Finite Group Databasebeta

← #27281651

Subgroups — #27281651

Order 13568.

508 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 99 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#20285172C211yesmaximal, characteristic
2#20273441C211yesmaximal, characteristic, complemented
3#20273261C212yesmaximal
4#20272579C212yesmaximal, complemented
5C53×#2381C211yesmaximal, characteristic, complemented
6C2×#13133248—22no—
7#16811173C2211yescharacteristic
8#16811015—22no—
9#16811002C412yescomplemented
10#16810997C412yescomplemented
11#16810680C2212yes—
12C4×#11719917C2211yescharacteristic
13#16810372—22no—
14#16810356—24no—
15C2×C4⋊Q8×C53C2211yescharacteristic
16C53×#391—22no—
17C53×#387—24no—
18C53×#383C2212yes—
19C2×#11719945—21no—
20#13133248—44no—
21#13133248—44no—
22C2×#11719925—24no—
23C2×#11719924—24no—
24C2×#11719917—21no—
25#13133217C2×C411yescharacteristic
26#13133214C2×C411yescharacteristic
27#13133205—21no—
28#13133191—22no—
29C4×D212—22no—
30C4⋊#88322—22no—
31C22⋊#88322—22no—
32#13133128—21no—
33#13133128—22no—
34C4×#88322C2×C412yes—
35C22×Q8×C53D412yes—
36C4⋊Q8×C53—22no—
37C4⋊Q8×C53C2×C412yes—
38C4⋊Q8×C53D412yes—
39C2×C4⋊C4×C53—22no—
40C2×C4⋊C4×C53—22no—
41C2×C42×C53—22no—
42C2×C42×C53C2311yescharacteristic
43#95×C53—24no—
44#95×C53—22no—
45#95×C53—22no—
46#95×C53D414yes—
47C2×Q8×C53—22no—
48C2×Q8×C53—42no—
49C2×Q8×C53—42no—
50C2×Q8×C53C22⋊C414yes—
51C22×C4×C53—22no—
52C22×C4×C53—24no—
53C22×C4×C53C2×D411yescharacteristic
54C22×C4×C53C2×D412yes—
55C2×C22⋊D53—41no—
56#11719945—42no—
57#11719945—42no—
58C2×D212—21no—
59C2×D212—22no—
60C2×C4×D53—41no—
61C2×Dic106—21no—
62C4⋊C4×C53—24no—
63C4⋊C4×C53—24no—
64C4⋊C4×C53—82no—
65C42×C53—82no—
66C42×C53C22×C411yescharacteristic
67C42×C53C2×D411yescharacteristic
68C42×C53#4112yes—
69#11719925—48no—
70#11719924—48no—
71#11719921—42no—
72C4⋊Dic53—22no—
73#11719917—24no—
74C2×#88322—22no—
75C2×#88322—42no—
76C23×C53#12011yescharacteristic
77C22×D53—41no—
78Q8×C53—44no—
79Q8×C53—82no—
80C2×C4×C53—22no—
81C2×C4×C53—24no—
82C2×C4×C53—42no—
83C2×C4×C53—42no—
84C2×C4×C53—42no—
85C2×C4×C53—48no—
86C2×C4×C53—44no—
87C2×C4×C53C2×C22⋊C411yescharacteristic
88C2×C4×C53C4×D412yes—
89C2×C4×C53#12112yes—
90C2×C4×C53#12311yescharacteristic
91C22⋊D53—82no—
92C2×Dic53—41no—
93D212—42no—
94D212—82no—
95C4×D53—81no—
96C4×D53—81no—
97Dic106—42no—
98#88322—24no—
99#88322—82no—
100#9429—531nomaximal, Sylow
101C22×C53—22no—
102C22×C53#38611yescharacteristic
103C22×C53#45312yes—
104C2×D53—81no—
105C2×D53—82no—
106C4×C53—21no—
107C4×C53—21no—
108C4×C53—22no—
109C4×C53—22no—
110C4×C53—44no—
111C4×C53—84no—
112Dic53—81no—
113#3152—531no—
114#2381D5311yescharacteristic, complemented
115#1558—531no—
116#1514—532no—
117#1288—532no—
118C2×C53—22no—
