Finite Group Databasebeta

← #27287339

Subgroups — #27287339

Order 13568.

406 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 75 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#20274814C212yesmaximal
3#20272623C211yesmaximal, characteristic
4#20272567C212yesmaximal
5C53×#1820C211yesmaximal, characteristic
6#16810587C2212yes—
7#16810546—22no—
8#16810520—22no—
9#16810506—22no—
10C4×C22⋊Dic53C2212yes—
11#16810382—22no—
12#16810382—22no—
13C2×#117×C53C2211yescharacteristic
14C53×#381C2211yescharacteristic
15C53×#381C2211yescharacteristic
16C53×#342—22no—
17C2×#11719917—22no—
18C2×C4⋊Dic53—22no—
19C2×#11719919—22no—
20C2×#11719919—22no—
21C2×C22⋊Dic53—22no—
22C2×C4×Dic53D412yes—
23C2×C4×Dic53D412yes—
24C22⋊#88322—22no—
25C22⋊#88322—22no—
26#13133124—22no—
27#13133124—22no—
28C23×C4×C53C2311yescharacteristic
29#117×C53—22no—
30#117×C53—22no—
31C2×C4⋊C4×C53Q812yes—
32C2×C22⋊C4×C53—21no—
33C2×C22⋊C4×C53—21no—
34C2×C22⋊C4×C53D412yes—
35C2×C42×C53—21no—
36#95×C53—22no—
37#95×C53—22no—
38C24×C53C2×D411yescharacteristic
39C22×C4×C53—22no—
40C22×C4×C53—22no—
41C22×C4×C53—42no—
42C22×C4×C53—41no—
43C22×C4×C53C2×D412yes—
44C22×C4×C53C2×Q811yescharacteristic
45C22×C4×C53#4112yes—
46C22×C4×C53#4111yescharacteristic
47C22×Dic53—22no—
48C22×Dic53—22no—
49C4⋊C4×C53—24no—
50C22⋊C4×C53—41no—
51C22⋊C4×C53—41no—
52C22⋊C4×C53—42no—
53C22⋊C4×C53—42no—
54C42×C53—42no—
55C22⋊Dic53—44no—
56C4⋊Dic53—44no—
57#11719919—44no—
58#11719919—82no—
59C4×Dic53—42no—
60C4×Dic53—42no—
61#11719917—42no—
62#11719917—42no—
63C2×#88322—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—41no—
73C2×C4×C53—42no—
74C2×C4×C53—42no—
75C2×C4×C53—42no—
76C2×C4×C53—42no—
77C2×C4×C53—42no—
78C2×C4×C53—82no—
79C2×C4×C53—82no—
80C2×C4×C53—81no—
81C2×C4×C53C22⋊Q812yes—
82C2×C4×C53#12412yes—
83C2×Dic53—24no—
84C2×Dic53—42no—
85C2×Dic53—42no—
86C2×Dic53—42no—
87C2×Dic53—42no—
88C2×Dic53—42no—
89#88322—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—
102C4×C53—82no—
103C4×C53—161no—
104Dic53—44no—
105Dic53—82no—
106#2608—531no—
107#1826—532no—
108#1820D5311yescharacteristic
109#1306—531no—
110#1284—532no—
111C2×C53—22no—
112C2×C53—41no—
113C2×C53—41no—
114C2×C53#203011yescharacteristic
115C2×C53#203112yes—
116C2×#117C2×D5311yescharacteristic
117#407—532no—
118#396—1062no—
119#389—1062no—
120#387—1062no—
121#381C2×D5311yescharacteristic
122#381C2×D5311yescharacteristic
123C4×C22⋊C4—532no—
124#342—22no—
125#339—1062no—
126#339—1062no—
127C53#1038911yescharacteristic, complemented, Sylow
128C23×C4C22×D5311yescharacteristic
129C2×M16—1062no—
130#117—22no—
131#117—22no—
132C2×C4⋊C4—1062no—
133C2×C4⋊C4—1062no—
134C2×C4⋊C4—1062no—
135C2×C4⋊C4Dic10612yes—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—1062no—
139C2×C22⋊C4C22⋊D5312yes—
140C2×C42—21no—
141C2×C42—532no—
142C2×C42—532no—
143C22⋊C8—1062no—
144C22⋊C8—1062no—
145#95—22no—
146#95—22no—
147#95—1062no—
148#95—1062no—
149C24C2×C22⋊D5311yescharacteristic
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—42no—
153C22×C4—41no—
154C22×C4—1062no—
155C22×C4—1062no—
156C22×C4C2×Dic10611yescharacteristic
157C22×C4#1171994511yescharacteristic
158C22×C4D4×D5312yes—
159C22×C4#1171994712yes—
160M16—2122no—
161M16—2122no—
162C2×C8—2122no—
163C4⋊C4—24no—
164C4⋊C4—4242no—
165C4⋊C4—2124no—
166C4⋊C4—2124no—
167C22⋊C4—41no—
168C22⋊C4—41no—
169C22⋊C4—42no—
170C22⋊C4—42no—
171C22⋊C4—2124no—
172C42—42no—
173C42—2122no—
174C42—2122no—
175C23—22no—
176C23—42no—
177C23#1313318511yescharacteristic
178C23#1313319611yescharacteristic
179C23#1313320411yescharacteristic
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—82no—
190C2×C4—81no—
191C2×C4—82no—
192C2×C4—1064no—
193C2×C4—2122no—
194C2×C4—2122no—
195C2×C4—2122no—
196C2×C4—2122no—
197C2×C4—2122no—
198C2×C4#1313319412yes—
199C2×C4#1313320312yes—
200C8—4242no—
201C22—22no—
202C22—21no—
203C22—21no—
204C22—22no—
205C22—22no—
206C22—82no—
207C22#1681053511yescharacteristic
208C22#1681081212yes—
209C4—22no—
210C4—22no—
211C4—82no—
212C4—82no—
213C4—161no—
214C4—4242no—
215C4—2124no—
216C2—22no—
217C2—41no—
218C2—41no—
219C2#2027588211yescharacteristic
220C2#2027588912yes—
subgroupG/H
centerC22#16810535
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1820C2
socleC22×C53#393
prime radicalC24×C53C2×D4
hypercenter#1820D53
hypocenterC53#10389
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1820

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
#202748142
#202726232
#202725672
C53×#18202
#1038953