Finite Group Databasebeta

← #27294938

Subgroups — #27294938

Order 13568.

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

normal subgroups only

This group has 324 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#20285453C211yesmaximal, characteristic
2#20276527C211yesmaximal, characteristic, complemented
3#20272705C212yesmaximal, complemented
4#20272699C212yesmaximal, complemented
5C53×#1841C211yesmaximal, characteristic, complemented
6#16811173—21no—
7#16811021C411yescharacteristic, complemented
8#16811000C411yescharacteristic, complemented
9#16810970—21no—
10#16810876—21no—
11#16810876C2212yes—
12#16810736—21no—
13#16810736C412yescomplemented
14#16810444—21no—
15#16810405C2212yes—
16#16810401—21no—
17C4×M16×C53C2211yescharacteristic
18C53×#344—21no—
19C53×#339—21no—
20C53×#338C2211yescharacteristic
21C53×H(3,ℤ/4)C2211yescharacteristic
22C2×#11719945—41no—
23C2×D424—21no—
24C2×D424C2×C412yes—
25#13133217—21no—
26#13133214—21no—
27#13133205—41no—
28#13133191—22no—
29#13133191—41no—
30C4×D212—22no—
31C4×D212—41no—
32#13133157—22no—
33#13133149—21no—
34#13133149C2×C412yes—
35#13133139—22no—
36#13133139—22no—
37C2×M16×C53—21no—
38C2×M16×C53D411yescharacteristic
39C2×M16×C53C2311yescharacteristic
40#117×C53—21no—
41C2×C42×C53D411yescharacteristic
42#97×C53C2×C411yescharacteristic
43#97×C53D411yescharacteristic
44C4×C8×C53C2×C411yescharacteristic
45C4×C8×C53D411yescharacteristic
46C22×C4×C53—21no—
47C22×C4×C53C2×D411yescharacteristic
48C2×C22⋊D53—42no—
49#11719945—84no—
50C2×D212—22no—
51C2×D212—41no—
52C2×D212—41no—
53C2×C4×D53—42no—
54C2×Dic106—41no—
55M16×C53—41no—
56M16×C53—41no—
57M16×C53—41no—
58M16×C53—41no—
59C2×C8×C53—41no—
60C2×C8×C53C22⋊C411yescharacteristic
61C2×C8×C53C22⋊C411yescharacteristic
62C2×C8×C53C22×C411yescharacteristic
63C2×C8×C53C2×D411yescharacteristic
64C4⋊C4×C53—41no—
65C22⋊C4×C53—41no—
66C42×C53—21no—
67C42×C53—41no—
68C42×C53C22⋊C411yescharacteristic
69C42×C53C22⋊C411yescharacteristic
70#11719921—42no—
71#11719921—42no—
72C4⋊Dic53—22no—
73C4⋊Dic53—41no—
74D424—42no—
75D424—42no—
76#11719908—42no—
77C23×C53AGL(1,ℤ/8)11yescharacteristic
78C22×D53—42no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—42no—
84C2×C4×C53—81no—
85C2×C4×C53C2×C22⋊C411yescharacteristic
86C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
87C22⋊D53—84no—
88C2×Dic53—42no—
89D212—42no—
90D212—42no—
91D212—82no—
92D212—82no—
93C4×D53—84no—
94Dic106—82no—
95C8×C53—21no—
96C8×C53—21no—
97C8×C53—21no—
98C8×C53—21no—
99C8×C53—81no—
100#11837—531nomaximal, Sylow
101C22×C53—21no—
102C22×C53—41no—
103C22×C53#41811yescharacteristic
104C2×D53—82no—
105C2×D53—82no—
106C2×D53—82no—
107C4×C53—22no—
108C4×C53—21no—
109C4×C53—41no—
110C4×C53—161no—
111C4×C53#41011yescharacteristic
112C4×C53#42111yescharacteristic
113Dic53—82no—
114#3184—531no—
115#2165—531no—
116#1841D5311yescharacteristic, complemented
117#1341—532no—
118#1339—532no—
119C2×C53—21no—
120C2×C53—41no—
