Finite Group Databasebeta

← #27279327

Subgroups — #27279327

Order 13568.

500 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285270C211yesmaximal, characteristic
2#20275659C211yesmaximal, characteristic, complemented
3#20273210C211yesmaximal, characteristic
4#20273117C211yesmaximal, characteristic, complemented
5#20272554C211yesmaximal, characteristic, complemented
6#20272553C211yesmaximal, characteristic, complemented
7C53×#2935C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9#16811019C411yescharacteristic, complemented
10#16810995C411yescharacteristic, complemented
11#16810975C411yescharacteristic, complemented
12#16810969C411yescharacteristic, complemented
13#16810850—22no—
14C4×#11719923—22no—
15#16810676C2211yescharacteristic
16C2×C4×#88322C2211yescharacteristic
17C4×C4⋊Dic53C2211yescharacteristic
18#16810384—21no—
19#16810384—21no—
20#16810357—22no—
21#16810353—21no—
22#16810353—21no—
23C2×#131×C53—21no—
24C53×#532C2211yescharacteristic
25C53×#436—21no—
26C53×#435—22no—
27C8×D4×C53—21no—
28C2×C4⋊C8×C53C2211yescharacteristic
29C53×#408—21no—
30C4×M16×C53C2211yescharacteristic
31C2×#11719923—22no—
32C22×#88322—21no—
33#13133217C2×C411yescharacteristic
34#13133217C2×C411yescharacteristic
35#13133209—21no—
36#13133196—21no—
37#13133196—21no—
38C4×Dic106—21no—
39C4×Dic106—21no—
40#13133176—21no—
41C2×C4⋊Dic53—21no—
42C2×C4×Dic53—21no—
43#13133148—22no—
44C4⋊#88322—21no—
45#13133140—22no—
46#13133138—22no—
47C22⋊#88322—21no—
48C4×#88322—21no—
49C4×#88322—21no—
50C4×#88322C2×C411yescharacteristic
51C4×#88322C2×C411yescharacteristic
52#13133124—21no—
53C2×#41×C53—21no—
54#131×C53—42no—
55#131×C53—42no—
56C2×M16×C53—22no—
57C2×M16×C53D411yescharacteristic
58C2×M16×C53D411yescharacteristic
59C22×C8×C53—22no—
60C22×C8×C53—21no—
61C4⋊Q8×C53C2×C411yescharacteristic
62#127×C53C2×C411yescharacteristic
63#124×C53—21no—
64#121×C53—21no—
65#121×C53D412yes—
66C4×D4×C53—21no—
67C4×D4×C53D412yes—
68C2×C42×C53C2311yescharacteristic
69C4⋊C8×C53—21no—
70C4⋊C8×C53D411yescharacteristic
71C4⋊C8×C53D411yescharacteristic
72C22⋊C8×C53—22no—
73C22⋊C8×C53—22no—
74#97×C53—21no—
75C4×C8×C53—21no—
76#41×C53—41no—
77#41×C53—41no—
78#41×C53—41no—
79#41×C53—41no—
80C2×Q8×C53—21no—
81C2×D4×C53—22no—
82C2×D4×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53C22⋊C412yes—
85C22×C4×C53—22no—
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53#4111yescharacteristic
89C22×Dic53—41no—
90C2×Dic106—21no—
91C2×Dic106—21no—
92M16×C53—21no—
93M16×C53—21no—
94M16×C53—21no—
95M16×C53—44no—
96M16×C53SD1612yescomplemented
97C2×C8×C53—21no—
98C2×C8×C53—21no—
99C2×C8×C53—21no—
100C2×C8×C53—21no—
101C2×C8×C53—44no—
102C2×C8×C53—41no—
103C2×C8×C53—41no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53C22⋊C412yes—
106C22⋊C4×C53—22no—
107C22⋊C4×C53—22no—
108C42×C53C22×C411yescharacteristic
109C42×C53C2×D411yescharacteristic
110C42×C53C2×D411yescharacteristic
111C42×C53#4111yescharacteristic
112C22⋊Dic53—41no—
113#11719923—44no—
114C4⋊Dic53—22no—
115C4⋊Dic53—21no—
116C4⋊Dic53—21no—
117C4⋊Dic53—21no—
118C4⋊Dic53—21no—
119C4⋊Dic53—41no—
120#11719919—41no—
121#11719919—41no—
122C4×Dic53—41no—
123C4×Dic53—41no—
124C4×Dic53—41no—
125C2×#88322—21no—
