Finite Group Databasebeta

← #27296691

Subgroups — #27296691

Order 13568.

360 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 59 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#20282987C211yesmaximal, characteristic, complemented
2#20273752C212yesmaximal
3#20272741C212yesmaximal, complemented
4#20272659C211yesmaximal, characteristic, complemented
5C53×#2040C211yesmaximal, characteristic
6#16810939—22no—
7#16810565—22no—
8#16810562C2212yes—
9#16810544—22no—
10#16810543—21no—
11#16810489—21no—
12#16810357—22no—
13#16810349C2212yes—
14C53×#537C2211yescharacteristic
15C2×#104×C53—21no—
16C53×#417—21no—
17C53×#382C2211yescharacteristic
18C53×#343C2211yescharacteristic
19C22×Dic106—21no—
20C2×C4⋊Dic53—21no—
21C2×C4⋊Dic53—21no—
22C2×C4⋊Dic53D412yes—
23C2×#11719919—22no—
24#13133148—42no—
25#13133138—42no—
26C22⋊#88322—22no—
27#13133124—21no—
28#13133124—22no—
29#13133124—22no—
30#13133124—22no—
31#13133124D412yes—
32C2×#41×C53—21no—
33C2×M16×C53—21no—
34C22⋊Q8×C53—41no—
35#121×C53—22no—
36#121×C53—41no—
37C2×C4⋊C4×C53D411yescharacteristic
38C2×C4⋊C4×C53C2311yescharacteristic
39C2×C42×C53D411yescharacteristic
40#104×C53—42no—
41#103×C53—41no—
42#102×C53—41no—
43#95×C53—22no—
44#41×C53—41no—
45#41×C53—41no—
46#41×C53—41no—
47#41×C53—41no—
48C2×Q8×C53—21no—
49C2×D4×C53—21no—
50C2×D4×C53—22no—
51C2×D4×C53—41no—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—41no—
55C22×C4×C53C2×D411yescharacteristic
56C22×C4×C53C2×D412yes—
57C22×Dic53—22no—
58C22×Dic53—22no—
59C2×Dic106—82no—
60C2×Dic106—81no—
61M16×C53—41no—
62M16×C53—41no—
63C2×C8×C53—41no—
64C4⋊C4×C53—22no—
65C4⋊C4×C53—41no—
66C4⋊C4×C53—41no—
67C22⋊C4×C53—42no—
68C42×C53—21no—
69C42×C53SD1612yes—
70C4⋊Dic53—42no—
71C4⋊Dic53—81no—
72C4⋊Dic53—81no—
73#11719919—82no—
74C2×#88322—42no—
75C23×C53—41no—
76C23×C53#12011yescharacteristic
77Q8×C53—41no—
78Q8×C53—41no—
79D4×C53—41no—
80D4×C53—41no—
81D4×C53—82no—
82D4×C53—81no—
83D4×C53—81no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53—81no—
95C2×C4×C53—81no—
96C2×C4×C53C2×SD1611yescharacteristic
97C2×C4×C53#13711yescharacteristic
98C2×Dic53—42no—
99C2×Dic53—42no—
100C2×Dic53—42no—
101C2×Dic53—42no—
102C2×Dic53—42no—
103C2×Dic53—42no—
104Dic106—84no—
105C8×C53—81no—
106#88322—82no—
107#12185—531nomaximal, Sylow
108C22×C53—21no—
109C22×C53—21no—
110C22×C53—81no—
111C22×C53—81no—
112C22×C53—81no—
113C22×C53#44811yescharacteristic
114C22×C53#45411yescharacteristic
115C22×C53B(4,𝔽2)11yescharacteristic
116C4×C53—21no—
117C4×C53—21no—
118C4×C53—42no—
119C4×C53—42no—
120C4×C53—81no—
121C4×C53—81no—
122Dic53—82no—
123Dic53—82no—
124#2844—531no—
125#2040D5311yescharacteristic
126#1616—532no—
127#1353—532no—
128#1324—531no—
129C2×C53—21no—
130C2×C53—21no—
131C2×C53—161no—
132C2×C53#159911yescharacteristic
