Finite Group Databasebeta

← #28309611

Subgroups — #28309611

Order 13568.

274 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 60 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
2#20289720C214yesmaximal, complemented
3#20285095C211yesmaximal, characteristic
4#16811546—22no—
5#16811519—22no—
6#16811515C2214yes—
7#16811180C411yescharacteristic
8#16810991C412yes—
9#16810991C2212yes—
10#16810973—22no—
11#16810632—22no—
12C4×#11719917C2211yescharacteristic
13C2×C4⋊Q8×C53C411yescharacteristic
14#13133291—24no—
15C2×#11719917D412yes—
16#13133217—21no—
17#13133209C2×C412yes—
18#13133196—41no—
19C4×Dic106—41no—
20#13133176—21no—
21#13133141—24no—
22#13133141—24no—
23#13133128C2×C411yescharacteristic
24#13133128C2311yescharacteristic
25C4×#88322D412yes—
26C22×Q8×C53—21no—
27C4⋊Q8×C53—21no—
28C4⋊Q8×C53—41no—
29C4⋊Q8×C53C2×C412yes—
30C2×C4⋊C4×C53—41no—
31C2×C42×C53C2×C411yescharacteristic
32C2×Q8×C53—22no—
33C2×Q8×C53—44no—
34C2×Q8×C53—41no—
35C22×C4×C53—21no—
36C22×C4×C53—41no—
37C22×C4×C53C22⋊C411yescharacteristic
38C2×Dic106—41no—
39C4⋊C4×C53—42no—
40C4⋊C4×C53—42no—
41C42×C53—21no—
42C42×C53C22⋊C411yescharacteristic
43C42×C53C22×C411yescharacteristic
44C22⋊Dic53—41no—
45C4⋊Dic53—22no—
46C4⋊Dic53—41no—
47#11719919—42no—
48C4×Dic53—41no—
49#11719917—42no—
50C2×#88322C22⋊C412yes—
51C2×#88322C2×D412yes—
52#11719910—44no—
53C23×C53#10011yescharacteristic
54Q8×C53—44no—
55Q8×C53—82no—
56C2×C4×C53—21no—
57C2×C4×C53—22no—
58C2×C4×C53—21no—
59C2×C4×C53—42no—
60C2×C4×C53—42no—
61C2×C4×C53—82no—
62C2×C4×C53#10011yescharacteristic
63C2×C4×C53C2×C22⋊C411yescharacteristic
64C2×Dic53—42no—
65Dic106—82no—
66#88322—22no—
67#88322—22no—
68#11916—531nomaximal, Sylow
69C22×C53—21no—
70C22×C53—41no—
71C22×C53C2×#10011yescharacteristic
72C4×C53—22no—
73C4×C53—21no—
74C4×C53—41no—
75C4×C53—82no—
76C4×C53#42112yes—
77Dic53—82no—
78#3139—531no—
79#2165—532no—
80#1356—534no—
81C2×C53—21no—
82C2×C53—41no—
83C2×C53#153311yescharacteristic
84#533—531no—
85C2×C4⋊Q8#615711yescharacteristic
86#491—532no—
87#491—532no—
88#487—1062no—
89#419—1062no—
90C4×M16—531no—
91#364—1062no—
92#349—1062no—
93#347—534no—
94C53#1191611yescharacteristic, complemented, Sylow
95C22×Q8—21no—
96C2×M16—532no—
97C4⋊Q8—21no—
98C4⋊Q8—41no—
99C4⋊Q8—1061no—
100C4⋊Q8C2×#615712yes—
101#125—532no—
102C22⋊Q8—2121no—
103C4×Q8—2121no—
104#117—1061no—
105C2×C4⋊C4—41no—
106C2×C42C2×#615711yescharacteristic
107M32—1064no—
108#103—1064no—
109#103—1064no—
110#97—531no—
111#97—531no—
112C4×C8—532no—
113C2×Q8—22no—
114C2×Q8—44no—
115C2×Q8—41no—
116C2×Q8—2121no—
117C22×C4—21no—
118C22×C4—41no—
119C22×C4C22⋊#615711yescharacteristic
120M16—2122no—
121C2×C8—532no—
122C2×C8—532no—
123C4⋊C4—42no—
124C4⋊C4—42no—
125C4⋊C4—1062no—
126C4⋊C4—2122no—
127C4⋊C4—2121no—
128C22⋊C4—2121no—
129C42—21no—
130C42—2121no—
131C42C22⋊#615711yescharacteristic
132C42C22×#615711yescharacteristic
133C16—2124no—
134C23#1313327711yescharacteristic
135Q8—44no—
136Q8—82no—
137Q8—4242no—
138C2×C4—21no—
139C2×C4—22no—
140C2×C4—21no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—82no—
144C2×C4—2122no—
145C2×C4#1313327711yescharacteristic
146C2×C4C2×C22⋊#615711yescharacteristic
147C8—1062no—
148C8—1062no—
149C22—21no—
150C22—41no—
151C22C2×#1313327711yescharacteristic
152C4—22no—
153C4—21no—
154C4—41no—
155C4—82no—
156C4—4242no—
157C4#1681162112yes—
158C2—21no—
159C2—41no—
160C2#2029019411yescharacteristic
subgroupG/H
centerC2#20290194
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
FittingC2×C4⋊Q8×C53C4
socleC2×C53#1533
prime radicalC23×C53#100
cosocle#13133128C23
hypercenterC2×C4⋊Q8#6157
hypocenterC53#11916
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
2. #20289720C2semidirect
94. C53#11916semidirect

Maximal subgroups

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

subgroupindex
#202910842
#202897202
#202850952
#1191653