Finite Group Databasebeta

← #27278880

Subgroups — #27278880

Order 13568.

664 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810904C211yesmaximal, characteristic
2C4⋊C4×D212C211yesmaximal, characteristic
3#20275676C211yesmaximal, characteristic, complemented
4#20275638C211yesmaximal, characteristic, complemented
5C2×#16810352C211yesmaximal, characteristic
6#20272599C211yesmaximal, characteristic, complemented
7C53×#1279C211yesmaximal, characteristic, complemented
8C2×#13133217C2211yescharacteristic
9C2×C4×D212—21no—
10C2×C4×D212C2211yescharacteristic
11C2×C4⋊C4×D53—21no—
12#16810904—22no—
13#16810904C414yescomplemented
14C2×#13133151—21no—
15C2×C4⋊#88322C2211yescharacteristic
16C2×#13133140—21no—
17C2×#13133140—21no—
18C2×#13133139—21no—
19#16810521—21no—
20#16810497—22no—
21#16810493C2211yescharacteristic
22#16810480—21no—
23#16810386—22no—
24#16810352—22no—
25C2×C4⋊C8×C53C2211yescharacteristic
26C2×C4×C8×C53C2211yescharacteristic
27C4×C4⋊C4×C53C2211yescharacteristic
28C22×Dic106—21no—
29C22×D212D411yescharacteristic
30C22×C4×D53—21no—
31C22×C4×D53—22no—
32C22×#88322—21no—
33#13133217—21no—
34#13133217—21no—
35#13133217—21no—
36#13133217C2×C412yescomplemented
37C4×D212—21no—
38C4×D212—21no—
39C4×D212—42no—
40C4×D212—42no—
41C4×D212C2×C412yes—
42C4×D212D412yes—
43C2×C4⋊Dic53—21no—
44C2×C4⋊Dic53—21no—
45C2×C4⋊Dic53—21no—
46C2×C4⋊Dic53—21no—
47C2×C4⋊Dic53D411yescharacteristic
48C2×C4⋊Dic53D412yes—
49C2×#11719919—22no—
50C4⋊C4×D53—42no—
51C4⋊C4×D53—42no—
52C2×#11719921—22no—
53C2×#11719921—21no—
54#13133151—24no—
55C4⋊#88322—21no—
56C4⋊#88322—21no—
57#13133140—42no—
58#13133140—42no—
59#13133139—24no—
60#13133124—22no—
61C22×C8×C53—21no—
62C22×C8×C53—21no—
63C2×C4⋊C4×C53D411yescharacteristic
64C2×C4⋊C4×C53D411yescharacteristic
65C2×C42×C53—21no—
66C2×C42×C53C2311yescharacteristic
67C4⋊C8×C53—21no—
68C4⋊C8×C53C2×C412yescomplemented
69C4×C8×C53—21no—
70C4×C8×C53—21no—
71#95×C53—21no—
72C23×D53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×Dic53—21no—
79C22×Dic53—22no—
80C22×Dic53—41no—
81C2×D212—21no—
82C2×D212—21no—
83C2×D212—21no—
84C2×D212C22⋊C412yes—
85C2×D212SD1612yes—
86C2×D212Q1612yes—
87C2×C4×D53—22no—
88C2×C4×D53—22no—
89C2×C4×D53—41no—
90C2×C4×D53—41no—
91C2×C4×D53—41no—
92C2×C4×D53—41no—
93C2×C4×D53—44no—
94C2×C4×D53—44no—
95C2×C4×D53—82no—
96C2×Dic106—22no—
97C2×Dic106—41no—
98C2×Dic106—41no—
99C2×Dic106—41no—
100C2×Dic106—41no—
101C2×Dic106—41no—
102C2×C8×C53—22no—
103C2×C8×C53—22no—
104C2×C8×C53—41no—
105C2×C8×C53—41no—
106C2×C8×C53—41no—
107C2×C8×C53—41no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53D812yes—
112C42×C53—41no—
113C42×C53—41no—
114C42×C53C22×C411yescharacteristic
115C42×C53C2×D411yescharacteristic
116C42×C53#4111yescharacteristic
117C42×C53#4111yescharacteristic
118#11719921—22no—
119#11719921—22no—
120#11719921—44no—
121C4⋊Dic53—21no—
122C4⋊Dic53—22no—
123C4⋊Dic53—22no—
124C4⋊Dic53—41no—
125C4⋊Dic53—41no—
126C4⋊Dic53—41no—
127C4⋊Dic53—41no—
128C4⋊Dic53—41no—
129C4⋊Dic53—41no—
130C4⋊Dic53—42no—
131C4⋊Dic53C22⋊C412yes—
132#11719919—44no—
133C2×#88322—41no—
134C2×#88322—41no—
135C2×#88322—41no—
136C23×C53#12011yescharacteristic
137C22×D53—22no—
138C22×D53—22no—
139C22×D53—22no—
140C22×D53—41no—
141C22×D53—41no—
142C22×D53—41no—
143C22×D53—41no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—22no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53C2×C22⋊C411yescharacteristic
153C2×C4×C53C4×D411yescharacteristic
154C2×C4×C53C4×D411yescharacteristic
155C2×C4×C53#12111yescharacteristic
