Finite Group Databasebeta

← #28298848

Subgroups — #28298848

Order 13568.

374 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 58 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#20289963C212yesmaximal
2#20272982C211yesmaximal, characteristic
3#16811593—22no—
4#16811468—22no—
5#16811468C414yescomplemented
6C22×C4⋊Dic53C2211yescharacteristic
7C2×#13133124—22no—
8C2×#13133124C411yescharacteristic
9C2×#95×C53C411yescharacteristic
10C2×#11719940—42no—
11C22⋊#88324—24no—
12C22⋊#88324—42no—
13C23×Dic53—21no—
14C23×Dic53D412yes—
15C2×C4⋊Dic53—41no—
16C2×C4⋊Dic53—41no—
17C2×C4⋊Dic53C2×C412yes—
18C2×C4⋊Dic53D411yescharacteristic
19C2×C4⋊Dic53Q811yescharacteristic
20#13133124—44no—
21#13133124—41no—
22C23×C4×C53—21no—
23C23×C4×C53C2×C411yescharacteristic
24#95×C53—41no—
25C24×C53C22⋊C411yescharacteristic
26C22×C4×C53—21no—
27C22×C4×C53—21no—
28C22×C4×C53—22no—
29C22×C4×C53—42no—
30C22×C4×C53—42no—
31C22×C4×C53—41no—
32C22×C4×C53—42no—
33C22×C4×C53—41no—
34C22×C4×C53C4211yescharacteristic
35C22×C4×C53C4⋊C411yescharacteristic
36C22×Dic53—22no—
37C22×Dic53—22no—
38C22×Dic53—22no—
39C22×Dic53—42no—
40C22×Dic53—42no—
41C22×Dic53—42no—
42C22×Dic53—42no—
43C22×Dic53—41no—
44C22×Dic53—41no—
45C22×Dic53C22⋊C412yes—
46C22×Dic53C4⋊C412yes—
47#11719940—82no—
48#11719940—82no—
49C2×#88324—44no—
50C4⋊Dic53—42no—
51C4⋊Dic53—41no—
52C4⋊Dic53—41no—
53C23×C53—22no—
54C23×C53—21no—
55C23×C53—21no—
56C23×C53—21no—
57C23×C53—21no—
58C23×C53#9511yescharacteristic
59C23×C53#9911yescharacteristic
60C23×C53#9911yescharacteristic
61C2×C4×C53—21no—
62C2×C4×C53—21no—
63C2×C4×C53—22no—
64C2×C4×C53—22no—
65C2×C4×C53—42no—
66C2×C4×C53—42no—
67C2×C4×C53—44no—
68C2×C4×C53—42no—
69C2×C4×C53—82no—
70C2×C4×C53—81no—
71C2×C4×C53—81no—
72C2×Dic53—22no—
73C2×Dic53—22no—
74C2×Dic53—42no—
75C2×Dic53—42no—
76C2×Dic53—42no—
77C2×Dic53—42no—
78C2×Dic53—44no—
79C2×Dic53—42no—
80C2×Dic53—82no—
81C2×Dic53—82no—
82C2×Dic53—81no—
83C2×Dic53—81no—
84#88324—84no—
85#7326—531nomaximal, Sylow
86C22×C53—22no—
87C22×C53—22no—
88C22×C53—22no—
89C22×C53—22no—
90C22×C53—21no—
91C22×C53—21no—
92C22×C53—21no—
93C22×C53—21no—
94C22×C53—21no—
95C22×C53—21no—
96C22×C53#34211yescharacteristic
97C22×C53#34412yes—
98C22×C53#35311yescharacteristic
99C22×C53#35411yescharacteristic
100C22×C53#35511yescharacteristic
101C22×C53#35611yescharacteristic
102C4×C53—41no—
103C4×C53—41no—
104C4×C53—82no—
105Dic53—42no—
106Dic53—42no—
107Dic53—82no—
108#1463—532no—
109#1445—531no—
110C2×C53—21no—
111C2×C53—21no—
112C2×C53—22no—
113C2×C53—21no—
114C2×C53—21no—
115C2×C53#130611yescharacteristic
116C2×C53#139311yescharacteristic
117C2×C53#139411yescharacteristic
118C22×C4⋊C4—531no—
119#407—1062no—
120C2×#95—531no—
121C2×#95—1062no—
122C2×#95#615711yescharacteristic
123#324—534no—
124#324—1062no—
125C53#732611yescharacteristic, complemented, Sylow
126C23×C4—21no—
127C23×C4—532no—
128C23×C4—1061no—
129C23×C4C2×#615711yescharacteristic
130C2×M16—2122no—
131C2×C4⋊C4—532no—
132C2×C4⋊C4—531no—
133C2×C4⋊C4—531no—
134C2×C4⋊C4—2121no—
135C2×C4⋊C4—2121no—
136C22⋊C8—1064no—
137C22⋊C8—2122no—
138#95—41no—
139#95—2121no—
140#95—2124no—
141C24C22⋊#615711yescharacteristic
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—42no—
146C22×C4—41no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—41no—
150C22×C4—532no—
151C22×C4—532no—
152C22×C4—1062no—
153C22×C4—1062no—
154C22×C4—1062no—
155C22×C4—2122no—
156C22×C4—2121no—
157C22×C4—2122no—
158C22×C4—2121no—
159C22×C4—2122no—
160C22×C4—2122no—
161C22×C4C4×#615711yescharacteristic
162C22×C4C4⋊#615711yescharacteristic
163M16—4242no—
164M16—4242no—
165C2×C8—2124no—
166C4⋊C4—2122no—
167C4⋊C4—2121no—
168C4⋊C4—2121no—
169C23—21no—
170C23—21no—
171C23—21no—
172C23—21no—
173C23—22no—
174C23(C2×#6157)⋊C411yescharacteristic
175C23#1313327511yescharacteristic
176C23#1313327611yescharacteristic
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—44no—
184C2×C4—42no—
185C2×C4—82no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4—1062no—
189C2×C4—1062no—
190C2×C4—4242no—
191C2×C4—4241no—
192C2×C4—4241no—
193C2×C4—4242no—
194C2×C4—2124no—
195C2×C4—2122no—
196C2×C4—2122no—
197C2×C4—2122no—
198C2×C4—2122no—
199C2×C4—2122no—
200C8—4244no—
201C22—22no—
202C22—22no—
203C22—22no—
204C22—22no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22#1681150111yescharacteristic
212C22#1681150412yes—
213C22#1681152811yescharacteristic
214C22#1681153011yescharacteristic
215C22#1681153111yescharacteristic
216C22#1681153311yescharacteristic
217C4—41no—
218C4—41no—
219C4—82no—
220C4—4242no—
221C4—2122no—
222C4—2122no—
223C2—21no—
224C2—21no—
225C2—21no—
226C2—21no—
227C2—22no—
228C2#2028961811yescharacteristic
229C2#2028980911yescharacteristic
230C2#2028981311yescharacteristic
subgroupG/H
centerC22#16811501
commutatorC22×C4×C53C42
FrattiniC23×C4C2×#6157
FittingC2×#95×C53C4
socleC22×C53#342
prime radicalC24×C53C22⋊C4
cosocleC22×C4⋊Dic53C22
hypercenterC2×#95#6157
hypocenterC53#7326
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
5. #16811468C4semidirect
125. C53#7326semidirect

Maximal subgroups

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

subgroupindex
#202899632
#202729822
#732653