Finite Group Databasebeta

← D53×#1355 #27295261

Subgroups — D53×#1355 #27295261

Order 13568.

366 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 71 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#490×D53C211yesmaximal, characteristic
2#347×D53C212yesmaximal, complemented
3#20272748C211yesmaximal, characteristic, complemented
4#20272747C212yesmaximal, complemented
5C53×#1355C211yesmaximal, characteristic, complemented
6#127×D53C411yescharacteristic
7#125×D53C411yescharacteristic
8M32×D53—22no—
9#16810989C411yescharacteristic
10#16810988C2211yescharacteristic
11#16810987C411yescharacteristic
12#16810986—22no—
13#102×D53—22no—
14#97×D53C2211yescharacteristic
15#16810401C2212yescomplemented
16#16810400—22no—
17C53×#490C2211yescharacteristic
18C53×#347C2212yescomplemented
19C2×D4×D53—21no—
20C2×D4×D53—41no—
21C2×C8×D53D412yes—
22#13133214C2×C411yescharacteristic
23#13133213—21no—
24#13133213—41no—
25#13133209C2×C411yescharacteristic
26#13133208—21no—
27#13133207—41no—
28C4⋊C4×D53—21no—
29C4⋊C4×D53—41no—
30C42×D53C2×C411yescharacteristic
31#13133157—22no—
32#13133156—24no—
33C16×D53—24no—
34#13133139—22no—
35#13133138—22no—
36#13133137—22no—
37#13133128C2×C411yescharacteristic
38#13133127D412yes—
39#127×C53C2×C411yescharacteristic
40#125×C53C2×C411yescharacteristic
41M32×C53—22no—
42#102×C53—22no—
43#97×C53C2311yescharacteristic
44C23×D53—41no—
45C2×D4×C53—21no—
46C2×D4×C53—41no—
47C2×C22⋊D53—42no—
48C2×C22⋊D53—41no—
49D4×D53—42no—
50D4×D53—42no—
51D4×D53—84no—
52C2×D212—21no—
53C2×D212—41no—
54C2×C4×D53—21no—
55C2×C4×D53—41no—
56C2×C4×D53C22⋊C411yescharacteristic
57C2×C8×C53C2×D412yes—
58C4⋊C4×C53—21no—
59C4⋊C4×C53—41no—
60C42×C53C22×C411yescharacteristic
61C4⋊Dic53—21no—
62C4⋊Dic53—41no—
63#11719919—41no—
64#11719919—41no—
65#11719919—41no—
66C4×Dic53—21no—
67C4×Dic53C22⋊C411yescharacteristic
68C2×#88322C22⋊C412yes—
69C8×D53—22no—
70C8×D53—22no—
71C16×C53—24no—
72#11719908—24no—
73C23×C53—41no—
74C22×D53—41no—
75C22×D53—84no—
76C22×D53—81no—
77C22×D53—81no—
78C22×D53#10011yescharacteristic
79D4×C53—42no—
80D4×C53—82no—
81C2×C4×C53—21no—
82C2×C4×C53—41no—
83C2×C4×C53C2×C22⋊C411yescharacteristic
84C22⋊D53—84no—
85C22⋊D53—82no—
86C2×Dic53—21no—
87C2×Dic53—41no—
88C2×Dic53#10011yescharacteristic
89D212—42no—
90D212—82no—
91C4×D53—42no—
92C4×D53—82no—
93C4×D53#10412yes—
94C4×D53#10412yes—
95C8×C53—22no—
96#88322—22no—
97C2×#1355—531nomaximal, Sylow
98C22×C53—82no—
99C22×C53—81no—
100C22×C53C2×#10011yescharacteristic
101C2×D53—22no—
102C2×D53—82no—
103C2×D53—81no—
104C2×D53—162no—
105C2×D53—162no—
106C2×D53#32911yescharacteristic
107C2×D53#32911yescharacteristic
108C4×C53—41no—
109C4×C53—81no—
110C4×C53C2×#10412yes—
111Dic53—41no—
112Dic53—81no—
113Dic53#32912yes—
114C2×#490—531no—
115C2×#347—532no—
116#1355—531no—
117#1355—532no—
118#1355D5311yescharacteristic, direct product factor
