Finite Group Databasebeta

← #27367640

Subgroups — #27367640

Order 13568.

662 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286022C211yesmaximal, characteristic
2#20285840C211yesmaximal, characteristic, complemented
3#20285109C211yesmaximal, characteristic, complemented
4#20274290C212yesmaximal, complemented
5#20274248C212yesmaximal, complemented
6#20274166C212yesmaximal, complemented
7#20274124C212yesmaximal, complemented
8#20273467C211yesmaximal, characteristic, complemented
9C2×#16810371C211yesmaximal, characteristic, complemented
10#20273365C211yesmaximal, characteristic, complemented
11C53×#2935C211yesmaximal, characteristic, complemented
12#16811173C2211yescharacteristic
13C2×#13133217C2211yescharacteristic
14#16811019—21no—
15#16810983—22no—
16#16810974C2212yes—
17#16810970C2212yes—
18#16810969—21no—
19#16810939—22no—
20#16810913C2212yes—
21#16810913C2212yes—
22#16810903C2212yes—
23#16810900—21no—
24#16810857—22no—
25#16810818C2212yes—
26#16810815—21no—
27#16810676C2211yescharacteristic
28C2×C4⋊#88322C2211yescharacteristic
29#16810625—21no—
30#16810625—21no—
31#16810376C2212yes—
32#16810374C2212yes—
33#16810371C2212yes—
34#16810371C2212yes—
35#16810370C2211yescharacteristic
36#16810370C2211yescharacteristic
37#16810366C2211yescharacteristic
38#16810366C2211yescharacteristic
39C2×#131×C53—21no—
40C53×#532C2211yescharacteristic
41C53×#436C2212yes—
42C53×#435—22no—
43C8×D4×C53C2212yes—
44C2×C4⋊C8×C53C2211yescharacteristic
45C53×#408—21no—
46C4×M16×C53C2211yescharacteristic
47C2×#11719945—21no—
48C22×Dic106—21no—
49C2×#11719922—22no—
50C22×#88322—21no—
51#13133217—21no—
52#13133217—21no—
53#13133217—21no—
54#13133217C2311yescharacteristic
55#13133217C2312yes—
56#13133214C2311yescharacteristic
57#13133205—21no—
58#13133191—21no—
59#13133191—21no—
60C4×D212—21no—
61C4×D212—21no—
62C2×C4⋊Dic53—21no—
63C2×C4⋊Dic53—21no—
64#13133150—22no—
65#13133148—22no—
66#13133148—22no—
67C4⋊#88322—21no—
68C4⋊#88322C2312yes—
69C4⋊#88322C2311yescharacteristic
70C4⋊#88322C2311yescharacteristic
71#13133142—22no—
72#13133142—22no—
73#13133139—22no—
74#13133139—22no—
75#13133138—22no—
76#13133138—22no—
77#13133138—22no—
78#13133137—22no—
79C22⋊#88322—21no—
80C4×#88322—21no—
81C2×#41×C53—21no—
82#131×C53—44no—
83C2×M16×C53—22no—
84C2×M16×C53D411yescharacteristic
85C2×M16×C53D411yescharacteristic
86C22×C8×C53—22no—
87C22×C8×C53—21no—
88C4⋊Q8×C53D411yescharacteristic
89#127×C53D411yescharacteristic
90#124×C53—21no—
91#121×C53—21no—
92#121×C53D412yes—
93C4×D4×C53—21no—
94C4×D4×C53C2312yes—
95C2×C42×C53C2311yescharacteristic
96C4⋊C8×C53—21no—
97C4⋊C8×C53C2311yescharacteristic
98C4⋊C8×C53C2311yescharacteristic
99C22⋊C8×C53—22no—
100C22⋊C8×C53—22no—
101#97×C53D412yes—
102C4×C8×C53C2312yes—
103#41×C53—42no—
104#41×C53—42no—
105C2×Q8×C53—21no—
106C2×D4×C53—22no—
107C2×D4×C53—21no—
108C2×D4×C53—21no—
109C2×D4×C53C2×D412yes—
110C22×C4×C53—22no—
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53#4111yescharacteristic
114C2×C22⋊D53—41no—
115C22×Dic53—41no—
116#11719945—42no—
117#11719945—42no—
118C2×D212—21no—
119C2×D212—21no—
120C2×D212—21no—
121C2×C4×D53—41no—
122C2×Dic106—21no—
123C2×Dic106—22no—
124C2×Dic106—41no—
125C2×Dic106—82no—
126M16×C53—22no—
127M16×C53—22no—
128M16×C53—44no—
129C2×C8×C53—21no—
130C2×C8×C53—21no—
131C2×C8×C53—44no—
132C2×C8×C53—42no—
133C2×C8×C53C2×D412yes—
134C2×C8×C53C2×D412yes—
135C4⋊C4×C53—21no—
136C4⋊C4×C53C2×D412yes—
137C22⋊C4×C53—22no—
138C22⋊C4×C53—22no—
139C42×C53C2×D411yescharacteristic
