Finite Group Databasebeta

← #27287330

Subgroups — #27287330

Order 13568.

414 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 73 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#20280380C211yesmaximal, characteristic
2#20274812C212yesmaximal
3#20274790C211yesmaximal, characteristic
4#20272569C212yesmaximal
5C53×#1306C211yesmaximal, characteristic
6C2×#13133176C2211yescharacteristic
7#16810559—21no—
8#16810535—21no—
9#16810527—22no—
10#16810510—22no—
11#16810486C2211yescharacteristic
12C4×C22⋊Dic53C2212yes—
13#16810389—21no—
14#16810389—21no—
15#16810384—22no—
16#16810384—22no—
17C53×#407C2212yes—
18C53×#381C2211yescharacteristic
19C22×Dic106—21no—
20#13133176—22no—
21#13133176—22no—
22C2×C4⋊Dic53—21no—
23C2×C4⋊Dic53Q812yescomplemented
24C2×#11719919—22no—
25C2×C22⋊Dic53—22no—
26C2×C22⋊Dic53—21no—
27C2×C22⋊Dic53—21no—
28C2×C4×Dic53—21no—
29C2×C4×Dic53D412yes—
30C2×C4×Dic53D412yes—
31#13133124—21no—
32#13133124—22no—
33#13133124—21no—
34#13133124—22no—
35#13133124—22no—
36C23×C4×C53C2311yescharacteristic
37C2×M16×C53—22no—
38C2×C22⋊C4×C53D411yescharacteristic
39C2×C22⋊C4×C53D411yescharacteristic
40C22⋊C8×C53—22no—
41C22⋊C8×C53—22no—
42#95×C53—22no—
43C24×C53C2×D411yescharacteristic
44C22×C4×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53—41no—
47C22×C4×C53—41no—
48C22×C4×C53C2×D412yes—
49C22×C4×C53C2×Q811yescharacteristic
50C22×C4×C53#4112yes—
51C22×C4×C53#4111yescharacteristic
52C22×Dic53—22no—
53C22×Dic53—22no—
54C22×Dic53—22no—
55C22×Dic53—41no—
56C2×Dic106—44no—
57C2×Dic106—81no—
58M16×C53—42no—
59M16×C53—42no—
60C2×C8×C53—42no—
61C22⋊C4×C53—41no—
62C22⋊C4×C53—41no—
63C22⋊Dic53—44no—
64C22⋊Dic53—41no—
65C22⋊Dic53—41no—
66C22⋊Dic53—41no—
67C22⋊Dic53—41no—
68C4⋊Dic53—24no—
69C4⋊Dic53—81no—
70#11719919—44no—
71C4×Dic53—42no—
72C4×Dic53—42no—
73C4×Dic53—42no—
74C23×C53—21no—
75C23×C53—21no—
76C23×C53—42no—
77C23×C53#12011yescharacteristic
78C23×C53C22⋊Q811yescharacteristic
79C23×C53#12411yescharacteristic
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—41no—
83C2×C4×C53—42no—
84C2×C4×C53—41no—
85C2×C4×C53—42no—
86C2×C4×C53—42no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53C22⋊Q812yes—
90C2×C4×C53#12412yes—
91C2×Dic53—24no—
92C2×Dic53—42no—
93C2×Dic53—42no—
94C2×Dic53—42no—
95C2×Dic53—42no—
96C2×Dic53—42no—
97C2×Dic53—42no—
98C2×Dic53—82no—
99C2×Dic53—82no—
100C2×Dic53—81no—
101Dic106—84no—
102C8×C53—82no—
103#10387—531nomaximal, Sylow
104C22×C53—22no—
105C22×C53—21no—
106C22×C53—21no—
107C22×C53—22no—
108C22×C53—22no—
109C22×C53—81no—
110C22×C53—81no—
111C22×C53#39311yescharacteristic
112C22×C53#45811yescharacteristic
113C22×C53#45811yescharacteristic
114C4×C53—22no—
115C4×C53—22no—
116C4×C53—81no—
117C4×C53—81no—
118Dic53—44no—
119Dic53—82no—
120Dic53—82no—
121Dic53—161no—
122#2607—531no—
123#1826—532no—
124#1820—531no—
