Finite Group Databasebeta

← #27367550

Subgroups — #27367550

Order 13568.

584 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20286052C212yesmaximal
2#20285301C211yesmaximal, characteristic, complemented
3#20274233C211yesmaximal, characteristic, complemented
4#20274228C211yesmaximal, characteristic, complemented
5#20274209C211yesmaximal, characteristic, complemented
6#20274204C211yesmaximal, characteristic, complemented
7#20274145C212yesmaximal, complemented
8#20274138C212yesmaximal, complemented
9C2×#16810374C211yesmaximal, characteristic, complemented
10#20273416C212yesmaximal, complemented
11C53×#2966C211yesmaximal, characteristic
12#16811180C2212yes—
13#16811028C2211yescharacteristic
14#16811003C2211yescharacteristic
15#16810974C2212yes—
16#16810927—22no—
17#16810908C2212yes—
18#16810903C2212yes—
19#16810864—21no—
20#16810864—21no—
21#16810862—22no—
22#16810850—22no—
23#16810842—22no—
24#16810833C2212yes—
25#16810828C2212yes—
26C4×Dic212—21no—
27C4×Dic212—21no—
28C2×C4⋊#88322C2212yes—
29#16810374C2211yescharacteristic
30#16810374C2211yescharacteristic
31#16810374C2212yes—
32#16810370C2212yes—
33#16810370C2212yes—
34#16810368C2212yes—
35#16810368C2212yes—
36C53×#529C2211yescharacteristic
37C53×#446C2211yescharacteristic
38C8×Q8×C53C2211yescharacteristic
39C53×#436C2211yescharacteristic
40C53×#435—21no—
41C8×D4×C53—21no—
42C8×D4×C53C2211yescharacteristic
43C53×#433—21no—
44C53×#432C2211yescharacteristic
45C53×#405—21no—
46C2×C4×C8×C53C2211yescharacteristic
47C2×Dic212—21no—
48C2×Dic212—21no—
49C2×#11719925—22no—
50C2×#11719923—22no—
51C22×#88322—22no—
52#13133217C2312yes—
53#13133217C2312yes—
54#13133209—22no—
55#13133196—22no—
56#13133196—22no—
57C4×Dic106—22no—
58C4×Dic106—22no—
59#13133176—22no—
60#13133151—21no—
61#13133151—21no—
62#13133150—22no—
63#13133148—22no—
64C4⋊#88322—22no—
65C4⋊#88322C2312yes—
66C4⋊#88322C2312yes—
67#13133142—22no—
68#13133142—22no—
69#13133141—22no—
70#13133140—22no—
71#13133140—22no—
72#13133138—22no—
73C2×M16×C53—21no—
74C22×C8×C53—21no—
75C22×C8×C53D411yescharacteristic
76C22×C8×C53D411yescharacteristic
77#126×C53—21no—
78#125×C53D411yescharacteristic
79#124×C53D411yescharacteristic
80#123×C53—21no—
81C22⋊Q8×C53D411yescharacteristic
82#121×C53D411yescharacteristic
83C4×Q8×C53C2311yescharacteristic
84C4×D4×C53—21no—
85C4×D4×C53C2311yescharacteristic
86#117×C53—21no—
87C2×C42×C53C2311yescharacteristic
88C4⋊C8×C53—21no—
89C4⋊C8×C53—21no—
90C4⋊C8×C53C2311yescharacteristic
91C4⋊C8×C53C2311yescharacteristic
92C22⋊C8×C53—21no—
93C22⋊C8×C53—21no—
94C22⋊C8×C53—21no—
95#97×C53—21no—
96#97×C53—21no—
97C4×C8×C53—21no—
98C4×C8×C53D411yescharacteristic
99C4×C8×C53D411yescharacteristic
100C4×C8×C53C2311yescharacteristic
101C4×C8×C53C2311yescharacteristic
102C2×Q8×C53C2×D411yescharacteristic
103C2×D4×C53—21no—
104C2×D4×C53C2×D411yescharacteristic
105C22×C4×C53—21no—
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53#4111yescharacteristic
109C2×Dic106—22no—
110C2×Dic106—22no—
111M16×C53—42no—
112C2×C8×C53—22no—
113C2×C8×C53—21no—
114C2×C8×C53—21no—
115C2×C8×C53—41no—
116C2×C8×C53—42no—
117C2×C8×C53C2×D411yescharacteristic
118C2×C8×C53C2×D411yescharacteristic
119C2×C8×C53C2×D411yescharacteristic
120C2×C8×C53C2×D411yescharacteristic
121C4⋊C4×C53—21no—
122C4⋊C4×C53—21no—
123C4⋊C4×C53—21no—
124C4⋊C4×C53—21no—
125C4⋊C4×C53C2×D411yescharacteristic
126C4⋊C4×C53C2×D411yescharacteristic
