Finite Group Databasebeta

← #27297114

Subgroups — #27297114

Order 13568.

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

normal subgroups only

This group has 342 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#20276065C211yesmaximal, characteristic
2#20273722C212yesmaximal
3#20272721C212yesmaximal
4#20272659C211yesmaximal, characteristic
5C53×#2822C211yesmaximal, characteristic
6#16811254C2211yescharacteristic
7#16810953—22no—
8#16810663—21no—
9#16810392C2211yescharacteristic
10#16810383—21no—
11#16810359—22no—
12#16810349C2212yes—
13C2×#125×C53—21no—
14C53×#400—22no—
15C53×#398—21no—
16C53×#398—22no—
17C53×#398C2212yes—
18C53×#382C2211yescharacteristic
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—
29#125×C53—42no—
30#125×C53—42no—
31C2×C4⋊C4×C53—21no—
32C2×C4⋊C4×C53—22no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53—22no—
36C2×C4⋊C4×C53D411yescharacteristic
37C2×C4⋊C4×C53D412yes—
38C2×C4⋊C4×C53C2311yescharacteristic
39C2×C42×C53D411yescharacteristic
40#95×C53—22no—
41#95×C53—21no—
42#95×C53—22no—
43#95×C53—22no—
44#95×C53D412yes—
45C22×C4×C53—21no—
46C22×C4×C53—21no—
47C22×C4×C53—22no—
48C22×C4×C53—22no—
49C22×C4×C53C2×D411yescharacteristic
50C22×C4×C53C2×D412yes—
51C2×Dic106—22no—
52C4⋊C4×C53—41no—
53C4⋊C4×C53—42no—
54C4⋊C4×C53—44no—
55C4⋊C4×C53—42no—
56C4⋊C4×C53—42no—
57C4⋊C4×C53—41no—
58C4⋊C4×C53—42no—
59C4⋊C4×C53—42no—
60C4⋊C4×C53—81no—
61C42×C53—21no—
62C42×C53SD1612yes—
63C22⋊Dic53—41no—
64C4⋊Dic53—21no—
65C4⋊Dic53—22no—
66C4⋊Dic53—21no—
67#11719919—42no—
68C4×Dic53—41no—
69#11719917—22no—
70#11719917—41no—
71C2×#88322—41no—
72C2×#88322—42no—
73C23×C53#12011yescharacteristic
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—41no—
79C2×C4×C53—41no—
80C2×C4×C53—42no—
81C2×C4×C53—42no—
82C2×C4×C53—42no—
83C2×C4×C53—42no—
84C2×C4×C53—42no—
85C2×C4×C53—42no—
86C2×C4×C53—42no—
87C2×C4×C53—42no—
88C2×C4×C53—41no—
89C2×C4×C53C2×SD1611yescharacteristic
90C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
91C2×Dic53—41no—
92C2×Dic53—41no—
93Dic106—82no—
94#88322—42no—
95#88322—82no—
96#12251—531nomaximal, Sylow
97C22×C53—21no—
98C22×C53—21no—
99C22×C53#45011yescharacteristic
100C22×C53#45611yescharacteristic
101C22×C53B(4,𝔽2)11yescharacteristic
102C4×C53—21no—
103C4×C53—21no—
104C4×C53—42no—
105C4×C53—42no—
106C4×C53—82no—
107C4×C53—82no—
108C4×C53—81no—
109Dic53—81no—
110Dic53—81no—
111#2822D5311yescharacteristic
112#2061—531no—
113#1611—532no—
114#1346—532no—
115#1324—531no—
116C2×C53—21no—
117C2×C53—21no—
118C2×C53#160411yescharacteristic
119C2×C53#220111yescharacteristic
120C2×C53#220311yescharacteristic
121#544—531no—
122C2×#125—21no—
123#483—1062no—
124#428—1061no—
125#400—22no—
126#398—21no—
127#398—22no—
128#398C2×D5312yes—
129#382C2×D5311yescharacteristic
130#343—531no—
131#339—1061no—
132#328—1062no—
133#324—532no—
134C53#1225111yescharacteristic, complemented, Sylow
135C2×M16—1061no—
136C4⋊Q8—2121no—
137#125—42no—
138#125—42no—
139#125—2121no—
140C22⋊Q8—1062no—
141#117—1061no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4C22⋊D5311yescharacteristic
148C2×C4⋊C4C22⋊D5312yes—
149C2×C4⋊C4C22×D5311yescharacteristic
150C2×C42C22⋊D5311yescharacteristic
151#107—2121no—
152#106—2121no—
153#106—2122no—
154#103—2122no—
155C22⋊C8—1062no—
156#95—22no—
157#95—21no—
158#95—22no—
159#95—22no—
160#95C22⋊D5312yes—
161C2×Q8—1062no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4C2×C22⋊D5311yescharacteristic
167C22×C4C2×C22⋊D5312yes—
168M16—1062no—
169M16—2121no—
170C2×C8—2122no—
171C2×C8—2121no—
172C4⋊C4—41no—
173C4⋊C4—42no—
174C4⋊C4—41no—
175C4⋊C4—42no—
176C4⋊C4—44no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—81no—
181C4⋊C4—1061no—
182C4⋊C4—1062no—
183C4⋊C4—1061no—
184C4⋊C4—2122no—
185C22⋊C4—2121no—
186C42—21no—
187C42—2121no—
188C42#1171992312yes—
189C23#1313318611yescharacteristic
190Q8—4242no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—2121no—
207C2×C4—2121no—
208C2×C4C2×#1171992311yescharacteristic
209C2×C4#1313324311yescharacteristic
210C8—4242no—
211C8—2122no—
212C22—21no—
213C22—21no—
214C22#1681077411yescharacteristic
215C22#1681080611yescharacteristic
216C22#1681081011yescharacteristic
217C4—21no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—82no—
222C4—81no—
223C4—82no—
224C4—4241no—
225C4—4241no—
226C2—21no—
227C2—21no—
228C2#2027368411yescharacteristic
229C2#2027672811yescharacteristic
230C2#2027673611yescharacteristic
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2822C2
socleC22×C53B(4,𝔽2)
prime radicalC23×C53#120
hypercenter#2822D53
hypocenterC53#12251
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2822

Complements

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

NH ≅ G/Ntype
134. C53#12251semidirect

Maximal subgroups

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

subgroupindex
#202760652
#202737222
#202727212
#202726592
C53×#28222
#1225153