Finite Group Databasebeta

← #27278466

Subgroups — #27278466

Order 13568.

428 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 115 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#20275685C211yesmaximal, characteristic
2#20274884C212yesmaximal
3C2×#16810365C211yesmaximal, characteristic
4#20272596C212yesmaximal
5C53×#2382C211yesmaximal, characteristic
6C2×#13133150—22no—
7C2×#13133148—22no—
8C2×#13133128C2211yescharacteristic
9C2×#13133128C2212yes—
10#16810493C2211yescharacteristic
11#16810385—22no—
12#16810365C414yescomplemented
13C2×#125×C53C2211yescharacteristic
14C53×#389—22no—
15C53×#387—22no—
16C53×#387—22no—
17C53×#382C2212yes—
18C22×#88322—22no—
19C22×#88322—21no—
20C2×C4⋊Dic53—21no—
21C2×C4⋊Dic53—22no—
22#13133150—44no—
23#13133148—44no—
24#13133128—22no—
25#13133128—22no—
26#13133128C2×C414yes—
27#125×C53—22no—
28#125×C53C2×C412yes—
29#125×C53Q812yes—
30C2×C4⋊C4×C53—22no—
31C2×C4⋊C4×C53—22no—
32C2×C4⋊C4×C53D412yes—
33C2×C4⋊C4×C53D412yes—
34C2×C4⋊C4×C53D412yes—
35C2×C42×C53—22no—
36C2×C42×C53C2311yescharacteristic
37#95×C53—22no—
38#95×C53—22no—
39#95×C53—22no—
40#95×C53—22no—
41C22×C4×C53—22no—
42C22×C4×C53—22no—
43C22×C4×C53—22no—
44C22×C4×C53C2×D411yescharacteristic
45C22×C4×C53C2×D412yes—
46C22×Dic53—41no—
47C4⋊C4×C53—24no—
48C4⋊C4×C53—24no—
49C4⋊C4×C53—22no—
50C4⋊C4×C53—42no—
51C4⋊C4×C53—42no—
52C4⋊C4×C53—82no—
53C4⋊C4×C53C4⋊C414yes—
54C42×C53—82no—
55C42×C53C22×C411yescharacteristic
56C42×C53C2×Q811yescharacteristic
57C42×C53#4112yes—
58C4⋊Dic53—42no—
59C4⋊Dic53—42no—
60C4⋊Dic53—42no—
61C2×#88322—24no—
62C2×#88322—24no—
63C2×#88322—24no—
64C2×#88322—22no—
65C2×#88322—41no—
66C2×#88322—41no—
67C23×C53#12011yescharacteristic
68C2×C4×C53—22no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—22no—
72C2×C4×C53—24no—
73C2×C4×C53—42no—
74C2×C4×C53—42no—
75C2×C4×C53—42no—
76C2×C4×C53—42no—
77C2×C4×C53—42no—
78C2×C4×C53—42no—
79C2×C4×C53—42no—
80C2×C4×C53—42no—
81C2×C4×C53C2×C4⋊C411yescharacteristic
82C2×C4×C53C4×D412yes—
83C2×C4×C53C22⋊Q812yes—
84C2×C4×C53#12311yescharacteristic
85C2×Dic53—82no—
86C2×Dic53—81no—
87#88322—28no—
88#88322—42no—
89#8913—531nomaximal, Sylow
90C22×C53#38511yescharacteristic
91C22×C53#45312yes—
92C22×C53#45411yescharacteristic
93C22×C53#45611yescharacteristic
94C22×C53#45612yes—
95C4×C53—21no—
96C4×C53—21no—
97C4×C53—22no—
98C4×C53—22no—
99C4×C53—44no—
100C4×C53—82no—
101C4×C53—82no—
102Dic53—161no—
103#2382D5311yescharacteristic
104#1992—531no—
105#1847—532no—
106C2×#330—531no—
107#1295—532no—
108C2×C53#165611yescharacteristic
109C2×C53#166012yes—
110C2×C53#166311yescharacteristic
111C2×C53#185711yescharacteristic
112C2×C53#185912yes—
113C2×#125C2×D5311yescharacteristic
114C2×#107—1062no—
115C2×#106—1062no—
116C2×#97—532no—
117C2×#97—531no—
118#389—22no—
119#387—22no—
120#387—22no—
121#384—531no—
122#382C2×D5312yes—
123#340—1062no—
124#330—534no—
125C53#891311yescharacteristic, complemented, Sylow
126C22×C8—1062no—
127C22×C8—1061no—
128#125—22no—
129#125Dic10612yes—
130#125C4×D5312yes—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—1061no—
134C2×C4⋊C4—1062no—
135C2×C4⋊C4C22⋊D5312yes—
136C2×C4⋊C4C22⋊D5312yes—
137C2×C4⋊C4C22⋊D5312yes—
138C2×C42—22no—
139C2×C42C22×D5311yescharacteristic
140#107—2124no—
141#106—2124no—
142#97—534no—
143#97—1062no—
144#97—1062no—
145#95—22no—
146#95—22no—
147#95—22no—
148#95—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—2121no—
153C22×C4C2×C22⋊D5311yescharacteristic
154C22×C4C2×C22⋊D5312yes—
155C2×C8—1064no—
156C2×C8—1064no—
157C2×C8—1064no—
158C2×C8—1062no—
159C2×C8—2121no—
160C2×C8—2121no—
161C4⋊C4—24no—
162C4⋊C4—24no—
163C4⋊C4—22no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—82no—
167C4⋊C4—2122no—
168C4⋊C4—2122no—
169C4⋊C4—2122no—
170C4⋊C4#1171991914yes—
171C42—82no—
172C42C2×Dic10611yescharacteristic
173C42C2×C4×D5311yescharacteristic
174C42#1171994512yes—
175C23#1313318611yescharacteristic
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—4242no—
190C2×C4—4241no—
191C2×C4C2×#1171991911yescharacteristic
192C2×C4C4×C22⋊D5312yes—
193C2×C4#1313319612yes—
194C2×C4#1313320011yescharacteristic
195C8—1068no—
196C8—2122no—
197C22#1681049611yescharacteristic
198C22#1681079112yes—
199C22#1681079611yescharacteristic
200C22#1681080611yescharacteristic
201C22#1681080612yes—
202C4—21no—
203C4—21no—
204C4—22no—
205C4—22no—
206C4—44no—
207C4—82no—
208C4—82no—
209C4—8481no—
210C2#2027400811yescharacteristic
211C2#2027402612yes—
212C2#2027404311yescharacteristic
213C2#2027493011yescharacteristic
214C2#2027494112yes—
subgroupG/H
centerC23#13133186
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2382C2
socle, prime radicalC23×C53#120
cosocleC2×C42×C53C23
hypercenter#2382D53
hypocenterC53#8913
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2382

Complements

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

NH ≅ G/Ntype
12. #16810365C4semidirect
125. C53#8913semidirect

Maximal subgroups

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

subgroupindex
#202756852
#202748842
C2×#168103652
#202725962
C53×#23822
#891353