Finite Group Databasebeta

← #27286756

Subgroups — #27286756

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#20285845C212yesmaximal
2#20283093C211yesmaximal, characteristic, complemented
3#20275543C211yesmaximal, characteristic
4#20273399C212yesmaximal, complemented
5C53×#1276C211yesmaximal, characteristic, complemented
6#16811173C2211yescharacteristic
7C2×#13133217C2212yes—
8#16811021—22no—
9#16810970—22no—
10#16810942—24no—
11#16810914—24no—
12#16810647C2211yescharacteristic
13C2×#13133145—22no—
14#16810529—24no—
15#16810489—22no—
16#16810370—24no—
17C53×#431C2212yes—
18C43×C53C2211yescharacteristic
19C2×#11719945—21no—
20C22×Dic106—22no—
21C2×#11719917—21no—
22#13133217—22no—
23#13133217—22no—
24#13133217—22no—
25#13133217—22no—
26#13133217D411yescharacteristic
27#13133214D411yescharacteristic
28#13133205—21no—
29#13133191—22no—
30C4×D212—22no—
31C2×C4⋊Dic53—22no—
32C2×C4⋊Dic53D414yes—
33#13133149—28no—
34C4⋊#88322—22no—
35#13133145—44no—
36#13133139—28no—
37#13133124—28no—
38C2×C42×C53—22no—
39C2×C42×C53D412yes—
40C2×C42×C53C2311yescharacteristic
41C4⋊C8×C53D414yes—
42C22⋊C8×C53—22no—
43C4×C8×C53—22no—
44C22×C4×C53—22no—
45C22×C4×C53C2×D412yes—
46C22×C4×C53C2×D411yescharacteristic
47C2×C22⋊D53—41no—
48C22×Dic53—24no—
49#11719945—42no—
50#11719945—42no—
51C2×D212—22no—
52C2×D212—21no—
53C2×C4×D53—41no—
54C2×Dic106—21no—
55C2×Dic106—48no—
56C2×Dic106—42no—
57C2×Dic106—42no—
58C2×C8×C53—24no—
59C42×C53—22no—
60C42×C53—44no—
61C42×C53SD1614yes—
62C42×C53C2×D411yescharacteristic
63C42×C53C2×D412yes—
64C42×C53#4111yescharacteristic
65#11719921—42no—
66C4⋊Dic53—24no—
67C4⋊Dic53—22no—
68C4⋊Dic53—24no—
69C4⋊Dic53—42no—
70C4⋊Dic53—42no—
71#11719917—42no—
72C2×#88322—41no—
73C23×C53#12011yescharacteristic
74C22×D53—41no—
75C2×C4×C53—24no—
76C2×C4×C53—24no—
77C2×C4×C53—44no—
78C2×C4×C53#12012yes—
79C2×C4×C53#12112yes—
80C2×C4×C53#12111yescharacteristic
81C2×C4×C53#12711yescharacteristic
82C2×C4×C53C2×SD1612yes—
83C2×C4×C53#13712yes—
84C22⋊D53—82no—
85C2×Dic53—48no—
86C2×Dic53—41no—
87C2×Dic53—44no—
88D212—44no—
89D212—42no—
90C4×D53—81no—
91C4×D53—81no—
92Dic106—42no—
93Dic106—84no—
94Dic106—84no—
95C8×C53—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#45414yes—
106C4×C53#46112yes—
107C4×C53#49211yescharacteristic
108C4×C53#49711yescharacteristic
109Dic53—84no—
110Dic53—81no—
111#3227—532no—
112#2862—531no—
113#1962—531no—
114#1549—532no—
115#1276D5311yescharacteristic, complemented
116C2×C53—22no—
117C2×C53#200312yes—
118C2×C53#201311yescharacteristic
119D53—161no—
120#532—531no—
121C2×C4⋊Q8—532no—
122#498—1062no—
123#486—1062no—
124#481—1064no—
125#476—1064no—
126#431C2×D5312yes—
127#423—531no—
128C2×#104—1062no—
129#391—1064no—
130#382—1062no—
131C43C2×D5311yescharacteristic
132#332—1064no—
133C53#1028511yescharacteristic, complemented, Sylow
134C2×#41—1061no—
135C22×Q8—1062no—
136C2×M16—1061no—
137C4⋊Q8—531no—
138C4⋊Q8—1062no—
139C4⋊Q8—1062no—
140C4⋊Q8—1062no—
141C4⋊Q8—1062no—
142#127—531no—
143#124—1061no—
144#121—1062no—
145C4×D4—1062no—
146C2×C4⋊C4—534no—
147C2×C4⋊C4—1062no—
148C2×C42—22no—
149C2×C42D21212yes—
150C2×C42C22×D5311yescharacteristic
151#106—1068no—
152C4⋊C8—1062no—
153C4⋊C8D21214yes—
154#104—2124no—
155#102—1068no—
156C22⋊C8—22no—
157C4×C8—22no—
158#95—1068no—
159#41—2122no—
160#41—2122no—
161C2×Q8—1061no—
162C2×Q8—2128no—
163C2×Q8—2122no—
164C2×Q8—2122no—
165C2×D4—1062no—
166C2×D4—1061no—
167C2×D4—2121no—
168C22×C4—22no—
169C22×C4—1064no—
170C22×C4—2121no—
171C22×C4C2×D21211yescharacteristic
172C22×C4D4×D5312yes—
173M16—2122no—
174C2×C8—24no—
175C2×C8—2121no—
176C4⋊C4—1064no—
177C4⋊C4—1062no—
178C4⋊C4—1064no—
179C4⋊C4—2122no—
180C4⋊C4—2122no—
181C22⋊C4—2122no—
182C42—22no—
183C42—44no—
184C42#1171991314yes—
185C42C2×D21212yes—
186C42D4×D5311yescharacteristic
187C42#1171994911yescharacteristic
188C23—2121no—
189C23#1313318711yescharacteristic
190Q8—4244no—
191Q8—4244no—
192Q8—2122no—
193D4—4242no—
194D4—2124no—
195D4—2122no—
196C2×C4—24no—
197C2×C4—24no—
198C2×C4—44no—
199C2×C4—4241no—
200C2×C4—4241no—
201C2×C4—2128no—
202C2×C4—2121no—
203C2×C4—2124no—
204C2×C4#1313318712yes—
205C2×C4#1313319212yes—
206C2×C4#1313319211yescharacteristic
207C2×C4#1313321411yescharacteristic
208C2×C4C2×#1171991312yes—
209C2×C4#1313325012yes—
210C8—44no—
211C8—4241no—
212C22—22no—
213C22—4242no—
214C22—4241no—
215C22#1681053311yescharacteristic
216C22#1681076412yes—
217C4—22no—
218C4—44no—
219C4—4244no—
220C4—4241no—
221C4#1681079714yes—
222C4#1681082912yes—
223C4#1681099611yescharacteristic
224C4#1681101711yescharacteristic
225C2—22no—
226C2—8481no—
227C2#2027574012yes—
228C2#2027580011yescharacteristic
subgroupG/H
centerC22#16810533
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1276C2
socleC22×C53#392
prime radical#16811173C22
hypercenter#1276D53
hypocenterC53#10285
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
2. #20283093C2semidirect
4. #20273399C2semidirect
5. C53×#1276C2semidirect
115. #1276D53semidirect
133. C53#10285semidirect

Maximal subgroups

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

subgroupindex
#202858452
#202830932
#202755432
#202733992
C53×#12762
#1028553