Finite Group Databasebeta

← #27294944

Subgroups — #27294944

Order 13568.

280 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 79 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#20276527C211yesmaximal, characteristic
2#20274864C211yesmaximal, characteristic
3#20272709C212yesmaximal
4#20272703C212yesmaximal
5C53×#3185C211yesmaximal, characteristic
6#16810444—21no—
7#16810405C2212yes—
8#16810401—21no—
9#16810394—21no—
10#16810383—21no—
11#16810381C2211yescharacteristic
12#16810380C2211yescharacteristic
13C53×#533—21no—
14C53×#498C411yescharacteristic, complemented
15C53×#494C411yescharacteristic, complemented
16C53×#487—21no—
17C53×#470—21no—
18C53×#470C2212yes—
19C53×#441—21no—
20C53×#441C412yescomplemented
21C4×M16×C53C2211yescharacteristic
22C2×#11719917—21no—
23#13133176—21no—
24#13133157—22no—
25C2×Q16×C53—21no—
26C2×Q16×C53C2×C412yes—
27C2×M16×C53D411yescharacteristic
28C2×M16×C53C2311yescharacteristic
29C4⋊Q8×C53—21no—
30C4⋊Q8×C53—21no—
31#125×C53—41no—
32C22⋊Q8×C53—22no—
33C22⋊Q8×C53—41no—
34C4×Q8×C53—22no—
35C4×Q8×C53—41no—
36#117×C53—41no—
37C2×C42×C53D411yescharacteristic
38#106×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—
47C2×Q8×C53—41no—
48C22×C4×C53—21no—
49C22×C4×C53C2×D411yescharacteristic
50Q16×C53—42no—
51Q16×C53—42no—
52M16×C53—41no—
53M16×C53—41no—
54C2×C8×C53C22⋊C411yescharacteristic
55C2×C8×C53C22⋊C411yescharacteristic
56C2×C8×C53C22×C411yescharacteristic
57C2×C8×C53C2×D411yescharacteristic
58C4⋊C4×C53—22no—
59C4⋊C4×C53—41no—
60C4⋊C4×C53—41no—
61C4⋊C4×C53—42no—
62C4⋊C4×C53—41no—
63C4⋊C4×C53—42no—
64C22⋊C4×C53—42no—
65C42×C53—21no—
66C42×C53—42no—
67C42×C53C22⋊C411yescharacteristic
68C42×C53C22⋊C411yescharacteristic
69C22⋊Dic53—41no—
70C4⋊Dic53—41no—
71C4×Dic53—41no—
72#11719917—41no—
73#11719917—41no—
74C2×#88322—41no—
75#11719908—42no—
76C23×C53AGL(1,ℤ/8)11yescharacteristic
77Q8×C53—42no—
78Q8×C53—42no—
79Q8×C53—82no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—42no—
85C2×C4×C53—42no—
86C2×C4×C53—42no—
87C2×C4×C53C2×C22⋊C411yescharacteristic
88C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
89C2×Dic53—81no—
90C8×C53—21no—
91C8×C53—21no—
92C8×C53—21no—
93C8×C53—21no—
94#88322—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—82no—
103C4×C53—82no—
104C4×C53#41011yescharacteristic
105C4×C53#42111yescharacteristic
106Dic53—161no—
107#3185D5311yescharacteristic
108#2165—531no—
109#1841—531no—
110#1342—532no—
111#1340—532no—
112C2×C53—21no—
113C2×C53—41no—
114C2×C53#151411yescharacteristic
115#533—21no—
116#498Dic5311yescharacteristic, complemented
117#494Dic5311yescharacteristic, complemented
118#487—21no—
119#470—21no—
120#470C2×D5312yes—
121#441—21no—
122#441Dic5312yescomplemented
123C4×M16C2×D5311yescharacteristic
124#364—1061no—
125#349—532no—
126#347—1061no—
127#344—1061no—
128#339—1061no—
129#338—531no—
130H(3,ℤ/4)—531no—
131C53#1183811yescharacteristic, complemented, Sylow
132C2×Q16—21no—
133C2×Q16C2×Dic5312yes—
134C2×M16—1061no—
135C2×M16C22⋊D5311yescharacteristic
136C2×M16C22×D5311yescharacteristic
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139#125—41no—
140C22⋊Q8—22no—
141C22⋊Q8—41no—
142C4×Q8—22no—
143C4×Q8—41no—
144#117—41no—
145#117—1061no—
146C2×C42C22⋊D5311yescharacteristic
147M32—1062no—
148#106—21no—
149#106C2×Dic5312yes—
150#103—22no—
151#103—22no—
152#97C2×Dic5311yescharacteristic
153#97C22⋊D5311yescharacteristic
154C4×C8C2×Dic5311yescharacteristic
155C4×C8C22⋊D5311yescharacteristic
156C2×Q8—22no—
157C2×Q8—41no—
158C22×C4—21no—
159C22×C4C2×C22⋊D5311yescharacteristic
160Q16—42no—
161Q16—42no—
162M16—41no—
163M16—41no—
164M16—2121no—
165M16—2121no—
166C2×C8—2121no—
167C2×C8C22⋊Dic5311yescharacteristic
168C2×C8C22⋊Dic5311yescharacteristic
169C2×C8C22×Dic5311yescharacteristic
170C2×C8C2×C22⋊D5311yescharacteristic
171C4⋊C4—22no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—42no—
175C4⋊C4—41no—
176C4⋊C4—42no—
177C4⋊C4—2121no—
178C22⋊C4—42no—
179C22⋊C4—2121no—
180C42—21no—
181C42—42no—
182C42—2121no—
183C42C22⋊Dic5311yescharacteristic
184C42C22⋊Dic5311yescharacteristic
185C16—2122no—
186C23#1313324311yescharacteristic
187Q8—42no—
188Q8—42no—
189Q8—82no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—4241no—
198C2×C4C2×C22⋊Dic5311yescharacteristic
199C2×C4#1313324311yescharacteristic
200C8—21no—
201C8—21no—
202C8—21no—
203C8—21no—
204C8—4241no—
205C22—21no—
206C22—41no—
207C22#1681062811yescharacteristic
208C4—22no—
209C4—21no—
210C4—41no—
211C4—82no—
212C4—82no—
213C4—8481no—
214C4#1681059711yescharacteristic
215C4#1681064011yescharacteristic
216C2—21no—
217C2—41no—
218C2#2027326211yescharacteristic
subgroupG/H
centerC2#20273262
commutatorC2×C8×C53C22×C4
FrattiniC2×M16C22×D53
FittingC53×#3185C2
socleC2×C53#1514
prime radicalC23×C53AGL(1,ℤ/8)
cosocleC2×M16×C53C23
hypercenter#3185D53
hypocenterC53#11838
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #3185

Complements

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

NH ≅ G/Ntype
14. C53×#498C4semidirect
15. C53×#494C4semidirect
20. C53×#441C4semidirect
116. #498Dic53semidirect
117. #494Dic53semidirect
122. #441Dic53semidirect
131. C53#11838semidirect

Maximal subgroups

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

subgroupindex
#202765272
#202748642
#202727092
#202727032
C53×#31852
#1183853