Finite Group Databasebeta

← #27286383

Subgroups — #27286383

Order 13568.

466 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 103 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#20285859C212yesmaximal
2#20285239C211yesmaximal, characteristic
3#20273278C212yesmaximal
4#20272560C211yesmaximal, characteristic
5C53×#1960C211yesmaximal, characteristic
6#16811180C2212yes—
7#16811032—22no—
8#16811032—22no—
9#16811019—22no—
10#16810975—22no—
11#16810953—24no—
12#16810905—24no—
13#16810676C2212yes—
14#16810661—22no—
15#16810359—24no—
16C2×C4⋊Q8×C53C2211yescharacteristic
17C2×#103×C53—22no—
18C2×#97×C53C2211yescharacteristic
19C53×#382C2211yescharacteristic
20#13133217D412yes—
21#13133217D412yes—
22#13133209—22no—
23#13133196—22no—
24#13133196—22no—
25C4×Dic106—22no—
26C4×Dic106—22no—
27#13133176—22no—
28#13133151—24no—
29#13133149—22no—
30#13133149—22no—
31#13133148—28no—
32C4⋊#88322—22no—
33#13133140—28no—
34C22⋊#88322D414yes—
35C4×#88322—22no—
36C22×Q8×C53—21no—
37C22×C8×C53D412yes—
38C4⋊Q8×C53—21no—
39C4⋊Q8×C53—22no—
40C4⋊Q8×C53D412yes—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53—21no—
43C2×C42×C53C2311yescharacteristic
44#103×C53—44no—
45#97×C53—22no—
46#95×C53—22no—
47C2×Q8×C53—22no—
48C2×Q8×C53—41no—
49C2×Q8×C53—42no—
50C2×Q8×C53—42no—
51C22×C4×C53—41no—
52C22×C4×C53—41no—
53C22×C4×C53C2×D412yes—
54C22×C4×C53C2×D411yescharacteristic
55C2×Dic106—22no—
56C2×Dic106—22no—
57C2×C8×C53—22no—
58C2×C8×C53—22no—
59C2×C8×C53SD1614yes—
60C4⋊C4×C53—24no—
61C4⋊C4×C53—42no—
62C4⋊C4×C53—42no—
63C42×C53C2×D411yescharacteristic
64C42×C53C2×D412yes—
65C42×C53#4111yescharacteristic
66C22⋊Dic53—42no—
67C4⋊Dic53—22no—
68C4⋊Dic53—22no—
69C4⋊Dic53—22no—
70C4⋊Dic53—22no—
71#11719919—42no—
72#11719919—42no—
73C4×Dic53—42no—
74C2×#88322—24no—
75C23×C53#12711yescharacteristic
76Q8×C53—44no—
77Q8×C53—82no—
78C2×C4×C53—42no—
79C2×C4×C53—42no—
80C2×C4×C53—81no—
81C2×C4×C53—81no—
82C2×C4×C53—82no—
83C2×C4×C53—81no—
84C2×C4×C53#12011yescharacteristic
85C2×C4×C53#12012yes—
86C2×C4×C53#12112yes—
87C2×C4×C53#12111yescharacteristic
88C2×C4×C53C2×SD1612yes—
89C2×C4×C53#13712yes—
90C2×Dic53—42no—
91C2×Dic53—42no—
92Dic106—44no—
93Dic106—82no—
94C8×C53—24no—
95#88322—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—82no—
104C4×C53—161no—
105C4×C53#44812yes—
106C4×C53#46112yes—
107Dic53—82no—
108Dic53—82no—
109#3229—532no—
110#3162—531no—
111#1960D5311yescharacteristic
112#1518—532no—
113#1281—531no—
114C2×C53—21no—
115C2×C53—21no—
116C2×C53#200912yes—
117C2×C53#201311yescharacteristic
118#533—532no—
119C2×C4⋊Q8C2×D5311yescharacteristic
120#501—1062no—
121#501—1062no—
122#498—1062no—
123#487—1062no—
124#483—1064no—
125#475—1064no—
126#431—532no—
127#427—1062no—
128C2×#103—22no—
129C2×#97C2×D5311yescharacteristic
130#382C2×D5311yescharacteristic
131#328—1064no—
132C53#1022711yescharacteristic, complemented, Sylow
133C22×Q8—21no—
134C22×C8C22⋊D5312yes—
135C4⋊Q8—21no—
136C4⋊Q8—22no—
137C4⋊Q8—532no—
138C4⋊Q8—532no—
139C4⋊Q8D21212yes—
140#125—1062no—
141C22⋊Q8—1062no—
142C22⋊Q8—1062no—
143C4×Q8—1062no—
144C4×Q8—1062no—
145#117—1062no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C42C22×D5311yescharacteristic
149#107—1064no—
150#106—1068no—
151#106—1062no—
152#106—1062no—
153C4⋊C8—1062no—
154#103—44no—
155#103—1068no—
156C22⋊C8—534no—
157#97—22no—
158C4×C8—1062no—
159#95—22no—
160C2×Q8—22no—
161C2×Q8—41no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—1062no—
165C2×Q8—1062no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4D4×D5312yes—
169C22×C4C2×C22⋊D5311yescharacteristic
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—1064no—
173C2×C8#1171992314yes—
174C4⋊C4—24no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—1062no—
178C4⋊C4—1062no—
179C4⋊C4—1062no—
180C4⋊C4—1062no—
181C4⋊C4—2122no—
182C4⋊C4—2122no—
183C22⋊C4—2122no—
184C42—2122no—
185C42C2×D21211yescharacteristic
186C42#1171994511yescharacteristic
187C42D4×D5312yes—
188C23#1313321311yescharacteristic
189Q8—44no—
190Q8—82no—
191Q8—4242no—
192Q8—2124no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4—82no—
198C2×C4—81no—
199C2×C4—2122no—
200C2×C4—2122no—
201C2×C4#1313318511yescharacteristic
202C2×C4#1313318712yes—
203C2×C4#1313319012yes—
204C2×C4#1313319111yescharacteristic
205C2×C4C2×#1171992312yes—
206C2×C4#1313324712yes—
207C8—24no—
208C8—2124no—
209C22—22no—
210C22#1681053211yescharacteristic
211C22#1681099511yescharacteristic
212C22#1681101811yescharacteristic
213C4—22no—
214C4—22no—
215C4—82no—
216C4—161no—
217C4—4242no—
218C4—4242no—
219C4#1681076112yes—
220C4#1681082812yes—
221C2—21no—
222C2—21no—
223C2#2027577712yes—
224C2#2027579711yescharacteristic
subgroupG/H
centerC22#16810532
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1960C2
socleC22×C53#392
prime radicalC23×C53#127
hypercenter#1960D53
hypocenterC53#10227
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1960

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
#202858592
#202852392
#202732782
#202725602
C53×#19602
#1022753