Finite Group Databasebeta

← #27287352

Subgroups — #27287352

Order 13568.

526 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 77 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#20274787C212yesmaximal
2#20274787C211yesmaximal, characteristic
3#20272619C211yesmaximal, characteristic
4#20272619C212yesmaximal
5C53×#2842C211yesmaximal, characteristic
6C22×C22⋊Dic53C2211yescharacteristic
7#16810479C2212yes—
8#16810479C2211yescharacteristic
9#16810389—24no—
10#16810389—22no—
11C53×#399—22no—
12C53×#396—24no—
13C53×#379C2211yescharacteristic
14C53×#379C2212yes—
15C23×Dic53Q812yes—
16C2×C22⋊Dic53—22no—
17C2×C22⋊Dic53—21no—
18C2×C22⋊Dic53—22no—
19C2×C22⋊Dic53—21no—
20C2×C22⋊Dic53—22no—
21C2×C22⋊Dic53—21no—
22C2×C22⋊Dic53—22no—
23C2×C22⋊Dic53—22no—
24C2×C22⋊Dic53—22no—
25C25×C53C2311yescharacteristic
26C2×C4⋊C4×C53—24no—
27C2×C4⋊C4×C53—22no—
28C2×C22⋊C4×C53—22no—
29C2×C22⋊C4×C53—22no—
30C2×C22⋊C4×C53—22no—
31C2×C22⋊C4×C53D414yes—
32C2×C22⋊C4×C53D412yes—
33#95×C53—22no—
34C24×C53—42no—
35C24×C53—82no—
36C24×C53C2×D412yes—
37C24×C53C2×D411yescharacteristic
38C24×C53C2×Q811yescharacteristic
39C24×C53#4112yes—
40C24×C53#4111yescharacteristic
41C22×C4×C53—22no—
42C22×C4×C53—24no—
43C22×Dic53—22no—
44C22×Dic53—22no—
45C22×Dic53—22no—
46C22×Dic53—42no—
47C22×Dic53—41no—
48C22×Dic53—82no—
49C4⋊C4×C53—84no—
50C4⋊C4×C53—82no—
51C22⋊C4×C53—44no—
52C22⋊C4×C53—42no—
53C22⋊C4×C53—82no—
54C22⋊C4×C53—82no—
55C22⋊C4×C53—82no—
56C22⋊Dic53—41no—
57C22⋊Dic53—42no—
58C22⋊Dic53—41no—
59C22⋊Dic53—44no—
60C22⋊Dic53—42no—
61C22⋊Dic53—82no—
62C22⋊Dic53—82no—
63C22⋊Dic53—82no—
64C23×C53—24no—
65C23×C53—22no—
66C23×C53—42no—
67C23×C53—41no—
68C23×C53—41no—
69C23×C53—42no—
70C23×C53—42no—
71C23×C53—42no—
72C23×C53—42no—
73C23×C53—42no—
74C23×C53—42no—
75C23×C53—42no—
76C23×C53—88no—
77C23×C53#12011yescharacteristic
78C23×C53C22⋊Q812yes—
79C23×C53C22⋊Q811yescharacteristic
80C23×C53#12412yes—
81C23×C53#12411yescharacteristic
82C2×C4×C53—44no—
83C2×C4×C53—44no—
84C2×C4×C53—44no—
85C2×C4×C53—42no—
86C2×C4×C53—44no—
87C2×Dic53—42no—
88C2×Dic53—82no—
89C2×Dic53—82no—
90C2×Dic53—82no—
91C2×Dic53—82no—
92C2×Dic53—81no—
93C2×Dic53—82no—
94C2×Dic53—82no—
95C2×Dic53—82no—
96#10393—531nomaximal, Sylow
97C22×C53—22no—
98C22×C53—22no—
99C22×C53—21no—
100C22×C53—22no—
101C22×C53—21no—
102C22×C53—22no—
103C22×C53—22no—
104C22×C53—22no—
105C22×C53—22no—
106C22×C53—22no—
107C22×C53—22no—
108C22×C53—44no—
109C22×C53—84no—
110C22×C53—82no—
111C22×C53—82no—
112C22×C53—84no—
113C22×C53#39311yescharacteristic
114C22×C53#45812yes—
115C4×C53—82no—
116C4×C53—84no—
117Dic53—82no—
118Dic53—162no—
119Dic53—161no—
120#2842D5311yescharacteristic
121#1819—532no—
122#1819—531no—
123#1304—531no—
124#1304—532no—
125C2×C53—22no—
126C2×C53—42no—
