Finite Group Databasebeta

← #27279323

Subgroups — #27279323

Order 13568.

470 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285270C211yesmaximal, characteristic
2#20283658C211yesmaximal, characteristic, complemented
3#20275635C211yesmaximal, characteristic
4#20273215C211yesmaximal, characteristic, complemented
5#20272554C211yesmaximal, characteristic, complemented
6#20272551C211yesmaximal, characteristic, complemented
7C53×#1480C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9#16811019C411yescharacteristic, complemented
10#16810995C411yescharacteristic, complemented
11#16810975C411yescharacteristic, complemented
12#16810969C411yescharacteristic, complemented
13#16810850—22no—
14#16810752—22no—
15#16810750—21no—
16#16810747—21no—
17C4×#11719923—22no—
18#16810649—21no—
19C2×#13133144—21no—
20C2×#13133144—21no—
21#16810578—21no—
22C4×#11719917C2211yescharacteristic
23C2×C4×#88322C2211yescharacteristic
24C4×C4⋊Dic53C2211yescharacteristic
25#16810361—22no—
26C53×#532C2211yescharacteristic
27C53×#431C2211yescharacteristic
28C2×C4⋊C8×C53C2211yescharacteristic
29C53×D4⋊C8—21no—
30C53×D4⋊C8—21no—
31C2×#11719923—22no—
32C2×#11719917D411yescharacteristic
33C2×#11719917D411yescharacteristic
34C22×#88322—21no—
35#13133217C2×C411yescharacteristic
36#13133217C2×C411yescharacteristic
37#13133209—21no—
38#13133196—21no—
39#13133196D412yes—
40C4×Dic106—21no—
41C4×Dic106D412yes—
42#13133176—21no—
43C2×C4⋊Dic53—21no—
44C2×C4×Dic53—21no—
45#13133148—22no—
46#13133146—22no—
47#13133146—22no—
48#13133144—42no—
49#13133144—42no—
50#13133140—22no—
51#13133138—22no—
52#13133128—21no—
53#13133127—22no—
54C4×#88322—21no—
55C4×#88322—21no—
56C4×#88322C2×C411yescharacteristic
57C4×#88322C2×C411yescharacteristic
58C8×Dic53—22no—
59#13133124—21no—
60C2×#41×C53—21no—
61C22×C8×C53—21no—
62C4⋊Q8×C53C2×C411yescharacteristic
63#127×C53C2×C411yescharacteristic
64#124×C53—21no—
65#121×C53—21no—
66#121×C53—21no—
67C4×D4×C53—21no—
68C4×D4×C53—21no—
69C2×C42×C53C2311yescharacteristic
70C4⋊C8×C53—21no—
71C4⋊C8×C53—21no—
72C4⋊C8×C53D411yescharacteristic
73C4⋊C8×C53D411yescharacteristic
74C22⋊C8×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—21no—
82C2×D4×C53—21no—
83C2×D4×C53—21no—
84C2×D4×C53—41no—
85C22×C4×C53—41no—
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53#4111yescharacteristic
89C22×Dic53—41no—
90C2×Dic106—21no—
91C2×Dic106C22⋊C412yes—
92C2×C8×C53—21no—
93C2×C8×C53—21no—
94C2×C8×C53—41no—
95C2×C8×C53—41no—
96C2×C8×C53—41no—
97C4⋊C4×C53—21no—
98C4⋊C4×C53—21no—
99C22⋊C4×C53—41no—
100C22⋊C4×C53—41no—
101C42×C53C22×C411yescharacteristic
102C42×C53C2×D411yescharacteristic
103C42×C53C2×D411yescharacteristic
104C42×C53#4111yescharacteristic
105C22⋊Dic53—22no—
106#11719923—44no—
107C4⋊Dic53—21no—
108C4⋊Dic53—21no—
109C4⋊Dic53—21no—
110C4⋊Dic53—22no—
111C4⋊Dic53—41no—
112C4⋊Dic53C22⋊C412yes—
113#11719919—22no—
114#11719919—22no—
115C4×Dic53—22no—
116C4×Dic53—41no—
117C4×Dic53—41no—
118#11719917—21no—
119#11719917—21no—
120#11719917—21no—
121#11719917SD1612yescomplemented
122C2×#88322—21no—
123C2×#88322—21no—
