Finite Group Databasebeta

← #27279799

Subgroups — #27279799

Order 13568.

708 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810679C211yesmaximal, characteristic, complemented
2#20283057C211yesmaximal, characteristic
3#20274692C211yesmaximal, characteristic, complemented
4C2×#16810367C211yesmaximal, characteristic, complemented
5#20272599C211yesmaximal, characteristic, complemented
6#20272599C211yesmaximal, characteristic, complemented
7C2×C53×#331C211yesmaximal, characteristic, complemented
8C2×#13133188—21no—
9C2×#13133188—21no—
10C2×#13133188C412yes—
11C2×C42×D53C2211yescharacteristic
12#16810679—22no—
13#16810679—22no—
14C2×C4⋊#88322C2211yescharacteristic
15C2×C4⋊#88322C2211yescharacteristic
16C2×#13133129—21no—
17C2×#13133129—21no—
18C2×#13133127—21no—
19#16810526—21no—
20#16810526—21no—
21#16810526C412yes—
22#16810493C2211yescharacteristic
23#16810377—21no—
24#16810367—22no—
25C2×#127×C53C2211yescharacteristic
26C2×C4⋊C8×C53C2211yescharacteristic
27C2×C4⋊C8×C53C2211yescharacteristic
28C53×#331—22no—
29C22×C22⋊D53—22no—
30C22×C4×D53C2×C411yescharacteristic
31C22×C4×D53D411yescharacteristic
32C22×C4×D53D411yescharacteristic
33C22×#88322—21no—
34C22×#88322—21no—
35#13133188—42no—
36#13133188—42no—
37#13133188—42no—
38#13133188—42no—
39#13133188—44no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53—21no—
42C2×C4⋊Dic53C2×C412yes—
43C2×C22⋊Dic53—22no—
44C2×C22⋊Dic53—22no—
45C2×C22⋊Dic53—22no—
46C2×C4×Dic53C2×C411yescharacteristic
47C2×C4×Dic53D411yescharacteristic
48C2×C4×Dic53D411yescharacteristic
49C42×D53—21no—
50C42×D53—21no—
51C42×D53—21no—
52C42×D53—21no—
53C4⋊#88322—21no—
54C4⋊#88322—21no—
55C4⋊#88322—21no—
56C4⋊#88322D412yescomplemented
57#13133129—24no—
58#13133129—42no—
59#13133127—24no—
60C22×D4×C53—21no—
61C22×D4×C53—21no—
62C22×D4×C53C2×C412yes—
63C22×C8×C53—21no—
64C22×C8×C53—21no—
65#127×C53—21no—
66#127×C53—21no—
67#127×C53—21no—
68#127×C53—21no—
69C2×C42×C53C2311yescharacteristic
70C4⋊C8×C53—21no—
71C4⋊C8×C53—21no—
72C4⋊C8×C53—21no—
73C4⋊C8×C53D412yescomplemented
74C24×C53—22no—
75C23×D53C22⋊C411yescharacteristic
76C2×D4×C53—22no—
77C2×D4×C53—22no—
78C2×D4×C53—42no—
79C2×D4×C53—42no—
80C2×D4×C53—41no—
81C2×D4×C53—41no—
82C2×D4×C53—41no—
83C2×D4×C53—41no—
84C2×D4×C53—41no—
85C2×D4×C53—41no—
86C2×D4×C53—41no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—41no—
90C2×D4×C53—41no—
91C2×D4×C53—41no—
92C22×C4×C53C22×C411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C2×C22⋊D53—44no—
96C2×C22⋊D53—44no—
97C2×C22⋊D53—44no—
98C22×Dic53—22no—
99C22×Dic53C22⋊C411yescharacteristic
100C22×Dic53C22⋊C411yescharacteristic
101C22×Dic53C22⋊C411yescharacteristic
102C2×C4×D53—22no—
103C2×C4×D53—21no—
104C2×C4×D53—21no—
105C2×C4×D53—21no—
106C2×C4×D53—21no—
107C2×C4×D53—21no—
108C2×C4×D53—21no—
109C2×C4×D53—21no—
110C2×C4×D53—21no—
111C2×C4×D53—21no—
112C2×C4×D53—21no—
113C2×C4×D53—21no—
114C2×C4×D53—21no—
115C2×C4×D53—21no—
116C2×C4×D53—21no—
117C2×C4×D53—41no—
118C2×C8×C53—22no—
