Finite Group Databasebeta

← #27279041

Subgroups — #27279041

Order 13568.

416 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 99 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#20285060C211yesmaximal, characteristic
2#20273377C211yesmaximal, characteristic
3#20273187C212yesmaximal, complemented
4#20272613C212yesmaximal, complemented
5C53×#2980C211yesmaximal, characteristic
6#16811180C2211yescharacteristic
7#16810991—21no—
8#16810987—21no—
9#16810982C414yes—
10#16810676C2212yes—
11#16810619—22no—
12C2×#13133128C2211yescharacteristic
13#16810372—21no—
14#16810367—21no—
15#16810353—24no—
16C53×#532C2211yescharacteristic
17C53×#435—24no—
18C53×#432C2212yes—
19C53×#408—22no—
20C22×#88322—21no—
21#13133217C2×C412yes—
22#13133209—21no—
23#13133196—22no—
24C4×Dic106—22no—
25#13133176—21no—
26C4⋊#88322—22no—
27#13133141—24no—
28#13133138—24no—
29C22⋊#88322—22no—
30#13133128—21no—
31#13133128C2×C412yes—
32C4×#88322—22no—
33C2×#41×C53D412yes—
34C2×M16×C53—22no—
35C22×C8×C53—22no—
36C4⋊Q8×C53D411yescharacteristic
37#127×C53D411yescharacteristic
38#124×C53C2×C412yes—
39#121×C53—22no—
40C4×D4×C53—22no—
41C2×C42×C53C2311yescharacteristic
42C4⋊C8×C53D414yes—
43C22⋊C8×C53—24no—
44C22⋊C8×C53—22no—
45#97×C53—22no—
46#41×C53—42no—
47#41×C53—42no—
48C2×Q8×C53C22⋊C412yes—
49C2×D4×C53—22no—
50C2×D4×C53—22no—
51C2×D4×C53C22⋊C412yes—
52C22×C4×C53—22no—
53C22×C4×C53C2×D411yescharacteristic
54C22×C4×C53#4112yes—
55C2×Dic106—22no—
56M16×C53—82no—
57C2×C8×C53—24no—
58C2×C8×C53—82no—
59C4⋊C4×C53—22no—
60C22⋊C4×C53—24no—
61C42×C53C22×C411yescharacteristic
62C42×C53C2×D411yescharacteristic
63C42×C53C2×D412yes—
64C22⋊Dic53—41no—
65C4⋊Dic53—22no—
66C4⋊Dic53—22no—
67#11719919—42no—
68C4×Dic53—41no—
69C2×#88322—22no—
70C2×#88322—42no—
71C2×#88322—42no—
72C23×C53—22no—
73C23×C53#12311yescharacteristic
74Q8×C53—42no—
75D4×C53—42no—
76D4×C53—82no—
77D4×C53—82no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—42no—
81C2×C4×C53—42no—
82C2×C4×C53—42no—
83C2×C4×C53C2×C22⋊C411yescharacteristic
84C2×C4×C53C4×D412yes—
85C2×C4×C53#12011yescharacteristic
86C2×C4×C53#12112yes—
87C2×Dic53—42no—
88Dic106—44no—
89C8×C53—44no—
90#88322—42no—
91#88322—82no—
92#9005—531nomaximal, Sylow
93C22×C53—22no—
94C22×C53—21no—
95C22×C53—44no—
96C22×C53—42no—
97C22×C53#38611yescharacteristic
98C4×C53—22no—
99C4×C53—22no—
100C4×C53—21no—
101C4×C53—42no—
102C4×C53#45512yes—
103Dic53—82no—
104#3135—531no—
105#2980D5311yescharacteristic
106#1543—531no—
107#1497—532no—
108#1302—532no—
109C2×C53—41no—
110C2×C53—82no—
111C2×C53#188112yes—
112C2×C53#188511yescharacteristic
113#533—531no—
114#532C2×D5311yescharacteristic
115#491—1061no—
116#490—1061no—
117#489—534no—
118#435—24no—
119#432C2×D5312yes—
120#431—532no—
121#416—1062no—
122#408—22no—
123C2×#97—531no—
124#333—1061no—
125#331—1061no—
126Q8⋊C8—1064no—
127C53#900511yescharacteristic, complemented, Sylow
128C2×#41C22⋊D5312yes—
129C2×M16—22no—
130C22×C8—22no—
131C22×C8—1061no—
132C4⋊Q8—532no—
133C4⋊Q8D21211yescharacteristic
134#127D21211yescharacteristic
135#125—1061no—
136#124C4×D5312yes—
137C22⋊Q8—1062no—
138#121—22no—
139C4×Q8—1062no—
140C4×D4—22no—
141#117—1061no—
142C2×C42C22×D5311yescharacteristic
143C4⋊C8—1062no—
144C4⋊C8C22⋊D5314yes—
145#103—1064no—
146#102—1064no—
147C22⋊C8—24no—
148C22⋊C8—22no—
149C22⋊C8—1062no—
150#97—22no—
151#97—532no—
152#97—1061no—
153C4×C8—1062no—
154#41—42no—
155#41—42no—
156C2×Q8—1062no—
157C2×Q8#1171992112yes—
158C2×D4—22no—
159C2×D4—22no—
160C2×D4#1171992112yes—
161C22×C4—22no—
162C22×C4#1171994512yes—
163C22×C4C2×C22⋊D5311yescharacteristic
164M16—82no—
165C2×C8—24no—
166C2×C8—82no—
167C2×C8—1062no—
168C2×C8—2122no—
169C2×C8—2122no—
170C4⋊C4—22no—
171C4⋊C4—1062no—
172C4⋊C4—1062no—
173C4⋊C4—2122no—
174C22⋊C4—24no—
175C22⋊C4—2121no—
176C42—2121no—
177C42C2×C4×D5311yescharacteristic
178C42C2×D21211yescharacteristic
179C42C2×C22⋊D5312yes—
180C23—22no—
181C23#1313320011yescharacteristic
182Q8—42no—
183Q8—2124no—
184D4—42no—
185D4—82no—
186D4—82no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—2122no—
193C2×C4C2×#1171992111yescharacteristic
194C2×C4C4×C22⋊D5312yes—
195C2×C4#1313318611yescharacteristic
196C2×C4#1313319112yes—
197C8—44no—
198C8—4242no—
199C8—2122no—
200C22—22no—
201C22—21no—
202C22—44no—
203C22—42no—
204C22#1681050211yescharacteristic
205C4—22no—
206C4—22no—
207C4—21no—
208C4—42no—
209C4—4242no—
210C4#1681080112yes—
211C2—41no—
212C2—82no—
213C2#2027507712yes—
214C2#2027510311yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2980C2
socleC22×C53#386
prime radicalC53×#532C22
cosocleC2×C42×C53C23
hypercenter#2980D53
hypocenterC53#9005
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2980

Complements

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

NH ≅ G/Ntype
3. #20273187C2semidirect
4. #20272613C2semidirect
127. C53#9005semidirect

Maximal subgroups

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

subgroupindex
#202850602
#202733772
#202731872
#202726132
C53×#29802
#900553