Finite Group Databasebeta

← #28303443

Subgroups — #28303443

Order 13568.

546 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 80 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#20289960C212yesmaximal
2#20289616C214yesmaximal, complemented
3#20277327C211yesmaximal, characteristic
4#16811593C2214yes—
5#16811466—22no—
6#16811466—22no—
7C22×C22⋊Dic53C411yescharacteristic
8C22×C22⋊Dic53C2211yescharacteristic
9#16810496—22no—
10#16810495—22no—
11#16810481C412yes—
12#16810481C2212yes—
13C22×C22⋊C4×C53C411yescharacteristic
14C2×#11719940—24no—
15C22⋊#88324—24no—
16C22⋊#88324—24no—
17C23×Dic53—21no—
18C23×Dic53C2×C411yescharacteristic
19C23×Dic53C2311yescharacteristic
20C2×#11719919—22no—
21C2×#11719919—22no—
22C2×C22⋊Dic53—21no—
23C2×C22⋊Dic53—24no—
24C2×C22⋊Dic53—42no—
25C2×C22⋊Dic53C2×C412yes—
26C2×C22⋊Dic53D412yes—
27C2×C22⋊Dic53D412yes—
28#13133124—22no—
29#13133124—22no—
30C25×C53C2×C411yescharacteristic
31C23×C4×C53—21no—
32C2×C22⋊C4×C53—21no—
33C2×C22⋊C4×C53—42no—
34C2×C22⋊C4×C53C2×C412yes—
35C24×C53—22no—
36C24×C53—22no—
37C24×C53—22no—
38C24×C53—81no—
39C24×C53—81no—
40C24×C53C22⋊C411yescharacteristic
41C24×C53C22⋊C411yescharacteristic
42C24×C53C22×C411yescharacteristic
43C22×C4×C53—22no—
44C22×C4×C53—42no—
45C22×C4×C53—81no—
46C22×C4×C53—81no—
47C22×Dic53—22no—
48C22×Dic53—22no—
49C22×Dic53—42no—
50C22×Dic53—81no—
51C22×Dic53—81no—
52C22×Dic53—81no—
53C22×Dic53—81no—
54C22×Dic53C22⋊C412yes—
55C22×Dic53C2×D412yes—
56C22×Dic53#4114yes—
57#11719940—84no—
58C2×#88324—44no—
59C22⋊C4×C53—42no—
60C22⋊C4×C53—42no—
61C22⋊Dic53—24no—
62C22⋊Dic53—24no—
63C22⋊Dic53—42no—
64C22⋊Dic53—42no—
65#11719919—82no—
66#11719919—82no—
67C23×C53—28no—
68C23×C53—21no—
69C23×C53—21no—
70C23×C53—21no—
71C23×C53—21no—
72C23×C53—21no—
73C23×C53—21no—
74C23×C53—41no—
75C23×C53—41no—
76C23×C53—41no—
77C23×C53—41no—
78C23×C53—41no—
79C23×C53—41no—
80C23×C53—82no—
81C23×C53—82no—
82C23×C53—84no—
83C23×C53—84no—
84C23×C53#10012yes—
85C23×C53#10112yes—
86C23×C53C2×C22⋊C411yescharacteristic
87C23×C53#11712yes—
88C2×C4×C53—42no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—84no—
92C2×Dic53—22no—
93C2×Dic53—44no—
94C2×Dic53—42no—
95C2×Dic53—42no—
96C2×Dic53—42no—
97C2×Dic53—42no—
98C2×Dic53—42no—
99C2×Dic53—82no—
100C2×Dic53—84no—
101C2×Dic53#12314yes—
102#88324—84no—
103#8756—531nomaximal, Sylow
104C22×C53—21no—
105C22×C53—21no—
106C22×C53—22no—
107C22×C53—22no—
108C22×C53—22no—
109C22×C53—22no—
110C22×C53—22no—
111C22×C53—42no—
112C22×C53—42no—
113C22×C53—42no—
114C22×C53—42no—
115C22×C53—42no—
116C22×C53—42no—
117C22×C53—41no—
118C22×C53—41no—
119C22×C53—41no—
120C22×C53—41no—
121C22×C53—88no—
122C22×C53#38111yescharacteristic
123C22×C53C2×#10011yescharacteristic
124C22×C53C2×#10111yescharacteristic
125C4×C53—82no—
126Dic53—24no—
127Dic53—42no—
128Dic53—82no—
129#2315—531no—
130#1462—532no—