156C2×C4×C53#12111yescharacteristic
157C2×C4×C53#12311yescharacteristic
158C2×C4×C53C2×D811yescharacteristic
159C2×C4×C53C2×SD1611yescharacteristic
160C2×C4×C53C2×Q1611yescharacteristic
161C2×C4×C53#13511yescharacteristic
162C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C53#13711yescharacteristic
166C2×Dic53—22no—
167C2×Dic53—41no—
168C2×Dic53—41no—
169C2×Dic53—42no—
170C2×Dic53—42no—
171C2×Dic53—42no—
172C2×Dic53—42no—
173C2×Dic53—81no—
174C2×Dic53—81no—
175D212—22no—
176D212—22no—
177D212—22no—
178D212#10314yescomplemented
179C4×D53—42no—
180C4×D53—42no—
181C4×D53—42no—
182C4×D53—42no—
183C4×D53—88no—
184Dic106—42no—
185Dic106—42no—
186Dic106—81no—
187Dic106—81no—
188C8×C53—42no—
189C8×C53—42no—
190#88322—81no—
191#8981—531nomaximal, Sylow
192C22×C53#38611yescharacteristic
193C22×C53#44711yescharacteristic
194C22×C53#44911yescharacteristic
195C22×C53#45011yescharacteristic
196C22×C53#45111yescharacteristic
197C22×C53#45511yescharacteristic
198C22×C53#45611yescharacteristic
199C2×D53—24no—
200C2×D53—42no—
201C2×D53—42no—
202C2×D53—42no—
203C2×D53—42no—
204C2×D53—42no—
205C2×D53—42no—
206C4×C53—21no—
207C4×C53—21no—
208C4×C53—42no—
209C4×C53—81no—
210C4×C53C2×#10311yescharacteristic
211C4×C53#41711yescharacteristic
212C4×C53C4×D811yescharacteristic
213C4×C53#44211yescharacteristic
214C4×C53#45911yescharacteristic
215C4×C53#46311yescharacteristic
216C4×C53#48311yescharacteristic
217C4×C53#48411yescharacteristic
218Dic53—42no—
219Dic53—82no—
220Dic53—82no—
221C2×#474—531no—
222D4×C4⋊C4—531no—
223#1990—531no—
224#1981—531no—
225C2×D4⋊C8—531no—
226#1296—531no—
227#1279D5311yescharacteristic, complemented
228C2×C53#163911yescharacteristic
229C2×C53#164411yescharacteristic
230C2×C53#164911yescharacteristic
231C2×C53#165411yescharacteristic
232C2×C53#187211yescharacteristic
233C2×C53#187911yescharacteristic
234C2×C53#188211yescharacteristic
235D53—44no—
236C2×C4⋊Q8—531no—
237C2×C4×D4—531no—
238C2×C4×D4—1061no—
239C22×C4⋊C4—1061no—
240#474—534no—
241#474—1062no—
242C2×#107—1061no—
243C2×C4⋊C8—531no—
244C2×C4⋊C8C2×D5311yescharacteristic
245C2×#103—1061no—
246C2×#103—1061no—
247C2×#102—1061no—
248C2×C4×C8C2×D5311yescharacteristic
249#389—1061no—
250#385—1062no—
251#384—531no—
252#380—1061no—
253C4×C4⋊C4C2×D5311yescharacteristic
254#340—1062no—
255D4⋊C8—1062no—
256C53#898111yescharacteristic, complemented, Sylow
257C22×Q8—1061no—
258C22×D4—531no—
259C23×C4—1062no—
260C23×C4—1061no—
261C22×C8—21no—
262C22×C8—21no—
263C22×C8—1061no—
264C4⋊Q8—532no—
265C4⋊Q8—1061no—
266C4⋊Q8—1061no—
267C4⋊Q8—1061no—
268C4×D4—532no—
269C4×D4—532no—
270C4×D4—1061no—
271C4×D4—1061no—
272C4×D4—2122no—
273C4×D4—2122no—
274C2×C4⋊C4—531no—
275C2×C4⋊C4—532no—
276C2×C4⋊C4—1061no—
277C2×C4⋊C4—1061no—
278C2×C4⋊C4—1062no—
279C2×C4⋊C4—1061no—
280C2×C4⋊C4—1061no—
281C2×C4⋊C4—2122no—
282C2×C4⋊C4—2122no—
283C2×C4⋊C4D21211yescharacteristic
284C2×C4⋊C4D21211yescharacteristic
285C2×C22⋊C4—1061no—
286C2×C22⋊C4—1062no—
287C2×C42—21no—
288C2×C42C22×D5311yescharacteristic
289#107—1064no—
290C4⋊C8—21no—
291C4⋊C8—1061no—
292C4⋊C8—1061no—
293C4⋊C8C4×D5312yescomplemented
294#103—2122no—
295#103—2122no—
296#102—1064no—
297C4×C8—21no—
298C4×C8—21no—
299#95—21no—
300#95—1062no—
301C24—1061no—
302C2×Q8—1062no—
303C2×Q8—2121no—
304C2×Q8—2121no—
305C2×Q8—2121no—
306C2×Q8—2121no—
307C2×Q8—2121no—
308C2×D4—532no—
309C2×D4—532no—
310C2×D4—532no—
311C2×D4—1061no—
312C2×D4—1061no—
313C2×D4—1061no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—1061no—
317C22×C4—1062no—
318C22×C4—1062no—
319C22×C4—1062no—
320C22×C4—4242no—
