Finite Group Databasebeta

← #27276607

Subgroups — #27276607

Order 13568.

690 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 131 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#20274798C211yesmaximal, characteristic
2#20274792C211yesmaximal, characteristic
3#20273248C214yesmaximal
4C2×C53×#541C211yesmaximal, characteristic
5#16810587C2212yes—
6#16810486C2212yes—
7#16810392—22no—
8#16810390—22no—
9#16810360—28no—
10C53×#541—28no—
11C2×C4⋊Q8×C53C411yescharacteristic, complemented
12C2×#126×C53—22no—
13C2×#125×C53—22no—
14C2×#125×C53C411yescharacteristic, complemented
15C2×C22⋊Q8×C53C2212yes—
16C2×C4×Q8×C53C412yescomplemented
17C2×#117×C53C2211yescharacteristic
18C2×#11719917—22no—
19C2×C22⋊Dic53—22no—
20C22⋊#88322—24no—
21#13133124—24no—
22C22×Q8×C53C2×C412yes—
23C23×C4×C53C2311yescharacteristic
24C4⋊Q8×C53—24no—
25#126×C53—216no—
26#125×C53—216no—
27#125×C53—24no—
28C22⋊Q8×C53—24no—
29C22⋊Q8×C53D418yes—
30C4×Q8×C53—24no—
31C4×Q8×C53—24no—
32#117×C53—24no—
33C2×C4⋊C4×C53—22no—
34C2×C4⋊C4×C53—22no—
35C2×C4⋊C4×C53—22no—
36C2×C4⋊C4×C53—22no—
37C2×C4⋊C4×C53C2×C412yes—
38C2×C4⋊C4×C53D412yes—
39C2×C22⋊C4×C53—22no—
40C2×C22⋊C4×C53D412yes—
41C2×C42×C53—22no—
42C2×C42×C53C2×C411yescharacteristic
43C2×C42×C53C2×C411yescharacteristic
44C24×C53C2×D411yescharacteristic
45C2×Q8×C53—22no—
46C2×Q8×C53—44no—
47C2×Q8×C53C22⋊C414yes—
48C22×C4×C53—22no—
49C22×C4×C53—22no—
50C22×C4×C53—42no—
51C22×C4×C53C22⋊C412yes—
52C22×C4×C53C22⋊C412yes—
53C22×C4×C53C22×C411yescharacteristic
54C22×C4×C53C2×D411yescharacteristic
55C22×C4×C53C2×D414yes—
56C22×Dic53—42no—
57C4⋊C4×C53—28no—
58C4⋊C4×C53—28no—
59C4⋊C4×C53—28no—
60C4⋊C4×C53—22no—
61C4⋊C4×C53—28no—
62C4⋊C4×C53—24no—
63C4⋊C4×C53C22⋊C414yes—
64C22⋊C4×C53—28no—
65C22⋊C4×C53—24no—
66C42×C53—28no—
67C42×C53—22no—
68C42×C53—22no—
69C22⋊Dic53—44no—
70#11719917—44no—
71C2×#88322—42no—
72C23×C53—22no—
73C23×C53—42no—
74C23×C53C2×C22⋊C411yescharacteristic
75C23×C53#12012yes—
76Q8×C53—44no—
77Q8×C53—44no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—24no—
81C2×C4×C53—24no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—44no—
85C2×C4×C53—44no—
86C2×C4×C53—44no—
87C2×C4×C53—42no—
88C2×C4×C53—42no—
89C2×C4×C53C2×C22⋊C412yes—
90C2×C4×C53#12012yes—
91C2×Dic53—84no—
92C2×Dic53—82no—
93#88322—82no—
94#8521—531nomaximal, Sylow
95C22×C53—22no—
96C22×C53—24no—
97C22×C53—22no—
98C22×C53—82no—
99C22×C53#37911yescharacteristic
100C22×C53#41012yes—
101C4×C53—22no—
102C4×C53—22no—
103C4×C53—42no—
104C4×C53—42no—
105C4×C53—44no—
106C4×C53—44no—
107Dic53—162no—
108C2×#541D5311yescharacteristic
109#1822—531no—
110#1820—531no—
111#1511—534no—
112C2×C53—22no—
113C2×C53—42no—
