Finite Group Databasebeta

← #27297147

Subgroups — #27297147

Order 13568.

442 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 63 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#20276060C211yesmaximal, characteristic
2#20273688C212yesmaximal
3#20272745C212yesmaximal
4#20272661C211yesmaximal, characteristic
5C53×#2551C211yesmaximal, characteristic
6#16811254C2211yescharacteristic
7#16810779—22no—
8#16810663—21no—
9#16810392C2211yescharacteristic
10#16810383—21no—
11#16810359—22no—
12#16810349C2212yes—
13C2×C4⋊Q8×C53—22no—
14C53×#395—21no—
15C53×#395C2212yes—
16C53×#389—22no—
17C53×#384C2211yescharacteristic
18C4×C4⋊C4×C53—21no—
19C2×#11719925—42no—
20C2×#11719917—21no—
21#13133216—41no—
22#13133208—41no—
23#13133196—22no—
24#13133176—21no—
25#13133150—41no—
26#13133148—41no—
27#13133141—42no—
28C22⋊#88322—22no—
29C22×Q8×C53—21no—
30C22×Q8×C53D412yes—
31C4⋊Q8×C53—48no—
32C2×C4⋊C4×C53—21no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—22no—
35C2×C4⋊C4×C53—22no—
36C2×C4⋊C4×C53—22no—
37C2×C4⋊C4×C53D411yescharacteristic
38C2×C4⋊C4×C53C2311yescharacteristic
39C2×C42×C53—22no—
40C2×C42×C53D411yescharacteristic
41#95×C53—21no—
42#95×C53—22no—
43#95×C53D412yes—
44C2×Q8×C53—44no—
45C2×Q8×C53—44no—
46C2×Q8×C53—42no—
47C2×Q8×C53—81no—
48C22×C4×C53—21no—
49C22×C4×C53—22no—
50C22×C4×C53—22no—
51C22×C4×C53—21no—
52C22×C4×C53C2×D411yescharacteristic
53C22×C4×C53C2×D412yes—
54C2×Dic106—22no—
55C4⋊C4×C53—24no—
56C4⋊C4×C53—24no—
57C4⋊C4×C53—41no—
58C4⋊C4×C53—44no—
59C4⋊C4×C53—44no—
60C4⋊C4×C53—44no—
61C4⋊C4×C53—41no—
62C42×C53—21no—
63C42×C53—44no—
64C42×C53Q1612yescomplemented
65C22⋊Dic53—41no—
66#11719925—84no—
67C4⋊Dic53—22no—
68C4⋊Dic53—21no—
69C4⋊Dic53—21no—
70#11719919—42no—
71C4×Dic53—41no—
72#11719917—22no—
73#11719917—41no—
74C2×#88322—41no—
75C2×#88322—42no—
76C23×C53#12011yescharacteristic
77Q8×C53—44no—
78Q8×C53—82no—
79Q8×C53—84no—
80C2×C4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—24no—
86C2×C4×C53—24no—
87C2×C4×C53—22no—
88C2×C4×C53—42no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53—42no—
93C2×C4×C53—42no—
94C2×C4×C53—41no—
95C2×C4×C53C2×Q1611yescharacteristic
96C2×C4×C53#13711yescharacteristic
97C2×Dic53—41no—
98C2×Dic53—41no—
99Dic106—82no—
100#88322—42no—
101#88322—82no—
102#12256—531nomaximal, Sylow
103C22×C53—21no—
104C22×C53—21no—
105C22×C53#45111yescharacteristic
106C22×C53#45611yescharacteristic
107C22×C53B(4,𝔽2)11yescharacteristic
108C4×C53—24no—
109C4×C53—21no—
110C4×C53—21no—
111C4×C53—44no—
112C4×C53—42no—
113C4×C53—44no—
114C4×C53—81no—
115Dic53—81no—
116Dic53—81no—
117#2551D5311yescharacteristic
118#2060—531no—
119#1605—532no—
120#1354—532no—
121#1325—531no—
122C2×C53—21no—
123C2×C53—21no—
124C2×C53#160511yescharacteristic
125C2×C53#219611yescharacteristic
126C2×C53#220511yescharacteristic
127#544—531no—
128C2×C4⋊Q8—22no—
129#451—1062no—
130#428—1061no—
131#395—21no—
132#395C2×D5312yes—
133#389—22no—
134#384C2×D5311yescharacteristic
135C4×C4⋊C4—21no—
136#343—531no—
137#339—1061no—
138#328—1062no—
139#324—532no—
140C53#1225611yescharacteristic, complemented, Sylow
141C22×Q8—21no—
142C22×Q8C22⋊D5312yes—
143C2×Q16—2122no—
144C2×M16—1061no—
145C4⋊Q8—48no—
146C4⋊Q8—2121no—
147#125—2121no—
148C22⋊Q8—1062no—
149#117—1061no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4C22⋊D5311yescharacteristic
156C2×C4⋊C4C22×D5311yescharacteristic
157C2×C42—22no—
158C2×C42C22⋊D5311yescharacteristic
159#107—2121no—
160#106—2121no—
161#103—2122no—
162C22⋊C8—1062no—
163#95—21no—
164#95—22no—
165#95C22⋊D5312yes—
166C2×Q8—44no—
167C2×Q8—44no—
168C2×Q8—42no—
169C2×Q8—81no—
170C2×Q8—1062no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—21no—
175C22×C4C2×C22⋊D5311yescharacteristic
176C22×C4C2×C22⋊D5312yes—
177Q16—4244no—
178M16—1062no—
179M16—2121no—
180C2×C8—2122no—
181C2×C8—2121no—
182C4⋊C4—24no—
183C4⋊C4—24no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—44no—
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—1062no—
190C4⋊C4—1061no—
191C4⋊C4—1061no—
192C4⋊C4—2122no—
193C22⋊C4—2121no—
194C42—21no—
195C42—44no—
196C42—2121no—
197C42#1171992512yescomplemented
198C23#1313318611yescharacteristic
199Q8—44no—
200Q8—82no—
201Q8—84no—
202Q8—4242no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—22no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—2121no—
219C2×C4—2121no—
220C2×C4C2×#1171992511yescharacteristic
221C2×C4#1313324811yescharacteristic
222C8—4242no—
223C8—2122no—
224C22—21no—
225C22—21no—
226C22#1681077911yescharacteristic
227C22#1681080611yescharacteristic
228C22#1681081011yescharacteristic
229C4—24no—
230C4—21no—
231C4—21no—
232C4—44no—
233C4—42no—
234C4—44no—
235C4—81no—
236C4—4241no—
237C4—4241no—
238C2—21no—
239C2—21no—
240C2#2027368911yescharacteristic
241C2#2027669911yescharacteristic
242C2#2027674511yescharacteristic
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2551C2
socleC22×C53B(4,𝔽2)
prime radicalC23×C53#120
hypercenter#2551D53
hypocenterC53#12256
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2551

Complements

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

NH ≅ G/Ntype
64. C42×C53Q16semidirect
140. C53#12256semidirect
197. C42#11719925semidirect

Maximal subgroups

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

subgroupindex
#202760602
#202736882
#202727452
#202726612
C53×#25512
#1225653