Finite Group Databasebeta

← #27281956

Subgroups — #27281956

Order 13568.

454 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 115 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#20280077C211yesmaximal, characteristic
2#20274711C211yesmaximal, characteristic
3#20273145C212yesmaximal
4#20272574C212yesmaximal
5C53×#2983C211yesmaximal, characteristic
6C2×C4×Dic106C2211yescharacteristic
7#16810647C2212yes—
8C2×#13133136—21no—
9#16810562C412yes—
10#16810543C412yes—
11#16810530—21no—
12#16810519—21no—
13#16810507—22no—
14#16810489C2211yescharacteristic
15#16810392—21no—
16#16810355—24no—
17C2×C4×Q8×C53C2211yescharacteristic
18C53×#446—24no—
19C53×#432—22no—
20C53×#431C2212yes—
21C22×Dic106C2×C411yescharacteristic
22C2×#11719917—22no—
23C4×Dic106—22no—
24C4×Dic106—22no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×#11719919—22no—
28C2×#11719919—21no—
29C2×C4×Dic53—21no—
30C4⋊#88322—22no—
31C4⋊#88322—22no—
32#13133136—42no—
33#13133124—22no—
34#13133124—22no—
35#13133124—22no—
36#13133124—21no—
37#13133124C2×C412yes—
38C22×Q8×C53D411yescharacteristic
39C4×Q8×C53—22no—
40C4×Q8×C53—22no—
41C2×C4⋊C4×C53C2×C412yes—
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C42×C53D412yes—
44C2×C42×C53C2311yescharacteristic
45C4⋊C8×C53—24no—
46C4⋊C8×C53—22no—
47C22⋊C8×C53—22no—
48#97×C53—24no—
49C4×C8×C53D414yes—
50C2×Q8×C53—22no—
51C2×Q8×C53—21no—
52C2×Q8×C53—21no—
53C2×Q8×C53—22no—
54C22×C4×C53C22⋊C412yes—
55C22×C4×C53C22⋊C412yes—
56C22×C4×C53C22×C411yescharacteristic
57C22×C4×C53C2×D411yescharacteristic
58C22×C4×C53C2×D411yescharacteristic
59C22×Dic53—21no—
60C22×Dic53—22no—
61C22×Dic53—21no—
62C2×Dic106—22no—
63C2×Dic106—42no—
64C2×C8×C53—24no—
65C4⋊C4×C53—24no—
66C4⋊C4×C53—22no—
67C42×C53—22no—
68C42×C53—22no—
69C42×C53C2×D412yes—
70C42×C53#4112yes—
71C4⋊Dic53—22no—
72C4⋊Dic53—41no—
73C4⋊Dic53—41no—
74#11719919—44no—
75#11719919—41no—
76#11719919—41no—
77C4×Dic53—42no—
78#11719917—42no—
79#11719917—42no—
80C2×#88322—42no—
81C23×C53C2×C22⋊C411yescharacteristic
82Q8×C53—42no—
83Q8×C53—42no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53—42no—
91C2×C4×C53C4×D412yes—
92C2×C4×C53#12011yescharacteristic
93C2×C4×C53#12112yes—
94C2×C4×C53#12311yescharacteristic
95C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
96C2×C4×C53#13712yes—
97C2×C4×C53#13711yescharacteristic
98C2×Dic53—22no—
99C2×Dic53—41no—
100C2×Dic53—41no—
101C2×Dic53—42no—
102C2×Dic53—42no—
103C2×Dic53—42no—
104C2×Dic53—41no—
105C2×Dic53—42no—
106Dic106—24no—
107Dic106—42no—
108C8×C53—44no—
109#88322—82no—
110#9488—531nomaximal, Sylow
111C22×C53—21no—
112C22×C53—21no—
113C22×C53#38611yescharacteristic
114C22×C53#41711yescharacteristic
115C22×C53#41911yescharacteristic
116C4×C53—22no—
117C4×C53—21no—
118C4×C53—21no—
119C4×C53—42no—
120C4×C53—42no—
121C4×C53#46812yes—
122C4×C53#47012yes—
123Dic53—42no—
124Dic53—82no—
125Dic53—81no—
126#2983D5311yescharacteristic
127#2581—531no—
128#1801—531no—
129#1487—532no—
130#1286—532no—
131C2×C53—21no—
