Finite Group Databasebeta

← #27360354

Subgroups — #27360354

Order 13568.

1004 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 1004 conjugacy classes of subgroups — too many to draw. Only its 195 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#20285172C211yesmaximal, characteristic
2#20284837C212yesmaximal
3#20273855C214yesmaximal, complemented
4#20273793C214yesmaximal, complemented
5#20273441C211yesmaximal, characteristic, complemented
6#20273162C212yesmaximal, complemented
7C2×Q82×C53C211yesmaximal, characteristic, complemented
8C2×#13133248—22no—
9#16811173C2211yescharacteristic
10#16811015—22no—
11#16811002C2212yes—
12#16810997C2212yes—
13#16810936—22no—
14#16810918C2214yes—
15#16810823C2214yes—
16#16810762—22no—
17#16810721—24no—
18#16810680C2212yes—
19C4×#11719917C2211yescharacteristic
20#16810372C2214yes—
21Q8⋊#88322C2218yes—
22Q82×C53—22no—
23Q82×C53—24no—
24Q82×C53C2214yes—
25C2×C4⋊Q8×C53—22no—
26C2×C4⋊Q8×C53—22no—
27C2×C4⋊Q8×C53C2211yescharacteristic
28C2×C4×Q8×C53—22no—
29C2×C4×Q8×C53C2212yes—
30C2×#11719945—21no—
31#13133248—48no—
32C2×#11719925—24no—
33C2×#11719924—24no—
34C2×#11719924—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×C53—22no—
50C22×Q8×C53D412yes—
51C22×Q8×C53D412yes—
52C4⋊Q8×C53—28no—
53C4⋊Q8×C53—28no—
54C4⋊Q8×C53—28no—
55C4⋊Q8×C53—22no—
56C4⋊Q8×C53—28no—
57C4⋊Q8×C53D412yes—
58C4⋊Q8×C53C2312yes—
59C4×Q8×C53—28no—
60C4×Q8×C53—22no—
61C4×Q8×C53—22no—
62C4×Q8×C53—28no—
63C4×Q8×C53D414yes—
64C4×Q8×C53C2314yes—
65C2×C4⋊C4×C53—24no—
66C2×C4⋊C4×C53—22no—
67C2×C4⋊C4×C53—22no—
68C2×C4⋊C4×C53D412yes—
69C2×C42×C53—22no—
70C2×C42×C53—22no—
71C2×C42×C53C2311yescharacteristic
72C2×Q8×C53—28no—
73C2×Q8×C53—28no—
74C2×Q8×C53—22no—
75C2×Q8×C53—22no—
76C2×Q8×C53—44no—
77C2×Q8×C53—84no—
78C2×Q8×C53C2×D414yes—
79C2×Q8×C53C2×D414yes—
80C22×C4×C53—22no—
81C22×C4×C53—22no—
82C22×C4×C53—22no—
83C22×C4×C53C2×D412yes—
84C22×C4×C53C2×D411yescharacteristic
85C2×C22⋊D53—41no—
86#11719945—44no—
87C2×D212—22no—
88C2×D212—21no—
89C2×C4×D53—41no—
90C2×Dic106—21no—
91C4⋊C4×C53—216no—
92C4⋊C4×C53—24no—
93C4⋊C4×C53—24no—
94C4⋊C4×C53—22no—
95C4⋊C4×C53—24no—
96C4⋊C4×C53—24no—
97C4⋊C4×C53C2×D414yes—
98C42×C53—28no—
99C42×C53—24no—
100C42×C53—24no—
101C42×C53C2×D411yescharacteristic
102C42×C53C2×D412yes—
103C42×C53C2411yescharacteristic
104#11719925—48no—
105#11719924—48no—
106#11719924—48no—
107#11719921—42no—
108C4⋊Dic53—22no—
109#11719917—24no—
110C2×#88322—22no—
111C2×#88322—24no—
112C23×C53#12011yescharacteristic
113C22×D53—41no—
114Q8×C53—28no—
115Q8×C53—24no—
116Q8×C53—44no—
117Q8×C53—82no—
118Q8×C53—88no—
119Q8×C53#13718yescomplemented
120C2×C4×C53—28no—
121C2×C4×C53—24no—
122C2×C4×C53—22no—
123C2×C4×C53—24no—
124C2×C4×C53—24no—
125C2×C4×C53—24no—
126C2×C4×C53—44no—
127C2×C4×C53—44no—
128C2×C4×C53#12011yescharacteristic
129C2×C4×C53#12012yes—
130C2×C4×C53C22×D411yescharacteristic
131C2×C4×C53C22×D412yes—
132C22⋊D53—82no—
133C2×Dic53—41no—
134D212—42no—
135D212—44no—
136C4×D53—82no—
137Dic106—42no—
138#88322—24no—
139#88322—44no—
140#18546—531nomaximal, Sylow
141C22×C53—22no—
142C22×C53#45312yes—
143C22×C53C2×#12011yescharacteristic
144C2×D53—82no—
145C2×D53—81no—
146C4×C53—28no—
147C4×C53—22no—
148C4×C53—21no—
149C4×C53—21no—
150C4×C53—44no—
151C4×C53—44no—
152C4×C53C2×#13714yes—
153Dic53—81no—
154C2×Q82D5311yescharacteristic, complemented
155#3152—531no—
156#3112—532no—
157#1633—534no—
158#1623—534no—
159#1558—531no—
