Finite Group Databasebeta

← #27287636

Subgroups — #27287636

Order 13568.

530 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279638C211yesmaximal, characteristic
2#20274717C211yesmaximal, characteristic, complemented
3#20274712C211yesmaximal, characteristic
4C2×#16810355C211yesmaximal, characteristic, complemented
5C2×#16810352C211yesmaximal, characteristic
6#20272598C211yesmaximal, characteristic, complemented
7C53×#1834C211yesmaximal, characteristic, complemented
8C2×#13133197—21no—
9C2×C4×Dic106C2211yescharacteristic
10C2×C4×D212C2211yescharacteristic
11C2×C4⋊#88322C2211yescharacteristic
12C2×C4⋊#88322C2211yescharacteristic
13C2×#13133140—21no—
14C2×#13133137—21no—
15#16810545—22no—
16#16810524—21no—
17#16810522—21no—
18#16810480—21no—
19#16810385—21no—
20#16810385—21no—
21#16810355—22no—
22#16810352—22no—
23C2×C4×C8×C53C2211yescharacteristic
24C53×#384C2211yescharacteristic
25C53×#384C2211yescharacteristic
26C53×#340—22no—
27C22×Dic106D411yescharacteristic
28C22×D212D411yescharacteristic
29C22×C4×D53—21no—
30C22×#88322—21no—
31C22×#88322—21no—
32#13133197—42no—
33#13133197—42no—
34C4×Dic106—21no—
35C4×Dic106—21no—
36C4×Dic106—21no—
37C4×Dic106—21no—
38C4×D212—21no—
39C4×D212—21no—
40C4×D212—21no—
41C4×D212—21no—
42C2×C4⋊Dic53—22no—
43C2×C4⋊Dic53D411yescharacteristic
44C2×C4⋊Dic53D411yescharacteristic
45C2×#11719919—21no—
46C2×#11719921—22no—
47C2×#11719921—22no—
48C2×#11719921—21no—
49C2×C4×Dic53—21no—
50C4⋊#88322—21no—
51C4⋊#88322—21no—
52C4⋊#88322—21no—
53C4⋊#88322—21no—
54#13133140—42no—
55#13133137—42no—
56#13133124—22no—
57C22×C8×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53D412yes—
64C2×C42×C53C2311yescharacteristic
65C4×C8×C53—21no—
66C4×C8×C53D412yes—
67C23×D53—21no—
68C22×C4×C53—22no—
69C22×C4×C53—41no—
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76C2×D212—21no—
77C2×D212—21no—
78C2×D212—21no—
79C2×D212—21no—
80C2×D212—21no—
81C2×D212Q1612yes—
82C2×C4×D53—41no—
83C2×C4×D53—41no—
84C2×C4×D53—41no—
85C2×C4×D53—41no—
86C2×C4×D53—41no—
87C2×C4×D53—41no—
88C2×Dic106—21no—
89C2×Dic106—21no—
90C2×Dic106—21no—
91C2×Dic106—21no—
92C2×Dic106—21no—
93C2×Dic106SD1612yes—
94C2×C8×C53—22no—
95C2×C8×C53—41no—
96C2×C8×C53—41no—
97C4⋊C4×C53—42no—
98C4⋊C4×C53—42no—
99C4⋊C4×C53—41no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—41no—
102C4⋊C4×C53—41no—
103C4⋊C4×C53—41no—
104C4⋊C4×C53—41no—
105C4⋊C4×C53—41no—
106C4⋊C4×C53—41no—
107C42×C53C2×D411yescharacteristic
108C42×C53#4111yescharacteristic
109C42×C53#4111yescharacteristic
110C42×C53#4111yescharacteristic
111#11719921—44no—
112#11719921—41no—
113#11719921—41no—
114#11719921—82no—
115C4⋊Dic53—21no—
116C4⋊Dic53—21no—
117C4⋊Dic53—21no—
118C4⋊Dic53—21no—
119C4⋊Dic53—44no—
120#11719919—41no—
121#11719919—41no—
122C4×Dic53—41no—
123C4×Dic53—41no—
124C2×#88322—41no—
125C2×#88322—41no—
126C2×#88322—41no—
127C2×#88322—41no—
128C2×#88322—41no—
129C2×#88322—41no—
130C23×C53#12011yescharacteristic
131C22×D53—22no—
132C22×D53—41no—
133C22×D53—41no—
134C22×D53—41no—
135C22×D53—41no—
136C22×D53—41no—
137C22×D53—41no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—42no—
141C2×C4×C53—42no—
142C2×C4×C53—42no—
143C2×C4×C53—81no—
144C2×C4×C53—81no—
145C2×C4×C53—81no—
146C2×C4×C53#12111yescharacteristic
147C2×C4×C53#12111yescharacteristic
148C2×C4×C53#12111yescharacteristic
149C2×C4×C53#12311yescharacteristic
