Finite Group Databasebeta

← #28298663

Subgroups — #28298663

Order 13568.

336 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 68 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#20290595C211yesmaximal, characteristic
2#20290401C211yesmaximal, characteristic
3C22⋊C16×D53C211yesmaximal, characteristic
4C2×#13133296—21no—
5C2×#13133288C412yescomplemented
6#16811594—21no—
7#16811499—21no—
8#16811497—21no—
9#16811497C412yes—
10C22×C8×D53C2211yescharacteristic
11C2×C16×D53—21no—
12C22⋊#11719908C411yescharacteristic, complemented
13#16810402—21no—
14C53×C22⋊C16C411yescharacteristic, complemented
15C2×#11719936—22no—
16#13133296—24no—
17#13133296—42no—
18C2×C4⋊#6157—22no—
19C2×C22⋊#6157—41no—
20C2×C4×#6157—41no—
21#13133288—42no—
22#13133274—41no—
23C22⋊#88324—41no—
24C22×C4×D53D411yescharacteristic
25C22×#88322Q811yescharacteristic
26C2×C8×D53—42no—
27C2×C8×D53C2×C411yescharacteristic
28C2×C8×D53C2×C411yescharacteristic
29C2×C8×D53D411yescharacteristic
30C2×C8×D53D411yescharacteristic
31C2×#11719908—21no—
32C16×D53—42no—
33C22×C8×C53C2×C411yescharacteristic
34C2×C16×C53—21no—
35C23×D53SD1611yescharacteristic
36C22×#6157—42no—
37C22×C4×C53C4⋊C411yescharacteristic
38C22×Dic53SD1611yescharacteristic
39C2×C4×D53—22no—
40C2×C4×D53—22no—
41C2×C4×D53C22⋊C411yescharacteristic
42C2×C4×D53C22⋊C411yescharacteristic
43C2×C4×D53D811yescharacteristic
44C2×C4×D53Q1611yescharacteristic
45C22⋊#6157—44no—
46C2×#88324—42no—
47C4⋊#6157—24no—
48C4⋊#6157—42no—
49C4×#6157—44no—
50#11719936—42no—
51#11719936—42no—
52C2×C8×C53—41no—
53C2×C8×C53C4211yescharacteristic
54C2×C8×C53C22⋊C411yescharacteristic
55C2×#88322—41no—
56C2×#88322C4⋊C411yescharacteristic
57C2×#88322C4⋊C411yescharacteristic
58C8×D53—22no—
59C8×D53—22no—
60C8×D53—41no—
61C8×D53—41no—
62C16×C53—41no—
63#11719908—41no—
64C23×C53#10611yescharacteristic
65C22×D53—22no—
66C22×D53—22no—
67C22×D53—41no—
68C22×D53#10211yescharacteristic
69C22×D53#10311yescharacteristic
70C2×#6157—44no—
71C2×#6157—82no—
72C2×#6157—82no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53#9511yescharacteristic
76C2×C4×C53#10711yescharacteristic
77C2×Dic53—22no—
78C2×Dic53#10211yescharacteristic
79C2×Dic53#10311yescharacteristic
80C4×D53—22no—
81C4×D53—21no—
82C4×D53—21no—
83C4×D53—22no—
84C4×D53#9912yes—
85C4×D53#10412yes—
86#88324—82no—
87C8×C53—21no—
88C8×C53—21no—
89#88322—21no—
90#88322—21no—
91#7260—531nomaximal, Sylow
92C22×C53—21no—
93C22×C53—21no—
94C22×C53—21no—
95C22×C53#34011yescharacteristic
96C2×D53—21no—
97C2×D53—21no—
98C2×D53—21no—
99C2×D53—21no—
100C2×D53—44no—
101C2×D53#32812yes—
102C2×D53#35912yes—
103#6157—84no—
104C4×C53—21no—
105C4×C53#33911yescharacteristic
106C4×C53#34211yescharacteristic
107Dic53—21no—
108Dic53#32712yes—
109C2×C22⋊C16—531no—
110#1810—531no—
111#1735—531no—
112C2×C53—41no—
113C2×C53#129411yescharacteristic
