Finite Group Databasebeta

← #27279385

Subgroups — #27279385

Order 13568.

738 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×#16810996C211yesmaximal, characteristic
2#20277779C211yesmaximal, characteristic
3#20275663C211yesmaximal, characteristic, complemented
4#20275662C211yesmaximal, characteristic, complemented
5C2×#16810366C211yesmaximal, characteristic
6#20272555C211yesmaximal, characteristic, complemented
7C53×#1279C211yesmaximal, characteristic, complemented
8C22×#11719913—22no—
9C2×#13133217C2211yescharacteristic
10C2×#13133214C2211yescharacteristic
11C2×C4×D212—21no—
12C2×#13133179—21no—
13#16810996—22no—
14#16810996C414yescomplemented
15C2×C4⋊#88322C2211yescharacteristic
16#16810525—21no—
17#16810524—21no—
18#16810501—22no—
19C4×C4⋊Dic53C2211yescharacteristic
20#16810386—21no—
21#16810386—21no—
22#16810385—21no—
23#16810385—21no—
24#16810366—22no—
25C2×C4⋊C8×C53C2211yescharacteristic
26C2×C4×C8×C53C2211yescharacteristic
27C4×C4⋊C4×C53C2211yescharacteristic
28C22×Dic106—21no—
29C22×D212—21no—
30C22×D212—21no—
31C22×D212D412yes—
32C22×C4×D53—22no—
33C2×#11719913—28no—
34C2×#11719913—44no—
35C2×#11719913—44no—
36C22×#88322—21no—
37#13133217—21no—
38#13133217—21no—
39#13133217—21no—
40#13133217C2×C412yescomplemented
41#13133214—21no—
42#13133214—21no—
43#13133214C2×C412yes—
44#13133214D412yes—
45C4×D212—42no—
46C4×D212—42no—
47#13133179—42no—
48#13133179—42no—
49C2×C4⋊Dic53—21no—
50C2×C4⋊Dic53—21no—
51C2×C4⋊Dic53D411yescharacteristic
52C2×C4⋊Dic53D411yescharacteristic
53C2×#11719921—22no—
54C2×#11719921—22no—
55C2×#11719921—22no—
56C2×C4×Dic53—21no—
57C4⋊#88322—21no—
58C4⋊#88322—21no—
59#13133124—21no—
60C22×C8×C53—21no—
61C22×C8×C53—21no—
62C2×C4⋊C4×C53D411yescharacteristic
63C2×C4⋊C4×C53D411yescharacteristic
64C2×C42×C53—21no—
65C2×C42×C53C2311yescharacteristic
66C4⋊C8×C53—21no—
67C4⋊C8×C53—21no—
68C4×C8×C53—21no—
69C4×C8×C53C2×C412yescomplemented
70#95×C53—21no—
71C23×D53—22no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—41no—
80C2×D212—22no—
81C2×D212—22no—
82C2×D212—22no—
83C2×D212—22no—
84C2×D212—22no—
85C2×D212—41no—
86C2×D212—41no—
87C2×D212—41no—
88C2×D212—41no—
89C2×D212—41no—
90C2×D212—41no—
91C2×D212—41no—
92C2×D212—41no—
93C2×D212—42no—
94C2×D212—42no—
95C2×D212C22⋊C414yes—
96C2×C4×D53—44no—
97C2×C4×D53—44no—
98C2×C4×D53—82no—
99C2×Dic106—22no—
100C2×Dic106—42no—
101C2×Dic106—42no—
102C2×Dic106—41no—
103C2×C8×C53—22no—
104C2×C8×C53—22no—
105C2×C8×C53—22no—
106C2×C8×C53—41no—
107C2×C8×C53—41no—
108C2×C8×C53—41no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53SD1612yescomplemented
113C42×C53—41no—
114C42×C53—41no—
115C42×C53C22×C411yescharacteristic
116C42×C53C2×D411yescharacteristic
117C42×C53#4111yescharacteristic
118C42×C53#4111yescharacteristic
119#11719921—44no—
120#11719921—44no—
121#11719921—82no—
122C4⋊Dic53—21no—
123C4⋊Dic53—22no—
124C4⋊Dic53—22no—
125C4⋊Dic53—21no—
126C4⋊Dic53—21no—
127C4⋊Dic53—41no—
128C4⋊Dic53—41no—
129C4⋊Dic53SD1612yescomplemented
130C4×Dic53—41no—
131C4×Dic53—41no—
132C2×#88322—41no—
133C2×#88322—41no—
134C2×#88322—41no—
135#11719913—416no—
136C23×C53#12011yescharacteristic
137C22×D53—24no—
138C22×D53—42no—
139C22×D53—42no—
140C22×D53—42no—
141C22×D53—42no—
142C22×D53—42no—
143C22×D53—42no—
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×SD1611yescharacteristic
159C2×C4×C53C2×SD1611yescharacteristic
160C2×C4×C53#13511yescharacteristic
161C2×C4×C53#13511yescharacteristic
162C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C53#13711yescharacteristic
