Finite Group Databasebeta

← #28309812

Subgroups — #28309812

Order 13568.

360 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 64 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#20291084C212yesmaximal, complemented
2#20289726C212yesmaximal, complemented
3#20289724C212yesmaximal, complemented
4#20285172C211yesmaximal, characteristic
5#16811546C2214yes—
6#16811519—22no—
7#16811515—22no—
8C2×#13133248—22no—
9#16811173C411yescharacteristic
10#16811015—22no—
11#16811002C411yescharacteristic
12#16811002C2211yescharacteristic
13#16810997C411yescharacteristic
14#16810997C2211yescharacteristic
15C4×#11719917C2211yescharacteristic
16C2×C4⋊Q8×C53C411yescharacteristic
17#13133291—24no—
18C2×#11719945—21no—
19#13133248—48no—
20C2×#11719925—22no—
21C2×#11719925—22no—
22C2×#11719924—22no—
23C2×#11719924—22no—
24C2×#11719917D412yes—
25#13133217C2×C411yescharacteristic
26#13133214C2×C411yescharacteristic
27#13133205—21no—
28#13133191—41no—
29C4×D212—41no—
30#13133128D412yes—
31C4×#88322C2×C411yescharacteristic
32C4×#88322C2311yescharacteristic
33C22×Q8×C53—21no—
34C4⋊Q8×C53—21no—
35C4⋊Q8×C53—41no—
36C4⋊Q8×C53C2×C411yescharacteristic
37C4⋊Q8×C53C2×C411yescharacteristic
38C2×C4⋊C4×C53—41no—
39C2×C42×C53C2×C411yescharacteristic
40C2×Q8×C53—21no—
41C2×Q8×C53—21no—
42C2×Q8×C53—44no—
43C2×Q8×C53—41no—
44C22×C4×C53—21no—
45C22×C4×C53—41no—
46C22×C4×C53C22⋊C411yescharacteristic
47C2×C22⋊D53—41no—
48#11719945—44no—
49C2×D212—21no—
50C2×D212—41no—
51C2×C4×D53—41no—
52C2×Dic106—21no—
53C4⋊C4×C53—42no—
54C4⋊C4×C53—41no—
55C4⋊C4×C53—41no—
56C42×C53—21no—
57C42×C53C22⋊C411yescharacteristic
58C42×C53C22×C411yescharacteristic
59#11719925—44no—
60#11719925—44no—
61#11719924—44no—
62#11719924—44no—
63#11719921—42no—
64C4⋊Dic53—41no—
65#11719917—24no—
66C2×#88322C22⋊C412yes—
67C2×#88322C2×D412yes—
68#11719910—44no—
69C23×C53#10011yescharacteristic
70C22×D53—41no—
71Q8×C53—42no—
72Q8×C53—42no—
73Q8×C53—82no—
74C2×C4×C53—21no—
75C2×C4×C53—22no—
76C2×C4×C53—21no—
77C2×C4×C53—42no—
78C2×C4×C53—41no—
79C2×C4×C53—41no—
80C2×C4×C53—82no—
81C2×C4×C53#10011yescharacteristic
82C2×C4×C53C2×C22⋊C411yescharacteristic
83C22⋊D53—82no—
84C2×Dic53—41no—
85D212—42no—
86D212—82no—
87C4×D53—82no—
88Dic106—42no—
89#88322—22no—
90#88322AGL(1,ℤ/8)14yes—
91#11993—531nomaximal, Sylow
92C22×C53—21no—
93C22×C53—41no—
94C22×C53C2×#10011yescharacteristic
95C2×D53—82no—
96C2×D53—81no—
97C4×C53—22no—
98C4×C53—21no—
99C4×C53—41no—
100C4×C53—81no—
101C4×C53—81no—
102C4×C53#41812yes—
103Dic53—81no—
104#3152—531no—
105#2165—532no—
106#1359—532no—
107#1358—532no—
108C2×C53—21no—
109C2×C53—41no—
110C2×C53#155511yescharacteristic
111D53—161no—
112C2×#137—1062no—
113#532—531no—
114C2×C4⋊Q8#615711yescharacteristic
115#497—1062no—
116#494—531no—
117#494—531no—
118#492—531no—
119#492—531no—
120C4×M16—531no—
121#364—534no—
122#349—1062no—
123#347—1062no—
124C53#1199311yescharacteristic, complemented, Sylow
125C2×#41—1061no—
126C22×Q8—21no—
127#137—2128no—
128C2×Q16—1062no—
129C2×Q16—1062no—
130C2×SD16—1062no—
131C2×SD16—1062no—
132C2×M16—532no—
133C4⋊Q8—21no—
134C4⋊Q8—41no—
135C4⋊Q8—531no—
136C4⋊Q8C2×#615711yescharacteristic
137C4⋊Q8C2×#615711yescharacteristic
138#127—531no—
139#124—1061no—
140#121—2121no—
141C4×D4—2121no—
142C2×C4⋊C4—41no—
143C2×C42C2×#615711yescharacteristic
144M32—1064no—
145#97—532no—
146C4×C8—531no—
147C4×C8—531no—
148#41—2124no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—44no—
152C2×Q8—41no—
153C2×Q8—1061no—
154C2×D4—1061no—
155C2×D4—2121no—
156C2×D4—2121no—
157C22×C4—21no—
158C22×C4—41no—
159C22×C4—2121no—
160C22×C4C22⋊#615711yescharacteristic
161Q16—2124no—
162Q16—2124no—
163SD16—2124no—
164SD16—2124no—
165M16—1064no—
166C2×C8—532no—
167C2×C8—532no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—42no—
171C4⋊C4—2121no—
172C22⋊C4—2122no—
173C42—21no—
174C42C22⋊#615711yescharacteristic
175C42C22×#615711yescharacteristic
176C16—2124no—
177C23—2121no—
178C23#1313327711yescharacteristic
179Q8—42no—
180Q8—42no—
181Q8—82no—
182Q8—2122no—
183D4—4242no—
184D4—4242no—
185D4—2122no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—82no—
193C2×C4—4242no—
194C2×C4—2121no—
195C2×C4#1313327711yescharacteristic
196C2×C4C2×C22⋊#615711yescharacteristic
197C8—534no—
198C8—1062no—
199C22—21no—
200C22—41no—
201C22—4242no—
202C22—4241no—
203C22C2×#1313327711yescharacteristic
204C4—22no—
205C4—21no—
206C4—41no—
207C4—81no—
208C4—81no—
209C4—4241no—
210C4#1681161512yes—
211C2—21no—
212C2—41no—
213C2—8481no—
214C2#2029024211yescharacteristic
subgroupG/H
centerC2#20290242
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
FittingC2×C4⋊Q8×C53C4
socleC2×C53#1555
prime radical#16811173C4
cosocleC4×#88322C23
hypercenterC2×C4⋊Q8#6157
hypocenterC53#11993
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #20291084C2semidirect
2. #20289726C2semidirect
3. #20289724C2semidirect
124. C53#11993semidirect

Maximal subgroups

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

subgroupindex
#202910842
#202897262
#202897242
#202851722
#1199353