Finite Group Databasebeta

← #27489986

Subgroups — #27489986

Order 13568.

1148 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1148 conjugacy classes of subgroups — too many to draw. Only its 191 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#20278993C213yesmaximal
2#20276067C214yesmaximal, complemented
3#20275881C214yesmaximal, complemented
4#20274789C213yesmaximal
5C2×C53×#543C211yesmaximal, characteristic
6C22×C22⋊Dic53C2211yescharacteristic
7#16810551—26no—
8#16810496—23no—
9#16810495C2216yes—
10#16810486—23no—
11#16810484C2216yes—
12#16810479C2213yes—
13#16810390C22112yes—
14C53×#543—26no—
15C53×#543C2214yes—
16C2×C22⋊Q8×C53—23no—
17C2×C22⋊Q8×C53—23no—
18C22×C22⋊C4×C53C2213yes—
19C23×Dic53C2311yescharacteristic
20C23×Dic53C2311yescharacteristic
21C2×#11719919—26no—
22C2×C22⋊Dic53—23no—
23C2×C22⋊Dic53—23no—
24C2×C22⋊Dic53—23no—
25C2×C22⋊Dic53—23no—
26C2×C22⋊Dic53C2316yes—
27#13133124—26no—
28#13133124—26no—
29#13133124—26no—
30C25×C53C2311yescharacteristic
31C22×Q8×C53—21no—
32C22×Q8×C53—21no—
33C23×C4×C53D413yes—
34C23×C4×C53D413yes—
35C22⋊Q8×C53—212no—
36C22⋊Q8×C53—212no—
37C22⋊Q8×C53—212no—
38C22⋊Q8×C53—212no—
39C2×C4⋊C4×C53—23no—
40C2×C4⋊C4×C53—23no—
41C2×C22⋊C4×C53—26no—
42C2×C22⋊C4×C53—26no—
43C2×C22⋊C4×C53D416yes—
44C2×C22⋊C4×C53C2316yes—
45C24×C53—24no—
46C24×C53—42no—
47C24×C53—81no—
48C24×C53C2×D413yes—
49C24×C53C2×D413yes—
50C24×C53C2411yescharacteristic
51C2×Q8×C53—24no—
52C2×Q8×C53—24no—
53C2×Q8×C53—82no—
54C2×Q8×C53—82no—
55C22×C4×C53—23no—
56C22×C4×C53—23no—
57C22×C4×C53—23no—
58C22×C4×C53—23no—
59C22×C4×C53—46no—
60C22×C4×C53—83no—
61C22×C4×C53C2×D416yes—
62C22×C4×C53C2×D416yes—
63C22×Dic53—23no—
64C22×Dic53—26no—
65C22×Dic53—23no—
66C22×Dic53—24no—
67C22×Dic53—81no—
68C4⋊C4×C53—212no—
69C4⋊C4×C53—212no—
70C22⋊C4×C53—212no—
71C22⋊C4×C53—212no—
72C22⋊Dic53—212no—
73C22⋊Dic53—46no—
74C22⋊Dic53—43no—
75C22⋊Dic53—41no—
76C22⋊Dic53—83no—
77#11719919—86no—
78C23×C53—26no—
79C23×C53—26no—
80C23×C53—26no—
81C23×C53—41no—
82C23×C53—43no—
83C23×C53—41no—
84C23×C53—43no—
85C23×C53—43no—
86C23×C53—88no—
87C23×C53#12013yes—
88C23×C53#12011yescharacteristic
89C23×C53C22×D413yes—
90C23×C532−1+414yes—
91Q8×C53—84no—
92Q8×C53—84no—
93C2×C4×C53—26no—
94C2×C4×C53—26no—
95C2×C4×C53—46no—
96C2×C4×C53—46no—
97C2×C4×C53—46no—
98C2×C4×C53—46no—
99C2×C4×C53—83no—
100C2×C4×C53—83no—
101C2×Dic53—46no—
102C2×Dic53—46no—
103C2×Dic53—46no—
104C2×Dic53—41no—
105C2×Dic53—41no—
106C2×Dic53—46no—
107C2×Dic53—46no—
108#29657—531nomaximal, Sylow
109C22×C53—21no—
110C22×C53—23no—
111C22×C53—23no—
112C22×C53—21no—
113C22×C53—23no—
114C22×C53—23no—
115C22×C53—23no—
116C22×C53—44no—
117C22×C53—412no—
118C22×C53—86no—
119C22×C53#45812yes—
120C22×C53C2×#12011yescharacteristic
121C22×C53#53616yes—
122C4×C53—46no—
123C4×C53—46no—
124Dic53—86no—
125Dic53—81no—
126Dic53—81no—
127C2×#543D5311yescharacteristic
128#2483—533no—
129#2062—534no—
130#2030—534no—