140C42×C53C2×D411yescharacteristic
141C42×C53#4111yescharacteristic
142C42×C53C2411yescharacteristic
143#11719922—82no—
144#11719921—41no—
145#11719921—41no—
146C4⋊Dic53—22no—
147C4⋊Dic53—21no—
148C4⋊Dic53—22no—
149C4⋊Dic53—21no—
150C4⋊Dic53—41no—
151C4⋊Dic53—41no—
152C2×#88322—21no—
153C2×#88322—21no—
154C2×#88322—22no—
155C2×#88322—81no—
156C23×C53—22no—
157C23×C53#12111yescharacteristic
158C22×D53—41no—
159Q8×C53—42no—
160D4×C53—22no—
161D4×C53—22no—
162D4×C53—44no—
163D4×C53—42no—
164D4×C53—81no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—22no—
168C2×C4×C53—41no—
169C2×C4×C53—42no—
170C2×C4×C53—42no—
171C2×C4×C53#12111yescharacteristic
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53#12111yescharacteristic
174C2×C4×C53#13512yes—
175C2×C4×C53AGL(1,ℤ/8)12yes—
176C2×C4×C53C22×D411yescharacteristic
177C2×C4×C53C22×D411yescharacteristic
178C2×C4×C53C2×#4111yescharacteristic
179C22⋊D53—82no—
180C2×Dic53—42no—
181C2×Dic53—41no—
182C2×Dic53—82no—
183D212—41no—
184D212—41no—
185D212—41no—
186D212—41no—
187D212—81no—
188C4×D53—82no—
189Dic106—41no—
190Dic106—41no—
191Dic106—82no—
192Dic106—82no—
193C8×C53—22no—
194C8×C53—22no—
195C8×C53—41no—
196C8×C53—41no—
197#88322—41no—
198#88322—42no—
199#88322—41no—
200#19236—531nomaximal, Sylow
201C22×C53—21no—
202C22×C53—21no—
203C22×C53—42no—
204C22×C53—42no—
205C22×C53—42no—
206C22×C53#46312yes—
207C22×C53C2×#12111yescharacteristic
208C2×D53—81no—
209C2×D53—81no—
210C2×D53—81no—
211C4×C53—21no—
212C4×C53—21no—
213C4×C53—21no—
214C4×C53—42no—
215C4×C53#46312yes—
216C4×C53C2×#13511yescharacteristic
217C4×C53C2×AGL(1,ℤ/8)11yescharacteristic
218C4×C53#57711yescharacteristic
219C4×C53#57811yescharacteristic
220Dic53—81no—
221Dic53—82no—
222#3245—531no—
223#3227—531no—
224#3141—531no—
225#2935D5311yescharacteristic, complemented
226#1703—532no—
227#1697—532no—
228#1683—532no—
229#1677—532no—
230#1565—531no—
231C2×#333—531no—
232#1540—531no—
233C2×C53—22no—
234C2×C53—82no—
235C2×C53C2×#46311yescharacteristic
236C2×C53#306311yescharacteristic
237C2×C53#306811yescharacteristic
238D53—161no—
239C2×#131—21no—
240#532—531no—
241#532C2×D5311yescharacteristic
242C2×C4⋊Q8—531no—
243#498—1061no—
244#489—1062no—
245#487—532no—
246#486—532no—
247#486—1061no—
248#481—1062no—
249#476—532no—
250#476—532no—
251#474—532no—
252#474—1061no—
253#466—1062no—
254#459—532no—
255#459—1061no—
256#436C2×D5312yes—
257#435—22no—
258C8×D4C2×D5312yes—
259#431—531no—
260C2×C4⋊C8—531no—
261C2×C4⋊C8C2×D5311yescharacteristic
262#417—1061no—
263#417—1061no—
264#408—21no—
265C4×M16C2×D5311yescharacteristic
266#335—532no—
267#334—532no—
268#333—532no—
269#333—532no—
270#332—531no—
271#332—531no—
272#331—531no—
273#331—531no—
274C53#1923611yescharacteristic, complemented, Sylow
275C2×#41—21no—
276C2×#41—1061no—
277C22×Q8—1061no—
278C2×D8—1062no—
279#131—44no—
280C2×M16—22no—
281C2×M16C22⋊D5311yescharacteristic
282C2×M16C22⋊D5311yescharacteristic
283C22×C8—21no—
284C22×C8—22no—
285C22×C8—1061no—
286C4⋊Q8—532no—
287C4⋊Q8—531no—
288C4⋊Q8—1061no—
289C4⋊Q8—1061no—
290C4⋊Q8—1061no—
291C4⋊Q8D21211yescharacteristic
292#127—531no—
293#127D21211yescharacteristic
294#124—21no—
295#124—1061no—
296#121—21no—
297#121—1061no—
298#121—1061no—
299#121D21212yes—
300C4×D4—21no—
301C4×D4—1061no—
302C4×D4—1061no—
303C4×D4C22×D5312yes—
304C2×C4⋊C4—1061no—
305C2×C4⋊C4—1061no—
306C2×C42C22×D5311yescharacteristic
307#107—1062no—
308#106—1062no—
309#106—1062no—
310C4⋊C8—21no—
311C4⋊C8—531no—
312C4⋊C8—531no—
313C4⋊C8—532no—
