Finite Group Databasebeta

← #27297090

Subgroups — #27297090

Order 13568.

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

normal subgroups only

This group has 394 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#20273722C212yesmaximal
3#20272721C212yesmaximal
4#20272661C211yesmaximal, characteristic
5C53×#2582C211yesmaximal, characteristic
6#16811254C2211yescharacteristic
7#16810953—22no—
8#16810663—21no—
9#16810392C2211yescharacteristic
10#16810383—21no—
11#16810359—22no—
12#16810349C2212yes—
13C2×C4×Q8×C53—21no—
14C53×#398—21no—
15C53×#398C2212yes—
16C53×#391—22no—
17C53×#389—22no—
18C53×#384C2211yescharacteristic
19C2×#11719917—21no—
20#13133216—41no—
21#13133208—41no—
22#13133196—22no—
23#13133176—21no—
24#13133150—41no—
25#13133148—41no—
26#13133148—42no—
27#13133140—42no—
28C22⋊#88322—22no—
29C22×Q8×C53—21no—
30C4×Q8×C53—42no—
31C4×Q8×C53—42no—
32C2×C4⋊C4×C53—21no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—22no—
35C2×C4⋊C4×C53—22no—
36C2×C4⋊C4×C53D411yescharacteristic
37C2×C4⋊C4×C53D412yes—
38C2×C4⋊C4×C53C2311yescharacteristic
39C2×C42×C53—22no—
40C2×C42×C53D411yescharacteristic
41#95×C53—22no—
42#95×C53—21no—
43#95×C53—22no—
44#95×C53D412yes—
45C2×Q8×C53—24no—
46C2×Q8×C53—42no—
47C2×Q8×C53—42no—
48C22×C4×C53—21no—
49C22×C4×C53—22no—
50C22×C4×C53—21no—
51C22×C4×C53—22no—
52C22×C4×C53C2×D411yescharacteristic
53C22×C4×C53C2×D412yes—
54C2×Dic106—22no—
55C4⋊C4×C53—41no—
56C4⋊C4×C53—44no—
57C4⋊C4×C53—42no—
58C4⋊C4×C53—42no—
59C4⋊C4×C53—41no—
60C4⋊C4×C53—42no—
61C4⋊C4×C53—42no—
62C4⋊C4×C53—81no—
63C42×C53—21no—
64C42×C53—44no—
65C42×C53SD1612yes—
66C22⋊Dic53—41no—
67C4⋊Dic53—21no—
68C4⋊Dic53—22no—
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—48no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—24no—
84C2×C4×C53—22no—
85C2×C4×C53—42no—
86C2×C4×C53—42no—
87C2×C4×C53—42no—
88C2×C4×C53—42no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53—41no—
93C2×C4×C53C2×SD1611yescharacteristic
94C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
95C2×Dic53—41no—
96C2×Dic53—41no—
97Dic106—82no—
98#88322—42no—
99#88322—82no—
100#12247—531nomaximal, Sylow
101C22×C53—21no—
102C22×C53—21no—
103C22×C53#45011yescharacteristic
104C22×C53#45611yescharacteristic
105C22×C53B(4,𝔽2)11yescharacteristic
106C4×C53—24no—
107C4×C53—21no—
108C4×C53—21no—
109C4×C53—44no—
110C4×C53—42no—
111C4×C53—82no—
112C4×C53—81no—
113Dic53—81no—
114Dic53—81no—
115#2582D5311yescharacteristic
116#2060—531no—
117#1611—532no—
118#1346—532no—
119#1325—531no—
120C2×C53—21no—
121C2×C53—21no—
122C2×C53#160411yescharacteristic
123C2×C53#219811yescharacteristic
124C2×C53#220511yescharacteristic
125#544—531no—
126C2×C4×Q8—21no—
127#483—1062no—
128#428—1061no—
129#398—21no—
130#398C2×D5312yes—
131#391—22no—
132#389—22no—
133#384C2×D5311yescharacteristic
134#343—531no—
135#339—1061no—
136#328—1062no—
137#324—532no—
138C53#1224711yescharacteristic, complemented, Sylow
139C22×Q8—21no—
140C2×M16—1061no—
141C4⋊Q8—2121no—
142#125—2121no—
143C22⋊Q8—1062no—
144C4×Q8—42no—
145C4×Q8—42no—
146#117—1061no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4C22⋊D5311yescharacteristic
152C2×C4⋊C4C22⋊D5312yes—
153C2×C4⋊C4C22×D5311yescharacteristic
154C2×C42—22no—
155C2×C42C22⋊D5311yescharacteristic
156#107—2121no—
157#106—2121no—
158#106—2122no—
159#103—2122no—
160C22⋊C8—1062no—
161#95—22no—
162#95—21no—
163#95—22no—
164#95C22⋊D5312yes—
165C2×Q8—24no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—1062no—
169C22×C4—21no—
170C22×C4—22no—
171C22×C4—21no—
172C22×C4—22no—
173C22×C4C2×C22⋊D5311yescharacteristic
174C22×C4C2×C22⋊D5312yes—
175M16—1062no—
176M16—2121no—
177C2×C8—2122no—
178C2×C8—2121no—
179C4⋊C4—41no—
180C4⋊C4—42no—
181C4⋊C4—41no—
182C4⋊C4—42no—
183C4⋊C4—44no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—81no—
187C4⋊C4—1061no—
188C4⋊C4—1062no—
189C4⋊C4—1061no—
190C4⋊C4—2122no—
191C22⋊C4—2121no—
192C42—21no—
193C42—44no—
194C42—2121no—
195C42#1171992312yes—
196C23#1313318611yescharacteristic
197Q8—48no—
198Q8—4242no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—24no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—2121no—
215C2×C4—2121no—
216C2×C4C2×#1171992311yescharacteristic
217C2×C4#1313324311yescharacteristic
218C8—4242no—
219C8—2122no—
220C22—21no—
221C22—21no—
222C22#1681077411yescharacteristic
223C22#1681080611yescharacteristic
224C22#1681081011yescharacteristic
225C4—24no—
226C4—21no—
227C4—21no—
228C4—44no—
229C4—42no—
230C4—82no—
231C4—81no—
232C4—4241no—
233C4—4241no—
234C2—21no—
235C2—21no—
236C2#2027368411yescharacteristic
237C2#2027671111yescharacteristic
238C2#2027674511yescharacteristic
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2582C2
socleC22×C53B(4,𝔽2)
prime radicalC23×C53#120
hypercenter#2582D53
hypocenterC53#12247
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2582

Complements

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

NH ≅ G/Ntype
138. C53#12247semidirect

Maximal subgroups

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

subgroupindex
#202760602
#202737222
#202727212
#202726612
C53×#25822
#1224753