, complemented
8#20275451C212yesmaximal, complemented
9#20275402C212yesmaximal, complemented
10#20274720C211yesmaximal, characteristic, complemented
11C2×C53×#475C211yesmaximal, characteristic, complemented
12C2×#13133193—21no—
13C2×#13133193C2211yescharacteristic
14C2×C42×D53C2211yescharacteristic
15#16810679—24no—
16C2×#13133151C2212yes—
17C2×C4⋊#88322C2211yescharacteristic
18C2×#13133141C2212yes—
19C2×#13133129C2212yes—
20C2×#13133127C2212yes—
21#16810530—21no—
22#16810530C2211yescharacteristic
23#16810519—21no—
24#16810519C2212yes—
25#16810519C2211yescharacteristic
26#16810504—21no—
27#16810494—21no—
28#16810493C2211yescharacteristic
29#16810480—21no—
30#16810480C2211yescharacteristic
31#16810480C2212yes—
32C4×#11719919—21no—
33C4×C4⋊Dic53C2211yescharacteristic
34C4×#11719921—21no—
35#16810386C2212yes—
36#16810385C2212yes—
37#16810385C2212yes—
38#16810385C2212yes—
39C2×C4⋊Q8×C53C2211yescharacteristic
40C2×C4×Q8×C53C2211yescharacteristic
41C53×#475—24no—
42C2×#106×C53C2212yes—
43C2×C4⋊C8×C53C2211yescharacteristic
44C2×#103×C53C2212yes—
45C22×C4×D53D411yescharacteristic
46C22×C4×D53D411yescharacteristic
47C22×C4×D53C2311yescharacteristic
48C22×#88322C2312yes—
49#13133193—22no—
50#13133193—22no—
51#13133193—44no—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53C2312yes—
55C2×C4⋊Dic53C2311yescharacteristic
56C2×C4⋊Dic53C2311yescharacteristic
57C2×#11719919—21no—
58C2×#11719919—22no—
59C2×#11719919—21no—
60C2×#11719921—21no—
61C2×#11719921—22no—
62C2×#11719921—21no—
63C2×C4×Dic53—21no—
64C2×C4×Dic53D411yescharacteristic
65C2×C4×Dic53D411yescharacteristic
66C2×C4×Dic53C2311yescharacteristic
67C42×D53—22no—
68C42×D53—22no—
69#13133151—24no—
70C4⋊#88322—22no—
71#13133141—24no—
72#13133129—24no—
73#13133127—24no—
74#13133124—21no—
75#13133124—21no—
76#13133124—21no—
77#13133124—21no—
78#13133124—22no—
79#13133124—22no—
80#13133124—21no—
81C22×Q8×C53—21no—
82C22×Q8×C53C2311yescharacteristic
83C22×C8×C53C2312yes—
84C4⋊Q8×C53—22no—
85C4⋊Q8×C53—22no—
86C4×Q8×C53—22no—
87C4×Q8×C53—22no—
88C2×C4⋊C4×C53—21no—
89C2×C4⋊C4×C53—21no—
90C2×C4⋊C4×C53C2311yescharacteristic
91C2×C4⋊C4×C53C2312yes—
92C2×C42×C53—21no—
93C2×C42×C53C2311yescharacteristic
94#106×C53—24no—
95C4⋊C8×C53D414yes—
96#103×C53—24no—
97C23×D53C2×D411yescharacteristic
98C2×Q8×C53—21no—
99C2×Q8×C53—21no—
100C2×Q8×C53—21no—
101C2×Q8×C53—21no—
102C2×Q8×C53—21no—
103C2×Q8×C53—42no—
104C2×Q8×C53—42no—
105C2×Q8×C53—42no—
106C2×Q8×C53#4112yes—
107C22×C4×C53—21no—
108C22×C4×C53—21no—
109C22×C4×C53—22no—
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53C2411yescharacteristic
113C22×Dic53—21no—
114C22×Dic53—21no—
115C22×Dic53—22no—
116C22×Dic53C2×D411yescharacteristic
117C22×Dic53C2×D411yescharacteristic
118C22×Dic53C2×D411yescharacteristic
119C2×C4×D53—22no—
120C2×C4×D53—22no—
121C2×C4×D53—22no—
122C2×C4×D53—22no—
