Finite Group Databasebeta

← #27286818

Subgroups — #27286818

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#20285868C212yesmaximal
2#20283093C211yesmaximal, characteristic
3#20275692C211yesmaximal, characteristic
4#20273399C212yesmaximal
5C53×#1276C211yesmaximal, characteristic
6#16811180C2211yescharacteristic
7C2×#13133217C2212yes—
8#16811028—22no—
9#16810974—22no—
10#16810959—24no—
11#16810919—24no—
12#16810647C2211yescharacteristic
13#16810529—24no—
14#16810489—22no—
15#16810383—22no—
16#16810370—24no—
17C53×#431C2212yes—
18C43×C53C2211yescharacteristic
19C22×Dic106—22no—
20C2×#11719917—21no—
21#13133217—22no—
22#13133217—22no—
23#13133217—22no—
24#13133217—22no—
25#13133217D412yes—
26#13133209—21no—
27#13133196—22no—
28C4×Dic106—22no—
29#13133176—21no—
30C2×C4⋊Dic53—22no—
31C2×C4⋊Dic53D414yes—
32#13133151—28no—
33C4⋊#88322—22no—
34#13133142—28no—
35#13133124—28no—
36C2×C42×C53—22no—
37C2×C42×C53D412yes—
38C2×C42×C53C2311yescharacteristic
39C4⋊C8×C53D414yes—
40C22⋊C8×C53—22no—
41C4×C8×C53—22no—
42C22×C4×C53—22no—
43C22×C4×C53C2×D412yes—
44C22×C4×C53C2×D411yescharacteristic
45C22×Dic53—24no—
46C2×Dic106—22no—
47C2×Dic106—48no—
48C2×Dic106—42no—
49C2×Dic106—42no—
50C2×C8×C53—24no—
51C42×C53—22no—
52C42×C53—44no—
53C42×C53Q1614yes—
54C42×C53C2×D411yescharacteristic
55C42×C53C2×D412yes—
56C42×C53#4111yescharacteristic
57C22⋊Dic53—41no—
58C4⋊Dic53—24no—
59C4⋊Dic53—22no—
60C4⋊Dic53—24no—
61C4⋊Dic53—22no—
62C4⋊Dic53—42no—
63C4⋊Dic53—42no—
64#11719919—42no—
65C4×Dic53—41no—
66#11719917—42no—
67C2×#88322—41no—
68C23×C53#12011yescharacteristic
69C2×C4×C53—24no—
70C2×C4×C53—24no—
71C2×C4×C53—44no—
72C2×C4×C53#12012yes—
73C2×C4×C53#12112yes—
74C2×C4×C53#12111yescharacteristic
75C2×C4×C53#12711yescharacteristic
76C2×C4×C53C2×Q1612yes—
77C2×C4×C53#13712yes—
78C2×Dic53—48no—
79C2×Dic53—42no—
80C2×Dic53—44no—
81Dic106—44no—
82Dic106—84no—
83Dic106—84no—
84C8×C53—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#45414yes—
93C4×C53#46212yes—
94C4×C53#49411yescharacteristic
95C4×C53#49711yescharacteristic
96Dic53—82no—
97Dic53—84no—
98#3230—532no—
99#2862—531no—
100#1994—531no—
101#1549—532no—
102#1276D5311yescharacteristic
103C2×C53—22no—
104C2×C53#200312yes—
105C2×C53#201611yescharacteristic
106#533—531no—
107C2×C4⋊Q8—532no—
108#500—1062no—
109#487—1062no—
110#484—1064no—
111#477—1064no—
112#431C2×D5312yes—
113#423—531no—
114#391—1064no—
115#382—1062no—
116C43C2×D5311yescharacteristic
117#339—1062no—
118#332—1064no—
119C53#1029511yescharacteristic, complemented, Sylow
120C22×Q8—1062no—
121C2×M16—1061no—
122C4⋊Q8—532no—
123C4⋊Q8—1062no—
124C4⋊Q8—1062no—
125C4⋊Q8—1062no—
126C4⋊Q8—1062no—
127#125—1061no—
128C22⋊Q8—1062no—
129C4×Q8—1062no—
130#117—1061no—
131C2×C4⋊C4—534no—
132C2×C4⋊C4—1062no—
133C2×C42—22no—
134C2×C42D21212yes—
135C2×C42C22×D5311yescharacteristic
136#107—1068no—
137C4⋊C8—1062no—
138C4⋊C8D21214yes—
139#103—1068no—
140C22⋊C8—22no—
141C4×C8—22no—
142#95—1068no—
143C2×Q8—1062no—
144C2×Q8—2128no—
145C2×Q8—2122no—
146C2×Q8—2122no—
147C22×C4—22no—
148C22×C4—1064no—
149C22×C4C2×D21211yescharacteristic
150C22×C4D4×D5312yes—
151M16—2122no—
152C2×C8—24no—
153C2×C8—2121no—
154C4⋊C4—1064no—
155C4⋊C4—1062no—
156C4⋊C4—1064no—
157C4⋊C4—1062no—
158C4⋊C4—2122no—
159C4⋊C4—2122no—
160C4⋊C4—2122no—
161C22⋊C4—2121no—
162C42—22no—
163C42—44no—
164C42—2121no—
165C42Dic21214yes—
166C42C2×D21212yes—
167C42D4×D5311yescharacteristic
168C42#1171994911yescharacteristic
169C23#1313318711yescharacteristic
170Q8—4244no—
171Q8—4244no—
172Q8—2124no—
173C2×C4—24no—
174C2×C4—24no—
175C2×C4—44no—
176C2×C4—2128no—
177C2×C4—2122no—
178C2×C4—2124no—
179C2×C4#1313318712yes—
180C2×C4#1313319212yes—
181C2×C4#1313319211yescharacteristic
182C2×C4#1313321411yescharacteristic
183C2×C4C2×Dic21212yes—
184C2×C4#1313325012yes—
185C8—44no—
186C8—4241no—
187C22—22no—
188C22#1681053311yescharacteristic
189C22#1681078112yes—
190C4—22no—
191C4—44no—
192C4—4244no—
193C4—4242no—
194C4#1681079714yes—
195C4#1681083412yes—
196C4#1681100311yescharacteristic
197C4#1681101711yescharacteristic
198C2—22no—
199C2#2027574012yes—
200C2#2027581711yescharacteristic
subgroupG/H
centerC22#16810533
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1276C2
socleC22×C53#392
prime radicalC23×C53#120
hypercenter#1276D53
hypocenterC53#10295
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

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
#202858682
#202830932
#202756922
#202733992
C53×#12762
#1029553