131#1819—533no—
132C2×C53—26no—
133C2×C53—22no—
134C2×C53—43no—
135C2×C53#261411yescharacteristic
136C2×C53C2×#45811yescharacteristic
137C2×C53#302611yescharacteristic
138#543—26no—
139#543C2×D5314yes—
140C2×C22⋊Q8—23no—
141C2×C22⋊Q8—23no—
142C22×C22⋊C4—531no—
143C22×C22⋊C4C2×D5313yes—
144#397—1066no—
145#385—536no—
146#385—1063no—
147#381—536no—
148#381—1063no—
149#379—533no—
150#342—5312no—
151C53#2965711yescharacteristic, complemented, Sylow
152C25C22×D5311yescharacteristic
153C22×Q8—21no—
154C22×Q8—21no—
155C23×C4—531no—
156C23×C4—531no—
157C23×C4C22⋊D5313yes—
158C23×C4C22⋊D5313yes—
159C22⋊Q8—212no—
160C22⋊Q8—212no—
161C22⋊Q8—212no—
162C22⋊Q8—212no—
163C2×C4⋊C4—23no—
164C2×C4⋊C4—23no—
165C2×C4⋊C4—1066no—
166C2×C22⋊C4—26no—
167C2×C22⋊C4—26no—
168C2×C22⋊C4—536no—
169C2×C22⋊C4—1063no—
170C2×C22⋊C4—1063no—
171C2×C22⋊C4—1063no—
172C2×C22⋊C4—1063no—
173C2×C22⋊C4C22⋊D5316yes—
174C2×C22⋊C4C22×D5316yes—
175#95—1066no—
176#95—1066no—
177#95—1066no—
178C24—24no—
179C24—42no—
180C24—81no—
181C24C2×C22⋊D5313yes—
182C24C2×C22⋊D5313yes—
183C24C23×D5311yescharacteristic
184C2×Q8—24no—
185C2×Q8—24no—
186C2×Q8—82no—
187C2×Q8—82no—
188C22×C4—23no—
189C22×C4—23no—
190C22×C4—23no—
191C22×C4—23no—
192C22×C4—46no—
193C22×C4—83no—
194C22×C4—1063no—
195C22×C4—1066no—
196C22×C4—1063no—
197C22×C4—1064no—
198C22×C4—4241no—
199C22×C4C2×C22⋊D5316yes—
200C22×C4C2×C22⋊D5316yes—
201C4⋊C4—212no—
202C4⋊C4—212no—
203C4⋊C4—4246no—
204C22⋊C4—212no—
205C22⋊C4—212no—
206C22⋊C4—10612no—
207C22⋊C4—4243no—
208C22⋊C4—2126no—
209C22⋊C4—2123no—
210C22⋊C4—2121no—
211C23—26no—
212C23—26no—
213C23—26no—
214C23—43no—
215C23—41no—
216C23—41no—
217C23—43no—
218C23—43no—
219C23—88no—
220C23#1313318613yes—
221C23#1313318611yescharacteristic
222C23C22×C22⋊D5313yes—
223C23#1313326514yes—
224Q8—84no—
225Q8—84no—
226C2×C4—26no—
227C2×C4—26no—
228C2×C4—46no—
229C2×C4—46no—
230C2×C4—46no—
231C2×C4—46no—
232C2×C4—83no—
233C2×C4—83no—
234C2×C4—2126no—
235C2×C4—2126no—
236C2×C4—2126no—
237C2×C4—2121no—
238C2×C4—2121no—
239C2×C4—2126no—
240C2×C4—2126no—
241C22—21no—
242C22—23no—
243C22—21no—
244C22—23no—
245C22—23no—
246C22—23no—
247C22—23no—
248C22—44no—
249C22—412no—
250C22—86no—
251C22#1681081312yes—
252C22C2×#1313318611yescharacteristic
253C22#1681119716yes—
254C4—46no—
255C4—46no—
256C4—4246no—
257C4—4241no—
258C4—4241no—
259C2—26no—
260C2—22no—
261C2—43no—
262C2#2028043411yescharacteristic
263C2C2×#1681081311yescharacteristic
264C2#2028411211yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC24×C53C24
FrattiniC24C23×D53
FittingC2×C53×#543C2
socleC22×C53C2×#120
prime radicalC25×C53C23
hypercenterC2×#543D53
hypocenterC53#29657
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. #20276067C2semidirect
3. #20275881C2semidirect
151. C53#29657semidirect

Maximal subgroups

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

subgroupindex
#202789932
#202760672
#202758812
#202747892
C2×C53×#5432
#2965753