127C22⋊C4×C53—21no—
128C22⋊C4×C53—21no—
129C22⋊C4×C53—21no—
130C22⋊C4×C53—21no—
131C22⋊C4×C53—21no—
132C42×C53—21no—
133C42×C53C2×D411yescharacteristic
134C42×C53C2×D411yescharacteristic
135C42×C53#4111yescharacteristic
136C42×C53C2411yescharacteristic
137C22⋊Dic53—42no—
138#11719925—82no—
139#11719923—82no—
140C4⋊Dic53—22no—
141C4⋊Dic53—22no—
142C4⋊Dic53—22no—
143C4⋊Dic53—22no—
144#11719919—42no—
145#11719919—42no—
146C4×Dic53—42no—
147C2×#88322—22no—
148C2×#88322—22no—
149C2×#88322—82no—
150Dic212—42no—
151Dic212—42no—
152C23×C53—21no—
153C23×C53#12111yescharacteristic
154Q8×C53—22no—
155D4×C53—22no—
156D4×C53—42no—
157C2×C4×C53—22no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—22no—
161C2×C4×C53—21no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53#12111yescharacteristic
167C2×C4×C53#12111yescharacteristic
168C2×C4×C53#12111yescharacteristic
169C2×C4×C53C22×D411yescharacteristic
170C2×C4×C53C22×D411yescharacteristic
171C2×C4×C53C2×#4111yescharacteristic
172C2×Dic53—42no—
173C2×Dic53—42no—
174Dic106—42no—
175Dic106—42no—
176Dic106—42no—
177Dic106—42no—
178C8×C53—21no—
179C8×C53—21no—
180C8×C53—21no—
181C8×C53—41no—
182C8×C53—41no—
183C8×C53#13512yes—
184#88322—42no—
185#88322—42no—
186#19228—531nomaximal, Sylow
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—41no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53#47312yes—
193C22×C53C2×#12111yescharacteristic
194C4×C53—21no—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—21no—
198C4×C53—41no—
199C4×C53—41no—
200C4×C53—41no—
201C4×C53C2×#13511yescharacteristic
202C4×C53#57511yescharacteristic
203C4×C53#57712yes—
204Dic53—82no—
205Dic53—82no—
206#3248—532no—
207#3170—531no—
208#2966D5311yescharacteristic
209#1694—531no—
210#1693—531no—
211#1690—531no—
212#1689—531no—
213#1680—532no—
214#1679—532no—
215C2×#334—531no—
216#1552—532no—
217C2×C53—22no—
218C2×C53—81no—
219C2×C53#303511yescharacteristic
220C2×C53#306011yescharacteristic
221C2×C53C2×#47311yescharacteristic
222#533—532no—
223#529C2×D5311yescharacteristic
224#500—531no—
225#494—531no—
226#487—532no—
227#479—1062no—
228#475—532no—
229#474—532no—
230#467—1061no—
231#467—1061no—
232#467—1062no—
233#465—1062no—
234#464—1062no—
235#462—532no—
236#461—532no—
237#446C2×D5311yescharacteristic
238C8×Q8C2×D5311yescharacteristic
239C4×Q16—1061no—
240C4×Q16—1061no—
241#436C2×D5311yescharacteristic
242#435—21no—
243C8×D4—21no—
244C8×D4C2×D5311yescharacteristic
245#433—21no—
246#432C2×D5311yescharacteristic
247C2×C4⋊C8—532no—
248#405—21no—
249C2×C4×C8C2×D5311yescharacteristic
250#334—532no—
251#334—531no—
252#334—531no—
253#332—532no—
254#332—532no—
255#332—532no—
256#332—532no—
257C53#1922811yescharacteristic, complemented, Sylow
258C2×Q16—1062no—
259C2×Q16—1061no—
260C2×Q16—1061no—
261C2×SD16—1062no—
262C2×M16—21no—
263C22×C8—21no—
264C22×C8—1062no—
265C22×C8C22⋊D5311yescharacteristic
266C22×C8C22⋊D5311yescharacteristic
267C4⋊Q8—532no—
268C4⋊Q8—532no—
269#126—21no—
270#125—1062no—
271#125D21211yescharacteristic
272#124D21211yescharacteristic
273#123—21no—
274C22⋊Q8—1062no—
275C22⋊Q8—1062no—
276C22⋊Q8D21211yescharacteristic
277#121D21211yescharacteristic
278C4×Q8—1062no—
279C4×Q8—1062no—
280C4×Q8C22×D5311yescharacteristic
281C4×D4—21no—
282C4×D4C22×D5311yescharacteristic
283#117—21no—
284#117—1062no—
285C2×C42C22×D5311yescharacteristic
286#107—1061no—
287#107—1061no—
