Finite Group Databasebeta

← #27284721

Subgroups — #27284721

Order 13568.

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

normal subgroups only

This group has 492 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#20285490C211yesmaximal, characteristic
2#20284663C212yesmaximal
3#20273078C212yesmaximal
4#20272592C211yesmaximal, characteristic
5C53×#1818C211yesmaximal, characteristic
6C2×#13133196C2212yes—
7#16810964—24no—
8#16810959—22no—
9#16810958—22no—
10#16810953—22no—
11#16810864—22no—
12#16810781—22no—
13#16810666—22no—
14#16810631—22no—
15#16810587C2212yes—
16#16810348—24no—
17C2×#117×C53C2211yescharacteristic
18C2×C22⋊C8×C53C2211yescharacteristic
19C2×C22⋊C8×C53C2211yescharacteristic
20C53×#336—22no—
21C22×Dic106D412yes—
22C2×Dic212—22no—
23C2×#11719917—22no—
24#13133196—22no—
25#13133196—22no—
26#13133196—22no—
27#13133196—22no—
28C2×C4⋊Dic53D412yes—
29C2×#11719919—22no—
30C2×C22⋊Dic53—22no—
31#13133151—22no—
32#13133150—24no—
33#13133148—24no—
34#13133142—24no—
35#13133140—24no—
36#13133140—24no—
37C22⋊#88322—22no—
38C22⋊#88322D414yes—
39C23×C4×C53C2311yescharacteristic
40C22×C8×C53—21no—
41C22×C8×C53D412yes—
42#117×C53—22no—
43#117×C53—22no—
44C2×C4⋊C4×C53D411yescharacteristic
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C22⋊C4×C53—21no—
47C2×C42×C53—21no—
48C22⋊C8×C53—22no—
49C22⋊C8×C53—22no—
50C24×C53#4111yescharacteristic
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53—42no—
54C22×C4×C53C2×D412yes—
55C22×C4×C53C2×D411yescharacteristic
56C22×C4×C53C2×D411yescharacteristic
57C22×C4×C53C2×D412yes—
58C22×Dic53—22no—
59C22×Dic53—22no—
60C2×Dic106—22no—
61C2×Dic106—22no—
62C2×Dic106—44no—
63C2×C8×C53—22no—
64C2×C8×C53—42no—
65C2×C8×C53—42no—
66C2×C8×C53—41no—
67C4⋊C4×C53—22no—
68C4⋊C4×C53—22no—
69C22⋊C4×C53—42no—
70C42×C53—41no—
71C42×C53—41no—
72C22⋊Dic53—42no—
73C22⋊Dic53—42no—
74C4⋊Dic53—22no—
75C4⋊Dic53—22no—
76#11719919—42no—
77#11719919—42no—
78#11719917—44no—
79C2×#88322—24no—
80Dic212—82no—
81C23×C53—22no—
82C23×C53—41no—
83C23×C53—41no—
84C23×C53#12112yes—
85C23×C53#12111yescharacteristic
86C2×C4×C53—24no—
87C2×C4×C53—22no—
88C2×C4×C53—41no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—42no—
94C2×C4×C53#12011yescharacteristic
95C2×C4×C53#12012yes—
96C2×C4×C53#12711yescharacteristic
97C2×Dic53—44no—
98C2×Dic53—42no—
99C2×Dic53—42no—
100C2×Dic53—82no—
101Dic106—44no—
102Dic106—44no—
103C8×C53—42no—
104C8×C53—81no—
105#88322—44no—
106#9979—531nomaximal, Sylow
107C22×C53—21no—
108C22×C53—21no—
109C22×C53—22no—
110C22×C53—22no—
111C22×C53—22no—
112C22×C53—82no—
113C22×C53#39211yescharacteristic
114C22×C53#47312yes—
115C4×C53—22no—
116C4×C53—21no—
117C4×C53—81no—
118C4×C53—81no—
119C4×C53#45812yes—
120Dic53—82no—
121Dic53—82no—
122#3189—531no—
123#3095—532no—
124#1818D5311yescharacteristic
125#1472—532no—
126#1293—531no—
127C2×C53—22no—
128C2×C53—42no—
