Finite Group Databasebeta

← #27288236

Subgroups — #27288236

Order 13568.

370 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 73 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#20274855C211yesmaximal, characteristic, complemented
3#20273438C212yesmaximal, complemented
4#20273327C212yesmaximal, complemented
5C53×#1307C211yesmaximal, characteristic, complemented
6#16811173C2212yes—
7#16810914—22no—
8#16810876—22no—
9#16810819—22no—
10#16810647C2211yescharacteristic
11C2×#13133128C2211yescharacteristic
12#16810387—22no—
13#16810370—22no—
14#16810366—22no—
15#16810362—24no—
16C53×#432C2212yes—
17C53×#423C2211yescharacteristic
18C2×#11719945—22no—
19C2×D424—22no—
20C2×#11719917—21no—
21C22×#88322—21no—
22#13133217D412yes—
23#13133214D412yes—
24#13133205—22no—
25#13133191—22no—
26#13133191—22no—
27C4×D212—22no—
28C4×D212—22no—
29#13133149—22no—
30C4⋊#88322—22no—
31#13133139—24no—
32#13133128—21no—
33#13133128D412yes—
34C2×M16×C53D412yes—
35C2×C42×C53C2311yescharacteristic
36C4⋊C8×C53—22no—
37C4⋊C8×C53—22no—
38C22⋊C8×C53—22no—
39#97×C53—22no—
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—
50C2×C8×C53—22no—
51C2×C8×C53—42no—
52C42×C53C2×D412yes—
53C42×C53C2×D411yescharacteristic
54C42×C53#4111yescharacteristic
55#11719921—42no—
56#11719921—42no—
57C4⋊Dic53—22no—
58C4⋊Dic53—22no—
59#11719917—42no—
60C2×#88322—22no—
61C2×#88322—41no—
62C2×#88322—41no—
63C2×#88322—41no—
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#12112yes—
72C2×C4×C53#12111yescharacteristic
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—82no—
84#88322—42no—
85#88322—81no—
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#1838—531no—
101#1557—532no—
102#1530—532no—
103#1307D5311yescharacteristic, complemented
104C2×C53—41no—
105C2×C53#204011yescharacteristic
106C2×C53#204812yes—
107D53—162no—
108#532—532no—
109#476—1062no—
110#469—1062no—
111#459—1062no—
112#432C2×D5312yes—
113#423—531no—
114#423C2×D5311yescharacteristic
115C2×#97—531no—
116#341—1062no—
117#332—1062no—
118#331—1062no—
119#329—1064no—
120C53#1052511yescharacteristic, complemented, Sylow
121C2×#41—1062no—
122C2×D8—1062no—
123C2×M16—1061no—
124C2×M16D21212yes—
125C22×C8—1061no—
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—22no—
140#97—22no—
141#97—532no—
142#97—1061no—
143#41—4244no—
144C2×Q8—1062no—
145C2×D4—1062no—
146C2×D4—1062no—
147C2×D4—1062no—
148C2×D4—2122no—
149C22×C4—2122no—
150C22×C4C2×D21211yescharacteristic
151C22×C4#1171994512yes—
152D8—4242no—
153M16—42no—
154M16—2122no—
155C2×C8—22no—
156C2×C8—42no—
157C2×C8—1062no—
158C2×C8—2121no—
159C2×C8—2121no—
160C2×C8—2121no—
161C4⋊C4—1062no—
162C4⋊C4—1062no—
163C22⋊C4—2122no—
164C22⋊C4—2122no—
165C42D4×D5312yes—
166C42D4×D5311yescharacteristic
167C42#1171994911yescharacteristic
168C23—2122no—
169C23#1313320511yescharacteristic
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#1313318711yescharacteristic
182C2×C4#1313319012yes—
183C2×C4#1313319211yescharacteristic
184C2×C4#1313320212yes—
185C8—42no—
186C8—82no—
187C8—4241no—
188C8—2122no—
189C22—22no—
190C22—21no—
191C22—4242no—
192C22—4242no—
193C22—4242no—
194C22#1681054111yescharacteristic
195C4—22no—
196C4—22no—
197C4—21no—
198C4—4242no—
199C4#1681079811yescharacteristic
200C4#1681080311yescharacteristic
201C2—41no—
202C2—8482no—
203C2#2027594511yescharacteristic
204C2#2027599112yes—
subgroupG/H
centerC22#16810541
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1307C2
socleC22×C53#394
prime radical#20286096C2
hypercenter#1307D53
hypocenterC53#10525
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1307

Complements

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

NH ≅ G/Ntype
2. #20274855C2semidirect
3. #20273438C2semidirect
4. #20273327C2semidirect
5. C53×#1307C2semidirect
103. #1307D53semidirect
120. C53#10525semidirect

Maximal subgroups

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

subgroupindex
#202860962
#202748552
#202734382
#202733272
C53×#13072
#1052553