Finite Group Databasebeta

← #27270794

Subgroups — #27270794

Order 13568.

264 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 45 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#20275725C211yesmaximal, characteristic
2#20272639C211yesmaximal, characteristic
3C53×#1293C211yesmaximal, characteristic
4C22×Dic212C411yescharacteristic
5C2×#13133196—21no—
6C2×#13133142—21no—
7C22⋊#11719908—21no—
8#16810386C411yescharacteristic
9C53×#407—21no—
10C2×C22⋊C8×C53C2211yescharacteristic
11C22×Dic106—21no—
12C2×Dic212—42no—
13C2×Dic212—41no—
14#13133196—42no—
15#13133196—42no—
16C2×C4⋊Dic53—21no—
17C2×#11719919—41no—
18C2×C22⋊Dic53—41no—
19C2×#11719908—41no—
20#13133142—42no—
21C23×C4×C53D411yescharacteristic
22C2×M16×C53—41no—
23C22×C8×C53C2×C411yescharacteristic
24C22×C8×C53D411yescharacteristic
25C22⋊C8×C53—22no—
26C22⋊C8×C53—41no—
27C22⋊C8×C53—41no—
28C24×C53SD1611yescharacteristic
29C22×C4×C53—21no—
30C22×C4×C53—21no—
31C22×C4×C53—41no—
32C22×C4×C53—41no—
33C22×C4×C53C22⋊C411yescharacteristic
34C22×C4×C53D811yescharacteristic
35C22×Dic53—41no—
36C22×Dic53—41no—
37C2×Dic106—22no—
38C2×Dic106—41no—
39C2×Dic106—41no—
40C2×Dic106—41no—
41C2×Dic106—41no—
42C2×Dic106—41no—
43M16×C53—81no—
44M16×C53—81no—
45C2×C8×C53—21no—
46C2×C8×C53—21no—
47C2×C8×C53—41no—
48C2×C8×C53—41no—
49C2×C8×C53—42no—
50C22⋊Dic53—44no—
51C4⋊Dic53—22no—
52C4⋊Dic53—41no—
53#11719919—44no—
54Dic212—42no—
55Dic212—42no—
56#11719908—81no—
57C23×C53—21no—
58C23×C53—21no—
59C23×C53—21no—
60C23×C53—41no—
61C23×C53—41no—
62C23×C53#10211yescharacteristic
63C2×C4×C53—21no—
64C2×C4×C53—21no—
65C2×C4×C53—21no—
66C2×C4×C53—41no—
67C2×C4×C53—41no—
68C2×C4×C53—41no—
69C2×C4×C53—81no—
70C2×C4×C53#9911yescharacteristic
71C2×C4×C53#10411yescharacteristic
72C2×Dic53—42no—
73C2×Dic53—42no—
74C2×Dic53—81no—
75C2×Dic53—81no—
76C2×Dic53—81no—
77C2×Dic53—81no—
78Dic106—42no—
79Dic106—42no—
80Dic106—81no—
81Dic106—81no—
82C8×C53—41no—
83C8×C53—41no—
84C8×C53—82no—
85#7086—531nomaximal, Sylow
86C22×C53—21no—
87C22×C53—21no—
88C22×C53—42no—
89C22×C53—41no—
90C22×C53—41no—
91C22×C53—41no—
92C22×C53—81no—
93C22×C53#32711yescharacteristic
94C22×C53#35911yescharacteristic
95C22×C53#36211yescharacteristic
96C4×C53—21no—
97C4×C53—41no—
98C4×C53AΣL(1,𝔽16)11yescharacteristic
99C4×C53#35311yescharacteristic
100Dic53—82no—
101Dic53—82no—
102#2001—531no—
103#1314—531no—
104#1293D5311yescharacteristic
105C2×C53—21no—
106C2×C53—21no—
107C2×C53—81no—
108C2×C53#134211yescharacteristic
109C2×C53#134311yescharacteristic
110C2×C53#135011yescharacteristic
111C22×Q16—531no—
112C2×C22⋊Q8—1061no—
113C2×#103—1061no—
114#407—21no—
115C2×C22⋊C8C2×D5311yescharacteristic
116C22⋊C16—1061no—
117#340—531no—
118C53#708611yescharacteristic, complemented, Sylow
119C22×Q8—1061no—
120C23×C4C22⋊D5311yescharacteristic
121C2×Q16—2121no—
122C2×Q16—2122no—
123C2×M16—41no—
124C22×C8C4×D5311yescharacteristic
125C22×C8D21211yescharacteristic
126C22⋊Q8—2122no—
127C22⋊Q8—2122no—
128C2×C4⋊C4—1061no—
129C2×C4⋊C4—2121no—
130C2×C22⋊C4—2121no—
131C2×C16—2121no—
132#103—2122no—
133C22⋊C8—22no—
134C22⋊C8—41no—
135C22⋊C8—41no—
136C24#1171992411yescharacteristic
137C2×Q8—1062no—
138C2×Q8—2121no—
139C2×Q8—2121no—
140C2×Q8—2121no—
141C2×Q8—2121no—
142C2×Q8—2121no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—2121no—
148C22×C4—2121no—
149C22×C4#1171992111yescharacteristic
150C22×C4#1171992211yescharacteristic
151Q16—2122no—
152Q16—2122no—
153M16—81no—
154M16—81no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—42no—
160C4⋊C4—1062no—
161C4⋊C4—2121no—
162C4⋊C4—2124no—
163C22⋊C4—2124no—
164C16—4241no—
165C23—21no—
166C23—21no—
167C23—21no—
168C23—41no—
169C23—41no—
170C23#1313313711yescharacteristic
171Q8—4241no—
172Q8—4241no—
173Q8—2122no—
174Q8—2122no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—81no—
182C2×C4—4241no—
183C2×C4—4241no—
184C2×C4—4241no—
185C2×C4—4241no—
186C2×C4—2122no—
187C2×C4—2122no—
188C2×C4#1313313111yescharacteristic
189C2×C4#1313314511yescharacteristic
190C8—41no—
191C8—41no—
192C8—82no—
193C22—21no—
194C22—21no—
195C22—42no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22—81no—
200C22#1681035611yescharacteristic
201C22#1681042911yescharacteristic
202C22#1681043811yescharacteristic
203C4—21no—
204C4—41no—
205C4—4242no—
206C4—4242no—
207C4#1681040911yescharacteristic
208C4#1681041511yescharacteristic
209C2—21no—
210C2—21no—
211C2—81no—
212C2#2027270811yescharacteristic
213C2#2027271111yescharacteristic
214C2#2027273211yescharacteristic
subgroupG/H
centerC22#16810356
commutatorC22×C8×C53C2×C4
FrattiniC2×C22⋊C8C2×D53
FittingC53×#1293C2
socleC22×C53#327
prime radicalC24×C53SD16
cosocleC2×C22⋊C8×C53C22
hypercenter#1293D53
hypocenterC53#7086
LCSG ▷ C22×C8×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #1293

Complements

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

NH ≅ G/Ntype
118. C53#7086semidirect

Maximal subgroups

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

subgroupindex
#202757252
#202726392
C53×#12932
#708653