Finite Group Databasebeta

← #27285382

Subgroups — #27285382

Order 13568.

552 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 95 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×#16810987C212yesmaximal
2#20282783C211yesmaximal, characteristic, complemented
3C2×#16810364C212yesmaximal, complemented
4#20272596C211yesmaximal, characteristic, complemented
5C53×#1961C211yesmaximal, characteristic
6C2×#13133209C2212yes—
7#16810987—24no—
8#16810987—24no—
9C2×#13133138—22no—
10C2×#13133138—22no—
11C2×#13133128C2212yes—
12#16810563—22no—
13#16810555—22no—
14#16810553—22no—
15#16810364—24no—
16C2×#127×C53C2211yescharacteristic
17C2×#102×C53—22no—
18C2×#97×C53C2211yescharacteristic
19C53×#382C2211yescharacteristic
20C22×#88322—22no—
21#13133209—22no—
22#13133209—22no—
23#13133209—22no—
24#13133209—22no—
25C2×C4⋊Dic53—22no—
26C2×C4⋊Dic53D412yes—
27C2×C4⋊Dic53D412yes—
28C2×#11719919—22no—
29C2×#11719919—22no—
30#13133138—28no—
31#13133138—28no—
32#13133128—22no—
33#13133128D414yes—
34#13133124—24no—
35#13133124—22no—
36C22×D4×C53—22no—
37C22×D4×C53—21no—
38C22×C8×C53—21no—
39#127×C53—22no—
40#127×C53—21no—
41#127×C53D412yes—
42C2×C4⋊C4×C53D412yes—
43C2×C42×C53C2311yescharacteristic
44#102×C53—44no—
45#97×C53—22no—
46#95×C53—22no—
47C24×C53—41no—
48C2×D4×C53—24no—
49C2×D4×C53—22no—
50C2×D4×C53—42no—
51C2×D4×C53—48no—
52C2×D4×C53—42no—
53C2×D4×C53—42no—
54C2×D4×C53—41no—
55C22×C4×C53—22no—
56C22×C4×C53C2×D412yes—
57C22×C4×C53C2×D411yescharacteristic
58C22×Dic53—22no—
59C22×Dic53—22no—
60C2×C8×C53—22no—
61C2×C8×C53—41no—
62C2×C8×C53—41no—
63C4⋊C4×C53—42no—
64C42×C53C2×D411yescharacteristic
65C42×C53C2×D412yes—
66C42×C53#4111yescharacteristic
67C4⋊Dic53—22no—
68C4⋊Dic53—22no—
69C4⋊Dic53—22no—
70C4⋊Dic53—22no—
71C4⋊Dic53—82no—
72#11719919—42no—
73#11719919—42no—
74#11719919—42no—
75#11719919—42no—
76C2×#88322—24no—
77C2×#88322—44no—
78C23×C53—42no—
79C23×C53—81no—
80C23×C53—84no—
81C23×C53—81no—
82C23×C53#12011yescharacteristic
83D4×C53—48no—
84D4×C53—44no—
85D4×C53—84no—
86D4×C53—82no—
87C2×C4×C53—21no—
88C2×C4×C53—24no—
89C2×C4×C53—21no—
90C2×C4×C53—42no—
91C2×C4×C53—44no—
92C2×C4×C53#12012yes—
93C2×C4×C53#12112yes—
94C2×C4×C53#12111yescharacteristic
95C2×C4×C53#12711yescharacteristic
96C2×Dic53—42no—
97C2×Dic53—42no—
98C2×Dic53—42no—
99C2×Dic53—44no—
100C2×Dic53—42no—
101C8×C53—42no—
102#88322—44no—
103#10080—531nomaximal, Sylow
104C22×C53—84no—
105C22×C53—82no—
106C22×C53—162no—
107C22×C53—162no—
108C22×C53#39211yescharacteristic
109C22×C53#45311yescharacteristic
110C22×C53#45414yes—
111C22×C53#45411yescharacteristic
112C4×C53—22no—
113C4×C53—22no—
114C4×C53—21no—
115C4×C53—21no—
116C4×C53—82no—
117Dic53—82no—
118Dic53—82no—
119C2×#490—532no—
120#2822—531no—
121#1961D5311yescharacteristic
122C2×#330—532no—
123#1295—531no—
124C2×C53—162no—
125C2×C53#165712yes—
