Finite Group Databasebeta

← #27297802

Subgroups — #27297802

Order 13568.

282 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 69 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#20287604C211yesmaximal, characteristic
2#20276942C211yesmaximal, characteristic
3#20274082C211yesmaximal, characteristic
4#20272769C211yesmaximal, characteristic
5#20272693C211yesmaximal, characteristic
6#20272679C211yesmaximal, characteristic
7C53×#2209C211yesmaximal, characteristic
8#16811352—21no—
9#16811028C2211yescharacteristic
10#16811024—21no—
11#16810932—22no—
12#16810908—21no—
13#16810905—21no—
14C8×Dic106C2211yescharacteristic
15Q16×Dic53—21no—
16#16810427—22no—
17C4×#11719908C2211yescharacteristic
18#16810376C2211yescharacteristic
19#16810369—21no—
20C2×Q32×C53—21no—
21C53×#494C2211yescharacteristic
22C4×Q16×C53C2211yescharacteristic
23C53×C4⋊C16C2211yescharacteristic
24C53×#358—21no—
25#13133217—21no—
26#13133217—41no—
27#13133208—22no—
28#13133208—42no—
29Q8×Dic53—22no—
30C4×Dic106—41no—
31C4×Dic106D411yescharacteristic
32C2×#11719908—21no—
33#13133151—21no—
34#13133151D412yes—
35#13133148—22no—
36#13133148—41no—
37C4⋊#88322—21no—
38C4⋊#88322D411yescharacteristic
39#13133146—21no—
40#13133141—22no—
41#13133140—41no—
42C8×Dic53—21no—
43C2×Q16×C53—21no—
44C2×Q16×C53—21no—
45C2×Q16×C53D411yescharacteristic
46C4⋊Q8×C53—21no—
47C4×Q8×C53—21no—
48Q32×C53—42no—
49C2×C16×C53—21no—
50#107×C53D411yescharacteristic
51#103×C53—21no—
52C4×C8×C53C2311yescharacteristic
53C2×Q8×C53—21no—
54C2×Q8×C53—41no—
55C2×Dic106—41no—
56C2×Dic106D811yescharacteristic
57Q16×C53—21no—
58Q16×C53—42no—
59Q16×C53—41no—
60Q16×C53—41no—
61Q16×C53SD1612yes—
62C2×C8×C53C2×D411yescharacteristic
63C2×C8×C53C2×D411yescharacteristic
64C4⋊C4×C53—21no—
65C4⋊C4×C53—41no—
66C4⋊C4×C53—41no—
67C4⋊C4×C53—41no—
68C42×C53—41no—
69C42×C53C2×D411yescharacteristic
70C4⋊Dic53—22no—
71C4⋊Dic53—42no—
72C4⋊Dic53—42no—
73C4⋊Dic53—41no—
74C4⋊Dic53D811yescharacteristic
75#11719919—21no—
76#11719919—42no—
77C4×Dic53—21no—
78C4×Dic53—42no—
79C2×#88322—21no—
80C2×#88322—41no—
81C16×C53—41no—
82#11719908—41no—
83Q8×C53—22no—
84Q8×C53—41no—
85Q8×C53—81no—
86Q8×C53—81no—
87C2×C4×C53—41no—
88C2×C4×C53—41no—
89C2×C4×C53—81no—
90C2×C4×C53#12011yescharacteristic
91C2×C4×C53C2×D811yescharacteristic
92C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
93C2×Dic53—21no—
94C2×Dic53—21no—
95C2×Dic53—42no—
96Dic106—21no—
97Dic106—21no—
98Dic106—81no—
99Dic106—81no—
100C8×C53—21no—
101C8×C53C2×SD1611yescharacteristic
102C8×C53#13711yescharacteristic
103#88322—41no—
104#88322—81no—
105#12357—531nomaximal, Sylow
106C22×C53#44711yescharacteristic
107C4×C53—21no—
108C4×C53—42no—
109C4×C53—81no—
110C4×C53—161no—
111C4×C53#44811yescharacteristic
112C4×C53#45411yescharacteristic
113C4×C53#50811yescharacteristic
114C4×C53#51011yescharacteristic
115Dic53—41no—
116Dic53—41no—
117Dic53—82no—
118#3383—531no—
119#2241—531no—
