Finite Group Databasebeta

← #27393360

Subgroups — #27393360

Order 13568.

1334 conjugacy classes of subgroups; only the 315 normal subgroups are shown.

normal subgroups only

This group has 1334 conjugacy classes of subgroups — too many to draw. Only its 315 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#20277318C212yesmaximal, complemented
2#20275552C218yesmaximal, complemented
3#20274633C214yesmaximal, complemented
4C2×C53×#537C211yesmaximal, characteristic
5C2×#13133132C22116yes—
6#16810509—28no—
7#16810486C2214yes—
8#16810483C2214yes—
9C4×C22⋊Dic53C2214yes—
10C22×#41×C53C2212yes—
11C53×#537—28no—
12C2×C22⋊Q8×C53C414yescomplemented
13C2×#121×C53C414yescomplemented
14C2×#121×C53C2214yes—
15C22×C4⋊C4×C53C2211yescharacteristic
16C2×C4⋊Dic53—24no—
17C2×C22⋊Dic53C2318yes—
18C2×C4×Dic53—24no—
19#13133132—416no—
20#13133124—24no—
21#13133124—24no—
22#13133124—28no—
23C2×#41×C53—28no—
24C2×#41×C53—48no—
25C22×Q8×C53C2×C412yes—
26C22×D4×C53C2×C412yes—
27C22×D4×C53C2×C414yes—
28C22×D4×C53C2314yes—
29C23×C4×C53C2×C414yes—
30C23×C4×C53C2311yescharacteristic
31C23×C4×C53C2312yes—
32C22⋊Q8×C53—216no—
33#121×C53—216no—
34#121×C53—28no—
35#121×C53—28no—
36C2×C4⋊C4×C53C2×C414yes—
37C2×C4⋊C4×C53C2×C414yes—
38C2×C4⋊C4×C53D418yes—
39C2×C22⋊C4×C53C2×C418yes—
40C24×C53C22×C414yes—
41C24×C53C2411yescharacteristic
42#41×C53—432no—
43C2×Q8×C53—24no—
44C2×Q8×C53—44no—
45C2×D4×C53—28no—
46C2×D4×C53—216no—
47C2×D4×C53—24no—
48C2×D4×C53—28no—
49C2×D4×C53—44no—
50C2×D4×C53—84no—
51C22×C4×C53—24no—
52C22×C4×C53—28no—
53C22×C4×C53—48no—
54C22×C4×C53—82no—
55C22×C4×C53C22×C414yes—
56C22×C4×C53C22×C412yes—
57C22×C4×C53C22×C414yes—
58C22×C4×C53C2×D414yes—
59C22×C4×C53C2×D414yes—
60C22×C4×C53C2×D414yes—
61C22×Dic53—28no—
62C4⋊C4×C53—28no—
63C4⋊C4×C53C22⋊C4116yes—
64C22⋊C4×C53—216no—
65C22⋊Dic53—216no—
66C4⋊Dic53—84no—
67C4×Dic53—84no—
68C23×C53—24no—
69C23×C53—28no—
70C23×C53—41no—
71C23×C53—41no—
72C23×C53—44no—
73C23×C53—44no—
74C23×C53C23×C411yescharacteristic
75C23×C53C22×D412yes—
76C23×C53H(5,𝔽2)14yes—
77Q8×C53—48no—
78D4×C53—48no—
79D4×C53—416no—
80D4×C53—88no—
81C2×C4×C53—24no—
82C2×C4×C53—28no—
83C2×C4×C53—24no—
84C2×C4×C53—28no—
85C2×C4×C53—44no—
86C2×C4×C53—44no—
87C2×C4×C53—44no—
88C2×C4×C53—416no—
89C2×C4×C53C2×C22⋊C414yes—
90C2×C4×C53C2×C22⋊C418yes—
91C2×Dic53—48no—
92C2×Dic53—48no—
93C2×Dic53—48no—
94#21796—531nomaximal, Sylow
95C22×C53—22no—
96C22×C53—21no—
97C22×C53—22no—
98C22×C53—21no—
99C22×C53—44no—
100C22×C53—44no—
101C22×C53—44no—
102C22×C53—44no—
103C22×C53—88no—
104C22×C53C22×C22⋊C411yescharacteristic
105C22×C53#51812yes—
106C22×C53#53514yes—
107C4×C53—24no—
108C4×C53—28no—
109C4×C53—44no—
110Dic53—88no—
111C2×#537D5311yescharacteristic
112#2313—532no—
113#1964—538no—
114#1782—534no—
115C2×C53—24no—
116C2×C53—22no—
117C2×C53—84no—
118C2×C53#230911yescharacteristic
119C2×C53#288311yescharacteristic