121C2×C53#151411yescharacteristic
122D53—162no—
123#532—1061no—
124#498—531no—
125#493—531no—
126#486—1061no—
127#469—532no—
128#469—1061no—
129#442—532no—
130#442—1061no—
131C4×M16C2×D5311yescharacteristic
132#364—1061no—
133#349—532no—
134#347—1061no—
135#344—21no—
136#339—21no—
137#338C2×D5311yescharacteristic
138H(3,ℤ/4)C2×D5311yescharacteristic
139C53#1183711yescharacteristic, complemented, Sylow
140C2×#41—2121no—
141C2×D8—532no—
142C2×D8—1061no—
143C2×M16—21no—
144C2×M16C22⋊D5311yescharacteristic
145C2×M16C22×D5311yescharacteristic
146C4⋊Q8—1061no—
147#127—1061no—
148#124—2121no—
149#121—1062no—
150#121—2121no—
151C4×D4—1062no—
152C4×D4—2121no—
153#117—21no—
154C2×C42C22⋊D5311yescharacteristic
155M32—1062no—
156#106—532no—
157#106—1061no—
158#102—1062no—
159#102—1062no—
160#97C4×D5311yescharacteristic
161#97D21211yescharacteristic
162C4×C8C4×D5311yescharacteristic
163C4×C8D21211yescharacteristic
164#41—4244no—
165C2×Q8—2121no—
166C2×D4—1062no—
167C2×D4—2121no—
168C2×D4—2121no—
169C2×D4—2122no—
170C22×C4—21no—
171C22×C4—2122no—
172C22×C4C2×C22⋊D5311yescharacteristic
173D8—2122no—
174D8—2122no—
175M16—41no—
176M16—41no—
177M16—41no—
178M16—41no—
179C2×C8—41no—
180C2×C8#1171992111yescharacteristic
181C2×C8#1171992111yescharacteristic
182C2×C8C2×C4×D5311yescharacteristic
183C2×C8C2×D21211yescharacteristic
184C4⋊C4—41no—
185C4⋊C4—1062no—
186C4⋊C4—2121no—
187C22⋊C4—41no—
188C22⋊C4—2122no—
189C22⋊C4—2122no—
190C42—21no—
191C42—41no—
192C42#1171992111yescharacteristic
193C42#1171992111yescharacteristic
194C16—2122no—
195C23—2122no—
196C23#1313324211yescharacteristic
197Q8—4242no—
198D4—4242no—
199D4—4242no—
200D4—4244no—
201D4—2122no—
202D4—2122no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—42no—
208C2×C4—81no—
209C2×C4—4244no—
210C2×C4—2122no—
211C2×C4C2×#1171992111yescharacteristic
212C2×C4#1313324211yescharacteristic
213C8—21no—
214C8—21no—
215C8—21no—
216C8—21no—
217C8—81no—
218C22—21no—
219C22—41no—
220C22—4242no—
221C22—4242no—
222C22—4242no—
223C22#1681062711yescharacteristic
224C4—22no—
225C4—21no—
226C4—41no—
227C4—161no—
228C4—4242no—
229C4#1681059611yescharacteristic
230C4#1681064111yescharacteristic
231C2—21no—
232C2—41no—
233C2—8482no—
234C2#2027326111yescharacteristic
subgroupG/H
centerC2#20273261
commutatorC2×C8×C53C22×C4
FrattiniC2×M16C22×D53
FittingC53×#1841C2
socleC2×C53#1514
prime radical#20285453C2
cosocleC2×M16×C53C23
hypercenter#1841D53
hypocenterC53#11837
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #1841

Complements

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

NH ≅ G/Ntype
2. #20276527C2semidirect
3. #20272705C2semidirect
4. #20272699C2semidirect
5. C53×#1841C2semidirect
7. #16811021C4semidirect
8. #16811000C4semidirect
13. #16810736C4semidirect
116. #1841D53semidirect
139. C53#11837semidirect

Maximal subgroups

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

subgroupindex
#202854532
#202765272
#202727052
#202726992
C53×#18412
#1183753