Finite Group Databasebeta

← #27303779

Subgroups — #27303779

Order 13568.

384 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 81 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#20274929C212yesmaximal
2#20274929C211yesmaximal, characteristic
3#20272842C211yesmaximal, characteristic
4#20272842C212yesmaximal
5C53×#2809C211yesmaximal, characteristic
6#16811101C2211yescharacteristic
7#16810663—22no—
8#16810663—21no—
9C2×#13133136C2212yes—
10C2×#13133136C2211yescharacteristic
11#16810383—22no—
12#16810383—21no—
13C53×#400—21no—
14C53×#398—22no—
15C53×#391C2211yescharacteristic
16C53×#391C2212yes—
17C53×#389—21no—
18C53×#387—22no—
19C2×#11719917—22no—
20C2×#11719917—21no—
21Q8×Dic53—41no—
22Q8×Dic53—41no—
23#13133176—22no—
24#13133176—21no—
25#13133150—42no—
26#13133150—41no—
27#13133148—42no—
28#13133148—41no—
29#13133136—22no—
30#13133136—21no—
31#13133136Q814yes—
32#13133136Q812yes—
33C22×Q8×C53C2311yescharacteristic
34C2×C4⋊C4×C53—22no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53D412yes—
39C2×C4⋊C4×C53D411yescharacteristic
40C2×C42×C53D412yes—
41C2×C42×C53D411yescharacteristic
42#95×C53—21no—
43#95×C53—22no—
44#95×C53—22no—
45#95×C53—22no—
46#95×C53—22no—
47C2×Q8×C53—41no—
48C2×Q8×C53—41no—
49C2×Q8×C53C2×Q812yes—
50C2×Q8×C53C2×Q811yescharacteristic
51C2×Q8×C53#4111yescharacteristic
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—22no—
55C22×C4×C53—22no—
56C22×C4×C53C2×D412yes—
57C22×C4×C53C2×D411yescharacteristic
58C4⋊C4×C53—44no—
59C4⋊C4×C53—42no—
60C4⋊C4×C53—42no—
61C4⋊C4×C53—41no—
62C4⋊C4×C53—81no—
63C4⋊C4×C53—81no—
64C42×C53—41no—
65C42×C53—42no—
66C22⋊Dic53—41no—
67C4⋊Dic53—22no—
68C4⋊Dic53—21no—
69C4⋊Dic53—21no—
70C4⋊Dic53—22no—
71C4×Dic53—42no—
72C4×Dic53—41no—
73#11719917—24no—
74#11719917—22no—
75#11719917—42no—
76#11719917—41no—
77C2×#88322—42no—
78C2×#88322—41no—
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—42no—
92C2×C4×C53—42no—
93C2×C4×C53—42no—
94C2×C4×C53—42no—
95C2×C4×C53—41no—
96C2×C4×C53—42no—
97C2×C4×C53—42no—
98C2×C4×C53C22⋊Q812yes—
99C2×C4×C53C22⋊Q811yescharacteristic
100C2×C4×C53C4⋊Q812yes—
101C2×C4×C53C4⋊Q811yescharacteristic
102C2×Dic53—41no—
103C2×Dic53—41no—
104#88322—44no—
105#88322—42no—
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—22no—
115C4×C53—21no—
116C4×C53—42no—
117C4×C53—82no—
118C4×C53—81no—
119C4×C53—82no—
120Dic53—81no—
121Dic53—81no—
122#2809D5311yescharacteristic
123#1857—532no—
124#1857—531no—
125#1394—531no—
126#1394—532no—
127C2×C53—21no—
128C2×C53—21no—
129C2×C53#206211yescharacteristic
130C2×C53#220311yescharacteristic
131C2×C53#220411yescharacteristic
132#519—531no—
133#428—1062no—
134#428—1061no—
135C2×#101—532no—
136C2×#101—531no—
137#400—21no—
138#398—22no—
139#391C2×D5311yescharacteristic
140#391C2×D5312yes—
141#389—21no—
142#387—22no—
143#339—1062no—
144#339—1061no—
145C53#1346511yescharacteristic, complemented, Sylow
146C22×Q8C22×D5311yescharacteristic
147C2×M16—1062no—
148C2×M16—1061no—
149C4×Q8—2121no—
150C4×Q8—2121no—
151#117—1062no—
152#117—1061no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C22⋊D5312yes—
158C2×C4⋊C4C22⋊D5311yescharacteristic
159C2×C42C22⋊D5312yes—
160C2×C42C22⋊D5311yescharacteristic
161#107—2122no—
162#107—2121no—
163#106—2122no—
164#106—2121no—
165#101—534no—
166#101—532no—
167#101—1062no—
168#101—1061no—
169#95—21no—
170#95—22no—
171#95—22no—
172#95—22no—
173#95—22no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8Q8×D5312yes—
177C2×Q8Q8×D5311yescharacteristic
178C2×Q8#1171994911yescharacteristic
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4C2×C22⋊D5312yes—
184C22×C4C2×C22⋊D5311yescharacteristic
185M16—1064no—
186M16—1062no—
187M16—2121no—
188M16—2122no—
189C2×C8—2122no—
190C2×C8—2121no—
191C4⋊C4—42no—
192C4⋊C4—41no—
193C4⋊C4—44no—
194C4⋊C4—42no—
195C4⋊C4—81no—
196C4⋊C4—81no—
197C4⋊C4—1062no—
198C4⋊C4—1061no—
199C4⋊C4—1061no—
200C4⋊C4—1062no—
201C22⋊C4—2121no—
202C42—41no—
203C42—42no—
204C42—2122no—
205C42—2121no—
206C23#1313318611yescharacteristic
207Q8—42no—
208Q8—41no—
209Q8—41no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—2121no—
226C2×C4—2121no—
227C2×C4#1313319312yes—
228C2×C4#1313319311yescharacteristic
229C2×C4#1313321512yes—
230C2×C4#1313321511yescharacteristic
231C8—2124no—
232C8—2122no—
233C22—21no—
234C22—21no—
235C22#1681054211yescharacteristic
236C22#1681081011yescharacteristic
237C22#1681081311yescharacteristic
238C4—22no—
239C4—21no—
240C4—22no—
241C4—21no—
242C4—42no—
243C4—82no—
244C4—81no—
245C4—82no—
246C4—4241no—
247C4—4241no—
248C2—21no—
249C2—21no—
250C2#2027606711yescharacteristic
251C2#2027673611yescharacteristic
252C2#2027674111yescharacteristic
subgroupG/H
centerC22#16810810
commutator, cosocleC22×Q8×C53C23
FrattiniC22×Q8C22×D53
FittingC53×#2809C2
socleC22×C53B(4,𝔽2)
prime radicalC23×C53#120
hypercenter#2809D53
hypocenterC53#13465
LCSG ▷ C22×Q8×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #2809

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
#202749292
#202728422
C53×#28092
#1346553