Finite Group Databasebeta

← #27367603

Subgroups — #27367603

Order 13568.

640 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286027C211yesmaximal, characteristic
2#20285843C211yesmaximal, characteristic
3#20283461C211yesmaximal, characteristic
4#20274235C212yesmaximal, complemented
5#20274211C212yesmaximal, complemented
6#20274111C212yesmaximal, complemented
7#20274087C212yesmaximal, complemented
8#20273467C211yesmaximal, characteristic, complemented
9#20273372C211yesmaximal, characteristic, complemented
10C2×#16810372C211yesmaximal, characteristic, complemented
11C53×#3165C211yesmaximal, characteristic, complemented
12C2×#13133225—21no—
13#16811173C2211yescharacteristic
14#16811019—21no—
15#16810971—21no—
16#16810941—22no—
17#16810910C2212yes—
18#16810910C2212yes—
19#16810910C2212yes—
20#16810908—21no—
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#16810372C2212yes—
34#16810372C2211yescharacteristic
35#16810372C2212yes—
36#16810372C2211yescharacteristic
37#16810369C2211yescharacteristic
38#16810369C2211yescharacteristic
39C53×#533C2211yescharacteristic
40C2×C4⋊Q8×C53C2211yescharacteristic
41C53×#501—22no—
42C53×#500C2212yes—
43C53×#498C2212yes—
44C53×#428—21no—
45C53×#428—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#13133148—22no—
62#13133148—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×C53C2311yescharacteristic
86C4⋊Q8×C53C2312yes—
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#107×C53—22no—
98#107×C53—22no—
99#106×C53—22no—
100#106×C53—22no—
101#97×C53D412yes—
102C4×C8×C53C2312yes—
103C2×Q8×C53—21no—
104C2×Q8×C53—21no—
105C2×Q8×C53—22no—
106C2×Q8×C53—41no—
107C2×Q8×C53—82no—
108C22×C4×C53—41no—
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53#4111yescharacteristic
112C2×C22⋊D53—22no—
113#11719945—42no—
114#11719945—42no—
115C2×D212—21no—
116C2×D212—21no—
117C2×D212C2×D412yes—
118C2×C4×D53—22no—
119C2×Dic106—21no—
120M16×C53—22no—
121M16×C53SD1614yescomplemented
122C2×C8×C53C2×D412yes—
123C2×C8×C53C2×D412yes—
124C4⋊C4×C53—21no—
125C4⋊C4×C53—22no—
126C4⋊C4×C53—21no—
127C4⋊C4×C53—22no—
128C4⋊C4×C53—21no—
129C4⋊C4×C53—21no—
130C4⋊C4×C53—41no—
131C4⋊C4×C53—41no—
132C4⋊C4×C53—41no—
133C4⋊C4×C53—41no—
134C22⋊C4×C53—41no—
135C42×C53—41no—
136C42×C53C2×D411yescharacteristic
137C42×C53C2×D411yescharacteristic
138C42×C53#4111yescharacteristic
139C42×C53C2411yescharacteristic
140#11719924—82no—
141#11719921—22no—
142#11719921—22no—
143C4⋊Dic53—21no—
144C4⋊Dic53C2×D412yes—
145C2×#88322—21no—
146C2×#88322—21no—
147C2×#88322—21no—
148C2×#88322—21no—
149C2×#88322—22no—
150C2×#88322—42no—
151C2×#88322—81no—
152#11719912—44no—
153C8×D53—44no—
154C23×C53#12111yescharacteristic
155C22×D53—22no—
156Q8×C53—41no—
157Q8×C53—41no—
158Q8×C53—82no—
159Q8×C53—81no—
160Q8×C53—82no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—41no—
164C2×C4×C53—42no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—82no—
168C2×C4×C53#12111yescharacteristic
