Finite Group Databasebeta

← #27288533

Subgroups — #27288533

Order 13568.

494 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 137 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#20279744C211yesmaximal, characteristic
2#20275479C212yesmaximal
3#20274871C211yesmaximal, characteristic
4#20272554C212yesmaximal
5C53×#1296C211yesmaximal, characteristic
6C2×#13133216—22no—
7C2×#13133149—22no—
8C2×#13133148—22no—
9C2×C4×#88322C2211yescharacteristic
10#16810543—22no—
11#16810519—22no—
12#16810493C2211yescharacteristic
13C4×C4⋊Dic53C2212yes—
14#16810386—22no—
15C2×C4⋊C8×C53C2212yes—
16C53×#384C2211yescharacteristic
17C22×Dic106—22no—
18C22×#88322—21no—
19#13133216—44no—
20#13133216—44no—
21C2×C4⋊Dic53—22no—
22C2×C4⋊Dic53—21no—
23C2×C4⋊Dic53D412yes—
24C2×C4⋊Dic53D412yes—
25C2×#11719919—24no—
26C2×C4×Dic53—22no—
27#13133149—44no—
28#13133148—44no—
29C4×#88322—21no—
30C4×#88322Q812yescomplemented
31#13133124—22no—
32#13133124—22no—
33C22×C8×C53—22no—
34C2×C4⋊C4×C53—22no—
35C2×C4⋊C4×C53D412yes—
36C2×C42×C53C2311yescharacteristic
37C4⋊C8×C53—22no—
38C4⋊C8×C53Q814yescomplemented
39C22×C4×C53—22no—
40C22×C4×C53C2×D411yescharacteristic
41C22×C4×C53C2×D412yes—
42C22×Dic53—22no—
43C22×Dic53—22no—
44C22×Dic53—41no—
45C2×Dic106—48no—
46C2×Dic106—82no—
47C2×C8×C53—24no—
48C2×C8×C53—42no—
49C2×C8×C53—42no—
50C4⋊C4×C53—42no—
51C4⋊C4×C53—42no—
52C4⋊C4×C53—42no—
53C42×C53C2×Q812yes—
54C42×C53C2×Q811yescharacteristic
55C42×C53#4111yescharacteristic
56C4⋊Dic53—22no—
57C4⋊Dic53—22no—
58C4⋊Dic53—42no—
59C4⋊Dic53—42no—
60C4⋊Dic53—42no—
61C4⋊Dic53SD1614yes—
62C4⋊Dic53SD1614yes—
63#11719919—48no—
64C4×Dic53—42no—
65C4×Dic53—42no—
66C2×#88322—22no—
67C2×#88322—22no—
68C2×#88322—22no—
69C23×C53#12011yescharacteristic
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—44no—
73C2×C4×C53—42no—
74C2×C4×C53C22⋊Q812yes—
75C2×C4×C53C22⋊Q811yescharacteristic
76C2×C4×C53C4⋊Q812yes—
77C2×C4×C53C4⋊Q811yescharacteristic
78C2×C4×C53C2×SD1612yes—
79C2×C4×C53C2×SD1612yes—
80C2×C4×C53AGL(1,ℤ/8)12yes—
81C2×C4×C53#13712yes—
82C2×Dic53—24no—
83C2×Dic53—24no—
84C2×Dic53—42no—
85C2×Dic53—42no—
86C2×Dic53—42no—
87C2×Dic53—42no—
88C2×Dic53—82no—
89C2×Dic53—81no—
90Dic106—88no—
91C8×C53—44no—
92#88322—24no—
93#10574—531nomaximal, Sylow
94C22×C53#39511yescharacteristic
95C22×C53#44812yes—
96C22×C53#45012yes—
97C22×C53#45311yescharacteristic
98C22×C53#45611yescharacteristic
99C4×C53—21no—
100C4×C53—21no—
101C4×C53—82no—
102C4×C53#47512yes—
103C4×C53#47612yes—
104C4×C53#49812yes—
105C4×C53#50112yes—
106Dic53—44no—
107Dic53—44no—
108Dic53—161no—
109#2551—531no—
110#1948—532no—
111#1844—531no—
112#1296D5311yescharacteristic
113#1279—532no—
114C2×C53#162311yescharacteristic
115C2×C53#162911yescharacteristic
