Finite Group Databasebeta

← #27360359

Subgroups — #27360359

Order 13568.

1068 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 163 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
1C2×#16811339C211yesmaximal, characteristic, complemented
2#20284802C212yesmaximal, complemented
3#20273858C214yesmaximal, complemented
4#20273796C214yesmaximal, complemented
5#20273441C211yesmaximal, characteristic, complemented
6#20273159C212yesmaximal, complemented
7C53×#3152C211yesmaximal, characteristic, complemented
8#16811339—24no—
9#16811339—22no—
10#16811339C2214yes—
11C2×#13133206—22no—
12C2×#13133192—22no—
13C2×C4×D212C2212yes—
14C2×#13133175—22no—
15#16810939—22no—
16#16810910C2214yescomplemented
17#16810815C2214yescomplemented
18#16810756—22no—
19#16810733—24no—
20#16810680C2212yes—
21#16810372C2214yescomplemented
22#16810352C2218yescomplemented
23C2×#137×C53—22no—
24C53×#532C2211yescharacteristic
25C2×C4⋊Q8×C53C2211yescharacteristic
26C53×#497—22no—
27C53×#494C2212yescomplemented
28C53×#492C2212yescomplemented
29C4×M16×C53C2211yescharacteristic
30C22×D212—22no—
31C22×D212D412yes—
32C22×C4×D53—22no—
33C2×#11719922—24no—
34#13133206—28no—
35#13133206—28no—
36#13133192—28no—
37#13133192—28no—
38C4×D212—22no—
39C4×D212—22no—
40C4×D212D414yes—
41C4×D212C2314yes—
42#13133175—28no—
43#13133175—28no—
44C2×C4⋊Dic53D412yes—
45C2×#11719921—24no—
46C2×#11719921—22no—
47C2×C4×Dic53—22no—
48#13133148—24no—
49C4⋊#88322C2314yes—
50#13133138—24no—
51#13133137—24no—
52C22⋊#88322—22no—
53#13133128—22no—
54C2×#41×C53—21no—
55C22×Q8×C53D412yes—
56#137×C53—48no—
57C2×Q16×C53—24no—
58C2×SD16×C53—24no—
59C2×M16×C53—21no—
60C4⋊Q8×C53—22no—
61C4⋊Q8×C53D412yes—
62C4⋊Q8×C53C2311yescharacteristic
63C4⋊Q8×C53C2312yes—
64#127×C53C2311yescharacteristic
65#124×C53—21no—
66#121×C53—22no—
67C4×D4×C53—22no—
68C2×C4⋊C4×C53—22no—
69C2×C42×C53C2311yescharacteristic
70#97×C53—21no—
71C4×C8×C53C2312yes—
72C23×D53—22no—
73#41×C53—44no—
74C2×Q8×C53—22no—
75C2×Q8×C53—21no—
76C2×Q8×C53—44no—
77C2×Q8×C53C2×D414yes—
78C2×D4×C53—22no—
79C2×D4×C53—21no—
80C2×D4×C53—41no—
81C22×C4×C53—22no—
82C22×C4×C53—41no—
83C22×C4×C53C2×D412yes—
84C22×C4×C53C2×D411yescharacteristic
85C22×Dic53—22no—
86C2×D212—22no—
87C2×D212—28no—
88C2×D212—44no—
89C2×D212—84no—
90C2×D212C2×D414yes—
91C2×C4×D53—24no—
92C2×C4×D53—24no—
93C2×C4×D53—48no—
94Q16×C53—48no—
95SD16×C53—48no—
96M16×C53—24no—
97C2×C8×C53—22no—
98C4⋊C4×C53—24no—
99C4⋊C4×C53—24no—
100C4⋊C4×C53—22no—
101C22⋊C4×C53—42no—
102C42×C53C2×D412yes—
103C42×C53C2×D411yescharacteristic
104C42×C53C2411yescharacteristic
105#11719922—48no—
106#11719921—216no—
107#11719921—24no—
108#11719921—24no—
109C4⋊Dic53—22no—
110C4⋊Dic53C2×D414yes—
111C4×Dic53—28no—
112C2×#88322—24no—
113C23×C53—41no—
114C23×C53#12011yescharacteristic
115C22×D53—24no—
116C22×D53—44no—
117C22×D53—44no—
118C22×D53—44no—
119Q8×C53—44no—
120Q8×C53—42no—
121Q8×C53—82no—
122D4×C53—42no—
123D4×C53—44no—
124D4×C53—82no—
125C2×C4×C53—22no—
126C2×C4×C53—24no—
127C2×C4×C53—24no—
128C2×C4×C53—44no—
129C2×C4×C53—41no—
130C2×C4×C53—82no—
131C2×C4×C53#12012yes—
132C2×C4×C53#12011yescharacteristic
133C2×C4×C53C22×D412yes—
134C2×C4×C53C22×D411yescharacteristic
135C2×Dic53—24no—
136C2×Dic53—44no—
137D212—24no—
138D212—24no—
139D212—44no—
140D212—88no—
141C4×D53—48no—
142C4×D53—48no—
143C8×C53—24no—
144#88322—44no—
145#18547—531nomaximal, Sylow
146C22×C53—22no—
147C22×C53—82no—
148C22×C53—81no—
149C22×C53#45312yes—
150C22×C53C2×#12011yescharacteristic
151C2×D53—44no—
152C2×D53—44no—
153C2×D53—44no—
154C2×D53—88no—
155C4×C53—22no—
156C4×C53—21no—
157C4×C53—21no—
158C4×C53—44no—
159C4×C53—81no—
160C4×C53#57514yes—
161Dic53—44no—
162C2×#555—531no—
163#3152D5311yescharacteristic, complemented
164#3109—532no—
165#1634—534no—
166#1624—534no—
167#1558—531no—
168#1490—532no—
169C2×C53—22no—
170C2×C53—161no—
171C2×C53#300512yes—
172C2×C53C2×#45311yescharacteristic
173D53—84no—
