Finite Group Databasebeta

← #27367625

Subgroups — #27367625

Order 13568.

640 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286027C211yesmaximal, characteristic
2#20285836C211yesmaximal, characteristic
3#20283461C211yesmaximal, characteristic
4#20274287C212yesmaximal, complemented
5#20274259C212yesmaximal, complemented
6#20274163C212yesmaximal, complemented
7#20274135C212yesmaximal, complemented
8#20273467C211yesmaximal, characteristic, complemented
9#20273372C211yesmaximal, characteristic, complemented
10C2×#16810369C211yesmaximal, characteristic, complemented
11C53×#3139C211yesmaximal, characteristic, complemented
12C2×#13133225—21no—
13#16811173C2211yescharacteristic
14#16811026—21no—
15#16810971—21no—
16#16810936—22no—
17#16810910C2212yes—
18#16810908—21no—
19#16810900C2212yes—
20#16810900C2212yes—
21#16810854—22no—
22#16810825C2212yes—
23#16810823—21no—
24#16810702C2212yes—
25#16810696—22no—
26C8×D212C2212yes—
27#16810676C2211yescharacteristic
28C2×C4⋊#88322C2211yescharacteristic
29C2×C4⋊#88322C2211yescharacteristic
30#16810591—21no—
31#16810376C2212yes—
32#16810374C2212yes—
33#16810372C2211yescharacteristic
34#16810372C2211yescharacteristic
35#16810369C2211yescharacteristic
36#16810369C2212yes—
37#16810369C2212yes—
38#16810369C2211yescharacteristic
39C53×#533C2211yescharacteristic
40C2×C4⋊Q8×C53C2211yescharacteristic
41C53×#491—22no—
42C53×#487C2212yes—
43C53×#487C2212yes—
44C53×#419—21no—
45C53×#419—21no—
46C4×M16×C53C2211yescharacteristic
47C2×#11719945—21no—
48C2×#11719924—22no—
49#13133225—44no—
50C2×#11719912—22no—
51C22×#88322—21no—
52C22×#88322—21no—
53C2×C8×D53—22no—
54#13133217D411yescharacteristic
55#13133214D411yescharacteristic
56#13133205—21no—
57#13133191—21no—
58#13133191D412yes—
59C4×D212—21no—
60C4×D212C2312yes—
61#13133150—22no—
62#13133150—22no—
63#13133148—22no—
64C4⋊#88322—21no—
65C4⋊#88322—21no—
66C4⋊#88322C2312yes—
67C4⋊#88322C2311yescharacteristic
68C4⋊#88322C2311yescharacteristic
69C4⋊#88322C2311yescharacteristic
70C4⋊#88322C2311yescharacteristic
71#13133141—22no—
72#13133137—22no—
73#13133137—22no—
74#13133137—22no—
75C22⋊#88322—21no—
76#13133129—22no—
77#13133129—22no—
78C4×#88322—21no—
79C22×Q8×C53—21no—
80C2×M16×C53D411yescharacteristic
81C2×M16×C53D411yescharacteristic
82C4⋊Q8×C53—21no—
83C4⋊Q8×C53—21no—
84C4⋊Q8×C53—21no—
85C4⋊Q8×C53C2312yes—
86C4⋊Q8×C53C2311yescharacteristic
87C4⋊Q8×C53C2311yescharacteristic
88#125×C53—21no—
89C22⋊Q8×C53—21no—
90C22⋊Q8×C53—21no—
91C4×Q8×C53—21no—
92C4×Q8×C53—21no—
93#117×C53—21no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53—21no—
96C2×C42×C53C2311yescharacteristic
97#103×C53—22no—
98#103×C53—22no—
99#103×C53—22no—
100#103×C53—22no—
101#97×C53D412yes—
102C4×C8×C53C2312yes—
103C2×Q8×C53—22no—
104C2×Q8×C53—21no—
105C2×Q8×C53—21no—
106C2×Q8×C53—41no—
107C2×Q8×C53—42no—
108C2×Q8×C53—42no—
109C22×C4×C53—41no—
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53#4111yescharacteristic
113C2×C22⋊D53—22no—
114#11719945—42no—
115#11719945—42no—
116C2×D212—21no—
117C2×D212—21no—
118C2×D212C2×D412yes—
119C2×C4×D53—22no—
120C2×Dic106—21no—
121M16×C53—22no—
122M16×C53—22no—
123C2×C8×C53C2×D412yes—
124C2×C8×C53C2×D412yes—
125C4⋊C4×C53—21no—
126C4⋊C4×C53—21no—
127C4⋊C4×C53—22no—
128C4⋊C4×C53—21no—
129C4⋊C4×C53—22no—
130C4⋊C4×C53—21no—
131C4⋊C4×C53—41no—
132C4⋊C4×C53—41no—
133C4⋊C4×C53—41no—
