Finite Group Databasebeta

← #28299084

Subgroups — #28299084

Order 13568.

226 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 36 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#20289809C211yesmaximal, characteristic
2#20289644C211yesmaximal, characteristic
3#20275862C211yesmaximal, characteristic
4C2×#13133276—21no—
5#16811468—21no—
6C2×#13133209C411yescharacteristic, complemented
7C2×#13133139—21no—
8#16810525C2211yescharacteristic
9#16810386—21no—
10C53×#336C411yescharacteristic, complemented
11C2×C22⋊#6157—41no—
12#13133276—22no—
13#13133276—41no—
14C22⋊#88324—41no—
15C22×D212D411yescharacteristic
16#13133209—42no—
17C2×C4⋊Dic53—21no—
18C2×C4⋊Dic53D411yescharacteristic
19C2×#11719919—41no—
20C2×#11719921—41no—
21#13133139—42no—
22C22×C8×C53—21no—
23C2×C42×C53C2×C411yescharacteristic
24C23×D53—21no—
25C22×#6157—81no—
26C22×C4×C53—21no—
27C22×C4×C53C22⋊C411yescharacteristic
28C22×Dic53—21no—
29C22×Dic53—41no—
30C2×D212—21no—
31C2×D212—41no—
32C2×D212—41no—
33C2×D212SD1611yescharacteristic
34C2×D212Q1611yescharacteristic
35C22⋊#6157—42no—
36C22⋊#6157—81no—
37C2×#88324—81no—
38C2×C8×C53—41no—
39C2×C8×C53—41no—
40C2×C8×C53—41no—
41C42×C53—41no—
42#11719921—82no—
43C4⋊Dic53—22no—
44C4⋊Dic53—41no—
45C4⋊Dic53—41no—
46#11719919—42no—
47#11719919—42no—
48C23×C53#9911yescharacteristic
49C22×D53—21no—
50C22×D53—21no—
51C22×D53—41no—
52C22×D53—41no—
53C22×D53—81no—
54C22×D53—81no—
55C2×#6157—82no—
56C2×#6157—161no—
57C2×#6157—161no—
58C2×C4×C53—21no—
59C2×C4×C53—21no—
60C2×C4×C53—41no—
61C2×C4×C53—41no—
62C2×C4×C53—41no—
63C2×C4×C53#10311yescharacteristic
64C2×C4×C53#10411yescharacteristic
65C2×Dic53—42no—
66C2×Dic53—41no—
67C2×Dic53—81no—
68C2×Dic53—81no—
69C2×Dic53—81no—
70D212—41no—
71D212—41no—
72D212—81no—
73#88324—161no—
74C8×C53—81no—
75#7428—531nomaximal, Sylow
76C22×C53—21no—
77C22×C53—21no—
78C22×C53#32811yescharacteristic
79C22×C53AΣL(1,𝔽16)11yescharacteristic
80C22×C53#35411yescharacteristic
81C2×D53—41no—
82C2×D53—41no—
83C2×D53—81no—
84C2×D53—81no—
85C2×D53—81no—
86C2×D53—81no—
87C2×D53—161no—
88#6157—162no—
89C4×C53—21no—
90C4×C53—21no—
91C4×C53—81no—
92Dic53—82no—
93Dic53—81no—
94#2025—531no—
95#1393—531no—
96#1318—531no—
97C2×C53—21no—
98C2×C53—21no—
99C2×C53#135111yescharacteristic
100C2×C53#141211yescharacteristic
101C2×C53#141411yescharacteristic
102D53—81no—
103D53—81no—
104C2×#125—531no—
105C2×#102—1061no—
106C2×#99—1061no—
107#390—531no—
108#340—1061no—
109#336#615711yescharacteristic, complemented
110#324—1061no—
111C53#742811yescharacteristic, complemented, Sylow
112C22×D4—531no—
113C22×C8—21no—
114#125—2122no—
115C2×C4⋊C4—531no—
116C2×C4⋊C4—1061no—
117C2×C4⋊C4—2121no—
118C2×C22⋊C4—2121no—
119C2×C22⋊C4—2121no—
120C2×C42C2×#615711yescharacteristic
121#102—2122no—
122#99—1062no—
123#99—2121no—
124C22⋊C8—2121no—
125C24—1061no—
126C2×D4—531no—
127C2×D4—531no—
128C2×D4—1061no—
129C2×D4—2121no—
130C2×D4—2121no—
131C22×C4—21no—
132C22×C4—1061no—
133C22×C4—4241no—
134C22×C4—2121no—
135C22×C4C22⋊#615711yescharacteristic
136C2×C8—41no—
137C2×C8—41no—
138C2×C8—41no—
139C2×C8—4241no—
140C4⋊C4—1062no—
141C4⋊C4—2121no—
142C4⋊C4—2121no—
143C4⋊C4—2122no—
144C4⋊C4—2122no—
145C22⋊C4—4242no—
146C22⋊C4—4241no—
147C22⋊C4—2122no—
148C42—41no—
149C23—1061no—
150C23—1061no—
151C23—4241no—
152C23—4241no—
153C23—2121no—
154C23—2121no—
155C23#1313327611yescharacteristic
156D4—4241no—
157D4—2121no—
158D4—2121no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—8481no—
165C2×C4—8481no—
166C2×C4—4242no—
167C2×C4—4241no—
168C2×C4—4241no—
169C2×C4—4241no—
170C2×C4—2122no—
171C2×C4—2121no—
172C2×C4#1313328011yescharacteristic
173C2×C4#1313328111yescharacteristic
174C8—81no—
175C8—8481no—
176C22—21no—
177C22—21no—
178C22—8481no—
179C22—4241no—
180C22—4241no—
181C22—4241no—
182C22—4241no—
183C22—2121no—
184C22—2121no—
185C22#1681147611yescharacteristic
186C22#1681152411yescharacteristic
187C22#1681153011yescharacteristic
188C4—21no—
189C4—21no—
190C4—81no—
191C4—8482no—
192C4—4242no—
193C4—4241no—
194C2—21no—
195C2—21no—
196C2—4241no—
197C2—4241no—
198C2#2028970911yescharacteristic
199C2#2028985611yescharacteristic
200C2#2028986211yescharacteristic
subgroupG/H
centerC22#16811530
commutatorC2×C42×C53C2×C4
FrattiniC2×C42C2×#6157
FittingC53×#336C4
socleC22×C53#354
prime radicalC22×D212D4
cosocle#16810525C22
hypercenter#336#6157
hypocenterC53#7428
LCSG ▷ C2×C42×C53 ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #336

Complements

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

NH ≅ G/Ntype
6. C2×#13133209C4semidirect
10. C53×#336C4semidirect
109. #336#6157semidirect
111. C53#7428semidirect

Maximal subgroups

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

subgroupindex
#202898092
#202896442
#202758622
#742853