Finite Group Databasebeta

← #27294641

Subgroups — #27294641

Order 13568.

292 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 75 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#20285374C211yesmaximal, characteristic
2#20274470C211yesmaximal, characteristic
3#20272710C212yesmaximal
4#20272704C212yesmaximal
5C53×#2166C211yesmaximal, characteristic
6#16811235—21no—
7#16811025C411yescharacteristic, complemented
8#16811003C411yescharacteristic, complemented
9#16810992—21no—
10#16810881C2212yes—
11#16810864—21no—
12#16810731C412yescomplemented
13C4×Dic212—21no—
14#16810578—21no—
15#16810381C2211yescharacteristic
16#16810380C2211yescharacteristic
17#16810379—21no—
18C53×#405C2211yescharacteristic
19C53×#364—21no—
20C53×#349C2212yes—
21C53×#347—21no—
22C2×Dic212—21no—
23C2×Dic212C2×C412yes—
24C2×#11719917—21no—
25#13133217—21no—
26#13133210—42no—
27#13133209—21no—
28#13133207—42no—
29#13133196—22no—
30C4×Dic106—22no—
31#13133176—21no—
32#13133151—21no—
33#13133149C2×C412yes—
34#13133142—22no—
35#13133142—22no—
36#13133127—22no—
37C8×Dic53—22no—
38C2×M16×C53C2311yescharacteristic
39C22×C8×C53D411yescharacteristic
40#117×C53D411yescharacteristic
41M32×C53—22no—
42#97×C53C2×C411yescharacteristic
43#97×C53D411yescharacteristic
44C4×C8×C53C2×C411yescharacteristic
45C4×C8×C53D411yescharacteristic
46C22×C4×C53C2×D411yescharacteristic
47C2×Dic106—22no—
48M16×C53—41no—
49C2×C8×C53—21no—
50C2×C8×C53—21no—
51C2×C8×C53C22⋊C411yescharacteristic
52C2×C8×C53C22⋊C411yescharacteristic
53C2×C8×C53C22×C411yescharacteristic
54C2×C8×C53C2×D411yescharacteristic
55C4⋊C4×C53—21no—
56C22⋊C4×C53—21no—
57C42×C53C22⋊C411yescharacteristic
58C42×C53C22⋊C411yescharacteristic
59C22⋊Dic53—41no—
60C22⋊Dic53—42no—
61C4⋊Dic53—22no—
62C4⋊Dic53—41no—
63C4⋊Dic53—42no—
64#11719919—42no—
65#11719919—42no—
66#11719919—42no—
67C4×Dic53—22no—
68C4×Dic53—42no—
69#11719917—41no—
70#11719917—41no—
71C2×#88322—22no—
72Dic212—42no—
73Dic212—41no—
74Dic212—41no—
75C16×C53—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×C53C2×C22⋊C411yescharacteristic
82C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
83C2×Dic53—42no—
84C2×Dic53—42no—
85C2×Dic53—42no—
86Dic106—42no—
87Dic106—42no—
88C8×C53—21no—
89C8×C53—21no—
90C8×C53—21no—
91C8×C53#10411yescharacteristic
92C8×C53#10411yescharacteristic
93#88322—42no—
94#11771—531nomaximal, Sylow
95C22×C53—21no—
96C22×C53—41no—
97C22×C53#41811yescharacteristic
98C4×C53—21no—
99C4×C53—41no—
100C4×C53—41no—
101C4×C53#41011yescharacteristic
102C4×C53C2×#10411yescharacteristic
103Dic53—82no—
104Dic53—82no—
105Dic53—82no—
106#3177—531no—
107#2166D5311yescharacteristic
108#1747—531no—
109#1342—532no—
110#1340—532no—
111C2×C53—21no—
112C2×C53—41no—
113C2×C53#150511yescharacteristic
114#541—1061no—
115#499—531no—
116#494—531no—
117#491—1061no—
118#470—532no—
119#467—1061no—
120#441—532no—
121C4×Q16—1061no—
122#405—1061no—
123#405C2×D5311yescharacteristic
124#364—21no—
125#349C2×D5312yes—
126#347—21no—
127#338—531no—
128H(3,ℤ/4)—531no—
129H(3,ℤ/4)—1061no—
130C53#1177111yescharacteristic, complemented, Sylow
131C2×Q16—532no—
132C2×Q16—1061no—
133C2×M16—1061no—
134C2×M16C22×D5311yescharacteristic
135C22×C8D21211yescharacteristic
136C4⋊Q8—1061no—
137#126—2122no—
138#125—1061no—
139#125—2122no—
140C22⋊Q8—1062no—
141C4×Q8—1062no—
142#117—1061no—
143#117D21211yescharacteristic
144M32—22no—
145#107—1061no—
146#106—532no—
147#103—1062no—
148#103—1062no—
149#97—1062no—
150#97C4×D5311yescharacteristic
151#97C22⋊D5311yescharacteristic
152C4×C8—1062no—
153C4×C8C4×D5311yescharacteristic
154C4×C8C22⋊D5311yescharacteristic
155C2×Q8—1062no—
156C22×C4C2×D21211yescharacteristic
157Q16—2122no—
158Q16—2121no—
159Q16—2121no—
160M16—41no—
161M16—2121no—
162M16—2121no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—1062no—
166C2×C8#1171992111yescharacteristic
167C2×C8#1171992111yescharacteristic
168C2×C8C2×C4×D5311yescharacteristic
169C2×C8C2×C22⋊D5311yescharacteristic
170C4⋊C4—21no—
171C4⋊C4—1062no—
172C4⋊C4—2121no—
173C4⋊C4—2122no—
174C4⋊C4—2122no—
175C4⋊C4—2122no—
176C4⋊C4—2122no—
177C22⋊C4—21no—
178C22⋊C4—2121no—
179C22⋊C4—2122no—
180C42—1062no—
181C42—2122no—
182C42#1171992111yescharacteristic
183C42#1171992111yescharacteristic
184C16—42no—
185C23#1313324511yescharacteristic
186Q8—2122no—
187Q8—2122no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—2122no—
193C2×C4—2122no—
194C2×C4—2122no—
195C2×C4C2×#1171992111yescharacteristic
196C2×C4#1313324511yescharacteristic
197C8—21no—
198C8—21no—
199C8—21no—
200C8—2122no—
201C8#1313314311yescharacteristic
202C8#1313314311yescharacteristic
203C22—21no—
204C22—41no—
205C22#1681062911yescharacteristic
206C4—21no—
207C4—41no—
208C4—41no—
209C4—4242no—
210C4—4242no—
211C4—4242no—
212C4#1681059611yescharacteristic
213C4C2×#1313314311yescharacteristic
214C2—21no—
215C2—41no—
216C2#2027322411yescharacteristic
subgroupG/H
centerC2#20273224
commutatorC2×C8×C53C22×C4
FrattiniC2×M16C22×D53
FittingC53×#2166C2
socleC2×C53#1505
prime radicalC23×C53AGL(1,ℤ/8)
cosocleC2×M16×C53C23
hypercenter#2166D53
hypocenterC53#11771
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2166

Complements

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

NH ≅ G/Ntype
7. #16811025C4semidirect
8. #16811003C4semidirect
12. #16810731C4semidirect
130. C53#11771semidirect

Maximal subgroups

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

subgroupindex
#202853742
#202744702
#202727102
#202727042
C53×#21662
#1177153