Finite Group Databasebeta

← #28302792

Subgroups — #28302792

Order 13568.

690 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 92 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#20290616C211yesmaximal, characteristic
2#20290612C211yesmaximal, characteristic
3#20290125C214yesmaximal
4C2×#16811234C211yesmaximal, characteristic
5#16811595C2212yes—
6#16811566C2212yes—
7#16811503—22no—
8#16811501—22no—
9#16811477—28no—
10#16811234—28no—
11C2×#13133212—22no—
12C2×#13133209C411yescharacteristic, complemented
13C2×#13133207—22no—
14C2×#13133193C2212yes—
15C2×Q8×Dic53C412yescomplemented
16C2×#13133174C2211yescharacteristic
17C2×C4⋊Q8×C53C411yescharacteristic, complemented
18C2×#11719940—22no—
19C2×C22⋊#6157—22no—
20#13133274—24no—
21(C2×#6157)⋊C4—24no—
22C22×C4×D53C2311yescharacteristic
23#13133212—216no—
24#13133209—24no—
25#13133207—216no—
26#13133193—24no—
27#13133193D418yes—
28Q8×Dic53—24no—
29Q8×Dic53—24no—
30#13133174—24no—
31C2×C4⋊Dic53—22no—
32C2×C4⋊Dic53C2×C412yes—
33C2×#11719919—22no—
34C2×#11719919—22no—
35C2×#11719919D412yes—
36C2×#11719921—22no—
37C2×#11719921D412yes—
38C2×C4×Dic53—22no—
39C2×C4×Dic53C2×C411yescharacteristic
40C22×Q8×C53C2×C412yes—
41C4⋊Q8×C53—24no—
42C2×C4⋊C4×C53—22no—
43C2×C42×C53C2×C411yescharacteristic
44C23×D53C2×D411yescharacteristic
45C22×#6157—42no—
46C2×Q8×C53—22no—
47C2×Q8×C53—44no—
48C2×Q8×C53C22⋊C414yes—
49C22×C4×C53—22no—
50C22×C4×C53C22⋊C412yes—
51C22×C4×C53C22×C411yescharacteristic
52C22×Dic53—22no—
53C22×Dic53C22⋊C412yes—
54C22×Dic53C2×D411yescharacteristic
55C2×C4×D53—42no—
56C2×C4×D53C2×D414yes—
57C22⋊#6157—44no—
58#11719940—44no—
59C2×#88324—42no—
60C4⋊C4×C53—28no—
61C42×C53—22no—
62#11719921—28no—
63#11719921—24no—
64C4⋊Dic53—28no—
65C4⋊Dic53—22no—
66C4⋊Dic53C22⋊C414yes—
67#11719919—28no—
68#11719919—28no—
69#11719919—24no—
70C4×Dic53—28no—
71C4×Dic53—22no—
72C23×C53C2×C22⋊C411yescharacteristic
73C22×D53—22no—
74C22×D53—42no—
75C22×D53#12012yes—
76C2×#6157—82no—
77C2×#6157—84no—
78Q8×C53—44no—
79Q8×C53—44no—
80C2×C4×C53—22no—
81C2×C4×C53—24no—
82C2×C4×C53—22no—
83C2×C4×C53—44no—
84C2×C4×C53—42no—
85C2×C4×C53C2×C22⋊C412yes—
86C2×Dic53—22no—
87C2×Dic53—24no—
88C2×Dic53—22no—
89C2×Dic53—44no—
90C2×Dic53—42no—
91C2×Dic53#12012yes—
92C4×D53—44no—
93#88324—82no—
94#8521—531nomaximal, Sylow
95C22×C53—22no—
96C22×C53#37911yescharacteristic
97C22×C53#41012yes—
98C2×D53—24no—
99C2×D53—22no—
100C2×D53—82no—
101#6157—162no—
102C4×C53—22no—
103C4×C53—42no—
104C4×C53—44no—
105Dic53—22no—
106Dic53—42no—
107Dic53—44no—
108C2×#541—531no—
109#1822—531no—
110#1820—531no—
111#1511—534no—
112C2×C53—22no—
