Finite Group Databasebeta

← #27489991

Subgroups — #27489991

Order 13568.

1086 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1086 conjugacy classes of subgroups — too many to draw. Only its 187 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#20278985C213yesmaximal
2#20276067C214yesmaximal, complemented
3#20275878C214yesmaximal
4#20274792C213yesmaximal, complemented
5C53×#3458C211yesmaximal, characteristic, complemented
6C22×#11719921C2211yescharacteristic
7#16810558—23no—
8#16810549—23no—
9#16810514—26no—
10#16810495C2216yes—
11#16810486C2213yes—
12#16810478C2216yes—
13#16810390C22112yes—
14C53×#544—23no—
15C53×#544—23no—
16C53×#543C2214yes—
17C53×#533—26no—
18C2×#117×C53C2213yes—
19C24×D53C2311yescharacteristic
20C23×Dic53C2311yescharacteristic
21C2×#11719919—26no—
22C2×C22⋊Dic53C2316yes—
23C2×#11719921—23no—
24C2×#11719921—26no—
25C2×#11719921—23no—
26C2×#11719921—26no—
27#13133124—23no—
28#13133124—26no—
29#13133124—23no—
30C23×C4×C53C2311yescharacteristic
31C4⋊Q8×C53—26no—
32C4⋊Q8×C53—26no—
33#125×C53—212no—
34C22⋊Q8×C53—212no—
35C22⋊Q8×C53—212no—
36C4×Q8×C53—24no—
37C4×Q8×C53—24no—
38#117×C53—26no—
39#117×C53—26no—
40C2×C4⋊C4×C53D413yes—
41C2×C4⋊C4×C53D413yes—
42C2×C22⋊C4×C53C2316yes—
43C2×C42×C53D416yes—
44C24×C53C2411yescharacteristic
45C23×D53—23no—
46C23×D53—24no—
47C23×D53—42no—
48C23×D53—81no—
49C2×Q8×C53—24no—
50C2×Q8×C53—24no—
51C22×C4×C53—24no—
52C22×C4×C53C2×D413yes—
53C22×C4×C53C2×D413yes—
54C22×C4×C53C2×D416yes—
55C22×C4×C53C2×D416yes—
56C22×Dic53—26no—
57C22×Dic53—23no—
58C22×Dic53—81no—
59C4⋊C4×C53—212no—
60C4⋊C4×C53—212no—
61C4⋊C4×C53—26no—
62C4⋊C4×C53—26no—
63C22⋊C4×C53—212no—
64C42×C53—26no—
65C42×C53—26no—
66C22⋊Dic53—212no—
67#11719921—412no—
68#11719921—41no—
69#11719921—43no—
70#11719921—86no—
71#11719919—86no—
72C23×C53—41no—
73C23×C53—41no—
74C23×C53C22×D413yes—
75C23×C532−1+414yes—
76C22×D53—26no—
77C22×D53—43no—
78C22×D53—46no—
79C22×D53—41no—
80C22×D53—46no—
81C22×D53—88no—
82Q8×C53—84no—
83Q8×C53—84no—
84C2×C4×C53—26no—
85C2×C4×C53—26no—
86C2×C4×C53—26no—
87C2×C4×C53—26no—
88C2×C4×C53—46no—
89C2×C4×C53—46no—
90C2×C4×C53#12011yescharacteristic
91C2×C4×C53#12013yes—
92C2×Dic53—46no—
93C2×Dic53—46no—
94C2×Dic53—46no—
95C2×Dic53—41no—
96C2×Dic53—46no—
97#29658—531nomaximal, Sylow
98C22×C53—23no—
99C22×C53—23no—
100C22×C53—44no—
101C22×C53C2×#12011yescharacteristic
102C22×C53#53616yes—
103C2×D53—412no—
104C2×D53—42no—
105C2×D53—86no—
106C2×D53—81no—
107C2×D53—81no—
108C4×C53—21no—
109C4×C53—23no—
110C4×C53—46no—
111C4×C53—46no—
112Dic53—86no—
113Dic53—81no—
114#3458D5311yescharacteristic, complemented
115#2482—533no—
116#2062—534no—
117#2029—534no—
118#1820—533no—
119C2×C53—26no—
120C2×C53#261411yescharacteristic
