Finite Group Databasebeta

← #27287532

Subgroups — #27287532

Order 13568.

556 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279669C211yesmaximal, characteristic
2#20275476C211yesmaximal, characteristic, complemented
3#20274712C211yesmaximal, characteristic
4C2×#16810369C211yesmaximal, characteristic, complemented
5C2×#16810352C211yesmaximal, characteristic
6#20272555C211yesmaximal, characteristic, complemented
7C53×#1990C211yesmaximal, characteristic, complemented
8C2×#13133193—21no—
9C2×#13133193—21no—
10C2×C4×D212C2211yescharacteristic
11C2×#13133148—21no—
12C2×C4⋊#88322C2211yescharacteristic
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133137—21no—
15#16810558—22no—
16#16810524—21no—
17#16810492—21no—
18C4×C4⋊Dic53C2211yescharacteristic
19#16810385—21no—
20#16810385—21no—
21#16810369—22no—
22#16810352—22no—
23C2×C4⋊Q8×C53C2211yescharacteristic
24C2×C4×C8×C53C2211yescharacteristic
25C53×#384C2211yescharacteristic
26C53×#340—22no—
27C22×D212D411yescharacteristic
28C22×C4×D53—21no—
29C22×#88322—21no—
30C22×#88322—21no—
31#13133193—42no—
32#13133193—42no—
33#13133193—42no—
34#13133193—42no—
35C4×D212—21no—
36C4×D212—21no—
37C4×D212—21no—
38C4×D212—21no—
39C2×C4⋊Dic53D411yescharacteristic
40C2×C4⋊Dic53D411yescharacteristic
41C2×C4⋊Dic53D411yescharacteristic
42C2×#11719919—22no—
43C2×#11719919—22no—
44C2×#11719921—22no—
45C2×#11719921—22no—
46C2×#11719921—21no—
47C2×C4×Dic53—21no—
48#13133148—42no—
49C4⋊#88322—21no—
50C4⋊#88322—21no—
51C4⋊#88322—21no—
52C4⋊#88322D412yes—
53#13133137—42no—
54#13133124—21no—
55C22×Q8×C53—21no—
56C22×C8×C53—21no—
57C4⋊Q8×C53—21no—
58C4⋊Q8×C53—21no—
59C4⋊Q8×C53—21no—
60C4⋊Q8×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53D412yes—
66C2×C42×C53C2311yescharacteristic
67C4×C8×C53—21no—
68C4×C8×C53—21no—
69C23×D53—21no—
70C2×Q8×C53—41no—
71C2×Q8×C53—41no—
72C2×Q8×C53—41no—
73C2×Q8×C53—41no—
74C2×Q8×C53—41no—
75C2×Q8×C53—41no—
76C22×C4×C53—22no—
77C22×C4×C53—41no—
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×Dic53—21no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C2×D212—21no—
85C2×D212—21no—
86C2×D212—21no—
87C2×D212—21no—
88C2×D212—21no—
89C2×D212SD1612yes—
90C2×C4×D53—41no—
91C2×C4×D53—41no—
92C2×C4×D53—41no—
93C2×C4×D53—41no—
94C2×C4×D53—41no—
95C2×C4×D53—41no—
96C2×C8×C53—22no—
97C2×C8×C53—41no—
98C2×C8×C53—41no—
99C4⋊C4×C53—22no—
100C4⋊C4×C53—22no—
101C4⋊C4×C53—42no—
102C4⋊C4×C53—41no—
103C4⋊C4×C53—41no—
104C4⋊C4×C53—41no—
105C4⋊C4×C53—41no—
106C4⋊C4×C53—41no—
107C4⋊C4×C53—41no—
108C42×C53C2×D411yescharacteristic
109C42×C53#4111yescharacteristic
110C42×C53#4111yescharacteristic
111C42×C53#4111yescharacteristic
112#11719921—44no—
113#11719921—44no—
114#11719921—41no—
115#11719921—41no—
116C4⋊Dic53—21no—
117C4⋊Dic53—21no—
118C4⋊Dic53—21no—
119C4⋊Dic53—21no—
120C4⋊Dic53D812yes—
121C4⋊Dic53SD1612yes—
122#11719919—44no—
123#11719919—44no—
124C4×Dic53—41no—
125C4×Dic53—41no—
126C2×#88322—22no—
127C2×#88322—41no—
128C2×#88322—41no—
129C2×#88322—41no—
130C2×#88322—41no—
131C2×#88322—41no—
132C23×C53#12011yescharacteristic
133C22×D53—22no—
134C22×D53—41no—
135C22×D53—41no—
136C22×D53—41no—
137C22×D53—41no—
138C22×D53—41no—
139C22×D53—41no—
140Q8×C53—81no—
141Q8×C53—81no—
142Q8×C53—81no—
143Q8×C53—81no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—42no—
147C2×C4×C53—42no—
148C2×C4×C53—42no—
149C2×C4×C53—81no—
