Finite Group Databasebeta

← #27271677

Subgroups — #27271677

Order 13568.

226 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 43 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#20272840C211yesmaximal, characteristic
2#20272647C211yesmaximal, characteristic
3C53×#2025C211yesmaximal, characteristic
4C2×#13133132—21no—
5#16810349—21no—
6C2×#125×C53C411yescharacteristic, complemented
7C2×#102×C53—21no—
8C53×#390C2211yescharacteristic
9C53×#340—21no—
10C53×#336C411yescharacteristic, complemented
11C2×C22⋊Dic53—41no—
12#13133132—22no—
13#13133132—41no—
14C22⋊#88322—41no—
15C22×D4×C53D411yescharacteristic
16C22×C8×C53—21no—
17#125×C53—42no—
18C2×C4⋊C4×C53—21no—
19C2×C4⋊C4×C53—41no—
20C2×C4⋊C4×C53D411yescharacteristic
21C2×C22⋊C4×C53—41no—
22C2×C42×C53C2×C411yescharacteristic
23#102×C53—42no—
24C24×C53—21no—
25C2×D4×C53—21no—
26C2×D4×C53—41no—
27C2×D4×C53—41no—
28C2×D4×C53SD1611yescharacteristic
29C2×D4×C53Q1611yescharacteristic
30C22×C4×C53—21no—
31C22×C4×C53—21no—
32C22×C4×C53—41no—
33C22×C4×C53C22⋊C411yescharacteristic
34C22×Dic53—81no—
35C2×C8×C53—41no—
36C2×C8×C53—41no—
37C2×C8×C53—41no—
38C4⋊C4×C53—22no—
39C4⋊C4×C53—41no—
40C4⋊C4×C53—41no—
41C4⋊C4×C53—42no—
42C4⋊C4×C53—42no—
43C22⋊C4×C53—82no—
44C42×C53—41no—
45C22⋊Dic53—42no—
46C22⋊Dic53—81no—
47C2×#88322—81no—
48C23×C53—21no—
49C23×C53—21no—
50C23×C53—41no—
51C23×C53—41no—
52C23×C53—81no—
53C23×C53—81no—
54C23×C53#9911yescharacteristic
55D4×C53—41no—
56D4×C53—41no—
57D4×C53—81no—
58C2×C4×C53—21no—
59C2×C4×C53—21no—
60C2×C4×C53—41no—
61C2×C4×C53—41no—
62C2×C4×C53—41no—
63C2×C4×C53—42no—
64C2×C4×C53—41no—
65C2×C4×C53—81no—
66C2×C4×C53—81no—
67C2×C4×C53—81no—
68C2×C4×C53#10311yescharacteristic
69C2×C4×C53#10411yescharacteristic
70C2×Dic53—82no—
71C2×Dic53—161no—
72C2×Dic53—161no—
73C8×C53—81no—
74#88322—161no—
75#7428—531nomaximal, Sylow
76C22×C53—21no—
77C22×C53—21no—
78C22×C53—41no—
79C22×C53—41no—
80C22×C53—81no—
81C22×C53—81no—
82C22×C53—81no—
83C22×C53—81no—
84C22×C53—161no—
85C22×C53#32811yescharacteristic
86C22×C53AΣL(1,𝔽16)11yescharacteristic
87C22×C53#35411yescharacteristic
88C4×C53—21no—
89C4×C53—21no—
90C4×C53—81no—
91C4×C53—82no—
92C4×C53—81no—
93Dic53—162no—
94#2025D5311yescharacteristic
95#1393—531no—
96#1318—531no—
97C2×C53—21no—
98C2×C53—21no—
99C2×C53—81no—
100C2×C53—81no—
101C2×C53#135111yescharacteristic
102C2×C53#141211yescharacteristic
103C2×C53#141411yescharacteristic
104C2×#125Dic5311yescharacteristic, complemented
105C2×#102—21no—
106C2×#99—1061no—
107#390C2×D5311yescharacteristic
108#340—21no—
109#336Dic5311yescharacteristic, complemented
110#324—1061no—
111C53#742811yescharacteristic, complemented, Sylow
112C22×D4C22⋊D5311yescharacteristic
113C22×C8—21no—
114#125—42no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4C22⋊D5311yescharacteristic
118C2×C22⋊C4—41no—
119C2×C22⋊C4—2121no—
120C2×C42C2×Dic5311yescharacteristic
121#102—42no—
122#99—1062no—
123#99—2121no—
124C22⋊C8—2121no—
125C24—21no—
126C2×D4—21no—
127C2×D4—41no—
128C2×D4—41no—
129C2×D4#1171992411yescharacteristic
130C2×D4#1171992511yescharacteristic
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—41no—
134C22×C4—4241no—
135C22×C4C22⋊Dic5311yescharacteristic
136C2×C8—41no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—4241no—
140C4⋊C4—22no—
141C4⋊C4—41no—
142C4⋊C4—41no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C22⋊C4—82no—
146C22⋊C4—4241no—
147C22⋊C4—2122no—
148C42—41no—
149C23—21no—
150C23—21no—
151C23—41no—
152C23—41no—
153C23—81no—
154C23—81no—
155C23#1313313211yescharacteristic
156D4—41no—
157D4—41no—
158D4—81no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—81no—
167C2×C4—81no—
168C2×C4—81no—
169C2×C4—8481no—
170C2×C4—8481no—
171C2×C4—4242no—
172C2×C4#1313314111yescharacteristic
173C2×C4#1313314411yescharacteristic
174C8—81no—
175C8—8481no—
176C22—21no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—81no—
181C22—81no—
182C22—81no—
183C22—81no—
184C22—161no—
185C22#1681036011yescharacteristic
186C22#1681041011yescharacteristic
187C22#1681041811yescharacteristic
188C4—21no—
189C4—21no—
190C4—81no—
191C4—82no—
192C4—81no—
193C4—8482no—
194C2—21no—
195C2—21no—
196C2—81no—
197C2—81no—
198C2#2027273611yescharacteristic
199C2#2027288611yescharacteristic
200C2#2027289011yescharacteristic
subgroupG/H
centerC22#16810418
commutatorC2×C42×C53C2×C4
Frattini#390C2×D53
FittingC53×#2025C2
socleC22×C53#354
prime radicalC22×D4×C53D4
cosocleC53×#390C22
hypercenter#2025D53
hypocenterC53#7428
LCSG ▷ C2×C42×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #390 ◁ #2025

Complements

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

NH ≅ G/Ntype
6. C2×#125×C53C4semidirect
10. C53×#336C4semidirect
104. C2×#125Dic53semidirect
109. #336Dic53semidirect
111. C53#7428semidirect

Maximal subgroups

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

subgroupindex
#202728402
#202726472
C53×#20252
#742853