Finite Group Databasebeta

← #27293328

Subgroups — #27293328

Order 13568.

398 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 91 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#20283408C211yesmaximal, characteristic
2C22⋊C16×D53C212yesmaximal, complemented
3#20272657C211yesmaximal, characteristic, complemented
4#20272640C212yesmaximal, complemented
5C53×#1323C211yesmaximal, characteristic, complemented
6C2×#13133225C411yescharacteristic
7C22×C8×D53C2211yescharacteristic
8#16811135C411yescharacteristic
9C2×C16×D53—22no—
10#16810693—22no—
11#16810688—22no—
12#16810589C2211yescharacteristic
13#16810586C411yescharacteristic
14#16810579C411yescharacteristic
15C22⋊#11719908C2212yes—
16#16810402—22no—
17C53×#424C2211yescharacteristic
18C53×C22⋊C16C2212yes—
19C2×#11719945C2×C411yescharacteristic
20C22×C4×D53C2×C411yescharacteristic
21#13133225—42no—
22C2×#11719917C2×C411yescharacteristic
23C2×#11719912—21no—
24C22×#88322C2×C411yescharacteristic
25C2×C8×D53—21no—
26C2×C8×D53—42no—
27C2×C8×D53D412yes—
28C2×C8×D53D412yes—
29#13133201—21no—
30#13133197C812yes—
31#13133192C812yes—
32#13133187—21no—
33C4×D212—22no—
34C2×#11719908—22no—
35C16×D53—44no—
36#13133129—22no—
37#13133129—22no—
38#13133128—21no—
39C4×#88322—21no—
40C2×M16×C53C2×C411yescharacteristic
41C22×C8×C53C2311yescharacteristic
42#117×C53C2×C411yescharacteristic
43C2×C16×C53—22no—
44C4⋊C8×C53—22no—
45C23×D53M1611yescharacteristic
46C22×C4×C53C22×C411yescharacteristic
47C2×C22⋊D53—21no—
48C22×Dic53M1611yescharacteristic
49#11719945—41no—
50#11719945—41no—
51C2×D212—22no—
52C2×D212C2×C811yescharacteristic
53C2×C4×D53—21no—
54C2×C4×D53—41no—
55C2×C4×D53—41no—
56C2×C4×D53C22⋊C412yes—
57C2×C4×D53C2×C812yes—
58C2×Dic106C2×C811yescharacteristic
59M16×C53—41no—
60C2×C8×C53—21no—
61C2×C8×C53—41no—
62C2×C8×C53C2×D412yes—
63C4⋊C4×C53C2×C812yes—
64C22⋊C4×C53—21no—
65C42×C53—21no—
66#11719921—22no—
67#11719921—22no—
68C4⋊Dic53—22no—
69#11719917—41no—
70C2×#88322—22no—
71C2×#88322—21no—
72C2×#88322C22⋊C411yescharacteristic
73C2×#88322C22⋊C411yescharacteristic
74#11719912—42no—
75C8×D53—24no—
76C8×D53—42no—
77C8×D53—42no—
78C16×C53—42no—
79#11719908—42no—
80C23×C53C2×M1611yescharacteristic
81C22×D53—24no—
82C22×D53—21no—
83C22×D53—41no—
84C22×D53C22⋊C812yes—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53C2×C22⋊C411yescharacteristic
89C2×C4×C53C22×C811yescharacteristic
90C22⋊D53—42no—
91C2×Dic53—21no—
92C2×Dic53—41no—
93C2×Dic53C22⋊C812yes—
94D212—41no—
95D212—44no—
96C4×D53—22no—
97C4×D53—22no—
98C4×D53—42no—
99C4×D53—41no—
100C4×D53—41no—
101Dic106—41no—
102C8×C53—22no—
103C8×C53—41no—
104#88322—21no—
105#88322—41no—
106#88322#9912yes—
107#11468—531nomaximal, Sylow
108C22×C53—22no—
109C22×C53—21no—
110C22×C53C2×C22⋊C811yescharacteristic
111C2×D53—22no—
112C2×D53—22no—
113C2×D53—44no—
114C2×D53—42no—
115C2×D53—41no—
116C2×D53#35014yes—
117C4×C53—21no—
118C4×C53—42no—
119C4×C53C2×#9911yescharacteristic
120C4×C53#41311yescharacteristic
121Dic53—22no—
122Dic53—41no—
123#2929—531no—
124C2×C22⋊C16—532no—
125#1323—531no—
