Finite Group Databasebeta

← #27277045

Subgroups — #27277045

Order 13568.

450 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C2×#16810688C211yesmaximal, characteristic
2#20273476C212yesmaximal, complemented
3#20273055C212yesmaximal
4#20272585C211yesmaximal, characteristic, complemented
5C53×#1290C211yesmaximal, characteristic, complemented
6C22×C8×D53C412yes—
7C2×C4×D212C2211yescharacteristic
8#16810688—22no—
9#16810688—22no—
10#16810680C2212yes—
11C2×#13133129C412yes—
12C2×C4×#88322C2211yescharacteristic
13#16810375—22no—
14#16810373—22no—
15#16810346—24no—
16C53×#432C2212yes—
17C2×C4⋊C8×C53C2211yescharacteristic
18C22×D212Q811yescharacteristic
19C22×C4×D53C2×C412yes—
20C22×#88322C2×C411yescharacteristic
21C22×#88322C2×C411yescharacteristic
22C2×C8×D53—22no—
23C2×C8×D53—22no—
24C2×C8×D53—44no—
25C4×D212—22no—
26C4×D212—22no—
27C2×C4⋊Dic53Q811yescharacteristic
28C2×#11719921D412yes—
29C4⋊#88322—22no—
30C22⋊#88322—22no—
31#13133129—24no—
32#13133128—22no—
33C4×#88322D412yes—
34C4×#88322D412yes—
35C22×C8×C53C2×C412yes—
36C2×C42×C53C2311yescharacteristic
37C4⋊C8×C53—22no—
38C4⋊C8×C53—22no—
39C22⋊C8×C53—22no—
40#97×C53—22no—
41C23×D53C4⋊C412yes—
42C22×C4×C53C22×C411yescharacteristic
43C22×C4×C53C2×D411yescharacteristic
44C22×C4×C53C2×Q811yescharacteristic
45C22×Dic53C4⋊C412yes—
46C2×D212—24no—
47C2×D212—22no—
48C2×C4×D53—22no—
49C2×C4×D53—22no—
50C2×C4×D53—42no—
51C2×C4×D53—42no—
52C2×C8×C53—22no—
53C2×C8×C53—42no—
54C2×C8×C53—42no—
55C42×C53C2×D412yes—
56C42×C53#4112yes—
57#11719921—24no—
58C4⋊Dic53—22no—
59C2×#88322—22no—
60C2×#88322—41no—
61C2×#88322—42no—
62C2×#88322C22⋊C412yes—
63C2×#88322C22⋊C411yescharacteristic
64C2×#88322C22⋊C411yescharacteristic
65C8×D53—44no—
66C8×D53—44no—
67C23×C53C2×C4⋊C411yescharacteristic
68C22×D53—22no—
69C22×D53—22no—
70C22×D53—42no—
71C22×D53—42no—
72C22×D53#10814yes—
73C2×C4×C53—22no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—41no—
77C2×C4×C53C2×C22⋊C411yescharacteristic
78C2×C4×C53#11711yescharacteristic
79C2×C4×C53#12112yes—
80C2×C4×C53C22⋊Q812yes—
81C2×Dic53—22no—
82C2×Dic53—42no—
83D212—28no—
84C4×D53—44no—
85C4×D53—42no—
86C4×D53—42no—
87C8×C53—42no—
88C8×C53—82no—
89#88322—21no—
90#88322—22no—
91#88322—82no—
92#88322#10012yes—
93#8622—531nomaximal, Sylow
94C22×C53—21no—
95C22×C53—21no—
96C22×C53#38011yescharacteristic
97C22×C53C2×#10812yes—
98C2×D53—24no—
99C2×D53—44no—
100C2×D53—42no—
101C2×D53—42no—
102C2×D53—44no—
103C4×C53—21no—
104C4×C53—22no—
105C4×C53—22no—
106C4×C53C2×#10011yescharacteristic
107C4×C53#41311yescharacteristic
108Dic53—42no—
109C2×C8×D4—531no—
110#1568—532no—
111#1466—532no—
112#1290—531no—
113#1290D5311yescharacteristic, complemented
114C2×C53—22no—
115C2×C53#180011yescharacteristic
116C2×C53#181011yescharacteristic
