Finite Group Databasebeta

← #27281726

Subgroups — #27281726

Order 13568.

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

normal subgroups only

This group has 456 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#20285118C211yesmaximal, characteristic
2#20273342C211yesmaximal, characteristic, complemented
3#20273222C212yesmaximal
4#20272608C212yesmaximal, complemented
5C53×#2371C211yesmaximal, characteristic, complemented
6#16811173C2211yescharacteristic
7#16810993C412yescomplemented
8#16810989C412yescomplemented
9#16810980—22no—
10#16810680C2212yes—
11#16810622—22no—
12C4×#11719917C2211yescharacteristic
13#16810365—22no—
14#16810356—22no—
15#16810356—22no—
16C2×#125×C53C2211yescharacteristic
17C53×#387—22no—
18C53×#387—22no—
19C4×C4⋊C4×C53—22no—
20C53×#376C2212yes—
21C2×#11719945—21no—
22C2×#11719917—21no—
23#13133217C2×C411yescharacteristic
24#13133214C2×C411yescharacteristic
25#13133205—21no—
26#13133191—22no—
27C4×D212—22no—
28C4⋊#88322—22no—
29#13133140—24no—
30#13133137—24no—
31C22⋊#88322—22no—
32#13133128—22no—
33#13133128C2×C412yes—
34C4×#88322—21no—
35#125×C53—22no—
36#125×C53C2×C412yes—
37#125×C53D412yes—
38C2×C4⋊C4×C53—22no—
39C2×C4⋊C4×C53—22no—
40C2×C4⋊C4×C53D412yes—
41C2×C42×C53—22no—
42C2×C42×C53—22no—
43C2×C42×C53C2311yescharacteristic
44#95×C53—22no—
45#95×C53—22no—
46#95×C53—22no—
47#95×C53D412yes—
48#95×C53D412yes—
49C22×C4×C53—22no—
50C22×C4×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53C2×D411yescharacteristic
53C22×C4×C53C2×D412yes—
54C2×C22⋊D53—41no—
55#11719945—44no—
56C2×D212—21no—
57C2×D212—22no—
58C2×C4×D53—41no—
59C2×Dic106—21no—
60C4⋊C4×C53—24no—
61C4⋊C4×C53—24no—
62C4⋊C4×C53—22no—
63C4⋊C4×C53—44no—
64C4⋊C4×C53C22⋊C414yes—
65C42×C53—44no—
66C42×C53—44no—
67C42×C53C22×C411yescharacteristic
68C42×C53C2×D411yescharacteristic
69C42×C53#4112yes—
70#11719921—42no—
71C4⋊Dic53—22no—
72#11719917—42no—
73C2×#88322—22no—
74C2×#88322—42no—
75C23×C53#12011yescharacteristic
76C22×D53—41no—
77C2×C4×C53—22no—
78C2×C4×C53—24no—
79C2×C4×C53—24no—
80C2×C4×C53—42no—
81C2×C4×C53—42no—
82C2×C4×C53—42no—
83C2×C4×C53—42no—
84C2×C4×C53—42no—
85C2×C4×C53—42no—
86C2×C4×C53—42no—
87C2×C4×C53—42no—
88C2×C4×C53C2×C22⋊C411yescharacteristic
89C2×C4×C53C4×D412yes—
90C2×C4×C53#12112yes—
91C2×C4×C53#12311yescharacteristic
92C22⋊D53—82no—
93C2×Dic53—41no—
94D212—42no—
95D212—82no—
96C4×D53—82no—
97Dic106—42no—
98#88322—42no—
99#88322—82no—
100#9442—531nomaximal, Sylow
101C22×C53—22no—
102C22×C53#38611yescharacteristic
103C22×C53#45412yes—
104C2×D53—81no—
105C2×D53—82no—
106C4×C53—21no—
107C4×C53—21no—
108C4×C53—22no—
109C4×C53—22no—
110C4×C53—44no—
111C4×C53—44no—
112C4×C53—82no—
113Dic53—81no—
114#3142—531no—
115#2371D5311yescharacteristic, complemented
116#1534—531no—
117#1505—532no—
118#1300—532no—
119C2×C53—22no—
120C2×C53#188811yescharacteristic
121C2×C53#190612yes—
122D53—161no—
123#532—531no—
124C2×#125C2×D5311yescharacteristic
125#491—532no—
126#490—532no—
127#488—1062no—
128#432—532no—
129#417—1062no—
130C4×M16—531no—
131#387—22no—
132#387—22no—
133C4×C4⋊C4—22no—
134#376C2×D5312yes—
135#330—1062no—
136#327—1062no—
137#327—1062no—
138C53#944211yescharacteristic, complemented, Sylow
139C2×#41—1061no—
140C2×M16—1061no—
141C4⋊Q8—531no—
142#127—531no—
143#125—22no—
144#125C4×D5312yes—
145#125D21212yes—
146#124—1061no—
147#121—1062no—
148C4×D4—1062no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4C22⋊D5312yes—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42C22×D5311yescharacteristic
155C4⋊C8—1062no—
156#103—1064no—
157#102—1064no—
158C22⋊C8—1062no—
159#97—532no—
160#97—1062no—
161C4×C8—1061no—
162#95—22no—
163#95—22no—
164#95—22no—
165#95C22⋊D5312yes—
166#95C22⋊D5312yes—
167#41—2124no—
168C2×Q8—1061no—
169C2×D4—1061no—
170C2×D4—1062no—
171C2×D4—2121no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—2121no—
176C22×C4C2×C22⋊D5311yescharacteristic
177C22×C4C2×C22⋊D5312yes—
178M16—2122no—
179C2×C8—1062no—
180C2×C8—2122no—
181C4⋊C4—24no—
182C4⋊C4—24no—
183C4⋊C4—22no—
184C4⋊C4—44no—
185C4⋊C4—1062no—
186C4⋊C4#1171992114yes—
187C22⋊C4—2122no—
188C42—44no—
189C42—44no—
190C42C2×C4×D5311yescharacteristic
191C42C2×D21211yescharacteristic
192C42#1171994512yes—
193C23—2121no—
194C23#1313318611yescharacteristic
195Q8—2122no—
196D4—4242no—
197D4—4242no—
198D4—2122no—
199C2×C4—22no—
200C2×C4—24no—
201C2×C4—24no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—4242no—
211C2×C4—2121no—
212C2×C4C2×#1171992111yescharacteristic
213C2×C4C4×C22⋊D5312yes—
214C2×C4#1313319112yes—
215C2×C4#1313320011yescharacteristic
216C8—4242no—
217C8—2122no—
218C22—22no—
219C22—4242no—
220C22—4241no—
221C22#1681050211yescharacteristic
222C22#1681079612yes—
223C4—22no—
224C4—22no—
225C4—21no—
226C4—21no—
227C4—44no—
228C4—44no—
229C4—82no—
230C4—4241no—
231C2—22no—
232C2—8481no—
233C2#2027511811yescharacteristic
234C2#2027521612yes—
subgroupG/H
centerC22#16810502
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2371C2
socleC22×C53#386
prime radical#16811173C22
cosocleC2×C42×C53C23
hypercenter#2371D53
hypocenterC53#9442
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2371

Complements

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

NH ≅ G/Ntype
2. #20273342C2semidirect
4. #20272608C2semidirect
5. C53×#2371C2semidirect
7. #16810993C4semidirect
8. #16810989C4semidirect
115. #2371D53semidirect
138. C53#9442semidirect

Maximal subgroups

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

subgroupindex
#202851182
#202733422
#202732222
#202726082
C53×#23712
#944253