Finite Group Databasebeta

← #27278407

Subgroups — #27278407

Order 13568.

438 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283284C211yesmaximal, characteristic
2#20274736C211yesmaximal, characteristic
3#20273486C211yesmaximal, characteristic, complemented
4#20273470C211yesmaximal, characteristic, complemented
5#20273095C211yesmaximal, characteristic
6#20272556C211yesmaximal, characteristic, complemented
7C53×#1303C211yesmaximal, characteristic, complemented
8C2×#13133225—21no—
9C4×#11719945C2211yescharacteristic
10#16810651—21no—
11#16810648—22no—
12#16810647C2211yescharacteristic
13#16810645—21no—
14C2×C4⋊#88322C2211yescharacteristic
15C4⋊C8×D53—21no—
16C2×#13133145—21no—
17C2×C4×#88322C2211yescharacteristic
18#16810383—21no—
19#16810374—21no—
20#16810374—21no—
21#16810373—22no—
22#16810353C411yescharacteristic, complemented
23#16810353C411yescharacteristic, complemented
24#16810350C411yescharacteristic, complemented
25#16810350C411yescharacteristic, complemented
26C53×#431C2211yescharacteristic
27C53×#423C2211yescharacteristic
28C4×C4⋊C4×C53C2211yescharacteristic
29C2×#11719945D411yescharacteristic
30#13133225—42no—
31#13133225—42no—
32C2×#11719917—21no—
33C2×#11719912—22no—
34C22×#88322—21no—
35C22×#88322—21no—
36C2×C8×D53—22no—
37C4×Dic106C2×C411yescharacteristic
38C4×Dic106Q811yescharacteristic
39C4×D212C2×C411yescharacteristic
40C4×D212Q811yescharacteristic
41C4×C22⋊D53—21no—
42C4×C22⋊D53—21no—
43#13133176D411yescharacteristic
44#13133174—21no—
45C42×D53—21no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48C4⋊#88322—21no—
49C4⋊#88322D411yescharacteristic
50C4⋊#88322D411yescharacteristic
51#13133146—22no—
52#13133146—22no—
53#13133145—42no—
54C4×#88322—21no—
55C4×#88322C2×C411yescharacteristic
56C4×#88322C2×C411yescharacteristic
57C2×M16×C53—21no—
58C2×C4⋊C4×C53—21no—
59C2×C42×C53—21no—
60C2×C42×C53C2311yescharacteristic
61C4⋊C8×C53—21no—
62C4⋊C8×C53C2×C411yescharacteristic
63C4⋊C8×C53C2×C411yescharacteristic
64C4⋊C8×C53D412yes—
65C22⋊C8×C53—21no—
66C4×C8×C53—21no—
67#95×C53—21no—
68C22×C4×C53—41no—
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53#4111yescharacteristic
71C22×C4×C53#4111yescharacteristic
72C2×C22⋊D53—21no—
73#11719945—21no—
74#11719945—21no—
75#11719945SD1612yes—
76#11719945SD1612yes—
77C2×D212C4⋊C411yescharacteristic
78C2×C4×D53—21no—
79C2×C4×D53—21no—
80C2×C4×D53—21no—
81C2×Dic106C4⋊C411yescharacteristic
82M16×C53—41no—
83M16×C53—41no—
84C2×C8×C53—21no—
85C2×C8×C53—21no—
86C2×C8×C53—22no—
87C4⋊C4×C53—22no—
88C4⋊C4×C53—41no—
89C42×C53—41no—
90C42×C53—41no—
91C42×C53C22×C411yescharacteristic
92C42×C53C2×D411yescharacteristic
93C42×C53C2×D411yescharacteristic
94C42×C53C2×Q811yescharacteristic
95C22⋊Dic53—21no—
96#11719921—21no—
97#11719921—21no—
98C4⋊Dic53C4⋊C411yescharacteristic
99C4⋊Dic53C4⋊C411yescharacteristic
100#11719919—21no—
101#11719919—21no—
102C4×Dic53—21no—
103C4×Dic53—21no—
104C4×Dic53—21no—
105#11719917—41no—
106#11719917—41no—
107C2×#88322—21no—
108C2×#88322—21no—
109C2×#88322—21no—
110C2×#88322—21no—
