Finite Group Databasebeta

← #27276608

Subgroups — #27276608

Order 13568.

690 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 107 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×#16811235C211yesmaximal, characteristic
2#20274792C211yesmaximal, characteristic
3#20273249C214yesmaximal
4C53×#1822C211yesmaximal, characteristic
5#16811235—28no—
6C2×#13133217C411yescharacteristic, complemented
7C2×#13133210—22no—
8C2×#13133209C411yescharacteristic, complemented
9C2×#13133207—22no—
10C2×#13133196C2212yes—
11C2×C4×Dic106C412yescomplemented
12#16810486C2212yes—
13#16810390—22no—
14#16810361—28no—
15C2×#117×C53C2211yescharacteristic
16C53×#407C2212yes—
17C53×#343—22no—
18C22×Dic106C2×C412yes—
19#13133217—24no—
20#13133210—216no—
21#13133209—24no—
22#13133207—216no—
23#13133196—24no—
24#13133196D418yes—
25C4×Dic106—24no—
26C4×Dic106—24no—
27C2×C4⋊Dic53—22no—
28C2×C4⋊Dic53C2×C412yes—
29C2×#11719919—22no—
30C2×#11719919—22no—
31C2×#11719919—22no—
32C2×C22⋊Dic53—22no—
33C2×C22⋊Dic53—22no—
34C2×C4×Dic53—22no—
35#13133124—24no—
36C23×C4×C53C2311yescharacteristic
37C2×M16×C53—22no—
38#117×C53—24no—
39C2×C4⋊C4×C53D412yes—
40C2×C22⋊C4×C53D412yes—
41C2×C42×C53C2×C411yescharacteristic
42C2×C42×C53C2×C411yescharacteristic
43C22⋊C8×C53—24no—
44C24×C53C2×D411yescharacteristic
45C22×C4×C53—42no—
46C22×C4×C53C22⋊C412yes—
47C22×C4×C53C22⋊C412yes—
48C22×C4×C53C22×C411yescharacteristic
49C22×C4×C53C2×D411yescharacteristic
50C22×C4×C53C2×D414yes—
51C22×Dic53—22no—
52C22×Dic53—22no—
53C22×Dic53—42no—
54C2×Dic106—22no—
55C2×Dic106—44no—
56C2×Dic106C22⋊C414yes—
57M16×C53—44no—
58C2×C8×C53—42no—
59C4⋊C4×C53—24no—
60C22⋊C4×C53—24no—
61C42×C53—22no—
62C42×C53—22no—
63C22⋊Dic53—28no—
64C22⋊Dic53—44no—
65C4⋊Dic53—28no—
66C4⋊Dic53—22no—
67C4⋊Dic53C22⋊C414yes—
68#11719919—28no—
69#11719919—28no—
70#11719919—28no—
71C4×Dic53—28no—
72C23×C53—22no—
73C23×C53—42no—
74C23×C53C2×C22⋊C411yescharacteristic
75C23×C53#12012yes—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—44no—
81C2×C4×C53—42no—
82C2×C4×C53—42no—
83C2×C4×C53C2×C22⋊C412yes—
84C2×C4×C53#12012yes—
85C2×Dic53—24no—
86C2×Dic53—24no—
87C2×Dic53—44no—
88C2×Dic53—44no—
89C2×Dic53—84no—
90C2×Dic53—82no—
91Dic106—44no—
92Dic106—44no—
93C8×C53—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—
105Dic53—44no—
106Dic53—44no—
107Dic53—162no—
108C2×#541—531no—
109#1822D5311yescharacteristic
110#1820—531no—
111#1511—534no—
112C2×C53—22no—
113C2×C53—42no—
114C2×C53#178211yescharacteristic
115C2×C53#179612yes—
116#541—1068no—
117C2×C4⋊Q8—531no—
118C2×#126—1062no—
119C2×#125—531no—
120C2×#125—1062no—
121C2×C22⋊Q8—532no—
122C2×C4×Q8—532no—
123C2×#117C2×D5311yescharacteristic
124#407C2×D5312yes—
125#381—532no—
126#343—22no—
127#342—1062no—
128#328—1068no—
129C53#852111yescharacteristic, complemented, Sylow
130C22×Q8—532no—
131C23×C4C22×D5311yescharacteristic
132C2×M16—22no—
133C4⋊Q8—1064no—
134#126—10616no—
135#125—1064no—
136#125—10616no—
137C22⋊Q8—538no—
138C22⋊Q8—1064no—
139C4×Q8—1064no—
140C4×Q8—1064no—
141#117—24no—
142C2×C4⋊C4—532no—
143C2×C4⋊C4—1062no—
144C2×C4⋊C4—1062no—
145C2×C4⋊C4—1062no—
146C2×C4⋊C4—1062no—
147C2×C4⋊C4C22⋊D5312yes—
148C2×C22⋊C4—1062no—
149C2×C22⋊C4—1062no—
150C2×C22⋊C4D21212yes—
151C2×C42—1062no—
152C2×C42C4×D5311yescharacteristic
153C2×C42C4×D5311yescharacteristic
154C22⋊C8—24no—
155#95—1064no—
156C24C2×D21211yescharacteristic
157C2×Q8—534no—
158C2×Q8—1062no—
159C2×Q8—2124no—
160C22×C4—42no—
161C22×C4—1062no—
162C22×C4—1062no—
163C22×C4—2122no—
164C22×C4#1171992112yes—
165C22×C4#1171992112yes—
166C22×C4C2×C4×D5311yescharacteristic
167C22×C4D4×D5314yes—
168C22×C4C2×C22⋊D5311yescharacteristic
169M16—44no—
170C2×C8—42no—
171C4⋊C4—24no—
172C4⋊C4—534no—
173C4⋊C4—1068no—
174C4⋊C4—1068no—
175C4⋊C4—1068no—
176C4⋊C4—1062no—
177C4⋊C4—1068no—
178C22⋊C4—24no—
179C22⋊C4—1068no—
180C22⋊C4—2124no—
181C42—22no—
182C42—22no—
183C42—1068no—
184C23—22no—
185C23—42no—
186C23C2×#1171992111yescharacteristic
187C23#1313318712yes—
188Q8—2124no—
189Q8—2124no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—44no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—1064no—
198C2×C4—1064no—
199C2×C4—4242no—
200C2×C4—4244no—
201C2×C4—2124no—
202C2×C4—2124no—
203C2×C4C22⋊C4×D5312yes—
204C2×C4#1313318512yes—
205C8—82no—
206C22—22no—
207C22—24no—
208C22—22no—
209C22—82no—
210C22#1681047811yescharacteristic
211C22#1681059612yes—
212C4—22no—
213C4—22no—
214C4—42no—
215C4—42no—
216C4—8482no—
217C4—2124no—
218C4—2124no—
219C2—22no—
220C2—42no—
221C2#2027463211yescharacteristic
222C2#2027469112yes—
subgroupG/H
centerC22#16810478
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1822C2
socleC22×C53#379
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#1822D53
hypocenterC53#8521
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1822

Complements

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

NH ≅ G/Ntype
6. C2×#13133217C4semidirect
8. C2×#13133209C4semidirect
11. C2×C4×Dic106C4semidirect
129. C53#8521semidirect

Maximal subgroups

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

subgroupindex
C2×#168112352
#202747922
#202732492
C53×#18222
#852153