Finite Group Databasebeta

← #27297317

Subgroups — #27297317

Order 13568.

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

normal subgroups only

This group has 344 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#20285490C211yesmaximal, characteristic
2#20276918C212yesmaximal, complemented
3#20276143C211yesmaximal, characteristic
4#20272728C212yesmaximal, complemented
5C53×#1314C211yesmaximal, characteristic
6C2×#13133196—21no—
7#16810959—22no—
8#16810864—21no—
9#16810864C2212yes—
10#16810781—22no—
11C2×#13133151C2211yescharacteristic
12#16810483—21no—
13#16810449—22no—
14#16810428—22no—
15#16810386—21no—
16#16810386C2211yescharacteristic
17C2×C22⋊C8×C53C2211yescharacteristic
18C53×C22⋊C16C2212yes—
19C22×Dic106—21no—
20C2×Dic212—21no—
21C2×Dic212D412yes—
22#13133196—22no—
23#13133196—22no—
24#13133196—41no—
25#13133196—41no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53—21no—
29C2×#11719919—41no—
30C2×C22⋊Dic53—41no—
31C2×C22⋊Dic53—41no—
32#13133151—21no—
33#13133151—22no—
34#13133151D412yes—
35#13133142—22no—
36#13133142—22no—
37#13133124—41no—
38C23×C4×C53D411yescharacteristic
39C22×C8×C53D411yescharacteristic
40C22×C8×C53C2311yescharacteristic
41C2×C16×C53—22no—
42C22⋊C8×C53—22no—
43C24×C53D811yescharacteristic
44C22×C4×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53—41no—
47C22×C4×C53—41no—
48C22×C4×C53D811yescharacteristic
49C22×C4×C53C2×D411yescharacteristic
50C22×Dic53—41no—
51C22×Dic53—41no—
52C22×Dic53—41no—
53C22×Dic53—41no—
54C2×Dic106—22no—
55C2×Dic106—41no—
56C2×Dic106—81no—
57C2×Dic106—81no—
58C2×C8×C53—21no—
59C2×C8×C53—41no—
60C2×C8×C53—41no—
61C2×C8×C53C2×D412yes—
62C22⋊Dic53—42no—
63C22⋊Dic53—42no—
64C22⋊Dic53—82no—
65C4⋊Dic53—22no—
66C4⋊Dic53—41no—
67C4⋊Dic53—41no—
68C4⋊Dic53—41no—
69C4⋊Dic53—41no—
70C4⋊Dic53—41no—
71#11719919—42no—
72#11719919—42no—
73Dic212—42no—
74Dic212—81no—
75C16×C53—42no—
76C23×C53—21no—
77C23×C53—21no—
78C23×C53—41no—
79C23×C53—41no—
80C23×C53Q3212yes—
81C23×C53C2×D811yescharacteristic
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—41no—
86C2×C4×C53—41no—
87C2×C4×C53—41no—
88C2×C4×C53—81no—
89C2×C4×C53#12011yescharacteristic
90C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
91C2×Dic53—42no—
92C2×Dic53—42no—
93C2×Dic53—82no—
94C2×Dic53—82no—
95C2×Dic53—81no—
96C2×Dic53—81no—
97C2×Dic53—81no—
98C2×Dic53—81no—
99C2×Dic53—81no—
100C2×Dic53—81no—
101Dic106—42no—
102Dic106—42no—
103Dic106—82no—
104C8×C53—22no—
105C8×C53—41no—
106#12282—531nomaximal, Sylow
107C22×C53—21no—
108C22×C53—21no—
109C22×C53—22no—
110C22×C53—42no—
111C22×C53—41no—
112C22×C53—41no—
113C22×C53—81no—
114C22×C53#44711yescharacteristic
115C22×C53C2×Q3211yescharacteristic
116C22×C53#51011yescharacteristic
117C4×C53—21no—
118C4×C53—41no—
119C4×C53#45511yescharacteristic
120C4×C53#45811yescharacteristic
121Dic53—82no—
122Dic53—82no—
123Dic53—161no—
124Dic53—161no—
125#3189—531no—
126#2236—532no—
127#2076—531no—
128#1348—532no—
129#1314D5311yescharacteristic
130C2×C53—21no—
131C2×C53—21no—
