Finite Group Databasebeta

← #27286816

Subgroups — #27286816

Order 13568.

504 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 113 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#20285866C212yesmaximal
2#20275692C211yesmaximal, characteristic
3#20273400C212yesmaximal
4#20272548C211yesmaximal, characteristic
5C53×#2862C211yesmaximal, characteristic
6#16811180C2211yescharacteristic
7#16811026—22no—
8#16810975—22no—
9#16810958—24no—
10#16810915—24no—
11#16810676C2212yes—
12#16810647C2211yescharacteristic
13#16810383—22no—
14#16810369—24no—
15C2×C4⋊Q8×C53C2212yes—
16C53×#391—24no—
17C53×#382—22no—
18C43×C53C2211yescharacteristic
19C2×#11719917—21no—
20#13133217D412yes—
21#13133209—21no—
22#13133196—22no—
23C4×Dic106—22no—
24#13133176—21no—
25#13133150—28no—
26C4⋊#88322—22no—
27C4⋊#88322D414yes—
28#13133140—28no—
29C22⋊#88322—22no—
30C4×#88322—22no—
31C22×Q8×C53—22no—
32C4⋊Q8×C53—22no—
33C4⋊Q8×C53—22no—
34C4⋊Q8×C53—22no—
35C4⋊Q8×C53—22no—
36C2×C4⋊C4×C53—22no—
37C2×C4⋊C4×C53D414yes—
38C2×C42×C53—22no—
39C2×C42×C53D412yes—
40C2×C42×C53C2311yescharacteristic
41#95×C53—28no—
42C2×Q8×C53—48no—
43C2×Q8×C53—42no—
44C2×Q8×C53—42no—
45C22×C4×C53—22no—
46C22×C4×C53—24no—
47C22×C4×C53C2×D412yes—
48C22×C4×C53C2×D411yescharacteristic
49C2×Dic106—22no—
50C4⋊C4×C53—24no—
51C4⋊C4×C53—24no—
52C4⋊C4×C53—42no—
53C4⋊C4×C53—42no—
54C42×C53—22no—
55C42×C53—44no—
56C42×C53Q1614yes—
57C42×C53C2×D411yescharacteristic
58C42×C53C2×D412yes—
59C42×C53#4111yescharacteristic
60C22⋊Dic53—41no—
61C4⋊Dic53—22no—
62C4⋊Dic53—22no—
63#11719919—42no—
64C4×Dic53—41no—
65#11719917—42no—
66C2×#88322—24no—
67C2×#88322—41no—
68C23×C53#12011yescharacteristic
69Q8×C53—84no—
70Q8×C53—84no—
71C2×C4×C53—24no—
72C2×C4×C53—24no—
73C2×C4×C53—44no—
74C2×C4×C53—48no—
75C2×C4×C53—44no—
76C2×C4×C53#12012yes—
77C2×C4×C53#12111yescharacteristic
78C2×C4×C53#12112yes—
79C2×C4×C53#12711yescharacteristic
80C2×C4×C53C2×Q1612yes—
81C2×C4×C53#13712yes—
82C2×Dic53—42no—
83Dic106—44no—
84#88322—44no—
85#88322—81no—
86#10295—531nomaximal, Sylow
87C22×C53—22no—
88C22×C53#39211yescharacteristic
89C22×C53#45112yes—
90C4×C53—22no—
91C4×C53—44no—
92C4×C53—84no—
93C4×C53#45414yes—
94C4×C53#46212yes—
95C4×C53#49411yescharacteristic
96C4×C53#49711yescharacteristic
97Dic53—82no—
98#3230—532no—
99#2862D5311yescharacteristic
100#1994—531no—
101#1549—532no—
102#1276—531no—
103C2×C53—22no—
104C2×C53#200312yes—
105C2×C53#201611yescharacteristic
106#533—531no—
107C2×C4⋊Q8C2×D5312yes—
108#500—1062no—
109#487—1062no—
110#484—1064no—
111#477—1064no—
112#431—532no—
113#423—531no—
114#391—24no—
115#382—22no—
116C43C2×D5311yescharacteristic
117#339—1062no—
118#332—1064no—
119C53#1029511yescharacteristic, complemented, Sylow
120C22×Q8—22no—
121C2×M16—1061no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—532no—
127#125—1061no—
128C22⋊Q8—1062no—
129C4×Q8—1062no—
130#117—1061no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4C22⋊D5314yes—
133C2×C42—22no—
134C2×C42C22⋊D5312yes—
135C2×C42C22×D5311yescharacteristic
136#107—1068no—
137C4⋊C8—534no—
138C4⋊C8—1062no—
139#103—1068no—
140C22⋊C8—1062no—
141C4×C8—1062no—
142#95—28no—
143C2×Q8—48no—
144C2×Q8—42no—
145C2×Q8—42no—
146C2×Q8—1062no—
147C22×C4—22no—
148C22×C4—24no—
149C22×C4C2×C22⋊D5312yes—
150C22×C4C2×C22⋊D5311yescharacteristic
151M16—2122no—
152C2×C8—1064no—
153C2×C8—2121no—
154C4⋊C4—24no—
155C4⋊C4—24no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—1062no—
159C4⋊C4—1062no—
160C4⋊C4—2122no—
161C22⋊C4—2121no—
162C42—22no—
163C42—44no—
164C42—2121no—
165C42#1171992514yes—
166C42D4×D5311yescharacteristic
167C42D4×D5312yes—
168C42#1171994711yescharacteristic
169C23#1313318611yescharacteristic
170Q8—84no—
171Q8—84no—
172Q8—2124no—
173C2×C4—24no—
174C2×C4—24no—
175C2×C4—44no—
176C2×C4—48no—
177C2×C4—44no—
178C2×C4—2122no—
179C2×C4#1313318512yes—
180C2×C4#1313318811yescharacteristic
181C2×C4#1313318912yes—
182C2×C4#1313321311yescharacteristic
183C2×C4C2×#1171992512yes—
184C2×C4#1313324812yes—
185C8—4241no—
186C8—2124no—
187C22—22no—
188C22#1681053111yescharacteristic
189C22#1681077912yes—
190C4—22no—
191C4—44no—
192C4—84no—
193C4—4242no—
194C4#1681079514yes—
195C4#1681083012yes—
196C4#1681100211yescharacteristic
197C4#1681101511yescharacteristic
198C2—22no—
199C2#2027573712yes—
200C2#2027581511yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2862C2
socleC22×C53#392
prime radicalC23×C53#120
hypercenter#2862D53
hypocenterC53#10295
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2862

Complements

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

NH ≅ G/Ntype
119. C53#10295semidirect

Maximal subgroups

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

subgroupindex
#202858662
#202756922
#202734002
#202725482
C53×#28622
#1029553