Finite Group Databasebeta

← #28299017

Subgroups — #28299017

Order 13568.

238 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 32 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#20289803C211yesmaximal, characteristic
2#20289650C211yesmaximal, characteristic
3#20275834C211yesmaximal, characteristic
4#16811602—21no—
5#16811519—21no—
6C2×#13133248—21no—
7C2×#13133136—21no—
8#16810589C2211yescharacteristic
9#16810388C411yescharacteristic, complemented
10C22×Q16×C53C411yescharacteristic, complemented
11#13133296—41no—
12#13133291—41no—
13#13133276—41no—
14#13133275—22no—
15C2×#11719945D411yescharacteristic
16#13133248—42no—
17#13133248—42no—
18C2×#11719925—41no—
19C2×#11719924—41no—
20C2×#11719917—21no—
21C2×#11719917D411yescharacteristic
22#13133136—42no—
23#13133129—41no—
24C22×Q8×C53—21no—
25C2×Q16×C53—42no—
26C2×Q16×C53—41no—
27C22×C8×C53C2×C411yescharacteristic
28C2×Q8×C53—22no—
29C2×Q8×C53—41no—
30C2×Q8×C53—41no—
31C2×Q8×C53—41no—
32C2×Q8×C53—41no—
33C2×Q8×C53—41no—
34C22×C4×C53—41no—
35C22×C4×C53C22⋊C411yescharacteristic
36C2×C22⋊D53—21no—
37#11719945—41no—
38#11719945—41no—
39C2×D212D811yescharacteristic
40C2×C4×D53—21no—
41C2×Dic106SD1611yescharacteristic
42C22⋊#6157—42no—
43C4⋊#6157—81no—
44C4×#6157—81no—
45Q16×C53—42no—
46Q16×C53—42no—
47C2×C8×C53—21no—
48C2×C8×C53—41no—
49#11719925—44no—
50#11719924—44no—
51#11719917—22no—
52#11719917—41no—
53#11719917—41no—
54C2×#88322—21no—
55C2×#88322—41no—
56#11719910—81no—
57C23×C53#10411yescharacteristic
58C22×D53—21no—
59C2×#6157—82no—
60Q8×C53—42no—
61Q8×C53—42no—
62Q8×C53—81no—
63Q8×C53—81no—
64C2×C4×C53—21no—
65C2×C4×C53—21no—
66C2×C4×C53—42no—
67C2×C4×C53—81no—
68C2×C4×C53—81no—
69C2×C4×C53#9911yescharacteristic
70C2×C4×C53#10211yescharacteristic
71C22⋊D53—81no—
72C2×Dic53—21no—
73D212—41no—
74C4×D53—41no—
75C4×D53—41no—
76Dic106—41no—
77C8×C53—41no—
78#88322—42no—
79#88322—41no—
80#7399—531nomaximal, Sylow
81C22×C53—21no—
82C22×C53—41no—
83C22×C53#32711yescharacteristic
84C2×D53—41no—
85C2×D53—81no—
86#6157—162no—
87C4×C53—21no—
88C4×C53—82no—
89C4×C53AΣL(1,𝔽16)11yescharacteristic
90C4×C53#35311yescharacteristic
91Dic53—41no—
92#2020—531no—
93#1391—531no—
94#1321—531no—
95C2×C53—21no—
96C2×C53—41no—
97C2×C53#134311yescharacteristic
98D53—81no—
99C2×#137—1061no—
100C22×Q16#615711yescharacteristic, complemented
101C2×#101—1061no—
102#410—1061no—
103#408—531no—
104#349—1061no—
105#341—531no—
106C53#739911yescharacteristic, complemented, Sylow
107C2×#41—531no—
108C22×Q8—21no—
109#137—2122no—
110#137—2122no—
111C2×Q16—41no—
112C2×Q16—42no—
113C2×Q16—2121no—
114C2×SD16—2121no—
115C2×M16—531no—
116C2×M16—1061no—
117C22×C8C2×#615711yescharacteristic
118#117—2121no—
119M32—2121no—
120#101—2122no—
121#99—1062no—
122#99—2121no—
123C22⋊C8—2121no—
124#41—2121no—
125#41—2121no—
126C2×Q8—22no—
127C2×Q8—41no—
128C2×Q8—41no—
129C2×Q8—41no—
130C2×Q8—41no—
131C2×Q8—41no—
132C2×Q8—531no—
133C2×D4—531no—
134C2×D4—1061no—
135C22×C4—41no—
136C22×C4—1061no—
137C22×C4C22⋊#615711yescharacteristic
138Q16—42no—
139Q16—42no—
140Q16—2124no—
141SD16—2124no—
142M16—1062no—
143M16—2121no—
144M16—2121no—
145C2×C8—21no—
146C2×C8—41no—
147C2×C8—1061no—
148C2×C8—2121no—
149C4⋊C4—4241no—
150C22⋊C4—2122no—
151C42—4241no—
152C16—4241no—
153C23—1061no—
154C23#1313328111yescharacteristic
155Q8—42no—
156Q8—42no—
157Q8—81no—
158Q8—81no—
159Q8—2121no—
160D4—4241no—
161D4—2121no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—42no—
165C2×C4—81no—
166C2×C4—81no—
167C2×C4—1061no—
168C2×C4—4242no—
169C2×C4—2121no—
170C2×C4—2121no—
171C2×C4#1313327511yescharacteristic
172C2×C4#1313327911yescharacteristic
173C8—41no—
174C8—2122no—
175C8—2121no—
176C22—21no—
177C22—41no—
178C22—4241no—
179C22—2121no—
180C22#1681147411yescharacteristic
181C4—21no—
182C4—82no—
183C4—8482no—
184C4—2121no—
185C4#1681152311yescharacteristic
186C4#1681152711yescharacteristic
187C2—21no—
188C2—41no—
189C2—4241no—
190C2#2028969311yescharacteristic
subgroupG/H
centerC2#20289693
commutatorC22×C8×C53C2×C4
FrattiniC22×C8C2×#6157
FittingC22×Q16×C53C4
socleC2×C53#1343
prime radicalC2×#11719945D4
cosocle#16810589C22
hypercenterC22×Q16#6157
hypocenterC53#7399
LCSG ▷ C22×C8×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
9. #16810388C4semidirect
10. C22×Q16×C53C4semidirect
100. C22×Q16#6157semidirect
106. C53#7399semidirect

Maximal subgroups

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

subgroupindex
#202898032
#202896502
#202758342
#739953