Finite Group Databasebeta

← #27288191

Subgroups — #27288191

Order 13568.

344 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 79 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#20286060C211yesmaximal, characteristic
2#20273413C212yesmaximal
3#20273226C212yesmaximal
4#20272585C211yesmaximal, characteristic
5C53×#1832C211yesmaximal, characteristic
6#16811180C2212yes—
7#16810919—22no—
8#16810864—22no—
9#16810834—22no—
10#16810680C2212yes—
11#16810370—22no—
12#16810368—22no—
13#16810359—24no—
14C2×C4⋊C8×C53C2211yescharacteristic
15C2×C4⋊C8×C53C2211yescharacteristic
16C53×#376C2211yescharacteristic
17C53×#336—22no—
18C2×Dic212—22no—
19#13133217D412yes—
20#13133217D412yes—
21#13133209—22no—
22#13133196—22no—
23#13133196—22no—
24C4×Dic106—22no—
25C4×Dic106—22no—
26#13133176—22no—
27#13133151—22no—
28C4⋊#88322—22no—
29#13133142—24no—
30C22⋊#88322—22no—
31#13133128—22no—
32C22×C8×C53—21no—
33C22×C8×C53D412yes—
34C2×C42×C53—21no—
35C2×C42×C53C2311yescharacteristic
36C4⋊C8×C53—22no—
37C4⋊C8×C53—22no—
38#95×C53—21no—
39#95×C53D412yes—
40C22×C4×C53—22no—
41C22×C4×C53C2×D411yescharacteristic
42C22×C4×C53C2×D411yescharacteristic
43C22×C4×C53#4111yescharacteristic
44C2×Dic106—22no—
45C2×Dic106—22no—
46C2×C8×C53—22no—
47C2×C8×C53—42no—
48C2×C8×C53—42no—
49C2×C8×C53—41no—
50C42×C53—41no—
51C42×C53—41no—
52C42×C53C2×D412yes—
53C42×C53#4112yes—
54C22⋊Dic53—42no—
55C4⋊Dic53—22no—
56C4⋊Dic53—22no—
57C4⋊Dic53—22no—
58C4⋊Dic53—22no—
59#11719919—42no—
60#11719919—42no—
61C4×Dic53—42no—
62C2×#88322—42no—
63Dic212—82no—
64C23×C53#12111yescharacteristic
65C2×C4×C53—22no—
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—41no—
69C2×C4×C53—42no—
70C2×C4×C53—42no—
71C2×C4×C53—42no—
72C2×C4×C53#12011yescharacteristic
73C2×C4×C53#12112yes—
74C2×C4×C53#12312yes—
75C2×C4×C53#12411yescharacteristic
76C2×Dic53—42no—
77C2×Dic53—42no—
78Dic106—42no—
79Dic106—42no—
80Dic106—82no—
81C8×C53—42no—
82C8×C53—81no—
83#88322—82no—
84#10517—531nomaximal, Sylow
85C22×C53—21no—
86C22×C53—21no—
87C22×C53#39411yescharacteristic
88C22×C53#47312yes—
89C4×C53—22no—
90C4×C53—21no—
91C4×C53—22no—
92C4×C53—82no—
93C4×C53#45411yescharacteristic
94C4×C53#45511yescharacteristic
95Dic53—82no—
96Dic53—82no—
97#3249—531no—
98#1832D5311yescharacteristic
99#1552—532no—
100#1506—532no—
101#1290—531no—
102C2×C53—22no—
103C2×C53#204011yescharacteristic
104C2×C53#204711yescharacteristic
105C2×C53#205311yescharacteristic
106#533—532no—
107#477—1062no—
108#467—1062no—
109#462—1062no—
110#432—532no—
111C2×C4⋊C8C2×D5311yescharacteristic
112C2×C4⋊C8C2×D5311yescharacteristic
113#376C2×D5311yescharacteristic
114#336—22no—
115#332—1062no—
116#332—1062no—
117#328—1064no—
118C53#1051711yescharacteristic, complemented, Sylow
119C2×Q16—1062no—
120C22×C8—21no—
121C22×C8C22⋊D5312yes—
122C4⋊Q8—532no—
123C4⋊Q8—532no—
124#125—1062no—
125C22⋊Q8—1062no—
126C22⋊Q8—1062no—
127C4×Q8—1062no—
128C4×Q8—1062no—
129#117—1062no—
130C2×C42—21no—
131C2×C42C22×D5311yescharacteristic
132#107—1062no—
133C4⋊C8—22no—
134C4⋊C8—22no—
135C4⋊C8—1062no—
136#103—1064no—
137C22⋊C8—1062no—
138#97—1062no—
139#95—21no—
140#95D21212yes—
141C2×Q8—1062no—
142C2×Q8—1062no—
143C22×C4—22no—
144C22×C4C2×D21211yescharacteristic
145C22×C4#1171994511yescharacteristic
146C22×C4C2×C22⋊D5311yescharacteristic
147Q16—4242no—
148C2×C8—22no—
149C2×C8—42no—
150C2×C8—42no—
151C2×C8—41no—
152C2×C8—2122no—
153C4⋊C4—1062no—
154C4⋊C4—1062no—
155C4⋊C4—1062no—
156C4⋊C4—1062no—
157C4⋊C4—2122no—
158C4⋊C4—2122no—
159C22⋊C4—2122no—
160C42—41no—
161C42—41no—
162C42—2122no—
163C42D4×D5312yes—
164C42#1171994912yes—
165C23#1313319111yescharacteristic
166Q8—4242no—
167Q8—2122no—
168Q8—2122no—
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—2122no—
177C2×C4—2122no—
178C2×C4#1313318511yescharacteristic
179C2×C4#1313319212yes—
180C2×C4#1313320212yes—
181C2×C4#1313320611yescharacteristic
182C8—42no—
183C8—81no—
184C8—4242no—
185C22—21no—
186C22—21no—
187C22#1681054011yescharacteristic
188C22#1681089812yes—
189C4—22no—
190C4—22no—
191C4—21no—
192C4—82no—
193C4—4242no—
194C4—4242no—
195C4#1681079511yescharacteristic
196C4#1681080011yescharacteristic
197C2—22no—
198C2#2027594411yescharacteristic
199C2#2027598311yescharacteristic
200C2#2027602111yescharacteristic
subgroupG/H
centerC22#16810540
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1832C2
socleC22×C53#394
prime radicalC23×C53#121
hypercenter#1832D53
hypocenterC53#10517
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1832

Complements

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

NH ≅ G/Ntype
118. C53#10517semidirect

Maximal subgroups

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

subgroupindex
#202860602
#202734132
#202732262
#202725852
C53×#18322
#1051753