126C2×#88322—21no—
127C2×#88322—42no—
128C2×#88322—41no—
129C23×C53—22no—
130C23×C53#12111yescharacteristic
131Q8×C53—41no—
132Q8×C53—41no—
133D4×C53—42no—
134D4×C53—42no—
135D4×C53—42no—
136D4×C53—41no—
137D4×C53—41no—
138D4×C53—41no—
139D4×C53—41no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—22no—
143C2×C4×C53—41no—
144C2×C4×C53—42no—
145C2×C4×C53—42no—
146C2×C4×C53C2×C22⋊C411yescharacteristic
147C2×C4×C53C4×D411yescharacteristic
148C2×C4×C53C4×D411yescharacteristic
149C2×C4×C53#12011yescharacteristic
150C2×C4×C53#12111yescharacteristic
151C2×C4×C53#12311yescharacteristic
152C2×C4×C53C2×SD1611yescharacteristic
153C2×C4×C53#13511yescharacteristic
154C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C53#13711yescharacteristic
156C2×Dic53—42no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—81no—
160C2×Dic53—81no—
161Dic106—42no—
162Dic106—41no—
163Dic106—41no—
164C8×C53—22no—
165C8×C53—41no—
166C8×C53—41no—
167C8×C53—41no—
168#88322—22no—
169#88322—41no—
170#88322—81no—
171#9055—531nomaximal, Sylow
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—42no—
175C22×C53—42no—
176C22×C53—42no—
177C22×C53#38611yescharacteristic
178C22×C53#46111yescharacteristic
179C22×C53#46311yescharacteristic
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—42no—
184C4×C53C4×SD1611yescharacteristic
185C4×C53#44011yescharacteristic
186C4×C53#44311yescharacteristic
187C4×C53#44411yescharacteristic
188C4×C53#45011yescharacteristic
189C4×C53#45211yescharacteristic
190Dic53—82no—
191Dic53—81no—
192Dic53—81no—
193#3167—531no—
194#2935D5311yescharacteristic
195#1986—531no—
196#1502—531no—
197#1481—531no—
198#1279—531no—
199#1278—531no—
200C2×C53—21no—
201C2×C53—21no—
202C2×C53—82no—
203C2×C53#187611yescharacteristic
204C2×C53#187711yescharacteristic
205C2×C53#188411yescharacteristic
206C2×#131—21no—
207#533—531no—
208#532C2×D5311yescharacteristic
209#498—531no—
210#492—531no—
211#487—531no—
212#486—531no—
213#465—1062no—
214C4×SD16—1062no—
215#436—21no—
216#435—22no—
217C8×D4—21no—
218#431—531no—
219C2×C4⋊C8C2×D5311yescharacteristic
220#408—21no—
221C4×M16C2×D5311yescharacteristic
222C2×C4×C8—531no—
223C4×C4⋊C4—531no—
224#339—1061no—
225#339—1061no—
226#327—1062no—
227Q8⋊C8—1061no—
228Q8⋊C8—1061no—
229C53#905511yescharacteristic, complemented, Sylow
230C2×#41—21no—
231C2×SD16—1062no—
232#131—42no—
233#131—42no—
234C2×M16—22no—
235C2×M16C22⋊D5311yescharacteristic
236C2×M16C22⋊D5311yescharacteristic
237C22×C8—22no—
238C22×C8—21no—
239C22×C8—1061no—
240C4⋊Q8—531no—
241C4⋊Q8—531no—
242C4⋊Q8C4×D5311yescharacteristic
243#127C4×D5311yescharacteristic
244#125—1061no—
245#124—21no—
246C22⋊Q8—1061no—
247C22⋊Q8—1061no—
248#121—21no—
249#121D21212yes—
250C4×Q8—1061no—
251C4×Q8—1061no—
252C4×D4—21no—
253C4×D4C22⋊D5312yes—
254#117—1061no—
255C2×C4⋊C4—1061no—
256C2×C42—1061no—
257C2×C42C22×D5311yescharacteristic
258#106—1062no—
259C4⋊C8—21no—
260C4⋊C8—1061no—
261C4⋊C8C22⋊D5311yescharacteristic
262C4⋊C8C22⋊D5311yescharacteristic
263#103—1062no—
264#102—1062no—
265C22⋊C8—22no—
266C22⋊C8—22no—
267C22⋊C8—1061no—
268#97—21no—
269C4×C8—21no—
270C4×C8—531no—
271C4×C8—531no—
272C4×C8—1061no—
273C4×C8—1061no—
274#95—1061no—