133C2×C53#220111yescharacteristic
134C2×C53#220211yescharacteristic
135#537C2×D5311yescharacteristic
136#481—1062no—
137C2×#104—21no—
138#417—21no—
139#400—532no—
140#400—1062no—
141#395—1062no—
142#395—1061no—
143#382—1061no—
144#382C2×D5311yescharacteristic
145#343C2×D5311yescharacteristic
146#327—1062no—
147#324—532no—
148C53#1218511yescharacteristic, complemented, Sylow
149C2×#41—21no—
150C22×Q8—1061no—
151C2×M16—21no—
152C22⋊Q8—41no—
153#121—22no—
154#121—41no—
155C2×C4⋊C4—532no—
156C2×C4⋊C4—1062no—
157C2×C4⋊C4—1061no—
158C2×C4⋊C4—1061no—
159C2×C4⋊C4C22⋊D5311yescharacteristic
160C2×C4⋊C4C22×D5311yescharacteristic
161C2×C42C22⋊D5311yescharacteristic
162#106—2122no—
163#104—42no—
164#103—41no—
165#102—41no—
166#102—2122no—
167C22⋊C8—1062no—
168#95—22no—
169#95—532no—
170#95—1062no—
171#95—1061no—
172#95—1062no—
173#95—1062no—
174#41—41no—
175#41—41no—
176#41—41no—
177#41—41no—
178C2×Q8—21no—
179C2×Q8—4242no—
180C2×Q8—4241no—
181C2×D4—21no—
182C2×D4—22no—
183C2×D4—41no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—41no—
187C22×C4—1062no—
188C22×C4—1062no—
189C22×C4D4×D5312yes—
190C22×C4C2×C22⋊D5311yescharacteristic
191M16—41no—
192M16—41no—
193C2×C8—41no—
194C2×C8—2122no—
195C4⋊C4—22no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—4242no—
199C4⋊C4—4241no—
200C4⋊C4—4241no—
201C4⋊C4—2122no—
202C22⋊C4—42no—
203C42—21no—
204C42#1171992312yes—
205C23—41no—
206C23#1313318511yescharacteristic
207Q8—41no—
208Q8—41no—
209Q8—4244no—
210D4—41no—
211D4—41no—
212D4—82no—
213D4—81no—
214D4—81no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—2122no—
228C2×C4—2122no—
229C2×C4—2122no—
230C2×C4—2122no—
231C2×C4—2122no—
232C2×C4—2122no—
233C2×C4C2×#1171992311yescharacteristic
234C2×C4#1313324711yescharacteristic
235C8—81no—
236C8—4242no—
237C22—21no—
238C22—21no—
239C22—81no—
240C22—81no—
241C22—81no—
242C22#1681076111yescharacteristic
243C22#1681079511yescharacteristic
244C22#1681080911yescharacteristic
245C4—21no—
246C4—21no—
247C4—42no—
248C4—42no—
249C4—81no—
250C4—81no—
251C4—4242no—
252C4—4242no—
253C2—21no—
254C2—21no—
255C2—161no—
256C2#2027365411yescharacteristic
257C2#2027672711yescharacteristic
258C2#2027673111yescharacteristic
subgroupG/H
centerC22#16810809
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2040C2
socleC22×C53B(4,𝔽2)
prime radicalC53×#537C22
hypercenter#2040D53
hypocenterC53#12185
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2040

Complements

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

NH ≅ G/Ntype
1. #20282987C2semidirect
3. #20272741C2semidirect
4. #20272659C2semidirect
148. C53#12185semidirect

Maximal subgroups

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

subgroupindex
#202829872
#202737522
#202727412
#202726592
C53×#20402
#1218553