Finite Group Databasebeta

← #27297806

Subgroups — #27297806

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#20276943C211yesmaximal, characteristic
3#20276765C211yesmaximal, characteristic
4#20274086C211yesmaximal, characteristic
5#20272771C211yesmaximal, characteristic
6#20272695C211yesmaximal, characteristic
7C53×#1332C211yesmaximal, characteristic
8#16811352—21no—
9C2×#13133163—21no—
10#16811028C2211yescharacteristic
11#16811024—21no—
12#16811003C2211yescharacteristic
13#16810932—22no—
14#16810909—21no—
15#16810908—21no—
16C8×Dic106C2211yescharacteristic
17Q16×Dic53—21no—
18C4⋊#11719908C2211yescharacteristic
19#16810428—22no—
20#16810426—21no—
21#16810368—21no—
22C4×Q16×C53C2211yescharacteristic
23C4×C16×C53C2211yescharacteristic
24C53×#335C2211yescharacteristic
25C2×Dic212—21no—
26C2×Dic212—21no—
27#13133217—21no—
28#13133217—21no—
29#13133217—41no—
30#13133208—22no—
31#13133208—42no—
32Q8×Dic53—22no—
33C4×Dic106—41no—
34C4×Dic106D411yescharacteristic
35#13133163—42no—
36C2×#11719908—21no—
37#13133151—21no—
38#13133151D412yes—
39#13133149—41no—
40#13133148—22no—
41C4⋊#88322D411yescharacteristic
42#13133146—21no—
43#13133142—41no—
44#13133141—22no—
45C8×Dic53—21no—
46C2×Q16×C53D411yescharacteristic
47C4×Q8×C53—21no—
48C2×C16×C53—21no—
49#107×C53D411yescharacteristic
50C4⋊C8×C53—21no—
51#103×C53—21no—
52C4×C8×C53C2311yescharacteristic
53C2×Q8×C53—21no—
54C2×Dic106—41no—
55C2×Dic106—41no—
56C2×Dic106D811yescharacteristic
57Q16×C53—21no—
58Q16×C53SD1612yes—
59C2×C8×C53—41no—
60C2×C8×C53C2×D411yescharacteristic
61C2×C8×C53C2×D411yescharacteristic
62C4⋊C4×C53—21no—
63C4⋊C4×C53—41no—
64C42×C53—41no—
65C42×C53C2×D411yescharacteristic
66C4⋊Dic53—22no—
67C4⋊Dic53—42no—
68C4⋊Dic53—42no—
69C4⋊Dic53—41no—
70C4⋊Dic53—41no—
71C4⋊Dic53—41no—
72C4⋊Dic53D811yescharacteristic
73#11719919—21no—
74#11719919—42no—
75C4×Dic53—21no—
76C4×Dic53—42no—
77C2×#88322—21no—
78Dic212—42no—
79Dic212—41no—
80Dic212—41no—
81C16×C53—41no—
82#11719908—41no—
83Q8×C53—22no—
84Q8×C53—41no—
85C2×C4×C53—41no—
86C2×C4×C53—41no—
87C2×C4×C53#12011yescharacteristic
88C2×C4×C53C2×D811yescharacteristic
89C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
90C2×Dic53—21no—
91C2×Dic53—21no—
92C2×Dic53—42no—
93C2×Dic53—81no—
94Dic106—21no—
95Dic106—21no—
96Dic106—81no—
97Dic106—81no—
98Dic106—81no—
99Dic106—81no—
100C8×C53—21no—
101C8×C53—81no—
102C8×C53C2×SD1611yescharacteristic
103C8×C53#13711yescharacteristic
104#88322—41no—
105#12357—531nomaximal, Sylow
106C22×C53#44711yescharacteristic
107C4×C53—21no—
108C4×C53—42no—
109C4×C53—81no—
110C4×C53#44811yescharacteristic
111C4×C53#45411yescharacteristic
112C4×C53#50811yescharacteristic
113C4×C53#51011yescharacteristic
114Dic53—41no—
115Dic53—41no—
116Dic53—82no—
