Finite Group Databasebeta

← #27279408

Subgroups — #27279408

Order 13568.

404 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 107 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#20285225C211yesmaximal, characteristic
2#20273310C211yesmaximal, characteristic
3#20273135C212yesmaximal
4#20272551C212yesmaximal
5C53×#2986C211yesmaximal, characteristic
6#16811180C2211yescharacteristic
7#16811026C412yes—
8#16811022—22no—
9#16811019C412yes—
10#16810659—22no—
11C4×#11719917C2212yes—
12C2×C4×#88322C2211yescharacteristic
13#16810365—22no—
14#16810353—22no—
15#16810353—22no—
16C53×#533C2211yescharacteristic
17C53×#446—22no—
18C8×Q8×C53—22no—
19C53×#433—22no—
20C53×#431C2212yes—
21C2×#11719917—22no—
22C22×#88322—21no—
23#13133217C2×C412yes—
24#13133209—21no—
25#13133196—22no—
26C4×Dic106—22no—
27#13133176—21no—
28#13133150—24no—
29#13133148—24no—
30#13133128—22no—
31C4×#88322—21no—
32C4×#88322—22no—
33C4×#88322C2×C411yescharacteristic
34C4×#88322C2×C411yescharacteristic
35C4⋊Q8×C53C2×C412yes—
36#125×C53D412yes—
37C22⋊Q8×C53—22no—
38C4×Q8×C53—22no—
39#117×C53D412yes—
40C2×C42×C53C2311yescharacteristic
41C4⋊C8×C53—24no—
42C4⋊C8×C53D412yes—
43C4⋊C8×C53D412yes—
44C22⋊C8×C53—22no—
45#97×C53—22no—
46C4×C8×C53—22no—
47C4×C8×C53—22no—
48C2×Q8×C53—22no—
49C22×C4×C53C2×D411yescharacteristic
50C22×C4×C53#4112yes—
51C2×Dic106—22no—
52C2×C8×C53—22no—
53C2×C8×C53—22no—
54C4⋊C4×C53—22no—
55C4⋊C4×C53—24no—
56C4⋊C4×C53C22⋊C414yes—
57C22⋊C4×C53—22no—
58C42×C53—22no—
59C42×C53C22×C411yescharacteristic
60C42×C53C2×D411yescharacteristic
61C42×C53C2×D412yes—
62C22⋊Dic53—41no—
63C4⋊Dic53—22no—
64C4⋊Dic53—22no—
65#11719919—42no—
66C4×Dic53—41no—
67#11719917—24no—
68#11719917—42no—
69C2×#88322—24no—
70C2×#88322—21no—
71C2×#88322—21no—
72C2×#88322—42no—
73C23×C53#12311yescharacteristic
74Q8×C53—42no—
75Q8×C53—42no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—24no—
80C2×C4×C53C2×C22⋊C411yescharacteristic
81C2×C4×C53C4×D412yes—
82C2×C4×C53#12011yescharacteristic
83C2×C4×C53#12112yes—
84C2×Dic53—42no—
85Dic106—42no—
86Dic106—42no—
87C8×C53—42no—
88C8×C53—42no—
89#88322—22no—
90#88322—22no—
91#88322—44no—
92#9068—531nomaximal, Sylow
93C22×C53—22no—
94C22×C53—21no—
95C22×C53#38611yescharacteristic
96C4×C53—22no—
97C4×C53—21no—
98C4×C53—44no—
99C4×C53#44312yes—
100C4×C53#44412yes—
101C4×C53#45612yes—
102Dic53—82no—
103#3160—531no—
104#2986D5311yescharacteristic
105#1526—531no—
106#1485—532no—
107#1278—532no—
108C2×C53—41no—
109C2×C53#187612yes—
110C2×C53#188611yescharacteristic
111#533—531no—
112#533C2×D5311yescharacteristic
113#500—532no—
114#499—1062no—
115#498—532no—
116#446—22no—
117C8×Q8—22no—
118#433—22no—
119#431C2×D5312yes—
120#427—1062no—
121C4×M16—532no—
122C2×C4×C8—531no—
123#330—1062no—
124Q8⋊C8—1062no—
125Q8⋊C8—1062no—
126C53#906811yescharacteristic, complemented, Sylow
127C2×M16—1062no—
128C22×C8—1061no—
129C4⋊Q8—532no—
130C4⋊Q8C4×D5312yes—
131#125—1061no—
132#125D21212yes—
133C22⋊Q8—22no—
134C22⋊Q8—1062no—
135C4×Q8—22no—
136C4×Q8—1062no—
137#117—1061no—
138#117C22⋊D5312yes—
139C2×C42C22×D5311yescharacteristic
140#107—1064no—
141#106—1064no—
142C4⋊C8—24no—
143C4⋊C8C22⋊D5312yes—
144C4⋊C8C22⋊D5312yes—
145C22⋊C8—22no—
146#97—22no—
147#97—1062no—
148C4×C8—22no—
149C4×C8—22no—
150C4×C8—531no—
151C4×C8—531no—
152C4×C8—1061no—
153C4×C8—1062no—
154C2×Q8—22no—
155C2×Q8—1062no—
156C22×C4#1171994512yes—
157C22×C4C2×C22⋊D5311yescharacteristic
158M16—1064no—
159M16—2122no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—1064no—
163C2×C8—1061no—
164C2×C8—1061no—
165C2×C8—2122no—
166C4⋊C4—22no—
167C4⋊C4—24no—
168C4⋊C4—1062no—
169C4⋊C4—1062no—
170C4⋊C4—2122no—
171C4⋊C4#1171992114yes—
172C22⋊C4—22no—
173C22⋊C4—2121no—
174C42—22no—
175C42—2121no—
176C42C2×C4×D5311yescharacteristic
177C42C2×D21211yescharacteristic
178C42C2×C22⋊D5312yes—
179C23#1313320011yescharacteristic
180Q8—42no—
181Q8—42no—
182Q8—2122no—
183Q8—2122no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—2122no—
189C2×C4C2×#1171992111yescharacteristic
190C2×C4C4×C22⋊D5312yes—
191C2×C4#1313318611yescharacteristic
192C2×C4#1313319112yes—
193C8—42no—
194C8—42no—
195C8—1062no—
196C8—1062no—
197C8—2124no—
198C22—22no—
199C22—21no—
200C22#1681050211yescharacteristic
201C4—22no—
202C4—21no—
203C4—44no—
204C4—4242no—
205C4#1681073812yes—
206C4#1681074312yes—
207C4#1681080612yes—
208C2—41no—
209C2#2027504212yes—
210C2#2027510811yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2986C2
socleC22×C53#386
prime radicalC23×C53#123
cosocleC2×C42×C53C23
hypercenter#2986D53
hypocenterC53#9068
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2986

Complements

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

NH ≅ G/Ntype
126. C53#9068semidirect

Maximal subgroups

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

subgroupindex
#202852252
#202733102
#202731352
#202725512
C53×#29862
#906853