123C2×C4×D53—22no—
124C2×C4×D53—21no—
125C2×C4×D53—21no—
126C2×C4×D53—21no—
127C2×C4×D53—21no—
128C2×C4×D53—21no—
129C2×C4×D53—21no—
130C2×C4×D53—41no—
131C2×C8×C53—22no—
132C2×C8×C53—22no—
133C2×C8×C53C2×D414yes—
134C4⋊C4×C53—22no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—22no—
138C4⋊C4×C53—42no—
139C4⋊C4×C53—42no—
140C42×C53—42no—
141C42×C53C2×D412yes—
142C42×C53#4112yes—
143#11719921—24no—
144#11719921—44no—
145#11719921—42no—
146C4⋊Dic53—21no—
147C4⋊Dic53—22no—
148C4⋊Dic53—22no—
149C4⋊Dic53—21no—
150C4⋊Dic53—21no—
151C4⋊Dic53—42no—
152C4⋊Dic53—42no—
153C4⋊Dic53#4112yes—
154#11719919—24no—
155#11719919—42no—
156#11719919—44no—
157C4×Dic53—22no—
158C4×Dic53—22no—
159C4×Dic53—21no—
160C4×Dic53—21no—
161C4×Dic53—42no—
162C2×#88322—22no—
163C2×#88322—22no—
164C2×#88322—22no—
165C23×C53C22×D411yescharacteristic
166C22×D53—21no—
167C22×D53—21no—
168C22×D53—21no—
169C22×D53—21no—
170C22×D53—21no—
171C22×D53—21no—
172C22×D53#13712yes—
173Q8×C53—22no—
174Q8×C53—22no—
175Q8×C53—41no—
176Q8×C53—41no—
177Q8×C53—84no—
178C2×C4×C53—21no—
179C2×C4×C53—21no—
180C2×C4×C53—22no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53—42no—
184C2×C4×C53—41no—
185C2×C4×C53—42no—
186C2×C4×C53—42no—
187C2×C4×C53—41no—
188C2×C4×C53—41no—
189C2×C4×C53#12112yes—
190C2×C4×C53#12112yes—
191C2×C4×C53AGL(1,ℤ/8)12yes—
192C2×C4×C53#13712yes—
193C2×C4×C53C22×D411yescharacteristic
194C2×C4×C53C2×#4111yescharacteristic
195C2×C4×C53C2×#4111yescharacteristic
196C2×C4×C53C2×#4111yescharacteristic
197C2×C4×C532−1+411yescharacteristic
198C2×C4×C532−1+411yescharacteristic
199C2×Dic53—21no—
200C2×Dic53—21no—
201C2×Dic53—21no—
202C2×Dic53—21no—
203C2×Dic53—22no—
204C2×Dic53—21no—
205C2×Dic53—21no—
206C2×Dic53—21no—
207C2×Dic53—21no—
208C2×Dic53—21no—
209C2×Dic53—41no—
210C2×Dic53—41no—
211C2×Dic53—42no—
212C2×Dic53—41no—
213C2×Dic53—41no—
214C2×Dic53—42no—
215C2×Dic53—41no—
216C2×Dic53—42no—
217C4×D53—44no—
218C4×D53—41no—
219C4×D53—41no—
220C4×D53—41no—
221C4×D53—41no—
222C4×D53—44no—
223C8×C53—24no—
224#88322—42no—
225#24710—531nomaximal, Sylow
226C22×C53C2×#12111yescharacteristic
227C22×C53#52511yescharacteristic
228C22×C53#53611yescharacteristic
229C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
230C22×C53C2×#13711yescharacteristic
231C22×C53C2×#13711yescharacteristic
232C22×C53#57511yescharacteristic
233C2×D53—22no—
234C2×D53—22no—
235C2×D53—22no—
236C2×D53—41no—
237C2×D53—41no—
238C2×D53—41no—
239C2×D53—41no—
240C4×C53—22no—
241C4×C53—42no—
242C4×C53—82no—
243C4×C53—81no—
244C4×C53#46814yes—
245C4×C53#54711yescharacteristic
246C4×C53#54811yescharacteristic
247C4×C53#55211yescharacteristic
248C4×C53#55611yescharacteristic
249Dic53—42no—
250Dic53—41no—
251Dic53—41no—
252Dic53—41no—
253Dic53—82no—
254Dic53—81no—
255C2×#475D5311yescharacteristic, complemented
256C2×#432—531no—
257#2857—531no—
