Finite Group Databasebeta

← #27281487

Subgroups — #27281487

Order 13568.

514 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 119 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#20279888C211yesmaximal, characteristic
2#20274707C211yesmaximal, characteristic
3#20273395C212yesmaximal, complemented
4#20272617C212yesmaximal, complemented
5C53×#2975C211yesmaximal, characteristic
6C2×#13133217C2211yescharacteristic
7#16810647C2212yes—
8C2×#13133134—21no—
9#16810535C414yes—
10#16810526—21no—
11#16810488—21no—
12#16810483—22no—
13C4×C4⋊Dic53C2211yescharacteristic
14#16810392—21no—
15#16810371—24no—
16C2×C4×D4×C53C2211yescharacteristic
17C53×#436—24no—
18C53×#431C2212yes—
19C53×#407—22no—
20C22×Dic106C2×C412yes—
21C2×#11719917—22no—
22#13133217—22no—
23#13133217—22no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53—21no—
27C2×C22⋊Dic53—24no—
28C2×C4×Dic53—21no—
29C4⋊#88322—22no—
30C4⋊#88322—22no—
31#13133134—42no—
32#13133124—24no—
33#13133124C2×C412yes—
34C22×D4×C53D411yescharacteristic
35C23×C4×C53D412yes—
36C2×M16×C53—24no—
37C4×D4×C53—22no—
38C4×D4×C53—22no—
39C2×C4⋊C4×C53D411yescharacteristic
40C2×C22⋊C4×C53C2×C412yes—
41C2×C42×C53C2311yescharacteristic
42C4⋊C8×C53D414yes—
43C22⋊C8×C53—24no—
44C22⋊C8×C53—22no—
45C4×C8×C53—22no—
46C24×C53C22⋊C412yes—
47C2×D4×C53—22no—
48C2×D4×C53—21no—
49C2×D4×C53—22no—
50C2×D4×C53—21no—
51C22×C4×C53—22no—
52C22×C4×C53—22no—
53C22×C4×C53—44no—
54C22×C4×C53C22⋊C412yes—
55C22×C4×C53C22×C411yescharacteristic
56C22×C4×C53C2×D411yescharacteristic
57C22×C4×C53C2×D411yescharacteristic
58C22×Dic53—22no—
59C22×Dic53—22no—
60C2×Dic106—24no—
61C2×Dic106—82no—
62M16×C53—84no—
63C2×C8×C53—24no—
64C4⋊C4×C53—22no—
65C22⋊C4×C53—24no—
66C42×C53C2×D412yes—
67C42×C53#4112yes—
68C22⋊Dic53—84no—
69C4⋊Dic53—42no—
70C4⋊Dic53—41no—
71C4⋊Dic53—42no—
72C4⋊Dic53—41no—
73C4×Dic53—41no—
74C4×Dic53—41no—
75#11719917—42no—
76#11719917—42no—
77C2×#88322—42no—
78C23×C53—22no—
79C23×C53—22no—
80C23×C53—22no—
81C23×C53—42no—
82C23×C53—42no—
83C23×C53C2×C22⋊C411yescharacteristic
84D4×C53—42no—
85D4×C53—42no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—41no—
89C2×C4×C53—44no—
90C2×C4×C53—42no—
91C2×C4×C53—82no—
92C2×C4×C53C4×D412yes—
93C2×C4×C53#12011yescharacteristic
94C2×C4×C53#12112yes—
95C2×C4×C53#12311yescharacteristic
96C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
97C2×C4×C53#13712yes—
98C2×C4×C53#13711yescharacteristic
99C2×Dic53—44no—
100C2×Dic53—42no—
101C2×Dic53—42no—
102C2×Dic53—42no—
103C2×Dic53—42no—
104Dic106—84no—
105C8×C53—44no—
106#88322—82no—
107#9399—531nomaximal, Sylow
108C22×C53—21no—
109C22×C53—21no—
110C22×C53—44no—
111C22×C53—42no—
112C22×C53—42no—
113C22×C53—42no—
114C22×C53—82no—
115C22×C53#38611yescharacteristic
116C22×C53#41711yescharacteristic
117C22×C53#41911yescharacteristic
118C4×C53—21no—
119C4×C53—22no—
120C4×C53—42no—
121C4×C53#45512yes—
122C4×C53#46812yes—
123C4×C53#47012yes—
124Dic53—82no—
125Dic53—82no—
126#2975D5311yescharacteristic
127#2563—531no—
128#1800—531no—
129#1548—532no—
130#1303—532no—
131C2×C53—21no—
132C2×C53—21no—