119C2×C53#188711yescharacteristic
120C2×C53#190512yes—
121D53—161no—
122C2×#137—1062no—
123#532—531no—
124C2×C4⋊Q8C2×D5311yescharacteristic
125#497—1062no—
126#494—532no—
127#492—532no—
128#432—532no—
129C4×M16—531no—
130#391—22no—
131#387—24no—
132#383C2×D5312yes—
133#333—1062no—
134#327—1064no—
135C53#942911yescharacteristic, complemented, Sylow
136C2×#41—1061no—
137C22×Q8C22⋊D5312yes—
138#137—2124no—
139#137—2124no—
140C2×Q16—1064no—
141C2×SD16—1064no—
142C2×M16—1061no—
143C4⋊Q8—22no—
144C4⋊Q8—531no—
145C4⋊Q8C4×D5312yes—
146C4⋊Q8D21212yes—
147#127—531no—
148#124—1061no—
149#121—1062no—
150C4×D4—1062no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C42—22no—
154C2×C42C22×D5311yescharacteristic
155C4⋊C8—1062no—
156C22⋊C8—1062no—
157#97—1061no—
158#97—1062no—
159C4×C8—532no—
160#95—24no—
161#95—22no—
162#95—22no—
163#95C22⋊D5314yes—
164#41—2122no—
165#41—2122no—
166C2×Q8—22no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—1061no—
170C2×Q8#1171992114yes—
171C2×D4—1061no—
172C2×D4—1062no—
173C2×D4—2121no—
174C22×C4—22no—
175C22×C4—24no—
176C22×C4—2121no—
177C22×C4C2×C22⋊D5311yescharacteristic
178C22×C4C2×C22⋊D5312yes—
179Q16—2128no—
180SD16—2128no—
181M16—1064no—
182C2×C8—1062no—
183C2×C8—2122no—
184C4⋊C4—24no—
185C4⋊C4—24no—
186C4⋊C4—82no—
187C4⋊C4—1062no—
188C22⋊C4—2122no—
189C42—82no—
190C42C2×C4×D5311yescharacteristic
191C42C2×D21211yescharacteristic
192C42#1171994512yes—
193C23—2121no—
194C23#1313318611yescharacteristic
195Q8—44no—
196Q8—82no—
197Q8—2122no—
198D4—4242no—
199D4—4242no—
200D4—2122no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—48no—
207C2×C4—44no—
208C2×C4—4241no—
209C2×C4—4241no—
210C2×C4—2121no—
211C2×C4C2×#1171992111yescharacteristic
212C2×C4C4×C22⋊D5312yes—
213C2×C4#1313319112yes—
214C2×C4#1313320011yescharacteristic
215C8—1064no—
216C8—4242no—
217C22—22no—
218C22—4242no—
219C22—4241no—
220C22#1681050211yescharacteristic
221C22#1681079112yes—
222C4—22no—
223C4—22no—
224C4—21no—
225C4—21no—
226C4—44no—
227C4—84no—
228C4—4241no—
229C2—22no—
230C2—8481no—
231C2#2027511311yescharacteristic
232C2#2027520912yes—
subgroupG/H
centerC22#16810502
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2381C2
socleC22×C53#386
prime radical#16811173C22
cosocleC2×C42×C53C23
hypercenter#2381D53
hypocenterC53#9429
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2381

Complements

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

NH ≅ G/Ntype
2. #20273441C2semidirect
4. #20272579C2semidirect
5. C53×#2381C2semidirect
9. #16811002C4semidirect
10. #16810997C4semidirect
114. #2381D53semidirect
135. C53#9429semidirect

Maximal subgroups

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

subgroupindex
#202851722
#202734412
#202732612
#202725792
C53×#23812
#942953