169C2×C4×C53#12111yescharacteristic
170C2×C4×C53#12111yescharacteristic
171C2×C4×C53C2×SD1612yes—
172C2×C4×C53#13712yes—
173C2×C4×C53C22×D411yescharacteristic
174C2×C4×C53C22×D411yescharacteristic
175C2×C4×C53C2×#4111yescharacteristic
176C22⋊D53—44no—
177C2×Dic53—22no—
178D212—22no—
179D212—22no—
180D212—42no—
181D212—81no—
182C4×D53—42no—
183C4×D53—42no—
184Dic106—42no—
185C8×C53—22no—
186C8×C53C2×SD1614yes—
187#88322—41no—
188#88322—41no—
189#88322—41no—
190#88322—42no—
191#88322—41no—
192#19233—531nomaximal, Sylow
193C22×C53—21no—
194C22×C53—21no—
195C22×C53#46112yes—
196C22×C53C2×#12111yescharacteristic
197C2×D53—42no—
198C2×D53—42no—
199C2×D53—42no—
200C4×C53—21no—
201C4×C53—21no—
202C4×C53—21no—
203C4×C53—81no—
204C4×C53—82no—
205C4×C53—81no—
206C4×C53#46112yes—
207C4×C53C22×SD1611yescharacteristic
208C4×C53C2×#13711yescharacteristic
209C4×C53#57711yescharacteristic
210C4×C53#57811yescharacteristic
211Dic53—42no—
212#3246—531no—
213#3227—531no—
214#3165D5311yescharacteristic, complemented
215#2935—531no—
216#1695—532no—
217#1691—532no—
218#1675—532no—
219#1671—532no—
220#1565—531no—
221#1542—531no—
222C2×#333—531no—
223C2×C53—22no—
224C2×C53C2×#46111yescharacteristic
225C2×C53#306311yescharacteristic
226C2×C53#306811yescharacteristic
227D53—82no—
228C2×#131—1061no—
229#533C2×D5311yescharacteristic
230#532—531no—
231C2×C4⋊Q8C2×D5311yescharacteristic
232#501—22no—
233#500C2×D5312yes—
234#498—1061no—
235#498C2×D5312yes—
236#486—1061no—
237#481—1062no—
238#476—532no—
239#476—532no—
240#476—532no—
241#475—1061no—
242#465—1062no—
243#461—532no—
244#460—1061no—
245#436—532no—
246#435—1062no—
247C8×D4—532no—
248#431—531no—
249#428—21no—
250#428—21no—
251C2×C4⋊C8—531no—
252C2×C4⋊C8—531no—
253#408—1061no—
254C4×M16C2×D5311yescharacteristic
255#335—532no—
256#334—532no—
257#333—532no—
258#333—532no—
259#333—531no—
260#333—531no—
261#332—531no—
262#332—531no—
263C53#1923311yescharacteristic, complemented, Sylow
264C2×#41—1061no—
265C22×Q8—21no—
266C2×SD16—1062no—
267#131—2124no—
268C2×M16—1062no—
269C2×M16C22⋊D5311yescharacteristic
270C2×M16C22⋊D5311yescharacteristic
271C22×C8—1061no—
272C22×C8—1061no—
273C22×C8—1062no—
274C4⋊Q8—21no—
275C4⋊Q8—21no—
276C4⋊Q8—21no—
277C4⋊Q8—531no—
278C4⋊Q8C22×D5312yes—
279C4⋊Q8C22×D5311yescharacteristic
280C4⋊Q8C22×D5311yescharacteristic
281#127—531no—
282#125—21no—
283#124—1061no—
284C22⋊Q8—21no—
285C22⋊Q8—21no—
286#121—532no—
287#121—1061no—
288C4×Q8—21no—
289C4×Q8—21no—
290C4×D4—532no—
291C4×D4—1061no—
292#117—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C42C22×D5311yescharacteristic
296#107—22no—
297#107—22no—
298#106—22no—
299#106—22no—
300#106—1062no—
301#106—1062no—
302#106—1062no—
303C4⋊C8—531no—
304C4⋊C8—531no—
305C4⋊C8—531no—
306C4⋊C8—531no—
307C4⋊C8—532no—
308C4⋊C8—1061no—
309C4⋊C8—1061no—
310#103—1062no—
311#102—1062no—
312#102—1062no—
313#102—1062no—
314C22⋊C8—1061no—
315C22⋊C8—1062no—
316C22⋊C8—1062no—
317#97C22⋊D5312yes—