133C2×C53—82no—
134C2×C53#188511yescharacteristic
135C2×C53#189411yescharacteristic
136C2×C53#191411yescharacteristic
137C2×C4⋊Q8—531no—
138C2×C4×D4C2×D5311yescharacteristic
139#436—24no—
140#431C2×D5312yes—
141#423—532no—
142C2×#100—1061no—
143#407—22no—
144#393—534no—
145#390—1061no—
146#382—1061no—
147#380—1062no—
148C4×C4⋊C4—531no—
149#343—1061no—
150#333—1064no—
151C53#939911yescharacteristic, complemented, Sylow
152C22×Q8—532no—
153C22×D4D21211yescharacteristic
154C23×C4C22⋊D5312yes—
155C2×M16—24no—
156C2×M16—1062no—
157C4⋊Q8—1062no—
158C4⋊Q8—1062no—
159C4×D4—22no—
160C4×D4—22no—
161C2×C4⋊C4—1061no—
162C2×C4⋊C4—1061no—
163C2×C4⋊C4—1061no—
164C2×C4⋊C4D21211yescharacteristic
165C2×C22⋊C4—1064no—
166C2×C22⋊C4C4×D5312yes—
167C2×C42—1061no—
168C2×C42C22×D5311yescharacteristic
169C4⋊C8—1062no—
170C4⋊C8—1062no—
171C4⋊C8C22⋊D5314yes—
172#100—2122no—
173C22⋊C8—24no—
174C22⋊C8—22no—
175C4×C8—22no—
176#95—532no—
177#95—1064no—
178C24#1171992112yes—
179C2×Q8—1064no—
180C2×Q8—4242no—
181C2×D4—22no—
182C2×D4—21no—
183C2×D4—22no—
184C2×D4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—44no—
188C22×C4—1062no—
189C22×C4—1062no—
190C22×C4#1171992112yes—
191C22×C4C2×C4×D5311yescharacteristic
192C22×C4C2×D21211yescharacteristic
193C22×C4C2×C22⋊D5311yescharacteristic
194M16—84no—
195M16—2122no—
196M16—2122no—
197C2×C8—24no—
198C2×C8—2122no—
199C4⋊C4—22no—
200C4⋊C4—2122no—
201C4⋊C4—2121no—
202C4⋊C4—2122no—
203C4⋊C4—2121no—
204C22⋊C4—24no—
205C22⋊C4—4244no—
206C42—2121no—
207C42—2121no—
208C42#1171994512yes—
209C42C2×C22⋊D5312yes—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23C2×#1171992111yescharacteristic
216Q8—4244no—
217D4—42no—
218D4—42no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—44no—
223C2×C4—42no—
224C2×C4—82no—
225C2×C4—2122no—
226C2×C4—2124no—
227C2×C4—2122no—
228C2×C4—2122no—
229C2×C4—2122no—
230C2×C4C4×C22⋊D5312yes—
231C2×C4#1313318611yescharacteristic
232C2×C4#1313319112yes—
233C2×C4#1313320011yescharacteristic
234C2×C4#1313324511yescharacteristic
235C2×C4#1313325012yes—
236C2×C4#1313325011yescharacteristic
237C8—44no—
238C8—4242no—
239C22—21no—
240C22—21no—
241C22—44no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—82no—
246C22#1681050211yescharacteristic
247C22#1681062511yescharacteristic
248C22#1681063311yescharacteristic
249C4—22no—
250C4—21no—
251C4—42no—
252C4—4242no—
253C4—4242no—
254C4#1681080112yes—
255C4#1681086912yes—
256C4#1681088112yes—
257C2—21no—
258C2—21no—
259C2—82no—
260C2#2027510311yescharacteristic
261C2#2027514811yescharacteristic
262C2#2027526211yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2975C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#2975D53
hypocenterC53#9399
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2975

Complements

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

NH ≅ G/Ntype
3. #20273395C2semidirect
4. #20272617C2semidirect
151. C53#9399semidirect

Maximal subgroups

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

subgroupindex
#202798882
#202747072
#202733952
#202726172
C53×#29752
#939953