Finite Group Databasebeta

← #27286780

Subgroups — #27286780

Order 13568.

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

normal subgroups only

This group has 614 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#20285838C212yesmaximal
2#20283113C211yesmaximal, characteristic, complemented
3#20275543C211yesmaximal, characteristic
4#20273386C212yesmaximal, complemented
5C53×#1276C211yesmaximal, characteristic, complemented
6#16811173C2211yescharacteristic
7C2×#13133217—22no—
8C2×#13133217C2212yes—
9#16811028—22no—
10#16810970—22no—
11#16810937—24no—
12#16810904—24no—
13#16810647C2211yescharacteristic
14C2×#13133145—22no—
15#16810493—24no—
16#16810368—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#13133217—44no—
27#13133217—44no—
28#13133217D411yescharacteristic
29#13133214D411yescharacteristic
30#13133205—21no—
31#13133191—22no—
32C4×D212—22no—
33C2×C4⋊Dic53—28no—
34C2×C4⋊Dic53—22no—
35C2×C4⋊Dic53D414yes—
36#13133151—28no—
37C4⋊#88322—22no—
38#13133145—44no—
39#13133139—28no—
40C2×C42×C53—22no—
41C2×C42×C53D412yes—
42C2×C42×C53C2311yescharacteristic
43C4⋊C8×C53D414yes—
44C22⋊C8×C53—22no—
45C4×C8×C53—22no—
46C22×C4×C53—22no—
47C22×C4×C53C2×D412yes—
48C22×C4×C53C2×D411yescharacteristic
49C2×C22⋊D53—41no—
50C22×Dic53—24no—
51#11719945—42no—
52#11719945—42no—
53C2×D212—22no—
54C2×D212—21no—
55C2×C4×D53—41no—
56C2×Dic106—21no—
57C2×Dic106—42no—
58C2×Dic106—42no—
59C2×Dic106—44no—
60C2×Dic106—44no—
61C2×C8×C53—24no—
62C42×C53—28no—
63C42×C53—22no—
64C42×C53D814yes—
65C42×C53C2×D411yescharacteristic
66C42×C53C2×D412yes—
67C42×C53#4111yescharacteristic
68#11719921—42no—
69C4⋊Dic53—24no—
70C4⋊Dic53—22no—
71C4⋊Dic53—24no—
72C4⋊Dic53—416no—
73C4⋊Dic53—42no—
74C4⋊Dic53—42no—
75#11719917—42no—
76C2×#88322—41no—
77C23×C53#12011yescharacteristic
78C22×D53—41no—
79C2×C4×C53—28no—
80C2×C4×C53—24no—
81C2×C4×C53—24no—
82C2×C4×C53#12012yes—
83C2×C4×C53#12112yes—
84C2×C4×C53#12111yescharacteristic
85C2×C4×C53#12711yescharacteristic
86C2×C4×C53C2×D812yes—
87C2×C4×C53AGL(1,ℤ/8)12yes—
88C22⋊D53—82no—
89C2×Dic53—41no—
90C2×Dic53—44no—
91C2×Dic53—48no—
92D212—44no—
93D212—42no—
94C4×D53—81no—
95C4×D53—81no—
96Dic106—42no—
97Dic106—84no—
98Dic106—82no—
99Dic106—82no—
100C8×C53—44no—
101#88322—81no—
102#10289—531nomaximal, Sylow
103C22×C53—22no—
104C22×C53#39211yescharacteristic
105C22×C53#44712yes—
106C2×D53—82no—
107C2×D53—81no—
108C4×C53—28no—
109C4×C53—22no—
110C4×C53#45614yes—
111C4×C53#45912yes—
112C4×C53#49311yescharacteristic
113C4×C53#49611yescharacteristic
114Dic53—81no—
115Dic53—84no—
116#3226—532no—
117#2868—531no—
118#1962—531no—
119#1546—532no—
120#1276D5311yescharacteristic, complemented
121C2×C53—22no—
122C2×C53#200512yes—
