Finite Group Databasebeta

← #27279940

Subgroups — #27279940

Order 13568.

478 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284524C211yesmaximal, characteristic
2#20283284C211yesmaximal, characteristic
3C2×#16810369C211yesmaximal, characteristic, complemented
4#20273103C211yesmaximal, characteristic
5#20272611C211yesmaximal, characteristic, complemented
6#20272589C211yesmaximal, characteristic, complemented
7C53×#1548C211yesmaximal, characteristic, complemented
8C2×#13133225—21no—
9C4×#11719945C2211yescharacteristic
10#16810915C412yes—
11#16810910C412yes—
12#16810905—21no—
13#16810900—21no—
14#16810663—21no—
15#16810651—21no—
16#16810648—22no—
17#16810647C2211yescharacteristic
18#16810645—21no—
19C2×C4⋊#88322C2211yescharacteristic
20C2×C4⋊#88322C2211yescharacteristic
21C4⋊C8×D53—21no—
22#16810622—21no—
23#16810369—21no—
24#16810369—21no—
25#16810355—21no—
26#16810352—21no—
27C2×C4⋊Q8×C53C2211yescharacteristic
28C53×#431C2211yescharacteristic
29C53×#423C2211yescharacteristic
30C53×#333—22no—
31C2×#11719945D411yescharacteristic
32#13133225—42no—
33#13133225—42no—
34C2×#11719917—21no—
35C2×#11719912—22no—
36C22×#88322—21no—
37C22×#88322—21no—
38C2×C8×D53—22no—
39C4×Dic106C2×C411yescharacteristic
40C4×Dic106D411yescharacteristic
41C4×D212C2×C411yescharacteristic
42C4×D212D411yescharacteristic
43C4×C22⋊D53—21no—
44C4×C22⋊D53—21no—
45#13133176D411yescharacteristic
46#13133174—21no—
47C42×D53—21no—
48#13133150—22no—
49#13133148—22no—
50C4⋊#88322—21no—
51C4⋊#88322—21no—
52C4⋊#88322—21no—
53C4⋊#88322—21no—
54C4⋊#88322C2×C412yes—
55C4⋊#88322D411yescharacteristic
56C4⋊#88322D411yescharacteristic
57#13133146—22no—
58#13133146—22no—
59#13133140—22no—
60#13133137—22no—
61C22×Q8×C53—21no—
62C2×M16×C53—21no—
63C4⋊Q8×C53—21no—
64C4⋊Q8×C53—21no—
65C4⋊Q8×C53—21no—
66C4⋊Q8×C53C2×C412yes—
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53—21no—
69C2×C42×C53C2311yescharacteristic
70C4⋊C8×C53—21no—
71C4⋊C8×C53—21no—
72C4⋊C8×C53D412yes—
73C22⋊C8×C53—21no—
74C4×C8×C53—21no—
75C2×Q8×C53—22no—
76C2×Q8×C53—41no—
77C2×Q8×C53—81no—
78C2×Q8×C53—81no—
79C22×C4×C53—41no—
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53#4111yescharacteristic
82C22×C4×C53#4111yescharacteristic
83C2×C22⋊D53—21no—
84#11719945—21no—
85#11719945—21no—
86#11719945SD1612yes—
87#11719945Q1612yes—
88C2×D212C22⋊C411yescharacteristic
89C2×C4×D53—21no—
90C2×C4×D53—21no—
91C2×C4×D53—21no—
92C2×Dic106C22⋊C411yescharacteristic
93M16×C53—41no—
94M16×C53—41no—
95C2×C8×C53—22no—
96C2×C8×C53—41no—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—41no—
100C4⋊C4×C53—41no—
101C42×C53C22×C411yescharacteristic
102C42×C53C2×D411yescharacteristic
103C42×C53C2×D411yescharacteristic
104C42×C53C2×D411yescharacteristic
105C22⋊Dic53—21no—
106#11719921—21no—
107#11719921—21no—
108C4⋊Dic53C22⋊C411yescharacteristic
109C4⋊Dic53C22⋊C411yescharacteristic
110#11719919—21no—
111#11719919—21no—
112C4×Dic53—21no—
113C4×Dic53—21no—
114C4×Dic53—21no—
115#11719917—41no—
116#11719917—41no—
117C2×#88322—22no—
118C2×#88322—21no—