114C2×C53#178211yescharacteristic
115C2×C53#179612yes—
116#541—28no—
117C2×C4⋊Q8Dic5311yescharacteristic, complemented
118C2×#126—22no—
119C2×#125—22no—
120C2×#125Dic5311yescharacteristic, complemented
121C2×C22⋊Q8C2×D5312yes—
122C2×C4×Q8Dic5312yescomplemented
123C2×#117C2×D5311yescharacteristic
124#407—532no—
125#381—532no—
126#343—1062no—
127#342—1062no—
128#328—1068no—
129C53#852111yescharacteristic, complemented, Sylow
130C22×Q8C2×Dic5312yes—
131C23×C4C22×D5311yescharacteristic
132C2×M16—1062no—
133C4⋊Q8—24no—
134#126—216no—
135#125—216no—
136#125—24no—
137C22⋊Q8—24no—
138C22⋊Q8C22⋊D5318yes—
139C4×Q8—24no—
140C4×Q8—24no—
141#117—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4C2×Dic5312yes—
147C2×C4⋊C4C22⋊D5312yes—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—1062no—
150C2×C22⋊C4C22⋊D5312yes—
151C2×C42—22no—
152C2×C42C2×Dic5311yescharacteristic
153C2×C42C2×Dic5311yescharacteristic
154C22⋊C8—1064no—
155#95—1064no—
156C24C2×C22⋊D5311yescharacteristic
157C2×Q8—22no—
158C2×Q8—44no—
159C2×Q8C22⋊Dic5314yes—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—42no—
163C22×C4—2122no—
164C22×C4C22⋊Dic5312yes—
165C22×C4C22⋊Dic5312yes—
166C22×C4C22×Dic5311yescharacteristic
167C22×C4C2×C22⋊D5311yescharacteristic
168C22×C4C2×C22⋊D5314yes—
169M16—2124no—
170C2×C8—2122no—
171C4⋊C4—28no—
172C4⋊C4—28no—
173C4⋊C4—28no—
174C4⋊C4—22no—
175C4⋊C4—28no—
176C4⋊C4—24no—
177C4⋊C4C22⋊Dic5314yes—
178C22⋊C4—28no—
179C22⋊C4—24no—
180C22⋊C4—2124no—
181C42—28no—
182C42—22no—
183C42—22no—
184C23—22no—
185C23—42no—
186C23C2×C22⋊Dic5311yescharacteristic
187C23#1313318612yes—
188Q8—44no—
189Q8—44no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—44no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—4242no—
202C2×C4—4244no—
203C2×C4C2×C22⋊Dic5312yes—
204C2×C4#1313318612yes—
205C8—4242no—
206C22—22no—
207C22—24no—
208C22—22no—
209C22—82no—
210C22#1681047911yescharacteristic
211C22#1681059712yes—
212C4—22no—
213C4—22no—
214C4—44no—
215C4—42no—
216C4—42no—
217C4—44no—
218C4—8482no—
219C2—22no—
220C2—42no—
221C2#2027463311yescharacteristic
222C2#2027469012yes—
subgroupG/H
centerC22#16810479
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#541C2
socleC22×C53#379
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenterC2×#541D53
hypocenterC53#8521
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
11. C2×C4⋊Q8×C53C4semidirect
14. C2×#125×C53C4semidirect
16. C2×C4×Q8×C53C4semidirect
117. C2×C4⋊Q8Dic53semidirect
120. C2×#125Dic53semidirect
122. C2×C4×Q8Dic53semidirect
129. C53#8521semidirect

Maximal subgroups

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

subgroupindex
#202747982
#202747922
#202732482
C2×C53×#5412
#852153