120C2×C53#288411yescharacteristic
121C22×#41C2×D5312yes—
122#537—28no—
123C2×C22⋊Q8Dic5314yescomplemented
124C2×#121Dic5314yescomplemented
125C2×#121C2×D5314yes—
126C22×C4⋊C4C2×D5311yescharacteristic
127C2×#99—5316no—
128#387—1068no—
129#381—534no—
130#380—534no—
131C4×C22⋊C4—534no—
132C53#2179611yescharacteristic, complemented, Sylow
133C2×#41—28no—
134C2×#41—48no—
135C22×Q8C2×Dic5312yes—
136C22×D4C2×Dic5312yes—
137C22×D4C2×Dic5314yes—
138C22×D4C22×D5314yes—
139C23×C4C2×Dic5314yes—
140C23×C4C22×D5311yescharacteristic
141C23×C4C22×D5312yes—
142C22⋊Q8—216no—
143#121—28no—
144#121—28no—
145#121—216no—
146C2×C4⋊C4—1064no—
147C2×C4⋊C4C2×Dic5314yes—
148C2×C4⋊C4C2×Dic5314yes—
149C2×C4⋊C4C22⋊D5318yes—
150C2×C22⋊C4—538no—
151C2×C22⋊C4C2×Dic5318yes—
152C2×C42—1064no—
153#99—21216no—
154#95—1064no—
155#95—1064no—
156#95—1068no—
157C24C22×Dic5314yes—
158C24C23×D5311yescharacteristic
159#41—432no—
160C2×Q8—24no—
161C2×Q8—44no—
162C2×D4—28no—
163C2×D4—216no—
164C2×D4—24no—
165C2×D4—28no—
166C2×D4—44no—
167C2×D4—84no—
168C22×C4—24no—
169C22×C4—28no—
170C22×C4—48no—
171C22×C4—82no—
172C22×C4—1068no—
173C22×C4C22×Dic5314yes—
174C22×C4C22×Dic5312yes—
175C22×C4C22×Dic5314yes—
176C22×C4C2×C22⋊D5314yes—
177C22×C4C2×C22⋊D5314yes—
178C22×C4C2×C22⋊D5314yes—
179C4⋊C4—28no—
180C4⋊C4—4244no—
181C4⋊C4C22⋊Dic53116yes—
182C22⋊C4—216no—
183C22⋊C4—10616no—
184C42—4244no—
185C23—24no—
186C23—28no—
187C23—41no—
188C23—41no—
189C23—44no—
190C23—44no—
191C23C23×Dic5311yescharacteristic
192C23C22×C22⋊D5312yes—
193C23#1313326214yes—
194Q8—48no—
195D4—48no—
196D4—416no—
197D4—88no—
198C2×C4—24no—
199C2×C4—28no—
200C2×C4—24no—
201C2×C4—28no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—416no—
206C2×C4—2128no—
207C2×C4—2128no—
208C2×C4—2128no—
209C2×C4C2×C22⋊Dic5318yes—
210C2×C4C2×C22⋊Dic5314yes—
211C22—22no—
212C22—21no—
213C22—22no—
214C22—21no—
215C22—44no—
216C22—44no—
217C22—44no—
218C22—44no—
219C22—88no—
220C22C22×C22⋊Dic5311yescharacteristic
221C22#1681109812yes—
222C22#1681119014yes—
223C4—28no—
224C4—24no—
225C4—44no—
226C4—4248no—
227C2—24no—
228C2—22no—
229C2—84no—
230C2#2027729711yescharacteristic
231C2#2028317611yescharacteristic
232C2#2028317911yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic53
commutatorC23×C53C23×C4
FrattiniC24C23×D53
Fitting, prime radicalC2×C53×#537C2
socleC22×C53C22×C22⋊C4
cosocleC24×C53C24
hypercenterC2×#537D53
hypocenterC53#21796
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4⋊C4 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. #20277318C2semidirect
2. #20275552C2semidirect
3. #20274633C2semidirect
12. C2×C22⋊Q8×C53C4semidirect
13. C2×#121×C53C4semidirect
123. C2×C22⋊Q8Dic53semidirect
124. C2×#121Dic53semidirect
132. C53#21796semidirect

Maximal subgroups

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

subgroupindex
#202773182
#202755522
#202746332
C2×C53×#5372
#2179653