Finite Group Databasebeta

← #27293397

Subgroups — #27293397

Order 13568.

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

normal subgroups only

This group has 368 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#20283417C211yesmaximal, characteristic
2#20276334C212yesmaximal, complemented
3#20272668C211yesmaximal, characteristic, complemented
4#20272650C212yesmaximal, complemented
5C53×#1328C211yesmaximal, characteristic, complemented
6C2×#13133225C2211yescharacteristic
7C22×#11719912C411yescharacteristic
8#16811135C411yescharacteristic
9M32×D53—22no—
10#16810700—22no—
11#16810693—22no—
12#16810589C411yescharacteristic
13#16810586C2211yescharacteristic
14#16810579C411yescharacteristic
15#16810405C2212yes—
16#16810404—22no—
17C53×#424C2211yescharacteristic
18C53×#349C2212yes—
19C2×#11719945C2×C411yescharacteristic
20C22×C4×D53C2×C411yescharacteristic
21#13133225—42no—
22C2×#11719917C2×C411yescharacteristic
23C2×#11719912—21no—
24C2×#11719912—21no—
25C2×#11719912—42no—
26C2×#11719912D412yes—
27C22×#88322C2×C411yescharacteristic
28C2×C8×D53D412yes—
29#13133201C812yes—
30#13133197—21no—
31#13133192—21no—
32#13133187C812yes—
33C4×D212—22no—
34#13133157—22no—
35#13133156—42no—
36C16×D53—42no—
37#13133129—22no—
38#13133129—22no—
39#13133128—21no—
40C4×#88322—21no—
41C2×M16×C53C2311yescharacteristic
42C22×C8×C53C2×C411yescharacteristic
43#117×C53C2×C411yescharacteristic
44M32×C53—22no—
45C4⋊C8×C53—22no—
46C23×D53C2×C811yescharacteristic
47C22×C4×C53C22×C411yescharacteristic
48C2×C22⋊D53C2×C812yes—
49C22×Dic53C2×C811yescharacteristic
50#11719945—41no—
51#11719945—41no—
52C2×D212—22no—
53C2×D212M1611yescharacteristic
54C2×C4×D53—21no—
55C2×C4×D53—21no—
56C2×C4×D53—41no—
57C2×C4×D53—41no—
58C2×C4×D53C22⋊C412yes—
59C2×Dic106M1611yescharacteristic
60M16×C53—41no—
61C2×C8×C53—21no—
62C2×C8×C53—41no—
63C2×C8×C53C2×D412yes—
64C4⋊C4×C53—21no—
65C22⋊C4×C53C2×C812yes—
66C42×C53—21no—
67#11719921—22no—
68#11719921—22no—
69C4⋊Dic53—22no—
70#11719917—41no—
71C2×#88322—22no—
72C2×#88322—21no—
73C2×#88322C22⋊C411yescharacteristic
74C2×#88322C22⋊C411yescharacteristic
75#11719912—42no—
76#11719912—42no—
77#11719912—42no—
78C8×D53—22no—
79C8×D53—22no—
80C16×C53—42no—
81#11719908—42no—
82C23×C53C22×C811yescharacteristic
83C22×D53—21no—
84C22×D53—41no—
85C22×D53—81no—
86C22×D53C22⋊C812yes—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—22no—
90C2×C4×C53C2×C22⋊C411yescharacteristic
91C2×C4×C53C2×M1611yescharacteristic
92C22⋊D53—42no—
93C2×Dic53—21no—
94C2×Dic53—41no—
95C2×Dic53C22⋊C812yes—
96D212—41no—
97D212—82no—
98C4×D53—22no—
99C4×D53—22no—
100C4×D53—42no—
101C4×D53—41no—
102C4×D53—41no—
103Dic106—41no—
104C8×C53—22no—
105C8×C53—41no—
106#88322—21no—
107#88322—41no—
108#88322#9912yes—
109#11484—531nomaximal, Sylow
110C22×C53—21no—
111C22×C53—41no—
112C22×C53C2×C22⋊C811yescharacteristic
113C2×D53—22no—
114C2×D53—42no—
115C2×D53—41no—
116C2×D53—41no—
117C2×D53—41no—
118C2×D53—82no—
119C4×C53—21no—
120C4×C53—42no—
121C4×C53C2×#9911yescharacteristic
122C4×C53#41311yescharacteristic
123Dic53—22no—
124Dic53—41no—
