Finite Group Databasebeta

← #27286386

Subgroups — #27286386

Order 13568.

466 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 105 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#20285861C212yesmaximal
2#20285239C211yesmaximal, characteristic
3#20275535C211yesmaximal, characteristic
4#20273280C212yesmaximal
5C53×#1281C211yesmaximal, characteristic
6#16811180C2212yes—
7C2×#13133217C2211yescharacteristic
8#16811032—22no—
9#16811032—22no—
10#16811021—22no—
11#16810974—22no—
12#16810954—24no—
13#16810909—24no—
14#16810661—22no—
15C2×#13133142—22no—
16#16810489C2211yescharacteristic
17#16810361—24no—
18C53×#431C2212yes—
19C2×#97×C53C2211yescharacteristic
20C22×Dic106—21no—
21#13133217—21no—
22#13133217—22no—
23#13133217D412yes—
24#13133217D412yes—
25#13133217D412yes—
26#13133209—22no—
27#13133196—22no—
28#13133196—22no—
29C4×Dic106—22no—
30C4×Dic106—22no—
31#13133176—22no—
32C2×C4⋊Dic53—22no—
33C2×C4⋊Dic53—21no—
34#13133151—24no—
35#13133149—28no—
36#13133149—22no—
37#13133149—22no—
38#13133142—28no—
39#13133142—44no—
40#13133124—22no—
41C22×C8×C53D412yes—
42C2×C42×C53C2311yescharacteristic
43C4⋊C8×C53—22no—
44C22⋊C8×C53D414yes—
45#97×C53—22no—
46C4×C8×C53—22no—
47C22×C4×C53C2×D412yes—
48C22×C4×C53C2×D411yescharacteristic
49C22×Dic53—41no—
50C22×Dic53—41no—
51C2×Dic106—22no—
52C2×Dic106—22no—
53C2×Dic106—22no—
54C2×Dic106—42no—
55C2×Dic106—42no—
56C2×Dic106—41no—
57C2×C8×C53—22no—
58C2×C8×C53—22no—
59C2×C8×C53—24no—
60C2×C8×C53SD1614yes—
61C42×C53C2×D412yes—
62C42×C53C2×D411yescharacteristic
63C42×C53#4111yescharacteristic
64C22⋊Dic53—42no—
65C4⋊Dic53—24no—
66C4⋊Dic53—22no—
67C4⋊Dic53—22no—
68C4⋊Dic53—22no—
69C4⋊Dic53—22no—
70C4⋊Dic53—42no—
71C4⋊Dic53—42no—
72#11719919—42no—
73#11719919—42no—
74C4×Dic53—42no—
75C23×C53#12711yescharacteristic
76C2×C4×C53—42no—
77C2×C4×C53#12012yes—
78C2×C4×C53#12011yescharacteristic
79C2×C4×C53#12112yes—
80C2×C4×C53#12111yescharacteristic
81C2×C4×C53C2×SD1612yes—
82C2×C4×C53#13712yes—
83C2×Dic53—42no—
84C2×Dic53—42no—
85C2×Dic53—42no—
86C2×Dic53—81no—
87C2×Dic53—81no—
88C2×Dic53—82no—
89C2×Dic53—81no—
90Dic106—44no—
91Dic106—44no—
92Dic106—82no—
93Dic106—82no—
94C8×C53—24no—
95C8×C53—44no—
96#10227—531nomaximal, Sylow
97C22×C53—22no—
98C22×C53#39211yescharacteristic
99C22×C53#49211yescharacteristic
100C22×C53#49711yescharacteristic
101C4×C53—22no—
102C4×C53—22no—
103C4×C53#44812yes—
104C4×C53#46112yes—
105Dic53—82no—
106Dic53—82no—
107Dic53—82no—
108Dic53—161no—
109#3229—532no—
110#3162—531no—
111#1960—531no—
112#1518—532no—
113#1281D5311yescharacteristic
114C2×C53—21no—
115C2×C53—21no—
116C2×C53#200912yes—
117C2×C53#201311yescharacteristic
118#533—532no—
119C2×C4⋊Q8—531no—
120#501—1062no—
121#501—1062no—
122#498—1062no—
123#487—1062no—
124#483—1064no—
125#475—1064no—
126#431C2×D5312yes—
127#427—1062no—
128C2×#103—1062no—
129C2×#97C2×D5311yescharacteristic
130#382—531no—
131#328—1064no—
132C53#1022711yescharacteristic, complemented, Sylow
133C22×Q8—1061no—
134C22×C8D21212yes—
135C4⋊Q8—532no—
136C4⋊Q8—532no—
137C4⋊Q8—532no—
138C4⋊Q8—1062no—
139C4⋊Q8—1061no—
140#125—1062no—
141C22⋊Q8—1062no—
142C22⋊Q8—1062no—
143C4×Q8—1062no—
144C4×Q8—1062no—
145#117—1062no—
146C2×C4⋊C4—1062no—
147C2×C4⋊C4—1061no—
148C2×C42C22×D5311yescharacteristic
149#107—1064no—
150#106—1062no—
151#106—1062no—
152#106—1068no—
153C4⋊C8—22no—
154#103—1068no—
155#103—2124no—
156C22⋊C8D21214yes—
157#97—22no—
158C4×C8—22no—
159#95—1062no—
160C2×Q8—1062no—
161C2×Q8—1062no—
162C2×Q8—1062no—
163C2×Q8—2122no—
164C2×Q8—2122no—
165C2×Q8—2121no—
166C22×C4—2121no—
167C22×C4—2121no—
168C22×C4C2×D21212yes—
169C22×C4C2×D21211yescharacteristic
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—24no—
173C2×C8#1171991314yes—
174C4⋊C4—1064no—
175C4⋊C4—1062no—
176C4⋊C4—1062no—
177C4⋊C4—1062no—
178C4⋊C4—1062no—
179C4⋊C4—2122no—
180C4⋊C4—2122no—
181C4⋊C4—2122no—
182C4⋊C4—2122no—
183C22⋊C4—2122no—
184C42—2122no—
185C42D4×D5312yes—
186C42D4×D5311yescharacteristic
187C42#1171994911yescharacteristic
188C23#1313321411yescharacteristic
189Q8—4242no—
190Q8—4242no—
191Q8—2124no—
192Q8—2124no—
193C2×C4—42no—
194C2×C4—4242no—
195C2×C4—4241no—
196C2×C4—4241no—
197C2×C4—4241no—
198C2×C4—2122no—
199C2×C4—2122no—
200C2×C4—2122no—
201C2×C4#1313318712yes—
202C2×C4#1313318711yescharacteristic
203C2×C4#1313319212yes—
204C2×C4#1313319211yescharacteristic
205C2×C4C2×#1171991312yes—
206C2×C4#1313325012yes—
207C8—24no—
208C8—44no—
209C22—22no—
210C22#1681053311yescharacteristic
211C22#1681099611yescharacteristic
212C22#1681101711yescharacteristic
213C4—22no—
214C4—22no—
215C4—8481no—
216C4—4242no—
217C4—4242no—
218C4—4242no—
219C4#1681076412yes—
220C4#1681082912yes—
221C2—21no—
222C2—21no—
223C2#2027577812yes—
224C2#2027580011yescharacteristic
subgroupG/H
centerC22#16810533
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1281C2
socleC22×C53#392
prime radicalC23×C53#127
hypercenter#1281D53
hypocenterC53#10227
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1281

Complements

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

NH ≅ G/Ntype
132. C53#10227semidirect

Maximal subgroups

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

subgroupindex
#202858612
#202852392
#202755352
#202732802
C53×#12812
#1022753