150C2×C4×C53#12311yescharacteristic
151C2×C4×C53#12411yescharacteristic
152C2×C4×C53C2×SD1611yescharacteristic
153C2×C4×C53C2×Q1611yescharacteristic
154C2×C4×C53#13511yescharacteristic
155C2×C4×C53#13511yescharacteristic
156C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
159C2×C4×C53#13711yescharacteristic
160C2×Dic53—22no—
161C2×Dic53—41no—
162C2×Dic53—41no—
163C2×Dic53—41no—
164C2×Dic53—41no—
165C2×Dic53—41no—
166C2×Dic53—41no—
167C2×Dic53—41no—
168C2×Dic53—41no—
169D212—22no—
170D212—22no—
171D212—41no—
172D212—41no—
173C4×D53—81no—
174C4×D53—81no—
175C4×D53—81no—
176C4×D53—81no—
177Dic106—22no—
178Dic106—22no—
179Dic106—41no—
180Dic106—41no—
181C8×C53—42no—
182#88322—81no—
183#88322—81no—
184#10436—531nomaximal, Sylow
185C22×C53#39411yescharacteristic
186C22×C53#44911yescharacteristic
187C22×C53#44911yescharacteristic
188C22×C53#45011yescharacteristic
189C22×C53#45111yescharacteristic
190C22×C53#45411yescharacteristic
191C22×C53#45511yescharacteristic
192C2×D53—42no—
193C2×D53—42no—
194C2×D53—42no—
195C2×D53—81no—
196C2×D53—81no—
197C2×D53—81no—
198C2×D53—81no—
199C4×C53—21no—
200C4×C53—21no—
201C4×C53—82no—
202C4×C53—161no—
203C4×C53#46511yescharacteristic
204C4×C53#46711yescharacteristic
205C4×C53#46811yescharacteristic
206C4×C53#46911yescharacteristic
207C4×C53#48111yescharacteristic
208C4×C53#48211yescharacteristic
209C4×C53#48411yescharacteristic
210C4×C53#48511yescharacteristic
211Dic53—42no—
212Dic53—81no—
213Dic53—81no—
214#2538—531no—
215#1834D5311yescharacteristic, complemented
216#1803—531no—
217#1802—531no—
218C2×Q8⋊C8—531no—
219C2×D4⋊C8—531no—
220#1296—531no—
221C2×C53#162811yescharacteristic
222C2×C53#163711yescharacteristic
223C2×C53#164211yescharacteristic
224C2×C53#164411yescharacteristic
225C2×C53#203411yescharacteristic
226C2×C53#203611yescharacteristic
227C2×C53#204011yescharacteristic
228D53—82no—
229C2×C22⋊Q8—1061no—
230C2×C4×Q8—531no—
231C2×C4×D4—531no—
232C2×C4⋊C8—531no—
233C2×C4⋊C8—531no—
234C2×#103—1061no—
235C2×#102—1061no—
236C2×C4×C8C2×D5311yescharacteristic
237#396—1062no—
238#390—1061no—
239#389—1061no—
240#384C2×D5311yescharacteristic
241#384C2×D5311yescharacteristic
242#380—1061no—
243#340—22no—
244#340—1061no—
245#340—1061no—
246Q8⋊C8—1062no—
247D4⋊C8—1062no—
248C53#1043611yescharacteristic, complemented, Sylow
249C22×Q8—531no—
250C22×D4—531no—
251C23×C4—1061no—
252C22×C8—21no—
253C22×C8—1061no—
254C22×C8—1061no—
255C22⋊Q8—2122no—
256C22⋊Q8—2122no—
257C4×Q8—1061no—
258C4×Q8—1061no—
259C4×Q8—1061no—
260C4×Q8—1061no—
261C4×D4—1061no—
262C4×D4—1061no—
263C4×D4—1061no—
264C4×D4—1061no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—531no—
271C2×C4⋊C4—531no—
272C2×C4⋊C4—1062no—
273C2×C4⋊C4—1061no—
274C2×C4⋊C4C22⋊D5312yes—
275C2×C22⋊C4—1062no—
276C2×C22⋊C4—1062no—
277C2×C22⋊C4—1061no—
278C2×C42—1061no—
279C2×C42C22×D5311yescharacteristic
280C4⋊C8—1061no—
281C4⋊C8—1061no—
282C4⋊C8—1061no—
283C4⋊C8—1061no—
284#103—2122no—
285#102—2122no—
286C4×C8—21no—
287C4×C8D21212yes—
288#95—1062no—
289C24—1061no—
290C2×Q8—532no—
291C2×Q8—1061no—
292C2×Q8—1061no—
293C2×Q8—1061no—
294C2×Q8—1061no—
295C2×Q8—1061no—
296C2×D4—532no—
297C2×D4—1061no—
298C2×D4—1061no—
299C2×D4—1061no—
300C2×D4—1061no—
301C2×D4—1061no—
302C22×C4—22no—
303C22×C4—41no—
304C22×C4—1061no—
305C22×C4—1061no—
306C22×C4—1061no—
307C22×C4—2121no—
308C22×C4—2121no—
309C22×C4—2121no—
310C22×C4—2121no—