318C4×C8—1061no—
319C4×C8C22×D5312yes—
320#41—2122no—
321#41—2122no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—22no—
325C2×Q8—41no—
326C2×Q8—82no—
327C2×Q8—1061no—
328C2×D4—532no—
329C2×D4—1062no—
330C2×D4—1061no—
331C2×D4—1061no—
332C22×C4—41no—
333C22×C4—1062no—
334C22×C4D4×D5311yescharacteristic
335C22×C4#1171994711yescharacteristic
336C22×C4C2×C22⋊D5311yescharacteristic
337SD16—4242no—
338M16—22no—
339M16—2124no—
340M16#1171992414yescomplemented
341C2×C8—1061no—
342C2×C8—1061no—
343C2×C8—1062no—
344C2×C8—1061no—
345C2×C8—1061no—
346C2×C8—4241no—
347C2×C8—2122no—
348C2×C8—2124no—
349C2×C8C2×C22⋊D5312yes—
350C2×C8C2×C22⋊D5312yes—
351C4⋊C4—22no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—22no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—532no—
362C4⋊C4—1061no—
363C22⋊C4—41no—
364C22⋊C4—1062no—
365C22⋊C4—1062no—
366C42—41no—
367C42D4×D5311yescharacteristic
368C42#1171994711yescharacteristic
369C42C2×C22⋊D5311yescharacteristic
370C42C23×D5311yescharacteristic
371C23—1062no—
372C23#1313318811yescharacteristic
373Q8—41no—
374Q8—41no—
375Q8—81no—
376Q8—82no—
377Q8—82no—
378Q8—2122no—
379D4—1062no—
380D4—1062no—
381D4—4241no—
382D4—2124no—
383D4—2122no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—82no—
391C2×C4—1062no—
392C2×C4—2122no—
393C2×C4—2122no—
394C2×C4#1313318811yescharacteristic
395C2×C4#1313318811yescharacteristic
396C2×C4#1313318811yescharacteristic
397C2×C4C2×#1171992412yes—
398C2×C4#1313324812yes—
399C2×C4C2×D4×D5311yescharacteristic
400C2×C4C22×C22⋊D5311yescharacteristic
401C2×C4C2×#1171994711yescharacteristic
402C8—22no—
403C8—2121no—
404C8—2121no—
405C8—2121no—
406C8—2122no—
407C8—2121no—
408C8C2×#1171992414yes—
409C22—21no—
410C22—21no—
411C22—2122no—
412C22—2122no—
413C22—2122no—
414C22#1681082512yes—
415C22C2×#1313318811yescharacteristic
416C4—21no—
417C4—21no—
418C4—21no—
419C4—82no—
420C4—81no—
421C4—81no—
422C4—2122no—
423C4#1681082512yes—
424C4C22×#1171992411yescharacteristic
425C4C2×#1313324811yescharacteristic
426C4#1681143211yescharacteristic
427C4#1681143911yescharacteristic
428C2—22no—
429C2—4242no—
430C2C2×#1681082511yescharacteristic
431C2#2028440011yescharacteristic
432C2#2028443911yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3165C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenter#3165D53
hypocenterC53#19233
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
4. #20274235C2semidirect
5. #20274211C2semidirect
6. #20274111C2semidirect
7. #20274087C2semidirect
8. #20273467C2semidirect
9. #20273372C2semidirect
10. C2×#16810372C2semidirect
11. C53×#3165C2semidirect
121. M16×C53SD16semidirect
214. #3165D53semidirect
263. C53#19233semidirect
340. M16#11719924semidirect

Maximal subgroups

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

subgroupindex
#202860272
#202858432
#202834612
#202742352
#202742112
#202741112
#202740872
#202734672
#202733722
C2×#168103722
C53×#31652
#1923353