Finite Group Databasebeta

← #27411500

Subgroups — #27411500

Order 13568.

1506 conjugacy classes of subgroups; only the 275 normal subgroups are shown.

normal subgroups only

This group has 1506 conjugacy classes of subgroups — too many to draw. Only its 275 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#20277290C213yesmaximal
2#20275881C214yesmaximal, complemented
3#20275878C214yesmaximal, complemented
4C2×#16810390C213yesmaximal, complemented
5C2×C53×#543C211yesmaximal, characteristic, complemented
6C23×C22⋊D53C2211yescharacteristic
7C22×C22⋊Dic53C2213yes—
8#16810502—26no—
9#16810500—26no—
10#16810484C2216yes—
11#16810478C2216yes—
12#16810390C22112yes—
13C53×#543—26no—
14C53×#543C2214yes—
15C2×C22⋊Q8×C53—26no—
16C22×C22⋊C4×C53C2213yes—
17C24×D53C2311yescharacteristic
18C22×C22⋊D53—23no—
19C22×C22⋊D53—23no—
20C22×C22⋊D53—26no—
21C22×C22⋊D53Q814yescomplemented
22C23×Dic53—23no—
23C23×Dic53C2311yescharacteristic
24C2×C22⋊Dic53—23no—
25C2×C22⋊Dic53—212no—
26C2×C22⋊Dic53C2316yes—
27C2×#11719921—212no—
28#13133124—212no—
29C25×C53C2311yescharacteristic
30C22×Q8×C53—22no—
31C23×C4×C53D416yes—
32C22⋊Q8×C53—224no—
33C22⋊Q8×C53—224no—
34C2×C4⋊C4×C53—26no—
35C2×C22⋊C4×C53—26no—
36C2×C22⋊C4×C53—26no—
37C2×C22⋊C4×C53D416yes—
38C2×C22⋊C4×C53C2316yes—
39C24×C53—82no—
40C24×C53C2×D413yes—
41C24×C53C2×D413yes—
42C24×C53C2×Q812yes—
43C24×C53#4116yes—
44C24×C53C2411yescharacteristic
45C23×D53—23no—
46C23×D53—23no—
47C23×D53—82no—
48C23×D53C2×Q812yescomplemented
49C2×Q8×C53—28no—
50C2×Q8×C53—84no—
51C22×C4×C53—26no—
52C22×C4×C53—26no—
53C22×C4×C53—86no—
54C22×C4×C53C2×D4112yes—
55C2×C22⋊D53—212no—
56C2×C22⋊D53—212no—
57C2×C22⋊D53—41no—
58C2×C22⋊D53—43no—
59C2×C22⋊D53—46no—
60C2×C22⋊D53—46no—
61C22×Dic53—23no—
62C22×Dic53—23no—
63C22×Dic53—26no—
64C22×Dic53—46no—
65C22×Dic53—86no—
66C22×Dic53C2×Q812yescomplemented
67C4⋊C4×C53—224no—
68C22⋊C4×C53—212no—
69C22⋊C4×C53—212no—
70C22⋊Dic53—212no—
71C22⋊Dic53—46no—
72#11719921—812no—
73C23×C53—46no—
74C23×C53—46no—
75C23×C53—42no—
76C23×C53—82no—
77C23×C53—86no—
78C23×C53#12011yescharacteristic
79C23×C53#12013yes—
80C23×C53C22⋊Q816yes—
81C23×C53C22⋊Q816yes—
82C23×C53#12416yes—
83C23×C53#12416yes—
84C23×C53C22×D413yes—
85C23×C53C22×Q811yescharacteristic
86C23×C53C2×#4113yes—
87C22×D53—26no—
88C22×D53—412no—
89C22×D53—41no—
90C22×D53—43no—
91C22×D53—82no—
92C22×D53—86no—
93Q8×C53—88no—
94C2×C4×C53—212no—
95C2×C4×C53—412no—
96C2×C4×C53—412no—
97C2×C4×C53—86no—
98C22⋊D53—44no—
99C22⋊D53—412no—
100C2×Dic53—26no—
101C2×Dic53—41no—
102C2×Dic53—43no—
103C2×Dic53—412no—
104C2×Dic53—46no—
105C2×Dic53—812no—
106#23468—531nomaximal, Sylow
107C22×C53—26no—
108C22×C53—22no—
109C22×C53—26no—
110C22×C53—46no—
111C22×C53—41no—
112C22×C53—43no—
113C22×C53—46no—
114C22×C53—46no—
115C22×C53—46no—
116C22×C53#39312yes—
117C22×C53#39316yes—
118C22×C53C2×#12011yescharacteristic
119C22×C53C2×C22⋊Q813yes—
120C22×C53C2×#12413yes—
121C2×D53—412no—
122C2×D53—42no—
123C2×D53—86no—
124C2×D53—82no—
125C4×C53—412no—
126Dic53—42no—
127Dic53—86no—
128C2×#543D5311yescharacteristic, complemented
129#2308—533no—
130#2030—534no—
131#2029—534no—
132C2×#342—533no—
133C2×C53—23no—
134C2×C53—23no—
135C2×C53—22no—
136C2×C53—26no—