113C2×C53#178211yescharacteristic
114C2×C53#179612yes—
115D53—42no—
116#541—1068no—
117C2×C4⋊Q8#615711yescharacteristic, complemented
118C2×#126—1062no—
119C2×#125—531no—
120C2×#125—1062no—
121C2×C22⋊Q8—532no—
122C2×C4×Q8—532no—
123C2×#117—531no—
124#407—532no—
125#381—532no—
126#343—1062no—
127#342—1062no—
128#328—1068no—
129C53#852111yescharacteristic, complemented, Sylow
130C22×Q8C2×#615712yes—
131C23×C4—531no—
132C2×M16—1062no—
133C4⋊Q8—24no—
134#126—10616no—
135#125—1064no—
136#125—10616no—
137C22⋊Q8—538no—
138C22⋊Q8—1064no—
139C4×Q8—1064no—
140C4×Q8—1064no—
141#117—1064no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—532no—
144C2×C4⋊C4—532no—
145C2×C4⋊C4—1062no—
146C2×C4⋊C4—1062no—
147C2×C4⋊C4—1062no—
148C2×C22⋊C4—532no—
149C2×C22⋊C4—1062no—
150C2×C22⋊C4—1062no—
151C2×C42—531no—
152C2×C42—1062no—
153C2×C42C2×#615711yescharacteristic
154C22⋊C8—1064no—
155#95—1064no—
156C24—531no—
157C2×Q8—22no—
158C2×Q8—44no—
159C2×Q8C22⋊#615714yes—
160C22×C4—22no—
161C22×C4—531no—
162C22×C4—534no—
163C22×C4—532no—
164C22×C4—1062no—
165C22×C4—2122no—
166C22×C4—2122no—
167C22×C4C22⋊#615712yes—
168C22×C4C22×#615711yescharacteristic
169M16—2124no—
170C2×C8—2122no—
171C4⋊C4—28no—
172C4⋊C4—534no—
173C4⋊C4—1064no—
174C4⋊C4—1068no—
175C4⋊C4—1068no—
176C4⋊C4—1068no—
177C4⋊C4—1062no—
178C22⋊C4—1064no—
179C22⋊C4—1068no—
180C22⋊C4—2124no—
181C42—22no—
182C42—1068no—
183C42—1062no—
184C23—532no—
185C23—1062no—
186C23—2122no—
187C23C2×C22⋊#615711yescharacteristic
188Q8—44no—
189Q8—44no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—24no—
193C2×C4—44no—
194C2×C4—42no—
195C2×C4—532no—
196C2×C4—1062no—
197C2×C4—1062no—
198C2×C4—1064no—
199C2×C4—4242no—
200C2×C4—4244no—
201C2×C4—2124no—
202C2×C4—2124no—
203C2×C4—2122no—
204C2×C4C2×C22⋊#615712yes—
205C8—4242no—
206C22—22no—
207C22—1064no—
208C22—1062no—
209C22—4242no—
210C22#1681156411yescharacteristic
211C22#1681160312yes—
212C4—22no—
213C4—42no—
214C4—44no—
215C4—8482no—
216C4—1062no—
217C4—2124no—
218C4—2122no—
219C2—22no—
220C2—2122no—
221C2#2029054511yescharacteristic
222C2#2029056712yes—
subgroupG/H
centerC22#16811564
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4⋊Q8×C53C4
socleC22×C53#379
prime radicalC23×D53C2×D4
cosocleC22×C4×D53C23
hypercenterC2×C4⋊Q8#6157
hypocenterC53#8521
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
12. C2×#13133209C4semidirect
15. C2×Q8×Dic53C4semidirect
17. C2×C4⋊Q8×C53C4semidirect
117. C2×C4⋊Q8#6157semidirect
129. C53#8521semidirect

Maximal subgroups

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

subgroupindex
#202906162
#202906122
#202901252
C2×#168112342
#852153