Finite Group Databasebeta

← #27279056

Subgroups — #27279056

Order 13568.

398 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 99 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#20285063C211yesmaximal, characteristic
2#20273380C211yesmaximal, characteristic, complemented
3#20273183C212yesmaximal
4#20272613C212yesmaximal, complemented
5C53×#2987C211yesmaximal, characteristic, complemented
6#16811173C2211yescharacteristic
7#16810993C412yes—
8#16810989C412yes—
9#16810985—22no—
10#16810676C2212yes—
11#16810617—22no—
12C2×#13133128C2211yescharacteristic
13#16810369—22no—
14#16810350—24no—
15C53×#533C2211yescharacteristic
16C53×#446—24no—
17C53×#433—22no—
18C53×#432C2212yes—
19C2×#11719945—21no—
20C22×#88322—21no—
21#13133217C2×C411yescharacteristic
22#13133214C2×C411yescharacteristic
23#13133205—21no—
24#13133191—22no—
25C4×D212—22no—
26C4⋊#88322—22no—
27#13133140—24no—
28#13133137—24no—
29C22⋊#88322—22no—
30#13133128—21no—
31#13133128C2×C412yes—
32C4×#88322—22no—
33C4⋊Q8×C53D412yes—
34#125×C53C2×C412yes—
35C22⋊Q8×C53—22no—
36C4×Q8×C53—22no—
37#117×C53D412yes—
38C2×C42×C53C2311yescharacteristic
39C4⋊C8×C53—24no—
40C4⋊C8×C53D414yes—
41C22⋊C8×C53—22no—
42#97×C53—22no—
43#97×C53—22no—
44C4×C8×C53—22no—
45C2×Q8×C53—22no—
46C22×C4×C53C2×D411yescharacteristic
47C22×C4×C53#4112yes—
48C2×C22⋊D53—41no—
49#11719945—44no—
50C2×D212—21no—
51C2×D212—22no—
52C2×C4×D53—41no—
53C2×Dic106—21no—
54C2×C8×C53—24no—
55C4⋊C4×C53—24no—
56C4⋊C4×C53—22no—
57C4⋊C4×C53C22⋊C414yes—
58C22⋊C4×C53—22no—
59C42×C53—22no—
60C42×C53C22×C411yescharacteristic
61C42×C53C2×D411yescharacteristic
62C42×C53C2×D412yes—
63#11719921—42no—
64C4⋊Dic53—22no—
65C2×#88322—22no—
66C2×#88322—42no—
67C2×#88322—42no—
68C23×C53#12311yescharacteristic
69C22×D53—41no—
70Q8×C53—82no—
71C2×C4×C53—22no—
72C2×C4×C53—24no—
73C2×C4×C53—42no—
74C2×C4×C53C2×C22⋊C411yescharacteristic
75C2×C4×C53C4×D412yes—
76C2×C4×C53#12011yescharacteristic
77C2×C4×C53#12112yes—
78C22⋊D53—82no—
79C2×Dic53—41no—
80D212—44no—
81D212—42no—
82C4×D53—82no—
83Dic106—42no—
84C8×C53—44no—
85#88322—42no—
86#88322—82no—
87#9008—531nomaximal, Sylow
88C22×C53—22no—
89C22×C53—21no—
90C22×C53#38611yescharacteristic
91C2×D53—82no—
92C2×D53—81no—
93C4×C53—22no—
94C4×C53—22no—
95C4×C53—21no—
96C4×C53—44no—
97C4×C53#45412yes—
98Dic53—81no—
99#3135—531no—
100#2987D5311yescharacteristic, complemented
101#1544—531no—
102#1496—532no—
103#1302—532no—
104C2×C53—41no—
105C2×C53#188112yes—
106C2×C53#188811yescharacteristic
107D53—161no—
108#533C2×D5311yescharacteristic
109#532—531no—
110#491—532no—
111#490—532no—
112#489—1062no—
113#446—24no—
114#433—22no—
115#432C2×D5312yes—
116#431—532no—
117#416—1062no—
118C2×#97—531no—
119#332—1062no—
120D4⋊C8—1064no—
121C53#900811yescharacteristic, complemented, Sylow
122C2×#41—1061no—
123C22×C8—1061no—
124C4⋊Q8—531no—
125C4⋊Q8D21212yes—
126#127—531no—
127#125C4×D5312yes—
128#124—1061no—
129C22⋊Q8—22no—
130#121—1062no—
131C4×Q8—22no—
132C4×D4—1062no—
133#117C22⋊D5312yes—
134C2×C42C22×D5311yescharacteristic
135C4⋊C8—24no—
136C4⋊C8—1062no—
137C4⋊C8C22⋊D5314yes—
138#103—1064no—
139#102—1064no—
140C22⋊C8—22no—
141C22⋊C8—1062no—
142#97—22no—
143#97—22no—
144#97—532no—
145#97—1061no—
146C4×C8—22no—
147C4×C8—1062no—
148#41—2124no—
149C2×Q8—22no—
150C2×Q8—1061no—
151C2×D4—1061no—
152C2×D4—1062no—
153C2×D4—2121no—
154C22×C4—2121no—
155C22×C4#1171994512yes—
156C22×C4C2×C22⋊D5311yescharacteristic
157C2×C8—24no—
158C2×C8—1062no—
159C2×C8—2122no—
160C2×C8—2122no—
161C4⋊C4—24no—
162C4⋊C4—22no—
163C4⋊C4—1062no—
164C4⋊C4#1171992114yes—
165C22⋊C4—22no—
166C22⋊C4—2122no—
167C42—22no—
168C42C2×C4×D5311yescharacteristic
169C42C2×D21211yescharacteristic
170C42C2×C22⋊D5312yes—
171C23—2121no—
172C23#1313320011yescharacteristic
173Q8—82no—
174Q8—2122no—
175D4—4242no—
176D4—2124no—
177D4—2122no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—42no—
181C2×C4—4242no—
182C2×C4—2121no—
183C2×C4C2×#1171992111yescharacteristic
184C2×C4C4×C22⋊D5312yes—
185C2×C4#1313318611yescharacteristic
186C2×C4#1313319112yes—
187C8—44no—
188C8—4242no—
189C8—2122no—
190C22—22no—
191C22—21no—
192C22—4242no—
193C22—4241no—
194C22#1681050211yescharacteristic
195C4—22no—
196C4—22no—
197C4—21no—
198C4—44no—
199C4—4241no—
200C4#1681079612yes—
201C2—41no—
202C2—8481no—
203C2#2027507712yes—
204C2#2027511811yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2987C2
socleC22×C53#386
prime radical#16811173C22
cosocleC2×C42×C53C23
hypercenter#2987D53
hypocenterC53#9008
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2987

Complements

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

NH ≅ G/Ntype
2. #20273380C2semidirect
4. #20272613C2semidirect
5. C53×#2987C2semidirect
100. #2987D53semidirect
121. C53#9008semidirect

Maximal subgroups

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

subgroupindex
#202850632
#202733802
#202731832
#202726132
C53×#29872
#900853