Finite Group Databasebeta

← #27295267

Subgroups — #27295267

Order 13568.

318 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 69 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#20285059C211yesmaximal, characteristic
2#347×D53C212yesmaximal, complemented
3#20272749C212yesmaximal, complemented
4#20272748C211yesmaximal, characteristic, complemented
5C53×#1355C211yesmaximal, characteristic, complemented
6#16811179C411yescharacteristic
7#16811171C411yescharacteristic
8M32×D53—22no—
9#16810993C411yescharacteristic
10#16810992C2211yescharacteristic
11#16810987C411yescharacteristic
12#16810981—22no—
13#16810616—22no—
14#97×D53C2211yescharacteristic
15#16810401C2212yes—
16#16810400—22no—
17C53×#490C2211yescharacteristic
18C53×#347C2212yes—
19C2×#11719947—21no—
20C2×C8×D53D412yes—
21#13133217C2×C411yescharacteristic
22#13133216—21no—
23#13133209C2×C411yescharacteristic
24#13133204—21no—
25#13133195—41no—
26#13133188—41no—
27#13133182—41no—
28D4×Dic53—41no—
29#13133175—21no—
30C42×D53C2×C411yescharacteristic
31#13133157—22no—
32#13133156—24no—
33C16×D53—24no—
34#13133142—22no—
35#13133140—22no—
36#13133138—22no—
37#13133128C2×C411yescharacteristic
38#13133127D412yes—
39#127×C53C2×C411yescharacteristic
40#125×C53C2×C411yescharacteristic
41M32×C53—22no—
42#102×C53—22no—
43#97×C53C2311yescharacteristic
44C2×D4×C53—21no—
45C2×D4×C53—41no—
46C2×C22⋊D53—41no—
47C22×Dic53—41no—
48#11719947—42no—
49#11719947—42no—
50C2×C4×D53—21no—
51C2×C4×D53C22⋊C411yescharacteristic
52C2×Dic106—21no—
53C2×Dic106—41no—
54C2×C8×C53C2×D412yes—
55C4⋊C4×C53—21no—
56C4⋊C4×C53—41no—
57C42×C53C22×C411yescharacteristic
58C22⋊Dic53—42no—
59#11719921—41no—
60C4⋊Dic53—21no—
61C4⋊Dic53—41no—
62#11719919—41no—
63#11719919—41no—
64C4×Dic53—21no—
65C4×Dic53—41no—
66C4×Dic53C22⋊C411yescharacteristic
67C2×#88322C22⋊C412yes—
68C8×D53—22no—
69C8×D53—22no—
70C16×C53—24no—
71#11719908—24no—
72C23×C53—41no—
73C22×D53#10011yescharacteristic
74D4×C53—42no—
75D4×C53—82no—
76C2×C4×C53—21no—
77C2×C4×C53—41no—
78C2×C4×C53C2×C22⋊C411yescharacteristic
79C22⋊D53—82no—
80C2×Dic53—21no—
81C2×Dic53—41no—
82C2×Dic53—41no—
83C2×Dic53—81no—
84C2×Dic53—81no—
85C2×Dic53#10011yescharacteristic
86C4×D53—81no—
87C4×D53#10412yes—
88C4×D53#10412yes—
89Dic106—42no—
90Dic106—82no—
91C8×C53—22no—
92#88322—22no—
93#11911—531nomaximal, Sylow
94C22×C53—82no—
95C22×C53—81no—
96C22×C53C2×#10011yescharacteristic
97C2×D53—41no—
98C2×D53#32912yes—
99C4×C53—41no—
100C4×C53—81no—
101C4×C53C2×#10412yes—
102Dic53—41no—
103Dic53—81no—
104Dic53—81no—
105Dic53#32911yescharacteristic
106Dic53#32911yescharacteristic
107#3135—531no—
108C2×#347—532no—
109#1356—532no—
110#1355—531no—
111#1355D5311yescharacteristic, complemented
112C2×C53—21no—
113C2×C53—161no—
114C2×C53C2×#32911yescharacteristic
115D53—22no—
116#533—531no—
117#532—531no—
118C2×M32—1062no—
119#491—531no—
120#491—531no—
121#490—531no—
122#490C2×D5311yescharacteristic
123#489—1062no—
124#416—1062no—
125C2×#97—531no—
126#347—532no—
127#347—1062no—
128#347C2×D5312yes—
129C53#1191111yescharacteristic, complemented, Sylow
130C2×#41—1061no—
131C22×C8—532no—
132C4⋊Q8—531no—
133C4⋊Q8—1061no—
134#127C4×D5311yescharacteristic
135#125—531no—
136#125C4×D5311yescharacteristic
137#124—1061no—
138C22⋊Q8—2121no—
139#121—2121no—
140C4×Q8—2121no—
141C4×D4—2121no—
142#117—1061no—
143C2×C42—531no—
144M32—22no—
145M32—1064no—
146M32—1062no—
147C2×C16—1064no—
148#103—1062no—
149#103—1062no—
150#102—22no—
151#102—1062no—
152#97—532no—
153#97—531no—
154#97C22×D5311yescharacteristic
155#41—2122no—
156#41—2122no—
157C2×Q8—1061no—
158C2×Q8—2121no—
159C2×D4—21no—
160C2×D4—41no—
161C2×D4—2121no—
162C22×C4—531no—
163C22×C4—1061no—
164C22×C4—2121no—
165C2×C8—532no—
166C2×C8—1062no—
167C2×C8—1062no—
168C2×C8D4×D5312yes—
169C4⋊C4—21no—
170C4⋊C4—41no—
171C4⋊C4—1061no—
172C4⋊C4—2121no—
173C4⋊C4—2121no—
174C4⋊C4—2121no—
175C22⋊C4—2121no—
176C22⋊C4—2122no—
177C42—531no—
178C42—1061no—
179C42—2121no—
180C42C2×C4×D5311yescharacteristic
181C16—24no—
182C16—1064no—
183C23—41no—
184C23—531no—
185Q8—4242no—
186Q8—2122no—
187D4—42no—
188D4—82no—
189D4—4242no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—531no—
193C2×C4—532no—
194C2×C4—532no—
195C2×C4—1061no—
196C2×C4—4241no—
197C2×C4—4241no—
198C2×C4—4241no—
199C2×C4—2121no—
200C2×C4—2121no—
201C2×C4C22⋊C4×D5311yescharacteristic
202C8—22no—
203C8—1062no—
204C22—82no—
205C22—81no—
206C22—532no—
207C22—2121no—
208C22#100×D5311yescharacteristic
209C4—41no—
210C4—81no—
211C4—531no—
212C4—531no—
213C4—4241no—
214C4—4241no—
215C4—2121no—
216C4#104×D5312yes—
217C2—21no—
218C2—161no—
219C2—1062no—
220C2#329×D5311yescharacteristic
subgroupG/H
centerC2#329×D53
commutatorC42×C53C22×C4
Frattini#97C22×D53
FittingC53×#1355C2
socleC2×C53C2×#329
prime radical#16811171C4
cosocle#97×C53C23
hypercenter#1355D53
hypocenterC53#11911
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. #20272749C2semidirect
4. #20272748C2semidirect
5. C53×#1355C2semidirect
111. #1355D53semidirect
129. C53#11911semidirect

Maximal subgroups

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

subgroupindex
#202850592
#347×D532
#202727492
#202727482
C53×#13552
#1191153