137C2×C53C2×#39311yescharacteristic
138C2×C53#302612yes—
139D53—82no—
140C23×D4—531no—
141#543—26no—
142#543C2×D5314yes—
143C2×C22⋊Q8—26no—
144C22×C22⋊C4—533no—
145C22×C22⋊C4C2×D5313yes—
146#386—1066no—
147#386—1066no—
148#381—536no—
149#379—536no—
150#342—5312no—
151C53#2346811yescharacteristic, complemented, Sylow
152C25—531no—
153C25C22×D5311yescharacteristic
154C22×Q8—22no—
155C22×D4—534no—
156C22×D4—1066no—
157C22×D4—1063no—
158C22×D4—1063no—
159C23×C4—531no—
160C23×C4—1063no—
161C23×C4C22⋊D5316yes—
162C22⋊Q8—224no—
163C22⋊Q8—224no—
164C2×C4⋊C4—26no—
165C2×C22⋊C4—26no—
166C2×C22⋊C4—26no—
167C2×C22⋊C4—536no—
168C2×C22⋊C4—10612no—
169C2×C22⋊C4—10612no—
170C2×C22⋊C4—1063no—
171C2×C22⋊C4C22⋊D5316yes—
172C2×C22⋊C4C22×D5316yes—
173#95—10612no—
174C24—82no—
175C24—532no—
176C24—1063no—
177C24—1063no—
178C24—4242no—
179C24Q8×D5312yes—
180C24#1171994916yes—
181C24C2×C22⋊D5313yes—
182C24C2×C22⋊D5313yes—
183C24C23×D5311yescharacteristic
184C2×Q8—28no—
185C2×Q8—84no—
186C2×D4—10612no—
187C2×D4—10612no—
188C2×D4—2123no—
189C2×D4—2121no—
190C2×D4—2126no—
191C2×D4—2126no—
192C22×C4—26no—
193C22×C4—26no—
194C22×C4—86no—
195C22×C4—532no—
196C22×C4—1063no—
197C22×C4—1063no—
198C22×C4—1066no—
199C22×C4—4246no—
200C22×C4—2126no—
201C22×C4C2×C22⋊D53112yes—
202C4⋊C4—224no—
203C22⋊C4—212no—
204C22⋊C4—212no—
205C22⋊C4—10612no—
206C22⋊C4—42412no—
207C22⋊C4—2126no—
208C23—46no—
209C23—46no—
210C23—42no—
211C23—82no—
212C23—86no—
213C23—1066no—
214C23—4246no—
215C23—4242no—
216C23—21212no—
217C23—2121no—
218C23—2123no—
219C23#1313318613yes—
220C23#1313318611yescharacteristic
221C23#1313319316yes—
222C23#1313319316yes—
223C23#1313320616yes—
224C23#1313320616yes—
225C23C22×C22⋊D5313yes—
226C23C2×Q8×D5311yescharacteristic
227C23C2×#1171994913yes—
228Q8—88no—
229D4—21212no—
230D4—2124no—
231C2×C4—212no—
232C2×C4—412no—
233C2×C4—412no—
234C2×C4—86no—
235C2×C4—1066no—
236C2×C4—42412no—
237C2×C4—21212no—
238C2×C4—2126no—
239C2×C4—2121no—
240C2×C4—2123no—
241C22—22no—
242C22—26no—
243C22—26no—
244C22—46no—
245C22—41no—
246C22—43no—
247C22—46no—
248C22—46no—
249C22—46no—
250C22—4246no—
251C22—4242no—
252C22—21212no—
253C22—2122no—
254C22#1681053416yes—
255C22#1681053412yes—
256C22C2×#1313318611yescharacteristic
257C22C2×#1313319313yes—
258C22C2×#1313320613yes—
259C4—412no—
260C4—4246no—
261C4—2122no—
262C2—23no—
263C2—23no—
264C2—22no—
265C2—26no—
266C2—4242no—
267C2C2×#1681053411yescharacteristic
268C2#2028411212yes—
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC24×C53C24
FrattiniC24C23×D53
FittingC2×C53×#543C2
socleC22×C53C2×#120
prime radicalC23×C22⋊D53C22
hypercenterC2×#543D53
hypocenterC53#23468
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C25 ◁ C2×#543

Complements

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

NH ≅ G/Ntype
2. #20275881C2semidirect
3. #20275878C2semidirect
4. C2×#16810390C2semidirect
5. C2×C53×#543C2semidirect
21. C22×C22⋊D53Q8semidirect
48. C23×D53C2×Q8semidirect
66. C22×Dic53C2×Q8semidirect
128. C2×#543D53semidirect
151. C53#23468semidirect

Maximal subgroups

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

subgroupindex
#202772902
#202758812
#202758782
C2×#168103902
C2×C53×#5432
#2346853