131#1305—534no—
132C2×C53—21no—
133C2×C53—21no—
134C2×C53—42no—
135C2×C53—42no—
136C2×C53—42no—
137C2×C53#182111yescharacteristic
138C2×C53#182811yescharacteristic
139C2×C53#182911yescharacteristic
140C22×C22⋊C4—531no—
141C22×C22⋊C4—531no—
142C22×C22⋊C4#615711yescharacteristic
143#407—534no—
144#385—1062no—
145#385—1062no—
146#380—532no—
147#380—532no—
148#323—1062no—
149#323—1062no—
150C53#875611yescharacteristic, complemented, Sylow
151C25C2×#615711yescharacteristic
152C23×C4—21no—
153C23×C4—531no—
154C23×C4—531no—
155C23×C4—1061no—
156C2×M16—1064no—
157C2×C4⋊C4—1062no—
158C2×C4⋊C4—1062no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4—532no—
162C2×C22⋊C4—532no—
163C2×C22⋊C4—532no—
164C2×C22⋊C4—1064no—
165C2×C22⋊C4—1061no—
166C2×C22⋊C4—2122no—
167C2×C22⋊C4C2×#615712yes—
168C22⋊C8—1064no—
169C22⋊C8—1064no—
170#95—1062no—
171#95—1062no—
172C24—22no—
173C24—22no—
174C24—22no—
175C24—81no—
176C24—81no—
177C24C22⋊#615711yescharacteristic
178C24C22⋊#615711yescharacteristic
179C24C22×#615711yescharacteristic
180C22×C4—22no—
181C22×C4—42no—
182C22×C4—81no—
183C22×C4—81no—
184C22×C4—532no—
185C22×C4—534no—
186C22×C4—532no—
187C22×C4—1062no—
188C22×C4—1062no—
189C22×C4—4241no—
190C22×C4—4241no—
191C22×C4—4241no—
192C22×C4—4241no—
193C22×C4—2122no—
194M16—4244no—
195C2×C8—2124no—
196C4⋊C4—4242no—
197C4⋊C4—4242no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—1064no—
201C22⋊C4—1064no—
202C22⋊C4—2122no—
203C22⋊C4—2122no—
204C23—28no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—41no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—82no—
218C23—82no—
219C23—84no—
220C23—84no—
221C23#1313327712yes—
222C23#1313327812yes—
223C23C2×C22⋊#615711yescharacteristic
224C23#1313329612yes—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—84no—
229C2×C4—534no—
230C2×C4—1062no—
231C2×C4—4244no—
232C2×C4—4242no—
233C2×C4—2122no—
234C2×C4—2124no—
235C2×C4—2122no—
236C2×C4—2122no—
237C2×C4—2122no—
238C2×C4—2122no—
239C8—4244no—
240C22—21no—
241C22—21no—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—41no—
257C22—88no—
258C22#1681156611yescharacteristic
259C22C2×#1313327711yescharacteristic
260C22C2×#1313327811yescharacteristic
261C4—82no—
262C4—1064no—
263C4—4242no—
264C4—2122no—
265C2—21no—
266C2—21no—
267C2—42no—
268C2—42no—
269C2—42no—
270C2#2029061411yescharacteristic
271C2#2029062611yescharacteristic
272C2#2029062811yescharacteristic
subgroupG/H
centerC22#16811566
commutatorC24×C53C22×C4
FrattiniC24C22×#6157
FittingC22×C22⋊C4×C53C4
socleC22×C53#381
prime radicalC25×C53C2×C4
cosocleC23×Dic53C23
hypercenterC22×C22⋊C4#6157
hypocenterC53#8756
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #20289616C2semidirect
150. C53#8756semidirect

Maximal subgroups

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

subgroupindex
#202899602
#202896162
#202773272
#875653