288#107—1062no—
289#106—1062no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—532no—
293C4⋊C8—532no—
294C4⋊C8—1062no—
295C4⋊C8C22×D5311yescharacteristic
296C4⋊C8C22×D5311yescharacteristic
297#103—1062no—
298#103—1062no—
299#103—1062no—
300#103—1062no—
301#103—1062no—
302#102—1062no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8—21no—
306#97—21no—
307#97—21no—
308C4×C8—21no—
309C4×C8C22⋊D5311yescharacteristic
310C4×C8C22⋊D5311yescharacteristic
311C4×C8C22×D5311yescharacteristic
312C4×C8C22×D5311yescharacteristic
313C2×Q8—1062no—
314C2×Q8—1062no—
315C2×Q8C2×D21211yescharacteristic
316C2×D4—21no—
317C2×D4C2×D21211yescharacteristic
318C22×C4—21no—
319C22×C4C2×D21211yescharacteristic
320C22×C4#1171994511yescharacteristic
321C22×C4C2×C22⋊D5311yescharacteristic
322Q16—4242no—
323Q16—2122no—
324Q16—2122no—
325SD16—4242no—
326M16—42no—
327C2×C8—22no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—41no—
331C2×C8—42no—
332C2×C8—1062no—
333C2×C8—1062no—
334C2×C8—4242no—
335C2×C8C2×C22⋊D5311yescharacteristic
336C2×C8C2×C22⋊D5311yescharacteristic
337C2×C8C2×C22⋊D5311yescharacteristic
338C2×C8C2×C22⋊D5311yescharacteristic
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—1062no—
344C4⋊C4—1062no—
345C4⋊C4—1062no—
346C4⋊C4—1062no—
347C4⋊C4—2122no—
348C4⋊C4—2122no—
349C4⋊C4C2×D21211yescharacteristic
350C4⋊C4C2×D21211yescharacteristic
351C22⋊C4—21no—
352C22⋊C4—21no—
353C22⋊C4—21no—
354C22⋊C4—21no—
355C22⋊C4—21no—
356C22⋊C4—2122no—
357C42—21no—
358C42—2122no—
359C42C2×D21211yescharacteristic
360C42#1171994511yescharacteristic
361C42C2×C22⋊D5311yescharacteristic
362C42C23×D5311yescharacteristic
363C23—21no—
364C23#1313319111yescharacteristic
365Q8—22no—
366Q8—2122no—
367Q8—2122no—
368Q8—2122no—
369Q8—2122no—
370D4—22no—
371D4—42no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—2122no—
382C2×C4—2122no—
383C2×C4#1313319111yescharacteristic
384C2×C4#1313319111yescharacteristic
385C2×C4#1313319111yescharacteristic
386C2×C4C22×C22⋊D5311yescharacteristic
387C2×C4C22×D21211yescharacteristic
388C2×C4C2×#1171994511yescharacteristic
389C8—21no—
390C8—21no—
391C8—21no—
392C8—41no—
393C8—41no—
394C8—2122no—
395C8—2122no—
396C8#1313323712yes—
397C22—21no—
398C22—21no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22#1681089812yes—
403C22C2×#1313319111yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—41no—
409C4—41no—
410C4—41no—
411C4—4242no—
412C4—4242no—
413C4C2×#1313323711yescharacteristic
414C4#1681142111yescharacteristic
415C4#1681143512yes—
416C2—22no—
417C2—81no—
418C2#2028417011yescharacteristic
419C2#2028437711yescharacteristic
420C2C2×#1681089811yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2966C2
socleC22×C53C2×#121
prime radical#121×C53D4
hypercenter#2966D53
hypocenterC53#19228
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2966

Complements

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

NH ≅ G/Ntype
2. #20285301C2semidirect
3. #20274233C2semidirect
4. #20274228C2semidirect
5. #20274209C2semidirect
6. #20274204C2semidirect
7. #20274145C2semidirect
8. #20274138C2semidirect
9. C2×#16810374C2semidirect
10. #20273416C2semidirect
257. C53#19228semidirect

Maximal subgroups

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

subgroupindex
#202860522
#202853012
#202742332
#202742282
#202742092
#202742042
#202741452
#202741382
C2×#168103742
#202734162
C53×#29662
#1922853