Finite Group Databasebeta

← #27284753

Subgroups — #27284753

Order 13568.

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

normal subgroups only

This group has 502 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#20284659C212yesmaximal
2#20274790C211yesmaximal, characteristic
3#20273248C212yesmaximal
4#20272623C211yesmaximal, characteristic
5C53×#2817C211yesmaximal, characteristic
6C2×#13133176C2211yescharacteristic
7#16810961—24no—
8#16810956—22no—
9#16810951—22no—
10#16810663—22no—
11#16810632—22no—
12#16810587C2212yes—
13#16810486C2211yescharacteristic
14#16810389—22no—
15#16810360—24no—
16C2×C22⋊Q8×C53C2212yes—
17C53×#400—22no—
18C53×#397—22no—
19C53×#395—22no—
20C53×#381C2211yescharacteristic
21C2×#11719917—22no—
22#13133176—22no—
23#13133176—22no—
24C2×C4⋊Dic53D412yes—
25C2×C22⋊Dic53—21no—
26C2×C22⋊Dic53—21no—
27C2×C4×Dic53—21no—
28#13133150—24no—
29#13133148—24no—
30#13133141—24no—
31#13133141—24no—
32C22⋊#88322—22no—
33C22⋊#88322D414yes—
34#13133124—22no—
35C22×Q8×C53D412yes—
36C23×C4×C53C2311yescharacteristic
37C22⋊Q8×C53—22no—
38C22⋊Q8×C53—22no—
39C22⋊Q8×C53—22no—
40C22⋊Q8×C53—22no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53—22no—
43C2×C4⋊C4×C53D412yes—
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53D412yes—
46#95×C53—24no—
47#95×C53—22no—
48#95×C53—22no—
49C24×C53C2×D411yescharacteristic
50C2×Q8×C53—22no—
51C2×Q8×C53—22no—
52C2×Q8×C53—44no—
53C22×C4×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53—22no—
56C22×C4×C53—42no—
57C22×C4×C53C2×D412yes—
58C22×C4×C53C2×D411yescharacteristic
59C22×C4×C53C2×D412yes—
60C22×C4×C53#4111yescharacteristic
61C22×Dic53—22no—
62C22×Dic53—41no—
63C4⋊C4×C53—22no—
64C4⋊C4×C53—22no—
65C4⋊C4×C53—42no—
66C4⋊C4×C53—42no—
67C4⋊C4×C53—82no—
68C22⋊C4×C53—42no—
69C22⋊C4×C53—42no—
70C22⋊C4×C53—42no—
71C22⋊Dic53—41no—
72C22⋊Dic53—41no—
73C22⋊Dic53—42no—
74C4⋊Dic53—22no—
75C4⋊Dic53—22no—
76C4×Dic53—42no—
77#11719917—44no—
78C2×#88322—24no—
79C23×C53—22no—
80C23×C53—42no—
81C23×C53#12011yescharacteristic
82C23×C53#12111yescharacteristic
83C23×C53#12711yescharacteristic
84Q8×C53—44no—
85Q8×C53—82no—
86C2×C4×C53—24no—
87C2×C4×C53—41no—
88C2×C4×C53—42no—
89C2×C4×C53—41no—
90C2×C4×C53—44no—
91C2×C4×C53—44no—
92C2×C4×C53—42no—
93C2×C4×C53—42no—
94C2×C4×C53—44no—
95C2×C4×C53—42no—
96C2×C4×C53#12012yes—
97C2×C4×C53#12112yes—
98C2×Dic53—42no—
99C2×Dic53—82no—
100C2×Dic53—82no—
101C2×Dic53—81no—
102#88322—44no—
103#9985—531nomaximal, Sylow
104C22×C53—22no—
105C22×C53—22no—
106C22×C53—22no—
107C22×C53—21no—
108C22×C53—21no—
109C22×C53—82no—
110C22×C53#39211yescharacteristic
111C22×C53#45812yes—
112C4×C53—22no—
113C4×C53—22no—
114C4×C53—82no—
115C4×C53—82no—
116C4×C53—82no—
117Dic53—82no—
118Dic53—161no—
119#3095—532no—
120#2817D5311yescharacteristic
121#1820—531no—
122#1511—532no—
123#1306—531no—
124C2×C53—22no—
125C2×C53—41no—
126C2×C53—41no—
127C2×C53#200912yes—
128C2×C53#202211yescharacteristic
129C2×C22⋊Q8C2×D5312yes—
130C2×#117—531no—
131#485—1064no—
132#484—1062no—
133#483—1062no—
134#428—1062no—
135#419—1062no—
136#407—532no—
137#400—22no—
138#397—22no—
139#395—22no—
140#381—531no—
141#381C2×D5311yescharacteristic
142#342—1062no—
143#328—1064no—
144C53#998511yescharacteristic, complemented, Sylow
145C22×Q8C22⋊D5312yes—
146C23×C4C22×D5311yescharacteristic
147C2×M16—1062no—
148C22⋊Q8—22no—
149C22⋊Q8—22no—
150C22⋊Q8—22no—
151C22⋊Q8—22no—
152#117—1062no—
153#117—1062no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—532no—
157C2×C4⋊C4C22⋊D5312yes—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—1061no—
160C2×C22⋊C4—1061no—
161C2×C22⋊C4C22⋊D5312yes—
162C2×C42—1061no—
163#107—1064no—
164#106—1064no—
165#103—1064no—
166#103—1064no—
167C22⋊C8—534no—
168C22⋊C8—1062no—
169#95—24no—
170#95—22no—
171#95—22no—
172#95—1062no—
173C24C2×C22⋊D5311yescharacteristic
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—44no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—42no—
181C22×C4—1062no—
182C22×C4—2121no—
183C22×C4D4×D5311yescharacteristic
184C22×C4D4×D5312yes—
185C22×C4#1171994711yescharacteristic
186C22×C4C2×C22⋊D5312yes—
187M16—2124no—
188C2×C8—1064no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—82no—
194C4⋊C4—1062no—
195C4⋊C4—1062no—
196C22⋊C4—42no—
197C22⋊C4—42no—
198C22⋊C4—42no—
199C22⋊C4—2122no—
200C22⋊C4—2121no—
201C22⋊C4—2121no—
202C42—2122no—
203C23—22no—
204C23—42no—
205C23#1313318611yescharacteristic
206C23#1313318811yescharacteristic
207C23#1313321311yescharacteristic
208Q8—44no—
209Q8—82no—
210C2×C4—24no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—44no—
215C2×C4—44no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—44no—
220C2×C4—4241no—
221C2×C4—4242no—
222C2×C4—4242no—
223C2×C4—2122no—
224C2×C4#1313318512yes—
225C2×C4#1313318912yes—
226C8—2124no—
227C22—22no—
228C22—22no—
229C22—22no—
230C22—21no—
231C22—21no—
232C22—82no—
233C22#1681053111yescharacteristic
234C22#1681081312yes—
235C4—22no—
236C4—22no—
237C4—82no—
238C4—82no—
239C4—82no—
240C4—8481no—
241C4—4242no—
242C2—22no—
243C2—41no—
244C2—41no—
245C2#2027577512yes—
246C2#2027584411yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#2817C2
socleC22×C53#392
prime radicalC24×C53C2×D4
hypercenter#2817D53
hypocenterC53#9985
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2817

Complements

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

NH ≅ G/Ntype
144. C53#9985semidirect

Maximal subgroups

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

subgroupindex
#202846592
#202747902
#202732482
#202726232
C53×#28172
#998553