120#2209D5311yescharacteristic
121#1670—531no—
122#1364—531no—
123Q8⋊C16—531no—
124#1332—531no—
125C2×C53#162711yescharacteristic
126C2×C53#218911yescharacteristic
127C2×C53#219511yescharacteristic
128#557—1061no—
129C2×Q32—21no—
130#500—531no—
131#499—1061no—
132#494C2×D5311yescharacteristic
133#479—1062no—
134#475—1061no—
135#475—1061no—
136C8×Q8—531no—
137C4×Q16—1061no—
138C4×Q16C2×D5311yescharacteristic
139C4⋊C16C2×D5311yescharacteristic
140#358—21no—
141#358—1062no—
142C4×C16—531no—
143#335—531no—
144#332—1061no—
145C53#1235711yescharacteristic, complemented, Sylow
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16C22⋊D5311yescharacteristic
149C4⋊Q8—21no—
150C4⋊Q8—1061no—
151C4⋊Q8—2121no—
152#125—1062no—
153#125—2122no—
154C4×Q8—21no—
155C4×Q8—531no—
156C4×Q8—1062no—
157C4×Q8—2121no—
158Q32—42no—
159C2×C16—21no—
160C2×C16—1061no—
161#107—532no—
162#107—1061no—
163#107C22⋊D5311yescharacteristic
164#106—1062no—
165#106—2121no—
166C4⋊C8—531no—
167C4⋊C8—1061no—
168C4⋊C8—1061no—
169#103—21no—
170#103—1062no—
171#103—2121no—
172C4×C8—1061no—
173C4×C8C22×D5311yescharacteristic
174C2×Q8—21no—
175C2×Q8—41no—
176C2×Q8—531no—
177C2×Q8—2121no—
178Q16—21no—
179Q16—42no—
180Q16—41no—
181Q16—41no—
182Q16#1171992312yes—
183C2×C8—1061no—
184C2×C8—2121no—
185C2×C8D4×D5311yescharacteristic
186C2×C8C2×C22⋊D5311yescharacteristic
187C4⋊C4—21no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—531no—
192C4⋊C4—1061no—
193C4⋊C4—1062no—
194C4⋊C4—2122no—
195C4⋊C4—2122no—
196C4⋊C4—2122no—
197C4⋊C4—2121no—
198C42—41no—
199C42—1061no—
200C42—2122no—
201C42D4×D5311yescharacteristic
202C16—41no—
203C16—2121no—
204Q8—22no—
205Q8—41no—
206Q8—81no—
207Q8—81no—
208Q8—1061no—
209Q8—1061no—
210Q8—4241no—
211Q8—4241no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—81no—
215C2×C4—1061no—
216C2×C4—1061no—
217C2×C4—2122no—
218C2×C4#1313318511yescharacteristic
219C2×C4D8×D5311yescharacteristic
220C2×C4#1313324111yescharacteristic
221C8—21no—
222C8—4241no—
223C8—2121no—
224C8C2×#1171992311yescharacteristic
225C8#1313324711yescharacteristic
226C22#1681075511yescharacteristic
227C4—21no—
228C4—42no—
229C4—81no—
230C4—161no—
231C4—4242no—
232C4—2121no—
233C4—2121no—
234C4#1681076111yescharacteristic
235C4#1681079411yescharacteristic
236C4#1681105611yescharacteristic
237C4#1681106511yescharacteristic
238C2#2027381711yescharacteristic
239C2#2027665411yescharacteristic
240C2#2027669011yescharacteristic
subgroupG/H
centerC22#16810755
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#2209C2
socle, prime radicalC22×C53#447
hypercenter#2209D53
hypocenterC53#12357
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2209

Complements

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

NH ≅ G/Ntype
145. C53#12357semidirect

Maximal subgroups

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

subgroupindex
#202876042
#202769422
#202740822
#202727692
#202726932
#202726792
C53×#22092
#1235753