119C2×C53—21no—
120C2×C53—161no—
121C2×C53C2×#32911yescharacteristic
122D53—21no—
123D53—161no—
124D53#135512yesdirect product factor
125C2×#127—531no—
126C2×#125—531no—
127C2×M32—1062no—
128#490—531no—
129#490—531no—
130#490—531no—
131#490—1062no—
132#490C2×D5311yescharacteristic
133C2×#102—1062no—
134C2×#97—531no—
135#347—532no—
136#347—1062no—
137#347C2×D5312yescomplemented
138C53C2×#135511yescharacteristic, complemented, Sylow
139C22×D4—1061no—
140C22×D4—2121no—
141C22×C8—532no—
142#127—531no—
143#127—1061no—
144#127—2121no—
145#127C4×D5311yescharacteristic
146#125—531no—
147#125—1061no—
148#125—2121no—
149#125C4×D5311yescharacteristic
150C2×C4⋊C4—1061no—
151C2×C4⋊C4—2121no—
152C2×C42—531no—
153M32—22no—
154M32—1064no—
155M32—1062no—
156C2×C16—1064no—
157#102—22no—
158#102—1062no—
159#102—1062no—
160#102—1062no—
161#97—532no—
162#97—531no—
163#97C22×D5311yescharacteristic
164C24—2121no—
165C2×D4—21no—
166C2×D4—41no—
167C2×D4—1061no—
168C2×D4—4244no—
169C2×D4—2121no—
170C2×D4—2122no—
171C2×D4—2122no—
172C2×D4—2121no—
173C2×D4—2122no—
174C22×C4—531no—
175C22×C4—1061no—
176C22×C4—2121no—
177C2×C8—532no—
178C2×C8—1062no—
179C2×C8—1062no—
180C2×C8D4×D5312yes—
181C4⋊C4—21no—
182C4⋊C4—41no—
183C4⋊C4—1061no—
184C4⋊C4—2121no—
185C4⋊C4—2121no—
186C4⋊C4—2121no—
187C4⋊C4—2121no—
188C42—531no—
189C42—1061no—
190C42C2×C4×D5311yescharacteristic
191C16—24no—
192C16—1064no—
193C23—41no—
194C23—531no—
195C23—4241no—
196C23—4241no—
197C23—4244no—
198C23—2121no—
199D4—42no—
200D4—82no—
201D4—4242no—
202D4—4242no—
203D4—4244no—
204D4—2122no—
205C2×C4—21no—
206C2×C4—41no—
207C2×C4—531no—
208C2×C4—532no—
209C2×C4—532no—
210C2×C4—1061no—
211C2×C4—4242no—
212C2×C4—2122no—
213C2×C4—2121no—
214C2×C4C22⋊C4×D5311yescharacteristic
215C8—22no—
216C8—1062no—
217C22—82no—
218C22—81no—
219C22—531no—
220C22—531no—
221C22—8482no—
222C22—8482no—
223C22—1062no—
224C22—4242no—
225C22—4241no—
226C22#100×D5311yescharacteristic
227C4—41no—
228C4—81no—
229C4—532no—
230C4—4241no—
231C4—2121no—
232C4#104×D5312yes—
233C2—21no—
234C2—161no—
235C2—532no—
236C2—8481no—
237C2—1061no—
238C2#329×D5311yescharacteristic
subgroupG/H
centerC2#329×D53
commutatorC42×C53C22×C4
Frattini#97C22×D53
FittingC53×#1355C2
socleC2×C53C2×#329
prime radical#127×D53C4
cosocle#97×C53C23
hypercenter#1355D53
hypocenterC53C2×#1355
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #1355

Complements

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

NH ≅ G/Ntype
2. #347×D53C2semidirect
3. #20272748C2semidirect
4. #20272747C2semidirect
5. C53×#1355C2semidirect
15. #16810401C22semidirect
18. C53×#347C22semidirect
118. #1355D53direct (N × H)
124. D53#1355direct (N × H)
137. #347C2×D53semidirect
138. C53C2×#1355semidirect

Maximal subgroups

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

subgroupindex
#490×D532
#347×D532
#202727482
#202727472
C53×#13552
C2×#135553