Finite Group Databasebeta

← #27412229

Subgroups — #27412229

Order 13568.

1334 conjugacy classes of subgroups; only the 291 normal subgroups are shown.

normal subgroups only

This group has 1334 conjugacy classes of subgroups — too many to draw. Only its 291 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#20277327C213yesmaximal
2#20276067C218yesmaximal, complemented
3C2×#16810390C213yesmaximal
4C2×C53×#543C211yesmaximal, characteristic
5C22×C22⋊Dic53C2211yescharacteristic
6C22×C22⋊Dic53C2213yes—
7#16810496—26no—
8#16810495C22112yes—
9#16810481—26no—
10#16810390C22112yes—
11C53×#543—26no—
12C53×#543C2214yes—
13C2×C22⋊Q8×C53—26no—
14C22×C22⋊C4×C53C2213yes—
15C23×Dic53—23no—
16C23×Dic53C2312yes—
17C2×#11719919—212no—
18C2×C22⋊Dic53—23no—
19C2×C22⋊Dic53—23no—
20C2×C22⋊Dic53—23no—
21C2×C22⋊Dic53—26no—
22C2×C22⋊Dic53Q8112yescomplemented
23C2×C22⋊Dic53C2316yes—
24#13133124—212no—
25C25×C53C2311yescharacteristic
26C22×Q8×C53—22no—
27C23×C4×C53D416yes—
28C22⋊Q8×C53—224no—
29C22⋊Q8×C53—224no—
30C2×C4⋊C4×C53—26no—
31C2×C22⋊C4×C53—26no—
32C2×C22⋊C4×C53—26no—
33C2×C22⋊C4×C53D416yes—
34C2×C22⋊C4×C53C2316yes—
35C24×C53—82no—
36C24×C53C2×D413yes—
37C24×C53C2×D413yes—
38C24×C53C2×Q816yes—
39C24×C53#4112yes—
40C24×C53C2411yescharacteristic
41C2×Q8×C53—28no—
42C2×Q8×C53—84no—
43C22×C4×C53—26no—
44C22×C4×C53—26no—
45C22×C4×C53—86no—
46C22×C4×C53C2×D4112yes—
47C22×Dic53—26no—
48C22×Dic53—212no—
49C22×Dic53—26no—
50C22×Dic53—82no—
51C22×Dic53—86no—
52C4⋊C4×C53—224no—
53C22⋊C4×C53—212no—
54C22⋊C4×C53—212no—
55C22⋊Dic53—212no—
56C22⋊Dic53—26no—
57C22⋊Dic53—41no—
58C22⋊Dic53—43no—
59C22⋊Dic53C2×Q8112yescomplemented
60#11719919—812no—
61C23×C53—46no—
62C23×C53—42no—
63C23×C53—46no—
64C23×C53—86no—
65C23×C53—82no—
66C23×C53#12011yescharacteristic
67C23×C53#12013yes—
68C23×C53C22⋊Q816yes—
69C23×C53C22⋊Q816yes—
70C23×C53C4⋊Q816yes—
71C23×C53C4⋊Q816yes—
72C23×C53C22×D413yes—
73C23×C53C22×Q813yes—
74C23×C53C2×#4111yescharacteristic
75Q8×C53—88no—
76C2×C4×C53—212no—
77C2×C4×C53—412no—
78C2×C4×C53—412no—
79C2×C4×C53—86no—
80C2×Dic53—212no—
81C2×Dic53—412no—
82C2×Dic53—412no—
83C2×Dic53—42no—
84C2×Dic53—812no—
85#23565—531nomaximal, Sylow
86C22×C53—26no—
87C22×C53—22no—
88C22×C53—26no—
89C22×C53—46no—
90C22×C53—46no—
91C22×C53—46no—
92C22×C53—46no—
93C22×C53—43no—
94C22×C53—41no—
95C22×C53#39512yes—
96C22×C53#39516yes—
97C22×C53C2×#12011yescharacteristic
98C22×C53C2×C22⋊Q813yes—
99C22×C53C2×C4⋊Q813yes—
100C4×C53—412no—
101Dic53—412no—
102Dic53—82no—
103C2×#543D5311yescharacteristic
104#2315—533no—
105#2062—538no—
106C2×#342—533no—
107C2×C53—23no—
