Finite Group Databasebeta

← #27297152

Subgroups — #27297152

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#20276060C211yesmaximal, characteristic
2#20273756C212yesmaximal
3#20272742C212yesmaximal
4#20272661C211yesmaximal, characteristic
5C53×#2571C211yesmaximal, characteristic
6#16811254C2211yescharacteristic
7#16810958—22no—
8#16810663—21no—
9#16810392C2211yescharacteristic
10#16810383—21no—
11#16810359—22no—
12#16810349C2212yes—
13C53×#400—21no—
14C53×#400C2212yes—
15C53×#389—22no—
16C53×#384C2211yescharacteristic
17C53×#382—22no—
18C4×C4⋊C4×C53—21no—
19C2×#11719917—21no—
20#13133216—41no—
21#13133208—41no—
22#13133196—22no—
23#13133176—21no—
24#13133150—41no—
25#13133150—42no—
26#13133148—41no—
27#13133140—42no—
28C22⋊#88322—22no—
29C2×C4⋊C4×C53—21no—
30C2×C4⋊C4×C53—21no—
31C2×C4⋊C4×C53—21no—
32C2×C4⋊C4×C53—22no—
33C2×C4⋊C4×C53D411yescharacteristic
34C2×C4⋊C4×C53D412yes—
35C2×C4⋊C4×C53C2311yescharacteristic
36C2×C42×C53—22no—
37C2×C42×C53D411yescharacteristic
38#95×C53—21no—
39#95×C53—22no—
40#95×C53—22no—
41#95×C53—22no—
42#95×C53D412yes—
43C22×C4×C53—21no—
44C22×C4×C53—22no—
45C22×C4×C53—22no—
46C22×C4×C53—21no—
47C22×C4×C53C2×D411yescharacteristic
48C22×C4×C53C2×D412yes—
49C2×Dic106—22no—
50C4⋊C4×C53—41no—
51C4⋊C4×C53—42no—
52C4⋊C4×C53—44no—
53C4⋊C4×C53—42no—
54C4⋊C4×C53—42no—
55C4⋊C4×C53—41no—
56C4⋊C4×C53—81no—
57C42×C53—21no—
58C42×C53—44no—
59C42×C53Q1612yes—
60C22⋊Dic53—41no—
61C4⋊Dic53—21no—
62C4⋊Dic53—21no—
63C4⋊Dic53—22no—
64#11719919—42no—
65C4×Dic53—41no—
66#11719917—22no—
67#11719917—41no—
68C2×#88322—41no—
69C2×#88322—42no—
70C23×C53#12011yescharacteristic
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—22no—
77C2×C4×C53—42no—
78C2×C4×C53—42no—
79C2×C4×C53—42no—
80C2×C4×C53—42no—
81C2×C4×C53—42no—
82C2×C4×C53—42no—
83C2×C4×C53—42no—
84C2×C4×C53—41no—
85C2×C4×C53—42no—
86C2×C4×C53C2×Q1611yescharacteristic
87C2×C4×C53#13711yescharacteristic
88C2×Dic53—41no—
89C2×Dic53—41no—
90Dic106—82no—
91#88322—42no—
92#88322—82no—
93#12257—531nomaximal, Sylow
94C22×C53—21no—
95C22×C53—21no—
96C22×C53#45111yescharacteristic
97C22×C53#45611yescharacteristic
98C22×C53B(4,𝔽2)11yescharacteristic
99C4×C53—24no—
100C4×C53—21no—
101C4×C53—21no—
102C4×C53—42no—
103C4×C53—82no—
104C4×C53—81no—
105C4×C53—82no—
106Dic53—81no—
107Dic53—81no—
108#2571D5311yescharacteristic
109#2060—531no—
110#1617—532no—
111#1353—532no—
112#1325—531no—
113C2×C53—21no—
114C2×C53—21no—
115C2×C53#160511yescharacteristic
116C2×C53#219811yescharacteristic
117C2×C53#220411yescharacteristic
118#544—531no—
119#484—1062no—
120#428—1061no—
121#400—21no—
122#400C2×D5312yes—
123#389—22no—
124#384C2×D5311yescharacteristic
125#382—22no—
126C4×C4⋊C4—21no—
127#343—531no—
128#339—1061no—
129#328—1062no—
130#324—532no—
131C53#1225711yescharacteristic, complemented, Sylow
132C2×M16—1061no—
133C4⋊Q8—2121no—
134#125—2121no—
135C22⋊Q8—1062no—
136#117—1061no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4C22⋊D5311yescharacteristic
142C2×C4⋊C4C22⋊D5312yes—
143C2×C4⋊C4C22×D5311yescharacteristic
144C2×C42—22no—
145C2×C42C22⋊D5311yescharacteristic
146#107—2121no—
147#107—2122no—
148#106—2121no—
149#103—2122no—
150C22⋊C8—1062no—
151#95—21no—
152#95—22no—
153#95—22no—
154#95—22no—
155#95C22⋊D5312yes—
156C2×Q8—1062no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—21no—
161C22×C4C2×C22⋊D5311yescharacteristic
162C22×C4C2×C22⋊D5312yes—
163M16—1062no—
164M16—2121no—
165C2×C8—2122no—
166C2×C8—2121no—
167C4⋊C4—41no—
168C4⋊C4—42no—
169C4⋊C4—41no—
170C4⋊C4—42no—
171C4⋊C4—44no—
172C4⋊C4—42no—
173C4⋊C4—81no—
174C4⋊C4—1061no—
175C4⋊C4—1061no—
176C4⋊C4—1062no—
177C4⋊C4—2122no—
178C22⋊C4—2121no—
179C42—21no—
180C42—44no—
181C42—2121no—
182C42#1171992512yes—
183C23#1313318611yescharacteristic
184Q8—4242no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—2121no—
201C2×C4—2121no—
202C2×C4C2×#1171992511yescharacteristic
203C2×C4#1313324811yescharacteristic
204C8—4242no—
205C8—2122no—
206C22—21no—
207C22—21no—
208C22#1681077911yescharacteristic
209C22#1681080611yescharacteristic
210C22#1681081011yescharacteristic
211C4—24no—
212C4—21no—
213C4—21no—
214C4—42no—
215C4—82no—
216C4—82no—
217C4—81no—
218C4—4241no—
219C4—4241no—
220C2—21no—
221C2—21no—
222C2#2027368911yescharacteristic
223C2#2027671111yescharacteristic
224C2#2027674111yescharacteristic
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2571C2
socleC22×C53B(4,𝔽2)
prime radicalC23×C53#120
hypercenter#2571D53
hypocenterC53#12257
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2571

Complements

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

NH ≅ G/Ntype
131. C53#12257semidirect

Maximal subgroups

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

subgroupindex
#202760602
#202737562
#202727422
#202726612
C53×#25712
#1225753