150C2×C4×C53—81no—
151C2×C4×C53—81no—
152C2×C4×C53#12111yescharacteristic
153C2×C4×C53#12111yescharacteristic
154C2×C4×C53#12111yescharacteristic
155C2×C4×C53#12311yescharacteristic
156C2×C4×C53#12311yescharacteristic
157C2×C4×C53#12411yescharacteristic
158C2×C4×C53C2×D811yescharacteristic
159C2×C4×C53C2×SD1611yescharacteristic
160C2×C4×C53C2×SD1611yescharacteristic
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—22no—
168C2×Dic53—41no—
169C2×Dic53—41no—
170C2×Dic53—41no—
171C2×Dic53—41no—
172C2×Dic53—41no—
173C2×Dic53—41no—
174C2×Dic53—41no—
175D212—22no—
176D212—22no—
177D212—41no—
178D212—41no—
179C4×D53—81no—
180C4×D53—81no—
181C4×D53—81no—
182C4×D53—81no—
183C8×C53—42no—
184#88322—42no—
185#88322—81no—
186#10420—531nomaximal, Sylow
187C22×C53#39411yescharacteristic
188C22×C53#44711yescharacteristic
189C22×C53#44811yescharacteristic
190C22×C53#45011yescharacteristic
191C22×C53#45211yescharacteristic
192C22×C53#45411yescharacteristic
193C22×C53#45611yescharacteristic
194C2×D53—42no—
195C2×D53—42no—
196C2×D53—42no—
197C2×D53—81no—
198C2×D53—81no—
199C2×D53—81no—
200C2×D53—81no—
201C4×C53—21no—
202C4×C53—21no—
203C4×C53—82no—
204C4×C53—161no—
205C4×C53#45911yescharacteristic
206C4×C53#46011yescharacteristic
207C4×C53#46511yescharacteristic
208C4×C53#47011yescharacteristic
209C4×C53#48311yescharacteristic
210C4×C53#48511yescharacteristic
211C4×C53#48611yescharacteristic
212C4×C53#49011yescharacteristic
213Dic53—42no—
214Dic53—42no—
215Dic53—81no—
216#2542—531no—
217#1990D5311yescharacteristic, complemented
218#1948—531no—
219#1802—531no—
220C2×#332—531no—
221C2×D4⋊C8—531no—
222#1279—531no—
223C2×C53#162711yescharacteristic
224C2×C53#163511yescharacteristic
225C2×C53#164711yescharacteristic
226C2×C53#165111yescharacteristic
227C2×C53#203511yescharacteristic
228C2×C53#203811yescharacteristic
229C2×C53#204111yescharacteristic
230D53—82no—
231C2×C4⋊Q8C2×D5311yescharacteristic
232C2×C22⋊Q8—1061no—
233C2×C22⋊Q8—1061no—
234C2×C4×D4—531no—
235C2×#106—1061no—
236C2×C4⋊C8—531no—
237C2×C4⋊C8—531no—
238C2×#102—1061no—
239C2×C4×C8C2×D5311yescharacteristic
240#399—1062no—
241#390—1061no—
242#384—1061no—
243#384C2×D5311yescharacteristic
244C4×C4⋊C4—531no—
245#340—22no—
246#340—1061no—
247#340—1061no—
248#332—1062no—
249D4⋊C8—1062no—
250C53#1042011yescharacteristic, complemented, Sylow
251C22×Q8—21no—
252C22×D4—531no—
253C23×C4—1061no—
254C22×C8—21no—
255C22×C8—1061no—
256C22×C8—1061no—
257C4⋊Q8—21no—
258C4⋊Q8—21no—
259C4⋊Q8—21no—
260C4⋊Q8—21no—
261C22⋊Q8—2122no—
262C22⋊Q8—2122no—
263C22⋊Q8—2122no—
264C22⋊Q8—2122no—
265C4×D4—1061no—
266C4×D4—1061no—
267C4×D4—1061no—
268C4×D4—1061no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—531no—
274C2×C4⋊C4—531no—
275C2×C4⋊C4—531no—
276C2×C4⋊C4—1062no—
277C2×C4⋊C4—1062no—
278C2×C4⋊C4C22⋊D5312yes—
279C2×C22⋊C4—1062no—
280C2×C22⋊C4—1062no—
281C2×C22⋊C4—1061no—
282C2×C42—1061no—
283C2×C42C22×D5311yescharacteristic
284#106—2122no—
285C4⋊C8—532no—
286C4⋊C8—1061no—
287C4⋊C8—1061no—
288C4⋊C8—1061no—
289#102—2122no—
290C4×C8—21no—
291C4×C8—21no—
292#95—1061no—
293C24—1061no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×D4—532no—
301C2×D4—1061no—
302C2×D4—1061no—
303C2×D4—1061no—
304C2×D4—1061no—
305C2×D4—1061no—
306C22×C4—22no—
307C22×C4—41no—
308C22×C4—1061no—
309C22×C4—1061no—
310C22×C4—1061no—
311C22×C4—2121no—
312C22×C4—2121no—
313C22×C4—2121no—