111C2×#88322—42no—
112C2×#88322—41no—
113C2×#88322—41no—
114C2×#88322—41no—
115#11719912—44no—
116C8×D53—44no—
117C23×C53#12311yescharacteristic
118C22×D53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—41no—
122C2×C4×C53—42no—
123C2×C4×C53—81no—
124C2×C4×C53—81no—
125C2×C4×C53C2×C4⋊C411yescharacteristic
126C2×C4×C53C4×D411yescharacteristic
127C2×C4×C53C4×D411yescharacteristic
128C2×C4×C53#12011yescharacteristic
129C2×C4×C53C22⋊Q811yescharacteristic
130C2×C4×C53C22⋊Q811yescharacteristic
131C2×C4×C53C2×SD1611yescharacteristic
132C2×C4×C53C2×SD1611yescharacteristic
133C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C53#13711yescharacteristic
135C22⋊D53—22no—
136C22⋊D53—22no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140D212—21no—
141D212#10612yes—
142C4×D53—22no—
143C4×D53—22no—
144C4×D53—41no—
145C4×D53—41no—
146C4×D53—81no—
147Dic106—21no—
148Dic106#10612yes—
149C8×C53—41no—
150C8×C53—41no—
151C8×C53—42no—
152#88322—22no—
153#88322—41no—
154#88322—41no—
155#88322—41no—
156#88322—81no—
157#8904—531nomaximal, Sylow
158C22×C53—21no—
159C22×C53—21no—
160C22×C53#38511yescharacteristic
161C22×C53#48111yescharacteristic
162C22×C53#48311yescharacteristic
163C2×D53—22no—
164C2×D53—41no—
165C2×D53—41no—
166C4×C53—21no—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—82no—
170C4×C53C2×#10611yescharacteristic
171C4×C53#42811yescharacteristic
172C4×C53#44311yescharacteristic
173C4×C53#44411yescharacteristic
174C4×C53#44811yescharacteristic
175C4×C53#45011yescharacteristic
176Dic53—22no—
177Dic53—41no—
178Dic53—41no—
179#2906—531no—
180#1806—531no—
181#1570—531no—
182#1566—531no—
183#1476—531no—
184#1303D5311yescharacteristic, complemented
185#1279—531no—
186C2×C53—21no—
187C2×C53—21no—
188C2×C53#185311yescharacteristic
189C2×C53#186111yescharacteristic
190C2×C53#186611yescharacteristic
191D53—42no—
192C2×#131—1061no—
193C4×#41—531no—
194#431C2×D5311yescharacteristic
195#424—1061no—
196#424—1062no—
197#423—531no—
198#423—1061no—
199#423C2×D5311yescharacteristic
200C2×C4⋊C8—531no—
201C2×C4⋊C8—1061no—
202C2×#104—1061no—
203C2×C4×C8—531no—
204C4×C4⋊C4C2×D5311yescharacteristic
205#339—1061no—
206#334—1061no—
207#334—1061no—
208#334—1062no—
209Q8⋊C8—531no—
210Q8⋊C8—531no—
211D4⋊C8—531no—
212D4⋊C8—531no—
213C53#890411yescharacteristic, complemented, Sylow
214C2×#41—531no—
215#131—2122no—
216#131—2122no—
217C2×M16—21no—
218C2×M16—1062no—
219C2×M16—1061no—
220C22×C8—1062no—
221C22×C8—1061no—
222C22×C8—1061no—
223C4×Q8—531no—
224C4×Q8—531no—
225C4×D4—531no—
226C4×D4—531no—
227C4×D4—1061no—
228C4×D4—1061no—
229#117—531no—
230#117—1061no—
231C2×C4⋊C4—21no—
232C2×C42—21no—
233C2×C42—1061no—
234C2×C42C22×D5311yescharacteristic
235C4⋊C8—21no—
236C4⋊C8—531no—
237C4⋊C8—531no—
238C4⋊C8—1062no—
239C4⋊C8—1062no—
240C4⋊C8—1061no—
241C4⋊C8—1061no—
242C4⋊C8—1061no—
243C4⋊C8C4×D5311yescharacteristic
244C4⋊C8C4×D5311yescharacteristic
245C4⋊C8D21212yes—
246#104—2122no—
247C22⋊C8—21no—
248C4×C8—21no—
249C4×C8—531no—
250C4×C8—531no—
251C4×C8—1061no—
252#95—21no—
253#41—532no—
