Finite Group Databasebeta

← #27281936

Subgroups — #27281936

Order 13568.

474 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 131 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×#2982C211yesmaximal, 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
18C8×Q8×C53—24no—
19C53×#431—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—
48C4×C8×C53—24no—
49C4×C8×C53D414yes—
50C2×Q8×C53—22no—
51C2×Q8×C53—22no—
52C2×Q8×C53SD1612yes—
53C2×Q8×C53Q1612yes—
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—42no—
73#11719919—44no—
74#11719919—41no—
75#11719919—41no—
76C4×Dic53—42no—
77#11719917—42no—
78#11719917—42no—
79C2×#88322—42no—
80C23×C53C2×C22⋊C411yescharacteristic
81Q8×C53—24no—
82Q8×C53—24no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—41no—
87C2×C4×C53—41no—
88C2×C4×C53—42no—
89C2×C4×C53#10314yes—
90C2×C4×C53C4×D412yes—
91C2×C4×C53#12011yescharacteristic
92C2×C4×C53#12112yes—
93C2×C4×C53#12311yescharacteristic
94C2×C4×C53C2×SD1611yescharacteristic
95C2×C4×C53C2×Q1611yescharacteristic
96C2×C4×C53#13512yes—
97C2×Dic53—22no—
98C2×Dic53—41no—
99C2×Dic53—41no—
100C2×Dic53—42no—
101C2×Dic53—42no—
102C2×Dic53—42no—
103C2×Dic53—41no—
104C2×Dic53—42no—
105Dic106—24no—
106Dic106—42no—
107C8×C53—44no—
108#88322—82no—
109#9484—531nomaximal, Sylow
110C22×C53—21no—
111C22×C53—21no—
112C22×C53#38611yescharacteristic
113C22×C53C2×#10311yescharacteristic
114C22×C53#41611yescharacteristic
115C4×C53—22no—
116C4×C53—24no—
117C4×C53—21no—
118C4×C53—21no—
119C4×C53—42no—
120C4×C53#46512yes—
121C4×C53#46712yes—
122Dic53—42no—
123Dic53—82no—
124Dic53—81no—
125#2982D5311yescharacteristic
126#2581—531no—
127#1801—531no—
128#1487—532no—
129#1286—532no—
130C2×C53—21no—
131C2×C53—21no—
132C2×C53#189111yescharacteristic
133C2×C53#189211yescharacteristic
134C2×C53#191211yescharacteristic
135C2×C4×Q8—531no—
136C2×C4×Q8C2×D5311yescharacteristic
137C8×Q8—24no—
138#431—22no—
139#431C2×D5312yes—
140#423—532no—
141C2×#101—1061no—
142#400—532no—
143#395—532no—
144#391—1061no—
145#389—1061no—
146#387—1062no—
147#382—531no—
148#343—1061no—
149Q8⋊C8—1064no—
150C53#948411yescharacteristic, complemented, Sylow
151C22×Q8—531no—
152C22×Q8D21211yescharacteristic
153C2×M16—1062no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—1062no—
157C4×Q8—1062no—
158C2×C4⋊C4—531no—
159C2×C4⋊C4—1061no—
160C2×C4⋊C4—1061no—
161C2×C4⋊C4—1062no—
162C2×C4⋊C4C4×D5312yes—
163C2×C4⋊C4D21211yescharacteristic
164C2×C42—1061no—
165C2×C42C22⋊D5312yes—
166C2×C42C22×D5311yescharacteristic
167C4⋊C8—24no—
168C4⋊C8—22no—
169C4⋊C8—1062no—
170C4⋊C8—1062no—
171#101—2122no—
172C22⋊C8—22no—
173C4×C8—24no—
174C4×C8C22⋊D5314yes—
175#95—532no—
176#95—1061no—
177#95—1062no—
178#95—1062no—
179#95—1062no—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—1062no—
183C2×Q8—2122no—
184C2×Q8#1171991312yes—
185C2×Q8Dic21212yes—
186C22×C4—1061no—
187C22×C4—1062no—
188C22×C4—1061no—
189C22×C4#1171992112yes—
190C22×C4#1171992112yes—
191C22×C4C2×C4×D5311yescharacteristic
192C22×C4C2×D21211yescharacteristic
193C22×C4C2×C22⋊D5311yescharacteristic
194M16—2122no—
195M16—2122no—
196C2×C8—24no—
197C2×C8—2122no—
198C4⋊C4—24no—
199C4⋊C4—22no—
200C4⋊C4—1062no—
201C4⋊C4—2124no—
202C4⋊C4—2121no—
203C4⋊C4—2121no—
204C4⋊C4—2122no—
205C42—22no—
206C42—22no—
207C42—2122no—
208C42#1171994512yes—
209C42C2×C22⋊D5312yes—
210C23C2×#1171992111yescharacteristic
211Q8—24no—
212Q8—24no—
213Q8—1064no—
214Q8—2122no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—1062no—
222C2×C4—2122no—
223C2×C4—2122no—
224C2×C4—2122no—
225C2×C4—2121no—
226C2×C4—2122no—
227C2×C4—2121no—
228C2×C4—2121no—
229C2×C4#1313314214yes—
230C2×C4C4×C22⋊D5312yes—
231C2×C4#1313318611yescharacteristic
232C2×C4#1313319112yes—
233C2×C4#1313320011yescharacteristic
234C2×C4C2×#1171991311yescharacteristic
235C2×C4C2×Dic21211yescharacteristic
236C2×C4#1313323912yes—
237C8—44no—
238C8—4242no—
239C22—21no—
240C22—21no—
241C22#1681050211yescharacteristic
242C22C2×#1313314211yescharacteristic
243C22#1681062011yescharacteristic
244C4—21no—
245C4—21no—
246C4—22no—
247C4—24no—
248C4—42no—
249C4—4242no—
250C4—4241no—
251C4—2122no—
252C4#1681085212yes—
253C4#1681086412yes—
254C2—21no—
255C2—21no—
256C2#2027513311yescharacteristic
257C2#2027513811yescharacteristic
258C2#2027525211yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2982C2
socleC22×C53#386
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#2982D53
hypocenterC53#9484
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
150. C53#9484semidirect

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×#29822
#948453