314C22×C4—2121no—
315C22×C4—2121no—
316C22×C4—2121no—
317C22×C4D4×D5311yescharacteristic
318C22×C4D4×D5311yescharacteristic
319C22×C4C2×C22⋊D5311yescharacteristic
320C2×C8—22no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—1062no—
324C2×C8—2121no—
325C2×C8—2121no—
326C2×C8—2121no—
327C2×C8—2121no—
328C2×C8—2121no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—42no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—532no—
339C4⋊C4—532no—
340C4⋊C4—1061no—
341C4⋊C4—1061no—
342C4⋊C4—1061no—
343C4⋊C4—1061no—
344C4⋊C4—2124no—
345C4⋊C4—2124no—
346C22⋊C4—2124no—
347C22⋊C4—2124no—
348C22⋊C4—2121no—
349C22⋊C4—2121no—
350C42—2121no—
351C42—2121no—
352C42#1171994511yescharacteristic
353C42#1171994511yescharacteristic
354C42D4×D5311yescharacteristic
355C42#1171994711yescharacteristic
356C23—1062no—
357C23—2121no—
358C23—2121no—
359C23—2121no—
360C23—2121no—
361C23—2121no—
362C23—2121no—
363C23#1313318511yescharacteristic
364Q8—81no—
365Q8—81no—
366Q8—81no—
367Q8—81no—
368D4—1062no—
369D4—1062no—
370D4—2121no—
371D4—2121no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—1062no—
381C2×C4—1062no—
382C2×C4—4241no—
383C2×C4—4241no—
384C2×C4—4241no—
385C2×C4—4241no—
386C2×C4—2121no—
387C2×C4—2121no—
388C2×C4—2121no—
389C2×C4—2121no—
390C2×C4—2121no—
391C2×C4—2121no—
392C2×C4—2121no—
393C2×C4#1313318911yescharacteristic
394C2×C4#1313319011yescharacteristic
395C2×C4#1313319011yescharacteristic
396C2×C4#1313319911yescharacteristic
397C2×C4#1313320011yescharacteristic
398C2×C4#1313320311yescharacteristic
399C2×C4D8×D5311yescharacteristic
400C2×C4SD16×D5311yescharacteristic
401C2×C4SD16×D5311yescharacteristic
402C2×C4#1313324011yescharacteristic
403C2×C4#1313324111yescharacteristic
404C2×C4#1313324411yescharacteristic
405C2×C4#1313324611yescharacteristic
406C2×C4#1313324911yescharacteristic
407C8—42no—
408C8—4241no—
409C8—2122no—
410C22—4241no—
411C22—4241no—
412C22—4241no—
413C22—4241no—
414C22—2122no—
415C22—2122no—
416C22—2122no—
417C22#1681053811yescharacteristic
418C22#1681075511yescharacteristic
419C22#1681076011yescharacteristic
420C22#1681077211yescharacteristic
421C22#1681078411yescharacteristic
422C22#1681079511yescharacteristic
423C22#1681080511yescharacteristic
424C4—21no—
425C4—21no—
426C4—82no—
427C4—161no—
428C4—4241no—
429C4—2122no—
430C4—2122no—
431C4#1681081711yescharacteristic
432C4#1681082211yescharacteristic
433C4#1681085511yescharacteristic
434C4#1681088011yescharacteristic
435C4#1681095211yescharacteristic
436C4#1681096311yescharacteristic
437C4#1681096811yescharacteristic
438C4#1681098611yescharacteristic
439C2—4242no—
440C2#2027381511yescharacteristic
441C2#2027386311yescharacteristic
442C2#2027394711yescharacteristic
443C2#2027397511yescharacteristic
444C2#2027590911yescharacteristic
445C2#2027593211yescharacteristic
446C2#2027594711yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1990C2
socleC23×C53#120
prime radicalC22×D212D4
hypercenter#1990D53
hypocenterC53#10420
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
2. #20275476C2semidirect
4. C2×#16810369C2semidirect
6. #20272555C2semidirect
7. C53×#1990C2semidirect
217. #1990D53semidirect
250. C53#10420semidirect

Maximal subgroups

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

subgroupindex
#202796692
#202754762
#202747122
C2×#168103692
C2×#168103522
#202725552
C53×#19902
#1042053