Finite Group Databasebeta

← #27296528

Subgroups — #27296528

Order 13568.

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

normal subgroups only

This group has 372 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#20275944C211yesmaximal, characteristic
2#20273717C212yesmaximal
3#20272721C212yesmaximal, complemented
4#20272659C211yesmaximal, characteristic, complemented
5C53×#2822C211yesmaximal, characteristic, complemented
6#16811204C2211yescharacteristic
7#16810936—22no—
8C2×#13133145—21no—
9#16810622—21no—
10#16810392C2211yescharacteristic
11#16810356—22no—
12#16810349C2212yes—
13C2×#125×C53—21no—
14C53×#400—22no—
15C53×#398—21no—
16C53×#398—22no—
17C53×#398C2212yes—
18C53×#382C2211yescharacteristic
19C2×#11719945—21no—
20C2×#11719917—21no—
21#13133197—41no—
22#13133192—41no—
23#13133191—22no—
24#13133150—42no—
25#13133145—42no—
26#13133140—41no—
27#13133137—41no—
28#13133137—42no—
29C22⋊#88322—22no—
30#125×C53—42no—
31#125×C53—42no—
32C2×C4⋊C4×C53—21no—
33C2×C4⋊C4×C53—22no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—22no—
37C2×C4⋊C4×C53D411yescharacteristic
38C2×C4⋊C4×C53D412yes—
39C2×C4⋊C4×C53C2311yescharacteristic
40C2×C42×C53D411yescharacteristic
41#95×C53—22no—
42#95×C53—21no—
43#95×C53—22no—
44#95×C53—22no—
45#95×C53D412yes—
46C22×C4×C53—21no—
47C22×C4×C53—21no—
48C22×C4×C53—22no—
49C22×C4×C53—22no—
50C22×C4×C53C2×D411yescharacteristic
51C22×C4×C53C2×D412yes—
52C2×C22⋊D53—41no—
53#11719945—41no—
54#11719945—41no—
55#11719945—41no—
56#11719945—41no—
57C2×D212—21no—
58C2×D212—22no—
59C2×C4×D53—41no—
60C2×Dic106—21no—
61C4⋊C4×C53—41no—
62C4⋊C4×C53—42no—
63C4⋊C4×C53—44no—
64C4⋊C4×C53—42no—
65C4⋊C4×C53—41no—
66C4⋊C4×C53—42no—
67C4⋊C4×C53—42no—
68C4⋊C4×C53—42no—
69C4⋊C4×C53—81no—
70C42×C53—21no—
71C42×C53D812yes—
72#11719921—42no—
73C4⋊Dic53—22no—
74#11719917—41no—
75#11719917—41no—
76C2×#88322—42no—
77C2×#88322—41no—
78C23×C53#12011yescharacteristic
79C22×D53—41no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—41no—
85C2×C4×C53—41no—
86C2×C4×C53—42no—
87C2×C4×C53—42no—
88C2×C4×C53—42no—
89C2×C4×C53—42no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53—42no—
93C2×C4×C53—42no—
94C2×C4×C53—41no—
95C2×C4×C53C2×D811yescharacteristic
96C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
97C22⋊D53—81no—
98C22⋊D53—81no—
99C2×Dic53—41no—
100D212—41no—
101D212—41no—
102D212—82no—
103C4×D53—81no—
104C4×D53—81no—
105Dic106—41no—
106Dic106—41no—
107#88322—81no—
108#88322—82no—
109#12157—531nomaximal, Sylow
110C22×C53—21no—
111C22×C53—21no—
112C22×C53#44711yescharacteristic
113C22×C53#45411yescharacteristic
114C22×C53B(4,𝔽2)11yescharacteristic
115C2×D53—81no—
116C2×D53—81no—
117C2×D53—81no—
118C4×C53—21no—
119C4×C53—21no—
120C4×C53—42no—
121C4×C53—42no—
122C4×C53—82no—
123C4×C53—82no—
124C4×C53—81no—
125Dic53—81no—
126#2822D5311yescharacteristic, complemented
127#2040—531no—
128#1610—532no—
129#1346—532no—
130#1324—531no—
131C2×C53—21no—
132C2×C53—21no—
133C2×C53#159811yescharacteristic
134C2×C53#220111yescharacteristic
135C2×C53#220311yescharacteristic
136D53—161no—
137#537—531no—
138C2×#125—21no—
139#480—1062no—
140C2×#104—1061no—
141#417—1061no—
142#400—22no—
143#398—21no—
144#398—22no—
145#398C2×D5312yes—
146#382C2×D5311yescharacteristic
147#343—531no—
148#327—1062no—
149#324—532no—
150C53#1215711yescharacteristic, complemented, Sylow
151C2×#41—1061no—
152C2×M16—1061no—
153#125—42no—
154#125—42no—
155C22⋊Q8—2121no—
156#121—1062no—
157#121—2121no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4C22⋊D5311yescharacteristic
164C2×C4⋊C4C22⋊D5312yes—
165C2×C4⋊C4C22×D5311yescharacteristic
166C2×C42C22⋊D5311yescharacteristic
167#107—2122no—
168#104—2122no—
169#103—2121no—
170#102—2121no—
171#102—2122no—
172C22⋊C8—1062no—
173#95—22no—
174#95—21no—
175#95—22no—
176#95—22no—
177#95C22⋊D5312yes—
178#41—2121no—
179#41—2121no—
180#41—2121no—
181#41—2121no—
182C2×Q8—1061no—
183C2×D4—1062no—
184C2×D4—1061no—
185C2×D4—2121no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—2121no—
191C22×C4C2×C22⋊D5311yescharacteristic
192C22×C4C2×C22⋊D5312yes—
193M16—2121no—
194M16—2121no—
195C2×C8—2122no—
196C2×C8—2121no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—44no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—81no—
206C4⋊C4—1062no—
207C22⋊C4—2122no—
208C42—21no—
209C42#1171992212yes—
210C23—2121no—
211C23#1313318611yescharacteristic
212Q8—2121no—
213Q8—2121no—
214D4—4242no—
215D4—4241no—
216D4—4241no—
217D4—2121no—
218D4—2121no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—4241no—
235C2×C4—4241no—
236C2×C4—2121no—
237C2×C4C2×#1171992211yescharacteristic
238C2×C4#1313324311yescharacteristic
239C8—4241no—
240C8—4242no—
241C22—21no—
242C22—21no—
243C22—4241no—
244C22—4241no—
245C22—4241no—
246C22#1681075711yescharacteristic
247C22#1681079611yescharacteristic
248C22#1681081011yescharacteristic
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—82no—
254C4—81no—
255C4—82no—
256C4—4241no—
257C2—21no—
258C2—21no—
259C2—8481no—
260C2#2027365011yescharacteristic
261C2#2027672811yescharacteristic
262C2#2027673611yescharacteristic
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2822C2
socleC22×C53B(4,𝔽2)
prime radical#16811204C22
hypercenter#2822D53
hypocenterC53#12157
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2822

Complements

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

NH ≅ G/Ntype
3. #20272721C2semidirect
4. #20272659C2semidirect
5. C53×#2822C2semidirect
126. #2822D53semidirect
150. C53#12157semidirect

Maximal subgroups

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

subgroupindex
#202759442
#202737172
#202727212
#202726592
C53×#28222
#1215753