Finite Group Databasebeta

← #27279317

Subgroups — #27279317

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#20283662C211yesmaximal, characteristic, complemented
3#20273213C211yesmaximal, characteristic, complemented
4#20273114C211yesmaximal, characteristic
5#20272556C211yesmaximal, characteristic, complemented
6#20272553C211yesmaximal, characteristic, complemented
7C53×#1980C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9#16811019C411yescharacteristic, complemented
10#16810995C411yescharacteristic, complemented
11#16810975C411yescharacteristic, complemented
12#16810969C411yescharacteristic, complemented
13#16810850—22no—
14#16810753—21no—
15#16810751—22no—
16C8×Dic106—21no—
17C4×#11719923—22no—
18#16810676C2211yescharacteristic
19#16810651—21no—
20C2×C4⋊#88322C2211yescharacteristic
21#16810581—21no—
22C2×C4×#88322C2211yescharacteristic
23#16810359—22no—
24D4⋊#88322—21no—
25D4⋊#88322—21no—
26C53×#532C2211yescharacteristic
27C2×#104×C53—21no—
28C2×#104×C53—21no—
29C4×M16×C53C2211yescharacteristic
30C4×C4⋊C4×C53C2211yescharacteristic
31C2×#11719923—22no—
32C22×#88322—21no—
33C22×#88322—21no—
34#13133217C2×C411yescharacteristic
35#13133217C2×C411yescharacteristic
36#13133209—21no—
37#13133196—21no—
38#13133196D412yes—
39C4×Dic106—21no—
40C4×Dic106D412yes—
41#13133176—21no—
42#13133148—22no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C4⋊#88322D411yescharacteristic
46C4⋊#88322D411yescharacteristic
47#13133146—22no—
48#13133146—22no—
49#13133140—22no—
50#13133138—22no—
51C22⋊#88322—21no—
52#13133127—22no—
53C4×#88322—21no—
54C4×#88322—21no—
55C4×#88322C2×C411yescharacteristic
56C4×#88322C2×C411yescharacteristic
57C8×Dic53—22no—
58C2×#41×C53—21no—
59C2×M16×C53D411yescharacteristic
60C2×M16×C53D411yescharacteristic
61C4⋊Q8×C53C2×C411yescharacteristic
62#127×C53C2×C411yescharacteristic
63#124×C53—21no—
64#121×C53—21no—
65#121×C53—21no—
66C4×D4×C53—21no—
67C4×D4×C53—21no—
68C2×C4⋊C4×C53—21no—
69C2×C42×C53—21no—
70C2×C42×C53C2311yescharacteristic
71#104×C53—42no—
72#104×C53—42no—
73#97×C53—21no—
74C4×C8×C53—21no—
75#95×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×C53—41no—
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53#4111yescharacteristic
90C2×Dic106—21no—
91C2×Dic106C22⋊C412yes—
92M16×C53—21no—
93M16×C53—21no—
94M16×C53—21no—
95M16×C53SD1612yescomplemented
96C2×C8×C53—21no—
97C2×C8×C53—21no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—41no—
102C22⋊C4×C53—41no—
103C22⋊C4×C53—41no—
104C42×C53—41no—
105C42×C53—41no—
106C42×C53C22×C411yescharacteristic
107C42×C53C2×D411yescharacteristic
108C42×C53C2×D411yescharacteristic
109C42×C53#4111yescharacteristic
110C22⋊Dic53—22no—
111#11719923—44no—
112C4⋊Dic53—21no—
113C4⋊Dic53—21no—
114C4⋊Dic53—21no—
115C4⋊Dic53C22⋊C412yes—
116#11719919—22no—
117#11719919—22no—
118C4×Dic53—22no—
119C2×#88322—21no—
120C2×#88322—21no—
121C2×#88322—21no—
122C2×#88322—21no—
123C2×#88322—42no—
124C2×#88322—41no—
125C2×#88322—41no—
126C2×#88322—41no—
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—42no—
142C2×C4×C53—41no—
143C2×C4×C53—81no—
144C2×C4×C53—81no—
145C2×C4×C53—81no—
146C2×C4×C53—81no—
147C2×C4×C53C2×C22⋊C411yescharacteristic
148C2×C4×C53C4×D411yescharacteristic
149C2×C4×C53C4×D411yescharacteristic
150C2×C4×C53#12011yescharacteristic
151C2×C4×C53#12111yescharacteristic
152C2×C4×C53#12311yescharacteristic
153C2×C4×C53C2×SD1611yescharacteristic
154C2×C4×C53#13511yescharacteristic
155C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C53#13711yescharacteristic
157C2×Dic53—22no—
158C2×Dic53—22no—
159Dic106—41no—
160Dic106—41no—
161Dic106—42no—
162C8×C53—22no—
163C8×C53—41no—
164#88322—22no—
165#88322—41no—
166#88322—41no—
167#88322—41no—
168#88322—81no—
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—82no—
182C4×C53—81no—
183C4×C53C4×SD1611yescharacteristic
184C4×C53#44011yescharacteristic
185C4×C53#44311yescharacteristic
186C4×C53#44411yescharacteristic
187C4×C53#44811yescharacteristic
