Finite Group Databasebeta

← #27287632

Subgroups — #27287632

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#20279633C211yesmaximal, characteristic
2#20274842C211yesmaximal, characteristic, complemented
3#20274712C211yesmaximal, characteristic
4C2×Q8⋊#88322C211yesmaximal, characteristic, complemented
5C2×#16810350C211yesmaximal, characteristic
6#20272599C211yesmaximal, characteristic, complemented
7C53×#1803C211yesmaximal, characteristic, complemented
8C2×#13133193—21no—
9C2×C4×D212C2211yescharacteristic
10C2×C4⋊#88322C2211yescharacteristic
11C2×#13133137—21no—
12C2×C4×#88322C2211yescharacteristic
13#16810545—22no—
14#16810525—21no—
15#16810521—21no—
16#16810493C2211yescharacteristic
17#16810480—21no—
18#16810385—21no—
19#16810385—22no—
20Q8⋊#88322—22no—
21#16810350—22no—
22C2×C4×Q8×C53C2211yescharacteristic
23C2×C4⋊C8×C53C2211yescharacteristic
24C2×#103×C53—21no—
25C53×#384C2211yescharacteristic
26C53×#340—21no—
27C22×D212D411yescharacteristic
28C22×C4×D53—21no—
29C22×#88322—21no—
30C22×#88322—21no—
31#13133193—42no—
32#13133193—42no—
33C4×D212—21no—
34C4×D212—21no—
35C4×D212—21no—
36C4×D212—21no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53D411yescharacteristic
40C2×C4⋊Dic53D412yes—
41C2×#11719919—22no—
42C2×#11719921—22no—
43C2×#11719921—22no—
44C2×#11719921—21no—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47#13133137—42no—
48C4×#88322—21no—
49C4×#88322D412yes—
50#13133124—22no—
51C22×Q8×C53D411yescharacteristic
52C22×C8×C53—21no—
53C4×Q8×C53—21no—
54C4×Q8×C53—21no—
55C4×Q8×C53—21no—
56C4×Q8×C53—21no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53D411yescharacteristic
62C2×C42×C53—21no—
63C2×C42×C53C2311yescharacteristic
64C4⋊C8×C53—21no—
65C4⋊C8×C53—21no—
66#103×C53—42no—
67C23×D53—21no—
68C2×Q8×C53—21no—
69C2×Q8×C53—21no—
70C2×Q8×C53—21no—
71C2×Q8×C53—21no—
72C2×Q8×C53—21no—
73C2×Q8×C53SD1612yes—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—41no—
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×Dic53—21no—
81C22×Dic53—22no—
82C2×D212—21no—
83C2×D212—21no—
84C2×D212—21no—
85C2×D212—21no—
86C2×D212—21no—
87C2×D212Q1612yes—
88C2×C4×D53—41no—
89C2×C4×D53—41no—
90C2×C4×D53—41no—
91C2×C4×D53—41no—
92C2×C4×D53—41no—
93C2×C4×D53—41no—
94C2×C8×C53—41no—
95C2×C8×C53—41no—
96C2×C8×C53—41no—
97C4⋊C4×C53—21no—
98C4⋊C4×C53—21no—
99C4⋊C4×C53—42no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—41no—
102C4⋊C4×C53—41no—
103C4⋊C4×C53—41no—
104C4⋊C4×C53—41no—
105C4⋊C4×C53—41no—
106C42×C53—41no—
107C42×C53—41no—
108C42×C53C2×D411yescharacteristic
109C42×C53#4111yescharacteristic
110C42×C53#4111yescharacteristic
111C42×C53#4111yescharacteristic
112#11719921—44no—
113#11719921—41no—
114#11719921—41no—
115#11719921—82no—
116C4⋊Dic53—21no—
117C4⋊Dic53—21no—
118C4⋊Dic53—41no—
119C4⋊Dic53—41no—
120C4⋊Dic53—41no—
121C4⋊Dic53—41no—
122C4⋊Dic53—42no—
123#11719919—44no—
124C2×#88322—22no—
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—
138Q8×C53—22no—
139Q8×C53—22no—
140Q8×C53—41no—
141Q8×C53—41no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—22no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—81no—
151C2×C4×C53—81no—
152C2×C4×C53—81no—
153C2×C4×C53#12111yescharacteristic
154C2×C4×C53#12111yescharacteristic
155C2×C4×C53#12111yescharacteristic
156C2×C4×C53#12311yescharacteristic
157C2×C4×C53#12311yescharacteristic
158C2×C4×C53#12411yescharacteristic
159C2×C4×C53C2×SD1611yescharacteristic
160C2×C4×C53C2×Q1611yescharacteristic
161C2×C4×C53#13511yescharacteristic
162C2×C4×C53#13511yescharacteristic
163C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
166C2×C4×C53#13711yescharacteristic
167C2×Dic53—41no—
168C2×Dic53—41no—
169C2×Dic53—42no—
170C2×Dic53—41no—
171C2×Dic53—42no—
172C2×Dic53—42no—
173D212—22no—
174D212—22no—
175D212—41no—
176D212—41no—
177C4×D53—81no—
178C4×D53—81no—
179C4×D53—81no—
180C4×D53—81no—
181C8×C53—81no—
182#88322—42no—
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—42no—
202C4×C53—81no—
203C4×C53—161no—
204C4×C53#46511yescharacteristic
205C4×C53#46711yescharacteristic
206C4×C53#46811yescharacteristic
207C4×C53#46911yescharacteristic
208C4×C53#48111yescharacteristic
209C4×C53#48211yescharacteristic
210C4×C53#48411yescharacteristic
211C4×C53#48511yescharacteristic
212Dic53—82no—
213Dic53—81no—
214#2538—531no—
215#1834—531no—
216#1803D5311yescharacteristic, complemented
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×Q8C2×D5311yescharacteristic
231C2×C4×D4—531no—
232C2×C4⋊C8—531no—
