Finite Group Databasebeta

← #28309608

Subgroups — #28309608

Order 13568.

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

normal subgroups only

This group has 326 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, complemented
2#20289717C214yesmaximal, complemented
3#20285089C211yesmaximal, characteristic
4#16811546—22no—
5#16811519—22no—
6#16811515C2214yescomplemented
7#16811180C411yescharacteristic
8#16810987C412yes—
9#16810987C2212yes—
10#16810969—22no—
11#16810628—22no—
12C4×#11719917C2211yescharacteristic
13C2×#127×C53C411yescharacteristic
14#13133291—24no—
15C2×#11719917D412yes—
16#13133217—21no—
17#13133209C2×C412yes—
18#13133196—41no—
19C4×Dic106—41no—
20#13133176—21no—
21#13133138—24no—
22#13133138—24no—
23#13133128C2×C411yescharacteristic
24#13133128C2311yescharacteristic
25C4×#88322D412yes—
26C22×D4×C53—21no—
27C22×D4×C53—41no—
28#127×C53—21no—
29#127×C53—41no—
30#127×C53C2×C412yes—
31C2×C42×C53C2×C411yescharacteristic
32C24×C53—41no—
33C2×D4×C53—22no—
34C2×D4×C53—44no—
35C2×D4×C53—42no—
36C2×D4×C53—42no—
37C2×D4×C53—41no—
38C2×D4×C53—84no—
39C22×C4×C53—21no—
40C22×C4×C53C22⋊C411yescharacteristic
41C2×Dic106—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—42no—
51C2×#88322C22⋊C412yes—
52C2×#88322C2×D412yes—
53#11719910—44no—
54C23×C53—42no—
55C23×C53—82no—
56C23×C53—84no—
57C23×C53#10011yescharacteristic
58D4×C53—44no—
59D4×C53—82no—
60D4×C53—84no—
61D4×C53—84no—
62C2×C4×C53—21no—
63C2×C4×C53—22no—
64C2×C4×C53—21no—
65C2×C4×C53—42no—
66C2×C4×C53#10011yescharacteristic
67C2×C4×C53C2×C22⋊C411yescharacteristic
68C2×Dic53—42no—
69Dic106—82no—
70#88322—22no—
71#88322—22no—
72#11915—531nomaximal, Sylow
73C22×C53—21no—
74C22×C53—41no—
75C22×C53—84no—
76C22×C53—82no—
77C22×C53—164no—
78C22×C53C2×#10011yescharacteristic
79C4×C53—22no—
80C4×C53—21no—
81C4×C53—41no—
82C4×C53#42112yes—
83Dic53—82no—
84#3138—531no—
85#2165—532no—
86#1355—534no—
87C2×C53—21no—
88C2×C53—41no—
89C2×C53—162no—
90C2×C53#153311yescharacteristic
91#533—531no—
92C2×#127#615711yescharacteristic
93#490—532no—
94#490—532no—
95#486—1062no—
96#418—1062no—
97C4×M16—531no—
98#364—1062no—
99#349—1062no—
100#347—534no—
101C53#1191511yescharacteristic, complemented, Sylow
102C22×D4—21no—
103C22×D4—41no—
104C2×M16—532no—
105C4⋊Q8—1061no—
106#127—21no—
107#127—41no—
108#127C2×#615712yes—
109#125—532no—
110C22⋊Q8—2121no—
111C4×Q8—2121no—
112#117—1061no—
113C2×C42C2×#615711yescharacteristic
114M32—1064no—
115#102—1064no—
116#102—1064no—
117#97—531no—
118#97—531no—
119C4×C8—532no—
120C24—41no—
121C2×Q8—2121no—
122C2×D4—22no—
123C2×D4—44no—
124C2×D4—42no—
125C2×D4—42no—
126C2×D4—41no—
127C2×D4—84no—
128C22×C4—21no—
129C22×C4C22⋊#615711yescharacteristic
130M16—2122no—
131C2×C8—532no—
132C2×C8—532no—
133C4⋊C4—1062no—
134C4⋊C4—2122no—
135C4⋊C4—2121no—
136C22⋊C4—2121no—
137C42—21no—
138C42—2121no—
139C42C22⋊#615711yescharacteristic
140C42C22×#615711yescharacteristic
141C16—2124no—
142C23—42no—
143C23—82no—
144C23—84no—
145C23#1313327711yescharacteristic
146Q8—4242no—
147D4—44no—
148D4—82no—
149D4—84no—
150D4—84no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—42no—
155C2×C4—2122no—
156C2×C4#1313327711yescharacteristic
157C2×C4C2×C22⋊#615711yescharacteristic
158C8—1062no—
159C8—1062no—
160C22—21no—
161C22—41no—
162C22—84no—
163C22—82no—
164C22—164no—
165C22C2×#1313327711yescharacteristic
166C4—22no—
167C4—21no—
168C4—41no—
169C4—4242no—
170C4#1681162112yes—
171C2—21no—
172C2—41no—
173C2—162no—
174C2#2029019411yescharacteristic
subgroupG/H
centerC2#20290194
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
Fitting, prime radicalC2×#127×C53C4
socleC2×C53#1533
cosocle#13133128C23
hypercenterC2×#127#6157
hypocenterC53#11915
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #20291084C2semidirect
2. #20289717C2semidirect
6. #16811515C22semidirect
101. C53#11915semidirect

Maximal subgroups

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

subgroupindex
#202910842
#202897172
#202850892
#1191553