Finite Group Databasebeta

← #27285036

Subgroups — #27285036

Order 13568.

512 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 83 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#20286025C211yesmaximal, characteristic
2#20284837C212yesmaximal, complemented
3#20273325C212yesmaximal, complemented
4#20272585C211yesmaximal, characteristic, complemented
5C53×#1985C211yesmaximal, characteristic, complemented
6#16811173C2212yes—
7#16810936—22no—
8#16810918—22no—
9#16810904—22no—
10#16810859—22no—
11#16810823—22no—
12#16810819—22no—
13#16810762—22no—
14#16810721—24no—
15#16810680C2212yes—
16#16810362—24no—
17C2×C4×Q8×C53C2211yescharacteristic
18C2×C4⋊C8×C53C2211yescharacteristic
19C2×#97×C53C2211yescharacteristic
20C53×#336—22no—
21C2×#11719945—22no—
22C2×#11719924—24no—
23C2×D424—22no—
24#13133217D412yes—
25#13133214D412yes—
26#13133205—22no—
27#13133191—22no—
28#13133191—22no—
29C4×D212—22no—
30C4×D212—22no—
31#13133151—22no—
32#13133150—24no—
33C4⋊#88322—22no—
34#13133141—24no—
35#13133139—24no—
36#13133137—24no—
37C22⋊#88322—22no—
38#13133128D414yes—
39C22×Q8×C53D411yescharacteristic
40C22×C8×C53—21no—
41C22×C8×C53D412yes—
42C4×Q8×C53—22no—
43C4×Q8×C53—22no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C42×C53—21no—
47C2×C42×C53C2311yescharacteristic
48C4⋊C8×C53—22no—
49#97×C53—22no—
50C2×Q8×C53—22no—
51C2×Q8×C53—42no—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53C2×D411yescharacteristic
55C22×C4×C53C2×D411yescharacteristic
56C22×C4×C53#4111yescharacteristic
57C2×C22⋊D53—42no—
58#11719945—84no—
59C2×D212—22no—
60C2×D212—22no—
61C2×D212—22no—
62C2×C4×D53—42no—
63C2×Dic106—22no—
64C2×C8×C53—22no—
65C2×C8×C53—22no—
66C2×C8×C53—41no—
67C2×C8×C53—41no—
68C2×C8×C53—42no—
69C4⋊C4×C53—22no—
70C4⋊C4×C53—42no—
71C42×C53—41no—
72C42×C53—41no—
73C42×C53C2×D412yes—
74C42×C53C2×D412yes—
75#11719924—48no—
76#11719921—42no—
77#11719921—42no—
78C4⋊Dic53—22no—
79C4⋊Dic53—22no—
80C2×#88322—24no—
81D424—82no—
82C23×C53#12111yescharacteristic
83C22×D53—42no—
84Q8×C53—24no—
85Q8×C53—42no—
86C2×C4×C53—22no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—22no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53—42no—
95C2×C4×C53#12011yescharacteristic
96C2×C4×C53#12012yes—
97C2×C4×C53#12112yes—
98C2×C4×C53#12711yescharacteristic
99C22⋊D53—84no—
100C2×Dic53—42no—
101D212—44no—
102D212—42no—
103D212—42no—
104D212—82no—
105C4×D53—84no—
106Dic106—82no—
107C8×C53—42no—
108C8×C53—42no—
109#88322—44no—
110#10028—531nomaximal, Sylow
111C22×C53—21no—
112C22×C53—21no—
113C22×C53#39211yescharacteristic
114C22×C53#47212yes—
115C2×D53—82no—
116C2×D53—82no—
117C2×D53—82no—
118C4×C53—21no—
119C4×C53—22no—
120C4×C53—22no—
121C4×C53—42no—
122C4×C53—81no—
123C4×C53#45511yescharacteristic
124C4×C53#45611yescharacteristic
125Dic53—82no—
126#3245—531no—
127#3112—532no—
128#1985D5311yescharacteristic, complemented
129#1530—532no—
130#1290—531no—
131C2×C53—22no—
132C2×C53#200711yescharacteristic
133C2×C53#201011yescharacteristic