119C2×C8×C53—41no—
120C2×C8×C53—41no—
121C2×C8×C53—41no—
122C2×C8×C53—41no—
123C2×C8×C53—41no—
124C42×C53C2×D411yescharacteristic
125C42×C53C2×D411yescharacteristic
126C42×C53#4111yescharacteristic
127C42×C53#4111yescharacteristic
128C22⋊Dic53—44no—
129C22⋊Dic53—44no—
130C22⋊Dic53—44no—
131C4⋊Dic53—41no—
132C4⋊Dic53—41no—
133C4⋊Dic53—41no—
134C4⋊Dic53—41no—
135C4⋊Dic53—42no—
136C4×Dic53—21no—
137C4×Dic53—21no—
138C4×Dic53—21no—
139C4×Dic53—21no—
140C4×Dic53—21no—
141C4×Dic53—21no—
142C2×#88322—22no—
143C2×#88322—41no—
144C2×#88322—41no—
145C2×#88322—41no—
146C2×#88322—41no—
147C2×#88322—41no—
148C23×C53—42no—
149C23×C53—42no—
150C23×C53—42no—
151C23×C53—42no—
152C23×C53—42no—
153C23×C53—42no—
154C23×C53—42no—
155C23×C53C2×C22⋊C411yescharacteristic
156C22×D53—21no—
157C22×D53—21no—
158C22×D53—21no—
159C22×D53—21no—
160C22×D53—21no—
161C22×D53—21no—
162C22×D53#10012yes—
163D4×C53—82no—
164D4×C53—82no—
165D4×C53—81no—
166D4×C53—81no—
167D4×C53—81no—
168D4×C53—81no—
169D4×C53—81no—
170D4×C53—81no—
171D4×C53—81no—
172D4×C53—81no—
173C2×C4×C53—21no—
174C2×C4×C53—21no—
175C2×C4×C53C4×D411yescharacteristic
176C2×C4×C53C4×D411yescharacteristic
177C2×C4×C53#12011yescharacteristic
178C2×C4×C53#12111yescharacteristic
179C2×C4×C53#12111yescharacteristic
180C2×C4×C53#12311yescharacteristic
181C2×C4×C53#13511yescharacteristic
182C2×C4×C53#13511yescharacteristic
183C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
184C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
185C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
186C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
187C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
188C2×C4×C53#13711yescharacteristic
189C22⋊D53—88no—
190C2×Dic53—21no—
191C2×Dic53—21no—
192C2×Dic53—21no—
193C2×Dic53—21no—
194C2×Dic53—21no—
195C2×Dic53—21no—
196C2×Dic53—21no—
197C2×Dic53—21no—
198C2×Dic53—21no—
199C2×Dic53—42no—
200C2×Dic53—42no—
201C2×Dic53—42no—
202C4×D53—42no—
203C4×D53—42no—
204C4×D53—41no—
205C4×D53—41no—
206C4×D53—41no—
207C4×D53—41no—
208C4×D53—41no—
209C4×D53—41no—
210C4×D53—41no—
211C4×D53—41no—
212C8×C53—42no—
213C8×C53—81no—
214#88322—42no—
215#88322—81no—
216#9142—531nomaximal, Sylow
217C22×C53—82no—
218C22×C53—82no—
219C22×C53—82no—
220C22×C53—82no—
221C22×C53—82no—
222C22×C53—82no—
223C22×C53—82no—
224C22×C53#38611yescharacteristic
225C22×C53C2×#10011yescharacteristic
226C22×C53#41311yescharacteristic
227C22×C53#41611yescharacteristic
228C22×C53#41711yescharacteristic
229C22×C53#41811yescharacteristic
230C22×C53#41811yescharacteristic
231C2×D53—22no—
232C2×D53—22no—
233C2×D53—22no—
234C2×D53—41no—
235C2×D53—41no—
236C2×D53—41no—
237C2×D53—41no—
238C4×C53—21no—
239C4×C53—21no—
240C4×C53#44911yescharacteristic
241C4×C53#44911yescharacteristic
242C4×C53#46811yescharacteristic
243C4×C53#46811yescharacteristic
244C4×C53#46911yescharacteristic
245C4×C53#46911yescharacteristic
246C4×C53#48211yescharacteristic
247C4×C53#48211yescharacteristic
248Dic53—41no—
