Finite Group Databasebeta

← #27277547

Subgroups — #27277547

Order 13568.

404 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 99 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#20277212C211yesmaximal, characteristic
2#20273027C212yesmaximal
3#20272548C211yesmaximal, characteristic, complemented
4#20272543C212yesmaximal, complemented
5C53×#1276C211yesmaximal, characteristic, complemented
6C2×#13133174C2211yescharacteristic
7#16810676C2212yes—
8#16810514—22no—
9#16810507—22no—
10C4×#11719919C412yes—
11C4×#11719921C412yes—
12#16810469C2211yescharacteristic
13#16810348—22no—
14#16810346—22no—
15C53×#431C2212yes—
16C43×C53C2211yescharacteristic
17C22×C4×D53C2×C411yescharacteristic
18#13133174—22no—
19#13133174—21no—
20#13133174—21no—
21C2×#11719919—22no—
22C2×#11719919D412yes—
23C2×#11719921—22no—
24C2×#11719921D412yes—
25C2×C4×Dic53C2×C412yes—
26C2×C4×Dic53C2×C411yescharacteristic
27C4⋊#88322—22no—
28C22⋊#88322—22no—
29C4×#88322—22no—
30#13133124—22no—
31#13133124—22no—
32#13133124—22no—
33C2×C42×C53—22no—
34C2×C42×C53C2×C412yes—
35C2×C42×C53C2311yescharacteristic
36C4⋊C8×C53—22no—
37C22⋊C8×C53—22no—
38C4×C8×C53—22no—
39C23×D53C22⋊C411yescharacteristic
40C22×C4×C53—22no—
41C22×C4×C53C22×C411yescharacteristic
42C22×C4×C53C2×D412yes—
43C22×Dic53—22no—
44C22×Dic53C22⋊C411yescharacteristic
45C22×Dic53C22⋊C412yes—
46C2×C4×D53—21no—
47C2×C4×D53—21no—
48C2×C4×D53—22no—
49C2×C4×D53—22no—
50C2×C8×C53—42no—
51C42×C53—22no—
52C42×C53—22no—
53C42×C53—44no—
54C42×C53#4112yes—
55C42×C53#4111yescharacteristic
56C42×C53#4111yescharacteristic
57#11719921—22no—
58#11719921—22no—
59#11719921—44no—
60#11719919—22no—
61#11719919—22no—
62#11719919—44no—
63C4×Dic53—21no—
64C4×Dic53—21no—
65C4×Dic53—22no—
66C4×Dic53—22no—
67C2×#88322—42no—
68C23×C53C2×C22⋊C411yescharacteristic
69C22×D53—21no—
70C22×D53—22no—
71C22×D53—42no—
72C2×C4×C53—24no—
73C2×C4×C53—42no—
74C2×C4×C53—42no—
75C2×C4×C53—42no—
76C2×C4×C53#11711yescharacteristic
77C2×C4×C53#11711yescharacteristic
78C2×C4×C53#12312yes—
79C2×C4×C53#12312yes—
80C2×C4×C53#13112yes—
81C2×C4×C53#13112yes—
82C2×Dic53—21no—
83C2×Dic53—22no—
84C2×Dic53—22no—
85C2×Dic53—42no—
86C2×Dic53—42no—
87C2×Dic53—42no—
88C2×Dic53—42no—
89C4×D53—41no—
90C4×D53—42no—
91C4×D53—41no—
92C8×C53—82no—
93#88322—82no—
94#8733—531nomaximal, Sylow
95C22×C53—22no—
96C22×C53#38111yescharacteristic
97C22×C53#40812yes—
98C2×D53—41no—
99C2×D53—42no—
100C2×D53—82no—
101C4×C53—22no—
102C4×C53—22no—
103C4×C53—44no—
104C4×C53#43311yescharacteristic
105C4×C53#43312yes—
106C4×C53#43311yescharacteristic
107Dic53—41no—
108Dic53—42no—
109Dic53—82no—
110#2298—531no—
111#1458—532no—
112#1276—531no—
113#1276D5311yescharacteristic, complemented
114#1274—532no—
