Finite Group Databasebeta

← #27297814

Subgroups — #27297814

Order 13568.

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

normal subgroups only

This group has 296 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#20287584C211yesmaximal, characteristic
2#20276757C211yesmaximal, characteristic
3#20274082C211yesmaximal, characteristic, complemented
4#20272773C211yesmaximal, characteristic, complemented
5#20272695C211yesmaximal, characteristic, complemented
6#20272677C211yesmaximal, characteristic
7C53×#2241C211yesmaximal, characteristic, complemented
8#16811334—21no—
9C2×#13133160—21no—
10#16811024—21no—
11#16810926—22no—
12#16810910—21no—
13#16810905—21no—
14C8×Dic106C2211yescharacteristic
15C4×D424C2211yescharacteristic
16#16810705—21no—
17C4⋊#11719908C2211yescharacteristic
18#16810427—21no—
19#16810376C2211yescharacteristic
20#16810369—21no—
21C53×#500C2211yescharacteristic
22C53×#494C2211yescharacteristic
23C53×#358—22no—
24C4×C16×C53C2211yescharacteristic
25C2×D424D411yescharacteristic
26#13133208—22no—
27#13133197—41no—
28#13133193—42no—
29C4×Dic106D411yescharacteristic
30C4×D212—21no—
31#13133179—22no—
32C4⋊C4×D53—41no—
33#13133160—42no—
34C2×#11719908—21no—
35#13133151D411yescharacteristic
36#13133148—22no—
37#13133148—41no—
38C4⋊#88322—21no—
39C4⋊#88322D411yescharacteristic
40#13133146—21no—
41#13133140—41no—
42#13133139—21no—
43#13133137—22no—
44C8×Dic53—21no—
45C2×Q16×C53—21no—
46C2×Q16×C53—21no—
47C4⋊Q8×C53—21no—
48C4⋊Q8×C53—21no—
49C2×C16×C53—21no—
50#107×C53—21no—
51#107×C53D412yes—
52C4×C8×C53C2311yescharacteristic
53C2×Q8×C53—41no—
54C2×Q8×C53—41no—
55C2×D212—21no—
56C2×C4×D53—41no—
57C2×C4×D53—42no—
58C2×Dic106D811yescharacteristic
59Q16×C53—42no—
60Q16×C53—41no—
61Q16×C53—41no—
62C2×C8×C53C2×D411yescharacteristic
63C2×C8×C53C2×D411yescharacteristic
64C4⋊C4×C53—22no—
65C4⋊C4×C53—41no—
66C4⋊C4×C53—41no—
67C4⋊C4×C53—41no—
68C42×C53C2×D411yescharacteristic
69#11719921—42no—
70#11719921—41no—
71C4⋊Dic53—21no—
72C4⋊Dic53—42no—
73C4⋊Dic53D811yescharacteristic
74#11719919—21no—
75#11719919—42no—
76C4×Dic53—21no—
77C2×#88322—21no—
78C2×#88322—41no—
79D424—21no—
80D424SD1612yes—
81C16×C53—41no—
82#11719908—41no—
83C22×D53—41no—
84Q8×C53—81no—
85Q8×C53—81no—
86Q8×C53—161no—
87C2×C4×C53—42no—
88C2×C4×C53—81no—
89C2×C4×C53#12011yescharacteristic
90C2×C4×C53C2×D811yescharacteristic
91C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
92C2×Dic53—21no—
93C2×Dic53—21no—
94C2×Dic53—41no—
95D212—22no—
96D212—41no—
97C4×D53—81no—
98C4×D53—81no—
99C4×D53—162no—
100Dic106—21no—
101Dic106—21no—
102C8×C53—21no—
103C8×C53C2×SD1611yescharacteristic
104C8×C53#13711yescharacteristic
105#88322—41no—
106#88322—81no—
107#12359—531nomaximal, Sylow
108C22×C53#44711yescharacteristic
109C2×D53—42no—
110C2×D53—81no—
111C2×D53—81no—
112C4×C53—21no—
113C4×C53—82no—
114C4×C53—161no—
115C4×C53#44811yescharacteristic
116C4×C53#45411yescharacteristic
117C4×C53#50811yescharacteristic
118C4×C53#51011yescharacteristic
119Dic53—41no—
120Dic53—41no—
121Dic53—81no—
122#3380—531no—
123#2241D5311yescharacteristic, complemented
124#2208—531no—
125#1670—531no—
