Finite Group Databasebeta

← #27286778

Subgroups — #27286778

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#20285836C212yesmaximal
2#20275543C211yesmaximal, characteristic
3#20273387C212yesmaximal, complemented
4#20272548C211yesmaximal, characteristic, complemented
5C53×#2868C211yesmaximal, characteristic, complemented
6#16811173C2211yescharacteristic
7#16811026—22no—
8#16810971—22no—
9#16810936—24no—
10#16810900—24no—
11#16810676C2212yes—
12#16810647C2211yescharacteristic
13C2×#13133145—22no—
14#16810369—24no—
15C2×C4⋊Q8×C53—22no—
16C2×C4⋊Q8×C53C2212yes—
17C53×#384—24no—
18C43×C53C2211yescharacteristic
19C2×#11719945—21no—
20C2×#11719917—21no—
21#13133217D411yescharacteristic
22#13133214D411yescharacteristic
23#13133205—21no—
24#13133191—22no—
25C4×D212—22no—
26#13133150—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—
38C4⋊Q8×C53—44no—
39C4⋊Q8×C53—44no—
40C2×C4⋊C4×C53—28no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53D414yes—
43C2×C42×C53—22no—
44C2×C42×C53D412yes—
45C2×C42×C53C2311yescharacteristic
46C2×Q8×C53—42no—
47C2×Q8×C53—42no—
48C2×Q8×C53—44no—
49C2×Q8×C53—44no—
50C22×C4×C53—22no—
51C22×C4×C53—24no—
52C22×C4×C53C2×D412yes—
53C22×C4×C53C2×D411yescharacteristic
54C2×C22⋊D53—41no—
55#11719945—42no—
56#11719945—42no—
57C2×D212—22no—
58C2×D212—21no—
59C2×C4×D53—41no—
60C2×Dic106—21no—
61C4⋊C4×C53—24no—
62C4⋊C4×C53—24no—
63C4⋊C4×C53—416no—
64C4⋊C4×C53—42no—
65C4⋊C4×C53—42no—
66C42×C53—28no—
67C42×C53—22no—
68C42×C53D814yes—
69C42×C53C2×D411yescharacteristic
70C42×C53C2×D412yes—
71C42×C53#4111yescharacteristic
72#11719921—42no—
73C4⋊Dic53—22no—
74#11719917—42no—
75C2×#88322—24no—
76C2×#88322—41no—
77C23×C53#12011yescharacteristic
78C22×D53—41no—
79Q8×C53—84no—
80Q8×C53—82no—
81Q8×C53—82no—
82C2×C4×C53—28no—
83C2×C4×C53—24no—
84C2×C4×C53—24no—
85C2×C4×C53—44no—
86C2×C4×C53—48no—
87C2×C4×C53#12012yes—
88C2×C4×C53#12111yescharacteristic
89C2×C4×C53#12112yes—
90C2×C4×C53#12711yescharacteristic
91C2×C4×C53C2×D812yes—
92C2×C4×C53AGL(1,ℤ/8)12yes—
93C22⋊D53—82no—
94C2×Dic53—41no—
95D212—44no—
96D212—42no—
97C4×D53—81no—
98C4×D53—81no—
99Dic106—42no—
100#88322—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—84no—
111C4×C53#45614yes—
112C4×C53#45912yes—
113C4×C53#49311yescharacteristic
114C4×C53#49611yescharacteristic
115Dic53—81no—
116#3226—532no—
117#2868D5311yescharacteristic, complemented
118#1962—531no—
119#1546—532no—
120#1276—531no—
121C2×C53—22no—
122C2×C53#200512yes—
123C2×C53#201111yescharacteristic
124D53—161no—
125#532—531no—
126C2×C4⋊Q8—22no—
127C2×C4⋊Q8C2×D5312yes—
128#500—1062no—
129#486—1062no—
130#480—1064no—
131#474—1064no—
132#431—532no—
133#423—531no—
134C2×#104—1062no—
135#384—24no—
136C43C2×D5311yescharacteristic
137#332—1064no—
138C53#1028911yescharacteristic, complemented, Sylow
139C2×#41—1061no—
140C22×Q8—22no—
141C2×M16—1061no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—44no—
147C4⋊Q8—44no—
148C4⋊Q8—531no—
149#127—531no—
150#124—1061no—
151#121—1062no—
152C4×D4—1062no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4C22⋊D5314yes—
156C2×C42—22no—
157C2×C42C22⋊D5312yes—
158C2×C42C22×D5311yescharacteristic
159#107—1068no—
160C4⋊C8—534no—
161C4⋊C8—1062no—
162#104—2124no—
163#102—1068no—
164C22⋊C8—1062no—
165C4×C8—1062no—
166#41—2122no—
167#41—2122no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—44no—
171C2×Q8—44no—
172C2×Q8—1061no—
173C2×D4—1062no—
174C2×D4—1061no—
175C2×D4—2121no—
176C22×C4—22no—
177C22×C4—24no—
178C22×C4—2121no—
179C22×C4C2×C22⋊D5312yes—
180C22×C4C2×C22⋊D5311yescharacteristic
181M16—2122no—
182C2×C8—1064no—
183C2×C8—2121no—
184C4⋊C4—24no—
185C4⋊C4—24no—
186C4⋊C4—416no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—1062no—
190C22⋊C4—2122no—
191C42—28no—
192C42—22no—
193C42#1171992214yes—
194C42D4×D5311yescharacteristic
195C42D4×D5312yes—
196C42#1171994711yescharacteristic
197C23—2121no—
198C23#1313318611yescharacteristic
199Q8—82no—
200Q8—84no—
201Q8—82no—
202Q8—2122no—
203D4—4242no—
204D4—2124no—
205D4—2122no—
206C2×C4—28no—
207C2×C4—24no—
208C2×C4—24no—
209C2×C4—48no—
210C2×C4—44no—
211C2×C4—4241no—
212C2×C4—4241no—
213C2×C4—2121no—
214C2×C4#1313318512yes—
215C2×C4#1313318811yescharacteristic
216C2×C4#1313318912yes—
217C2×C4#1313321311yescharacteristic
218C2×C4C2×#1171992212yes—
219C2×C4#1313324312yes—
220C8—4241no—
221C8—2124no—
222C22—22no—
223C22—4242no—
224C22—4241no—
225C22#1681053111yescharacteristic
226C22#1681075712yes—
227C4—28no—
228C4—22no—
229C4—84no—
230C4—4241no—
231C4#1681080514yes—
232C4#1681081512yes—
233C4#1681099911yescharacteristic
234C4#1681101011yescharacteristic
235C2—22no—
236C2—8481no—
237C2#2027575112yes—
238C2#2027578611yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2868C2
socleC22×C53#392
prime radical#16811173C22
hypercenter#2868D53
hypocenterC53#10289
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2868

Complements

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

NH ≅ G/Ntype
3. #20273387C2semidirect
4. #20272548C2semidirect
5. C53×#2868C2semidirect
117. #2868D53semidirect
138. C53#10289semidirect

Maximal subgroups

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

subgroupindex
#202858362
#202755432
#202733872
#202725482
C53×#28682
#1028953