Finite Group Databasebeta

← #27303782

Subgroups — #27303782

Order 13568.

384 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 79 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#20282703C211yesmaximal, characteristic
2#20274930C212yesmaximal
3#20272842C211yesmaximal, characteristic
4#20272841C212yesmaximal
5C53×#1857C211yesmaximal, characteristic
6#16810666—22no—
7C2×#13133136C2212yes—
8#16810562—21no—
9#16810555—22no—
10#16810530C2212yes—
11#16810521—21no—
12#16810510—22no—
13#16810382—22no—
14C53×#519C2211yescharacteristic
15C53×#428—21no—
16C2×#101×C53C2211yescharacteristic
17C53×#391C2211yescharacteristic
18C53×#339—21no—
19C2×#11719917—22no—
20C2×C4⋊Dic53—21no—
21C2×C4⋊Dic53—21no—
22C2×C4⋊Dic53—21no—
23C2×C4⋊Dic53D412yes—
24C2×#11719919—22no—
25C2×C4×Dic53D412yes—
26#13133150—42no—
27#13133148—42no—
28#13133136—22no—
29#13133136Q814yes—
30#13133124—21no—
31#13133124—22no—
32#13133124—22no—
33#13133124—22no—
34C22×Q8×C53C2311yescharacteristic
35C2×M16×C53—21no—
36C4×Q8×C53—41no—
37C4×Q8×C53—41no—
38#117×C53—21no—
39#117×C53—22no—
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C42×C53D411yescharacteristic
42#107×C53—41no—
43#106×C53—41no—
44#101×C53—21no—
45#101×C53Q812yes—
46#95×C53—22no—
47C2×Q8×C53—41no—
48C2×Q8×C53—41no—
49C2×Q8×C53C2×Q811yescharacteristic
50C2×Q8×C53C2×Q812yes—
51C2×Q8×C53#4111yescharacteristic
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53C2×D412yes—
55C22×C4×C53C2×D411yescharacteristic
56C22×Dic53—22no—
57C22×Dic53—22no—
58M16×C53—22no—
59M16×C53—41no—
60C2×C8×C53—41no—
61C4⋊C4×C53—21no—
62C4⋊C4×C53—21no—
63C4⋊C4×C53—22no—
64C4⋊C4×C53—22no—
65C4⋊C4×C53—41no—
66C22⋊C4×C53—41no—
67C42×C53—42no—
68C42×C53—41no—
69C42×C53—41no—
70C4⋊Dic53—42no—
71C4⋊Dic53—42no—
72C4⋊Dic53—81no—
73C4⋊Dic53—81no—
74#11719919—44no—
75C4×Dic53—42no—
76#11719917—24no—
77#11719917—42no—
78C2×#88322—42no—
79C23×C53#12011yescharacteristic
80Q8×C53—42no—
81Q8×C53—41no—
82Q8×C53—41no—
83C2×C4×C53—22no—
84C2×C4×C53—21no—
85C2×C4×C53—22no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53—42no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53C22⋊Q812yes—
95C2×C4×C53C22⋊Q811yescharacteristic
96C2×C4×C53C4⋊Q811yescharacteristic
97C2×C4×C53C4⋊Q812yes—
98C2×Dic53—42no—
99C2×Dic53—42no—
100C2×Dic53—42no—
101C2×Dic53—42no—
102C2×Dic53—42no—
103C2×Dic53—42no—
104C8×C53—42no—
105#88322—44no—
106#13465—531nomaximal, Sylow
107C22×C53—21no—
108C22×C53—21no—
109C22×C53#39511yescharacteristic
110C22×C53B(4,𝔽2)11yescharacteristic
111C22×C53#45811yescharacteristic
112C4×C53—22no—
113C4×C53—21no—
114C4×C53—21no—
115C4×C53—22no—
116C4×C53—42no—
117C4×C53—81no—
118C4×C53—81no—
119C4×C53—81no—
120Dic53—82no—
121Dic53—82no—
122#2809—531no—
123#1857—532no—
124#1857D5311yescharacteristic
125#1394—532no—
126#1394—531no—
127C2×C53—21no—