125#1306D5311yescharacteristic
126#1284—532no—
127C2×C53—21no—
128C2×C53—21no—
129C2×C53—41no—
130C2×C53—41no—
131C2×C53#203011yescharacteristic
132C2×C53#203112yes—
133C2×#117—531no—
134#407C2×D5312yes—
135#399—1061no—
136#393—1061no—
137#391—1062no—
138#387—1062no—
139#381—531no—
140#381C2×D5311yescharacteristic
141C4×C22⋊C4—532no—
142#342—1061no—
143#342—1061no—
144#339—1062no—
145#339—1062no—
146C53#1038711yescharacteristic, complemented, Sylow
147C22×Q8—1061no—
148C23×C4C22×D5311yescharacteristic
149C2×M16—22no—
150#117—1062no—
151#117—1062no—
152C2×C4⋊C4—532no—
153C2×C4⋊C4—1061no—
154C2×C4⋊C4—1062no—
155C2×C22⋊C4—1061no—
156C2×C22⋊C4—1062no—
157C2×C22⋊C4—1061no—
158C2×C22⋊C4D21211yescharacteristic
159C2×C22⋊C4D21211yescharacteristic
160C2×C42—532no—
161C2×C42—532no—
162C2×C42—1061no—
163C22⋊C8—22no—
164C22⋊C8—22no—
165#95—22no—
166#95—1061no—
167#95—1062no—
168#95—1061no—
169#95—1062no—
170#95—1062no—
171C24C2×D21211yescharacteristic
172C2×Q8—4241no—
173C2×Q8—2124no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—1062no—
179C22×C4—1062no—
180C22×C4—1062no—
181C22×C4—2121no—
182C22×C4#1171994512yes—
183C22×C4D4×D5312yes—
184C22×C4#1171994711yescharacteristic
185C22×C4Q8×D5311yescharacteristic
186M16—42no—
187M16—42no—
188C2×C8—42no—
189C4⋊C4—1064no—
190C4⋊C4—4241no—
191C4⋊C4—2124no—
192C22⋊C4—41no—
193C22⋊C4—41no—
194C22⋊C4—2124no—
195C22⋊C4—2121no—
196C22⋊C4—2121no—
197C22⋊C4—2121no—
198C22⋊C4—2121no—
199C42—2122no—
200C42—2122no—
201C42—2122no—
202C23—21no—
203C23—21no—
204C23—42no—
205C23#1313318711yescharacteristic
206C23#1313319711yescharacteristic
207C23#1313320511yescharacteristic
208Q8—4244no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—1064no—
219C2×C4—4241no—
220C2×C4—4242no—
221C2×C4—4242no—
222C2×C4—2122no—
223C2×C4—2122no—
224C2×C4—2122no—
225C2×C4—2122no—
226C2×C4—2122no—
227C2×C4—2122no—
228C2×C4#1313319512yes—
229C2×C4#1313320312yes—
230C8—82no—
231C22—22no—
232C22—21no—
233C22—21no—
234C22—22no—
235C22—22no—
236C22—81no—
237C22—81no—
238C22#1681053611yescharacteristic
239C22#1681081411yescharacteristic
240C22#1681081411yescharacteristic
241C4—22no—
242C4—22no—
243C4—81no—
244C4—81no—
245C4—8481no—
246C4—4242no—
247C4—4242no—
248C4—2124no—
249C2—21no—
250C2—21no—
251C2—41no—
252C2—41no—
253C2#2027588311yescharacteristic
254C2#2027589012yes—
subgroupG/H
centerC22#16810536
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1306C2
socleC22×C53#393
prime radicalC24×C53C2×D4
hypercenter#1306D53
hypocenterC53#10387
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1306

Complements

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

NH ≅ G/Ntype
23. C2×C4⋊Dic53Q8semidirect
146. C53#10387semidirect

Maximal subgroups

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

subgroupindex
#202803802
#202748122
#202747902
#202725692
C53×#13062
#1038753