Finite Group Databasebeta

← #27278609

Subgroups — #27278609

Order 13568.

356 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 97 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#20283816C211yesmaximal, characteristic
2#20275692C211yesmaximal, characteristic
3#20273504C211yesmaximal, characteristic
4#20273503C211yesmaximal, characteristic
5#20273379C211yesmaximal, characteristic
6#20272548C211yesmaximal, characteristic
7C53×#1276C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9#16810753—21no—
10#16810750—21no—
11C8×Dic106—21no—
12#16810747—21no—
13#16810681—22no—
14#16810676C2211yescharacteristic
15#16810676C2211yescharacteristic
16#16810647C2211yescharacteristic
17#16810383—22no—
18#16810376—21no—
19#16810375—21no—
20#16810374—21no—
21#16810373—21no—
22#16810370C411yescharacteristic, complemented
23#16810370C411yescharacteristic, complemented
24#16810368C411yescharacteristic, complemented
25#16810368C411yescharacteristic, complemented
26C53×#431C2211yescharacteristic
27C53×#431C2211yescharacteristic
28C43×C53C2211yescharacteristic
29C2×#11719917—21no—
30#13133217C2×C411yescharacteristic
31#13133217C2×C411yescharacteristic
32#13133209Q812yes—
33#13133196—21no—
34#13133196—21no—
35C4×Dic106—21no—
36C4×Dic106—21no—
37#13133176D412yes—
38C4⋊#88322—21no—
39C4⋊#88322—21no—
40C4⋊#88322C2×C411yescharacteristic
41C4⋊#88322C2×C411yescharacteristic
42C4⋊#88322D411yescharacteristic
43C4⋊#88322D411yescharacteristic
44#13133146—22no—
45#13133146—22no—
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48#13133127—22no—
49C4×#88322—21no—
50C4×#88322—21no—
51C8×Dic53—22no—
52C2×C42×C53—21no—
53C2×C42×C53—21no—
54C2×C42×C53—21no—
55C2×C42×C53C2311yescharacteristic
56C4⋊C8×C53C2×C411yescharacteristic
57C4⋊C8×C53C2×C411yescharacteristic
58C4⋊C8×C53D411yescharacteristic
59C4⋊C8×C53D411yescharacteristic
60C22⋊C8×C53—21no—
61C22⋊C8×C53—21no—
62C4×C8×C53—21no—
63C4×C8×C53—21no—
64C22×C4×C53—41no—
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53#4111yescharacteristic
67C22×C4×C53#4111yescharacteristic
68C2×Dic106—21no—
69C2×Dic106—21no—
70C2×C8×C53—21no—
71C2×C8×C53—21no—
72C2×C8×C53—21no—
73C2×C8×C53—21no—
74C42×C53—22no—
75C42×C53—22no—
76C42×C53—22no—
77C42×C53—41no—
78C42×C53—41no—
79C42×C53—41no—
80C42×C53C22×C411yescharacteristic
81C42×C53C2×D411yescharacteristic
82C42×C53C2×D411yescharacteristic
83C42×C53C2×Q811yescharacteristic
84C22⋊Dic53—22no—
85C4⋊Dic53—21no—
86C4⋊Dic53—21no—
87C4⋊Dic53C4⋊C412yes—
88C4⋊Dic53C4⋊C412yes—
89#11719919—22no—
90#11719919—22no—
91C4×Dic53—22no—
92#11719917—42no—
93C2×#88322—21no—
94C2×#88322—21no—
95C2×#88322—21no—
96C2×#88322—21no—
97C2×#88322—41no—
98C23×C53#12311yescharacteristic
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53C2×C4⋊C411yescharacteristic
109C2×C4×C53C4×D411yescharacteristic
110C2×C4×C53C4×D411yescharacteristic
111C2×C4×C53#12011yescharacteristic
112C2×C4×C53C22⋊Q811yescharacteristic
113C2×C4×C53C22⋊Q811yescharacteristic
114C2×Dic53—22no—
115C2×Dic53—22no—
116Dic106—41no—
117Dic106—41no—
118Dic106—41no—
119Dic106—41no—
120C8×C53—41no—
121C8×C53—41no—
122C8×C53—41no—
123C8×C53—41no—
124#88322—41no—
125#88322—41no—
126#88322—41no—
127#88322—41no—
128#88322—81no—
129#8936—531nomaximal, Sylow
130C22×C53—21no—
131C22×C53—21no—
132C22×C53—21no—
133C22×C53#38511yescharacteristic
134C4×C53—21no—
135C4×C53—21no—
136C4×C53—21no—
137C4×C53—44no—
138C4×C53#44311yescharacteristic
139C4×C53#44311yescharacteristic
140C4×C53#44411yescharacteristic
141C4×C53#44411yescharacteristic
142C4×C53#45411yescharacteristic
143C4×C53#45611yescharacteristic
