Finite Group Databasebeta

← #28303392

Subgroups — #28303392

Order 13568.

498 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C2×#16811466C212yesmaximal
2#20289558C214yesmaximal, complemented
3#20277311C211yesmaximal, characteristic
4C2×C22⋊#88324C2214yes—
5#16811466—22no—
6#16811466—22no—
7C22×C22⋊Dic53C2211yescharacteristic
8#16810517—22no—
9#16810511—22no—
10#16810481C412yes—
11#16810479C411yescharacteristic
12C22⋊C4×Dic53C2212yes—
13C53×#379C411yescharacteristic
14C22×#88324—24no—
15C22⋊#88324—24no—
16C22⋊#88324—24no—
17C23×Dic53C2×C411yescharacteristic
18C23×Dic53C2311yescharacteristic
19C2×#11719919—22no—
20C2×C22⋊Dic53—21no—
21C2×C22⋊Dic53—21no—
22C2×C22⋊Dic53—24no—
23C2×C22⋊Dic53C2×C412yes—
24C2×C22⋊Dic53D412yes—
25C2×C22⋊Dic53D412yes—
26C2×C4×Dic53—22no—
27#13133124—22no—
28#13133124—22no—
29C25×C53C2×C411yescharacteristic
30C2×C22⋊C4×C53—21no—
31C2×C22⋊C4×C53—21no—
32C2×C22⋊C4×C53C2×C412yes—
33C24×C53—22no—
34C24×C53—24no—
35C24×C53—81no—
36C24×C53—81no—
37C24×C53C22⋊C411yescharacteristic
38C24×C53C22⋊C411yescharacteristic
39C24×C53C22×C411yescharacteristic
40C22×C4×C53—22no—
41C22×Dic53—22no—
42C22×Dic53—22no—
43C22×Dic53—81no—
44C22×Dic53—81no—
45C22×Dic53C22⋊C412yes—
46C22×Dic53C2×D412yes—
47C22×Dic53#4114yes—
48C2×#88324—44no—
49C2×#88324—84no—
50C22⋊C4×C53—24no—
51C22⋊C4×C53—81no—
52C22⋊C4×C53—81no—
53C22⋊Dic53—24no—
54C22⋊Dic53—24no—
55C22⋊Dic53—24no—
56C22⋊Dic53—81no—
57C22⋊Dic53—81no—
58#11719919—82no—
59C4×Dic53—82no—
60C23×C53—21no—
61C23×C53—21no—
62C23×C53—22no—
63C23×C53—22no—
64C23×C53—41no—
65C23×C53—41no—
66C23×C53—41no—
67C23×C53—41no—
68C23×C53—41no—
69C23×C53—41no—
70C23×C53—44no—
71C23×C53—82no—
72C23×C53—82no—
73C23×C53—88no—
74C23×C53C2×C22⋊C411yescharacteristic
75C23×C53#11712yes—
76C23×C53#13114yes—
77C2×C4×C53—42no—
78C2×C4×C53—42no—
79C2×C4×C53—42no—
80C2×Dic53—22no—
81C2×Dic53—44no—
82C2×Dic53—42no—
83C2×Dic53—42no—
84C2×Dic53—42no—
85C2×Dic53—42no—
86C2×Dic53—42no—
87C2×Dic53—82no—
88C2×Dic53#12314yes—
89#88324—84no—
90#8736—531nomaximal, Sylow
91C22×C53—21no—
92C22×C53—22no—
93C22×C53—21no—
94C22×C53—22no—
95C22×C53—44no—
96C22×C53—41no—
97C22×C53—41no—
98C22×C53—41no—
99C22×C53—41no—
100C22×C53—41no—
101C22×C53—41no—
102C22×C53—82no—
103C22×C53—82no—
104C22×C53—88no—
105C22×C53#38111yescharacteristic
106C22×C53#40812yes—
107C22×C53#43314yes—
108C4×C53—82no—
109Dic53—24no—
110Dic53—42no—
111Dic53—82no—
112#2312—531no—
113C2×#323—532no—
114#1282—534no—
115C2×C53—22no—
116C2×C53—24no—
117C2×C53—81no—
118C2×C53—81no—
119C2×C53#181811yescharacteristic
120C2×C53#182811yescharacteristic
121C2×C53#182911yescharacteristic
122C22×C22⋊C4—531no—
123C2×C22⋊C8—534no—
124#388—1062no—
125#388—1062no—
126#380—532no—
127#379—531no—
128#379#615711yescharacteristic
129C4×C22⋊C4—532no—
130#323—1062no—
131#323—1062no—
132C53#873611yescharacteristic, complemented, Sylow
133C25C2×#615711yescharacteristic
134C23×C4—531no—
135C23×C4—531no—
136C22×C8—1064no—
137C2×C4⋊C4—1062no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—532no—
141C2×C22⋊C4—532no—
142C2×C22⋊C4—532no—
143C2×C22⋊C4—1064no—
144C2×C22⋊C4—1061no—
145C2×C22⋊C4—1061no—
146C2×C22⋊C4C2×#615712yes—
147C2×C42—1062no—
148C22⋊C8—1064no—
149C22⋊C8—1064no—
150#95—1062no—
151#95—1062no—
152C24—22no—
153C24—24no—
154C24—81no—
155C24—81no—
156C24C22⋊#615711yescharacteristic
157C24C22⋊#615711yescharacteristic
158C24C22×#615711yescharacteristic
159C22×C4—22no—
160C22×C4—532no—
161C22×C4—534no—
162C22×C4—532no—
163C22×C4—1062no—
164C22×C4—1062no—
165C22×C4—4241no—
166C22×C4—4241no—
167C2×C8—4244no—
168C2×C8—2124no—
169C4⋊C4—4242no—
170C22⋊C4—24no—
171C22⋊C4—81no—
172C22⋊C4—81no—
173C22⋊C4—1064no—
174C22⋊C4—1064no—
175C22⋊C4—1064no—
176C22⋊C4—4241no—
177C22⋊C4—4241no—
178C42—4242no—
179C23—21no—
180C23—21no—
181C23—22no—
182C23—22no—
183C23—41no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—44no—
190C23—82no—
191C23—82no—
192C23—88no—
193C23C2×C22⋊#615711yescharacteristic
194C23#1313329612yes—
195C23#1313330414yes—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—534no—
200C2×C4—1062no—
201C2×C4—4242no—
202C2×C4—2122no—
203C2×C4—2124no—
204C2×C4—2122no—
205C2×C4—2122no—
206C2×C4—2122no—
207C2×C4—2122no—
208C8—4244no—
209C22—21no—
210C22—22no—
211C22—21no—
212C22—22no—
213C22—44no—
214C22—41no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—82no—
221C22—82no—
222C22—88no—
223C22#1681156611yescharacteristic
224C22#1681159612yes—
225C22#1681164714yes—
226C4—82no—
227C4—1064no—
228C4—4242no—
229C4—2122no—
230C2—22no—
231C2—24no—
232C2—81no—
233C2—81no—
234C2#2029060811yescharacteristic
235C2#2029062611yescharacteristic
236C2#2029062811yescharacteristic
subgroupG/H
centerC22#16811566
commutatorC24×C53C22×C4
FrattiniC24C22×#6157
FittingC53×#379C4
socleC22×C53#381
prime radicalC25×C53C2×C4
cosocleC23×Dic53C23
hypercenter#379#6157
hypocenterC53#8736
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ #379

Complements

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

NH ≅ G/Ntype
2. #20289558C2semidirect
132. C53#8736semidirect

Maximal subgroups

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

subgroupindex
C2×#168114662
#202895582
#202773112
#873653