258C4×C22⋊Q8—531no—
259#2077—532no—
260#2036—532no—
261#1988—531no—
262#1947—531no—
263#1943—532no—
264#1935—532no—
265#1803—531no—
266C2×C53#241511yescharacteristic
267C2×C53C2×#46811yescharacteristic
268C2×C53#305611yescharacteristic
269C2×C53#311311yescharacteristic
270C2×C53#311511yescharacteristic
271C2×C53#323711yescharacteristic
272C2×C53#323911yescharacteristic
273D53—42no—
274C2×C4⋊Q8C2×D5311yescharacteristic
275C2×C22⋊Q8—531no—
276C2×C22⋊Q8—1061no—
277C2×C4×Q8C2×D5311yescharacteristic
278C22×C42—531no—
279#475—24no—
280#432—1064no—
281C2×#107—532no—
282C2×#106C2×D5312yes—
283C2×C4⋊C8—531no—
284C2×C4⋊C8C2×D5311yescharacteristic
285C2×#103—532no—
286C2×#103C2×D5312yes—
287C2×C22⋊C8—532no—
288C2×#97—532no—
289#391—531no—
290#391—1061no—
291#389—532no—
292#389—531no—
293#389—1061no—
294#387—1061no—
295#385—1061no—
296#384—531no—
297#380—531no—
298#380—532no—
299#380—1061no—
300C4×C4⋊C4—531no—
301C4×C4⋊C4—1061no—
302C4×C22⋊C4—1061no—
303#340—532no—
304#340—532no—
305#340—532no—
306#340—532no—
307C53#2471011yescharacteristic, complemented, Sylow
308C22×Q8—21no—
309C22×Q8C22×D5311yescharacteristic
310C23×C4—531no—
311C23×C4—531no—
312C23×C4—531no—
313C22×C8—532no—
314C22×C8C22×D5312yes—
315C4⋊Q8—22no—
316C4⋊Q8—22no—
317C22⋊Q8—1062no—
318C22⋊Q8—1062no—
319C22⋊Q8—2124no—
320C4×Q8—22no—
321C4×Q8—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—531no—
325C2×C4⋊C4—531no—
326C2×C4⋊C4—532no—
327C2×C4⋊C4—1061no—
328C2×C4⋊C4—1062no—
329C2×C4⋊C4—1061no—
330C2×C4⋊C4—1061no—
331C2×C4⋊C4—1061no—
332C2×C4⋊C4C22×D5311yescharacteristic
333C2×C4⋊C4C22×D5312yes—
334C2×C22⋊C4—1061no—
335C2×C22⋊C4—1062no—
336C2×C22⋊C4—1061no—
337C2×C42—21no—
338C2×C42—531no—
339C2×C42—531no—
340C2×C42—531no—
341C2×C42—1062no—
342C2×C42—1062no—
343C2×C42—1061no—
344C2×C42C22×D5311yescharacteristic
345#107—1064no—
346#106—24no—
347C4⋊C8—1062no—
348C4⋊C8C22⋊D5314yes—
349#103—24no—
350#103—1064no—
351C22⋊C8—1064no—
352#97—1064no—
353#95—1061no—
354#95—1061no—
355#95—1061no—
356#95—1061no—
357#95—1062no—
358#95—1062no—
359#95—1061no—
360C24—531no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8#1171994912yes—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—531no—
374C22×C4—531no—
375C22×C4—531no—
376C22×C4—1062no—
377C22×C4—1061no—
378C22×C4—1062no—
379C22×C4—1062no—
380C22×C4—1062no—
381C22×C4—1062no—
382C22×C4—1061no—
383C22×C4—1061no—
384C22×C4—1061no—
385C22×C4—1061no—
386C22×C4—1061no—
387C22×C4—1061no—
388C22×C4—1061no—
389C22×C4—1062no—
390C22×C4—2121no—
391C22×C4D4×D5311yescharacteristic
392C22×C4D4×D5311yescharacteristic
393C22×C4C23×D5311yescharacteristic
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—1062no—
397C2×C8—1062no—
398C2×C8—1062no—
399C2×C8C2×C22⋊D5314yes—
400C4⋊C4—22no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—532no—
407C4⋊C4—1064no—
408C4⋊C4—1061no—
409C4⋊C4—1062no—
