Finite Group Databasebeta

← #27288235

Subgroups — #27288235

Order 13568.

370 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 75 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#20286096C211yesmaximal, characteristic
2#20273435C212yesmaximal, complemented
3#20273325C212yesmaximal, complemented
4#20272626C211yesmaximal, characteristic, complemented
5C53×#1838C211yesmaximal, characteristic, complemented
6#16811173C2212yes—
7#16810914—22no—
8#16810876—22no—
9#16810819—22no—
10#16810680C2212yes—
11#16810368—22no—
12#16810366—22no—
13#16810362—24no—
14C53×#423C2211yescharacteristic
15C53×#423C2211yescharacteristic
16C2×#97×C53C2211yescharacteristic
17C53×#341—22no—
18C2×#11719945—22no—
19C2×D424—22no—
20#13133217D412yes—
21#13133214D412yes—
22#13133205—22no—
23#13133191—22no—
24#13133191—22no—
25C4×D212—22no—
26C4×D212—22no—
27#13133149—22no—
28C4⋊#88322—22no—
29#13133139—24no—
30C22⋊#88322—22no—
31#13133128—22no—
32C2×M16×C53—21no—
33C2×M16×C53D412yes—
34C22×C8×C53—21no—
35C2×C42×C53C2311yescharacteristic
36C4⋊C8×C53—22no—
37C4⋊C8×C53—22no—
38#97×C53—21no—
39#97×C53D412yes—
40C22×C4×C53C2×D411yescharacteristic
41C22×C4×C53#4112yes—
42C2×C22⋊D53—42no—
43#11719945—84no—
44C2×D212—22no—
45C2×D212—22no—
46C2×D212—22no—
47C2×C4×D53—42no—
48C2×Dic106—22no—
49M16×C53—42no—
50M16×C53—42no—
51C2×C8×C53—22no—
52C2×C8×C53—22no—
53C2×C8×C53—41no—
54C2×C8×C53—41no—
55C2×C8×C53—41no—
56C42×C53C2×D412yes—
57C42×C53C2×D411yescharacteristic
58C42×C53#4111yescharacteristic
59#11719921—42no—
60#11719921—42no—
61C4⋊Dic53—22no—
62C4⋊Dic53—22no—
63C2×#88322—42no—
64D424—82no—
65C23×C53#12411yescharacteristic
66C22×D53—42no—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—42no—
70C2×C4×C53#12011yescharacteristic
71C2×C4×C53#12111yescharacteristic
72C2×C4×C53#12112yes—
73C2×C4×C53#12312yes—
74C22⋊D53—84no—
75C2×Dic53—42no—
76D212—42no—
77D212—42no—
78D212—82no—
79D212—82no—
80C4×D53—84no—
81Dic106—82no—
82C8×C53—42no—
83C8×C53—42no—
84C8×C53—81no—
85#88322—82no—
86#10525—531nomaximal, Sylow
87C22×C53—22no—
88C22×C53—21no—
89C22×C53#39411yescharacteristic
90C2×D53—82no—
91C2×D53—82no—
92C2×D53—82no—
93C4×C53—22no—
94C4×C53—22no—
95C4×C53—21no—
96C4×C53#45411yescharacteristic
97C4×C53#45511yescharacteristic
98Dic53—82no—
99#3253—531no—
100#1838D5311yescharacteristic, complemented
101#1557—532no—
102#1530—532no—
103#1307—531no—
104C2×C53—41no—
105C2×C53#204011yescharacteristic
106C2×C53#204812yes—
107D53—162no—
108#532—532no—
109#476—1062no—
110#469—1062no—
111#459—1062no—
112#432—532no—
113#423C2×D5311yescharacteristic
114#423C2×D5311yescharacteristic
115C2×#97C2×D5311yescharacteristic
116#341—22no—
117#332—1062no—
118#331—1062no—
119#329—1064no—
120C53#1052511yescharacteristic, complemented, Sylow
121C2×#41—1062no—
122C2×D8—1062no—
123C2×M16—21no—
124C2×M16C22⋊D5312yes—
125C22×C8—21no—
126C4⋊Q8—532no—
127#127—532no—
128#124—1062no—
129#121—1062no—
130#121—1062no—
131C4×D4—1062no—
132C4×D4—1062no—
133C2×C42C22×D5311yescharacteristic
134#106—1062no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—1062no—
138#102—1064no—
139C22⋊C8—1062no—
140#97—21no—
141#97—1062no—
142#97D21212yes—
143#41—4244no—
144C2×Q8—1062no—
145C2×D4—1062no—
146C2×D4—1062no—
147C2×D4—1062no—
148C2×D4—2122no—
149C22×C4—2122no—
150C22×C4#1171994912yes—
151C22×C4C2×C22⋊D5311yescharacteristic
152D8—4242no—
153M16—42no—
154M16—42no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—2122no—
161C4⋊C4—1062no—
162C4⋊C4—1062no—
163C22⋊C4—2122no—
164C22⋊C4—2122no—
165C42C2×D21211yescharacteristic
166C42#1171994511yescharacteristic
167C42D4×D5312yes—
168C23—2122no—
169C23#1313320611yescharacteristic
170Q8—4242no—
171D4—4242no—
172D4—4242no—
173D4—4244no—
174D4—2122no—
175D4—2122no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—42no—
179C2×C4—4244no—
180C2×C4—2122no—
181C2×C4#1313318511yescharacteristic
182C2×C4#1313319111yescharacteristic
183C2×C4#1313319212yes—
184C2×C4#1313320212yes—
185C8—42no—
186C8—42no—
187C8—81no—
188C8—4242no—
189C22—22no—
190C22—21no—
191C22—4242no—
192C22—4242no—
193C22—4242no—
194C22#1681054011yescharacteristic
195C4—22no—
196C4—22no—
197C4—21no—
198C4—4242no—
199C4#1681079511yescharacteristic
200C4#1681080011yescharacteristic
201C2—41no—
202C2—8482no—
203C2#2027594411yescharacteristic
204C2#2027599012yes—
subgroupG/H
centerC22#16810540
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1838C2
socleC22×C53#394
prime radical#20286096C2
hypercenter#1838D53
hypocenterC53#10525
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1838

Complements

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

NH ≅ G/Ntype
2. #20273435C2semidirect
3. #20273325C2semidirect
4. #20272626C2semidirect
5. C53×#1838C2semidirect
100. #1838D53semidirect
120. C53#10525semidirect

Maximal subgroups

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

subgroupindex
#202860962
#202734352
#202733252
#202726262
C53×#18382
#1052553