Finite Group Databasebeta

← #27296538

Subgroups — #27296538

Order 13568.

390 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 71 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#20276643C212yesmaximal
2#20276026C211yesmaximal, characteristic
3#20272727C212yesmaximal, complemented
4#20272655C211yesmaximal, characteristic, complemented
5C53×#3249C211yesmaximal, characteristic, complemented
6C22×D424C2211yescharacteristic
7C2×#13133160—22no—
8C2×#13133139—21no—
9#16810525—21no—
10#16810423—22no—
11C22⋊#11719908C2212yes—
12#16810386C2211yescharacteristic
13C53×#533—21no—
14C53×#477—22no—
15C53×#467—21no—
16C53×#467C2212yes—
17C53×#462—22no—
18C2×C4⋊C8×C53C2211yescharacteristic
19C22×D212—21no—
20C2×D424—22no—
21C2×D424—42no—
22C2×C4⋊Dic53—21no—
23C2×#11719921—41no—
24C2×#11719921—41no—
25#13133160—44no—
26C2×#11719908—22no—
27#13133139—42no—
28C2×Q16×C53—21no—
29C2×Q16×C53D412yes—
30C22×C8×C53D411yescharacteristic
31C22×C8×C53C2311yescharacteristic
32C4⋊Q8×C53—21no—
33C4⋊Q8×C53—21no—
34#125×C53—41no—
35C22⋊Q8×C53—22no—
36C22⋊Q8×C53—41no—
37C4×Q8×C53—22no—
38C4×Q8×C53—41no—
39#117×C53—41no—
40C2×C42×C53D411yescharacteristic
41#107×C53—21no—
42#107×C53D412yes—
43C4⋊C8×C53—22no—
44#103×C53—22no—
45#103×C53—22no—
46C23×D53—41no—
47C2×Q8×C53—22no—
48C2×Q8×C53—41no—
49C22×C4×C53D811yescharacteristic
50C22×C4×C53D811yescharacteristic
51C22×C4×C53C2×D411yescharacteristic
52C22×Dic53—41no—
53C2×D212—22no—
54C2×D212—41no—
55C2×D212—41no—
56C2×D212—41no—
57C2×D212—41no—
58C2×D212—41no—
59Q16×C53—42no—
60Q16×C53—81no—
61C2×C8×C53—21no—
62C2×C8×C53—41no—
63C2×C8×C53—41no—
64C2×C8×C53C2×D412yes—
65C4⋊C4×C53—22no—
66C4⋊C4×C53—41no—
67C4⋊C4×C53—41no—
68C4⋊C4×C53—42no—
69C4⋊C4×C53—41no—
70C4⋊C4×C53—42no—
71C22⋊C4×C53—42no—
72C42×C53—21no—
73C42×C53—21no—
74C42×C53—42no—
75#11719921—82no—
76#11719921—82no—
77C4⋊Dic53—41no—
78C4⋊Dic53—41no—
79D424—24no—
80D424—42no—
81#11719908—42no—
82C23×C53C2×D811yescharacteristic
83C22×D53—42no—
84C22×D53—81no—
85C22×D53—81no—
86C22×D53—81no—
87C22×D53—81no—
88C22×D53—81no—
89C22×D53—81no—
90Q8×C53—42no—
91Q8×C53—42no—
92Q8×C53—82no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—41no—
99C2×C4×C53—42no—
100C2×C4×C53—42no—
101C2×C4×C53SD3212yescomplemented
102C2×C4×C53#12011yescharacteristic
103C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
104C2×Dic53—81no—
105C2×Dic53—81no—
106C2×Dic53—81no—
107D212—42no—
108D212—42no—
109D212—81no—
110D212—81no—
111C8×C53—22no—
112C8×C53—41no—
113#12159—531nomaximal, Sylow
114C22×C53—21no—
115C22×C53—21no—
116C22×C53#44711yescharacteristic
117C22×C53C2×SD3211yescharacteristic
118C22×C53#51011yescharacteristic
119C2×D53—82no—
120C2×D53—82no—
121C2×D53—82no—
122C2×D53—161no—
123C2×D53—161no—
124C2×D53—161no—
125C2×D53—161no—
126C4×C53—21no—
127C4×C53—22no—
128C4×C53—41no—
129C4×C53—82no—
130C4×C53—82no—
131C4×C53#45411yescharacteristic
132C4×C53B(4,𝔽2)11yescharacteristic
133Dic53—161no—
134#3249D5311yescharacteristic, complemented
135#2187—532no—
136#2054—531no—
137#1348—532no—
138#1322—531no—
139C2×C53—21no—
140C2×C53—21no—
141C2×C53#159811yescharacteristic
142C2×C53#218511yescharacteristic
143C2×C53#219011yescharacteristic
