Finite Group Databasebeta

← #27294943

Subgroups — #27294943

Order 13568.

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

normal subgroups only

This group has 280 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#20285460C211yesmaximal, characteristic
2#20276527C211yesmaximal, characteristic
3#20272708C212yesmaximal
4#20272702C212yesmaximal
5C53×#1841C211yesmaximal, characteristic
6#16811180—21no—
7#16811021C411yescharacteristic, complemented
8#16811003C411yescharacteristic, complemented
9#16810974—21no—
10#16810881—21no—
11#16810881C2212yes—
12#16810731—21no—
13#16810731C412yescomplemented
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×Dic212—21no—
23C2×Dic212C2×C412yes—
24#13133217—21no—
25#13133217—21no—
26#13133209—41no—
27#13133196—22no—
28#13133196—41no—
29C4×Dic106—22no—
30C4×Dic106—41no—
31#13133176—41no—
32#13133157—22no—
33#13133149—21no—
34#13133149C2×C412yes—
35#13133142—22no—
36#13133142—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×Dic106—22no—
49C2×Dic106—41no—
50M16×C53—41no—
51M16×C53—41no—
52M16×C53—41no—
53M16×C53—41no—
54C2×C8×C53—41no—
55C2×C8×C53C22⋊C411yescharacteristic
56C2×C8×C53C22⋊C411yescharacteristic
57C2×C8×C53C22×C411yescharacteristic
58C2×C8×C53C2×D411yescharacteristic
59C4⋊C4×C53—41no—
60C22⋊C4×C53—41no—
61C42×C53—21no—
62C42×C53—41no—
63C42×C53C22⋊C411yescharacteristic
64C42×C53C22⋊C411yescharacteristic
65C22⋊Dic53—42no—
66C4⋊Dic53—22no—
67C4⋊Dic53—41no—
68C4⋊Dic53—41no—
69C4⋊Dic53—41no—
70#11719919—42no—
71#11719919—42no—
72C4×Dic53—42no—
73Dic212—42no—
74Dic212—42no—
75#11719908—42no—
76C23×C53AGL(1,ℤ/8)11yescharacteristic
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—42no—
82C2×C4×C53—81no—
83C2×C4×C53C2×C22⋊C411yescharacteristic
84C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
85C2×Dic53—42no—
86C2×Dic53—42no—
87Dic106—42no—
88Dic106—42no—
89Dic106—82no—
90C8×C53—21no—
91C8×C53—21no—
92C8×C53—21no—
93C8×C53—21no—
94C8×C53—81no—
95#11838—531nomaximal, Sylow
96C22×C53—21no—
97C22×C53—41no—
98C22×C53#41811yescharacteristic
99C4×C53—22no—
100C4×C53—21no—
101C4×C53—41no—
102C4×C53—161no—
103C4×C53#41011yescharacteristic
104C4×C53#42111yescharacteristic
105Dic53—82no—
106Dic53—82no—
107#3185—531no—
108#2165—531no—
109#1841D5311yescharacteristic
110#1342—532no—
111#1340—532no—
112C2×C53—21no—
113C2×C53—41no—
114C2×C53#151411yescharacteristic
115#533—1061no—
116#498—531no—
117#494—531no—
118#487—1061no—
119#470—532no—
120#470—1061no—
121#441—532no—
122#441—1061no—
123C4×M16C2×D5311yescharacteristic
124#364—1061no—
125#349—532no—
126#347—1061no—
127#344—21no—
128#339—21no—
129#338C2×D5311yescharacteristic
130H(3,ℤ/4)C2×D5311yescharacteristic
131C53#1183811yescharacteristic, complemented, Sylow
132C2×Q16—532no—
133C2×Q16—1061no—
134C2×M16—21no—
135C2×M16C22⋊D5311yescharacteristic
136C2×M16C22×D5311yescharacteristic
137C4⋊Q8—1061no—
138C4⋊Q8—1061no—
139#125—2121no—
140C22⋊Q8—1062no—
141C22⋊Q8—2121no—
142C4×Q8—1062no—
143C4×Q8—2121no—
144#117—21no—
145#117—2121no—
146C2×C42C22⋊D5311yescharacteristic
147M32—1062no—
148#106—532no—
149#106—1061no—
150#103—1062no—
151#103—1062no—
152#97C4×D5311yescharacteristic
153#97D21211yescharacteristic
154C4×C8C4×D5311yescharacteristic
155C4×C8D21211yescharacteristic
156C2×Q8—1062no—
157C2×Q8—2121no—
158C22×C4—21no—
159C22×C4C2×C22⋊D5311yescharacteristic
160Q16—2122no—
161Q16—2122no—
162M16—41no—
163M16—41no—
164M16—41no—
165M16—41no—
166C2×C8—41no—
167C2×C8#1171992111yescharacteristic
168C2×C8#1171992111yescharacteristic
169C2×C8C2×C4×D5311yescharacteristic
170C2×C8C2×D21211yescharacteristic
171C4⋊C4—41no—
172C4⋊C4—1062no—
173C4⋊C4—2121no—
174C4⋊C4—2121no—
175C4⋊C4—2122no—
176C4⋊C4—2121no—
177C4⋊C4—2122no—
178C22⋊C4—41no—
179C22⋊C4—2122no—
180C42—21no—
181C42—41no—
182C42—2122no—
183C42#1171992111yescharacteristic
184C42#1171992111yescharacteristic
185C16—2122no—
186C23#1313324211yescharacteristic
187Q8—4242no—
188Q8—2122no—
189Q8—2122no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—81no—
196C2×C4—2122no—
197C2×C4—2122no—
198C2×C4C2×#1171992111yescharacteristic
199C2×C4#1313324211yescharacteristic
200C8—21no—
201C8—21no—
202C8—21no—
203C8—21no—
204C8—81no—
205C22—21no—
206C22—41no—
207C22#1681062711yescharacteristic
208C4—22no—
209C4—21no—
210C4—41no—
211C4—161no—
212C4—4242no—
213C4—4242no—
214C4#1681059611yescharacteristic
215C4#1681064111yescharacteristic
216C2—21no—
217C2—41no—
218C2#2027326111yescharacteristic
subgroupG/H
centerC2#20273261
commutatorC2×C8×C53C22×C4
FrattiniC2×M16C22×D53
FittingC53×#1841C2
socleC2×C53#1514
prime radicalC23×C53AGL(1,ℤ/8)
cosocleC2×M16×C53C23
hypercenter#1841D53
hypocenterC53#11838
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
7. #16811021C4semidirect
8. #16811003C4semidirect
13. #16810731C4semidirect
131. C53#11838semidirect

Maximal subgroups

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

subgroupindex
#202854602
#202765272
#202727082
#202727022
C53×#18412
#1183853