Finite Group Databasebeta

← #27294639

Subgroups — #27294639

Order 13568.

292 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 83 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#20276531C211yesmaximal, characteristic
2#20274470C211yesmaximal, characteristic
3#20272709C212yesmaximal
4#20272703C212yesmaximal
5C53×#3177C211yesmaximal, characteristic
6#16810578—21no—
7#16810444—21no—
8#16810405C2212yes—
9#16810401—21no—
10#16810381C2211yescharacteristic
11#16810380C2211yescharacteristic
12#16810379—21no—
13C53×#541—21no—
14C53×#499C411yescharacteristic, complemented
15C53×#494C411yescharacteristic, complemented
16C53×#491—21no—
17C53×#470C2212yes—
18C53×#467—21no—
19C53×#441C412yescomplemented
20C4×Q16×C53—21no—
21C53×#405C2211yescharacteristic
22C2×#11719917—21no—
23#13133176—21no—
24#13133157—22no—
25#13133127—22no—
26C8×Dic53—22no—
27C2×Q16×C53—21no—
28C2×Q16×C53C2×C412yes—
29C2×M16×C53C2311yescharacteristic
30C22×C8×C53D411yescharacteristic
31C4⋊Q8×C53—21no—
32#126×C53—42no—
33#125×C53—21no—
34#125×C53—42no—
35C22⋊Q8×C53—22no—
36C4×Q8×C53—22no—
37#117×C53D411yescharacteristic
38#107×C53—21no—
39#106×C53C2×C412yes—
40#103×C53—22no—
41#103×C53—22no—
42#97×C53C2×C411yescharacteristic
43#97×C53D411yescharacteristic
44C4×C8×C53C2×C411yescharacteristic
45C4×C8×C53D411yescharacteristic
46C2×Q8×C53—22no—
47C22×C4×C53C2×D411yescharacteristic
48Q16×C53—42no—
49Q16×C53—41no—
50Q16×C53—41no—
51M16×C53—41no—
52C2×C8×C53—21no—
53C2×C8×C53—21no—
54C2×C8×C53C22⋊C411yescharacteristic
55C2×C8×C53C22⋊C411yescharacteristic
56C2×C8×C53C22×C411yescharacteristic
57C2×C8×C53C2×D411yescharacteristic
58C4⋊C4×C53—21no—
59C4⋊C4×C53—22no—
60C4⋊C4×C53—42no—
61C4⋊C4×C53—42no—
62C4⋊C4×C53—42no—
63C4⋊C4×C53—42no—
64C22⋊C4×C53—21no—
65C22⋊C4×C53—42no—
66C42×C53—42no—
67C42×C53C22⋊C411yescharacteristic
68C42×C53C22⋊C411yescharacteristic
69C22⋊Dic53—41no—
70C4⋊Dic53—41no—
71C4×Dic53—22no—
72#11719917—41no—
73#11719917—41no—
74C2×#88322—22no—
75#11719908—42no—
76C23×C53AGL(1,ℤ/8)11yescharacteristic
77Q8×C53—42no—
78Q8×C53—42no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—42no—
84C2×C4×C53—42no—
85C2×C4×C53C2×C22⋊C411yescharacteristic
86C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
87C2×Dic53—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—82no—
102C4×C53—82no—
103C4×C53#41011yescharacteristic
104C4×C53C2×#10411yescharacteristic
105Dic53—82no—
106#3177D5311yescharacteristic
107#2166—531no—
108#1747—531no—
109#1342—532no—
110#1340—532no—
111C2×C53—21no—
112C2×C53—41no—
113C2×C53#150511yescharacteristic
114#541—21no—
115#499Dic5311yescharacteristic, complemented
116#494Dic5311yescharacteristic, complemented
117#491—21no—
118#470C2×D5312yes—
119#467—21no—
120#441Dic5312yescomplemented
121C4×Q16—21no—
122#405—1061no—
123#405C2×D5311yescharacteristic
124#364—1061no—
125#349—532no—
126#347—1061no—
127#338—531no—
128H(3,ℤ/4)—531no—
129H(3,ℤ/4)—1061no—
130C53#1177111yescharacteristic, complemented, Sylow
131C2×Q16—21no—
132C2×Q16C2×Dic5312yes—
133C2×M16—1061no—
134C2×M16C22×D5311yescharacteristic
135C22×C8C22⋊D5311yescharacteristic
136C4⋊Q8—21no—
137#126—42no—
138#125—21no—
139#125—42no—
140C22⋊Q8—22no—
141C4×Q8—22no—
142#117—1061no—
143#117C22⋊D5311yescharacteristic
144M32—1062no—
145#107—21no—
146#106C2×Dic5312yes—
147#103—22no—
148#103—22no—
149#97—1062no—
150#97C2×Dic5311yescharacteristic
151#97C22⋊D5311yescharacteristic
152C4×C8—1062no—
153C4×C8C2×Dic5311yescharacteristic
154C4×C8C22⋊D5311yescharacteristic
155C2×Q8—22no—
156C22×C4C2×C22⋊D5311yescharacteristic
157Q16—42no—
158Q16—41no—
159Q16—41no—
160M16—41no—
161M16—2121no—
162M16—2121no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—1062no—
166C2×C8C22⋊Dic5311yescharacteristic
167C2×C8C22⋊Dic5311yescharacteristic
168C2×C8C22×Dic5311yescharacteristic
169C2×C8C2×C22⋊D5311yescharacteristic
170C4⋊C4—21no—
171C4⋊C4—22no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—2121no—
177C22⋊C4—21no—
178C22⋊C4—42no—
179C22⋊C4—2121no—
180C42—42no—
181C42—1062no—
182C42C22⋊Dic5311yescharacteristic
183C42C22⋊Dic5311yescharacteristic
184C16—2122no—
185C23#1313324311yescharacteristic
186Q8—42no—
187Q8—42no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—2122no—
195C2×C4C2×C22⋊Dic5311yescharacteristic
196C2×C4#1313324311yescharacteristic
197C8—21no—
198C8—21no—
199C8—21no—
200C8—2122no—
201C8#1313314411yescharacteristic
202C8#1313314411yescharacteristic
203C22—21no—
204C22—41no—
205C22#1681062811yescharacteristic
206C4—21no—
207C4—41no—
208C4—41no—
209C4—82no—
210C4—82no—
211C4—4242no—
212C4#1681059711yescharacteristic
213C4C2×#1313314411yescharacteristic
214C2—21no—
215C2—41no—
216C2#2027322311yescharacteristic
subgroupG/H
centerC2#20273223
commutatorC2×C8×C53C22×C4
FrattiniC2×M16C22×D53
FittingC53×#3177C2
socleC2×C53#1505
prime radicalC23×C53AGL(1,ℤ/8)
cosocleC2×M16×C53C23
hypercenter#3177D53
hypocenterC53#11771
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #3177

Complements

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

NH ≅ G/Ntype
14. C53×#499C4semidirect
15. C53×#494C4semidirect
19. C53×#441C4semidirect
115. #499Dic53semidirect
116. #494Dic53semidirect
120. #441Dic53semidirect
130. C53#11771semidirect

Maximal subgroups

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

subgroupindex
#202765312
#202744702
#202727092
#202727032
C53×#31772
#1177153