115C2×C53—22no—
116C2×C53#181811yescharacteristic
117C2×C53#182712yes—
118D53—81no—
119C2×#117—531no—
120#431—532no—
121#431C2×D5312yes—
122#388—1062no—
123#387—1062no—
124C4×C4⋊C4—532no—
125C4×C22⋊C4—532no—
126#376—531no—
127C43C2×D5311yescharacteristic
128#324—1062no—
129#323—1062no—
130C53#873311yescharacteristic, complemented, Sylow
131C23×C4—531no—
132#117—1062no—
133#117—1061no—
134#117—1061no—
135C2×C4⋊C4—532no—
136C2×C4⋊C4—1062no—
137C2×C22⋊C4—532no—
138C2×C22⋊C4—1062no—
139C2×C42—22no—
140C2×C42—531no—
141C2×C42—532no—
142C2×C42C4×D5312yes—
143C2×C42C22×D5311yescharacteristic
144C4⋊C8—22no—
145C4⋊C8—1062no—
146C22⋊C8—22no—
147C22⋊C8—1062no—
148C4×C8—22no—
149C4×C8—1062no—
150#95—1062no—
151#95—1062no—
152#95—1062no—
153C24—531no—
154C22×C4—22no—
155C22×C4—531no—
156C22×C4—532no—
157C22×C4—1061no—
158C22×C4—1061no—
159C22×C4—1062no—
160C22×C4—1062no—
161C22×C4—1062no—
162C22×C4C2×C4×D5311yescharacteristic
163C22×C4D4×D5312yes—
164C2×C8—42no—
165C2×C8—2122no—
166C4⋊C4—1062no—
167C4⋊C4—1062no—
168C4⋊C4—2124no—
169C22⋊C4—1062no—
170C22⋊C4—1062no—
171C22⋊C4—2124no—
172C42—22no—
173C42—22no—
174C42—44no—
175C42—1061no—
176C42—1061no—
177C42—1062no—
178C42—1062no—
179C42#1171994512yes—
180C42#1171994711yescharacteristic
181C42#1171994911yescharacteristic
182C23—1062no—
183C23—1061no—
184C23—2122no—
185C23C22⋊C4×D5311yescharacteristic
186C2×C4—24no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—1062no—
191C2×C4—1062no—
192C2×C4—1061no—
193C2×C4—2122no—
194C2×C4—2122no—
195C2×C4—2122no—
196C2×C4—2121no—
197C2×C4—2122no—
198C2×C4—2121no—
199C2×C4—2122no—
200C2×C4#1313317411yescharacteristic
201C2×C4#1313317511yescharacteristic
202C2×C4#1313319912yes—
203C2×C4#1313320212yes—
204C2×C4#1313322312yes—
205C2×C4#1313322512yes—
206C8—82no—
207C8—4242no—
208C22—22no—
209C22—4242no—
210C22—2121no—
211C22—2122no—
212C22#1681048511yescharacteristic
213C22#1681058812yes—
214C4—22no—
215C4—22no—
216C4—44no—
217C4—4242no—
218C4—2121no—
219C4—2122no—
220C4#1681068211yescharacteristic
221C4#1681068312yes—
222C4#1681068511yescharacteristic
223C2—22no—
224C2—4241no—
225C2#2027478511yescharacteristic
226C2#2027481812yes—
subgroupG/H
centerC2×C4#13133175
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1276C2
socleC22×C53#381
prime radicalC23×D53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1276D53
hypocenterC53#8733
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
3. #20272548C2semidirect
4. #20272543C2semidirect
5. C53×#1276C2semidirect
113. #1276D53semidirect
130. C53#8733semidirect

Maximal subgroups

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

subgroupindex
#202772122
#202730272
#202725482
#202725432
C53×#12762
#873353