114C2×C53#138211yescharacteristic
115C2×C53#138911yescharacteristic
116D53—22no—
117D53—22no—
118C23×C8—531no—
119C2×#117—1061no—
120C22×C16—1061no—
121C2×#108—532no—
122#407—1061no—
123C22⋊C16—531no—
124C22⋊C16—1061no—
125C22⋊C16#615711yescharacteristic, complemented
126#341—1061no—
127#340—532no—
128#340—1061no—
129C53#726011yescharacteristic, complemented, Sylow
130C23×C4—531no—
131C2×M16—1062no—
132C22×C8—531no—
133C22×C8—531no—
134C22×C8—531no—
135C22×C8—531no—
136C22×C8—531no—
137C22×C8—2122no—
138C22×C8C2×#615711yescharacteristic
139#117—1064no—
140#117—2122no—
141C2×C4⋊C4—1062no—
142C2×C22⋊C4—2121no—
143C2×C42—2121no—
144C2×C16—21no—
145C2×C16—1061no—
146C2×C16—2122no—
147#108—2122no—
148C22⋊C8—2121no—
149C22⋊C8—2121no—
150C24—531no—
151C22×C4—531no—
152C22×C4—531no—
153C22×C4—531no—
154C22×C4—531no—
155C22×C4—531no—
156C22×C4—1062no—
157C22×C4—1062no—
158C22×C4—2122no—
159C22×C4C4⋊#615711yescharacteristic
160M16—2122no—
161M16—2122no—
162C2×C8—41no—
163C2×C8—531no—
164C2×C8—531no—
165C2×C8—1062no—
166C2×C8—1062no—
167C2×C8—2121no—
168C2×C8—2121no—
169C2×C8—2121no—
170C2×C8—2122no—
171C2×C8C4×#615711yescharacteristic
172C2×C8C22⋊#615711yescharacteristic
173C4⋊C4—1064no—
174C4⋊C4—2122no—
175C22⋊C4—2124no—
176C42—2124no—
177C16—41no—
178C16—2121no—
179C23—531no—
180C23—531no—
181C23—1062no—
182C23—1062no—
183C23—2121no—
184C23#1313328511yescharacteristic
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—531no—
188C2×C4—532no—
189C2×C4—532no—
190C2×C4—531no—
191C2×C4—1062no—
192C2×C4—1062no—
193C2×C4—1062no—
194C2×C4—1061no—
195C2×C4—1061no—
196C2×C4—4242no—
197C2×C4—4242no—
198C2×C4—2124no—
199C2×C4(C2×#6157)⋊C411yescharacteristic
200C2×C4#1313328611yescharacteristic
201C8—21no—
202C8—21no—
203C8—1061no—
204C8—1061no—
205C8—4242no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—532no—
210C22—532no—
211C22—1061no—
212C22—1061no—
213C22—1061no—
214C22—1061no—
215C22—2124no—
216C22#1681149711yescharacteristic
217C4—21no—
218C4—532no—
219C4—1061no—
220C4—4244no—
221C4#1681149511yescharacteristic
222C4#1681150011yescharacteristic
223C2—41no—
224C2—1062no—
225C2—1062no—
226C2#2028959011yescharacteristic
227C2#2028977711yescharacteristic
228C2#2028979611yescharacteristic
subgroupG/H
centerC22#16811497
commutatorC2×C8×C53C42
FrattiniC22×C8C2×#6157
FittingC53×C22⋊C16C4
socleC22×C53#340
prime radicalC23×D53SD16
cosocleC22×C8×D53C22
hypercenterC22⋊C16#6157
hypocenterC53#7260
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22⋊C16

Complements

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

NH ≅ G/Ntype
5. C2×#13133288C4semidirect
12. C22⋊#11719908C4semidirect
14. C53×C22⋊C16C4semidirect
125. C22⋊C16#6157semidirect
129. C53#7260semidirect

Maximal subgroups

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

subgroupindex
#202905952
#202904012
C22⋊C16×D532
#726053