275#41—41no—
276#41—41no—
277#41—41no—
278#41—41no—
279C2×Q8—21no—
280C2×Q8—1061no—
281C2×Q8—1061no—
282C2×D4—22no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4#1171992112yes—
286C22×C4—22no—
287C22×C4—2121no—
288C22×C4C2×D21211yescharacteristic
289C22×C4#1171994511yescharacteristic
290C22×C4C2×C22⋊D5311yescharacteristic
291SD16—2124no—
292M16—21no—
293M16—21no—
294M16—21no—
295M16—44no—
296M16#1171992312yescomplemented
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—44no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—1061no—
305C2×C8—1061no—
306C2×C8—2122no—
307C2×C8—2121no—
308C4⋊C4—21no—
309C4⋊C4—1062no—
310C4⋊C4—1061no—
311C4⋊C4—1061no—
312C4⋊C4—1061no—
313C4⋊C4—1061no—
314C4⋊C4—2121no—
315C4⋊C4—2121no—
316C4⋊C4—2121no—
317C4⋊C4#1171992112yes—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—2121no—
321C42—2121no—
322C42—2121no—
323C42—2121no—
324C42C2×C4×D5311yescharacteristic
325C42#1171994511yescharacteristic
326C42C2×C22⋊D5311yescharacteristic
327C42C2×C22⋊D5311yescharacteristic
328C23—22no—
329C23#1313319111yescharacteristic
330Q8—41no—
331Q8—41no—
332Q8—2122no—
333Q8—2121no—
334Q8—2121no—
335D4—42no—
336D4—42no—
337D4—42no—
338D4—41no—
339D4—41no—
340D4—41no—
341D4—41no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—4241no—
349C2×C4—4241no—
350C2×C4—2122no—
351C2×C4—2121no—
352C2×C4—2121no—
353C2×C4C2×#1171992111yescharacteristic
354C2×C4C4×C22⋊D5311yescharacteristic
355C2×C4C4×C22⋊D5311yescharacteristic
356C2×C4#1313318611yescharacteristic
357C2×C4#1313319111yescharacteristic
358C2×C4#1313320011yescharacteristic
359C2×C4C2×#1171992311yescharacteristic
360C2×C4#1313323711yescharacteristic
361C2×C4#1313324311yescharacteristic
362C2×C4#1313324711yescharacteristic
363C8—22no—
364C8—41no—
365C8—41no—
366C8—41no—
367C8—1062no—
368C8—4241no—
369C8—2121no—
370C22—21no—
371C22—21no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22#1681050211yescharacteristic
376C22#1681082811yescharacteristic
377C22#1681083811yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—42no—
382C4—4242no—
383C4—4241no—
384C4—4241no—
385C4C4×#1171992311yescharacteristic
386C4#1681072311yescharacteristic
387C4#1681073811yescharacteristic
388C4#1681074311yescharacteristic
389C4#1681077411yescharacteristic
390C4#1681078511yescharacteristic
391C2—21no—
392C2—21no—
393C2—82no—
394C2#2027504211yescharacteristic
395C2#2027504911yescharacteristic
396C2#2027509811yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2935C2
socleC22×C53#386
prime radicalC53×#532C22
cosocleC2×C42×C53C23
hypercenter#2935D53
hypocenterC53#9055
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2935

Complements

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

NH ≅ G/Ntype
2. #20275659C2semidirect
4. #20273117C2semidirect
5. #20272554C2semidirect
6. #20272553C2semidirect
9. #16811019C4semidirect
10. #16810995C4semidirect
11. #16810975C4semidirect
12. #16810969C4semidirect
96. M16×C53SD16semidirect
229. C53#9055semidirect
296. M16#11719923semidirect

Maximal subgroups

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

subgroupindex
#202852702
#202756592
#202732102
#202731172
#202725542
#202725532
C53×#29352
#905553