129C2×C53#201711yescharacteristic
130C2×C53#202011yescharacteristic
131C2×C53#202211yescharacteristic
132C2×C22⋊Q8—532no—
133C2×#117C2×D5311yescharacteristic
134#485—1064no—
135#484—1062no—
136#484—1062no—
137#483—1062no—
138#467—1062no—
139#451—1062no—
140#428—1062no—
141#419—1062no—
142#407—532no—
143C2×C22⋊C8C2×D5311yescharacteristic
144C2×C22⋊C8C2×D5311yescharacteristic
145#336—22no—
146#324—1064no—
147C53#997911yescharacteristic, complemented, Sylow
148C22×Q8—532no—
149C23×C4C22×D5311yescharacteristic
150C2×Q16—1062no—
151C2×M16—1062no—
152C22×C8—21no—
153C22×C8C22⋊D5312yes—
154C22⋊Q8—1062no—
155C22⋊Q8—1062no—
156C22⋊Q8—1062no—
157C22⋊Q8—1062no—
158#117—22no—
159#117—22no—
160C2×C4⋊C4—532no—
161C2×C4⋊C4—1062no—
162C2×C4⋊C4D21211yescharacteristic
163C2×C4⋊C4D21211yescharacteristic
164C2×C22⋊C4—21no—
165C2×C22⋊C4—1062no—
166C2×C42—21no—
167#107—1064no—
168#107—1062no—
169#106—1064no—
170#103—1064no—
171#103—1064no—
172#103—1064no—
173C22⋊C8—22no—
174C22⋊C8—22no—
175C22⋊C8—534no—
176C22⋊C8—1062no—
177C24#1171994511yescharacteristic
178C2×Q8—1062no—
179C2×Q8—1062no—
180C2×Q8—2124no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—42no—
184C22×C4—1062no—
185C22×C4—1062no—
186C22×C4C2×D21211yescharacteristic
187C22×C4D4×D5312yes—
188C22×C4D4×D5312yes—
189C22×C4C2×C22⋊D5311yescharacteristic
190Q16—4242no—
191M16—2124no—
192C2×C8—22no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—41no—
196C2×C8—1064no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—1062no—
200C4⋊C4—1062no—
201C4⋊C4—2122no—
202C4⋊C4—2122no—
203C22⋊C4—42no—
204C22⋊C4—2122no—
205C22⋊C4—2122no—
206C42—41no—
207C42—41no—
208C23—22no—
209C23—41no—
210C23—41no—
211C23#1313319012yes—
212C23#1313319111yescharacteristic
213Q8—2124no—
214Q8—2124no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—4242no—
224C2×C4—2124no—
225C2×C4—2122no—
226C2×C4—2122no—
227C2×C4#1313318511yescharacteristic
228C2×C4#1313318712yes—
229C2×C4#1313321311yescharacteristic
230C8—42no—
231C8—81no—
232C8—2124no—
233C22—21no—
234C22—21no—
235C22—22no—
236C22—22no—
237C22—22no—
238C22—82no—
239C22#1681053211yescharacteristic
240C22#1681089812yes—
241C4—21no—
242C4—22no—
243C4—81no—
244C4—81no—
245C4—4242no—
246C4—4242no—
247C4#1681081212yes—
248C2—22no—
249C2—42no—
250C2#2027581911yescharacteristic
251C2#2027583611yescharacteristic
252C2#2027584611yescharacteristic
subgroupG/H
centerC22#16810532
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1818C2
socleC22×C53#392
prime radicalC24×C53#41
hypercenter#1818D53
hypocenterC53#9979
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1818

Complements

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

NH ≅ G/Ntype
147. C53#9979semidirect

Maximal subgroups

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

subgroupindex
#202854902
#202846632
#202730782
#202725922
C53×#18182
#997953