Finite Group Databasebeta

← #28309100

Subgroups — #28309100

Order 13568.

292 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 62 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#20291090C211yesmaximal, characteristic
2#20290436C211yesmaximal, characteristic
3#20289691C212yesmaximal
4#20289687C212yesmaximal
5#20285371C211yesmaximal, characteristic
6#16811586—21no—
7#16811545—21no—
8#16811520C2212yes—
9#16811515—21no—
10#16811492C2211yescharacteristic
11#16811491C2211yescharacteristic
12#16811490—21no—
13#16811234—21no—
14#16811025C411yescharacteristic, complemented
15#16810991—21no—
16#16810878C2212yes—
17#16810861—21no—
18#16810730C412yescomplemented
19Q16×Dic53—21no—
20#16810577C2211yescharacteristic
21C53×#494C411yescharacteristic, complemented
22D53⋊M16—21no—
23#13133296—21no—
24#13133291—22no—
25#13133271—22no—
26C8×#6157—22no—
27C2×#11719912C2311yescharacteristic
28C2×C8×D53D411yescharacteristic
29#13133212—42no—
30#13133209—21no—
31#13133207—42no—
32#13133193—22no—
33Q8×Dic53—22no—
34#13133174D411yescharacteristic
35#13133151—21no—
36#13133149C2×C412yes—
37#13133141—22no—
38#13133141—22no—
39#13133128D411yescharacteristic
40#13133127C2×C411yescharacteristic
41C8×Dic53D411yescharacteristic
42C2×Q16×C53—21no—
43C2×Q16×C53C2×C412yes—
44C4⋊Q8×C53—21no—
45C4×C8×C53C2×C411yescharacteristic
46C2×Q8×C53—22no—
47C2×C4×D53C2×D411yescharacteristic
48C22⋊#6157—41no—
49#11719940—41no—
50C4⋊#6157—41no—
51C4×#6157—22no—
52#11719936—41no—
53D53⋊C8—22no—
54Q16×C53—42no—
55Q16×C53—41no—
56Q16×C53—41no—
57C2×C8×C53C22⋊C411yescharacteristic
58C2×C8×C53C22×C411yescharacteristic
59C4⋊C4×C53—42no—
60C42×C53C22⋊C411yescharacteristic
61#11719921—21no—
62#11719921—42no—
63C4⋊Dic53—22no—
64C4⋊Dic53—42no—
65#11719919—21no—
66#11719919—42no—
67#11719919—42no—
68C4×Dic53—42no—
69C4×Dic53C22⋊C411yescharacteristic
70C2×#88322C22⋊C411yescharacteristic
71C2×#88322C2×D411yescharacteristic
72#11719912—41no—
73C8×D53—21no—
74C8×D53—21no—
75#11719910—42no—
76C22×D53AGL(1,ℤ/8)11yescharacteristic
77C2×#6157—42no—
78Q8×C53—42no—
79Q8×C53—42no—
80C2×C4×C53—21no—
81C2×C4×C53—42no—
82C2×C4×C53C2×C22⋊C411yescharacteristic
83C2×Dic53—21no—
84C2×Dic53—42no—
85C2×Dic53AGL(1,ℤ/8)11yescharacteristic
86C4×D53—21no—
87C4×D53—21no—
88#88324—42no—
89C8×C53—21no—
90C8×C53#10411yescharacteristic
91C8×C53#10411yescharacteristic
92#88322—21no—
93#88322—21no—
94#11771—531nomaximal, Sylow
95C22×C53#41811yescharacteristic
96C2×D53—21no—
97C2×D53—41no—
98#6157—82no—
99C4×C53—41no—
100C4×C53—82no—
101C4×C53#41011yescharacteristic
102C4×C53C2×#10411yescharacteristic
103Dic53—21no—
104Dic53—41no—
105Dic53—82no—
106#3177—531no—
107#2166—531no—
108#1747—531no—
109#1342—532no—
110#1340—532no—
111C2×C53—21no—
112C2×C53#150511yescharacteristic
113D53—41no—
114#541—1061no—
115#499—531no—
116#494#615711yescharacteristic, complemented
117#491—1061no—
118#470—532no—
119#467—1061no—
120#441—532no—
121C4×Q16—1061no—
122#405—531no—
123#405—1061no—
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×#615712yes—
133C2×M16—531no—
134C2×M16—1061no—
135C22×C8—531no—
136C4⋊Q8—21no—
137#126—2122no—
138#125—1061no—
139#125—2122no—
140C22⋊Q8—1062no—
141C4×Q8—1062no—
142#117—531no—
143#117—1061no—
144M32—1062no—
145#107—1061no—
146#106—532no—
147#103—1062no—
148#103—1062no—
149#97—531no—
150#97—531no—
151#97—1062no—
152C4×C8—531no—
153C4×C8—1062no—
154C4×C8C2×#615711yescharacteristic
155C2×Q8—22no—
156C22×C4—531no—
157Q16—42no—
158Q16—41no—
159Q16—41no—
160M16—2121no—
161M16—2121no—
162M16—2121no—
163C2×C8—531no—
164C2×C8—531no—
165C2×C8—1061no—
166C2×C8—1061no—
167C2×C8—1062no—
168C2×C8C22⋊#615711yescharacteristic
169C2×C8C22×#615711yescharacteristic
170C4⋊C4—42no—
171C4⋊C4—1061no—
172C4⋊C4—1062no—
173C4⋊C4—2122no—
174C4⋊C4—2122no—
175C4⋊C4—2122no—
176C4⋊C4—2121no—
177C22⋊C4—1061no—
178C22⋊C4—2122no—
179C22⋊C4—2121no—
180C42—531no—
181C42—1062no—
182C42—2122no—
183C42C22⋊#615711yescharacteristic
184C16—2122no—
185C23—531no—
186Q8—42no—
187Q8—42no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—531no—
191C2×C4—1061no—
192C2×C4—1061no—
193C2×C4—1061no—
194C2×C4—2122no—
195C2×C4—2122no—
196C2×C4C2×C22⋊#615711yescharacteristic
197C8—21no—
198C8—1061no—
199C8—1061no—
200C8—2122no—
201C8#1313328111yescharacteristic
202C8#1313328111yescharacteristic
203C22—1061no—
204C22—2121no—
205C22#1681161511yescharacteristic
206C4—41no—
207C4—82no—
208C4—1061no—
209C4—4242no—
210C4—4242no—
211C4—2121no—
212C4#1681160311yescharacteristic
213C4C2×#1313328111yescharacteristic
214C2—21no—
215C2—2121no—
216C2#2029010411yescharacteristic
subgroupG/H
centerC2#20290104
commutatorC2×C8×C53C22×C4
FrattiniC2×C8C22×#6157
FittingC53×#494C4
socleC2×C53#1505
prime radicalC22×D53AGL(1,ℤ/8)
cosocleC2×#11719912C23
hypercenter#494#6157
hypocenterC53#11771
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #494

Complements

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

NH ≅ G/Ntype
14. #16811025C4semidirect
18. #16810730C4semidirect
21. C53×#494C4semidirect
116. #494#6157semidirect
130. C53#11771semidirect

Maximal subgroups

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

subgroupindex
#202910902
#202904362
#202896912
#202896872
#202853712
#1177153