Finite Group Databasebeta

← #27285053

Subgroups — #27285053

Order 13568.

468 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 83 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#20286060C211yesmaximal, characteristic
2#20284845C212yesmaximal
3#20273329C212yesmaximal
4#20272585C211yesmaximal, characteristic
5C53×#1985C211yesmaximal, characteristic
6#16811180C2212yes—
7#16810953—22no—
8#16810919—22no—
9#16810908—22no—
10#16810864—22no—
11#16810834—22no—
12#16810833—22no—
13#16810779—22no—
14#16810728—24no—
15#16810680C2212yes—
16#16810364—24no—
17C2×C4×Q8×C53C2211yescharacteristic
18C2×C4⋊C8×C53C2211yescharacteristic
19C2×#97×C53C2211yescharacteristic
20C53×#336—22no—
21C2×Dic212—22no—
22C2×#11719925—24no—
23#13133217D412yes—
24#13133217D412yes—
25#13133209—22no—
26#13133196—22no—
27#13133196—22no—
28C4×Dic106—22no—
29C4×Dic106—22no—
30#13133176—22no—
31#13133151—22no—
32#13133148—24no—
33C4⋊#88322—22no—
34#13133142—24no—
35#13133141—24no—
36#13133140—24no—
37C22⋊#88322—22no—
38#13133128D414yes—
39C22×Q8×C53D411yescharacteristic
40C22×C8×C53—21no—
41C22×C8×C53D412yes—
42C4×Q8×C53—22no—
43C4×Q8×C53—22no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C42×C53—21no—
47C2×C42×C53C2311yescharacteristic
48C4⋊C8×C53—22no—
49#97×C53—22no—
50C2×Q8×C53—22no—
51C2×Q8×C53—42no—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53C2×D411yescharacteristic
55C22×C4×C53C2×D411yescharacteristic
56C22×C4×C53#4111yescharacteristic
57C2×Dic106—22no—
58C2×Dic106—22no—
59C2×C8×C53—22no—
60C2×C8×C53—22no—
61C2×C8×C53—41no—
62C2×C8×C53—41no—
63C2×C8×C53—42no—
64C4⋊C4×C53—22no—
65C4⋊C4×C53—42no—
66C42×C53—41no—
67C42×C53—41no—
68C42×C53C2×D412yes—
69C42×C53C2×D412yes—
70C22⋊Dic53—42no—
71#11719925—48no—
72C4⋊Dic53—22no—
73C4⋊Dic53—22no—
74C4⋊Dic53—22no—
75C4⋊Dic53—22no—
76#11719919—42no—
77#11719919—42no—
78C4×Dic53—42no—
79C2×#88322—24no—
80Dic212—82no—
81C23×C53#12111yescharacteristic
82Q8×C53—24no—
83Q8×C53—42no—
84C2×C4×C53—22no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53—42no—
93C2×C4×C53#12011yescharacteristic
94C2×C4×C53#12012yes—
95C2×C4×C53#12112yes—
96C2×C4×C53#12711yescharacteristic
97C2×Dic53—42no—
98C2×Dic53—42no—
99Dic106—44no—
100Dic106—42no—
101Dic106—42no—
102C8×C53—42no—
103C8×C53—42no—
104#88322—44no—
105#10031—531nomaximal, Sylow
106C22×C53—21no—
107C22×C53—21no—
108C22×C53#39211yescharacteristic
109C22×C53#47312yes—
110C4×C53—21no—
111C4×C53—22no—
112C4×C53—22no—
113C4×C53—42no—
114C4×C53—81no—
115C4×C53#45511yescharacteristic
116C4×C53#45611yescharacteristic
117Dic53—82no—
118Dic53—82no—
119#3249—531no—
120#3113—532no—
121#1985D5311yescharacteristic
122#1531—532no—
123#1290—531no—
124C2×C53—22no—
125C2×C53#200711yescharacteristic
126C2×C53#201011yescharacteristic
127C2×C53#202011yescharacteristic
128#533—532no—
129C2×C4×Q8C2×D5311yescharacteristic
130#483—1062no—
131#477—1062no—
132#475—1062no—
133#467—1062no—
134#462—1062no—
135#462—1062no—
136#451—1062no—
137#441—1064no—
138#432—532no—
139C2×C4⋊C8C2×D5311yescharacteristic
140C2×#97C2×D5311yescharacteristic
141#336—22no—
142#330—1064no—
143C53#1003111yescharacteristic, complemented, Sylow
144C22×Q8D21211yescharacteristic
145C2×Q16—1062no—
146C2×Q16—1064no—
147C22×C8—21no—
148C22×C8C22⋊D5312yes—
149C4⋊Q8—532no—
150C4⋊Q8—532no—
151#125—1062no—
152C22⋊Q8—1062no—
153C22⋊Q8—1062no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—1062no—
157C4×Q8—1062no—
158#117—1062no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4D21211yescharacteristic
161C2×C42—21no—
162C2×C42C22×D5311yescharacteristic
163#107—1062no—
164#106—1064no—
165C4⋊C8—22no—
166C4⋊C8—1062no—
167#103—1064no—
168#103—1064no—
169#103—1064no—
170C22⋊C8—1062no—
171#97—22no—
172#97—534no—
173C2×Q8—22no—
174C2×Q8—42no—
175C2×Q8—1062no—
176C2×Q8—1062no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4C2×D21211yescharacteristic
180C22×C4#1171994511yescharacteristic
181C22×C4C2×C22⋊D5311yescharacteristic
182Q16—4242no—
183Q16—2128no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—42no—
189C2×C8—1064no—
190C4⋊C4—22no—
191C4⋊C4—42no—
192C4⋊C4—1062no—
193C4⋊C4—1062no—
194C4⋊C4—1062no—
195C4⋊C4—1062no—
196C4⋊C4—2122no—
197C4⋊C4—2122no—
198C22⋊C4—2122no—
199C42—41no—
200C42—41no—
201C42—2122no—
202C42D4×D5312yes—
203C42D4×D5312yes—
204C23#1313319111yescharacteristic
205Q8—24no—
206Q8—42no—
207Q8—2124no—
208Q8—2122no—
209Q8—2122no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—2122no—
220C2×C4—2122no—
221C2×C4#1313318511yescharacteristic
222C2×C4#1313318712yes—
223C2×C4#1313319012yes—
224C2×C4#1313321311yescharacteristic
225C8—42no—
226C8—42no—
227C8—2124no—
228C22—21no—
229C22—21no—
230C22#1681053211yescharacteristic
231C22#1681089812yes—
232C4—21no—
233C4—22no—
234C4—22no—
235C4—42no—
236C4—81no—
237C4—4242no—
238C4—4242no—
239C4#1681080011yescharacteristic
240C4#1681080511yescharacteristic
241C2—22no—
242C2#2027576111yescharacteristic
243C2#2027578211yescharacteristic
244C2#2027583611yescharacteristic
subgroupG/H
centerC22#16810532
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1985C2
socleC22×C53#392
prime radicalC23×C53#121
hypercenter#1985D53
hypocenterC53#10031
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1985

Complements

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

NH ≅ G/Ntype
143. C53#10031semidirect

Maximal subgroups

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

subgroupindex
#202860602
#202848452
#202733292
#202725852
C53×#19852
#1003153