125#2930—531no—
126#2115—532no—
127#1328—531no—
128#1328D5311yescharacteristic, complemented
129#1320—532no—
130C2×C53—21no—
131C2×C53—41no—
132C2×C53#145911yescharacteristic
133D53—42no—
134D53—81no—
135C2×#131—531no—
136C22×M16—531no—
137#525—531no—
138C2×M32—1062no—
139#436—1062no—
140#435—1062no—
141#424C2×D5311yescharacteristic
142#408—531no—
143#407—531no—
144#405—531no—
145#349—532no—
146#349—1062no—
147#349C2×D5312yes—
148C53#1148411yescharacteristic, complemented, Sylow
149C2×#41—531no—
150C23×C4—531no—
151#131—2122no—
152C2×M16—532no—
153C2×M16—531no—
154C2×M16—1061no—
155C2×M16—1061no—
156C2×M16—2122no—
157C2×M16C22×D5311yescharacteristic
158C22×C8—531no—
159C22×C8—532no—
160C22×C8C4×D5311yescharacteristic
161#123—532no—
162C22⋊Q8—1061no—
163#121—1061no—
164#120—532no—
165C4×D4—1062no—
166#117C4×D5311yescharacteristic
167M32—22no—
168M32—1062no—
169M32—2122no—
170C2×C16—2122no—
171C4⋊C8—22no—
172C22⋊C8—1062no—
173C22⋊C8—1062no—
174#97—1061no—
175C4×C8—1061no—
176C24—531no—
177#41—2121no—
178#41—2121no—
179C2×Q8—531no—
180C2×D4—532no—
181C2×D4—531no—
182C2×D4—1062no—
183C22×C4—532no—
184C22×C4—531no—
185C22×C4—1061no—
186C22×C4—1061no—
187C22×C4—2121no—
188C22×C4—2121no—
189C22×C4C2×C4×D5311yescharacteristic
190M16—41no—
191M16—2122no—
192M16—2122no—
193M16—2122no—
194M16—2121no—
195C2×C8—21no—
196C2×C8—41no—
197C2×C8—531no—
198C2×C8—531no—
199C2×C8—1062no—
200C2×C8—1062no—
201C2×C8—1062no—
202C2×C8—1061no—
203C2×C8D4×D5312yes—
204C4⋊C4—21no—
205C4⋊C4—1062no—
206C22⋊C4—1062no—
207C22⋊C4—1062no—
208C22⋊C4C8×D5312yes—
209C42—21no—
210C16—42no—
211C16—2122no—
212C23—532no—
213C23—1061no—
214C23—4241no—
215C23—2121no—
216C23C2×C8×D5311yescharacteristic
217Q8—2121no—
218D4—4242no—
219D4—2121no—
220D4—2122no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—532no—
225C2×C4—1062no—
226C2×C4—1062no—
227C2×C4—1061no—
228C2×C4—2122no—
229C2×C4—2121no—
230C2×C4—2121no—
231C2×C4—2121no—
232C2×C4C22⋊C4×D5311yescharacteristic
233C2×C4M16×D5311yescharacteristic
234C8—22no—
235C8—41no—
236C8—532no—
237C8—1061no—
238C8—2121no—
239C22—21no—
240C22—41no—
241C22—1062no—
242C22—4242no—
243C22—2121no—
244C22—2121no—
245C22—2122no—
246C22—2121no—
247C22C22⋊C8×D5311yescharacteristic
248C4—21no—
249C4—42no—
250C4—1062no—
251C4—2121no—
252C4#99×D5311yescharacteristic
253C4#1681060611yescharacteristic
254C2—21no—
255C2—41no—
256C2—4241no—
257C2—2122no—
258C2#2027303211yescharacteristic
subgroupG/H
centerC4#99×D53
commutatorC23×C53C22×C8
FrattiniC2×M16C22×D53
FittingC53×#1328C2
socleC2×C53#1459
prime radical#16811135C4
cosocleC2×M16×C53C23
hypercenter#1328D53
hypocenterC53#11484
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1328

Complements

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

NH ≅ G/Ntype
2. #20276334C2semidirect
3. #20272668C2semidirect
4. #20272650C2semidirect
5. C53×#1328C2semidirect
128. #1328D53semidirect
148. C53#11484semidirect

Maximal subgroups

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

subgroupindex
#202834172
#202763342
#202726682
#202726502
C53×#13282
#1148453