132C2×C53—42no—
133C2×C53#160711yescharacteristic
134C2×C53#218811yescharacteristic
135C2×C53#219011yescharacteristic
136C2×C22⋊Q8—1061no—
137#484—1062no—
138#467—532no—
139#467—1061no—
140#451—1062no—
141C2×#107—531no—
142C2×C22⋊C8C2×D5311yescharacteristic
143#380—1061no—
144#366—1062no—
145#358—1062no—
146C22⋊C16C2×D5312yes—
147#340—531no—
148#340—1061no—
149C53#1228211yescharacteristic, complemented, Sylow
150C22×Q8—1061no—
151C23×C4D21211yescharacteristic
152C2×Q16—532no—
153C2×Q16—1061no—
154C22×C8D21211yescharacteristic
155C22×C8C22×D5311yescharacteristic
156C22⋊Q8—1062no—
157C22⋊Q8—1062no—
158C22⋊Q8—2121no—
159C22⋊Q8—2121no—
160C2×C4⋊C4—1061no—
161C2×C4⋊C4—1061no—
162C2×C4⋊C4—1061no—
163C2×C4⋊C4—2121no—
164C2×C22⋊C4—2121no—
165C2×C22⋊C4—2121no—
166C2×C16—22no—
167#107—532no—
168#107—1061no—
169#107—1062no—
170#103—1062no—
171#103—1062no—
172C22⋊C8—22no—
173#95—2121no—
174C24D42411yescharacteristic
175C2×Q8—1062no—
176C2×Q8—4241no—
177C2×Q8—4241no—
178C2×Q8—2121no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—2121no—
184C22×C4—2121no—
185C22×C4—2121no—
186C22×C4—2121no—
187C22×C4D42411yescharacteristic
188C22×C4C2×D21211yescharacteristic
189Q16—4241no—
190Q16—2122no—
191C2×C8—21no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8D4×D5312yes—
195C4⋊C4—1062no—
196C4⋊C4—2121no—
197C4⋊C4—2121no—
198C4⋊C4—2121no—
199C4⋊C4—2122no—
200C4⋊C4—2121no—
201C4⋊C4—2121no—
202C4⋊C4—2122no—
203C22⋊C4—4242no—
204C22⋊C4—2122no—
205C22⋊C4—2122no—
206C16—42no—
207C23—21no—
208C23—21no—
209C23—41no—
210C23—41no—
211C23Dic42412yes—
212C23C2×D42411yescharacteristic
213Q8—4242no—
214Q8—2122no—
215Q8—2122no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—81no—
223C2×C4—4241no—
224C2×C4—4241no—
225C2×C4—4241no—
226C2×C4—4241no—
227C2×C4—4241no—
228C2×C4—4241no—
229C2×C4—4242no—
230C2×C4—4242no—
231C2×C4—2122no—
232C2×C4—2122no—
233C2×C4#1313318711yescharacteristic
234C2×C4#1313324511yescharacteristic
235C8—22no—
236C8—41no—
237C22—21no—
238C22—21no—
239C22—22no—
240C22—42no—
241C22—41no—
242C22—41no—
243C22—81no—
244C22#1681075911yescharacteristic
245C22C2×Dic42411yescharacteristic
246C22#1681106611yescharacteristic
247C4—21no—
248C4—41no—
249C4—8481no—
250C4—8481no—
251C4—4242no—
252C4—4242no—
253C4#1681080311yescharacteristic
254C4#1681081411yescharacteristic
255C2—21no—
256C2—21no—
257C2—42no—
258C2#2027370311yescharacteristic
259C2#2027665111yescharacteristic
260C2#2027666311yescharacteristic
subgroupG/H
centerC22#16810759
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
FittingC53×#1314C2
socleC22×C53#447
prime radicalC24×C53D8
hypercenter#1314D53
hypocenterC53#12282
LCSG ▷ C22×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #1314

Complements

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

NH ≅ G/Ntype
2. #20276918C2semidirect
4. #20272728C2semidirect
149. C53#12282semidirect

Maximal subgroups

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

subgroupindex
#202854902
#202769182
#202761432
#202727282
C53×#13142
#1228253