121C2×C53#302511yescharacteristic
122C2×C53#302711yescharacteristic
123D53—82no—
124#544—23no—
125#544—23no—
126#543C2×D5314yes—
127#533—26no—
128C2×#117C2×D5313yes—
129C22×C22⋊C4—531no—
130#399—1063no—
131#397—1063no—
132#388—1066no—
133#385—536no—
134#381—533no—
135#379—536no—
136#342—5312no—
137C53#2965811yescharacteristic, complemented, Sylow
138C25—531no—
139C23×C4—531no—
140C23×C4C22×D5311yescharacteristic
141C4⋊Q8—26no—
142C4⋊Q8—26no—
143#125—212no—
144C22⋊Q8—212no—
145C22⋊Q8—212no—
146C4×Q8—24no—
147C4×Q8—24no—
148#117—26no—
149#117—26no—
150C2×C4⋊C4—1066no—
151C2×C4⋊C4C22⋊D5313yes—
152C2×C4⋊C4C22⋊D5313yes—
153C2×C22⋊C4—536no—
154C2×C22⋊C4—1066no—
155C2×C22⋊C4—1063no—
156C2×C22⋊C4—1063no—
157C2×C22⋊C4—1066no—
158C2×C22⋊C4C22×D5316yes—
159C2×C42C22⋊D5316yes—
160#95—1063no—
161#95—1063no—
162#95—1066no—
163C24—1063no—
164C24—1064no—
165C24—4241no—
166C24—2122no—
167C24C23×D5311yescharacteristic
168C2×Q8—24no—
169C2×Q8—24no—
170C22×C4—24no—
171C22×C4—1066no—
172C22×C4—1063no—
173C22×C4—4241no—
174C22×C4C2×C22⋊D5313yes—
175C22×C4C2×C22⋊D5313yes—
176C22×C4C2×C22⋊D5316yes—
177C22×C4C2×C22⋊D5316yes—
178C4⋊C4—212no—
179C4⋊C4—212no—
180C4⋊C4—26no—
181C4⋊C4—26no—
182C4⋊C4—4246no—
183C22⋊C4—212no—
184C22⋊C4—10612no—
185C22⋊C4—4246no—
186C22⋊C4—21212no—
187C22⋊C4—2121no—
188C22⋊C4—2123no—
189C42—26no—
190C42—26no—
191C23—41no—
192C23—41no—
193C23—1066no—
194C23—4248no—
195C23—2126no—
196C23—2123no—
197C23—2121no—
198C23—2126no—
199C23C22×C22⋊D5313yes—
200C23#1313326514yes—
201Q8—84no—
202Q8—84no—
203C2×C4—26no—
204C2×C4—26no—
205C2×C4—26no—
206C2×C4—26no—
207C2×C4—46no—
208C2×C4—46no—
209C2×C4—2126no—
210C2×C4—2126no—
211C2×C4—2126no—
212C2×C4—2121no—
213C2×C4—2126no—
214C2×C4#1313318611yescharacteristic
215C2×C4#1313318613yes—
216C22—23no—
217C22—23no—
218C22—44no—
219C22—4246no—
220C22—4241no—
221C22—4241no—
222C22—21212no—
223C22—2122no—
224C22C2×#1313318611yescharacteristic
225C22#1681119716yes—
226C4—21no—
227C4—23no—
228C4—46no—
229C4—46no—
230C4—4246no—
231C4—4241no—
232C2—26no—
233C2—4242no—
234C2#2028043411yescharacteristic
235C2#2028410711yescharacteristic
236C2#2028411711yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC24×C53C24
FrattiniC24C23×D53
FittingC53×#3458C2
socleC22×C53C2×#120
prime radicalC24×D53C23
hypercenter#3458D53
hypocenterC53#29658
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3458

Complements

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

NH ≅ G/Ntype
2. #20276067C2semidirect
4. #20274792C2semidirect
5. C53×#3458C2semidirect
114. #3458D53semidirect
137. C53#29658semidirect

Maximal subgroups

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

subgroupindex
#202789852
#202760672
#202758782
#202747922
C53×#34582
#2965853