Finite Group Databasebeta

← #27271685

Subgroups — #27271685

Order 13568.

206 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 41 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#20275876C211yesmaximal, characteristic
2#20272541C211yesmaximal, characteristic
3C53×#1394C211yesmaximal, characteristic
4#16811254C411yescharacteristic
5#16810632—21no—
6#16810392C411yescharacteristic
7#16810383—21no—
8#16810378—21no—
9C2×#101×C53—21no—
10C53×#391C2211yescharacteristic
11C2×#11719917—21no—
12C22×#88322—41no—
13#13133216—41no—
14#13133208—41no—
15#13133196—41no—
16#13133176—21no—
17#13133141—41no—
18#13133141—41no—
19C22×Q8×C53D411yescharacteristic
20C2×M16×C53—41no—
21C2×C4⋊C4×C53C2×C411yescharacteristic
22C2×C42×C53D411yescharacteristic
23#101×C53—22no—
24#101×C53—41no—
25#95×C53—41no—
26C2×Q8×C53—21no—
27C2×Q8×C53—41no—
28C2×Q8×C53—41no—
29C2×Q8×C53SD1611yescharacteristic
30C2×Q8×C53Q1611yescharacteristic
31C22×C4×C53—21no—
32C22×C4×C53—21no—
33C22×C4×C53—21no—
34C22×C4×C53C22⋊C411yescharacteristic
35C2×Dic106—41no—
36M16×C53—42no—
37M16×C53—81no—
38C2×C8×C53—81no—
39C4⋊C4×C53—41no—
40C42×C53—41no—
41C22⋊Dic53—41no—
42C4⋊Dic53—21no—
43C4⋊Dic53—21no—
44C4⋊Dic53—41no—
45#11719919—42no—
46C4×Dic53—41no—
47#11719917—41no—
48#11719917—41no—
49C2×#88322—41no—
50C2×#88322—42no—
51C2×#88322—81no—
52C2×#88322—81no—
53C23×C53#9911yescharacteristic
54Q8×C53—41no—
55Q8×C53—41no—
56Q8×C53—81no—
57C2×C4×C53—21no—
58C2×C4×C53—21no—
59C2×C4×C53—21no—
60C2×C4×C53—21no—
61C2×C4×C53—21no—
62C2×C4×C53—21no—
63C2×C4×C53—41no—
64C2×C4×C53—41no—
65C2×C4×C53—41no—
66C2×C4×C53—41no—
67C2×C4×C53—41no—
68C2×C4×C53—81no—
69C2×C4×C53#10311yescharacteristic
70C2×C4×C53#10411yescharacteristic
71C2×Dic53—41no—
72C2×Dic53—41no—
73Dic106—161no—
74C8×C53—82no—
75#88322—81no—
76#88322—82no—
77#7431—531nomaximal, Sylow
78C22×C53—21no—
79C22×C53—21no—
80C22×C53#32811yescharacteristic
81C22×C53AΣL(1,𝔽16)11yescharacteristic
82C22×C53#35411yescharacteristic
83C4×C53—21no—
84C4×C53—21no—
85C4×C53—41no—
86C4×C53—41no—
87C4×C53—41no—
88C4×C53—41no—
89C4×C53—81no—
90Dic53—81no—
91Dic53—81no—
92#2028—531no—
93#1394D5311yescharacteristic
94#1272—531no—
95C2×C53—21no—
96C2×C53—21no—
97C2×C53#135111yescharacteristic
98C2×C53#140311yescharacteristic
99C2×C53#140511yescharacteristic
100#544—531no—
101#419—1061no—
102C2×#101—21no—
103#391C2×D5311yescharacteristic
104#343—531no—
105#339—1061no—
106#336—1061no—
107C53#743111yescharacteristic, complemented, Sylow
108C22×Q8C22⋊D5311yescharacteristic
109C2×M16—41no—
110C2×M16—1061no—
111C22×C8—2121no—
112C4⋊Q8—2121no—
113#125—2121no—
114C22⋊Q8—2121no—
115#117—1061no—
116C2×C4⋊C4C4×D5311yescharacteristic
117C2×C42D21211yescharacteristic
118#103—2121no—
119#103—2121no—
120#101—22no—
121#101—41no—
122#95—41no—
123C2×Q8—21no—
124C2×Q8—41no—
125C2×Q8—41no—
126C2×Q8—2121no—
127C2×Q8#1171992311yescharacteristic
128C2×Q8#1171992511yescharacteristic
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4#1171992111yescharacteristic
133M16—42no—
134M16—81no—
135M16—2121no—
136M16—2121no—
137C2×C8—81no—
138C2×C8—4241no—
139C2×C8—4241no—
140C2×C8—2121no—
141C2×C8—2122no—
142C4⋊C4—41no—
143C4⋊C4—1061no—
144C4⋊C4—1061no—
145C4⋊C4—2121no—
146C4⋊C4—2122no—
147C22⋊C4—2121no—
148C42—41no—
149C42—2121no—
150C23#1313313111yescharacteristic
151Q8—41no—
152Q8—41no—
153Q8—81no—
154Q8—8481no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—81no—
167C2×C4—2121no—
168C2×C4—2121no—
169C2×C4#1313314011yescharacteristic
170C2×C4#1313314511yescharacteristic
171C8—82no—
172C8—4241no—
173C8—4242no—
174C22—21no—
175C22—21no—
176C22#1681035911yescharacteristic
177C22#1681040911yescharacteristic
178C22#1681041711yescharacteristic
179C4—21no—
180C4—21no—
181C4—41no—
182C4—41no—
183C4—41no—
184C4—41no—
185C4—81no—
186C4—4241no—
187C4—4241no—
188C2—21no—
189C2—21no—
190C2#2027273511yescharacteristic
191C2#2027286211yescharacteristic
192C2#2027286811yescharacteristic
subgroupG/H
centerC22#16810409
commutatorC2×C4⋊C4×C53C2×C4
Frattini#391C2×D53
FittingC53×#1394C2
socleC22×C53AΣL(1,𝔽16)
prime radicalC23×C53#99
cosocleC53×#391C22
hypercenter#1394D53
hypocenterC53#7431
LCSG ▷ C2×C4⋊C4×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #391 ◁ #1394

Complements

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

NH ≅ G/Ntype
107. C53#7431semidirect

Maximal subgroups

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

subgroupindex
#202758762
#202725412
C53×#13942
#743153