126#1365—531no—
127Q8⋊C16—531no—
128#1331—531no—
129C2×C53#162711yescharacteristic
130C2×C53#218911yescharacteristic
131C2×C53#219411yescharacteristic
132D53—82no—
133#554—1061no—
134C2×SD32—1061no—
135#500C2×D5311yescharacteristic
136#499—1061no—
137#494C2×D5311yescharacteristic
138#478—1062no—
139#476—1061no—
140#475—1061no—
141C8×Q8—531no—
142C4×D8—531no—
143C4×D8—1061no—
144C4⋊C16—531no—
145#358—22no—
146#358—1061no—
147C4×C16C2×D5311yescharacteristic
148#335—531no—
149#332—1061no—
150C53#1235911yescharacteristic, complemented, Sylow
151C2×Q16—21no—
152C2×Q16—21no—
153C2×D8—531no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156#125—1062no—
157C22⋊Q8—2121no—
158C22⋊Q8—2122no—
159C4×Q8—531no—
160C4×D4—1062no—
161C4×D4—1061no—
162C2×C4⋊C4—2121no—
163SD32—2122no—
164C2×C16—21no—
165C2×C16—1061no—
166#107—21no—
167#107—531no—
168#107C22⋊D5312yes—
169#106—1062no—
170#106—2121no—
171C4⋊C8—531no—
172C4⋊C8—1061no—
173C4⋊C8—1061no—
174#103—2121no—
175#102—1062no—
176#102—1061no—
177C4×C8—1061no—
178C4×C8C22×D5311yescharacteristic
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—531no—
182C2×D4—1061no—
183C22×C4—2122no—
184C22×C4—2121no—
185Q16—42no—
186Q16—41no—
187Q16—41no—
188D8—532no—
189D8—1061no—
190C2×C8—1061no—
191C2×C8—2121no—
192C2×C8D4×D5311yescharacteristic
193C2×C8C2×C22⋊D5311yescharacteristic
194C4⋊C4—22no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—531no—
199C4⋊C4—1061no—
200C4⋊C4—1061no—
201C4⋊C4—2122no—
202C4⋊C4—2122no—
203C22⋊C4—2122no—
204C22⋊C4—2121no—
205C42—1061no—
206C42D4×D5311yescharacteristic
207C16—41no—
208C16—2121no—
209C23—2121no—
210Q8—81no—
211Q8—81no—
212Q8—161no—
213Q8—1061no—
214Q8—1061no—
215D4—1062no—
216D4—2121no—
217C2×C4—42no—
218C2×C4—81no—
219C2×C4—8482no—
220C2×C4—1061no—
221C2×C4—1061no—
222C2×C4—4241no—
223C2×C4—4241no—
224C2×C4—2121no—
225C2×C4#1313318511yescharacteristic
226C2×C4D8×D5311yescharacteristic
227C2×C4#1313324111yescharacteristic
228C8—21no—
229C8—4241no—
230C8—2121no—
231C8SD16×D5311yescharacteristic
232C8#1313324611yescharacteristic
233C22—4241no—
234C22—4241no—
235C22—2122no—
236C22#1681075511yescharacteristic
237C4—21no—
238C4—82no—
239C4—161no—
240C4—4241no—
241C4—2121no—
242C4—2121no—
243C4#1681076011yescharacteristic
244C4#1681079511yescharacteristic
245C4#1681105611yescharacteristic
246C4#1681106511yescharacteristic
247C2—4242no—
248C2#2027381511yescharacteristic
249C2#2027665411yescharacteristic
250C2#2027668511yescharacteristic
subgroupG/H
centerC22#16810755
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#2241C2
socleC22×C53#447
prime radicalC2×D424D4
hypercenter#2241D53
hypocenterC53#12359
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2241

Complements

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

NH ≅ G/Ntype
3. #20274082C2semidirect
4. #20272773C2semidirect
5. #20272695C2semidirect
7. C53×#2241C2semidirect
123. #2241D53semidirect
150. C53#12359semidirect

Maximal subgroups

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

subgroupindex
#202875842
#202767572
#202740822
#202727732
#202726952
#202726772
C53×#22412
#1235953