Finite Group Databasebeta

← #27365379

Subgroups — #27365379

Order 13568.

960 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 960 conjugacy classes of subgroups — too many to draw. Only its 171 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#20285702C212yesmaximal, complemented
2#20285118C211yesmaximal, characteristic, complemented
3#20274038C214yesmaximal, complemented
4#20274008C214yesmaximal, complemented
5#20273342C211yesmaximal, characteristic, complemented
6#20273313C212yesmaximal, complemented
7C2×C53×#562C211yesmaximal, characteristic, complemented
8#16811173C2211yescharacteristic
9#16811013C2214yescomplemented
10#16810993C2212yescomplemented
11#16810989C2212yescomplemented
12#16810982C2214yescomplemented
13#16810980—22no—
14#16810835—24no—
15#16810762—22no—
16#16810757—22no—
17#16810680C2212yes—
18#16810622—22no—
19C4×#11719917C2211yescharacteristic
20#16810365C2214yescomplemented
21#16810363C2218yescomplemented
22C53×#562—22no—
23C53×#562—24no—
24C53×#562C2214yes—
25C2×#125×C53C2211yescharacteristic
26C2×#124×C53C2212yes—
27C2×#123×C53—22no—
28C2×C22⋊Q8×C53—22no—
29C2×#121×C53—22no—
30C2×#11719945—21no—
31C2×#11719924—24no—
32C2×#11719922—24no—
33C2×#11719917—21no—
34#13133217C2311yescharacteristic
35#13133214C2311yescharacteristic
36#13133205—21no—
37#13133191—22no—
38C4×D212—22no—
39C4⋊#88322—22no—
40#13133141—24no—
41#13133140—24no—
42#13133138—24no—
43#13133137—24no—
44C22⋊#88322—22no—
45#13133128C2314yes—
46#13133128C2312yes—
47C4×#88322—21no—
48C22×Q8×C53D412yes—
49C22×D4×C53—22no—
50C22×D4×C53D412yes—
51C23×C4×C53—22no—
52#125×C53—22no—
53#125×C53D412yes—
54#125×C53C2312yes—
55#124×C53—22no—
56#124×C53—22no—
57#124×C53D414yes—
58#124×C53C2314yes—
59#123×C53—28no—
60#123×C53—28no—
61C22⋊Q8×C53—28no—
62C22⋊Q8×C53—28no—
63#121×C53—28no—
64#121×C53—28no—
65C2×C4⋊C4×C53—22no—
66C2×C4⋊C4×C53—22no—
67C2×C4⋊C4×C53D412yes—
68C2×C22⋊C4×C53—22no—
69C2×C22⋊C4×C53—22no—
70C2×C22⋊C4×C53—22no—
71C2×C42×C53C2311yescharacteristic
72C24×C53—22no—
73C2×Q8×C53—22no—
74C2×Q8×C53—82no—
75C2×Q8×C53C2×D414yes—
76C2×D4×C53—28no—
77C2×D4×C53—22no—
78C2×D4×C53—82no—
79C2×D4×C53—84no—
80C2×D4×C53C2×D414yes—
81C22×C4×C53—22no—
82C22×C4×C53—22no—
83C22×C4×C53—28no—
84C22×C4×C53—84no—
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53C2×D412yes—
87C2×C22⋊D53—41no—
88#11719945—44no—
89C2×D212—21no—
90C2×D212—22no—
91C2×C4×D53—41no—
92C2×Dic106—21no—
93C4⋊C4×C53—28no—
94C4⋊C4×C53—24no—
95C4⋊C4×C53—22no—
96C4⋊C4×C53—24no—
97C4⋊C4×C53C2×D414yes—
98C22⋊C4×C53—28no—
99C22⋊C4×C53—24no—
100C22⋊C4×C53—24no—
101C22⋊C4×C53—24no—
102C22⋊C4×C53—24no—
103C42×C53C2×D411yescharacteristic
104C42×C53C2×D412yes—
105C42×C53C2411yescharacteristic
106#11719924—84no—
107#11719922—84no—
108#11719921—42no—
109C4⋊Dic53—22no—
110#11719917—42no—
111C2×#88322—22no—
112C2×#88322—24no—
113C23×C53—24no—
114C23×C53—44no—
115C23×C53—44no—
116C23×C53—44no—
117C23×C53#12011yescharacteristic
118C22×D53—41no—
119Q8×C53—44no—
120Q8×C53—82no—
121D4×C53—44no—
122D4×C53—82no—
123D4×C53—88no—
124C2×C4×C53—22no—
125C2×C4×C53—24no—
126C2×C4×C53—24no—
127C2×C4×C53—42no—
128C2×C4×C53—44no—
129C2×C4×C53—44no—
130C2×C4×C53—88no—
131C2×C4×C53#12011yescharacteristic
132C2×C4×C53#12012yes—
133C2×C4×C53C22×D411yescharacteristic
134C2×C4×C53C22×D412yes—
135C22⋊D53—82no—
136C2×Dic53—41no—
137D212—42no—
138D212—44no—
139C4×D53—82no—
140Dic106—42no—
141#88322—42no—
142#88322—44no—
143#19001—531nomaximal, Sylow
144C22×C53—22no—
145C22×C53—44no—
146C22×C53—44no—
147C22×C53—44no—
148C22×C53—88no—
149C22×C53#45412yes—
150C22×C53C2×#12011yescharacteristic
151C2×D53—81no—
152C2×D53—82no—
153C4×C53—21no—
154C4×C53—21no—
155C4×C53—22no—
156C4×C53—22no—
157C4×C53—44no—
158C4×C53—44no—
159Dic53—81no—
160C2×#562D5311yescharacteristic, complemented