144Dic53—42no—
145Dic53—42no—
146#2986—531no—
147#1994—531no—
148#1574—531no—
149#1574—531no—
150#1544—531no—
151#1276—531no—
152#1276D5311yescharacteristic
153C2×C53—41no—
154C2×C53#185711yescharacteristic
155C2×C53#186611yescharacteristic
156C2×C53#186611yescharacteristic
157#533—531no—
158#446—1061no—
159#446—1061no—
160C8×Q8—1061no—
161C8×Q8—1061no—
162#433—1062no—
163#431—531no—
164#431—531no—
165#431C2×D5311yescharacteristic
166#431C2×D5311yescharacteristic
167#423—531no—
168C43C2×D5311yescharacteristic
169#339—1062no—
170#335—1061no—
171#335—1061no—
172#334—1061no—
173#334—1061no—
174#332—531no—
175#332—531no—
176#332—531no—
177#332—531no—
178C53#893611yescharacteristic, complemented, Sylow
179C2×M16—1061no—
180C4⋊Q8—531no—
181C4⋊Q8—531no—
182#125—532no—
183C22⋊Q8—1061no—
184C22⋊Q8—1061no—
185C4×Q8—1061no—
186C4×Q8—1061no—
187#117—532no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42C22×D5311yescharacteristic
192C4⋊C8—531no—
193C4⋊C8—531no—
194C4⋊C8—531no—
195C4⋊C8—531no—
196C4⋊C8—1062no—
197C4⋊C8—1062no—
198C4⋊C8—1061no—
199C4⋊C8—1061no—
200C4⋊C8C4×D5311yescharacteristic
201C4⋊C8C4×D5311yescharacteristic
202C4⋊C8D21211yescharacteristic
203C4⋊C8D21211yescharacteristic
204C22⋊C8—21no—
205C22⋊C8—21no—
206C22⋊C8—1061no—
207C22⋊C8—1061no—
208#97—1062no—
209C4×C8—21no—
210C4×C8—21no—
211C4×C8—1062no—
212C4×C8—1061no—
213C4×C8—1061no—
214C2×Q8—1061no—
215C2×Q8—1061no—
216C22×C4—41no—
217C22×C4#1171994511yescharacteristic
218C22×C4D4×D5311yescharacteristic
219C22×C4#1171994711yescharacteristic
220M16—2122no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—1061no—
226C2×C8—1061no—
227C2×C8—1061no—
228C2×C8—1061no—
229C2×C8—2121no—
230C4⋊C4—532no—
231C4⋊C4—532no—
232C4⋊C4—1061no—
233C4⋊C4—1061no—
234C4⋊C4—1062no—
235C4⋊C4—1062no—
236C22⋊C4—1062no—
237C42—22no—
238C42—22no—
239C42—22no—
240C42—41no—
241C42—41no—
242C42—41no—
243C42—1062no—
244C42C2×C4×D5311yescharacteristic
245C42C2×D21211yescharacteristic
246C42D4×D5311yescharacteristic
247C42Q8×D5311yescharacteristic
248C23#1313319911yescharacteristic
249Q8—2121no—
250Q8—2121no—
251Q8—2121no—
252Q8—2121no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—1062no—
263C2×C4—1062no—
264C2×C4C4⋊C4×D5311yescharacteristic
265C2×C4#1313317711yescharacteristic
266C2×C4C4×D21211yescharacteristic
267C2×C4#1313318711yescharacteristic
268C2×C4#1313319511yescharacteristic
269C2×C4#1313319711yescharacteristic
270C8—41no—
271C8—41no—
272C8—41no—
273C8—41no—
274C8—4241no—
275C8—2121no—
276C8—2121no—
277C8—2121no—
278C8—2121no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22#1681049711yescharacteristic
283C4—21no—
284C4—21no—
285C4—21no—
286C4—44no—
287C4—2122no—
288C4—2122no—
289C4#1681073711yescharacteristic
290C4#1681074111yescharacteristic
291C4#1681074211yescharacteristic
292C4#1681074611yescharacteristic
293C4#1681079711yescharacteristic
294C4#1681080711yescharacteristic
295C2—41no—
296C2#2027493111yescharacteristic
297C2#2027499111yescharacteristic
298C2#2027499311yescharacteristic
subgroupG/H
centerC2×C4#13133187
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1276C2
socleC22×C53#385
prime radicalC23×C53#123
cosocleC2×C42×C53C23
hypercenter#1276D53
hypocenterC53#8936
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
22. #16810370C4semidirect
23. #16810370C4semidirect
24. #16810368C4semidirect
25. #16810368C4semidirect
178. C53#8936semidirect

Maximal subgroups

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

subgroupindex
#202838162
#202756922
#202735042
#202735032
#202733792
#202725482
C53×#12762
#893653