311C22×C4—2121no—
312C22×C4—2121no—
313C22×C4D4×D5311yescharacteristic
314C22×C4D4×D5311yescharacteristic
315C22×C4C2×C22⋊D5311yescharacteristic
316C2×C8—22no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—2121no—
320C2×C8—2121no—
321C2×C8—2121no—
322C2×C8—2121no—
323C2×C8—2121no—
324C2×C8—2121no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—1061no—
336C4⋊C4—1061no—
337C4⋊C4—1061no—
338C4⋊C4—1061no—
339C4⋊C4—2124no—
340C4⋊C4—2121no—
341C4⋊C4—2121no—
342C22⋊C4—4242no—
343C22⋊C4—2124no—
344C22⋊C4—2121no—
345C22⋊C4—2121no—
346C42—2121no—
347C42—2121no—
348C42C2×D21211yescharacteristic
349C42#1171994511yescharacteristic
350C42#1171994911yescharacteristic
351C42#1171994911yescharacteristic
352C23—1062no—
353C23—2121no—
354C23—2121no—
355C23—2121no—
356C23—2121no—
357C23—2121no—
358C23—2121no—
359C23#1313318511yescharacteristic
360Q8—1062no—
361Q8—1062no—
362Q8—2121no—
363Q8—2121no—
364D4—1062no—
365D4—1062no—
366D4—2121no—
367D4—2121no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—1062no—
377C2×C4—4241no—
378C2×C4—4241no—
379C2×C4—4241no—
380C2×C4—4241no—
381C2×C4—2121no—
382C2×C4—2121no—
383C2×C4—2121no—
384C2×C4—2121no—
385C2×C4—2121no—
386C2×C4—2121no—
387C2×C4—2121no—
388C2×C4—2121no—
389C2×C4#1313319111yescharacteristic
390C2×C4#1313319211yescharacteristic
391C2×C4#1313319211yescharacteristic
392C2×C4#1313320211yescharacteristic
393C2×C4#1313320211yescharacteristic
394C2×C4#1313320611yescharacteristic
395C2×C4SD16×D5311yescharacteristic
396C2×C4Q16×D5311yescharacteristic
397C2×C4#1313323611yescharacteristic
398C2×C4#1313323811yescharacteristic
399C2×C4#1313324111yescharacteristic
400C2×C4#1313324411yescharacteristic
401C2×C4#1313324411yescharacteristic
402C2×C4#1313324911yescharacteristic
403C8—42no—
404C8—4241no—
405C8—4241no—
406C22—4241no—
407C22—4241no—
408C22—4241no—
409C22—4241no—
410C22—2122no—
411C22—2122no—
412C22—2122no—
413C22#1681054011yescharacteristic
414C22#1681076511yescharacteristic
415C22#1681076711yescharacteristic
416C22#1681077211yescharacteristic
417C22#1681077711yescharacteristic
418C22#1681079511yescharacteristic
419C22#1681080011yescharacteristic
420C4—21no—
421C4—21no—
422C4—82no—
423C4—161no—
424C4—4241no—
425C4—4241no—
426C4—2122no—
427C4#1681085311yescharacteristic
428C4#1681086511yescharacteristic
429C4#1681087011yescharacteristic
430C4#1681087711yescharacteristic
431C4#1681094311yescharacteristic
432C4#1681094911yescharacteristic
433C4#1681096011yescharacteristic
434C4#1681096611yescharacteristic
435C2—4242no—
436C2#2027382211yescharacteristic
437C2#2027387711yescharacteristic
438C2#2027391211yescharacteristic
439C2#2027392611yescharacteristic
440C2#2027590611yescharacteristic
441C2#2027592011yescharacteristic
442C2#2027594411yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1834C2
socleC23×C53#120
prime radicalC22×D212D4
hypercenter#1834D53
hypocenterC53#10436
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1834

Complements

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

NH ≅ G/Ntype
2. #20274717C2semidirect
4. C2×#16810355C2semidirect
6. #20272598C2semidirect
7. C53×#1834C2semidirect
215. #1834D53semidirect
248. C53#10436semidirect

Maximal subgroups

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

subgroupindex
#202796382
#202747172
#202747122
C2×#168103552
C2×#168103522
#202725982
C53×#18342
#1043653