Finite Group Databasebeta

← #27286766

Subgroups — #27286766

Order 13568.

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

normal subgroups only

This group has 566 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#20285640C212yesmaximal
2#20275543C211yesmaximal, characteristic
3#20273396C212yesmaximal, complemented
4#20272548C211yesmaximal, characteristic, complemented
5C53×#2862C211yesmaximal, characteristic, complemented
6#16811173C2211yescharacteristic
7#16810997—22no—
8#16810973—22no—
9#16810825—24no—
10#16810762—24no—
11#16810676C2212yes—
12#16810647C2211yescharacteristic
13C2×#13133145—22no—
14#16810372—24no—
15C2×C4⋊Q8×C53C2212yes—
16C53×#391—24no—
17C53×#382—22no—
18C43×C53C2211yescharacteristic
19C2×#11719945—21no—
20C2×#11719924—28no—
21C2×#11719917—21no—
22#13133217D411yescharacteristic
23#13133214D411yescharacteristic
24#13133205—21no—
25#13133191—22no—
26C4×D212—22no—
27C4⋊#88322—22no—
28C4⋊#88322D414yes—
29#13133145—44no—
30#13133141—28no—
31C22⋊#88322—22no—
32C4×#88322—22no—
33C22×Q8×C53D414yes—
34C4⋊Q8×C53—22no—
35C4⋊Q8×C53—22no—
36C4⋊Q8×C53—22no—
37C4⋊Q8×C53—22no—
38C2×C4⋊C4×C53—22no—
39C2×C4⋊C4×C53—22no—
40C2×C42×C53—22no—
41C2×C42×C53D412yes—
42C2×C42×C53C2311yescharacteristic
43#95×C53—28no—
44C2×Q8×C53—24no—
45C2×Q8×C53—24no—
46C2×Q8×C53—48no—
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—42no—
59C4⋊C4×C53—42no—
60C4⋊C4×C53—42no—
61C4⋊C4×C53—42no—
62C42×C53—22no—
63C42×C53—44no—
64C42×C53SD1614yescomplemented
65C42×C53C2×D411yescharacteristic
66C42×C53C2×D412yes—
67C42×C53#4111yescharacteristic
68#11719924—88no—
69#11719921—42no—
70C4⋊Dic53—22no—
71#11719917—42no—
72C2×#88322—24no—
73C2×#88322—41no—
74C23×C53#12011yescharacteristic
75C22×D53—41no—
76Q8×C53—48no—
77Q8×C53—84no—
78C2×C4×C53—24no—
79C2×C4×C53—24no—
80C2×C4×C53—44no—
81C2×C4×C53—48no—
82C2×C4×C53—44no—
83C2×C4×C53#12012yes—
84C2×C4×C53#12111yescharacteristic
85C2×C4×C53#12112yes—
86C2×C4×C53#12711yescharacteristic
87C2×C4×C53C2×SD1612yes—
88C2×C4×C53#13712yes—
89C22⋊D53—82no—
90C2×Dic53—41no—
91D212—44no—
92D212—42no—
93C4×D53—81no—
94C4×D53—81no—
95Dic106—42no—
96#88322—44no—
97#88322—81no—
98#10287—531nomaximal, Sylow
99C22×C53—22no—
100C22×C53#39211yescharacteristic
101C22×C53#44812yes—
102C2×D53—82no—
103C2×D53—81no—
104C4×C53—22no—
105C4×C53—44no—
106C4×C53—84no—
107C4×C53#45514yes—
108C4×C53#46112yes—
109C4×C53#49211yescharacteristic
110C4×C53#49711yescharacteristic
111Dic53—81no—
112#3206—532no—
113#2862D5311yescharacteristic, complemented
114#1962—531no—
115#1548—532no—
116#1276—531no—
117C2×C53—22no—
118C2×C53#200412yes—
119C2×C53#201311yescharacteristic
120D53—161no—
121#532—531no—
122C2×C4⋊Q8C2×D5312yes—
123#492—1062no—
124#487—1062no—
125#461—1064no—
126#448—1064no—
127#431—532no—
128#423—531no—
129C2×#104—1062no—
130#391—24no—
131#382—22no—
132C43C2×D5311yescharacteristic
133#333—1064no—
134C53#1028711yescharacteristic, complemented, Sylow
135C2×#41—1061no—
136C22×Q8C22⋊D5314yes—
137C2×SD16—1068no—
138C2×M16—1061no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—531no—
144#127—531no—
145#124—1061no—
146#121—1062no—
147C4×D4—1062no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C42—22no—
151C2×C42C22⋊D5312yes—
152C2×C42C22×D5311yescharacteristic
153C4⋊C8—534no—
154C4⋊C8—1062no—
155#104—2124no—
156#103—1068no—
157C22⋊C8—1062no—
158C4×C8—1062no—
159#95—28no—
160#41—2122no—
161#41—2122no—
162C2×Q8—24no—
163C2×Q8—24no—
164C2×Q8—48no—
165C2×Q8—1061no—
166C2×D4—1062no—
167C2×D4—1061no—
168C2×D4—2121no—
169C22×C4—22no—
170C22×C4—24no—
171C22×C4—2121no—
172C22×C4C2×C22⋊D5312yes—
173C22×C4C2×C22⋊D5311yescharacteristic
174SD16—4248no—
175M16—2122no—
176C2×C8—1064no—
177C2×C8—2121no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—1062no—
183C22⋊C4—2122no—
184C42—22no—
185C42—44no—
186C42#1171992414yescomplemented
187C42D4×D5311yescharacteristic
188C42D4×D5312yes—
189C42#1171994711yescharacteristic
190C23—2121no—
191C23#1313318611yescharacteristic
192Q8—48no—
193Q8—84no—
194Q8—2122no—
195D4—4242no—
196D4—2124no—
197D4—2122no—
198C2×C4—24no—
199C2×C4—24no—
200C2×C4—44no—
201C2×C4—48no—
202C2×C4—44no—
203C2×C4—4241no—
204C2×C4—4241no—
205C2×C4—2121no—
206C2×C4#1313318512yes—
207C2×C4#1313318811yescharacteristic
208C2×C4#1313318912yes—
209C2×C4#1313321311yescharacteristic
210C2×C4C2×#1171992412yes—
211C2×C4#1313324812yes—
212C8—4241no—
213C8—2124no—
214C22—22no—
215C22—4242no—
216C22—4241no—
217C22#1681053111yescharacteristic
218C22#1681076212yes—
219C4—22no—
220C4—44no—
221C4—84no—
222C4—4241no—
223C4#1681080014yes—
224C4#1681082512yes—
225C4#1681099711yescharacteristic
226C4#1681101511yescharacteristic
227C2—22no—
228C2—8481no—
229C2#2027574412yes—
230C2#2027579811yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2862C2
socleC22×C53#392
prime radical#16811173C22
hypercenter#2862D53
hypocenterC53#10287
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. #20273396C2semidirect
4. #20272548C2semidirect
5. C53×#2862C2semidirect
64. C42×C53SD16semidirect
113. #2862D53semidirect
134. C53#10287semidirect
186. C42#11719924semidirect

Maximal subgroups

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

subgroupindex
#202856402
#202755432
#202733962
#202725482
C53×#28622
#1028753