Finite Group Databasebeta

← #27270774

Subgroups — #27270774

Order 13568.

218 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 51 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#20272639C211yesmaximal, characteristic
2#20272583C211yesmaximal, characteristic
3C53×#2075C211yesmaximal, characteristic
4#16810483—21no—
5C22⋊#11719908—21no—
6C2×#107×C53C411yescharacteristic, complemented
7C2×C22⋊C8×C53C2211yescharacteristic
8C53×#380—21no—
9C53×#340—21no—
10C53×#340C411yescharacteristic, complemented
11C2×C4⋊Dic53—41no—
12C2×C22⋊Dic53—41no—
13C2×#11719908—41no—
14#13133124—22no—
15C23×C4×C53D411yescharacteristic
16C22×C8×C53C2×C411yescharacteristic
17C22×C8×C53D411yescharacteristic
18C2×C4⋊C4×C53—21no—
19C2×C4⋊C4×C53—21no—
20C2×C22⋊C4×C53—41no—
21#107×C53—41no—
22C22⋊C8×C53—41no—
23#95×C53—41no—
24C24×C53D811yescharacteristic
25C22×C4×C53—21no—
26C22×C4×C53—21no—
27C22×C4×C53—41no—
28C22×C4×C53—41no—
29C22×C4×C53—41no—
30C22×C4×C53—41no—
31C22×C4×C53C22⋊C411yescharacteristic
32C22×C4×C53SD1611yescharacteristic
33C22×Dic53—42no—
34C2×C8×C53—21no—
35C2×C8×C53—21no—
36C2×C8×C53—41no—
37C2×C8×C53—41no—
38C4⋊C4×C53—41no—
39C4⋊C4×C53—41no—
40C4⋊C4×C53—41no—
41C4⋊C4×C53—41no—
42C22⋊C4×C53—81no—
43C22⋊C4×C53—81no—
44C22⋊Dic53—81no—
45C22⋊Dic53—81no—
46C4⋊Dic53—81no—
47C4⋊Dic53—81no—
48#11719908—81no—
49C23×C53—21no—
50C23×C53—21no—
51C23×C53—41no—
52C23×C53—41no—
53C23×C53#10211yescharacteristic
54C23×C53D1611yescharacteristic
55C23×C53SD3211yescharacteristic
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53—21no—
59C2×C4×C53—41no—
60C2×C4×C53—41no—
61C2×C4×C53—41no—
62C2×C4×C53—81no—
63C2×C4×C53—81no—
64C2×C4×C53—81no—
65C2×C4×C53—81no—
66C2×C4×C53—81no—
67C2×C4×C53—81no—
68C2×C4×C53—81no—
69C2×C4×C53#9911yescharacteristic
70C2×C4×C53#10411yescharacteristic
71C2×Dic53—82no—
72C2×Dic53—82no—
73C2×Dic53—82no—
74C8×C53—41no—
75C8×C53—41no—
76#7079—531nomaximal, Sylow
77C22×C53—21no—
78C22×C53—21no—
79C22×C53—21no—
80C22×C53—21no—
81C22×C53—41no—
82C22×C53—41no—
83C22×C53—41no—
84C22×C53—41no—
85C22×C53—81no—
86C22×C53#32711yescharacteristic
87C22×C53#35711yescharacteristic
88C22×C53#36011yescharacteristic
89C4×C53—21no—
90C4×C53—41no—
91C4×C53—161no—
92C4×C53—161no—
93C4×C53#35211yescharacteristic
94C4×C53#35411yescharacteristic
95Dic53—162no—
96#2075D5311yescharacteristic
97#1314—531no—
98#1289—531no—
99C2×C53—21no—
100C2×C53—21no—
101C2×C53—41no—
102C2×C53—41no—
103C2×C53#134011yescharacteristic
104C2×C53#134411yescharacteristic
105C2×C53#134711yescharacteristic
106C2×#107Dic5311yescharacteristic, complemented
107C2×C22⋊C8C2×D5311yescharacteristic
108#380—21no—
109#380—1061no—
110C22⋊C16—1061no—
111#340—21no—
112#340Dic5311yescharacteristic, complemented
113C53#707911yescharacteristic, complemented, Sylow
114C23×C4C22⋊D5311yescharacteristic
115C22×C8C2×Dic5311yescharacteristic
116C22×C8C22⋊D5311yescharacteristic
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—2121no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—2121no—
122C2×C16—2121no—
123#107—41no—
124C22⋊C8—41no—
125#95—41no—
126#95—1062no—
127C24#1171992211yescharacteristic
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—2122no—
135C22×C4#1171992311yescharacteristic
136C22×C4C22⋊Dic5311yescharacteristic
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—41no—
140C2×C8—41no—
141C4⋊C4—41no—
142C4⋊C4—41no—
143C4⋊C4—41no—
144C4⋊C4—41no—
145C4⋊C4—4241no—
146C4⋊C4—4241no—
147C22⋊C4—81no—
148C22⋊C4—81no—
149C22⋊C4—4241no—
150C22⋊C4—4241no—
151C16—4241no—
152C23—21no—
153C23—21no—
154C23—41no—
155C23—41no—
156C23#1313313811yescharacteristic
157C23#1313315811yescharacteristic
158C23#1313316111yescharacteristic
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—81no—
166C2×C4—81no—
167C2×C4—81no—
168C2×C4—81no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—81no—
172C2×C4—4242no—
173C2×C4—4242no—
174C2×C4—4242no—
175C2×C4#1313313211yescharacteristic
176C2×C4#1313314411yescharacteristic
177C8—41no—
178C8—41no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—41no—
184C22—41no—
185C22—41no—
186C22—41no—
187C22—81no—
188C22#1681035711yescharacteristic
189C22#1681042411yescharacteristic
190C22#1681043311yescharacteristic
191C4—21no—
192C4—41no—
193C4—161no—
194C4—161no—
195C4—8482no—
196C4#1681041311yescharacteristic
197C4#1681041811yescharacteristic
198C2—21no—
199C2—21no—
200C2—41no—
201C2—41no—
202C2#2027270311yescharacteristic
203C2#2027271511yescharacteristic
204C2#2027272411yescharacteristic
subgroupG/H
centerC22#16810357
commutatorC22×C8×C53C2×C4
FrattiniC2×C22⋊C8C2×D53
FittingC53×#2075C2
socleC22×C53#327
prime radicalC24×C53D8
cosocleC2×C22⋊C8×C53C22
hypercenter#2075D53
hypocenterC53#7079
LCSG ▷ C22×C8×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #2075

Complements

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

NH ≅ G/Ntype
6. C2×#107×C53C4semidirect
10. C53×#340C4semidirect
106. C2×#107Dic53semidirect
112. #340Dic53semidirect
113. C53#7079semidirect

Maximal subgroups

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

subgroupindex
#202726392
#202725832
C53×#20752
#707953