108C2×C53—23no—
109C2×C53—22no—
110C2×C53—26no—
111C2×C53C2×#39511yescharacteristic
112C2×C53#302612yes—
113#543—26no—
114#543C2×D5314yes—
115C2×C22⋊Q8—26no—
116C22×C22⋊C4—533no—
117C22×C22⋊C4—531no—
118C22×C22⋊C4C2×D5313yes—
119#385—5312no—
120#385—1066no—
121#380—1066no—
122#342—5312no—
123C53#2356511yescharacteristic, complemented, Sylow
124C25C22×D5311yescharacteristic
125C22×Q8—22no—
126C23×C4—532no—
127C23×C4—1063no—
128C23×C4C22⋊D5316yes—
129C22⋊Q8—224no—
130C22⋊Q8—224no—
131C2×C4⋊C4—26no—
132C2×C4⋊C4—10612no—
133C2×C22⋊C4—26no—
134C2×C22⋊C4—26no—
135C2×C22⋊C4—5312no—
136C2×C22⋊C4—536no—
137C2×C22⋊C4—1066no—
138C2×C22⋊C4—1063no—
139C2×C22⋊C4—1063no—
140C2×C22⋊C4—1063no—
141C2×C22⋊C4C22⋊D5316yes—
142C2×C22⋊C4C22×D5316yes—
143#95—10612no—
144C24—82no—
145C24Q8×D5316yes—
146C24#1171994912yes—
147C24C2×C22⋊D5313yes—
148C24C2×C22⋊D5313yes—
149C24C23×D5311yescharacteristic
150C2×Q8—28no—
151C2×Q8—84no—
152C22×C4—26no—
153C22×C4—26no—
154C22×C4—86no—
155C22×C4—10612no—
156C22×C4—1066no—
157C22×C4—1066no—
158C22×C4—4246no—
159C22×C4—4242no—
160C22×C4C2×C22⋊D53112yes—
161C4⋊C4—224no—
162C4⋊C4—42412no—
163C22⋊C4—212no—
164C22⋊C4—212no—
165C22⋊C4—5312no—
166C22⋊C4—10612no—
167C22⋊C4—1066no—
168C22⋊C4—2123no—
169C22⋊C4—2121no—
170C23—46no—
171C23—46no—
172C23—42no—
173C23—86no—
174C23—82no—
175C23#1313318613yes—
176C23#1313318611yescharacteristic
177C23#1313319316yes—
178C23#1313319316yes—
179C23#1313321516yes—
180C23#1313321516yes—
181C23C22×C22⋊D5313yes—
182C23C2×Q8×D5313yes—
183C23C2×#1171994911yescharacteristic
184Q8—88no—
185C2×C4—212no—
186C2×C4—412no—
187C2×C4—412no—
188C2×C4—86no—
189C2×C4—10612no—
190C2×C4—42412no—
191C2×C4—21212no—
192C2×C4—2122no—
193C2×C4—21212no—
194C22—22no—
195C22—26no—
196C22—26no—
197C22—46no—
198C22—46no—
199C22—46no—
200C22—46no—
201C22—43no—
202C22—41no—
203C22#1681054216yes—
204C22#1681054212yes—
205C22C2×#1313318611yescharacteristic
206C22C2×#1313319313yes—
207C22C2×#1313321513yes—
208C4—412no—
209C4—4242no—
210C4—21212no—
211C2—23no—
212C2—22no—
213C2—26no—
214C2—23no—
215C2C2×#1681054211yescharacteristic
216C2#2028411212yes—
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC24×C53C24
FrattiniC24C23×D53
FittingC2×C53×#543C2
socleC22×C53C2×#120
prime radicalC25×C53C23
hypercenterC2×#543D53
hypocenterC53#23565
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
22. C2×C22⋊Dic53Q8semidirect
59. C22⋊Dic53C2×Q8semidirect
123. C53#23565semidirect

Maximal subgroups

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

subgroupindex
#202773272
#202760672
C2×#168103902
C2×C53×#5432
#2356553