249Dic53—41no—
250Dic53—41no—
251Dic53—82no—
252C2×#432—531no—
253#2856—531no—
254#1797—531no—
255C2×#331—531no—
256C2×#331D5311yescharacteristic, complemented
257#1296—531no—
258#1296—531no—
259C2×C53—162no—
260C2×C53#155111yescharacteristic
261C2×C53#155311yescharacteristic
262C2×C53#155511yescharacteristic
263C2×C53#155711yescharacteristic
264C2×C53#187511yescharacteristic
265C2×C53#189511yescharacteristic
266C2×C53#190011yescharacteristic
267D53—42no—
268C2×#127C2×D5311yescharacteristic
269C2×#121—532no—
270C2×#121—1061no—
271C2×#121—1061no—
272C22×C42—531no—
273#432—1062no—
274#432—1062no—
275C2×C4⋊C8—531no—
276C2×C4⋊C8—531no—
277C2×C4⋊C8C2×D5311yescharacteristic
278C2×C4⋊C8C2×D5311yescharacteristic
279C2×C22⋊C8—1061no—
280C2×C22⋊C8—1061no—
281C2×#97—1061no—
282#390—532no—
283#390—1061no—
284#390—1061no—
285#384—531no—
286#336—1061no—
287#331—22no—
288#331—1062no—
289C53#914211yescharacteristic, complemented, Sylow
290C22×D4—21no—
291C22×D4—21no—
292C22×D4—1062no—
293C22×D4C4×D5312yes—
294C23×C4—531no—
295C23×C4—531no—
296C23×C4—531no—
297C22×C8—21no—
298C22×C8—21no—
299C22×C8—1061no—
300C22×C8—1061no—
301#127—21no—
302#127—21no—
303#127—21no—
304#127—21no—
305#121—2124no—
306#121—2122no—
307#121—2122no—
308#121—2122no—
309#121—2122no—
310C2×C4⋊C4—532no—
311C2×C4⋊C4—1061no—
312C2×C4⋊C4—1061no—
313C2×C22⋊C4—1062no—
314C2×C22⋊C4—1062no—
315C2×C22⋊C4—1062no—
316C2×C42—531no—
317C2×C42—531no—
318C2×C42—531no—
319C2×C42—1061no—
320C2×C42—1061no—
321C2×C42—1061no—
322C2×C42—1061no—
323C2×C42C22×D5311yescharacteristic
324C4⋊C8—21no—
325C4⋊C8—21no—
326C4⋊C8—21no—
327C4⋊C8—532no—
328C4⋊C8—1061no—
329C4⋊C8—1061no—
330C4⋊C8—1061no—
331C4⋊C8C22⋊D5312yescomplemented
332C22⋊C8—1064no—
333C22⋊C8—2122no—
334#97—1064no—
335C24—22no—
336C24—531no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—2124no—
354C2×D4—2124no—
355C2×D4—2124no—
356C22×C4—531no—
357C22×C4—531no—
358C22×C4—531no—
359C22×C4—1062no—
360C22×C4—1061no—
361C22×C4—1061no—
362C22×C4—1061no—
363C22×C4—1061no—
364C22×C4—1061no—
365C22×C4—1061no—
366C22×C4—1061no—
367C22×C4—1061no—
368C22×C4—1061no—
369C22×C4—1061no—
370C22×C4—1061no—
371C22×C4—1061no—
372C22×C4—1061no—
373C22×C4—1061no—
374C22×C4—1062no—
375C22×C4—2121no—
376C22×C4C2×C4×D5311yescharacteristic
377C22×C4D4×D5311yescharacteristic
378C22×C4D4×D5311yescharacteristic
379C2×C8—22no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—1062no—
386C2×C8—2121no—
387C2×C8—2121no—
388C2×C8—2121no—
389C2×C8—2121no—
390C2×C8—2121no—
391C4⋊C4—2122no—
392C4⋊C4—2121no—
393C4⋊C4—2121no—
394C4⋊C4—2121no—
395C4⋊C4—2121no—
396C22⋊C4—2124no—
397C22⋊C4—2124no—
398C22⋊C4—2124no—
399C42—1061no—
400C42—1061no—
401C42—1061no—
402C42—1061no—
403C42—1061no—
404C42—1061no—
405C42#1171994511yescharacteristic
406C42D4×D5311yescharacteristic
407C42#1171994711yescharacteristic
408C42C2×C22⋊D5311yescharacteristic
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—532no—
417C23—1061no—
418C23—1061no—
419C23—1061no—