117Dic53—161no—
118#3383—531no—
119#2241—531no—
120#2209—531no—
121#1670—531no—
122#1364—531no—
123Q8⋊C16—531no—
124#1332D5311yescharacteristic
125C2×C53#162711yescharacteristic
126C2×C53#218911yescharacteristic
127C2×C53#219511yescharacteristic
128#557—1061no—
129C2×Q32—1061no—
130#500—531no—
131#499—1061no—
132#494—531no—
133#479—1062no—
134#475—1061no—
135#475—1061no—
136C8×Q8—531no—
137C4×Q16—1061no—
138C4×Q16C2×D5311yescharacteristic
139C4⋊C16—531no—
140#358—1062no—
141#358—1061no—
142C4×C16C2×D5311yescharacteristic
143#335C2×D5311yescharacteristic
144#332—1061no—
145C53#1235711yescharacteristic, complemented, Sylow
146C2×Q16—1061no—
147C2×Q16—1061no—
148C2×Q16D21211yescharacteristic
149C4⋊Q8—1061no—
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—2122no—
159C2×C16—21no—
160C2×C16—1061no—
161#107—532no—
162#107—1061no—
163#107D21211yescharacteristic
164#106—1062no—
165#106—2121no—
166C4⋊C8—21no—
167C4⋊C8—531no—
168C4⋊C8—1061no—
169#103—21no—
170#103—1062no—
171#103—2121no—
172C4×C8—1061no—
173C4×C8C22×D5311yescharacteristic
174C2×Q8—21no—
175C2×Q8—531no—
176C2×Q8—2121no—
177C2×Q8—2121no—
178Q16—21no—
179Q16—2122no—
180Q16—2121no—
181Q16—2121no—
182Q16#1171991312yes—
183C2×C8—41no—
184C2×C8—1061no—
185C2×C8C2×D21211yescharacteristic
186C2×C8D4×D5311yescharacteristic
187C4⋊C4—21no—
188C4⋊C4—41no—
189C4⋊C4—531no—
190C4⋊C4—1061no—
191C4⋊C4—1062no—
192C4⋊C4—2122no—
193C4⋊C4—2122no—
194C4⋊C4—2122no—
195C4⋊C4—2121no—
196C4⋊C4—2121no—
197C4⋊C4—2121no—
198C42—41no—
199C42—1061no—
200C42—2122no—
201C42D4×D5311yescharacteristic
202C16—41no—
203C16—2121no—
204Q8—22no—
205Q8—41no—
206Q8—1061no—
207Q8—1061no—
208Q8—4241no—
209Q8—4241no—
210Q8—4241no—
211Q8—4241no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—1061no—
215C2×C4—1061no—
216C2×C4—4241no—
217C2×C4—2122no—
218C2×C4#1313318711yescharacteristic
219C2×C4D8×D5311yescharacteristic
220C2×C4#1313324411yescharacteristic
221C8—21no—
222C8—81no—
223C8—2121no—
224C8C2×#1171991311yescharacteristic
225C8#1313325011yescharacteristic
226C22#1681075811yescharacteristic
227C4—21no—
228C4—42no—
229C4—81no—
230C4—8481no—
231C4—4242no—
232C4—2121no—
233C4—2121no—
234C4#1681076411yescharacteristic
235C4#1681079711yescharacteristic
236C4#1681105411yescharacteristic
237C4#1681106511yescharacteristic
238C2#2027382111yescharacteristic
239C2#2027665811yescharacteristic
240C2#2027669411yescharacteristic
subgroupG/H
centerC22#16810758
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#1332C2
socle, prime radicalC22×C53#447
hypercenter#1332D53
hypocenterC53#12357
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1332

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
#202769432
#202767652
#202740862
#202727712
#202726952
C53×#13322
#1235753