144D53—162no—
145C22×D8—531no—
146#533—21no—
147C2×SD32—1062no—
148#477—22no—
149#467—21no—
150#467C2×D5312yes—
151#462—22no—
152C2×C4⋊C8C2×D5311yescharacteristic
153C2×#102—1061no—
154#390—1061no—
155#357—1062no—
156C22⋊C16—532no—
157#340—531no—
158C53#1215911yescharacteristic, complemented, Sylow
159C22×D4—1061no—
160C2×Q16—21no—
161C2×Q16C22⋊D5312yes—
162C2×D8—1062no—
163C2×D8—2122no—
164C22×C8C22⋊D5311yescharacteristic
165C22×C8C22×D5311yescharacteristic
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168#125—41no—
169C22⋊Q8—22no—
170C22⋊Q8—41no—
171C4×Q8—22no—
172C4×Q8—41no—
173#117—41no—
174C2×C4⋊C4—1061no—
175C2×C22⋊C4—2121no—
176C2×C22⋊C4—2121no—
177C2×C42C22⋊D5311yescharacteristic
178SD32—2124no—
179C2×C16—1062no—
180#107—21no—
181#107C22⋊D5312yes—
182C4⋊C8—22no—
183#103—22no—
184#103—22no—
185#102—2122no—
186C24—2121no—
187C2×Q8—22no—
188C2×Q8—41no—
189C2×D4—1062no—
190C2×D4—2121no—
191C2×D4—2121no—
192C2×D4—2121no—
193C2×D4—2121no—
194C2×D4—2121no—
195C22×C4—2121no—
196C22×C4#1171992211yescharacteristic
197C22×C4#1171992211yescharacteristic
198C22×C4C2×C22⋊D5311yescharacteristic
199Q16—42no—
200Q16—81no—
201D8—1064no—
202D8—2122no—
203C2×C8—21no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8C2×C22⋊D5312yes—
207C4⋊C4—22no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—42no—
211C4⋊C4—41no—
212C4⋊C4—42no—
213C4⋊C4—2121no—
214C4⋊C4—2121no—
215C22⋊C4—42no—
216C22⋊C4—4242no—
217C22⋊C4—4242no—
218C42—21no—
219C42—21no—
220C42—42no—
221C16—2122no—
222C23—4241no—
223C23—4241no—
224C23—4241no—
225C23—4241no—
226C23—4241no—
227C23—4241no—
228C23—2122no—
229C23C2×#1171992211yescharacteristic
230Q8—42no—
231Q8—42no—
232Q8—82no—
233D4—4241no—
234D4—4241no—
235D4—2122no—
236D4—2122no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—4241no—
246C2×C4—4241no—
247C2×C4—4241no—
248C2×C4#1313316012yescomplemented
249C2×C4#1313318611yescharacteristic
250C2×C4#1313324311yescharacteristic
251C8—22no—
252C8—41no—
253C22—21no—
254C22—21no—
255C22—8481no—
256C22—8481no—
257C22—8481no—
258C22—8481no—
259C22—4242no—
260C22—4242no—
261C22—4242no—
262C22#1681075711yescharacteristic
263C22C2×#1313316011yescharacteristic
264C22#1681106411yescharacteristic
265C4—22no—
266C4—21no—
267C4—41no—
268C4—82no—
269C4—82no—
270C4—8481no—
271C4#1681079611yescharacteristic
272C4#1681081011yescharacteristic
273C2—21no—
274C2—21no—
275C2—8482no—
276C2#2027365011yescharacteristic
277C2#2027663211yescharacteristic
278C2#2027666111yescharacteristic
subgroupG/H
centerC22#16810757
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
FittingC53×#3249C2
socleC22×C53#447
prime radicalC22×D424C22
hypercenter#3249D53
hypocenterC53#12159
LCSG ▷ C22×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #3249

Complements

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

NH ≅ G/Ntype
3. #20272727C2semidirect
4. #20272655C2semidirect
5. C53×#3249C2semidirect
101. C2×C4×C53SD32semidirect
134. #3249D53semidirect
158. C53#12159semidirect
248. C2×C4#13133160semidirect

Maximal subgroups

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

subgroupindex
#202766432
#202760262
#202727272
#202726552
C53×#32492
#1215953