Finite Group Databasebeta

← #27360174

Subgroups — #27360174

Order 13568.

1068 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 179 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285165C211yesmaximal, characteristic
2#20284837C212yesmaximal, complemented
3#20273836C214yesmaximal, complemented
4#20273774C214yesmaximal, complemented
5#20273429C211yesmaximal, characteristic, complemented
6#20273162C212yesmaximal, complemented
7C2×C53×#559C211yesmaximal, characteristic, complemented
8C2×#13133243—22no—
9#16811173C2211yescharacteristic
10#16811010—22no—
11#16810999C2212yes—
12#16810995C2212yes—
13#16810936—22no—
14#16810918C2214yescomplemented
15#16810823C2214yescomplemented
16#16810762—22no—
17#16810721—24no—
18#16810680C2212yes—
19C4×#11719917C2211yescharacteristic
20#16810367C2214yes—
21Q8⋊#88322C2218yescomplemented
22C53×#559—22no—
23C53×#559—24no—
24C53×#559C2214yes—
25C2×#127×C53C2211yescharacteristic
26C2×#124×C53—22no—
27C2×#123×C53—22no—
28C2×C4×Q8×C53C2212yes—
29C2×C4×D4×C53—22no—
30C2×#11719945—21no—
31#13133243—48no—
32C2×#11719923—24no—
33C2×#11719924—24no—
34C2×#11719922—24no—
35C2×#11719917—21no—
36#13133217C2311yescharacteristic
37#13133214C2311yescharacteristic
38#13133205—21no—
39#13133191—22no—
40C4×D212—22no—
41#13133150—24no—
42C4⋊#88322C2314yes—
43#13133141—24no—
44#13133137—24no—
45C22⋊#88322—22no—
46#13133128—21no—
47#13133128—22no—
48C4×#88322C2312yes—
49C22×Q8×C53D412yes—
50C22×D4×C53—22no—
51C22×D4×C53D412yes—
52C23×C4×C53—22no—
53#127×C53—22no—
54#127×C53D412yes—
55#127×C53C2312yes—
56#124×C53—28no—
57#124×C53—28no—
58#123×C53—28no—
59#123×C53—28no—
60C4×Q8×C53—22no—
61C4×Q8×C53—22no—
62C4×Q8×C53D414yes—
63C4×Q8×C53C2314yes—
64C4×D4×C53—28no—
65C4×D4×C53—28no—
66C2×C4⋊C4×C53—22no—
67C2×C4⋊C4×C53D412yes—
68C2×C22⋊C4×C53—24no—
69C2×C22⋊C4×C53—22no—
70C2×C42×C53—22no—
71C2×C42×C53C2311yescharacteristic
72C24×C53—22no—
73C2×Q8×C53—22no—
74C2×Q8×C53—44no—
75C2×Q8×C53C2×D414yes—
76C2×D4×C53—24no—
77C2×D4×C53—22no—
78C2×D4×C53—24no—
79C2×D4×C53—48no—
80C2×D4×C53—44no—
81C2×D4×C53C2×D414yes—
82C22×C4×C53—22no—
83C22×C4×C53—22no—
84C22×C4×C53—28no—
85C22×C4×C53—84no—
86C22×C4×C53C2×D412yes—
87C22×C4×C53C2×D411yescharacteristic
88C2×C22⋊D53—41no—
89#11719945—44no—
90C2×D212—22no—
91C2×D212—21no—
92C2×C4×D53—41no—
93C2×Dic106—21no—
94C4⋊C4×C53—24no—
95C4⋊C4×C53—22no—
96C4⋊C4×C53—24no—
97C4⋊C4×C53C2×D414yes—
98C22⋊C4×C53—216no—
99C22⋊C4×C53—28no—
100C42×C53—24no—
101C42×C53—24no—
102C42×C53C2×D411yescharacteristic
103C42×C53C2×D412yes—
104C42×C53C2411yescharacteristic
105#11719924—48no—
106#11719923—48no—
107#11719922—48no—
108#11719921—42no—
109C4⋊Dic53—22no—
110#11719917—24no—
111C2×#88322—22no—
112C2×#88322—24no—
113C23×C53—24no—
114C23×C53—48no—
115C23×C53—44no—
116C23×C53#12011yescharacteristic
117C22×D53—41no—
118Q8×C53—24no—
119Q8×C53—24no—
120Q8×C53—44no—
121D4×C53—44no—
122D4×C53—48no—
123D4×C53—48no—
124D4×C53—82no—
125C2×C4×C53—22no—
126C2×C4×C53—24no—
127C2×C4×C53—24no—
128C2×C4×C53—24no—
129C2×C4×C53—44no—
130C2×C4×C53—44no—
131C2×C4×C53—88no—
132C2×C4×C53#12011yescharacteristic
133C2×C4×C53#12012yes—
134C2×C4×C53C22×D411yescharacteristic
135C2×C4×C53C22×D412yes—
136C22⋊D53—82no—
137C2×Dic53—41no—
138D212—42no—
139D212—44no—
140C4×D53—82no—
141Dic106—42no—
142#88322—24no—
143#88322—44no—
144#18526—531nomaximal, Sylow
145C22×C53—22no—
146C22×C53—48no—
147C22×C53—44no—
148C22×C53—88no—
149C22×C53#45312yes—
150C22×C53C2×#12011yescharacteristic
151C2×D53—82no—
152C2×D53—81no—
153C4×C53—22no—
154C4×C53—21no—
155C4×C53—21no—
156C4×C53—44no—
157C4×C53—44no—
158C4×C53#57514yes—
159Dic53—81no—
160C2×#559D5311yescharacteristic, complemented
