Finite Group Databasebeta

← #27297379

Subgroups — #27297379

Order 13568.

310 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 67 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#20286060C211yesmaximal, characteristic
2#20276918C212yesmaximal
3#20276208C211yesmaximal, characteristic
4#20272728C212yesmaximal
5C53×#1322C211yesmaximal, characteristic
6#16811180—21no—
7#16810919—22no—
8#16810864—21no—
9#16810864C2212yes—
10#16810834—22no—
11C2×#13133151C2211yescharacteristic
12#16810521—21no—
13#16810449—22no—
14#16810428—22no—
15#16810386—21no—
16#16810386C2211yescharacteristic
17C2×C4⋊C8×C53C2211yescharacteristic
18C53×C22⋊C16C2212yes—
19C2×Dic212—21no—
20C2×Dic212D412yes—
21#13133217—21no—
22#13133217—21no—
23#13133209—41no—
24#13133196—22no—
25#13133196—41no—
26C4×Dic106—22no—
27C4×Dic106—41no—
28#13133176—41no—
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53—21no—
31C2×#11719919—41no—
32#13133151—21no—
33#13133151—22no—
34#13133151D412yes—
35#13133142—22no—
36#13133142—22no—
37#13133124—41no—
38C22×C8×C53D411yescharacteristic
39C22×C8×C53C2311yescharacteristic
40C2×C42×C53D411yescharacteristic
41C2×C16×C53—22no—
42C4⋊C8×C53—22no—
43C22×C4×C53D811yescharacteristic
44C22×C4×C53D811yescharacteristic
45C22×C4×C53C2×D411yescharacteristic
46C22×Dic53—41no—
47C22×Dic53—41no—
48C2×Dic106—22no—
49C2×Dic106—41no—
50C2×C8×C53—21no—
51C2×C8×C53—41no—
52C2×C8×C53—41no—
53C2×C8×C53C2×D412yes—
54C42×C53—21no—
55C42×C53—21no—
56C22⋊Dic53—42no—
57C4⋊Dic53—22no—
58C4⋊Dic53—22no—
59C4⋊Dic53—41no—
60C4⋊Dic53—41no—
61C4⋊Dic53—41no—
62C4⋊Dic53—41no—
63C4⋊Dic53—41no—
64C4⋊Dic53—41no—
65#11719919—42no—
66#11719919—42no—
67#11719919—82no—
68C4×Dic53—42no—
69Dic212—42no—
70Dic212—81no—
71C16×C53—42no—
72C23×C53C2×D811yescharacteristic
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—41no—
79C2×C4×C53Q3212yes—
80C2×C4×C53#12011yescharacteristic
81C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
82C2×Dic53—42no—
83C2×Dic53—42no—
84C2×Dic53—42no—
85C2×Dic53—81no—
86C2×Dic53—81no—
87C2×Dic53—81no—
88C2×Dic53—81no—
89C2×Dic53—81no—
90Dic106—42no—
91Dic106—42no—
92Dic106—82no—
93C8×C53—22no—
94C8×C53—41no—
95#12292—531nomaximal, Sylow
96C22×C53—21no—
97C22×C53—21no—
98C22×C53#44711yescharacteristic
99C22×C53C2×Q3211yescharacteristic
100C22×C53#51011yescharacteristic
101C4×C53—21no—
102C4×C53—22no—
103C4×C53—41no—
104C4×C53#45611yescharacteristic
105C4×C53#45811yescharacteristic
106Dic53—82no—
107Dic53—82no—
108Dic53—82no—
109Dic53—161no—
110#3249—531no—
111#2236—532no—
112#2087—531no—
113#1348—532no—
114#1322D5311yescharacteristic
115C2×C53—21no—
116C2×C53—21no—
117C2×C53#161011yescharacteristic
118C2×C53#218811yescharacteristic
119C2×C53#219011yescharacteristic
120#533—1061no—
121#477—1062no—
122#467—532no—
123#467—1061no—
124#462—1062no—
125C2×#107—531no—
126C2×C4⋊C8C2×D5311yescharacteristic
127#389—1061no—
128#366—1062no—
129#358—1062no—
130C22⋊C16C2×D5312yes—
131#340—531no—
132#340—1061no—
133C53#1229211yescharacteristic, complemented, Sylow
134C2×Q16—532no—
135C2×Q16—1061no—
136C22×C8D21211yescharacteristic
137C22×C8C22×D5311yescharacteristic
138C4⋊Q8—1061no—
139C4⋊Q8—1061no—
140#125—2121no—
141C22⋊Q8—1062no—
142C22⋊Q8—2121no—
143C4×Q8—1062no—
144C4×Q8—2121no—
145#117—2121no—
146C2×C4⋊C4—1061no—
147C2×C4⋊C4—1061no—
148C2×C4⋊C4—2121no—
149C2×C42D21211yescharacteristic
150C2×C16—22no—
151#107—532no—
152#107—1061no—
153#107—1062no—
154C4⋊C8—22no—
155#103—1062no—
156#103—1062no—
157#95—2121no—
158C2×Q8—1062no—
159C2×Q8—2121no—
160C22×C4—2121no—
161C22×C4—2121no—
162C22×C4D42411yescharacteristic
163C22×C4D42411yescharacteristic
164C22×C4C2×D21211yescharacteristic
165Q16—4241no—
166Q16—2122no—
167C2×C8—21no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8D4×D5312yes—
171C4⋊C4—1062no—
172C4⋊C4—1062no—
173C4⋊C4—4242no—
174C4⋊C4—2121no—
175C4⋊C4—2121no—
176C4⋊C4—2121no—
177C4⋊C4—2122no—
178C4⋊C4—2121no—
179C4⋊C4—2121no—
180C4⋊C4—2121no—
181C4⋊C4—2122no—
182C22⋊C4—2122no—
183C42—21no—
184C42—21no—
185C42—2122no—
186C16—42no—
187C23C2×D42411yescharacteristic
188Q8—4242no—
189Q8—2122no—
190Q8—2122no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—41no—
197C2×C4—4241no—
198C2×C4—4241no—
199C2×C4—4241no—
200C2×C4—4241no—
201C2×C4—4241no—
202C2×C4—2122no—
203C2×C4—2122no—
204C2×C4—2122no—
205C2×C4Dic42412yes—
206C2×C4#1313318711yescharacteristic
207C2×C4#1313324511yescharacteristic
208C8—22no—
209C8—41no—
210C22—21no—
211C22—21no—
212C22#1681075911yescharacteristic
213C22C2×Dic42411yescharacteristic
214C22#1681106611yescharacteristic
215C4—21no—
216C4—22no—
217C4—41no—
218C4—8481no—
219C4—4242no—
220C4—4242no—
221C4—4242no—
222C4#1681080811yescharacteristic
223C4#1681081411yescharacteristic
224C2—21no—
225C2—21no—
226C2#2027372011yescharacteristic
227C2#2027665111yescharacteristic
228C2#2027666311yescharacteristic
subgroupG/H
centerC22#16810759
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
FittingC53×#1322C2
socleC22×C53#447
prime radicalC23×C53C2×D8
hypercenter#1322D53
hypocenterC53#12292
LCSG ▷ C22×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #1322

Complements

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

NH ≅ G/Ntype
133. C53#12292semidirect

Maximal subgroups

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

subgroupindex
#202860602
#202769182
#202762082
#202727282
C53×#13222
#1229253