Finite Group Databasebeta

← #28312474

Subgroups — #28312474

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#20290988C212yesmaximal
2#20289855C214yesmaximal, complemented
3#20285118C211yesmaximal, characteristic
4#16811602—22no—
5#16811530—22no—
6#16811528C2214yes—
7#16811173C2211yescharacteristic
8#16810993C412yescomplemented
9#16810989C2212yes—
10#16810980—22no—
11#16810622—22no—
12C4×#11719917C411yescharacteristic, complemented
13C2×#125×C53C411yescharacteristic, complemented
14#13133296—42no—
15#13133276—24no—
16#13133275—42no—
17C2×#11719945D412yes—
18C2×#11719917—21no—
19#13133217C2×C411yescharacteristic
20#13133214C2311yescharacteristic
21#13133205D412yes—
22#13133191—41no—
23C4×D212—41no—
24#13133140—24no—
25#13133137—24no—
26#13133128C2×C412yes—
27C4×#88322—21no—
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×C22⋊D53—22no—
38#11719945—42no—
39#11719945—42no—
40C2×D212—41no—
41C2×D212C2×D412yes—
42C2×C4×D53—22no—
43C2×Dic106C22⋊C412yes—
44C22⋊#6157—44no—
45C4⋊#6157—82no—
46C4×#6157—82no—
47C4⋊C4×C53—22no—
48C4⋊C4×C53—42no—
49C4⋊C4×C53—42no—
50C4⋊C4×C53—41no—
51C42×C53—21no—
52C42×C53C22⋊C411yescharacteristic
53C42×C53C22×C411yescharacteristic
54#11719921—42no—
55C4⋊Dic53—41no—
56#11719917—42no—
57C2×#88322—22no—
58C23×C53#9911yescharacteristic
59C22×D53—22no—
60C2×#6157—84no—
61C2×C4×C53—22no—
62C2×C4×C53—21no—
63C2×C4×C53—21no—
64C2×C4×C53—42no—
65C2×C4×C53—81no—
66C2×C4×C53—82no—
67C2×C4×C53#9911yescharacteristic
68C2×C4×C53C2×C22⋊C411yescharacteristic
69C22⋊D53—82no—
70C2×Dic53—22no—
71D212—42no—
72D212—82no—
73C4×D53—42no—
74C4×D53—42no—
75Dic106—42no—
76#88322—42no—
77#13596—531nomaximal, Sylow
78C22×C53—21no—
79C22×C53—41no—
80C22×C53C2×#9911yescharacteristic
81C2×D53—42no—
82C2×D53—82no—
83#6157—164no—
84C4×C53—21no—
85C4×C53—41no—
86C4×C53—41no—
87C4×C53—82no—
88C4×C53#35412yes—
89Dic53—42no—
90#3142—531no—
91#2126—532no—
92#1412—534no—
93C2×C53—21no—
94C2×C53—41no—
95C2×C53C2×#35411yescharacteristic
96D53—82no—
97#532—531no—
98C2×#125#615711yescharacteristic, complemented
99#491—532no—
100#490—532no—
101#488—1062no—
102#417—1062no—
103#410—1062no—
104C4×M16—531no—
105#354—1062no—
106#353—534no—
107C53#1359611yescharacteristic, complemented, Sylow
108C2×#41—532no—
109C2×M16—1061no—
110C4⋊Q8—531no—
111#127—531no—
112#125—21no—
113#125—41no—
114#125C2×#615712yes—
115#124—532no—
116#121—2121no—
117C4×D4—2121no—
118#117—2122no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—41no—
121C2×C42C2×#615711yescharacteristic
122#103—1064no—
123#102—1064no—
124#99—1064no—
125#99—2122no—
126#97—532no—
127C4×C8—1061no—
128#41—2122no—
129#41—2122no—
130C2×Q8—532no—
131C2×D4—532no—
132C2×D4—1062no—
133C2×D4—2121no—
134C22×C4—21no—
135C22×C4—41no—
136C22×C4—1062no—
137C22×C4C22⋊#615711yescharacteristic
138M16—2122no—
139C2×C8—1062no—
140C4⋊C4—22no—
141C4⋊C4—42no—
142C4⋊C4—42no—
143C4⋊C4—41no—
144C4⋊C4—4242no—
145C4⋊C4—2121no—
146C22⋊C4—2122no—
147C22⋊C4—2124no—
148C42—21no—
149C42—4242no—
150C42C22⋊#615711yescharacteristic
151C42C22×#615711yescharacteristic
152C23—1062no—
153C23#1313327511yescharacteristic
154Q8—2122no—
155D4—4242no—
156D4—4242no—
157D4—2122no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—42no—
162C2×C4—81no—
163C2×C4—82no—
164C2×C4—1062no—
165C2×C4—4244no—
166C2×C4—2122no—
167C2×C4—2122no—
168C2×C4#1313327511yescharacteristic
169C2×C4C2×C22⋊#615711yescharacteristic
170C8—2122no—
171C22—21no—
172C22—41no—
173C22—4242no—
174C22—2122no—
175C22C2×#1313327511yescharacteristic
176C4—21no—
177C4—41no—
178C4—41no—
179C4—82no—
180C4—8484no—
181C4—2122no—
182C4#1681152912yes—
183C2—21no—
184C2—41no—
185C2—4242no—
186C2C2×#1681152911yescharacteristic
subgroupG/H
centerC2C2×#16811529
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
FittingC2×#125×C53C4
socleC2×C53C2×#354
prime radical#16811173C22
cosocle#13133214C23
hypercenterC2×#125#6157
hypocenterC53#13596
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
2. #20289855C2semidirect
8. #16810993C4semidirect
12. C4×#11719917C4semidirect
13. C2×#125×C53C4semidirect
98. C2×#125#6157semidirect
107. C53#13596semidirect

Maximal subgroups

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

subgroupindex
#202909882
#202898552
#202851182
#1359653