Finite Group Databasebeta

← #27287345

Subgroups — #27287345

Order 13568.

542 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 75 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#20282968C211yesmaximal, characteristic
2#20274789C212yesmaximal
3#20272619C212yesmaximal
4#20272619C211yesmaximal, characteristic
5C53×#1819C211yesmaximal, characteristic
6#16810535—24no—
7#16810535—22no—
8#16810479C2212yes—
9#16810479C2212yes—
10#16810390—24no—
11C22×C22⋊C4×C53C2211yescharacteristic
12C53×#379C2211yescharacteristic
13C53×#379C2211yescharacteristic
14C53×#342—22no—
15C22×Dic106—22no—
16C2×C22⋊Dic53—22no—
17C2×C22⋊Dic53—22no—
18C2×C22⋊Dic53—22no—
19C2×C22⋊Dic53—22no—
20C2×C22⋊Dic53—22no—
21C2×C22⋊Dic53D414yes—
22#13133124—26no—
23C25×C53C2311yescharacteristic
24C23×C4×C53Q812yescomplemented
25C2×C22⋊C4×C53—21no—
26C2×C22⋊C4×C53—22no—
27C2×C22⋊C4×C53—22no—
28C2×C22⋊C4×C53—21no—
29C2×C22⋊C4×C53—22no—
30C2×C22⋊C4×C53—21no—
31C2×C22⋊C4×C53—22no—
32C2×C22⋊C4×C53D412yes—
33C24×C53—42no—
34C24×C53—82no—
35C24×C53C2×D412yes—
36C24×C53C2×D411yescharacteristic
37C24×C53C2×Q811yescharacteristic
38C24×C53#4111yescharacteristic
39C24×C53#4112yes—
40C22×C4×C53—22no—
41C22×C4×C53—22no—
42C22×C4×C53—22no—
43C22×C4×C53—22no—
44C22×C4×C53—41no—
45C22×C4×C53—82no—
46C22×Dic53—24no—
47C22×Dic53—42no—
48C2×Dic106—84no—
49C2×Dic106—82no—
50C22⋊C4×C53—41no—
51C22⋊C4×C53—42no—
52C22⋊C4×C53—41no—
53C22⋊C4×C53—42no—
54C22⋊C4×C53—42no—
55C22⋊C4×C53—82no—
56C22⋊C4×C53—82no—
57C22⋊Dic53—44no—
58C22⋊Dic53—44no—
59C22⋊Dic53—82no—
60C22⋊Dic53—82no—
61C22⋊Dic53—82no—
62C22⋊Dic53—82no—
63C23×C53—24no—
64C23×C53—22no—
65C23×C53—42no—
66C23×C53—41no—
67C23×C53—41no—
68C23×C53—42no—
69C23×C53—42no—
70C23×C53—42no—
71C23×C53—42no—
72C23×C53—42no—
73C23×C53—42no—
74C23×C53—42no—
75C23×C53—88no—
76C23×C53#12011yescharacteristic
77C23×C53C22⋊Q812yes—
78C23×C53C22⋊Q811yescharacteristic
79C23×C53#12412yes—
80C23×C53#12411yescharacteristic
81C2×C4×C53—44no—
82C2×C4×C53—42no—
83C2×C4×C53—42no—
84C2×C4×C53—82no—
85C2×C4×C53—82no—
86C2×C4×C53—82no—
87C2×C4×C53—81no—
88C2×C4×C53—82no—
89C2×Dic53—44no—
90C2×Dic53—44no—
91C2×Dic53—44no—
92C2×Dic53—82no—
93C2×Dic53—82no—
94C2×Dic53—82no—
95Dic106—88no—
96#10391—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—82no—
117C4×C53—161no—
118Dic53—84no—
119Dic53—162no—
120#2840—531no—
121#1819—532no—
122#1819D5311yescharacteristic
123#1304—532no—
124#1304—531no—
125C2×C53—22no—
126C2×C53—42no—
127C2×C53—41no—
128C2×C53—42no—
129C2×C53—41no—
130C2×C53#203011yescharacteristic
131C2×C53#203212yes—
132C22×C22⋊C4C2×D5311yescharacteristic
133#393—1064no—
134#393—1062no—
135#379—532no—
136#379—532no—
137#379C2×D5311yescharacteristic
138#379C2×D5311yescharacteristic
139#342—22no—
140#342—1064no—
141C53#1039111yescharacteristic, complemented, Sylow
142C25C22×D5311yescharacteristic
143C22×Q8—1062no—
144C23×C4Dic10612yescomplemented
145C2×C22⋊C4—21no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—22no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—534no—
153C2×C22⋊C4—1062no—
154C2×C22⋊C4—1062no—
155C2×C22⋊C4—1062no—
156C2×C22⋊C4—1062no—
157C2×C22⋊C4—1062no—
158C2×C22⋊C4C22⋊D5312yes—
159#95—1066no—
160C24—42no—
161C24—82no—
162C24C2×Dic10611yescharacteristic
163C24#1171994511yescharacteristic
164C24D4×D5312yes—
165C24#1171994712yes—
166C24C2×C22⋊D5311yescharacteristic
167C2×Q8—4244no—
168C2×Q8—4242no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—41no—
174C22×C4—82no—
175C22×C4—1064no—
176C22×C4—2122no—
177C22⋊C4—41no—
178C22⋊C4—42no—
179C22⋊C4—41no—
180C22⋊C4—42no—
181C22⋊C4—42no—
182C22⋊C4—82no—
183C22⋊C4—82no—
184C22⋊C4—4242no—
185C22⋊C4—4242no—
186C22⋊C4—4242no—
187C22⋊C4—4242no—
188C22⋊C4—2124no—
189C22⋊C4—2124no—
190C23—22no—
191C23—24no—
192C23—42no—
193C23—41no—
194C23—42no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—41no—
201C23—42no—
202C23—88no—
203C23#1313318511yescharacteristic
204C23#1313319412yes—
205C23#1313319611yescharacteristic
206C23#1313320312yes—
207C23#1313320411yescharacteristic
208Q8—4248no—
209C2×C4—44no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—81no—
216C2×C4—82no—
217C2×C4—4242no—
218C2×C4—4242no—
219C2×C4—4242no—
220C2×C4—2124no—
221C2×C4—2124no—
222C2×C4—2124no—
223C22—22no—
224C22—21no—
225C22—22no—
226C22—22no—
227C22—21no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—22no—
233C22—22no—
234C22—44no—
235C22—82no—
236C22—84no—
237C22—84no—
238C22—82no—
239C22#1681053511yescharacteristic
240C22#1681081212yes—
241C4—82no—
242C4—82no—
243C4—161no—
244C4—8482no—
245C4—4244no—
246C2—22no—
247C2—41no—
248C2—42no—
249C2—42no—
250C2—41no—
251C2#2027588211yescharacteristic
252C2#2027589212yes—
subgroupG/H
centerC22#16810535
commutator, prime radical, cosocleC25×C53C23
FrattiniC25C22×D53
FittingC53×#1819C2
socleC22×C53#393
hypercenter#1819D53
hypocenterC53#10391
LCSG ▷ C25×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C25 ◁ #1819

Complements

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

NH ≅ G/Ntype
24. C23×C4×C53Q8semidirect
141. C53#10391semidirect
144. C23×C4Dic106semidirect

Maximal subgroups

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

subgroupindex
#202829682
#202747892
#202726192
C53×#18192
#1039153