117C2×C53#181211yescharacteristic
118D53—44no—
119C23×C8—532no—
120C2×C4×D4—531no—
121C8×D4—1062no—
122C8×D4—1062no—
123#432—532no—
124#432C2×D5312yes—
125C2×C4⋊C8C2×D5311yescharacteristic
126C2×C22⋊C8—532no—
127C2×C4×C8—531no—
128#335—1062no—
129#334—1062no—
130#323—1064no—
131C53#862211yescharacteristic, complemented, Sylow
132C22×D4—531no—
133C23×C4—532no—
134C22×C8—531no—
135C22×C8—531no—
136C22×C8—1062no—
137C22×C8—1062no—
138C22×C8—2124no—
139C22×C8C4×D5312yes—
140C4×D4—1062no—
141C4×D4—1062no—
142C2×C4⋊C4—531no—
143C2×C22⋊C4—532no—
144C2×C42C22×D5311yescharacteristic
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—1062no—
148C22⋊C8—22no—
149C22⋊C8—1064no—
150C22⋊C8—1062no—
151#97—22no—
152#97—1062no—
153C4×C8—532no—
154C4×C8—532no—
155C24—532no—
156C2×D4—1064no—
157C2×D4—1062no—
158C22×C4—532no—
159C22×C4—1062no—
160C22×C4—1062no—
161C22×C4—2122no—
162C22×C4—2122no—
163C22×C4C2×C4×D5311yescharacteristic
164C22×C4D4×D5311yescharacteristic
165C22×C4Q8×D5311yescharacteristic
166C2×C8—22no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—531no—
170C2×C8—532no—
171C2×C8—531no—
172C2×C8—1062no—
173C2×C8—2124no—
174C2×C8—2124no—
175C2×C8—2121no—
176C2×C8—2122no—
177C4⋊C4—1062no—
178C22⋊C4—1064no—
179C42#1171994512yes—
180C42D4×D5312yes—
181C23—534no—
182C23—1062no—
183C23—1062no—
184C23—2122no—
185C23—2122no—
186C23C4⋊C4×D5311yescharacteristic
187D4—1068no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—1062no—
193C2×C4—2124no—
194C2×C4—2122no—
195C2×C4—2122no—
196C2×C4—2122no—
197C2×C4C22⋊C4×D5311yescharacteristic
198C2×C4#1313317411yescharacteristic
199C2×C4#1313319012yes—
200C2×C4#1313319512yes—
201C8—42no—
202C8—82no—
203C8—532no—
204C8—1061no—
205C8—1062no—
206C8—4242no—
207C22—21no—
208C22—21no—
209C22—1064no—
210C22—2124no—
211C22—2124no—
212C22—2122no—
213C22—2122no—
214C22#1681048211yescharacteristic
215C22#108×D5312yes—
216C4—21no—
217C4—22no—
218C4—22no—
219C4—2122no—
220C4#100×D5311yescharacteristic
221C4#1681060611yescharacteristic
222C2—22no—
223C2—2124no—
224C2#2027470611yescharacteristic
225C2#2027475211yescharacteristic
226C2#2027476011yescharacteristic
subgroupG/H
centerC22#16810482
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1290C2
socleC22×C53#380
prime radicalC22×D212Q8
cosocleC2×C42×C53C23
hypercenter#1290D53
hypocenterC53#8622
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1290

Complements

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

NH ≅ G/Ntype
2. #20273476C2semidirect
4. #20272585C2semidirect
5. C53×#1290C2semidirect
113. #1290D53semidirect
131. C53#8622semidirect

Maximal subgroups

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

subgroupindex
C2×#168106882
#202734762
#202730552
#202725852
C53×#12902
#862253