119C2×#88322—21no—
120C2×#88322—41no—
121C2×#88322—41no—
122C2×#88322—41no—
123C2×#88322—81no—
124#11719912—44no—
125C8×D53—44no—
126C23×C53#12311yescharacteristic
127C22×D53—21no—
128Q8×C53—82no—
129Q8×C53—81no—
130Q8×C53—81no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—41no—
134C2×C4×C53—42no—
135C2×C4×C53—82no—
136C2×C4×C53C2×C22⋊C411yescharacteristic
137C2×C4×C53C4×D411yescharacteristic
138C2×C4×C53C4×D411yescharacteristic
139C2×C4×C53#12011yescharacteristic
140C2×C4×C53#12111yescharacteristic
141C2×C4×C53#12111yescharacteristic
142C2×C4×C53C2×SD1611yescharacteristic
143C2×C4×C53C2×Q1611yescharacteristic
144C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C53#13711yescharacteristic
146C22⋊D53—22no—
147C22⋊D53—22no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151D212—21no—
152D212#10312yes—
153C4×D53—22no—
154C4×D53—22no—
155C4×D53—41no—
156C4×D53—41no—
157C4×D53—81no—
158Dic106—21no—
159Dic106#10312yes—
160C8×C53—42no—
161C8×C53—81no—
162#88322—42no—
163#88322—41no—
164#88322—41no—
165#88322—81no—
166#9167—531nomaximal, Sylow
167C22×C53—21no—
168C22×C53—21no—
169C22×C53#38611yescharacteristic
170C22×C53#48311yescharacteristic
171C22×C53#48411yescharacteristic
172C2×D53—22no—
173C2×D53—41no—
174C2×D53—41no—
175C4×C53—21no—
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—82no—
179C4×C53C2×#10311yescharacteristic
180C4×C53#41711yescharacteristic
181C4×C53#45011yescharacteristic
182C4×C53#45111yescharacteristic
183C4×C53#47111yescharacteristic
184C4×C53#47211yescharacteristic
185Dic53—22no—
186Dic53—41no—
187Dic53—41no—
188#3079—531no—
189#2906—531no—
190#1548D5311yescharacteristic, complemented
191C2×#332—531no—
192#1477—531no—
193#1301—531no—
194#1292—531no—
195C2×C53—21no—
196C2×C53—21no—
197C2×C53#187211yescharacteristic
198C2×C53#190211yescharacteristic
199C2×C53#190711yescharacteristic
200D53—42no—
201C2×#131—1061no—
202C2×C4⋊Q8C2×D5311yescharacteristic
203C4×#41—531no—
204#477—532no—
205#476—532no—
206#475—1061no—
207#474—1061no—
208#431C2×D5311yescharacteristic
209#428—1061no—
210#424—1061no—
211#424—1062no—
212#423—531no—
213#423—1061no—
214#423C2×D5311yescharacteristic
215C2×C4⋊C8—531no—
216C2×C4⋊C8—531no—
217C2×C4⋊C8—1061no—
218#417—1061no—
219#333—22no—
220#332—1061no—
221#332—1061no—
222Q8⋊C8—1061no—
223D4⋊C8—1061no—
224C53#916711yescharacteristic, complemented, Sylow
225C2×#41—531no—
226C22×Q8—21no—
227#131—2122no—
228#131—2122no—
229C2×M16—21no—
230C2×M16—1062no—
231C2×M16—1061no—
232C22×C8—1062no—
233C22×C8—1061no—
234C22×C8—1061no—
235C4⋊Q8—21no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8C4×D5312yes—
239C4×Q8—531no—
240C4×Q8—531no—
241C4×D4—531no—
242C4×D4—531no—
243C4×D4—1061no—
244C4×D4—1061no—
245#117—531no—
246#117—1061no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C42—1061no—
250C2×C42C22×D5311yescharacteristic
251#107—1062no—
252#106—1062no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—532no—
256C4⋊C8—531no—
257C4⋊C8—531no—
258C4⋊C8—1062no—
259C4⋊C8—1062no—
260C4⋊C8—1061no—