165C2×C4×C53#13711yescharacteristic
166C2×Dic53—22no—
167C2×Dic53—41no—
168C2×Dic53—41no—
169C2×Dic53—42no—
170C2×Dic53—41no—
171C2×Dic53—41no—
172C2×Dic53—41no—
173C2×Dic53—81no—
174C2×Dic53—81no—
175D212—44no—
176D212—42no—
177D212—42no—
178D212—42no—
179D212—42no—
180D212—42no—
181D212—42no—
182D212—42no—
183D212—42no—
184D212—82no—
185C4×D53—88no—
186Dic106—44no—
187Dic106—82no—
188C8×C53—24no—
189C8×C53—81no—
190#88322—81no—
191#9064—531nomaximal, Sylow
192C22×C53#38611yescharacteristic
193C22×C53#45011yescharacteristic
194C22×C53#45011yescharacteristic
195C22×C53#45211yescharacteristic
196C22×C53#45211yescharacteristic
197C22×C53#45311yescharacteristic
198C22×C53#45311yescharacteristic
199C2×D53—44no—
200C2×D53—44no—
201C2×D53—44no—
202C2×D53—82no—
203C2×D53—82no—
204C2×D53—82no—
205C2×D53—82no—
206C4×C53—21no—
207C4×C53—21no—
208C4×C53—42no—
209C4×C53—81no—
210C4×C53C4×SD1611yescharacteristic
211C4×C53C4×SD1611yescharacteristic
212C4×C53#44011yescharacteristic
213C4×C53#44011yescharacteristic
214C4×C53#46111yescharacteristic
215C4×C53#46111yescharacteristic
216C4×C53#46311yescharacteristic
217C4×C53#46311yescharacteristic
218Dic53—42no—
219Dic53—81no—
220Dic53—82no—
221C2×#492—531no—
222#2366—531no—
223#1987—531no—
224#1987—531no—
225C2×#331—531no—
226#1279—531no—
227#1279D5311yescharacteristic, complemented
228C2×C53#162911yescharacteristic
229C2×C53#163011yescharacteristic
230C2×C53#163111yescharacteristic
231C2×C53#163111yescharacteristic
232C2×C53#187711yescharacteristic
233C2×C53#187711yescharacteristic
234C2×C53#188711yescharacteristic
235D53—84no—
236C22×SD16—1062no—
237C2×C4⋊Q8—531no—
238C2×#127—531no—
239C2×C4×D4—1061no—
240C2×C4×D4—1061no—
241#492—534no—
242#492—1062no—
243C2×C4⋊C8—531no—
244C2×C4⋊C8C2×D5311yescharacteristic
245C2×C4×C8C2×D5311yescharacteristic
246#390—1061no—
247#390—1061no—
248#386—1062no—
249C4×C4⋊C4—531no—
250C4×C4⋊C4C2×D5311yescharacteristic
251#340—1061no—
252#340—1061no—
253#340—1061no—
254#340—1061no—
255#331—1062no—
256C53#906411yescharacteristic, complemented, Sylow
257C22×Q8—1061no—
258C22×D4—532no—
259C22×D4—1061no—
260C22×D4—1061no—
261C23×C4—1062no—
262C2×SD16—1068no—
263C2×SD16—2124no—
264C2×SD16—2124no—
265C22×C8—21no—
266C22×C8—21no—
267C22×C8—1061no—
268C4⋊Q8—532no—
269C4⋊Q8—1061no—
270C4⋊Q8—1061no—
271C4⋊Q8—1061no—
272#127—532no—
273#127—532no—
274#127—1061no—
275#127—1061no—
276C4×D4—2122no—
277C4×D4—2122no—
278C4×D4—2122no—
279C4×D4—2122no—
280C2×C4⋊C4—531no—
281C2×C4⋊C4—531no—
282C2×C4⋊C4—1061no—
283C2×C4⋊C4—1061no—
284C2×C4⋊C4D21211yescharacteristic
285C2×C4⋊C4D21211yescharacteristic
286C2×C22⋊C4—1062no—
287C2×C22⋊C4—1062no—
288C2×C22⋊C4—1062no—
289C2×C42—21no—
290C2×C42—1061no—
291C2×C42C22×D5311yescharacteristic
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—1061no—
295C4⋊C8—1061no—
296C4×C8—21no—
297C4×C8C4×D5312yescomplemented
298#95—21no—
299#95—1061no—
300C24—1062no—
301C2×Q8—1062no—
302C2×Q8—2122no—
303C2×Q8—2122no—
304C2×Q8—2121no—
305C2×D4—534no—
306C2×D4—1062no—
307C2×D4—1062no—
308C2×D4—1062no—
309C2×D4—1062no—
310C2×D4—1062no—
311C2×D4—2122no—
312C2×D4—2122no—
313C2×D4—2121no—
314C2×D4—2121no—
315C2×D4—2121no—
316C2×D4—2121no—
317C2×D4—2121no—
318C2×D4—2121no—
319C2×D4—2121no—
320C2×D4—2121no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—1061no—
324C22×C4—1061no—
325C22×C4—4242no—
326C22×C4—2124no—
327C22×C4—2124no—
328C22×C4—2121no—
329C22×C4C2×D21211yescharacteristic
