Finite Group Databasebeta

← #27294638

Subgroups — #27294638

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#20276531C211yesmaximal, characteristic
3#20272708C212yesmaximal
4#20272702C212yesmaximal
5C53×#1747C211yesmaximal, characteristic
6#16811235—21no—
7#16811025C411yescharacteristic, complemented
8#16811003C411yescharacteristic, complemented
9#16810992—21no—
10#16810881C2212yes—
11#16810864—21no—
12#16810731C412yescomplemented
13C4×Dic212—21no—
14#16810444—21no—
15#16810405C2212yes—
16#16810401—21no—
17C53×#405—21no—
18C53×#405C2211yescharacteristic
19C53×#338C2211yescharacteristic
20C53×H(3,ℤ/4)—21no—
21C53×H(3,ℤ/4)C2211yescharacteristic
22C2×Dic212—21no—
23C2×Dic212C2×C412yes—
24#13133217—21no—
25#13133210—42no—
26#13133209—21no—
27#13133207—42no—
28#13133196—22no—
29C4×Dic106—22no—
30#13133157—22no—
31#13133151—21no—
32#13133149C2×C412yes—
33#13133142—22no—
34#13133142—22no—
35C2×M16×C53—21no—
36C2×M16×C53C2311yescharacteristic
37C22×C8×C53D411yescharacteristic
38#117×C53—21no—
39#117×C53D411yescharacteristic
40#97×C53—22no—
41#97×C53C2×C411yescharacteristic
42#97×C53D411yescharacteristic
43C4×C8×C53—22no—
44C4×C8×C53C2×C411yescharacteristic
45C4×C8×C53D411yescharacteristic
46C22×C4×C53C2×D411yescharacteristic
47C2×Dic106—22no—
48M16×C53—41no—
49M16×C53—41no—
50M16×C53—41no—
51C2×C8×C53—21no—
52C2×C8×C53—21no—
53C2×C8×C53—22no—
54C2×C8×C53C22⋊C411yescharacteristic
55C2×C8×C53C22⋊C411yescharacteristic
56C2×C8×C53C22×C411yescharacteristic
57C2×C8×C53C2×D411yescharacteristic
58C4⋊C4×C53—21no—
59C4⋊C4×C53—41no—
60C22⋊C4×C53—21no—
61C22⋊C4×C53—41no—
62C42×C53—22no—
63C42×C53C22⋊C411yescharacteristic
64C42×C53C22⋊C411yescharacteristic
65C22⋊Dic53—42no—
66C4⋊Dic53—22no—
67C4⋊Dic53—42no—
68#11719919—42no—
69#11719919—42no—
70#11719919—42no—
71C4×Dic53—42no—
72Dic212—42no—
73Dic212—41no—
74Dic212—41no—
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×C53C2×C22⋊C411yescharacteristic
83C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
84C2×Dic53—42no—
85C2×Dic53—42no—
86Dic106—42no—
87Dic106—42no—
88C8×C53—21no—
89C8×C53—21no—
90C8×C53—21no—
91C8×C53—42no—
92C8×C53#10411yescharacteristic
93C8×C53#10411yescharacteristic
94#11771—531nomaximal, Sylow
95C22×C53—21no—
96C22×C53—41no—
97C22×C53#41811yescharacteristic
98C4×C53—21no—
99C4×C53—41no—
100C4×C53—41no—
101C4×C53—82no—
102C4×C53#41011yescharacteristic
103C4×C53C2×#10411yescharacteristic
104Dic53—82no—
105Dic53—82no—
106#3177—531no—
107#2166—531no—
108#1747D5311yescharacteristic
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—21no—
123#405C2×D5311yescharacteristic
124#364—1061no—
125#349—532no—
126#347—1061no—
127#338C2×D5311yescharacteristic
128H(3,ℤ/4)—21no—
129H(3,ℤ/4)C2×D5311yescharacteristic
130C53#1177111yescharacteristic, complemented, Sylow
131C2×Q16—532no—
132C2×Q16—1061no—
133C2×M16—21no—
134C2×M16C22×D5311yescharacteristic
135C22×C8C22⋊D5311yescharacteristic
136C4⋊Q8—1061no—
137#126—2122no—
138#125—1061no—
139#125—2122no—
140C22⋊Q8—1062no—
141C4×Q8—1062no—
142#117—21no—
143#117C22⋊D5311yescharacteristic
144M32—1062no—
145#107—1061no—
146#106—532no—
147#103—1062no—
148#103—1062no—
149#97—22no—
150#97C4×D5311yescharacteristic
151#97D21211yescharacteristic
152C4×C8—22no—
153C4×C8C4×D5311yescharacteristic
154C4×C8D21211yescharacteristic
155C2×Q8—1062no—
156C22×C4C2×C22⋊D5311yescharacteristic
157Q16—2122no—
158Q16—2121no—
159Q16—2121no—
160M16—41no—
161M16—41no—
162M16—41no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—22no—
166C2×C8#1171992111yescharacteristic
167C2×C8#1171992111yescharacteristic
168C2×C8C2×C4×D5311yescharacteristic
169C2×C8C2×D21211yescharacteristic
170C4⋊C4—21no—
171C4⋊C4—41no—
172C4⋊C4—1062no—
173C4⋊C4—2122no—
174C4⋊C4—2122no—
175C4⋊C4—2122no—
176C4⋊C4—2122no—
177C22⋊C4—21no—
178C22⋊C4—41no—
179C22⋊C4—2122no—
180C42—22no—
181C42—2122no—
182C42#1171992111yescharacteristic
183C42#1171992111yescharacteristic
184C16—2122no—
185C23#1313324211yescharacteristic
186Q8—2122no—
187Q8—2122no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—42no—
193C2×C4—2122no—
194C2×C4—2122no—
195C2×C4C2×#1171992111yescharacteristic
196C2×C4#1313324211yescharacteristic
197C8—21no—
198C8—21no—
199C8—21no—
200C8—42no—
201C8#1313314511yescharacteristic
202C8#1313314511yescharacteristic
203C22—21no—
204C22—41no—
205C22#1681062711yescharacteristic
206C4—21no—
207C4—41no—
208C4—41no—
209C4—82no—
210C4—4242no—
211C4—4242no—
212C4#1681059611yescharacteristic
213C4C2×#1313314511yescharacteristic
214C2—21no—
215C2—41no—
216C2#2027322211yescharacteristic
subgroupG/H
centerC2#20273222
commutatorC2×C8×C53C22×C4
FrattiniC2×M16C22×D53
FittingC53×#1747C2
socleC2×C53#1505
prime radicalC23×C53AGL(1,ℤ/8)
cosocleC2×M16×C53C23
hypercenter#1747D53
hypocenterC53#11771
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #1747

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
#202765312
#202727082
#202727022
C53×#17472
#1177153