126C2×C53#165812yes—
127C2×C53#200312yes—
128C2×C53#200611yescharacteristic
129C2×#127C2×D5311yescharacteristic
130C2×#125—532no—
131#490—1064no—
132#490—1064no—
133C2×#102—22no—
134C2×#102—1062no—
135C2×#102—1062no—
136C2×#97—532no—
137C2×#97C2×D5311yescharacteristic
138#400—1062no—
139#398—1062no—
140#398—1062no—
141#382C2×D5311yescharacteristic
142#330—1064no—
143C53#1008011yescharacteristic, complemented, Sylow
144C22×D4—22no—
145C22×D4—21no—
146C22×C8—21no—
147C22×C8—1062no—
148#127—21no—
149#127—22no—
150#127D21212yes—
151#125—1062no—
152#125—1062no—
153#125—1062no—
154#125—1062no—
155C2×C4⋊C4—532no—
156C2×C4⋊C4—532no—
157C2×C4⋊C4—1062no—
158C2×C4⋊C4—1062no—
159C2×C4⋊C4—1062no—
160C2×C4⋊C4C22⋊D5312yes—
161C2×C42C22×D5311yescharacteristic
162#102—44no—
163#102—1068no—
164#102—1068no—
165#97—22no—
166#97—534no—
167#97—1062no—
168#95—22no—
169#95—1064no—
170#95—1062no—
171C24—41no—
172C2×D4—24no—
173C2×D4—22no—
174C2×D4—42no—
175C2×D4—48no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—41no—
179C22×C4—22no—
180C22×C4—1062no—
181C22×C4—1062no—
182C22×C4D4×D5312yes—
183C22×C4C2×C22⋊D5311yescharacteristic
184C2×C8—22no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—1064no—
188C2×C8—2124no—
189C4⋊C4—42no—
190C4⋊C4—1062no—
191C4⋊C4—1062no—
192C4⋊C4—1062no—
193C4⋊C4—1062no—
194C4⋊C4—4242no—
195C4⋊C4—2122no—
196C4⋊C4—2122no—
197C4⋊C4—2122no—
198C4⋊C4—2122no—
199C42C2×D21211yescharacteristic
200C42#1171994511yescharacteristic
201C42D4×D5312yes—
202C23—42no—
203C23—81no—
204C23—84no—
205C23—81no—
206C23#1313318511yescharacteristic
207D4—48no—
208D4—44no—
209D4—84no—
210D4—82no—
211C2×C4—21no—
212C2×C4—24no—
213C2×C4—21no—
214C2×C4—42no—
215C2×C4—44no—
216C2×C4—2122no—
217C2×C4—2122no—
218C2×C4—2122no—
219C2×C4—2124no—
220C2×C4—2122no—
221C2×C4#1313318712yes—
222C2×C4#1313319012yes—
223C2×C4#1313319111yescharacteristic
224C2×C4#1313321311yescharacteristic
225C8—42no—
226C8—2124no—
227C22—84no—
228C22—82no—
229C22—162no—
230C22—162no—
231C22#1681053211yescharacteristic
232C22#1681079011yescharacteristic
233C22#1681079414yes—
234C22#1681079511yescharacteristic
235C4—22no—
236C4—22no—
237C4—21no—
238C4—21no—
239C4—82no—
240C4—4242no—
241C4—4242no—
242C2—162no—
243C2#2027401112yes—
244C2#2027401612yes—
245C2#2027573812yes—
246C2#2027575611yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1961C2
socleC23×C53#120
prime radicalC2×#127×C53C22
hypercenter#1961D53
hypocenterC53#10080
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1961

Complements

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

NH ≅ G/Ntype
2. #20282783C2semidirect
3. C2×#16810364C2semidirect
4. #20272596C2semidirect
143. C53#10080semidirect

Maximal subgroups

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

subgroupindex
C2×#168109872
#202827832
C2×#168103642
#202725962
C53×#19612
#1008053