160#1491—532no—
161C2×C53—22no—
162C2×C53#300412yes—
163C2×C53C2×#45311yescharacteristic
164D53—161no—
165C2×#137—1062no—
166Q82—22no—
167Q82—24no—
168Q82C2×D5314yes—
169#532—531no—
170C2×C4⋊Q8—22no—
171C2×C4⋊Q8—22no—
172C2×C4⋊Q8C2×D5311yescharacteristic
173C2×C4×Q8—22no—
174C2×C4×Q8C2×D5312yes—
175#497—1062no—
176#494—532no—
177#492—532no—
178#480—1062no—
179#477—534no—
180#460—534no—
181#448—1062no—
182#440—1064no—
183#432—532no—
184C4×M16—531no—
185#333—534no—
186Q8⋊C8—538no—
187C53#1854611yescharacteristic, complemented, Sylow
188C2×#41—1061no—
189C22×Q8—22no—
190C22×Q8C22⋊D5312yes—
191C22×Q8C22⋊D5312yes—
192#137—2128no—
193C2×Q16—1064no—
194C2×SD16—1064no—
195C2×SD16—1064no—
196C2×M16—1061no—
197C4⋊Q8—28no—
198C4⋊Q8—28no—
199C4⋊Q8—28no—
200C4⋊Q8—28no—
201C4⋊Q8—22no—
202C4⋊Q8—531no—
203C4⋊Q8C22⋊D5312yes—
204C4⋊Q8C22×D5312yes—
205#127—531no—
206#124—1061no—
207#121—1062no—
208C4×Q8—28no—
209C4×Q8—28no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8C22⋊D5314yes—
213C4×Q8C22×D5314yes—
214C4×D4—1062no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4C22⋊D5312yes—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42C22×D5311yescharacteristic
222#107—1064no—
223C4⋊C8—534no—
224#103—1064no—
225#102—1064no—
226C22⋊C8—1062no—
227#97—1061no—
228#97—1062no—
229C4×C8—532no—
230#41—2124no—
231C2×Q8—28no—
232C2×Q8—28no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—44no—
236C2×Q8—84no—
237C2×Q8—1061no—
238C2×Q8C2×C22⋊D5314yes—
239C2×Q8C2×C22⋊D5314yes—
240C2×D4—1062no—
241C2×D4—1061no—
242C2×D4—2121no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—2121no—
247C22×C4C2×C22⋊D5312yes—
248C22×C4C2×C22⋊D5311yescharacteristic
249Q16—2128no—
250SD16—2128no—
251SD16—2128no—
252M16—1064no—
253C2×C8—1062no—
254C2×C8—1064no—
255C4⋊C4—216no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—22no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—1062no—
262C4⋊C4C2×C22⋊D5314yes—
263C22⋊C4—2122no—
264C42—28no—
265C42—24no—
266C42—24no—
267C42C2×C22⋊D5312yes—
268C42C2×C22⋊D5311yescharacteristic
269C42C23×D5311yescharacteristic
270C23—2121no—
271C23#1313318611yescharacteristic
272Q8—28no—
273Q8—24no—
274Q8—44no—
275Q8—82no—
276Q8—88no—
277Q8—2122no—
278Q8#1313324818yescomplemented
279D4—4242no—
280D4—2122no—
281D4—2124no—
282C2×C4—28no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—4242no—
291C2×C4—2121no—
292C2×C4#1313318612yes—
293C2×C4#1313318611yescharacteristic
294C2×C4C22×C22⋊D5311yescharacteristic
295C2×C4C22×C22⋊D5312yes—
296C8—1064no—
297C8—2124no—
298C22—22no—
299C22—4242no—
300C22—4241no—
301C22#1681079112yes—
302C22C2×#1313318611yescharacteristic
303C4—28no—
304C4—22no—
305C4—21no—
306C4—21no—
307C4—44no—
308C4—44no—
309C4—4241no—
310C4C2×#1313324814yes—
311C2—22no—
312C2—8481no—
313C2#2028394912yes—
314C2C2×#1681079111yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×Q82×C53C2
socleC22×C53C2×#120
prime radical#16811173C22
hypercenterC2×Q82D53
hypocenterC53#18546
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
3. #20273855C2semidirect
4. #20273793C2semidirect
5. #20273441C2semidirect
6. #20273162C2semidirect
7. C2×Q82×C53C2semidirect
119. Q8×C53#137semidirect
154. C2×Q82D53semidirect
187. C53#18546semidirect
278. Q8#13133248semidirect

Maximal subgroups

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

subgroupindex
#202851722
#202848372
#202738552
#202737932
#202734412
#202731622
C2×Q82×C532
#1854653