174C2×#137—22no—
175#555—534no—
176#555—1064no—
177#555—1062no—
178#532C2×D5311yescharacteristic
179C2×C4⋊Q8C2×D5311yescharacteristic
180C2×#124—1062no—
181C2×#121—1062no—
182C2×C4×D4—532no—
183C2×#117—1062no—
184#497—22no—
185#494C2×D5312yescomplemented
186#492C2×D5312yescomplemented
187#481—1062no—
188#476—534no—
189#459—534no—
190#447—1062no—
191#442—1064no—
192#432—532no—
193C4×M16C2×D5311yescharacteristic
194#333—534no—
195D4⋊C8—538no—
196C53#1854711yescharacteristic, complemented, Sylow
197C2×#41—21no—
198C22×Q8C22⋊D5312yes—
199C22×D4—532no—
200C22×D4—1062no—
201C23×C4—1062no—
202#137—48no—
203C2×Q16—24no—
204C2×SD16—24no—
205C2×D8—1064no—
206C2×M16—21no—
207C4⋊Q8—22no—
208C4⋊Q8C22⋊D5312yes—
209C4⋊Q8C22×D5311yescharacteristic
210C4⋊Q8C22×D5312yes—
211#127C22×D5311yescharacteristic
212#124—21no—
213#124—1068no—
214#124—1068no—
215#121—22no—
216#121—1068no—
217#121—1068no—
218C4×D4—22no—
219C4×D4—534no—
220C4×D4—534no—
221C4×D4—1062no—
222C4×D4—1062no—
223#117—1068no—
224#117—1068no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—532no—
227C2×C22⋊C4—1064no—
228C2×C22⋊C4—1062no—
229C2×C42—1062no—
230C2×C42C22×D5311yescharacteristic
231#106—1064no—
232C4⋊C8—534no—
233#102—1064no—
234#102—1064no—
235C22⋊C8—1062no—
236#97—21no—
237#97—1062no—
238C4×C8C22×D5312yes—
239C24—1062no—
240#41—44no—
241C2×Q8—22no—
242C2×Q8—21no—
243C2×Q8—44no—
244C2×Q8C2×C22⋊D5314yes—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—41no—
248C2×D4—534no—
249C2×D4—1068no—
250C2×D4—1062no—
251C2×D4—4244no—
252C2×D4—2124no—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—1062no—
256C22×C4—1064no—
257C22×C4—1064no—
258C22×C4—2128no—
259C22×C4D4×D5312yes—
260C22×C4C2×C22⋊D5311yescharacteristic
261Q16—48no—
262SD16—48no—
263D8—2128no—
264M16—24no—
265C2×C8—22no—
266C2×C8—1064no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—22no—
270C4⋊C4—534no—
271C4⋊C4—1062no—
272C22⋊C4—42no—
273C22⋊C4—10616no—
274C22⋊C4—1064no—
275C22⋊C4—1064no—
276C42—1068no—
277C42D4×D5312yes—
278C42C2×C22⋊D5311yescharacteristic
279C42C23×D5311yescharacteristic
280C23—41no—
281C23—1064no—
282C23—2124no—
283C23—2124no—
284C23—2124no—
285C23#1313318511yescharacteristic
286Q8—44no—
287Q8—42no—
288Q8—82no—
289D4—42no—
290D4—44no—
291D4—82no—
292D4—1064no—
293D4—1064no—
294D4—4248no—
295D4—2124no—
296C2×C4—22no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—44no—
300C2×C4—41no—
301C2×C4—82no—
302C2×C4—1064no—
303C2×C4—2128no—
304C2×C4—2124no—
305C2×C4—2128no—
306C2×C4#1313318511yescharacteristic
307C2×C4#1313318512yes—
308C2×C4C2×D4×D5312yes—
309C2×C4C22×C22⋊D5311yescharacteristic
310C8—24no—
311C8—2124no—
312C22—22no—
313C22—82no—
314C22—81no—
315C22—4248no—
316C22—2124no—
317C22—2124no—
318C22—2124no—
319C22#1681079012yes—
320C22C2×#1313318511yescharacteristic
321C4—22no—
322C4—21no—
323C4—21no—
324C4—44no—
325C4—81no—
326C4—2124no—
327C4#1681142214yes—
328C2—22no—
329C2—161no—
330C2—4244no—
331C2#2028395612yes—
332C2C2×#1681079011yescharacteristic
subgroupG/H
centerC22C2×#13133185
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3152C2
socleC22×C53C2×#120
hypercenter#3152D53
hypocenterC53#18547
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3152

Complements

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

NH ≅ G/Ntype
1. C2×#16811339C2semidirect
2. #20284802C2semidirect
3. #20273858C2semidirect
4. #20273796C2semidirect
5. #20273441C2semidirect
6. #20273159C2semidirect
7. C53×#3152C2semidirect
16. #16810910C22semidirect
17. #16810815C22semidirect
21. #16810372C22semidirect
22. #16810352C22semidirect
27. C53×#494C22semidirect
28. C53×#492C22semidirect
163. #3152D53semidirect
185. #494C2×D53semidirect
186. #492C2×D53semidirect
196. C53#18547semidirect

Maximal subgroups

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

subgroupindex
C2×#168113392
#202848022
#202738582
#202737962
#202734412
#202731592
C53×#31522
#1854753