Finite Group Databasebeta

← #27281706

Subgroups — #27281706

Order 13568.

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

normal subgroups only

This group has 452 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
1#20285091C211yesmaximal, characteristic
2#20273453C211yesmaximal, characteristic, complemented
3#20273382C212yesmaximal
4#20272606C212yesmaximal, complemented
5C53×#2986C211yesmaximal, characteristic, complemented
6C2×#13133214C2211yescharacteristic
7#16810989C414yes—
8#16810971—22no—
9#16810647C2212yes—
10#16810627—22no—
11C4×#11719917C2211yescharacteristic
12#16810369—22no—
13#16810366—22no—
14#16810366—22no—
15C53×#533C2211yescharacteristic
16C53×#446—22no—
17C8×Q8×C53—22no—
18C53×#433—22no—
19C53×#431C2212yes—
20C22×D212—21no—
21C22×D212—22no—
22C2×#11719917—21no—
23C2×#11719917—22no—
24#13133214—22no—
25#13133214—21no—
26#13133214C2×C412yes—
27C4⋊#88322—22no—
28C4⋊#88322—22no—
29#13133137—28no—
30#13133128C2×C412yes—
31C4×#88322—21no—
32C4⋊Q8×C53D412yes—
33#125×C53C2×C412yes—
34C22⋊Q8×C53—22no—
35C4×Q8×C53—22no—
36#117×C53D412yes—
37C2×C42×C53C2311yescharacteristic
38C4⋊C8×C53—24no—
39C4⋊C8×C53D412yes—
40C4⋊C8×C53D412yes—
41C22⋊C8×C53—22no—
42#97×C53—22no—
43C4×C8×C53—22no—
44C4×C8×C53—22no—
45C23×D53—41no—
46C2×Q8×C53—22no—
47C22×C4×C53C2×D411yescharacteristic
48C22×C4×C53#4112yes—
49C2×D212—22no—
50C2×D212—24no—
51C2×D212—44no—
52C2×D212—41no—
53C2×D212—42no—
54C2×D212—82no—
55C2×D212—82no—
56C2×C8×C53—22no—
57C2×C8×C53—22no—
58C4⋊C4×C53—24no—
59C4⋊C4×C53—22no—
60C4⋊C4×C53C22⋊C414yes—
61C22⋊C4×C53—22no—
62C42×C53—22no—
63C42×C53C22×C411yescharacteristic
64C42×C53C2×D411yescharacteristic
65C42×C53C2×D412yes—
66#11719917—42no—
67#11719917—82no—
68C2×#88322—22no—
69C2×#88322—42no—
70C23×C53#12311yescharacteristic
71C22×D53—42no—
72C22×D53—82no—
73C22×D53—84no—
74Q8×C53—42no—
75Q8×C53—42no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—24no—
80C2×C4×C53C2×C22⋊C411yescharacteristic
81C2×C4×C53C4×D412yes—
82C2×C4×C53#12011yescharacteristic
83C2×C4×C53#12112yes—
84D212—44no—
85D212—84no—
86D212—82no—
87D212—82no—
88D212—82no—
89C8×C53—42no—
90C8×C53—42no—
91#88322—42no—
92#88322—82no—
93#9438—531nomaximal, Sylow
94C22×C53—21no—
95C22×C53—22no—
96C22×C53#38611yescharacteristic
97C2×D53—82no—
98C2×D53—84no—
99C2×D53—164no—
100C4×C53—21no—
101C4×C53—22no—
102C4×C53—44no—
103C4×C53#45412yes—
104C4×C53#47112yes—
105C4×C53#47212yes—
106#3138—531no—
107#2986D5311yescharacteristic, complemented
108#1561—531no—
109#1545—532no—
110#1299—532no—
111C2×C53—41no—
112C2×C53#188811yescharacteristic
113C2×C53#190712yes—
114D53—162no—
115#533C2×D5311yescharacteristic
116C2×#127—531no—
117#490—534no—
118#486—1062no—
119#446—22no—
120C8×Q8—22no—
121#433—22no—
122#431C2×D5312yes—
123#423—532no—
124#418—1062no—
125C4×M16—531no—
126#332—1062no—
127#331—1062no—
128#331—1062no—
129C53#943811yescharacteristic, complemented, Sylow
130C22×D4—1062no—
131C22×D4—1061no—
132C2×M16—1061no—
133C2×M16—1062no—
134C4⋊Q8D21212yes—
135#127—532no—
136#127—1061no—
137#127—1062no—
138#125C4×D5312yes—
139C22⋊Q8—22no—
140C4×Q8—22no—
141#117C22⋊D5312yes—
142C2×C42C22×D5311yescharacteristic
143C4⋊C8—24no—
144C4⋊C8—1062no—
145C4⋊C8—1062no—
146C4⋊C8C22⋊D5312yes—
147C4⋊C8C22⋊D5312yes—
148#102—1068no—
149C22⋊C8—22no—
150#97—22no—
151#97—532no—
152C4×C8—22no—
153C4×C8—22no—
154C4×C8—1061no—
155C24—2121no—
156C2×Q8—22no—
157C2×D4—1062no—
158C2×D4—1064no—
159C2×D4—4242no—
160C2×D4—4242no—
161C2×D4—2124no—
162C2×D4—2121no—
163C2×D4—2122no—
164C22×C4#1171994512yes—
165C22×C4C2×C22⋊D5311yescharacteristic
166M16—4242no—
167M16—2122no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—1062no—
171C2×C8—2122no—
172C4⋊C4—24no—
173C4⋊C4—22no—
174C4⋊C4#1171992114yes—
175C22⋊C4—22no—
176C42—22no—
177C42C2×C4×D5311yescharacteristic
178C42C2×D21211yescharacteristic
179C42C2×C22⋊D5312yes—
180C23—4244no—
181C23—4242no—
182C23—2122no—
183C23#1313320011yescharacteristic
184Q8—42no—
185Q8—42no—
186D4—4244no—
187D4—4242no—
188D4—4242no—
189D4—4242no—
190D4—2124no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4C2×#1171992111yescharacteristic
196C2×C4C4×C22⋊D5312yes—
197C2×C4#1313318611yescharacteristic
198C2×C4#1313319112yes—
199C8—42no—
200C8—42no—
201C8—4242no—
202C8—2122no—
203C22—22no—
204C22—21no—
205C22—8484no—
206C22—4244no—
207C22—4242no—
208C22#1681050211yescharacteristic
209C4—22no—
210C4—21no—
211C4—44no—
212C4#1681079612yes—
213C4#1681088612yes—
214C4#1681089312yes—
215C2—41no—
216C2—8482no—
217C2#2027511811yescharacteristic
218C2#2027522312yes—
subgroupG/H
centerC22#16810502
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2986C2
socleC22×C53#386
prime radicalC2×#13133214C22
cosocleC2×C42×C53C23
hypercenter#2986D53
hypocenterC53#9438
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2986

Complements

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

NH ≅ G/Ntype
2. #20273453C2semidirect
4. #20272606C2semidirect
5. C53×#2986C2semidirect
107. #2986D53semidirect
129. C53#9438semidirect

Maximal subgroups

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

subgroupindex
#202850912
#202734532
#202733822
#202726062
C53×#29862
#943853