116C2×C53#165112yes—
117C2×C53#205812yes—
118C2×C53#206011yescharacteristic
119C2×C4⋊Q8—1062no—
120C2×#106—1062no—
121C2×#106—1062no—
122C2×C4⋊C8C2×D5312yes—
123C2×C4×C8—531no—
124#395—1062no—
125#389—1062no—
126#384—531no—
127#384C2×D5311yescharacteristic
128C4×C4⋊C4—532no—
129#340—1062no—
130C53#1057411yescharacteristic, complemented, Sylow
131C22×Q8—1062no—
132C22×C8—22no—
133C22×C8—1061no—
134C4⋊Q8—2124no—
135C4⋊Q8—2124no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—532no—
138C2×C4⋊C4—532no—
139C2×C4⋊C4—1064no—
140C2×C4⋊C4—1061no—
141C2×C4⋊C4—1062no—
142C2×C4⋊C4D21212yes—
143C2×C42—1062no—
144C2×C42C22×D5311yescharacteristic
145#106—2124no—
146#106—2124no—
147C4⋊C8—22no—
148C4⋊C8Dic10614yescomplemented
149C4×C8—532no—
150C4×C8—1061no—
151#95—1062no—
152#95—1062no—
153C2×Q8—4242no—
154C2×Q8—2128no—
155C22×C4—22no—
156C22×C4—1062no—
157C22×C4—1062no—
158C22×C4—2121no—
159C22×C4C2×D21211yescharacteristic
160C22×C4D4×D5312yes—
161C2×C8—24no—
162C2×C8—42no—
163C2×C8—42no—
164C2×C8—1062no—
165C2×C8—1062no—
166C2×C8—1062no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—534no—
171C4⋊C4—534no—
172C4⋊C4—1062no—
173C4⋊C4—1062no—
174C4⋊C4—2122no—
175C4⋊C4—2128no—
176C4⋊C4—2122no—
177C4⋊C4—2122no—
178C42—2122no—
179C42—2122no—
180C42C2×Dic10612yes—
181C42#1171994711yescharacteristic
182C42Q8×D5311yescharacteristic
183C23#1313318711yescharacteristic
184Q8—4248no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—44no—
188C2×C4—42no—
189C2×C4—1064no—
190C2×C4—1064no—
191C2×C4—4242no—
192C2×C4—4241no—
193C2×C4—2122no—
194C2×C4—2122no—
195C2×C4—2122no—
196C2×C4—2122no—
197C2×C4#1313319412yes—
198C2×C4#1313319711yescharacteristic
199C2×C4#1313321612yes—
200C2×C4#1313321711yescharacteristic
201C2×C4SD16×D5312yes—
202C2×C4SD16×D5312yes—
203C2×C4#1313324112yes—
204C2×C4#1313324912yes—
205C8—44no—
206C8—1064no—
207C22#1681054411yescharacteristic
208C22#1681076312yes—
209C22#1681077512yes—
210C22#1681079311yescharacteristic
211C22#1681080811yescharacteristic
212C4—21no—
213C4—21no—
214C4—82no—
215C4—8481no—
216C4—2124no—
217C4—2124no—
218C4#1681090612yes—
219C4#1681091112yes—
220C4#1681102012yes—
221C4#1681103112yes—
222C2#2027379411yescharacteristic
223C2#2027383211yescharacteristic
224C2#2027397912yes—
225C2#2027604712yes—
226C2#2027606111yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1296C2
socle, prime radicalC23×C53#120
hypercenter#1296D53
hypocenterC53#10574
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
30. C4×#88322Q8semidirect
38. C4⋊C8×C53Q8semidirect
130. C53#10574semidirect
148. C4⋊C8Dic106semidirect

Maximal subgroups

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

subgroupindex
#202797442
#202754792
#202748712
#202725542
C53×#12962
#1057453