261C4⋊C8—1061no—
262C4⋊C8—1061no—
263C4⋊C8—1061no—
264C4⋊C8C22⋊D5312yes—
265#103—1062no—
266#102—1062no—
267C22⋊C8—21no—
268C4×C8—21no—
269#41—532no—
270#41—532no—
271#41—1061no—
272#41—1061no—
273C2×Q8—22no—
274C2×Q8—41no—
275C2×Q8—81no—
276C2×Q8—81no—
277C2×Q8—531no—
278C2×D4—531no—
279C2×D4—1061no—
280C22×C4—41no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4—1061no—
284C22×C4#1171994511yescharacteristic
285C22×C4D4×D5311yescharacteristic
286C22×C4#1171994711yescharacteristic
287M16—41no—
288M16—41no—
289M16—2124no—
290M16—2121no—
291M16—2121no—
292C2×C8—22no—
293C2×C8—41no—
294C2×C8—1062no—
295C2×C8—1061no—
296C2×C8—1061no—
297C2×C8—4241no—
298C2×C8—2124no—
299C2×C8—2121no—
300C2×C8—2121no—
301C2×C8—2121no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—531no—
307C4⋊C4—531no—
308C4⋊C4—1061no—
309C4⋊C4—1061no—
310C22⋊C4—1061no—
311C22⋊C4—1061no—
312C22⋊C4—1061no—
313C42—1061no—
314C42—1061no—
315C42—1061no—
316C42C2×C4×D5311yescharacteristic
317C42D4×D5311yescharacteristic
318C42D4×D5311yescharacteristic
319C42C2×C22⋊D5311yescharacteristic
320C23—1061no—
321C23#1313319911yescharacteristic
322Q8—82no—
323Q8—81no—
324Q8—81no—
325Q8—532no—
326Q8—1061no—
327D4—532no—
328D4—1062no—
329D4—1062no—
330D4—1061no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—82no—
336C2×C4—1061no—
337C2×C4—1062no—
338C2×C4—1062no—
339C2×C4—1061no—
340C2×C4—1061no—
341C2×C4—4241no—
342C2×C4—2121no—
343C2×C4—2121no—
344C2×C4C22⋊C4×D5311yescharacteristic
345C2×C4#1313317711yescharacteristic
346C2×C4C4×C22⋊D5311yescharacteristic
347C2×C4#1313318511yescharacteristic
348C2×C4#1313318811yescharacteristic
349C2×C4#1313319011yescharacteristic
350C2×C4SD16×D5311yescharacteristic
351C2×C4Q16×D5311yescharacteristic
352C2×C4#1313324411yescharacteristic
353C2×C4#1313324911yescharacteristic
354C8—42no—
355C8—81no—
356C8—4241no—
357C8—2121no—
358C8—2121no—
359C8—2122no—
360C22—21no—
361C22—21no—
362C22—1062no—
363C22—2121no—
364C22—2121no—
365C22#1681049911yescharacteristic
366C22#1681095211yescharacteristic
367C22#1681095711yescharacteristic
368C4—21no—
369C4—21no—
370C4—21no—
371C4—82no—
372C4—1062no—
373C4—2121no—
374C4—2121no—
375C4#103×D5311yescharacteristic
376C4#1681062311yescharacteristic
377C4#1681077211yescharacteristic
378C4#1681077711yescharacteristic
379C4#1681088511yescharacteristic
380C4#1681089211yescharacteristic
381C2—21no—
382C2—21no—
383C2—2122no—
384C2#2027501911yescharacteristic
385C2#2027518511yescharacteristic
386C2#2027522011yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1548C2
socleC22×C53#386
prime radicalC2×#11719945D4
cosocleC2×C42×C53C23
hypercenter#1548D53
hypocenterC53#9167
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
3. C2×#16810369C2semidirect
5. #20272611C2semidirect
6. #20272589C2semidirect
7. C53×#1548C2semidirect
190. #1548D53semidirect
224. C53#9167semidirect

Maximal subgroups

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

subgroupindex
#202845242
#202832842
C2×#168103692
#202731032
#202726112
#202725892
C53×#15482
#916753