410C4⋊C4—1062no—
411C4⋊C4—1061no—
412C4⋊C4—1061no—
413C4⋊C4—2122no—
414C4⋊C4—2124no—
415C4⋊C4—2122no—
416C4⋊C4—2122no—
417C22⋊C4—1064no—
418C22⋊C4—2122no—
419C22⋊C4—2124no—
420C42—42no—
421C42—1062no—
422C42—1062no—
423C42—1061no—
424C42—1061no—
425C42—2122no—
426C42#1171994712yes—
427C42C2×C22⋊D5312yes—
428C23—532no—
429C23—1061no—
430C23—1061no—
431C23—1061no—
432C23—1061no—
433C23—1061no—
434C23—1061no—
435C23C2×D4×D5311yescharacteristic
436Q8—22no—
437Q8—22no—
438Q8—41no—
439Q8—41no—
440Q8—84no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—1061no—
453C2×C4—1061no—
454C2×C4—1061no—
455C2×C4—1062no—
456C2×C4—1061no—
457C2×C4—1061no—
458C2×C4—1061no—
459C2×C4—1061no—
460C2×C4—1061no—
461C2×C4—1061no—
462C2×C4—2124no—
463C2×C4—2122no—
464C2×C4—2121no—
465C2×C4—2122no—
466C2×C4—2121no—
467C2×C4—2122no—
468C2×C4—2121no—
469C2×C4—2121no—
470C2×C4—2121no—
471C2×C4—2121no—
472C2×C4—2121no—
473C2×C4—2121no—
474C2×C4—2121no—
475C2×C4—2124no—
476C2×C4#1313318812yes—
477C2×C4#1313318812yes—
478C2×C4#1313324412yes—
479C2×C4#1313324612yes—
480C2×C4C22×C22⋊D5311yescharacteristic
481C2×C4#41×D5311yescharacteristic
482C2×C4C2×#1171994711yescharacteristic
483C2×C4C2×#1171994911yescharacteristic
484C2×C4#1313326411yescharacteristic
485C2×C4#1313326511yescharacteristic
486C8—24no—
487C8—2122no—
488C22—1062no—
489C22—1062no—
490C22—1062no—
491C22—2121no—
492C22—2121no—
493C22—2121no—
494C22—2121no—
495C22C2×#1313318811yescharacteristic
496C22#1681113611yescharacteristic
497C22#1681119611yescharacteristic
498C22C2×#1313324411yescharacteristic
499C22#137×D5311yescharacteristic
500C22C2×#1313324611yescharacteristic
501C22#1681142211yescharacteristic
502C4—22no—
503C4—42no—
504C4—82no—
505C4—81no—
506C4—4242no—
507C4—4241no—
508C4—2121no—
509C4—2121no—
510C4—2121no—
511C4—2122no—
512C4#1681086614yes—
513C4#1681127211yescharacteristic
514C4#1681128411yescharacteristic
515C4#1681131511yescharacteristic
516C4#1681134411yescharacteristic
517C2—2122no—
518C2#2027814011yescharacteristic
519C2C2×#1681086611yescharacteristic
520C2#2028433911yescharacteristic
521C2#2028484711yescharacteristic
522C2#2028486911yescharacteristic
523C2#2028592811yescharacteristic
524C2#2028594911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#475C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenterC2×#475D53
hypocenterC53#24710
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
6. #20275668C2semidirect
7. #20275474C2semidirect
8. #20275451C2semidirect
9. #20275402C2semidirect
10. #20274720C2semidirect
11. C2×C53×#475C2semidirect
255. C2×#475D53semidirect
307. C53#24710semidirect

Maximal subgroups

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

subgroupindex
C2×#168106792
#202830622
C4×#131331932
#202761492
#202759172
#202756682
#202754742
#202754512
#202754022
#202747202
C2×C53×#4752
#2471053