134C4⋊C4×C53—41no—
135C22⋊C4×C53—41no—
136C42×C53—41no—
137C42×C53C2×D411yescharacteristic
138C42×C53C2×D411yescharacteristic
139C42×C53#4111yescharacteristic
140C42×C53C2411yescharacteristic
141#11719924—82no—
142#11719921—22no—
143#11719921—22no—
144C4⋊Dic53—21no—
145C4⋊Dic53C2×D412yes—
146C2×#88322—21no—
147C2×#88322—21no—
148C2×#88322—21no—
149C2×#88322—21no—
150C2×#88322—22no—
151C2×#88322—42no—
152C2×#88322—81no—
153#11719912—44no—
154C8×D53—44no—
155C23×C53#12111yescharacteristic
156C22×D53—22no—
157Q8×C53—42no—
158Q8×C53—42no—
159Q8×C53—41no—
160Q8×C53—41no—
161Q8×C53—81no—
162Q8×C53—82no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—41no—
166C2×C4×C53—42no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—82no—
170C2×C4×C53#12111yescharacteristic
171C2×C4×C53#12111yescharacteristic
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53#13512yes—
174C2×C4×C53AGL(1,ℤ/8)12yes—
175C2×C4×C53C22×D411yescharacteristic
176C2×C4×C53C22×D411yescharacteristic
177C2×C4×C53C2×#4111yescharacteristic
178C22⋊D53—44no—
179C2×Dic53—22no—
180D212—22no—
181D212—22no—
182D212—42no—
183D212—81no—
184C4×D53—42no—
185C4×D53—42no—
186Dic106—42no—
187C8×C53—22no—
188C8×C53—22no—
189#88322—41no—
190#88322—41no—
191#88322—41no—
192#88322—42no—
193#88322—41no—
194#19235—531nomaximal, Sylow
195C22×C53—21no—
196C22×C53—21no—
197C22×C53#46312yes—
198C22×C53C2×#12111yescharacteristic
199C2×D53—42no—
200C2×D53—42no—
201C2×D53—42no—
202C4×C53—21no—
203C4×C53—21no—
204C4×C53—21no—
205C4×C53—82no—
206C4×C53—81no—
207C4×C53—81no—
208C4×C53#46312yes—
209C4×C53C2×#13511yescharacteristic
210C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
211C4×C53#57711yescharacteristic
212C4×C53#57811yescharacteristic
213Dic53—42no—
214#3246—531no—
215#3226—531no—
216#3139D5311yescharacteristic, complemented
217#2935—531no—
218#1703—532no—
219#1699—532no—
220#1683—532no—
221#1679—532no—
222#1565—531no—
223#1542—531no—
224C2×#332—531no—
225C2×C53—22no—
226C2×C53C2×#46311yescharacteristic
227C2×C53#306311yescharacteristic
228C2×C53#306911yescharacteristic
229D53—82no—
230C2×#131—1061no—
231#533C2×D5311yescharacteristic
232#532—531no—
233C2×C4⋊Q8C2×D5311yescharacteristic
234#500—1061no—
235#491—22no—
236#487C2×D5312yes—
237#487C2×D5312yes—
238#486—1061no—
239#480—1062no—
240#476—532no—
241#475—1061no—
242#474—532no—
243#474—532no—
244#465—1062no—
245#461—532no—
246#460—1061no—
247#436—532no—
248#435—1062no—
249C8×D4—532no—
250#431—531no—
251C2×C4⋊C8—531no—
252C2×C4⋊C8—531no—
253#419—21no—
254#419—21no—
255#408—1061no—
256C4×M16C2×D5311yescharacteristic
257#335—532no—
258#334—532no—
259#333—531no—
260#333—531no—
261#332—532no—
262#332—532no—
263#332—531no—
264#332—531no—
265C53#1923511yescharacteristic, complemented, Sylow
266C2×#41—1061no—
267C22×Q8—21no—
268C2×SD16—1062no—
269#131—2124no—
270C2×M16—1062no—
271C2×M16C22⋊D5311yescharacteristic
272C2×M16C22⋊D5311yescharacteristic
273C22×C8—1061no—
274C22×C8—1061no—
275C22×C8—1062no—
276C4⋊Q8—21no—
277C4⋊Q8—21no—
278C4⋊Q8—21no—
279C4⋊Q8—531no—
280C4⋊Q8C22×D5312yes—
281C4⋊Q8C22×D5311yescharacteristic
282C4⋊Q8C22×D5311yescharacteristic
283#127—531no—
284#125—21no—
285#124—1061no—
286C22⋊Q8—21no—
287C22⋊Q8—21no—
288#121—532no—
289#121—1061no—
290C4×Q8—21no—
291C4×Q8—21no—
292C4×D4—532no—
293C4×D4—1061no—
294#117—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C42C22×D5311yescharacteristic