254#41—532no—
255#41—1061no—
256#41—1061no—
257C2×Q8—531no—
258C2×D4—531no—
259C2×D4—1061no—
260C22×C4—41no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4—1061no—
264C22×C4#1171994511yescharacteristic
265C22×C4D4×D5311yescharacteristic
266C22×C4#1171994711yescharacteristic
267M16—41no—
268M16—41no—
269M16—2124no—
270M16—2121no—
271M16—2121no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—22no—
275C2×C8—1061no—
276C2×C8—1061no—
277C2×C8—1061no—
278C2×C8—1061no—
279C2×C8—2124no—
280C2×C8—2122no—
281C2×C8—2121no—
282C2×C8—2121no—
283C2×C8—2121no—
284C4⋊C4—22no—
285C4⋊C4—41no—
286C4⋊C4—531no—
287C4⋊C4—531no—
288C4⋊C4—1061no—
289C4⋊C4—1061no—
290C22⋊C4—1061no—
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C42—41no—
294C42—41no—
295C42—1061no—
296C42—1061no—
297C42—1061no—
298C42C2×C4×D5311yescharacteristic
299C42C2×D21211yescharacteristic
300C42D4×D5311yescharacteristic
301C42Q8×D5311yescharacteristic
302C23—1061no—
303C23#1313319911yescharacteristic
304Q8—532no—
305Q8—1061no—
306D4—532no—
307D4—1062no—
308D4—1062no—
309D4—1061no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—1061no—
317C2×C4—1062no—
318C2×C4—1062no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—4241no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4C4⋊C4×D5311yescharacteristic
325C2×C4#1313317711yescharacteristic
326C2×C4C4×D21211yescharacteristic
327C2×C4#1313318711yescharacteristic
328C2×C4#1313319511yescharacteristic
329C2×C4#1313319711yescharacteristic
330C2×C4SD16×D5311yescharacteristic
331C2×C4SD16×D5311yescharacteristic
332C2×C4#1313324111yescharacteristic
333C2×C4#1313324911yescharacteristic
334C8—42no—
335C8—41no—
336C8—41no—
337C8—1062no—
338C8—4241no—
339C8—2121no—
340C8—2121no—
341C8—2121no—
342C22—21no—
343C22—21no—
344C22—1062no—
345C22—2121no—
346C22—2121no—
347C22#1681049711yescharacteristic
348C22#1681094011yescharacteristic
349C22#1681095211yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—82no—
354C4—1062no—
355C4—2121no—
356C4—2121no—
357C4#106×D5311yescharacteristic
358C4#1681066411yescharacteristic
359C4#1681073711yescharacteristic
360C4#1681074211yescharacteristic
361C4#1681076311yescharacteristic
362C4#1681077511yescharacteristic
363C2—21no—
364C2—21no—
365C2—2122no—
366C2#2027490911yescharacteristic
367C2#2027495811yescharacteristic
368C2#2027499111yescharacteristic
subgroupG/H
centerC2×C4#13133187
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1303C2
socleC22×C53#385
prime radicalC2×#11719945D4
cosocleC2×C42×C53C23
hypercenter#1303D53
hypocenterC53#8904
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
3. #20273486C2semidirect
4. #20273470C2semidirect
6. #20272556C2semidirect
7. C53×#1303C2semidirect
22. #16810353C4semidirect
23. #16810353C4semidirect
24. #16810350C4semidirect
25. #16810350C4semidirect
184. #1303D53semidirect
213. C53#8904semidirect

Maximal subgroups

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

subgroupindex
#202832842
#202747362
#202734862
#202734702
#202730952
#202725562
C53×#13032
#890453