Finite Group Databasebeta

← #27286754

Subgroups — #27286754

Order 13568.

542 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 113 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#20285843C212yesmaximal
2#20275543C211yesmaximal, characteristic
3#20273400C212yesmaximal, complemented
4#20272548C211yesmaximal, characteristic, complemented
5C53×#2862C211yesmaximal, characteristic, complemented
6#16811173C2211yescharacteristic
7#16811019—22no—
8#16810971—22no—
9#16810941—24no—
10#16810910—24no—
11#16810676C2212yes—
12#16810647C2211yescharacteristic
13C2×#13133145—22no—
14#16810369—24no—
15C2×C4⋊Q8×C53C2212yes—
16C53×#391—24no—
17C53×#382—22no—
18C43×C53C2211yescharacteristic
19C2×#11719945—21no—
20C2×#11719917—21no—
21#13133217D411yescharacteristic
22#13133214D411yescharacteristic
23#13133205—21no—
24#13133191—22no—
25C4×D212—22no—
26#13133148—28no—
27C4⋊#88322—22no—
28C4⋊#88322D414yes—
29#13133145—44no—
30#13133137—28no—
31C22⋊#88322—22no—
32C4×#88322—22no—
33C22×Q8×C53—22no—
34C4⋊Q8×C53—22no—
35C4⋊Q8×C53—22no—
36C4⋊Q8×C53—22no—
37C4⋊Q8×C53—22no—
38C2×C4⋊C4×C53—22no—
39C2×C4⋊C4×C53D414yes—
40C2×C42×C53—22no—
41C2×C42×C53D412yes—
42C2×C42×C53C2311yescharacteristic
43#95×C53—28no—
44C2×Q8×C53—48no—
45C2×Q8×C53—42no—
46C2×Q8×C53—42no—
47C22×C4×C53—22no—
48C22×C4×C53—24no—
49C22×C4×C53C2×D412yes—
50C22×C4×C53C2×D411yescharacteristic
51C2×C22⋊D53—41no—
52#11719945—42no—
53#11719945—42no—
54C2×D212—22no—
55C2×D212—21no—
56C2×C4×D53—41no—
57C2×Dic106—21no—
58C4⋊C4×C53—24no—
59C4⋊C4×C53—24no—
60C4⋊C4×C53—42no—
61C4⋊C4×C53—42no—
62C42×C53—22no—
63C42×C53—44no—
64C42×C53SD1614yes—
65C42×C53C2×D411yescharacteristic
66C42×C53C2×D412yes—
67C42×C53#4111yescharacteristic
68#11719921—42no—
69C4⋊Dic53—22no—
70#11719917—42no—
71C2×#88322—24no—
72C2×#88322—41no—
73C23×C53#12011yescharacteristic
74C22×D53—41no—
75Q8×C53—84no—
76Q8×C53—84no—
77C2×C4×C53—24no—
78C2×C4×C53—24no—
79C2×C4×C53—44no—
80C2×C4×C53—48no—
81C2×C4×C53—44no—
82C2×C4×C53#12012yes—
83C2×C4×C53#12111yescharacteristic
84C2×C4×C53#12112yes—
85C2×C4×C53#12711yescharacteristic
86C2×C4×C53C2×SD1612yes—
87C2×C4×C53#13712yes—
88C22⋊D53—82no—
89C2×Dic53—41no—
90D212—44no—
91D212—42no—
92C4×D53—81no—
93C4×D53—81no—
94Dic106—42no—
95#88322—44no—
96#88322—81no—
97#10285—531nomaximal, Sylow
98C22×C53—22no—
99C22×C53#39211yescharacteristic
100C22×C53#44812yes—
101C2×D53—82no—
102C2×D53—81no—
103C4×C53—22no—
104C4×C53—44no—
105C4×C53—84no—
106C4×C53#45414yes—
107C4×C53#46112yes—
108C4×C53#49211yescharacteristic
109C4×C53#49711yescharacteristic
110Dic53—81no—
111#3227—532no—
112#2862D5311yescharacteristic, complemented
113#1962—531no—
114#1549—532no—
115#1276—531no—
116C2×C53—22no—
117C2×C53#200312yes—
118C2×C53#201311yescharacteristic
119D53—161no—
120#532—531no—
121C2×C4⋊Q8C2×D5312yes—
122#498—1062no—
123#486—1062no—
124#481—1064no—
125#476—1064no—
126#431—532no—
127#423—531no—
128C2×#104—1062no—
129#391—24no—
130#382—22no—
131C43C2×D5311yescharacteristic
132#332—1064no—
133C53#1028511yescharacteristic, complemented, Sylow
134C2×#41—1061no—
135C22×Q8—22no—
136C2×M16—1061no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—531no—
142#127—531no—
143#124—1061no—
144#121—1062no—
145C4×D4—1062no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4C22⋊D5314yes—
148C2×C42—22no—
149C2×C42C22⋊D5312yes—
150C2×C42C22×D5311yescharacteristic
151#106—1068no—
152C4⋊C8—534no—
153C4⋊C8—1062no—
154#104—2124no—
155#102—1068no—
156C22⋊C8—1062no—
157C4×C8—1062no—
158#95—28no—
159#41—2122no—
160#41—2122no—
161C2×Q8—48no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—1061no—
165C2×D4—1062no—
166C2×D4—1061no—
167C2×D4—2121no—
168C22×C4—22no—
169C22×C4—24no—
170C22×C4—2121no—
171C22×C4C2×C22⋊D5312yes—
172C22×C4C2×C22⋊D5311yescharacteristic
173M16—2122no—
174C2×C8—1064no—
175C2×C8—2121no—
176C4⋊C4—24no—
177C4⋊C4—24no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—1062no—
181C22⋊C4—2122no—
182C42—22no—
183C42—44no—
184C42#1171992414yes—
185C42D4×D5311yescharacteristic
186C42D4×D5312yes—
187C42#1171994711yescharacteristic
188C23—2121no—
189C23#1313318611yescharacteristic
190Q8—84no—
191Q8—84no—
192Q8—2122no—
193D4—4242no—
194D4—2124no—
195D4—2122no—
196C2×C4—24no—
197C2×C4—24no—
198C2×C4—44no—
199C2×C4—48no—
200C2×C4—44no—
201C2×C4—4241no—
202C2×C4—4241no—
203C2×C4—2121no—
204C2×C4#1313318512yes—
205C2×C4#1313318811yescharacteristic
206C2×C4#1313318912yes—
207C2×C4#1313321311yescharacteristic
208C2×C4C2×#1171992412yes—
209C2×C4#1313324812yes—
210C8—4241no—
211C8—2124no—
212C22—22no—
213C22—4242no—
214C22—4241no—
215C22#1681053111yescharacteristic
216C22#1681076212yes—
217C4—22no—
218C4—44no—
219C4—84no—
220C4—4241no—
221C4#1681079514yes—
222C4#1681082512yes—
223C4#1681099711yescharacteristic
224C4#1681101511yescharacteristic
225C2—22no—
226C2—8481no—
227C2#2027573712yes—
228C2#2027579811yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2862C2
socleC22×C53#392
prime radical#16811173C22
hypercenter#2862D53
hypocenterC53#10285
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2862

Complements

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

NH ≅ G/Ntype
3. #20273400C2semidirect
4. #20272548C2semidirect
5. C53×#2862C2semidirect
112. #2862D53semidirect
133. C53#10285semidirect

Maximal subgroups

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

subgroupindex
#202858432
#202755432
#202734002
#202725482
C53×#28622
#1028553