161#3151—531no—
162#3112—532no—
163#1630—534no—
164#1620—534no—
165#1555—531no—
166#1491—532no—
167C2×C53—22no—
168C2×C53—84no—
169C2×C53#300512yes—
170C2×C53C2×#45311yescharacteristic
171D53—161no—
172C2×AGL(1,ℤ/8)—1062no—
173#559—22no—
174#559—24no—
175#559C2×D5314yes—
176#532—531no—
177C2×#127C2×D5311yescharacteristic
178C2×#124—22no—
179C2×#123—22no—
180C2×C4×Q8C2×D5312yes—
181C2×C4×D4—22no—
182#496—1062no—
183#493—532no—
184#492—532no—
185#480—1062no—
186#477—534no—
187#460—534no—
188#448—1062no—
189#440—1064no—
190#432—532no—
191C4×M16—531no—
192#331—534no—
193Q8⋊C8—538no—
194C53#1852611yescharacteristic, complemented, Sylow
195C2×#41—1061no—
196C22×Q8C22⋊D5312yes—
197C22×D4—22no—
198C22×D4C22⋊D5312yes—
199C23×C4—22no—
200AGL(1,ℤ/8)—2128no—
201C2×SD16—1064no—
202C2×SD16—1064no—
203C2×D8—1064no—
204C2×M16—1061no—
205C4⋊Q8—531no—
206#127—22no—
207#127—531no—
208#127C22⋊D5312yes—
209#127C22×D5312yes—
210#124—28no—
211#124—28no—
212#124—1061no—
213#123—28no—
214#123—28no—
215#121—1062no—
216C4×Q8—22no—
217C4×Q8—22no—
218C4×Q8C22⋊D5314yes—
219C4×Q8C22×D5314yes—
220C4×D4—28no—
221C4×D4—28no—
222C4×D4—1062no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4C22⋊D5312yes—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—22no—
227C2×C42—22no—
228C2×C42C22×D5311yescharacteristic
229#107—1064no—
230C4⋊C8—534no—
231#103—1064no—
232#102—1064no—
233C22⋊C8—1062no—
234#97—1061no—
235#97—1062no—
236C4×C8—532no—
237C24—22no—
238#41—2124no—
239C2×Q8—22no—
240C2×Q8—44no—
241C2×Q8—1061no—
242C2×Q8C2×C22⋊D5314yes—
243C2×D4—24no—
244C2×D4—22no—
245C2×D4—24no—
246C2×D4—48no—
247C2×D4—44no—
248C2×D4—1062no—
249C2×D4—1061no—
250C2×D4—2121no—
251C2×D4C2×C22⋊D5314yes—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—28no—
255C22×C4—84no—
256C22×C4—2121no—
257C22×C4C2×C22⋊D5312yes—
258C22×C4C2×C22⋊D5311yescharacteristic
259SD16—2128no—
260SD16—2128no—
261D8—2128no—
262M16—1064no—
263C2×C8—1062no—
264C2×C8—1064no—
265C4⋊C4—24no—
266C4⋊C4—22no—
267C4⋊C4—24no—
268C4⋊C4—1062no—
269C4⋊C4C2×C22⋊D5314yes—
270C22⋊C4—216no—
271C22⋊C4—28no—
272C22⋊C4—2122no—
273C42—24no—
274C42—24no—
275C42C2×C22⋊D5312yes—
276C42C2×C22⋊D5311yescharacteristic
277C42C23×D5311yescharacteristic
278C23—24no—
279C23—48no—
280C23—44no—
281C23—2121no—
282C23#1313318611yescharacteristic
283Q8—24no—
284Q8—24no—
285Q8—44no—
286Q8—2122no—
287D4—44no—
288D4—48no—
289D4—48no—
290D4—82no—
291D4—4242no—
292D4—2122no—
293D4—2124no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—88no—
301C2×C4—4242no—
302C2×C4—2121no—
303C2×C4#1313318612yes—
304C2×C4#1313318611yescharacteristic
305C2×C4C22×C22⋊D5311yescharacteristic
306C2×C4C22×C22⋊D5312yes—
307C8—1064no—
308C8—2124no—
309C22—22no—
310C22—48no—
311C22—44no—
312C22—88no—
313C22—4242no—
314C22—4241no—
315C22#1681079112yes—
316C22C2×#1313318611yescharacteristic
317C4—22no—
318C4—21no—
319C4—21no—
320C4—44no—
321C4—44no—
322C4—4241no—
323C4#1681142114yes—
324C2—22no—
325C2—84no—
326C2—8481no—
327C2#2028395712yes—
328C2C2×#1681079111yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#559C2
socleC22×C53C2×#120
prime radical#20285165C2
hypercenterC2×#559D53
hypocenterC53#18526
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
2. #20284837C2semidirect
3. #20273836C2semidirect
4. #20273774C2semidirect
5. #20273429C2semidirect
6. #20273162C2semidirect
7. C2×C53×#559C2semidirect
14. #16810918C22semidirect
15. #16810823C22semidirect
21. Q8⋊#88322C22semidirect
160. C2×#559D53semidirect
194. C53#18526semidirect

Maximal subgroups

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

subgroupindex
#202851652
#202848372
#202738362
#202737742
#202734292
#202731622
C2×C53×#5592
#1852653