134C2×C53#201811yescharacteristic
135D53—162no—
136#532—532no—
137C2×C4×Q8C2×D5311yescharacteristic
138#480—1062no—
139#477—1062no—
140#474—1062no—
141#466—1062no—
142#460—1062no—
143#459—1062no—
144#448—1062no—
145#440—1064no—
146#432—532no—
147C2×C4⋊C8C2×D5311yescharacteristic
148C2×#97C2×D5311yescharacteristic
149#336—22no—
150#329—1064no—
151C53#1002811yescharacteristic, complemented, Sylow
152C2×#41—1062no—
153C22×Q8D21211yescharacteristic
154C2×SD16—1064no—
155C2×D8—1062no—
156C22×C8—21no—
157C22×C8C22⋊D5312yes—
158C4⋊Q8—532no—
159#127—532no—
160#124—1062no—
161#121—1062no—
162#121—1062no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×D4—1062no—
166C4×D4—1062no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4D21211yescharacteristic
169C2×C42—21no—
170C2×C42C22×D5311yescharacteristic
171#107—1064no—
172#107—1062no—
173C4⋊C8—22no—
174C4⋊C8—1062no—
175#103—1064no—
176#102—1064no—
177#102—1064no—
178C22⋊C8—1062no—
179#97—22no—
180#97—534no—
181#41—4244no—
182C2×Q8—22no—
183C2×Q8—42no—
184C2×Q8—1062no—
185C2×D4—1062no—
186C2×D4—1062no—
187C2×D4—1062no—
188C2×D4—2122no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—2122no—
192C22×C4C2×D21211yescharacteristic
193C22×C4#1171994511yescharacteristic
194C22×C4C2×C22⋊D5311yescharacteristic
195SD16—2128no—
196D8—4242no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—42no—
202C2×C8—1064no—
203C4⋊C4—22no—
204C4⋊C4—42no—
205C4⋊C4—1062no—
206C4⋊C4—1062no—
207C22⋊C4—2122no—
208C22⋊C4—2122no—
209C42—41no—
210C42—41no—
211C42D4×D5312yes—
212C42D4×D5312yes—
213C23—2122no—
214C23#1313319111yescharacteristic
215Q8—24no—
216Q8—42no—
217Q8—4242no—
218D4—4242no—
219D4—4244no—
220D4—2124no—
221D4—2122no—
222D4—2122no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—4244no—
233C2×C4—2122no—
234C2×C4#1313318511yescharacteristic
235C2×C4#1313318712yes—
236C2×C4#1313319012yes—
237C2×C4#1313321311yescharacteristic
238C8—42no—
239C8—42no—
240C8—2124no—
241C22—21no—
242C22—21no—
243C22—4242no—
244C22—4242no—
245C22—4242no—
246C22#1681053211yescharacteristic
247C22#1681089312yes—
248C4—21no—
249C4—22no—
250C4—22no—
251C4—42no—
252C4—81no—
253C4—4242no—
254C4#1681080011yescharacteristic
255C4#1681080511yescharacteristic
256C2—22no—
257C2—8482no—
258C2#2027576111yescharacteristic
259C2#2027578211yescharacteristic
260C2#2027582411yescharacteristic
subgroupG/H
centerC22#16810532
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1985C2
socleC22×C53#392
prime radical#20286025C2
hypercenter#1985D53
hypocenterC53#10028
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1985

Complements

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

NH ≅ G/Ntype
2. #20284837C2semidirect
3. #20273325C2semidirect
4. #20272585C2semidirect
5. C53×#1985C2semidirect
128. #1985D53semidirect
151. C53#10028semidirect

Maximal subgroups

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

subgroupindex
#202860252
#202848372
#202733252
#202725852
C53×#19852
#1002853