188C4×C53#44911yescharacteristic
189Dic53—42no—
190Dic53—42no—
191#3167—531no—
192#2962—531no—
193#1980D5311yescharacteristic
194#1503—531no—
195#1480—531no—
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—1061no—
212#446—1062no—
213C8×Q8—1061no—
214C4×SD16—1062no—
215#431—531no—
216#424—1061no—
217C2×C4⋊C8—531no—
218C2×#104—21no—
219C2×#104—21no—
220#405—1061no—
221C4×M16C2×D5311yescharacteristic
222C2×C4×C8—531no—
223C4×C4⋊C4C2×D5311yescharacteristic
224#328—1062no—
225D4⋊C8—1061no—
226D4⋊C8—1061no—
227C53#905411yescharacteristic, complemented, Sylow
228C2×#41—21no—
229C2×SD16—1062no—
230C2×M16C22⋊D5311yescharacteristic
231C2×M16C22⋊D5311yescharacteristic
232C22×C8—1061no—
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—21no—
250C2×C42—21no—
251C2×C42C22×D5311yescharacteristic
252#106—1062no—
253C4⋊C8—531no—
254C4⋊C8—531no—
255C4⋊C8—1062no—
256C4⋊C8—1062no—
257C4⋊C8—1061no—
258C4⋊C8—1061no—
259#104—42no—
260#104—42no—
261#103—1062no—
262#102—1062no—
263C22⋊C8—1061no—
264#97—21no—
265#97—1062no—
266C4×C8—21no—
267C4×C8—531no—
268C4×C8—531no—
269C4×C8—1062no—
270C4×C8—1061no—
271C4×C8—1061no—
272#95—21no—
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—41no—
286C22×C4D4×D5311yescharacteristic
287C22×C4#1171994711yescharacteristic
288C22×C4C2×C22⋊D5311yescharacteristic
289SD16—2124no—
290M16—21no—
291M16—21no—
292M16—21no—
293M16#1171992312yescomplemented
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—1061no—
297C2×C8—1061no—
298C2×C8—1061no—
299C2×C8—1061no—
300C2×C8—2122no—
301C2×C8—2121no—
302C2×C8—2121no—
303C2×C8—2121no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—532no—
309C4⋊C4—1061no—
310C4⋊C4—1061no—
311C4⋊C4—1061no—
312C4⋊C4—1062no—
313C4⋊C4—1062no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—1062no—
317C42—41no—
318C42—41no—
319C42—1062no—
320C42C2×C4×D5311yescharacteristic
321C42#1171994511yescharacteristic
322C42D4×D5311yescharacteristic
323C42D4×D5311yescharacteristic
324C23—41no—
325C23#1313318811yescharacteristic
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—42no—
342C2×C4—41no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—1062no—
348C2×C4—1062no—
349C2×C4C22⋊C4×D5311yescharacteristic
350C2×C4#1313317711yescharacteristic
351C2×C4C4×C22⋊D5311yescharacteristic
352C2×C4#1313318511yescharacteristic
353C2×C4#1313319011yescharacteristic
354C2×C4#1313319911yescharacteristic
355C2×C4C2×#1171992311yescharacteristic
356C2×C4#1313323711yescharacteristic
357C2×C4#1313324311yescharacteristic
358C2×C4#1313324711yescharacteristic
359C8—22no—
360C8—41no—
361C8—1062no—
362C8—4241no—
363C8—2121no—
364C8—2121no—
365C8—2121no—
366C22—21no—
367C22—21no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22#1681049911yescharacteristic
372C22#1681082011yescharacteristic
373C22#1681084211yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—82no—
378C4—81no—
379C4—2122no—
380C4—2122no—
381C4C4×#1171992311yescharacteristic
382C4#1681072311yescharacteristic
383C4#1681073711yescharacteristic
384C4#1681074211yescharacteristic
385C4#1681076111yescharacteristic
386C4#1681076611yescharacteristic
387C2—21no—
388C2—21no—
389C2—161no—
390C2#2027503911yescharacteristic
391C2#2027507311yescharacteristic
392C2#2027509511yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1980C2
socleC22×C53#386
prime radicalC53×#532C22
cosocleC2×C42×C53C23
hypercenter#1980D53
hypocenterC53#9054
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1980

Complements

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

NH ≅ G/Ntype
2. #20283662C2semidirect
3. #20273213C2semidirect
5. #20272556C2semidirect
6. #20272553C2semidirect
9. #16811019C4semidirect
10. #16810995C4semidirect
11. #16810975C4semidirect
12. #16810969C4semidirect
95. M16×C53SD16semidirect
227. C53#9054semidirect
293. M16#11719923semidirect

Maximal subgroups

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

subgroupindex
#202852702
#202836622
#202732132
#202731142
#202725562
#202725532
C53×#19802
#905453