127C2×C53—41no—
128C2×C53—42no—
129C2×C53—41no—
130C2×C53#203011yescharacteristic
131C2×C53#203212yes—
132C22×C22⋊C4—531no—
133#399—22no—
134#396—24no—
135#379—532no—
136#379—531no—
137#379C2×D5311yescharacteristic
138#379C2×D5312yes—
139#342—1064no—
140#342—1062no—
141C53#1039311yescharacteristic, complemented, Sylow
142C25C22×D5311yescharacteristic
143C23×C4—532no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—22no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—1062no—
150C2×C22⋊C4—1062no—
151C2×C22⋊C4—1062no—
152C2×C22⋊C4—1061no—
153C2×C22⋊C4—1062no—
154C2×C22⋊C4—1062no—
155C2×C22⋊C4—1061no—
156C2×C22⋊C4—1062no—
157C2×C22⋊C4—1061no—
158C2×C22⋊C4C22⋊D5314yes—
159C2×C22⋊C4C22⋊D5312yes—
160#95—22no—
161C24—42no—
162C24—82no—
163C24Q8×D5311yescharacteristic
164C24#1171994912yes—
165C24#1171994911yescharacteristic
166C24C2×C22⋊D5312yes—
167C24C2×C22⋊D5311yescharacteristic
168C22×C4—22no—
169C22×C4—24no—
170C22×C4—1062no—
171C22×C4—1062no—
172C22×C4—1062no—
173C22×C4—4242no—
174C22×C4—2122no—
175C22×C4—2121no—
176C4⋊C4—84no—
177C4⋊C4—82no—
178C22⋊C4—44no—
179C22⋊C4—42no—
180C22⋊C4—82no—
181C22⋊C4—82no—
182C22⋊C4—82no—
183C22⋊C4—4242no—
184C22⋊C4—4242no—
185C22⋊C4—4242no—
186C22⋊C4—2124no—
187C22⋊C4—2122no—
188C22⋊C4—2121no—
189C22⋊C4—2122no—
190C22⋊C4—2121no—
191C23—24no—
192C23—22no—
193C23—42no—
194C23—41no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23—41no—
202C23—42no—
203C23—88no—
204C23#1313318611yescharacteristic
205C23#1313319312yes—
206C23#1313319311yescharacteristic
207C23#1313320612yes—
208C23#1313320611yescharacteristic
209C2×C4—44no—
210C2×C4—44no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—42no—
214C2×C4—4242no—
215C2×C4—4242no—
216C2×C4—4242no—
217C2×C4—4242no—
218C2×C4—4241no—
219C2×C4—4242no—
220C2×C4—4242no—
221C2×C4—4242no—
222C2×C4—2122no—
223C22—22no—
224C22—22no—
225C22—21no—
226C22—22no—
227C22—21no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—22no—
233C22—22no—
234C22—44no—
235C22—84no—
236C22—82no—
237C22—82no—
238C22—84no—
239C22#1681053411yescharacteristic
240C22#1681081312yes—
241C4—82no—
242C4—84no—
243C4—8482no—
244C4—8481no—
245C4—4242no—
246C2—22no—
247C2—42no—
248C2—41no—
249C2—42no—
250C2—41no—
251C2#2027588111yescharacteristic
252C2#2027589112yes—
subgroupG/H
centerC22#16810534
commutator, prime radical, cosocleC25×C53C23
FrattiniC25C22×D53
FittingC53×#2842C2
socleC22×C53#393
hypercenter#2842D53
hypocenterC53#10393
LCSG ▷ C25×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C25 ◁ #2842

Complements

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

NH ≅ G/Ntype
141. C53#10393semidirect

Maximal subgroups

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

subgroupindex
#202747872
#202726192
C53×#28422
#1039353