123C2×C53#201111yescharacteristic
124D53—161no—
125#532—531no—
126C2×C4⋊Q8—532no—
127C2×C4⋊Q8—1062no—
128#500—1062no—
129#486—1062no—
130#480—1064no—
131#474—1064no—
132#431C2×D5312yes—
133#423—531no—
134C2×#104—1062no—
135#384—1064no—
136C43C2×D5311yescharacteristic
137#332—1064no—
138C53#1028911yescharacteristic, complemented, Sylow
139C2×#41—1061no—
140C22×Q8—1062no—
141C2×M16—1061no—
142C4⋊Q8—531no—
143C4⋊Q8—1062no—
144C4⋊Q8—1062no—
145C4⋊Q8—1062no—
146C4⋊Q8—1062no—
147C4⋊Q8—2124no—
148C4⋊Q8—2124no—
149#127—531no—
150#124—1061no—
151#121—1062no—
152C4×D4—1062no—
153C2×C4⋊C4—534no—
154C2×C4⋊C4—1068no—
155C2×C4⋊C4—1062no—
156C2×C42—22no—
157C2×C42D21212yes—
158C2×C42C22×D5311yescharacteristic
159#107—1068no—
160C4⋊C8—1062no—
161C4⋊C8D21214yes—
162#104—2124no—
163#102—1068no—
164C22⋊C8—22no—
165C4×C8—22no—
166#41—2122no—
167#41—2122no—
168C2×Q8—1061no—
169C2×Q8—2122no—
170C2×Q8—2122no—
171C2×Q8—2124no—
172C2×Q8—2124no—
173C2×D4—1062no—
174C2×D4—1061no—
175C2×D4—2121no—
176C22×C4—22no—
177C22×C4—1064no—
178C22×C4—2121no—
179C22×C4C2×D21211yescharacteristic
180C22×C4D4×D5312yes—
181M16—2122no—
182C2×C8—24no—
183C2×C8—2121no—
184C4⋊C4—1064no—
185C4⋊C4—1062no—
186C4⋊C4—1064no—
187C4⋊C4—21216no—
188C4⋊C4—2122no—
189C4⋊C4—2122no—
190C22⋊C4—2122no—
191C42—28no—
192C42—22no—
193C42D42414yes—
194C42C2×D21212yes—
195C42D4×D5311yescharacteristic
196C42#1171994911yescharacteristic
197C23—2121no—
198C23#1313318711yescharacteristic
199Q8—4242no—
200Q8—4244no—
201Q8—4242no—
202Q8—2122no—
203D4—4242no—
204D4—2124no—
205D4—2122no—
206C2×C4—28no—
207C2×C4—24no—
208C2×C4—24no—
209C2×C4—4241no—
210C2×C4—4241no—
211C2×C4—2128no—
212C2×C4—2121no—
213C2×C4—2124no—
214C2×C4#1313318712yes—
215C2×C4#1313319212yes—
216C2×C4#1313319211yescharacteristic
217C2×C4#1313321411yescharacteristic
218C2×C4C2×D42412yes—
219C2×C4#1313324512yes—
220C8—44no—
221C8—4241no—
222C22—22no—
223C22—4242no—
224C22—4241no—
225C22#1681053311yescharacteristic
226C22#1681075912yes—
227C4—28no—
228C4—22no—
229C4—4241no—
230C4—4244no—
231C4#1681080714yes—
232C4#1681081912yes—
233C4#1681100011yescharacteristic
234C4#1681101211yescharacteristic
235C2—22no—
236C2—8481no—
237C2#2027575412yes—
238C2#2027578811yescharacteristic
subgroupG/H
centerC22#16810533
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1276C2
socleC22×C53#392
prime radical#16811173C22
hypercenter#1276D53
hypocenterC53#10289
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. #20283113C2semidirect
4. #20273386C2semidirect
5. C53×#1276C2semidirect
120. #1276D53semidirect
138. C53#10289semidirect

Maximal subgroups

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

subgroupindex
#202858382
#202831132
#202755432
#202733862
C53×#12762
#1028953