Finite Group Databasebeta

← #27296872

Subgroups — #27296872

Order 13568.

380 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 63 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#20275876C211yesmaximal, characteristic
2#20273722C212yesmaximal
3#20272721C212yesmaximal
4#20272649C211yesmaximal, characteristic
5C53×#2734C211yesmaximal, characteristic
6#16811254C2211yescharacteristic
7#16810953—22no—
8#16810632—21no—
9#16810392C2211yescharacteristic
10#16810383—21no—
11#16810359—22no—
12#16810349C2212yes—
13C2×C4⋊Q8×C53—21no—
14C53×#398C2212yes—
15C53×#395—22no—
16C53×#391C2211yescharacteristic
17C53×#389—21no—
18C53×#387—22no—
19C2×#11719917—21no—
20#13133216—41no—
21#13133208—41no—
22#13133196—22no—
23#13133176—21no—
24#13133148—42no—
25#13133141—41no—
26#13133141—41no—
27#13133140—42no—
28C22⋊#88322—22no—
29C22×Q8×C53—21no—
30C22×Q8×C53D411yescharacteristic
31C4⋊Q8×C53—42no—
32C4⋊Q8×C53—42no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—22no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—22no—
37C2×C4⋊C4×C53D412yes—
38C2×C4⋊C4×C53C2311yescharacteristic
39C2×C42×C53—21no—
40C2×C42×C53D411yescharacteristic
41#95×C53—22no—
42#95×C53—22no—
43#95×C53—22no—
44#95×C53D412yes—
45C2×Q8×C53—21no—
46C2×Q8×C53—42no—
47C2×Q8×C53—41no—
48C2×Q8×C53—41no—
49C2×Q8×C53—81no—
50C2×Q8×C53—81no—
51C2×Q8×C53SD1612yes—
52C22×C4×C53—21no—
53C22×C4×C53—22no—
54C22×C4×C53—21no—
55C22×C4×C53—22no—
56C22×C4×C53C2×D411yescharacteristic
57C22×C4×C53C2×D412yes—
58C2×Dic106—22no—
59C4⋊C4×C53—24no—
60C4⋊C4×C53—41no—
61C4⋊C4×C53—44no—
62C4⋊C4×C53—42no—
63C4⋊C4×C53—42no—
64C4⋊C4×C53—42no—
65C4⋊C4×C53—81no—
66C42×C53—42no—
67C42×C53—41no—
68C22⋊Dic53—41no—
69C4⋊Dic53—21no—
70C4⋊Dic53—21no—
71C4⋊Dic53—22no—
72#11719919—42no—
73C4×Dic53—41no—
74#11719917—41no—
75#11719917—41no—
76C2×#88322—42no—
77C2×#88322—41no—
78C23×C53#12011yescharacteristic
79Q8×C53—42no—
80Q8×C53—81no—
81Q8×C53—84no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—22no—
85C2×C4×C53—24no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—41no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53—42no—
98C2×C4×C53C2×SD1611yescharacteristic
99C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
100C2×Dic53—41no—
101C2×Dic53—41no—
102Dic106—82no—
103#88322—81no—
104#88322—82no—
105#12213—531nomaximal, Sylow
106C22×C53—21no—
107C22×C53—21no—
108C22×C53#45011yescharacteristic
109C22×C53#45511yescharacteristic
110C22×C53B(4,𝔽2)11yescharacteristic
111C4×C53—21no—
112C4×C53—21no—
113C4×C53—44no—
114C4×C53—42no—
115C4×C53—42no—
116C4×C53—42no—
117C4×C53—82no—
118Dic53—81no—
119Dic53—81no—
120#2734D5311yescharacteristic
121#2028—531no—
122#1611—532no—
123#1346—532no—
124#1319—531no—
125C2×C53—21no—
126C2×C53—21no—
127C2×C53#160111yescharacteristic
128C2×C53#220111yescharacteristic
129C2×C53#220511yescharacteristic
130#544—531no—
131C2×C4⋊Q8—21no—
132#483—1062no—
133#419—1061no—
134#398C2×D5312yes—
135#395—22no—
136#391C2×D5311yescharacteristic
137#389—21no—
138#387—22no—
139#343—531no—
140#339—1061no—
141#328—1062no—
142#324—532no—
143C53#1221311yescharacteristic, complemented, Sylow
144C22×Q8—21no—
145C22×Q8C22⋊D5311yescharacteristic
146C2×M16—1061no—
147C4⋊Q8—42no—
148C4⋊Q8—42no—
149C4⋊Q8—2121no—
150#125—2121no—
151C22⋊Q8—1062no—
152#117—1061no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C22⋊D5312yes—
158C2×C4⋊C4C22×D5311yescharacteristic
159C2×C42—21no—
160C2×C42C22⋊D5311yescharacteristic
161#106—2122no—
162#103—2122no—
163#103—2121no—
164#103—2121no—
165C22⋊C8—1062no—
166#95—22no—
167#95—22no—
168#95—22no—
169#95C22⋊D5312yes—
170C2×Q8—21no—
171C2×Q8—42no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—81no—
175C2×Q8—81no—
176C2×Q8—1062no—
177C2×Q8#1171992312yes—
178C22×C4—21no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4C2×C22⋊D5311yescharacteristic
183C22×C4C2×C22⋊D5312yes—
184M16—2121no—
185M16—2121no—
186C2×C8—2122no—
187C2×C8—2121no—
188C4⋊C4—24no—
189C4⋊C4—41no—
190C4⋊C4—42no—
191C4⋊C4—44no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—81no—
195C4⋊C4—1061no—
196C4⋊C4—1061no—
197C4⋊C4—1062no—
198C4⋊C4—2122no—
199C22⋊C4—2121no—
200C42—41no—
201C42—42no—
202C42—2121no—
203C23#1313318611yescharacteristic
204Q8—42no—
205Q8—81no—
206Q8—84no—
207Q8—4242no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—2121no—
225C2×C4—2121no—
226C2×C4C2×#1171992311yescharacteristic
227C2×C4#1313324311yescharacteristic
228C8—4241no—
229C8—4242no—
230C22—21no—
231C22—21no—
232C22#1681077411yescharacteristic
233C22#1681080111yescharacteristic
234C22#1681081011yescharacteristic
235C4—21no—
236C4—21no—
237C4—44no—
238C4—42no—
239C4—42no—
240C4—42no—
241C4—82no—
242C4—4241no—
243C4—4241no—
244C2—21no—
245C2—21no—
246C2#2027366711yescharacteristic
247C2#2027672811yescharacteristic
248C2#2027674511yescharacteristic
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2734C2
socleC22×C53B(4,𝔽2)
prime radicalC23×C53#120
hypercenter#2734D53
hypocenterC53#12213
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2734

Complements

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

NH ≅ G/Ntype
143. C53#12213semidirect

Maximal subgroups

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

subgroupindex
#202758762
#202737222
#202727212
#202726492
C53×#27342
#1221353