Finite Group Databasebeta

← #27287340

Subgroups — #27287340

Order 13568.

406 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 73 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#20280385C211yesmaximal, characteristic
2#20274812C212yesmaximal
3#20274790C211yesmaximal, characteristic
4#20272569C212yesmaximal
5C53×#1306C211yesmaximal, characteristic
6C2×#13133176C2211yescharacteristic
7#16810546—22no—
8#16810520—22no—
9#16810506—22no—
10#16810486C2211yescharacteristic
11C4×C22⋊Dic53C2212yes—
12#16810389—22no—
13#16810384—22no—
14#16810384—22no—
15C53×#407C2212yes—
16C53×#381C2211yescharacteristic
17#13133176—22no—
18#13133176—22no—
19C2×C4⋊Dic53—22no—
20C2×C4⋊Dic53Q812yes—
21C2×#11719919—22no—
22C2×#11719919—22no—
23C2×C22⋊Dic53—22no—
24C2×C22⋊Dic53—21no—
25C2×C22⋊Dic53—21no—
26C2×C4×Dic53—21no—
27C2×C4×Dic53D412yes—
28C2×C4×Dic53D412yes—
29#13133124—22no—
30#13133124—22no—
31#13133124—22no—
32C23×C4×C53C2311yescharacteristic
33C2×M16×C53—22no—
34C2×C22⋊C4×C53D412yes—
35C22⋊C8×C53—22no—
36C22⋊C8×C53—22no—
37#95×C53—22no—
38C24×C53C2×D411yescharacteristic
39C22×C4×C53—22no—
40C22×C4×C53—42no—
41C22×C4×C53C2×D412yes—
42C22×C4×C53C2×Q811yescharacteristic
43C22×C4×C53#4112yes—
44C22×C4×C53#4111yescharacteristic
45C22×Dic53—22no—
46C22×Dic53—22no—
47C22×Dic53—22no—
48C22×Dic53—41no—
49M16×C53—42no—
50M16×C53—42no—
51C2×C8×C53—42no—
52C22⋊C4×C53—42no—
53C22⋊Dic53—44no—
54C22⋊Dic53—41no—
55C22⋊Dic53—41no—
56C22⋊Dic53—42no—
57C4⋊Dic53—24no—
58C4⋊Dic53—44no—
59#11719919—44no—
60#11719919—82no—
61C4×Dic53—42no—
62C4×Dic53—42no—
63C4×Dic53—42no—
64C23×C53—22no—
65C23×C53—42no—
66C23×C53#12011yescharacteristic
67C23×C53C22⋊Q811yescharacteristic
68C23×C53#12411yescharacteristic
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—41no—
72C2×C4×C53—42no—
73C2×C4×C53—41no—
74C2×C4×C53—42no—
75C2×C4×C53—42no—
76C2×C4×C53—42no—
77C2×C4×C53C22⋊Q812yes—
78C2×C4×C53#12412yes—
79C2×Dic53—24no—
80C2×Dic53—42no—
81C2×Dic53—42no—
82C2×Dic53—42no—
83C2×Dic53—42no—
84C2×Dic53—42no—
85C2×Dic53—42no—
86C2×Dic53—82no—
87C2×Dic53—82no—
88C2×Dic53—81no—
89C8×C53—82no—
90#10389—531nomaximal, Sylow
91C22×C53—22no—
92C22×C53—21no—
93C22×C53—21no—
94C22×C53—22no—
95C22×C53—22no—
96C22×C53—82no—
97C22×C53#39311yescharacteristic
98C22×C53#45812yes—
99C4×C53—22no—
100C4×C53—22no—
101C4×C53—82no—
102Dic53—44no—
103Dic53—82no—
104Dic53—82no—
105Dic53—161no—
106#2608—531no—
107#1826—532no—
108#1820—531no—
109#1306D5311yescharacteristic
110#1284—532no—
111C2×C53—22no—
112C2×C53—41no—
113C2×C53—41no—
114C2×C53#203011yescharacteristic
115C2×C53#203112yes—
116C2×#117—531no—
117#407C2×D5312yes—
118#396—1062no—
119#389—1062no—
120#387—1062no—
121#381—531no—
122#381C2×D5311yescharacteristic
123C4×C22⋊C4—532no—
124#342—1062no—
125#339—1062no—
126#339—1062no—
127C53#1038911yescharacteristic, complemented, Sylow
128C23×C4C22×D5311yescharacteristic
129C2×M16—22no—
130#117—1062no—
131#117—1062no—
132C2×C4⋊C4—532no—
133C2×C4⋊C4—1062no—
134C2×C4⋊C4—1062no—
135C2×C4⋊C4—1062no—
136C2×C22⋊C4—1061no—
137C2×C22⋊C4—1062no—
138C2×C22⋊C4—1061no—
139C2×C22⋊C4D21212yes—
140C2×C42—532no—
141C2×C42—532no—
142C2×C42—1061no—
143C22⋊C8—22no—
144C22⋊C8—22no—
145#95—22no—
146#95—1062no—
147#95—1062no—
148#95—1062no—
149C24C2×D21211yescharacteristic
150C22×C4—22no—
151C22×C4—42no—
152C22×C4—1062no—
153C22×C4—1062no—
154C22×C4—1062no—
155C22×C4—2121no—
156C22×C4#1171994512yes—
157C22×C4D4×D5312yes—
158C22×C4#1171994711yescharacteristic
159C22×C4Q8×D5311yescharacteristic
160M16—42no—
161M16—42no—
162C2×C8—42no—
163C4⋊C4—1064no—
164C4⋊C4—4242no—
165C4⋊C4—2124no—
166C4⋊C4—2124no—
167C22⋊C4—42no—
168C22⋊C4—2124no—
169C22⋊C4—2122no—
170C22⋊C4—2121no—
171C22⋊C4—2121no—
172C42—2122no—
173C42—2122no—
174C42—2122no—
175C23—22no—
176C23—42no—
177C23#1313318711yescharacteristic
178C23#1313319711yescharacteristic
179C23#1313320511yescharacteristic
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—1064no—
189C2×C4—4241no—
190C2×C4—4242no—
191C2×C4—4242no—
192C2×C4—2122no—
193C2×C4—2122no—
194C2×C4—2122no—
195C2×C4—2122no—
196C2×C4—2122no—
197C2×C4—2122no—
198C2×C4#1313319512yes—
199C2×C4#1313320312yes—
200C8—82no—
201C22—22no—
202C22—21no—
203C22—21no—
204C22—22no—
205C22—22no—
206C22—82no—
207C22#1681053611yescharacteristic
208C22#1681081412yes—
209C4—22no—
210C4—22no—
211C4—82no—
212C4—8481no—
213C4—4242no—
214C4—4242no—
215C4—2124no—
216C2—22no—
217C2—41no—
218C2—41no—
219C2#2027588311yescharacteristic
220C2#2027589012yes—
subgroupG/H
centerC22#16810536
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1306C2
socleC22×C53#393
prime radicalC24×C53C2×D4
hypercenter#1306D53
hypocenterC53#10389
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1306

Complements

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

NH ≅ G/Ntype
127. C53#10389semidirect

Maximal subgroups

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

subgroupindex
#202803852
#202748122
#202747902
#202725692
C53×#13062
#1038953