Finite Group Databasebeta

← #27271696

Subgroups — #27271696

Order 13568.

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

normal subgroups only

This group has 224 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#20276243C211yesmaximal, characteristic
2#20272840C211yesmaximal, characteristic
3C53×#1318C211yesmaximal, characteristic
4C2×#13133150—21no—
5C2×#13133138—21no—
6C2×#13133132—21no—
7#16810493C411yescharacteristic, complemented
8#16810378C411yescharacteristic, complemented
9C53×#390C2211yescharacteristic
10C53×#324—21no—
11C22×#88322—21no—
12C2×C4⋊Dic53—21no—
13C2×C4⋊Dic53—41no—
14C2×C22⋊Dic53—41no—
15#13133150—42no—
16#13133138—42no—
17#13133132—22no—
18#13133132—41no—
19C22×D4×C53D411yescharacteristic
20C2×C4⋊C4×C53D411yescharacteristic
21C2×C22⋊C4×C53—41no—
22C2×C42×C53C2×C411yescharacteristic
23C22⋊C8×C53—41no—
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×C53C22⋊C411yescharacteristic
33C22×Dic53—41no—
34C22×Dic53—81no—
35C2×C8×C53—81no—
36C4⋊C4×C53—41no—
37C22⋊C4×C53—82no—
38C42×C53—41no—
39C22⋊Dic53—42no—
40C22⋊Dic53—81no—
41C4⋊Dic53—41no—
42C4⋊Dic53—41no—
43C4⋊Dic53—82no—
44C2×#88322—22no—
45C2×#88322—41no—
46C2×#88322—41no—
47C23×C53—21no—
48C23×C53—21no—
49C23×C53—41no—
50C23×C53—41no—
51C23×C53—81no—
52C23×C53—81no—
53C23×C53#9911yescharacteristic
54D4×C53—41no—
55D4×C53—41no—
56D4×C53—81no—
57C2×C4×C53—21no—
58C2×C4×C53—21no—
59C2×C4×C53—22no—
60C2×C4×C53—41no—
61C2×C4×C53—41no—
62C2×C4×C53—41no—
63C2×C4×C53—81no—
64C2×C4×C53#10311yescharacteristic
65C2×C4×C53#10411yescharacteristic
66C2×Dic53—82no—
67C2×Dic53—82no—
68C2×Dic53—81no—
69C2×Dic53—161no—
70C2×Dic53—161no—
71C8×C53—161no—
72#88322—42no—
73#7434—531nomaximal, Sylow
74C22×C53—21no—
75C22×C53—21no—
76C22×C53—41no—
77C22×C53—41no—
78C22×C53—81no—
79C22×C53—81no—
80C22×C53—81no—
81C22×C53—81no—
82C22×C53—161no—
83C22×C53#32811yescharacteristic
84C22×C53#35211yescharacteristic
85C22×C53#35311yescharacteristic
86C4×C53—21no—
87C4×C53—21no—
88C4×C53—42no—
89C4×C53—81no—
90Dic53—162no—
91Dic53—161no—
92#2093—531no—
93#1393—531no—
94#1318D5311yescharacteristic
95C2×C53—21no—
96C2×C53—21no—
97C2×C53—81no—
98C2×C53—81no—
99C2×C53#135211yescharacteristic
100C2×C53#140711yescharacteristic
101C2×C53#140811yescharacteristic
102C2×#107—1061no—
103C2×#102—1061no—
104C2×#99—1061no—
105#390C2×D5311yescharacteristic
106#384—531no—
107#336—531no—
108#324—21no—
109C53#743411yescharacteristic, complemented, Sylow
110C22×D4D21211yescharacteristic
111C22×C8—1061no—
112C2×C4⋊C4—1061no—
113C2×C4⋊C4—2121no—
114C2×C4⋊C4C22⋊D5311yescharacteristic
115C2×C22⋊C4—41no—
116C2×C22⋊C4—2121no—
117C2×C42C4×D5311yescharacteristic
118#107—2122no—
119#102—2122no—
120#99—1062no—
121#99—2121no—
122C22⋊C8—41no—
123C24—21no—
124C2×D4—21no—
125C2×D4—41no—
126C2×D4—41no—
127C2×D4#1171991311yescharacteristic
128C2×D4Dic21211yescharacteristic
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—4241no—
132C22×C4—2121no—
133C22×C4#1171992111yescharacteristic
134C2×C8—81no—
135C2×C8—1062no—
136C2×C8—2121no—
137C2×C8—2121no—
138C4⋊C4—41no—
139C4⋊C4—4242no—
140C4⋊C4—2121no—
141C4⋊C4—2121no—
142C22⋊C4—82no—
143C22⋊C4—4241no—
144C22⋊C4—2122no—
145C42—41no—
146C23—21no—
147C23—21no—
148C23—41no—
149C23—41no—
150C23—81no—
151C23—81no—
152C23#1313313111yescharacteristic
153D4—41no—
154D4—41no—
155D4—81no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—81no—
163C2×C4—8481no—
164C2×C4—8481no—
165C2×C4—4242no—
166C2×C4—4242no—
167C2×C4—4241no—
168C2×C4#1313314211yescharacteristic
169C2×C4#1313314311yescharacteristic
170C8—161no—
171C8—2122no—
172C22—21no—
173C22—21no—
174C22—41no—
175C22—41no—
176C22—81no—
177C22—81no—
178C22—81no—
179C22—81no—
180C22—161no—
181C22#1681036111yescharacteristic
182C22#1681041411yescharacteristic
183C22#1681041511yescharacteristic
184C4—21no—
185C4—21no—
186C4—42no—
187C4—81no—
188C4—8482no—
189C4—8481no—
190C2—21no—
191C2—21no—
192C2—81no—
193C2—81no—
194C2#2027274011yescharacteristic
195C2#2027287611yescharacteristic
196C2#2027287711yescharacteristic
subgroupG/H
centerC22#16810415
commutatorC2×C42×C53C2×C4
Frattini#390C2×D53
FittingC53×#1318C2
socleC22×C53#353
prime radicalC22×D4×C53D4
cosocleC53×#390C22
hypercenter#1318D53
hypocenterC53#7434
LCSG ▷ C2×C42×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #390 ◁ #1318

Complements

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

NH ≅ G/Ntype
7. #16810493C4semidirect
8. #16810378C4semidirect
109. C53#7434semidirect

Maximal subgroups

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

subgroupindex
#202762432
#202728402
C53×#13182
#743453