330C22×C4D4×D5311yescharacteristic
331C22×C4D4×D5311yescharacteristic
332SD16—21216no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—2121no—
340C2×C8—2121no—
341C2×C8—2121no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—532no—
346C4⋊C4—1061no—
347C4⋊C4—1062no—
348C4⋊C4—1062no—
349C4⋊C4—1061no—
350C4⋊C4—1061no—
351C4⋊C4—2121no—
352C4⋊C4—2121no—
353C4⋊C4#1171991312yescomplemented
354C22⋊C4—4242no—
355C22⋊C4—2124no—
356C22⋊C4—2124no—
357C42—41no—
358C42—41no—
359C42—2121no—
360C42—2121no—
361C42C2×C4×D5311yescharacteristic
362C42#1171994511yescharacteristic
363C42D4×D5311yescharacteristic
364C42#1171994911yescharacteristic
365C23—1064no—
366C23—2122no—
367C23—2122no—
368C23—2122no—
369C23—2122no—
370C23—2122no—
371C23—2122no—
372C23#1313318711yescharacteristic
373Q8—4242no—
374Q8—2124no—
375D4—4242no—
376D4—2124no—
377D4—2122no—
378D4—2122no—
379D4—2122no—
380D4—2122no—
381D4—2122no—
382D4—2122no—
383D4—2122no—
384D4—2122no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—1062no—
394C2×C4—4248no—
395C2×C4—4241no—
396C2×C4—4241no—
397C2×C4—2122no—
398C2×C4—2121no—
399C2×C4—2121no—
400C2×C4—2121no—
401C2×C4—2121no—
402C2×C4—2121no—
403C2×C4C22⋊C4×D5311yescharacteristic
404C2×C4#1313317911yescharacteristic
405C2×C4C4×D21211yescharacteristic
406C2×C4#1313319011yescharacteristic
407C2×C4#1313319211yescharacteristic
408C2×C4#1313320211yescharacteristic
409C2×C4SD16×D5311yescharacteristic
410C2×C4C2×#1171991311yescharacteristic
411C2×C4#1313323811yescharacteristic
412C2×C4#1313323911yescharacteristic
413C2×C4#1313324111yescharacteristic
414C2×C4#1313324511yescharacteristic
415C2×C4#1313324911yescharacteristic
416C2×C4#1313325011yescharacteristic
417C8—24no—
418C8—81no—
419C8—4241no—
420C22—4242no—
421C22—4242no—
422C22—4242no—
423C22—4242no—
424C22—2124no—
425C22—2124no—
426C22—2124no—
427C22#1681050111yescharacteristic
428C22#1681077511yescharacteristic
429C22#1681077611yescharacteristic
430C22#1681078711yescharacteristic
431C22#1681078811yescharacteristic
432C22#1681079211yescharacteristic
433C22#1681079211yescharacteristic
434C4—21no—
435C4—21no—
436C4—42no—
437C4—81no—
438C4—4242no—
439C4—4241no—
440C4—2122no—
441C4#1681071011yescharacteristic
442C4C4×#1171991311yescharacteristic
443C4#1681072211yescharacteristic
444C4#1681072511yescharacteristic
445C4#1681082711yescharacteristic
446C4#1681082911yescharacteristic
447C4#1681083711yescharacteristic
448C4#1681083911yescharacteristic
449C2—4244no—
450C2#2027383311yescharacteristic
451C2#2027383811yescharacteristic
452C2#2027384411yescharacteristic
453C2#2027384511yescharacteristic
454C2#2027504811yescharacteristic
455C2#2027505011yescharacteristic
456C2#2027511211yescharacteristic
subgroupG/H
centerC23#13133187
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1279C2
socleC23×C53#120
prime radicalC2×#13133214C22
cosocleC2×C42×C53C23
hypercenter#1279D53
hypocenterC53#9064
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. #20275663C2semidirect
4. #20275662C2semidirect
6. #20272555C2semidirect
7. C53×#1279C2semidirect
14. #16810996C4semidirect
40. #13133217C2×C4semidirect
69. C4×C8×C53C2×C4semidirect
112. C4⋊C4×C53SD16semidirect
129. C4⋊Dic53SD16semidirect
227. #1279D53semidirect
256. C53#9064semidirect
297. C4×C8C4×D53semidirect
353. C4⋊C4#11719913semidirect

Maximal subgroups

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

subgroupindex
C2×#168109962
#202777792
#202756632
#202756622
C2×#168103662
#202725552
C53×#12792
#906453