Finite Group Databasebeta

← #27281635

Subgroups — #27281635

Order 13568.

422 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 103 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#20277896C211yesmaximal, characteristic
2#20273441C211yesmaximal, characteristic
3#20273266C212yesmaximal
4#20272580C212yesmaximal
5C53×#3139C211yesmaximal, characteristic
6#16810680C2212yes—
7#16810530—22no—
8#16810509—24no—
9#16810491C2212yes—
10#16810372—22no—
11#16810360—24no—
12C53×#533C2211yescharacteristic
13C2×C4⋊Q8×C53C2211yescharacteristic
14C53×#491—22no—
15C53×#487C414yescomplemented
16C53×#419—22no—
17C4×M16×C53C2211yescharacteristic
18C2×C4⋊Dic53—22no—
19C2×C4×Dic53—22no—
20C4⋊#88322—22no—
21C22⋊#88322—22no—
22#13133128—22no—
23#13133124—24no—
24#13133124—22no—
25#13133124—22no—
26#13133124D414yes—
27C22×Q8×C53D412yes—
28C2×M16×C53—21no—
29C4⋊Q8×C53—22no—
30C4⋊Q8×C53C2×C412yes—
31C4⋊Q8×C53C2×C412yes—
32C4⋊Q8×C53D412yes—
33#125×C53—21no—
34C22⋊Q8×C53—22no—
35C4×Q8×C53—22no—
36#117×C53—21no—
37C2×C4⋊C4×C53—22no—
38C2×C42×C53C2311yescharacteristic
39#103×C53—28no—
40#97×C53—21no—
41C4×C8×C53C2×C412yes—
42C2×Q8×C53—22no—
43C2×Q8×C53—22no—
44C2×Q8×C53—42no—
45C2×Q8×C53—42no—
46C2×Q8×C53C22⋊C414yes—
47C22×C4×C53—22no—
48C22×C4×C53C2×D411yescharacteristic
49C22×C4×C53C2×D412yes—
50C22×Dic53—24no—
51M16×C53—42no—
52C2×C8×C53—22no—
53C4⋊C4×C53—24no—
54C4⋊C4×C53—24no—
55C4⋊C4×C53—22no—
56C4⋊C4×C53—22no—
57C4⋊C4×C53—42no—
58C22⋊C4×C53—41no—
59C42×C53—41no—
60C42×C53C22×C411yescharacteristic
61C42×C53C2×D411yescharacteristic
62C42×C53#4112yes—
63C4⋊Dic53—82no—
64C4×Dic53—82no—
65C2×#88322—42no—
66C23×C53#12011yescharacteristic
67Q8×C53—44no—
68Q8×C53—82no—
69Q8×C53—82no—
70C2×C4×C53—22no—
71C2×C4×C53—24no—
72C2×C4×C53—42no—
73C2×C4×C53—42no—
74C2×C4×C53—42no—
75C2×C4×C53—42no—
76C2×C4×C53C2×C22⋊C411yescharacteristic
77C2×C4×C53C4×D412yes—
78C2×C4×C53#12112yes—
79C2×C4×C53#12311yescharacteristic
80C2×Dic53—48no—
81C2×Dic53—44no—
82C8×C53—42no—
83#88322—82no—
84#9426—531nomaximal, Sylow
85C22×C53—22no—
86C22×C53#38611yescharacteristic
87C22×C53#45512yes—
88C4×C53—21no—
89C4×C53—21no—
90C4×C53—22no—
91C4×C53—22no—
92C4×C53—44no—
93C4×C53—82no—
94Dic53—84no—
95#3139D5311yescharacteristic
96#2381—531no—
97#1558—531no—
98#1515—532no—
99#1288—532no—
100C2×C53—22no—
101C2×C53#188511yescharacteristic
102C2×C53#190512yes—
103#533C2×D5311yescharacteristic
104C2×C4⋊Q8C2×D5311yescharacteristic
105#491—22no—
106#487Dic5314yescomplemented
107#432—532no—
108#419—22no—
109C4×M16C2×D5311yescharacteristic
110#391—1062no—
111#387—1064no—
112#383—532no—
113#333—1062no—
114#328—1064no—
115C53#942611yescharacteristic, complemented, Sylow
116C22×Q8C22⋊D5312yes—
117C2×M16—21no—
118C4⋊Q8—22no—
119C4⋊Q8C2×Dic5312yes—
120C4⋊Q8C2×Dic5312yes—
121C4⋊Q8C22⋊D5312yes—
122#125—21no—
123C22⋊Q8—22no—
124C4×Q8—22no—
125#117—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—1062no—
128C2×C42—1062no—
129C2×C42C22×D5311yescharacteristic
130C4⋊C8—1062no—
131#103—28no—
132C22⋊C8—1062no—
133#97—21no—
134#97—1062no—
135C4×C8C2×Dic5312yes—
136#95—534no—
137#95—1064no—
138#95—1062no—
139#95—1062no—
140C2×Q8—22no—
141C2×Q8—22no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8C22⋊Dic5314yes—
145C22×C4—22no—
146C22×C4—1064no—
147C22×C4D4×D5312yes—
148C22×C4C2×C22⋊D5311yescharacteristic
149M16—42no—
150C2×C8—22no—
151C2×C8—2122no—
152C4⋊C4—24no—
153C4⋊C4—24no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—42no—
157C4⋊C4—4242no—
158C22⋊C4—41no—
159C42—41no—
160C42—4242no—
161C42#1171994712yes—
162C42C22×Dic5311yescharacteristic
163C42C2×C22⋊D5311yescharacteristic
164C23#1313318511yescharacteristic
165Q8—44no—
166Q8—82no—
167Q8—82no—
168C2×C4—22no—
169C2×C4—24no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—2128no—
175C2×C4—2124no—
176C2×C4C2×C22⋊Dic5311yescharacteristic
177C2×C4D4×Dic5312yes—
178C2×C4#1313318912yes—
179C2×C4#1313319811yescharacteristic
180C8—42no—
181C8—4242no—
182C22—22no—
183C22#1681050311yescharacteristic
184C22#1681080012yes—
185C4—22no—
186C4—22no—
187C4—21no—
188C4—21no—
189C4—44no—
190C4—82no—
191C4—4244no—
192C2—22no—
193C2#2027510411yescharacteristic
194C2#2027521112yes—
subgroupG/H
centerC22#16810503
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3139C2
socleC22×C53#386
prime radicalC23×C53#120
cosocleC2×C42×C53C23
hypercenter#3139D53
hypocenterC53#9426
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3139

Complements

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

NH ≅ G/Ntype
15. C53×#487C4semidirect
106. #487Dic53semidirect
115. C53#9426semidirect

Maximal subgroups

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

subgroupindex
#202778962
#202734412
#202732662
#202725802
C53×#31392
#942653