Finite Group Databasebeta

← D53⋊#1356 #28309586

Subgroups — D53⋊#1356 #28309586

Order 13568.

314 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 68 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#20290932C212yesmaximal
2#20289722C211yesmaximal, characteristic, complemented
3#20289721C211yesmaximal, characteristic, complemented
4#20289720C212yesmaximal, complemented
5#491×D53C211yesmaximal, characteristic
6D53⋊M32—22no—
7#16811516—22no—
8#16811515C2212yes—
9#16811515C2212yes—
10C4⋊Q8×D53C411yescharacteristic
11#125×D53C411yescharacteristic
12#16810993C2211yescharacteristic
13#16810992C411yescharacteristic
14#16810991C2211yescharacteristic
15#16810990—22no—
16#103×D53—22no—
17#97×D53C2211yescharacteristic
18C53×#491C411yescharacteristic
19#13133292—24no—
20#13133291—22no—
21#13133291—22no—
22D53⋊C16—24no—
23C2×Q8×D53—21no—
24C2×C8×D53D412yes—
25#13133217C2×C411yescharacteristic
26#13133216—41no—
27#13133215—21no—
28#13133209C2×C411yescharacteristic
29#13133208—21no—
30#13133207—41no—
31C4⋊C4×D53—21no—
32C4⋊C4×D53—41no—
33C4⋊C4×D53—41no—
34C42×D53C2×C411yescharacteristic
35#13133142—22no—
36#13133141—22no—
37#13133140—22no—
38#13133128C2311yescharacteristic
39#13133127D412yes—
40C4⋊Q8×C53C2×C411yescharacteristic
41#125×C53C2×C411yescharacteristic
42#103×C53—22no—
43#97×C53C2×C411yescharacteristic
44C2×Q8×C53—21no—
45Q8×D53—42no—
46Q8×D53—42no—
47C2×C4×D53—21no—
48C2×C4×D53—41no—
49C2×C4×D53—41no—
50C2×C4×D53C22⋊C411yescharacteristic
51C2×Dic106—21no—
52C2×Dic106—41no—
53C2×C8×C53C22⋊C412yes—
54C4⋊C4×C53—21no—
55C4⋊C4×C53—41no—
56C4⋊C4×C53—41no—
57C42×C53C22×C411yescharacteristic
58C4⋊Dic53—21no—
59C4⋊Dic53—41no—
60C4⋊Dic53—41no—
61#11719919—41no—
62#11719919—42no—
63#11719919—41no—
64#11719919—41no—
65C4×Dic53—21no—
66C4×Dic53C22⋊C411yescharacteristic
67C2×#88322C2×D412yes—
68C8×D53—22no—
69C8×D53—22no—
70#11719910—24no—
71#11719910—24no—
72C22×D53#10011yescharacteristic
73Q8×C53—42no—
74C2×C4×C53—21no—
75C2×C4×C53—41no—
76C2×C4×C53—41no—
77C2×C4×C53C2×C22⋊C411yescharacteristic
78C2×Dic53—21no—
79C2×Dic53—41no—
80C2×Dic53—41no—
81C2×Dic53#10011yescharacteristic
82C4×D53—42no—
83C4×D53—82no—
84C4×D53—81no—
85C4×D53—81no—
86C4×D53#10412yes—
87C4×D53#10412yes—
88Dic106—42no—
89Dic106—82no—
90C8×C53—22no—
91#88322—22no—
92C2×#1356—531nomaximal, Sylow
93C22×C53C2×#10011yescharacteristic
94C2×D53—22no—
95C2×D53#32911yescharacteristic
96C2×D53#32911yescharacteristic
97C4×C53—41no—
98C4×C53—81no—
99C4×C53—81no—
100C4×C53C2×#10412yes—
101Dic53—41no—
102Dic53—81no—
103Dic53—81no—
104Dic53#32912yes—
105C2×#491—531no—
106C2×#347—532no—
107#1356—532no—
108#1356—531no—
109#1356—531no—
110C2×C53—21no—
111C2×C53C2×#32911yescharacteristic
112D53—21no—
113D53#135612yescomplemented
114C2×C4⋊Q8—531no—
115C2×#125—531no—
116C2×M32—1062no—
117#491—531no—
118#491—531no—
119#491—531no—
120#491—1062no—
121#491#615711yescharacteristic
122C2×#103—1062no—
123C2×#97—531no—
124#347—532no—
125#347—532no—
126#347—1062no—
127C53C2×#135611yescharacteristic, complemented, Sylow
128C22×Q8—1061no—
129C22×C8—532no—
130C4⋊Q8—531no—
131C4⋊Q8—1061no—
132C4⋊Q8—2121no—
133C4⋊Q8C2×#615711yescharacteristic
134#125—531no—
135#125—1061no—
136#125—2121no—
137#125C2×#615711yescharacteristic
138C2×C4⋊C4—1061no—
139C2×C4⋊C4—2121no—
140C2×C4⋊C4—2121no—
141C2×C42—531no—
142M32—1062no—
143M32—1062no—
144M32—1064no—
145C2×C16—1064no—
146#103—22no—
147#103—1062no—
148#103—1062no—
149#103—1062no—
150#97—532no—
151#97—531no—
152#97C2×#615711yescharacteristic
153C2×Q8—21no—
154C2×Q8—1061no—
155C2×Q8—2121no—
156C2×Q8—2122no—
157C2×Q8—2122no—
158C22×C4—531no—
159C22×C4—1061no—
160C22×C4—2121no—
161C22×C4—2121no—
162C2×C8—532no—
163C2×C8—1062no—
164C2×C8—1062no—
165C2×C8C22⋊#615712yes—
166C4⋊C4—21no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—1061no—
170C4⋊C4—2121no—
171C4⋊C4—2121no—
172C4⋊C4—2121no—
173C4⋊C4—2121no—
174C4⋊C4—2121no—
175C4⋊C4—2122no—
176C42—531no—
177C42—1061no—
178C42C22×#615711yescharacteristic
179C16—1064no—
180C16—1064no—
181C23—531no—
182Q8—42no—
183Q8—4242no—
184Q8—2122no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—531no—
189C2×C4—532no—
190C2×C4—532no—
191C2×C4—1061no—
192C2×C4—4241no—
193C2×C4—4241no—
194C2×C4—4242no—
195C2×C4—2122no—
196C2×C4—2121no—
197C2×C4—2121no—
198C2×C4C2×C22⋊#615711yescharacteristic
199C8—22no—
200C8—1062no—
201C22—531no—
202C22—531no—
203C22—1062no—
204C22D53⋊#10011yescharacteristic
205C4—41no—
206C4—81no—
207C4—81no—
208C4—532no—
209C4—4241no—
210C4—4241no—
211C4—2121no—
212C4D53⋊#10412yes—
213C2—21no—
214C2—532no—
215C2—1061no—
216C2D53⋊#32911yescharacteristic
subgroupG/H
centerC2D53⋊#329
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
FittingC53×#491C4
socleC2×C53C2×#329
prime radicalC22×D53#100
cosocle#13133128C23
hypercenter#491#6157
hypocenterC53C2×#1356
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #491

Complements

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

NH ≅ G/Ntype
2. #20289722C2semidirect
3. #20289721C2semidirect
4. #20289720C2semidirect
113. D53#1356semidirect
127. C53C2×#1356semidirect

Maximal subgroups

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

subgroupindex
#202909322
#202897222
#202897212
#202897202
#491×D532
C2×#135653