420C23—1061no—
421C23—1061no—
422C23—1061no—
423C23C22⋊C4×D5311yescharacteristic
424D4—82no—
425D4—82no—
426D4—81no—
427D4—81no—
428D4—81no—
429D4—81no—
430D4—81no—
431D4—81no—
432D4—81no—
433D4—81no—
434D4—4248no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—1061no—
438C2×C4—1061no—
439C2×C4—1061no—
440C2×C4—1061no—
441C2×C4—1061no—
442C2×C4—1061no—
443C2×C4—1061no—
444C2×C4—1061no—
445C2×C4—1061no—
446C2×C4—2122no—
447C2×C4—2122no—
448C2×C4—2122no—
449C2×C4—2122no—
450C2×C4—2122no—
451C2×C4—2121no—
452C2×C4—2121no—
453C2×C4—2121no—
454C2×C4—2121no—
455C2×C4—2121no—
456C2×C4—2121no—
457C2×C4—2121no—
458C2×C4—2121no—
459C2×C4#1313317711yescharacteristic
460C2×C4C4×C22⋊D5311yescharacteristic
461C2×C4#1313318511yescharacteristic
462C2×C4#1313318811yescharacteristic
463C2×C4#1313319011yescharacteristic
464C2×C4#1313319911yescharacteristic
465C2×C4#1313323611yescharacteristic
466C2×C4#1313323811yescharacteristic
467C2×C4#1313324111yescharacteristic
468C2×C4#1313324111yescharacteristic
469C2×C4#1313324111yescharacteristic
470C2×C4#1313324411yescharacteristic
471C2×C4#1313324411yescharacteristic
472C2×C4#1313324611yescharacteristic
473C8—42no—
474C8—81no—
475C8—4241no—
476C8—2122no—
477C22—82no—
478C22—82no—
479C22—82no—
480C22—82no—
481C22—82no—
482C22—82no—
483C22—82no—
484C22—1062no—
485C22—1062no—
486C22—1062no—
487C22—2121no—
488C22—2121no—
489C22—2121no—
490C22—2121no—
491C22#1681049911yescharacteristic
492C22#100×D5311yescharacteristic
493C22#1681060411yescharacteristic
494C22#1681061611yescharacteristic
495C22#1681062111yescharacteristic
496C22#1681062611yescharacteristic
497C22#1681062611yescharacteristic
498C4—21no—
499C4—21no—
500C4—4242no—
501C4—2121no—
502C4—2121no—
503C4—2121no—
504C4#1681076511yescharacteristic
505C4#1681076711yescharacteristic
506C4#1681086611yescharacteristic
507C4#1681086811yescharacteristic
508C4#1681087311yescharacteristic
509C4#1681087511yescharacteristic
510C4#1681094511yescharacteristic
511C4#1681094611yescharacteristic
512C2—162no—
513C2—2122no—
514C2#2027340711yescharacteristic
515C2#2027341711yescharacteristic
516C2#2027342711yescharacteristic
517C2#2027343411yescharacteristic
518C2#2027503411yescharacteristic
519C2#2027515011yescharacteristic
520C2#2027517511yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×#331C2
socleC23×C53C2×C22⋊C4
prime radical#20283057C2
cosocleC2×C42×C53C23
hypercenterC2×#331D53
hypocenterC53#9142
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#331

Complements

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

NH ≅ G/Ntype
1. C2×#16810679C2semidirect
3. #20274692C2semidirect
4. C2×#16810367C2semidirect
5. #20272599C2semidirect
6. #20272599C2semidirect
7. C2×C53×#331C2semidirect
56. C4⋊#88322D4semidirect
73. C4⋊C8×C53D4semidirect
256. C2×#331D53semidirect
289. C53#9142semidirect
331. C4⋊C8C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168106792
#202830572
#202746922
C2×#168103672
#202725992
C2×C53×#3312
#914253