321C22×C4—2124no—
322C22×C4—2124no—
323C22×C4—2121no—
324C22×C4—2121no—
325C22×C4—2121no—
326C22×C4—2121no—
327C22×C4—2121no—
328C22×C4C2×D21211yescharacteristic
329C22×C4D4×D5311yescharacteristic
330C22×C4D4×D5311yescharacteristic
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—2121no—
338C2×C8—2121no—
339C2×C8—2121no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—532no—
344C4⋊C4—1061no—
345C4⋊C4—1062no—
346C4⋊C4—1062no—
347C4⋊C4—2122no—
348C4⋊C4—2124no—
349C4⋊C4—2121no—
350C4⋊C4—2121no—
351C4⋊C4—2121no—
352C4⋊C4—2121no—
353C4⋊C4—2121no—
354C4⋊C4—2121no—
355C4⋊C4D42412yes—
356C22⋊C4—1062no—
357C22⋊C4—1062no—
358C22⋊C4—2124no—
359C42—41no—
360C42—41no—
361C42C2×C4×D5311yescharacteristic
362C42#1171994511yescharacteristic
363C42D4×D5311yescharacteristic
364C42#1171994911yescharacteristic
365C23—1062no—
366C23—1062no—
367C23—1062no—
368C23—2121no—
369C23—2121no—
370C23—2121no—
371C23—2121no—
372C23#1313318711yescharacteristic
373Q8—4241no—
374Q8—4241no—
375Q8—2122no—
376Q8—2122no—
377D4—534no—
378D4—1062no—
379D4—1062no—
380D4—1062no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—1062no—
390C2×C4—4248no—
391C2×C4—4241no—
392C2×C4—4241no—
393C2×C4—2122no—
394C2×C4—2122no—
395C2×C4—2122no—
396C2×C4—2122no—
397C2×C4—2121no—
398C2×C4—2121no—
399C2×C4—2122no—
400C2×C4—2122no—
401C2×C4—2122no—
402C2×C4—2122no—
403C2×C4C22⋊C4×D5311yescharacteristic
404C2×C4#1313317911yescharacteristic
405C2×C4C4×D21211yescharacteristic
406C2×C4#1313319011yescharacteristic
407C2×C4#1313319211yescharacteristic
408C2×C4#1313320211yescharacteristic
409C2×C4C2×D42411yescharacteristic
410C2×C4SD16×D5311yescharacteristic
411C2×C4Q16×D5311yescharacteristic
412C2×C4#1313323911yescharacteristic
413C2×C4#1313324111yescharacteristic
414C2×C4#1313324511yescharacteristic
415C2×C4#1313324511yescharacteristic
416C2×C4#1313324611yescharacteristic
417C8—42no—
418C8—42no—
419C8—4241no—
420C22—1064no—
421C22—2122no—
422C22—2122no—
423C22—2122no—
424C22—2122no—
425C22—2122no—
426C22—2122no—
427C22#1681050111yescharacteristic
428C22#1681075911yescharacteristic
429C22#1681077011yescharacteristic
430C22#1681077511yescharacteristic
431C22#1681078011yescharacteristic
432C22#1681080211yescharacteristic
433C22#1681080711yescharacteristic
434C4—21no—
435C4—21no—
436C4—42no—
437C4—81no—
438C4—4242no—
439C4—4242no—
440C4—2122no—
441C4#103×D5311yescharacteristic
442C4#1681062111yescharacteristic
443C4C4×D42411yescharacteristic
444C4#1681073611yescharacteristic
445C4#1681081911yescharacteristic
446C4#1681083911yescharacteristic
447C4#1681095511yescharacteristic
448C4#1681096011yescharacteristic
449C2—2124no—
450C2#2027389611yescharacteristic
451C2#2027393111yescharacteristic
452C2#2027396611yescharacteristic
453C2#2027400111yescharacteristic
454C2#2027502111yescharacteristic
455C2#2027506211yescharacteristic
456C2#2027508511yescharacteristic
subgroupG/H
centerC23#13133187
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1279C2
socleC23×C53#120
prime radicalC22×D212D4
cosocleC2×C42×C53C23
hypercenter#1279D53
hypocenterC53#8981
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
3. #20275676C2semidirect
4. #20275638C2semidirect
6. #20272599C2semidirect
7. C53×#1279C2semidirect
13. #16810904C4semidirect
36. #13133217C2×C4semidirect
68. C4⋊C8×C53C2×C4semidirect
178. D212#103semidirect
227. #1279D53semidirect
256. C53#8981semidirect
293. C4⋊C8C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109042
C4⋊C4×D2122
#202756762
#202756382
C2×#168103522
#202725992
C53×#12792
#898153