126#1323D5311yescharacteristic, complemented
127#1315—532no—
128C2×C53—41no—
129C2×C53#145911yescharacteristic
130C2×C53C2×#35012yes—
131D53—24no—
132D53—81no—
133C2×#131—531no—
134C23×C8—531no—
135#525—531no—
136C22×C16—1062no—
137#435—1062no—
138C8×D4—1062no—
139#424C2×D5311yescharacteristic
140#408—531no—
141#407—531no—
142#405—531no—
143C22⋊C16—532no—
144C22⋊C16—1062no—
145C22⋊C16C2×D5312yes—
146C53#1146811yescharacteristic, complemented, Sylow
147C2×#41—531no—
148C23×C4—531no—
149#131—2122no—
150C2×M16—531no—
151C2×M16—1061no—
152C2×M16C4×D5311yescharacteristic
153C22×C8—531no—
154C22×C8—532no—
155C22×C8—532no—
156C22×C8—1061no—
157C22×C8—2122no—
158C22×C8C22×D5311yescharacteristic
159#123—1061no—
160C22⋊Q8—532no—
161#121—532no—
162#120—1061no—
163C4×D4—1062no—
164#117C4×D5311yescharacteristic
165C2×C16—22no—
166C2×C16—1062no—
167C2×C16—2124no—
168C4⋊C8—22no—
169C22⋊C8—1062no—
170C22⋊C8—1062no—
171#97—1061no—
172C4×C8—1061no—
173C24—531no—
174#41—2121no—
175#41—2121no—
176C2×Q8—531no—
177C2×D4—531no—
178C2×D4—1062no—
179C2×D4—1061no—
180C22×C4—531no—
181C22×C4—532no—
182C22×C4—532no—
183C22×C4—1061no—
184C22×C4—2121no—
185C22×C4—2121no—
186C22×C4C2×C4×D5311yescharacteristic
187M16—41no—
188M16—2122no—
189M16—2121no—
190C2×C8—21no—
191C2×C8—41no—
192C2×C8—531no—
193C2×C8—531no—
194C2×C8—1064no—
195C2×C8—1062no—
196C2×C8—1061no—
197C2×C8—2122no—
198C2×C8—2122no—
199C2×C8D4×D5312yes—
200C4⋊C4—1062no—
201C4⋊C4C8×D5312yes—
202C22⋊C4—21no—
203C22⋊C4—1062no—
204C22⋊C4—1062no—
205C42—21no—
206C16—42no—
207C16—2122no—
208C23—532no—
209C23—1064no—
210C23—1061no—
211C23—2121no—
212C23M16×D5311yescharacteristic
213Q8—2121no—
214D4—2121no—
215D4—2124no—
216D4—2122no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—532no—
221C2×C4—1062no—
222C2×C4—1062no—
223C2×C4—1061no—
224C2×C4—2122no—
225C2×C4—2121no—
226C2×C4—2121no—
227C2×C4—2121no—
228C2×C4C22⋊C4×D5311yescharacteristic
229C2×C4C2×C8×D5311yescharacteristic
230C8—22no—
231C8—41no—
232C8—532no—
233C8—1061no—
234C8—2121no—
235C22—22no—
236C22—21no—
237C22—534no—
238C22—1062no—
239C22—1062no—
240C22—2124no—
241C22—2122no—
242C22—2121no—
243C22C22⋊C8×D5311yescharacteristic
244C4—21no—
245C4—42no—
246C4—1062no—
247C4—2121no—
248C4#99×D5311yescharacteristic
249C4#1681060611yescharacteristic
250C2—41no—
251C2—1064no—
252C2—4241no—
253C2#2027303211yescharacteristic
254C2#350×D5312yes—
subgroupG/H
centerC2×C4C22⋊C4×D53
commutatorC2×C4×C53C22×C8
FrattiniC22×C8C22×D53
FittingC53×#1323C2
socleC22×C53C2×C22⋊C8
prime radical#16811135C4
cosocleC22×C8×C53C23
hypercenter#1323D53
hypocenterC53#11468
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1323

Complements

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

NH ≅ G/Ntype
2. C22⋊C16×D53C2semidirect
3. #20272657C2semidirect
4. #20272640C2semidirect
5. C53×#1323C2semidirect
126. #1323D53semidirect
146. C53#11468semidirect

Maximal subgroups

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

subgroupindex
#202834082
C22⋊C16×D532
#202726572
#202726402
C53×#13232
#1146853