Finite Group Databasebeta

← #27281717

Subgroups — #27281717

Order 13568.

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

normal subgroups only

This group has 434 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#20277821C211yesmaximal, characteristic
2#20273342C211yesmaximal, characteristic
3#20273227C212yesmaximal
4#20272609C212yesmaximal
5C53×#3166C211yesmaximal, characteristic
6#16810680C2212yes—
7#16810509—22no—
8#16810509—22no—
9C4⋊C4×Dic53—22no—
10#16810469C2212yes—
11#16810365—22no—
12#16810360—22no—
13#16810360—22no—
14C53×#533C2211yescharacteristic
15C2×#125×C53C2211yescharacteristic
16C53×#501C414yescomplemented
17C53×#499—22no—
18C53×#428—22no—
19C4×M16×C53C2211yescharacteristic
20C2×C4⋊Dic53—22no—
21C2×C4×Dic53—22no—
22C2×C4×Dic53—22no—
23C4⋊#88322—22no—
24C22⋊#88322—22no—
25#13133128—22no—
26#13133124—22no—
27#13133124—22no—
28#13133124—22no—
29#13133124D412yes—
30#13133124D412yes—
31C2×M16×C53—21no—
32C4⋊Q8×C53C2×C412yes—
33#125×C53—21no—
34#125×C53—22no—
35#125×C53C2×C412yes—
36#125×C53D412yes—
37C22⋊Q8×C53—22no—
38C4×Q8×C53—22no—
39#117×C53—21no—
40C2×C4⋊C4×C53—22no—
41C2×C4⋊C4×C53D412yes—
42C2×C42×C53C2311yescharacteristic
43#107×C53—24no—
44#106×C53—24no—
45#97×C53C2×C412yes—
46C4×C8×C53—21no—
47C2×Q8×C53—22no—
48C22×C4×C53—22no—
49C22×C4×C53C2×D411yescharacteristic
50C22×C4×C53C2×D412yes—
51C22×Dic53—22no—
52C22×Dic53—22no—
53M16×C53—24no—
54C2×C8×C53—22no—
55C4⋊C4×C53—24no—
56C4⋊C4×C53—24no—
57C4⋊C4×C53—22no—
58C4⋊C4×C53—22no—
59C4⋊C4×C53—22no—
60C4⋊C4×C53—42no—
61C4⋊C4×C53C22⋊C414yes—
62C22⋊C4×C53—41no—
63C42×C53—41no—
64C42×C53C22×C411yescharacteristic
65C42×C53C2×D411yescharacteristic
66C42×C53#4112yes—
67C4⋊Dic53—44no—
68C4×Dic53—44no—
69C4×Dic53—44no—
70C2×#88322—42no—
71C23×C53#12011yescharacteristic
72Q8×C53—82no—
73C2×C4×C53—22no—
74C2×C4×C53—24no—
75C2×C4×C53—42no—
76C2×C4×C53—42no—
77C2×C4×C53—42no—
78C2×C4×C53—42no—
79C2×C4×C53C2×C22⋊C411yescharacteristic
80C2×C4×C53C4×D412yes—
81C2×C4×C53#12112yes—
82C2×C4×C53#12311yescharacteristic
83C2×Dic53—24no—
84C2×Dic53—42no—
85C2×Dic53—42no—
86C2×Dic53—42no—
87C2×Dic53—42no—
88C2×Dic53—42no—
89C8×C53—24no—
90#88322—82no—
91#9440—531nomaximal, Sylow
92C22×C53—22no—
93C22×C53#38611yescharacteristic
94C22×C53#45612yes—
95C4×C53—21no—
96C4×C53—21no—
97C4×C53—22no—
98C4×C53—22no—
99C4×C53—44no—
100C4×C53—82no—
101Dic53—44no—
102Dic53—82no—
103#3166D5311yescharacteristic
104#2371—531no—
105#1534—531no—
106#1506—532no—
107#1300—532no—
108C2×C53—22no—
109C2×C53#188611yescharacteristic
110C2×C53#190612yes—
111#533C2×D5311yescharacteristic
112C2×#125C2×D5311yescharacteristic
113#501Dic5314yescomplemented
114#499—22no—
115#432—532no—
116#428—22no—
117C4×M16C2×D5311yescharacteristic
118#387—1062no—
119#387—1062no—
120C4×C4⋊C4—1062no—
121#376—532no—
122#330—1062no—
123#328—1062no—
124#328—1062no—
125C53#944011yescharacteristic, complemented, Sylow
126C2×M16—21no—
127C4⋊Q8C2×Dic5312yes—
128#125—22no—
129#125—21no—
130#125C2×Dic5312yes—
131#125C22⋊D5312yes—
132C22⋊Q8—22no—
133C4×Q8—22no—
134#117—21no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—1062no—
137C2×C4⋊C4C22⋊D5312yes—
138C2×C42—1062no—
139C2×C42—1062no—
140C2×C42C22×D5311yescharacteristic
141#107—24no—
142#106—24no—
143C4⋊C8—1062no—
144C22⋊C8—1062no—
145#97—1062no—
146#97C2×Dic5312yes—
147C4×C8—21no—
148#95—532no—
149#95—532no—
150#95—1062no—
151#95—1062no—
152#95—1062no—
153C2×Q8—22no—
154C22×C4—22no—
155C22×C4—1062no—
156C22×C4—1062no—
157C22×C4D4×D5312yes—
158C22×C4C2×C22⋊D5311yescharacteristic
159M16—24no—
160C2×C8—22no—
161C2×C8—2122no—
162C4⋊C4—24no—
163C4⋊C4—24no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—42no—
168C4⋊C4—2124no—
169C4⋊C4C22⋊Dic5314yes—
170C22⋊C4—41no—
171C42—41no—
172C42—2124no—
173C42—2124no—
174C42#1171994712yes—
175C42C22×Dic5311yescharacteristic
176C42C2×C22⋊D5311yescharacteristic
177C23#1313318511yescharacteristic
178Q8—82no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—1064no—
186C2×C4—2122no—
187C2×C4—2122no—
188C2×C4—2122no—
189C2×C4—2122no—
190C2×C4—2122no—
191C2×C4C2×C22⋊Dic5311yescharacteristic
192C2×C4D4×Dic5312yes—
193C2×C4#1313318912yes—
194C2×C4#1313319811yescharacteristic
195C8—24no—
196C8—4242no—
197C22—22no—
198C22#1681050311yescharacteristic
199C22#1681080512yes—
200C4—22no—
201C4—22no—
202C4—21no—
203C4—21no—
204C4—44no—
205C4—82no—
206C4—4242no—
207C4—2124no—
208C2—22no—
209C2#2027510911yescharacteristic
210C2#2027521812yes—
subgroupG/H
centerC22#16810503
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3166C2
socleC22×C53#386
prime radicalC23×C53#120
cosocleC2×C42×C53C23
hypercenter#3166D53
hypocenterC53#9440
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3166

Complements

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

NH ≅ G/Ntype
16. C53×#501C4semidirect
113. #501Dic53semidirect
125. C53#9440semidirect

Maximal subgroups

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

subgroupindex
#202778212
#202733422
#202732272
#202726092
C53×#31662
#944053