124C2×#88322—21no—
125C2×#88322—21no—
126C2×#88322—42no—
127C23×C53—41no—
128C23×C53#12111yescharacteristic
129Q8×C53—41no—
130Q8×C53—41no—
131D4×C53—42no—
132D4×C53—41no—
133D4×C53—41no—
134D4×C53—41no—
135D4×C53—41no—
136D4×C53—81no—
137D4×C53—81no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—81no—
143C2×C4×C53—81no—
144C2×C4×C53C2×C22⋊C411yescharacteristic
145C2×C4×C53C4×D411yescharacteristic
146C2×C4×C53C4×D411yescharacteristic
147C2×C4×C53#12011yescharacteristic
148C2×C4×C53#12111yescharacteristic
149C2×C4×C53#12311yescharacteristic
150C2×C4×C53C2×SD1611yescharacteristic
151C2×C4×C53#13511yescharacteristic
152C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C53#13711yescharacteristic
154C2×Dic53—22no—
155C2×Dic53—22no—
156C2×Dic53—42no—
157C2×Dic53—81no—
158C2×Dic53—81no—
159Dic106—41no—
160Dic106—41no—
161Dic106—42no—
162C8×C53—41no—
163C8×C53—41no—
164C8×C53—81no—
165#88322—22no—
166#88322—22no—
167#88322—41no—
168#88322—41no—
169#9054—531nomaximal, Sylow
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—81no—
173C22×C53—81no—
174C22×C53—81no—
175C22×C53#38611yescharacteristic
176C22×C53#46011yescharacteristic
177C22×C53#46411yescharacteristic
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—81no—
182C4×C53C4×SD1611yescharacteristic
183C4×C53#44011yescharacteristic
184C4×C53#44311yescharacteristic
185C4×C53#44411yescharacteristic
186C4×C53#44811yescharacteristic
187C4×C53#44911yescharacteristic
188Dic53—42no—
189Dic53—42no—
190Dic53—82no—
191#3167—531no—
192#2962—531no—
193#1980—531no—
194#1503—531no—
195#1480D5311yescharacteristic
196#1279—531no—
197#1278—531no—
198C2×C53—21no—
199C2×C53—21no—
200C2×C53—161no—
201C2×C53#187611yescharacteristic
202C2×C53#188011yescharacteristic
203C2×C53#188411yescharacteristic
204#533—531no—
205#532C2×D5311yescharacteristic
206#498—531no—
207#492—531no—
208#487—531no—
209#486—531no—
210#465—1062no—
211#446—1062no—
212#446—1061no—
213C8×Q8—1061no—
214C4×SD16—1062no—
215#431C2×D5311yescharacteristic
216#424—1061no—
217C2×C4⋊C8C2×D5311yescharacteristic
218C2×#104—1061no—
219C2×#104—1061no—
220#405—1061no—
221C4×M16—531no—
222C2×C4×C8—531no—
223C4×C4⋊C4—531no—
224#328—1062no—
225D4⋊C8—21no—
226D4⋊C8—21no—
227C53#905411yescharacteristic, complemented, Sylow
228C2×#41—21no—
229C2×SD16—1062no—
230C2×M16—531no—
231C2×M16—531no—
232C22×C8—21no—
233C22×C8—1061no—
234C4⋊Q8—531no—
235C4⋊Q8—531no—
236C4⋊Q8C4×D5311yescharacteristic
237#127C4×D5311yescharacteristic
238#125—1061no—
239#124—21no—
240C22⋊Q8—532no—
241C22⋊Q8—1061no—
242#121—21no—
243#121—21no—
244C4×Q8—532no—
245C4×Q8—1061no—
246C4×D4—21no—
247C4×D4—21no—
248#117—1061no—
249C2×C4⋊C4—1061no—
250C2×C42—1061no—
251C2×C42C22×D5311yescharacteristic
252#106—1062no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—1062no—
256C4⋊C8—1062no—
257C4⋊C8C22⋊D5311yescharacteristic
258C4⋊C8C22⋊D5311yescharacteristic
259#104—2122no—
260#104—2122no—
261#103—1062no—
262#102—1062no—
263C22⋊C8—21no—
264#97—1062no—
265#97—1061no—
266C4×C8—21no—
267C4×C8—531no—
268C4×C8—531no—
269C4×C8—1062no—
270C4×C8—1061no—