132C2×C53—21no—
133C2×C53#189111yescharacteristic
134C2×C53#189411yescharacteristic
135C2×C53#191211yescharacteristic
136C2×C4×Q8—531no—
137C2×C4×Q8C2×D5311yescharacteristic
138#446—24no—
139#432—22no—
140#431C2×D5312yes—
141#423—532no—
142C2×#101—1061no—
143#400—532no—
144#395—532no—
145#391—1061no—
146#389—1061no—
147#387—1062no—
148#382—531no—
149#343—1061no—
150Q8⋊C8—1064no—
151C53#948811yescharacteristic, complemented, Sylow
152C22×Q8—531no—
153C22×Q8D21211yescharacteristic
154C2×M16—1062no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—1062no—
158C4×Q8—1062no—
159C2×C4⋊C4—531no—
160C2×C4⋊C4—1061no—
161C2×C4⋊C4—1061no—
162C2×C4⋊C4—1062no—
163C2×C4⋊C4C4×D5312yes—
164C2×C4⋊C4D21211yescharacteristic
165C2×C42—1061no—
166C2×C42C22⋊D5312yes—
167C2×C42C22×D5311yescharacteristic
168C4⋊C8—24no—
169C4⋊C8—22no—
170C4⋊C8—1062no—
171C4⋊C8—1062no—
172#101—2122no—
173C22⋊C8—22no—
174#97—24no—
175C4×C8C22⋊D5314yes—
176#95—532no—
177#95—1061no—
178#95—1062no—
179#95—1062no—
180#95—1062no—
181C2×Q8—22no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—22no—
185C2×Q8—1062no—
186C2×Q8—2122no—
187C22×C4—1061no—
188C22×C4—1062no—
189C22×C4—1061no—
190C22×C4#1171992112yes—
191C22×C4#1171992112yes—
192C22×C4C2×C4×D5311yescharacteristic
193C22×C4C2×D21211yescharacteristic
194C22×C4C2×C22⋊D5311yescharacteristic
195M16—2122no—
196M16—2122no—
197C2×C8—24no—
198C2×C8—2122no—
199C4⋊C4—24no—
200C4⋊C4—22no—
201C4⋊C4—1062no—
202C4⋊C4—2124no—
203C4⋊C4—2121no—
204C4⋊C4—2121no—
205C4⋊C4—2121no—
206C4⋊C4—2121no—
207C42—22no—
208C42—22no—
209C42—2122no—
210C42#1171994512yes—
211C42C2×C22⋊D5312yes—
212C23C2×#1171992111yescharacteristic
213Q8—42no—
214Q8—42no—
215Q8—1064no—
216Q8—2122no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—1062no—
225C2×C4—2122no—
226C2×C4—2122no—
227C2×C4—2122no—
228C2×C4—2121no—
229C2×C4—2122no—
230C2×C4—2121no—
231C2×C4—2121no—
232C2×C4C4×C22⋊D5312yes—
233C2×C4#1313318611yescharacteristic
234C2×C4#1313319112yes—
235C2×C4#1313320011yescharacteristic
236C2×C4#1313324511yescharacteristic
237C2×C4#1313325012yes—
238C2×C4#1313325011yescharacteristic
239C8—44no—
240C8—4242no—
241C22—21no—
242C22—21no—
243C22#1681050211yescharacteristic
244C22#1681062511yescharacteristic
245C22#1681063311yescharacteristic
246C4—21no—
247C4—21no—
248C4—22no—
249C4—42no—
250C4—42no—
251C4—4242no—
252C4—4241no—
253C4—2122no—
254C4#1681086912yes—
255C4#1681088112yes—
256C2—21no—
257C2—21no—
258C2#2027513311yescharacteristic
259C2#2027514811yescharacteristic
260C2#2027525211yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2983C2
socleC22×C53#386
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#2983D53
hypocenterC53#9488
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2983

Complements

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

NH ≅ G/Ntype
151. C53#9488semidirect

Maximal subgroups

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

subgroupindex
#202800772
#202747112
#202731452
#202725742
C53×#29832
#948853