161#3213—532no—
162#3142—531no—
163#1662—534no—
164#1656—534no—
165#1534—531no—
166#1527—532no—
167C2×C53—22no—
168C2×C53—84no—
169C2×C53C2×#45411yescharacteristic
170C2×C53#301912yes—
171D53—161no—
172#562—22no—
173#562—24no—
174#562C2×D5314yes—
175#532—531no—
176C2×#125C2×D5311yescharacteristic
177C2×#124C2×D5312yes—
178C2×#123—22no—
179C2×C22⋊Q8—22no—
180C2×#121—22no—
181#496—534no—
182#491—532no—
183#490—532no—
184#489—534no—
185#488—1062no—
186#463—1064no—
187#448—1062no—
188#447—1062no—
189#432—532no—
190#417—1062no—
191C4×M16—531no—
192#330—534no—
193#329—538no—
194C53#1900111yescharacteristic, complemented, Sylow
195C2×#41—1061no—
196C22×Q8C22⋊D5312yes—
197C22×D4—22no—
198C22×D4C22⋊D5312yes—
199C23×C4—22no—
200C2×SD16—1064no—
201C2×D8—1064no—
202C2×M16—1061no—
203C4⋊Q8—531no—
204#127—531no—
205#125—22no—
206#125C22⋊D5312yes—
207#125C22×D5312yes—
208#124—22no—
209#124—22no—
210#124—1061no—
211#124C22⋊D5314yes—
212#124C22×D5314yes—
213#123—28no—
214#123—28no—
215C22⋊Q8—28no—
216C22⋊Q8—28no—
217#121—28no—
218#121—28no—
219#121—1062no—
220C4×D4—1062no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4C22⋊D5312yes—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C42C22×D5311yescharacteristic
228C4⋊C8—1062no—
229#103—1064no—
230#103—1064no—
231#102—1064no—
232#102—1064no—
233C22⋊C8—1062no—
234#97—532no—
235#97—534no—
236C4×C8—1061no—
237C24—22no—
238#41—2124no—
239C2×Q8—22no—
240C2×Q8—82no—
241C2×Q8—1061no—
242C2×Q8C2×C22⋊D5314yes—
243C2×D4—28no—
244C2×D4—22no—
245C2×D4—82no—
246C2×D4—84no—
247C2×D4—1061no—
248C2×D4—1062no—
249C2×D4—2121no—
250C2×D4C2×C22⋊D5314yes—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—28no—
254C22×C4—84no—
255C22×C4—2121no—
256C22×C4C2×C22⋊D5311yescharacteristic
257C22×C4C2×C22⋊D5312yes—
258SD16—4244no—
259D8—4244no—
260M16—2122no—
261C2×C8—1062no—
262C2×C8—1064no—
263C4⋊C4—28no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—24no—
267C4⋊C4—1062no—
268C4⋊C4C2×C22⋊D5314yes—
269C22⋊C4—28no—
270C22⋊C4—24no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—2122no—
275C42C2×C22⋊D5311yescharacteristic
276C42C2×C22⋊D5312yes—
277C42C23×D5311yescharacteristic
278C23—24no—
279C23—44no—
280C23—44no—
281C23—44no—
282C23—2121no—
283C23#1313318611yescharacteristic
284Q8—44no—
285Q8—82no—
286Q8—2122no—
287D4—44no—
288D4—82no—
289D4—88no—
290D4—4242no—
291D4—2124no—
292D4—2122no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—88no—
300C2×C4—4242no—
301C2×C4—2121no—
302C2×C4#1313318611yescharacteristic
303C2×C4#1313318612yes—
304C2×C4C22×C22⋊D5311yescharacteristic
305C2×C4C22×C22⋊D5312yes—
306C8—2122no—
307C8—2124no—
308C22—22no—
309C22—44no—
310C22—44no—
311C22—44no—
312C22—88no—
313C22—4241no—
314C22—4242no—
315C22#1681079612yes—
316C22C2×#1313318611yescharacteristic
317C4—21no—
318C4—21no—
319C4—22no—
320C4—22no—
321C4—44no—
322C4—44no—
323C4—4241no—
324C2—22no—
325C2—84no—
326C2—8481no—
327C2C2×#1681079611yescharacteristic
328C2#2028407212yes—
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#562C2
socleC22×C53C2×#120
hypercenterC2×#562D53
hypocenterC53#19001
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#562

Complements

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

NH ≅ G/Ntype
1. #20285702C2semidirect
2. #20285118C2semidirect
3. #20274038C2semidirect
4. #20274008C2semidirect
5. #20273342C2semidirect
6. #20273313C2semidirect
7. C2×C53×#562C2semidirect
9. #16811013C22semidirect
10. #16810993C22semidirect
11. #16810989C22semidirect
12. #16810982C22semidirect
20. #16810365C22semidirect
21. #16810363C22semidirect
160. C2×#562D53semidirect
194. C53#19001semidirect

Maximal subgroups

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

subgroupindex
#202857022
#202851182
#202740382
#202740082
#202733422
#202733132
C2×C53×#5622
#1900153