Finite Group Databasebeta

← #27279406

Subgroups — #27279406

Order 13568.

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

normal subgroups only

This group has 404 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#20285225C211yesmaximal, characteristic
2#20283817C211yesmaximal, characteristic
3#20273137C212yesmaximal
4#20272553C212yesmaximal
5C53×#1526C211yesmaximal, characteristic
6#16811180C2211yescharacteristic
7#16811026C412yes—
8#16811022—22no—
9#16811019C412yes—
10#16810754—22no—
11Q8×#88322—22no—
12#16810684—22no—
13#16810676C2212yes—
14#16810659—22no—
15C2×C4×#88322C2211yescharacteristic
16#16810355—22no—
17#16810355—22no—
18C53×#533C2211yescharacteristic
19C4×M16×C53C2212yes—
20C53×#330—22no—
21C22×#88322—21no—
22#13133217C2×C412yes—
23#13133209—21no—
24#13133196—22no—
25C4×Dic106—22no—
26#13133176—21no—
27#13133150—24no—
28#13133148—24no—
29C4⋊#88322—24no—
30C4⋊#88322D412yes—
31C4⋊#88322D412yes—
32C22⋊#88322—22no—
33#13133128—22no—
34C4×#88322—22no—
35C4×#88322—22no—
36C4×#88322—21no—
37C4×#88322C2×C411yescharacteristic
38C4×#88322C2×C411yescharacteristic
39C2×M16×C53—22no—
40C4⋊Q8×C53C2×C412yes—
41#125×C53D412yes—
42C22⋊Q8×C53—22no—
43C4×Q8×C53—22no—
44#117×C53D412yes—
45C2×C42×C53C2311yescharacteristic
46#97×C53—22no—
47C4×C8×C53—22no—
48C2×Q8×C53—22no—
49C22×C4×C53C2×D411yescharacteristic
50C22×C4×C53#4112yes—
51C2×Dic106—22no—
52M16×C53—24no—
53M16×C53—42no—
54C2×C8×C53—42no—
55C4⋊C4×C53—22no—
56C4⋊C4×C53—24no—
57C4⋊C4×C53C22⋊C414yes—
58C22⋊C4×C53—22no—
59C42×C53—22no—
60C42×C53C22×C411yescharacteristic
61C42×C53C2×D411yescharacteristic
62C42×C53C2×D412yes—
63C22⋊Dic53—41no—
64C4⋊Dic53—22no—
65C4⋊Dic53—22no—
66#11719919—42no—
67C4×Dic53—41no—
68C2×#88322—24no—
69C2×#88322—21no—
70C2×#88322—21no—
71C2×#88322—22no—
72C2×#88322—22no—
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—44no—
88#88322—22no—
89#88322—22no—
90#88322—42no—
91#88322—42no—
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#2986—531no—
105#1526D5311yescharacteristic
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—1062no—
117C8×Q8—1062no—
118#433—1062no—
119#431—532no—
120#427—1062no—
121C4×M16C2×D5312yes—
122C2×C4×C8—531no—
123#330—22no—
124Q8⋊C8—1062no—
125Q8⋊C8—1062no—
126C53#906811yescharacteristic, complemented, Sylow
127C2×M16—22no—
128C22×C8—1061no—
129C4⋊Q8—532no—
130C4⋊Q8C4×D5312yes—
131#125—1061no—
132#125C22⋊D5312yes—
133C22⋊Q8—22no—
134C22⋊Q8—1062no—
135C4×Q8—22no—
136C4×Q8—1062no—
137#117—1061no—
138#117D21212yes—
139C2×C42C22×D5311yescharacteristic
140#107—1064no—
141#106—1064no—
142C4⋊C8—532no—
143C4⋊C8—532no—
144C4⋊C8—1064no—
145C22⋊C8—1062no—
146#97—22no—
147#97—1062no—
148C4×C8—22no—
149C4×C8—531no—
150C4×C8—531no—
151C4×C8—1062no—
152C4×C8—1062no—
153C4×C8—1061no—
154C2×Q8—22no—
155C2×Q8—1062no—
156C22×C4C2×D21211yescharacteristic
157C22×C4#1171994712yes—
158M16—24no—
159M16—42no—
160C2×C8—42no—
161C2×C8—1064no—
162C2×C8—1061no—
163C2×C8—1061no—
164C2×C8—1062no—
165C2×C8—1062no—
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
177C42D4×D5312yes—
178C42C2×C22⋊D5311yescharacteristic
179C23#1313320111yescharacteristic
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×C4#1313317712yes—
191C2×C4#1313318711yescharacteristic
192C2×C4#1313318912yes—
193C8—44no—
194C8—1062no—
195C8—1062no—
196C8—2122no—
197C8—2122no—
198C22—22no—
199C22—21no—
200C22#1681050011yescharacteristic
201C4—22no—
202C4—21no—
203C4—44no—
204C4—4242no—
205C4#1681073712yes—
206C4#1681074212yes—
207C4#1681080812yes—
208C2—41no—
209C2#2027504012yes—
210C2#2027510611yescharacteristic
subgroupG/H
centerC2×C4#13133187
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1526C2
socleC22×C53#386
prime radicalC23×C53#123
cosocleC2×C42×C53C23
hypercenter#1526D53
hypocenterC53#9068
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1526

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
#202838172
#202731372
#202725532
C53×#15262
#906853