298#107—1062no—
299#107—1062no—
300#106—1062no—
301C4⋊C8—531no—
302C4⋊C8—531no—
303C4⋊C8—532no—
304C4⋊C8—531no—
305C4⋊C8—531no—
306C4⋊C8—1061no—
307C4⋊C8—1061no—
308#103—22no—
309#103—22no—
310#103—22no—
311#103—22no—
312#103—1062no—
313#102—1062no—
314#102—1062no—
315#102—1062no—
316C22⋊C8—1061no—
317C22⋊C8—1062no—
318C22⋊C8—1062no—
319#97C22⋊D5312yes—
320C4×C8—1061no—
321C4×C8C22×D5312yes—
322#41—2122no—
323#41—2122no—
324C2×Q8—21no—
325C2×Q8—22no—
326C2×Q8—21no—
327C2×Q8—41no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—1061no—
331C2×D4—532no—
332C2×D4—1062no—
333C2×D4—1061no—
334C2×D4—1061no—
335C22×C4—41no—
336C22×C4—1062no—
337C22×C4D4×D5311yescharacteristic
338C22×C4#1171994711yescharacteristic
339C22×C4C2×C22⋊D5311yescharacteristic
340SD16—4242no—
341M16—22no—
342M16—22no—
343M16—2124no—
344C2×C8—1061no—
345C2×C8—1061no—
346C2×C8—1062no—
347C2×C8—1061no—
348C2×C8—1061no—
349C2×C8—4241no—
350C2×C8—2124no—
351C2×C8—2122no—
352C2×C8C2×C22⋊D5312yes—
353C2×C8C2×C22⋊D5312yes—
354C4⋊C4—21no—
355C4⋊C4—22no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—22no—
359C4⋊C4—21no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—532no—
365C4⋊C4—1061no—
366C22⋊C4—41no—
367C22⋊C4—1062no—
368C22⋊C4—1062no—
369C42—41no—
370C42D4×D5311yescharacteristic
371C42#1171994711yescharacteristic
372C42C2×C22⋊D5311yescharacteristic
373C42C23×D5311yescharacteristic
374C23—1062no—
375C23#1313318811yescharacteristic
376Q8—42no—
377Q8—42no—
378Q8—41no—
379Q8—41no—
380Q8—81no—
381Q8—82no—
382Q8—2122no—
383D4—1062no—
384D4—1062no—
385D4—4241no—
386D4—2124no—
387D4—2122no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—82no—
395C2×C4—1062no—
396C2×C4—2122no—
397C2×C4—2122no—
398C2×C4#1313318811yescharacteristic
399C2×C4#1313318811yescharacteristic
400C2×C4#1313318811yescharacteristic
401C2×C4#1313323712yes—
402C2×C4#1313324212yes—
403C2×C4C2×D4×D5311yescharacteristic
404C2×C4C22×C22⋊D5311yescharacteristic
405C2×C4C2×#1171994711yescharacteristic
406C8—22no—
407C8—22no—
408C8—2121no—
409C8—2121no—
410C8—2121no—
411C8—2122no—
412C8—2121no—
413C22—21no—
414C22—21no—
415C22—2122no—
416C22—2122no—
417C22—2122no—
418C22#1681083512yes—
419C22C2×#1313318811yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—81no—
424C4—82no—
425C4—81no—
426C4—2122no—
427C4#1681083512yes—
428C4C2×#1313323711yescharacteristic
429C4C2×#1313324211yescharacteristic
430C4#1681143211yescharacteristic
431C4#1681143911yescharacteristic
432C2—22no—
433C2—4242no—
434C2C2×#1681083511yescharacteristic
435C2#2028440011yescharacteristic
436C2#2028444811yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3139C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenter#3139D53
hypocenterC53#19235
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3139

Complements

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

NH ≅ G/Ntype
4. #20274287C2semidirect
5. #20274259C2semidirect
6. #20274163C2semidirect
7. #20274135C2semidirect
8. #20273467C2semidirect
9. #20273372C2semidirect
10. C2×#16810369C2semidirect
11. C53×#3139C2semidirect
216. #3139D53semidirect
265. C53#19235semidirect

Maximal subgroups

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

subgroupindex
#202860272
#202858362
#202834612
#202742872
#202742592
#202741632
#202741352
#202734672
#202733722
C2×#168103692
C53×#31392
#1923553