Finite Group Databasebeta

← #28309816

Subgroups — #28309816

Order 13568.

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

normal subgroups only

This group has 258 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
2#20289735C214yesmaximal, complemented
3#20285264C211yesmaximal, characteristic
4#16811546C2214yes—
5#16811519—22no—
6#16811515—22no—
7#16811180C411yescharacteristic
8#16811033—22no—
9#16811022C412yes—
10#16811022C2212yes—
11#16810663—22no—
12C4×#11719917C2211yescharacteristic
13C2×#125×C53C411yescharacteristic
14#13133291—24no—
15C2×#11719917D412yes—
16#13133217—21no—
17#13133209C2×C412yes—
18#13133196—41no—
19C4×Dic106—41no—
20#13133176—21no—
21#13133150—22no—
22#13133150—22no—
23#13133148—22no—
24#13133148—22no—
25#13133128D412yes—
26C4×#88322C2×C411yescharacteristic
27C4×#88322C2311yescharacteristic
28#125×C53—21no—
29#125×C53—41no—
30#125×C53C2×C412yes—
31C2×C4⋊C4×C53—21no—
32C2×C4⋊C4×C53—41no—
33C2×C42×C53C2×C411yescharacteristic
34C22×C4×C53—21no—
35C22×C4×C53—41no—
36C22×C4×C53C22⋊C411yescharacteristic
37C2×Dic106—41no—
38C4⋊C4×C53—22no—
39C4⋊C4×C53—42no—
40C4⋊C4×C53—42no—
41C4⋊C4×C53—41no—
42C42×C53—21no—
43C42×C53C22⋊C411yescharacteristic
44C42×C53C22×C411yescharacteristic
45C22⋊Dic53—41no—
46C4⋊Dic53—22no—
47C4⋊Dic53—41no—
48#11719919—42no—
49C4×Dic53—41no—
50#11719917—24no—
51C2×#88322C22⋊C412yes—
52C2×#88322C2×D412yes—
53#11719910—44no—
54C23×C53#10011yescharacteristic
55C2×C4×C53—22no—
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53—42no—
59C2×C4×C53—81no—
60C2×C4×C53—82no—
61C2×C4×C53#10011yescharacteristic
62C2×C4×C53C2×C22⋊C411yescharacteristic
63C2×Dic53—42no—
64Dic106—82no—
65#88322—22no—
66#88322#13714yes—
67#11994—531nomaximal, Sylow
68C22×C53—21no—
69C22×C53—41no—
70C22×C53C2×#10011yescharacteristic
71C4×C53—21no—
72C4×C53—41no—
73C4×C53—41no—
74C4×C53—82no—
75C4×C53#41912yes—
76Dic53—82no—
77#3166—531no—
78#2165—532no—
79#1366—534no—
80C2×C53—21no—
81C2×C53—41no—
82C2×C53#155811yescharacteristic
83#533—531no—
84C2×#125#615711yescharacteristic
85#501—1062no—
86#499—532no—
87#499—532no—
88#428—1062no—
89C4×M16—531no—
90#364—534no—
91#349—1062no—
92#347—1062no—
93C53#1199411yescharacteristic, complemented, Sylow
94C2×M16—532no—
95C4⋊Q8—1061no—
96#125—21no—
97#125—41no—
98#125—532no—
99#125C2×#615712yes—
100C22⋊Q8—2121no—
101C4×Q8—2121no—
102#117—1061no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—41no—
105C2×C42C2×#615711yescharacteristic
106M32—1064no—
107#107—1062no—
108#107—1062no—
109#106—1062no—
110#106—1062no—
111#97—532no—
112C4×C8—531no—
113C4×C8—531no—
114C2×Q8—2121no—
115C22×C4—21no—
116C22×C4—41no—
117C22×C4C22⋊#615711yescharacteristic
118M16—1064no—
119C2×C8—532no—
120C2×C8—532no—
121C4⋊C4—22no—
122C4⋊C4—42no—
123C4⋊C4—42no—
124C4⋊C4—41no—
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—4242no—
136C2×C4—22no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—42no—
140C2×C4—81no—
141C2×C4—82no—
142C2×C4—2122no—
143C2×C4#1313327711yescharacteristic
144C2×C4C2×C22⋊#615711yescharacteristic
145C8—534no—
146C8—1062no—
147C22—21no—
148C22—41no—
149C22C2×#1313327711yescharacteristic
150C4—21no—
151C4—41no—
152C4—41no—
153C4—82no—
154C4—4242no—
155C4#1681161612yes—
156C2—21no—
157C2—41no—
158C2#2029024811yescharacteristic
subgroupG/H
centerC2#20290248
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
FittingC2×#125×C53C4
socleC2×C53#1558
prime radicalC23×C53#100
cosocleC4×#88322C23
hypercenterC2×#125#6157
hypocenterC53#11994
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
2. #20289735C2semidirect
93. C53#11994semidirect

Maximal subgroups

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

subgroupindex
#202910842
#202897352
#202852642
#1199453