233C2×C4⋊C8C2×D5311yescharacteristic
234C2×#103—21no—
235C2×#102—1061no—
236C2×C4×C8—531no—
237#396—1062no—
238#390—1061no—
239#389—1061no—
240#384—531no—
241#384C2×D5311yescharacteristic
242#380—1061no—
243#340—21no—
244#340—1061no—
245#340—1062no—
246Q8⋊C8—1062no—
247D4⋊C8—1062no—
248C53#1043611yescharacteristic, complemented, Sylow
249C22×Q8C22⋊D5311yescharacteristic
250C22×D4—531no—
251C23×C4—1061no—
252C22×C8—21no—
253C22×C8—1061no—
254C22×C8—1061no—
255C22⋊Q8—2122no—
256C22⋊Q8—2122no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
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—531no—
270C2×C4⋊C4—532no—
271C2×C4⋊C4—1062no—
272C2×C4⋊C4—1061no—
273C2×C4⋊C4—1061no—
274C2×C4⋊C4C22⋊D5311yescharacteristic
275C2×C22⋊C4—1062no—
276C2×C22⋊C4—1062no—
277C2×C22⋊C4—1061no—
278C2×C42—21no—
279C2×C42C22×D5311yescharacteristic
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—1061no—
283C4⋊C8—1061no—
284#103—42no—
285#102—2122no—
286C4×C8—532no—
287C4×C8—1061no—
288#95—1062no—
289C24—1061no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8#1171992412yes—
296C2×D4—532no—
297C2×D4—1061no—
298C2×D4—1061no—
299C2×D4—1061no—
300C2×D4—1061no—
301C2×D4—1061no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—41no—
305C22×C4—1061no—
306C22×C4—1062no—
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—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—1062no—
320C2×C8—2121no—
321C2×C8—2121no—
322C2×C8—2121no—
323C2×C8—2121no—
324C2×C8—2121no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—42no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—1061no—
335C4⋊C4—1061no—
336C4⋊C4—2122no—
337C4⋊C4—2124no—
338C4⋊C4—2121no—
339C4⋊C4—2121no—
340C4⋊C4—2121no—
341C4⋊C4—2121no—
342C22⋊C4—4242no—
343C22⋊C4—2124no—
344C22⋊C4—2121no—
345C22⋊C4—2121no—
346C42—41no—
347C42—41no—
348C42#1171994511yescharacteristic
349C42#1171994511yescharacteristic
350C42D4×D5311yescharacteristic
351C42#1171994711yescharacteristic
352C23—1062no—
353C23—2121no—
354C23—2121no—
355C23—2121no—
356C23—2121no—
357C23—2121no—
358C23—2121no—
359C23#1313318511yescharacteristic
360Q8—22no—
361Q8—22no—
362Q8—41no—
363Q8—41no—
364D4—1062no—
365D4—1062no—
366D4—2121no—
367D4—2121no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—4241no—
380C2×C4—4241no—
381C2×C4—4241no—
382C2×C4—4241no—
383C2×C4—2121no—
384C2×C4—2122no—
385C2×C4—2122no—
386C2×C4—2121no—
387C2×C4—2121no—
388C2×C4—2122no—
389C2×C4#1313318911yescharacteristic
390C2×C4#1313319011yescharacteristic
391C2×C4#1313319011yescharacteristic
392C2×C4#1313319911yescharacteristic
393C2×C4#1313320011yescharacteristic
394C2×C4#1313320311yescharacteristic
395C2×C4C2×#1171992411yescharacteristic
396C2×C4Q16×D5311yescharacteristic
397C2×C4#1313323711yescharacteristic
398C2×C4#1313323811yescharacteristic
399C2×C4#1313324211yescharacteristic
400C2×C4#1313324311yescharacteristic
401C2×C4#1313324411yescharacteristic
402C2×C4#1313324911yescharacteristic
403C8—81no—
404C8—4241no—
405C8—2122no—
406C22—4241no—
407C22—4241no—
408C22—4241no—
409C22—4241no—
410C22—2122no—
411C22—2122no—
412C22—2122no—
413C22#1681053811yescharacteristic
414C22#1681076611yescharacteristic
415C22#1681076711yescharacteristic
416C22#1681077311yescharacteristic
417C22#1681077711yescharacteristic
418C22#1681079411yescharacteristic
419C22#1681079911yescharacteristic
420C4—21no—
421C4—21no—
422C4—42no—
423C4—81no—
424C4—161no—
425C4—4241no—
426C4—4242no—
427C4#1681085411yescharacteristic
428C4#1681086311yescharacteristic
429C4#1681087211yescharacteristic
430C4#1681087411yescharacteristic
431C4#1681094111yescharacteristic
432C4#1681094711yescharacteristic
433C4#1681095711yescharacteristic
434C4#1681096211yescharacteristic
435C2—4242no—
436C2#2027382411yescharacteristic
437C2#2027387811yescharacteristic
438C2#2027391411yescharacteristic
439C2#2027392711yescharacteristic
440C2#2027590411yescharacteristic
441C2#2027591611yescharacteristic
442C2#2027594211yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1803C2
socleC23×C53#120
prime radicalC22×D212D4
hypercenter#1803D53
hypocenterC53#10436
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. #20274842C2semidirect
4. C2×Q8⋊#88322C2semidirect
6. #20272599C2semidirect
7. C53×#1803C2semidirect
216. #1803D53semidirect
248. C53#10436semidirect

Maximal subgroups

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

subgroupindex
#202796332
#202748422
#202747122
C2×Q8⋊#883222
C2×#168103502
#202725992
C53×#18032
#1043653