128C2×C53—21no—
129C2×C53#206211yescharacteristic
130C2×C53#220311yescharacteristic
131C2×C53#220411yescharacteristic
132#519C2×D5311yescharacteristic
133#428—21no—
134#428—1062no—
135C2×#101—532no—
136C2×#101C2×D5311yescharacteristic
137#400—1061no—
138#398—1062no—
139#391—532no—
140#391C2×D5311yescharacteristic
141#389—1061no—
142#387—1062no—
143#339—21no—
144#339—1062no—
145C53#1346511yescharacteristic, complemented, Sylow
146C22×Q8C22×D5311yescharacteristic
147C2×M16—21no—
148C2×M16—1062no—
149C4×Q8—41no—
150C4×Q8—41no—
151#117—22no—
152#117—21no—
153C2×C4⋊C4—532no—
154C2×C4⋊C4—1061no—
155C2×C4⋊C4—1061no—
156C2×C4⋊C4—1062no—
157C2×C4⋊C4—1061no—
158C2×C4⋊C4C22⋊D5311yescharacteristic
159C2×C42—532no—
160C2×C42C22⋊D5311yescharacteristic
161#107—41no—
162#107—2122no—
163#106—41no—
164#106—2122no—
165#101—21no—
166#101—534no—
167#101—1062no—
168#101Dic10612yes—
169#95—22no—
170#95—1061no—
171#95—1062no—
172#95—1062no—
173#95—1062no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8C2×Dic10611yescharacteristic
177C2×Q8#1171994511yescharacteristic
178C2×Q8Q8×D5312yes—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—1062no—
182C22×C4—1062no—
183C22×C4D4×D5312yes—
184C22×C4C2×C22⋊D5311yescharacteristic
185M16—22no—
186M16—41no—
187M16—1064no—
188M16—2122no—
189C2×C8—41no—
190C2×C8—2122no—
191C4⋊C4—21no—
192C4⋊C4—22no—
193C4⋊C4—21no—
194C4⋊C4—22no—
195C4⋊C4—41no—
196C4⋊C4—4241no—
197C4⋊C4—4241no—
198C4⋊C4—2122no—
199C4⋊C4—2124no—
200C4⋊C4—2122no—
201C22⋊C4—41no—
202C42—41no—
203C42—41no—
204C42—42no—
205C42—2122no—
206C23#1313318511yescharacteristic
207Q8—41no—
208Q8—41no—
209Q8—42no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—2122no—
222C2×C4—2122no—
223C2×C4—2122no—
224C2×C4—2122no—
225C2×C4—2122no—
226C2×C4—2122no—
227C2×C4#1313319512yes—
228C2×C4#1313319611yescharacteristic
229C2×C4#1313321511yescharacteristic
230C2×C4#1313321612yes—
231C8—42no—
232C8—2124no—
233C22—21no—
234C22—21no—
235C22#1681054311yescharacteristic
236C22#1681080911yescharacteristic
237C22#1681081211yescharacteristic
238C4—21no—
239C4—22no—
240C4—21no—
241C4—22no—
242C4—42no—
243C4—81no—
244C4—81no—
245C4—81no—
246C4—4242no—
247C4—4242no—
248C2—21no—
249C2—21no—
250C2#2027606811yescharacteristic
251C2#2027673511yescharacteristic
252C2#2027674011yescharacteristic
subgroupG/H
centerC22#16810809
commutator, cosocleC22×Q8×C53C23
FrattiniC22×Q8C22×D53
FittingC53×#1857C2
socleC22×C53B(4,𝔽2)
prime radicalC23×C53#120
hypercenter#1857D53
hypocenterC53#13465
LCSG ▷ C22×Q8×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #1857

Complements

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

NH ≅ G/Ntype
145. C53#13465semidirect

Maximal subgroups

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

subgroupindex
#202827032
#202749302
#202728422
#202728412
C53×#18572
#1346553