271C4×C8—1061no—
272#95—1061no—
273#41—41no—
274#41—41no—
275#41—41no—
276#41—41no—
277C2×Q8—21no—
278C2×Q8—532no—
279C2×Q8—1061no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C22×C4—41no—
285C22×C4—2121no—
286C22×C4#1171994511yescharacteristic
287C22×C4D4×D5311yescharacteristic
288C22×C4D4×D5311yescharacteristic
289SD16—2124no—
290M16—532no—
291M16—1061no—
292M16—1061no—
293M16—1061no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—1061no—
300C2×C8—1061no—
301C2×C8—1061no—
302C2×C8—1061no—
303C2×C8—2122no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—532no—
307C4⋊C4—1062no—
308C4⋊C4—1061no—
309C4⋊C4—1061no—
310C4⋊C4—1061no—
311C4⋊C4—1062no—
312C4⋊C4—1062no—
313C4⋊C4—2121no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—1062no—
317C42—1062no—
318C42—2121no—
319C42—2121no—
320C42C2×C4×D5311yescharacteristic
321C42D4×D5311yescharacteristic
322C42#1171994711yescharacteristic
323C42C2×C22⋊D5311yescharacteristic
324C23—41no—
325C23#1313319011yescharacteristic
326Q8—41no—
327Q8—41no—
328Q8—2122no—
329Q8—2121no—
330Q8—2121no—
331D4—42no—
332D4—41no—
333D4—41no—
334D4—41no—
335D4—41no—
336D4—81no—
337D4—81no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—1062no—
345C2×C4—1062no—
346C2×C4—4241no—
347C2×C4—4241no—
348C2×C4—2122no—
349C2×C4C22⋊C4×D5311yescharacteristic
350C2×C4#1313317711yescharacteristic
351C2×C4C4×C22⋊D5311yescharacteristic
352C2×C4#1313318511yescharacteristic
353C2×C4#1313318811yescharacteristic
354C2×C4#1313319911yescharacteristic
355C2×C4SD16×D5311yescharacteristic
356C2×C4#1313323611yescharacteristic
357C2×C4#1313324111yescharacteristic
358C2×C4#1313324611yescharacteristic
359C8—41no—
360C8—41no—
361C8—81no—
362C8—1062no—
363C8—1062no—
364C8—2121no—
365C8—2121no—
366C22—21no—
367C22—21no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22#1681049911yescharacteristic
372C22#1681082211yescharacteristic
373C22#1681084411yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—81no—
378C4—4242no—
379C4—2122no—
380C4—2122no—
381C4#1681071411yescharacteristic
382C4#1681072011yescharacteristic
383C4#1681073811yescharacteristic
384C4#1681074311yescharacteristic
385C4#1681076011yescharacteristic
386C4#1681076511yescharacteristic
387C2—21no—
388C2—21no—
389C2—161no—
390C2#2027504511yescharacteristic
391C2#2027506711yescharacteristic
392C2#2027509511yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1480C2
socleC22×C53#386
prime radicalC53×#532C22
cosocleC2×C42×C53C23
hypercenter#1480D53
hypocenterC53#9054
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1480

Complements

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

NH ≅ G/Ntype
2. #20283658C2semidirect
4. #20273215C2semidirect
5. #20272554C2semidirect
6. #20272551C2semidirect
9. #16811019C4semidirect
10. #16810995C4semidirect
11. #16810975C4semidirect
12. #16810969C4semidirect
121. #11719917SD16semidirect
227. C53#9054semidirect

Maximal subgroups

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

subgroupindex
#202852702
#202836582
#202756352
#202732152
#202725542
#202725512
C53×#14802
#905453