314C4⋊C8—1061no—
315C4⋊C8C22×D5311yescharacteristic
316C4⋊C8C22×D5311yescharacteristic
317#103—1062no—
318#103—1062no—
319#102—1062no—
320#102—1062no—
321#102—1062no—
322#102—1062no—
323#102—1062no—
324#102—1062no—
325C22⋊C8—22no—
326C22⋊C8—22no—
327C22⋊C8—1061no—
328#97C22⋊D5312yes—
329C4×C8—1061no—
330C4×C8C22×D5312yes—
331#41—42no—
332#41—42no—
333#41—2122no—
334#41—2122no—
335C2×Q8—21no—
336C2×Q8—1061no—
337C2×Q8—1062no—
338C2×Q8—4242no—
339C2×Q8—2121no—
340C2×D4—22no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—1061no—
344C2×D4—1061no—
345C2×D4—1061no—
346C2×D4—2121no—
347C2×D4C2×D21212yes—
348C22×C4—22no—
349C22×C4—2121no—
350C22×C4—2121no—
351C22×C4C2×D21211yescharacteristic
352C22×C4#1171994511yescharacteristic
353C22×C4C2×C22⋊D5311yescharacteristic
354D8—4242no—
355M16—22no—
356M16—22no—
357M16—44no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—44no—
361C2×C8—42no—
362C2×C8—1061no—
363C2×C8—1061no—
364C2×C8—1062no—
365C2×C8—4241no—
366C2×C8C2×C22⋊D5312yes—
367C2×C8C2×C22⋊D5312yes—
368C4⋊C4—21no—
369C4⋊C4—1062no—
370C4⋊C4—1062no—
371C4⋊C4—1061no—
372C4⋊C4—1061no—
373C4⋊C4—2121no—
374C4⋊C4—2121no—
375C4⋊C4C2×D21212yes—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—2121no—
379C22⋊C4—2121no—
380C42C2×D21211yescharacteristic
381C42#1171994511yescharacteristic
382C42C2×C22⋊D5311yescharacteristic
383C42C23×D5311yescharacteristic
384C23—22no—
385C23—2121no—
386C23#1313319111yescharacteristic
387Q8—42no—
388Q8—4242no—
389Q8—4242no—
390Q8—2121no—
391Q8—2121no—
392D4—22no—
393D4—22no—
394D4—44no—
395D4—42no—
396D4—81no—
397D4—4241no—
398D4—4242no—
399D4—2121no—
400D4—2121no—
401D4—2121no—
402D4—2121no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—4242no—
410C2×C4—4242no—
411C2×C4—2121no—
412C2×C4—2122no—
413C2×C4#1313319111yescharacteristic
414C2×C4#1313319111yescharacteristic
415C2×C4#1313319111yescharacteristic
416C2×C4#1313323712yes—
417C2×C4#1313324312yes—
418C2×C4C22×C22⋊D5311yescharacteristic
419C2×C4C22×D21211yescharacteristic
420C2×C4C2×#1171994511yescharacteristic
421C8—22no—
422C8—22no—
423C8—41no—
424C8—41no—
425C8—2121no—
426C8—2122no—
427C8—2121no—
428C22—21no—
429C22—21no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—4241no—
434C22—4241no—
435C22—4241no—
436C22#1681083812yes—
437C22C2×#1313319111yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—42no—
442C4—4242no—
443C4—4241no—
444C4#1681083812yes—
445C4C2×#1313323711yescharacteristic
446C4C2×#1313324311yescharacteristic
447C4#1681143511yescharacteristic
448C4#1681144011yescharacteristic
449C2—22no—
450C2—82no—
451C2—8481no—
452C2C2×#1681083811yescharacteristic
453C2#2028440611yescharacteristic
454C2#2028444311yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2935C2
socleC22×C53C2×#121
prime radical#20286022C2
hypercenter#2935D53
hypocenterC53#19236
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2935

Complements

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

NH ≅ G/Ntype
2. #20285840C2semidirect
3. #20285109C2semidirect
4. #20274290C2semidirect
5. #20274248C2semidirect
6. #20274166C2semidirect
7. #20274124C2semidirect
8. #20273467C2semidirect
9. C2×#16810371C2semidirect
10. #20273365C2semidirect
11. C53×#2935C2semidirect
225. #2935D53semidirect
274. C53#19236semidirect

Maximal subgroups

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

subgroupindex
#202860222
#202858402
#202851092
#202742902
#202742482
#202741662
#202741242
#202734672
C2×#168103712
#202733652
C53×#29352
#1923653