Finite Group Databasebeta

← #27296667

Subgroups — #27296667

Order 13568.

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

normal subgroups only

This group has 372 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#20279957C211yesmaximal, characteristic, complemented
2#20273752C212yesmaximal
3#20272741C212yesmaximal, complemented
4#20272661C211yesmaximal, characteristic, complemented
5C53×#2039C211yesmaximal, characteristic
6#16810939—22no—
7#16810562—21no—
8#16810562C2212yes—
9#16810522—22no—
10#16810488—22no—
11C4×C4⋊Dic53—21no—
12#16810357—22no—
13#16810349C2212yes—
14C53×#537C2211yescharacteristic
15C2×#104×C53—21no—
16C53×#417—21no—
17C53×#384C2211yescharacteristic
18C53×#343C2211yescharacteristic
19C2×C4⋊Dic53—21no—
20C2×C4⋊Dic53—21no—
21C2×C4⋊Dic53—21no—
22C2×C4⋊Dic53D412yes—
23C2×C4×Dic53—22no—
24#13133148—42no—
25#13133138—42no—
26C22⋊#88322—22no—
27#13133124—22no—
28#13133124—21no—
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×C53—22no—
38C2×C4⋊C4×C53D411yescharacteristic
39C2×C4⋊C4×C53C2311yescharacteristic
40C2×C42×C53D411yescharacteristic
41#104×C53—42no—
42#103×C53—41no—
43#102×C53—41no—
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—
59M16×C53—41no—
60M16×C53—41no—
61C2×C8×C53—41no—
62C4⋊C4×C53—22no—
63C4⋊C4×C53—41no—
64C4⋊C4×C53—44no—
65C4⋊C4×C53—41no—
66C22⋊C4×C53—42no—
67C42×C53—21no—
68C42×C53SD1612yes—
69C4⋊Dic53—42no—
70C4⋊Dic53—42no—
71C4⋊Dic53—42no—
72C4⋊Dic53—81no—
73C4×Dic53—44no—
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—22no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—22no—
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—
104C8×C53—81no—
105#88322—82no—
106#12181—531nomaximal, Sylow
107C22×C53—21no—
108C22×C53—21no—
109C22×C53—81no—
110C22×C53—81no—
111C22×C53—81no—
112C22×C53#44811yescharacteristic
113C22×C53#45411yescharacteristic
114C22×C53B(4,𝔽2)11yescharacteristic
115C4×C53—24no—
116C4×C53—21no—
117C4×C53—21no—
118C4×C53—42no—
119C4×C53—81no—
120C4×C53—81no—
121Dic53—82no—
122Dic53—82no—
123#2571—531no—
124#2039D5311yescharacteristic
125#1616—532no—
126#1353—532no—
127#1325—531no—
128C2×C53—21no—
129C2×C53—21no—
130C2×C53—161no—
131C2×C53#159911yescharacteristic
132C2×C53#219811yescharacteristic
133C2×C53#220411yescharacteristic
134#537C2×D5311yescharacteristic
135#481—1062no—
136C2×#104—21no—
137#417—21no—
138#400—532no—
139#400—1061no—
140#389—1062no—
141#384C2×D5311yescharacteristic
142#382—1062no—
143C4×C4⋊C4—1061no—
144#343C2×D5311yescharacteristic
145#327—1062no—
146#324—532no—
147C53#1218111yescharacteristic, complemented, Sylow
148C2×#41—21no—
149C2×M16—21no—
150C22⋊Q8—41no—
151#121—22no—
152#121—41no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—532no—
155C2×C4⋊C4—1061no—
156C2×C4⋊C4—1061no—
157C2×C4⋊C4—1061no—
158C2×C4⋊C4C22⋊D5311yescharacteristic
159C2×C4⋊C4C22×D5311yescharacteristic
160C2×C42—1062no—
161C2×C42C22⋊D5311yescharacteristic
162#106—2122no—
163#104—42no—
164#103—41no—
165#102—41no—
166#102—2122no—
167C22⋊C8—1062no—
168#95—532no—
169#95—1061no—
170#95—1062no—
171#95—1062no—
172#95—1062no—
173#41—41no—
174#41—41no—
175#41—41no—
176#41—41no—
177C2×Q8—21no—
178C2×D4—21no—
179C2×D4—22no—
180C2×D4—41no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—41no—
184C22×C4—1062no—
185C22×C4—1062no—
186C22×C4D4×D5312yes—
187C22×C4C2×C22⋊D5311yescharacteristic
188M16—41no—
189M16—41no—
190C2×C8—41no—
191C2×C8—2122no—
192C4⋊C4—22no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—44no—
196C4⋊C4—4241no—
197C4⋊C4—2122no—
198C4⋊C4—2122no—
199C4⋊C4—2122no—
200C22⋊C4—42no—
201C42—21no—
202C42—2124no—
203C42#1171992312yes—
204C23—41no—
205C23#1313318511yescharacteristic
206Q8—41no—
207Q8—41no—
208D4—41no—
209D4—41no—
210D4—82no—
211D4—81no—
212D4—81no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—2122no—
226C2×C4—2122no—
227C2×C4—2122no—
228C2×C4—2122no—
229C2×C4—2122no—
230C2×C4—2122no—
231C2×C4C2×#1171992311yescharacteristic
232C2×C4#1313324711yescharacteristic
233C8—81no—
234C8—4242no—
235C22—21no—
236C22—21no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22#1681076111yescharacteristic
241C22#1681079511yescharacteristic
242C22#1681080911yescharacteristic
243C4—24no—
244C4—21no—
245C4—21no—
246C4—42no—
247C4—81no—
248C4—81no—
249C4—4242no—
250C4—4242no—
251C2—21no—
252C2—21no—
253C2—161no—
254C2#2027365411yescharacteristic
255C2#2027671011yescharacteristic
256C2#2027674011yescharacteristic
subgroupG/H
centerC22#16810809
commutator, cosocleC2×C4⋊C4×C53C23
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2039C2
socleC22×C53B(4,𝔽2)
prime radicalC53×#537C22
hypercenter#2039D53
hypocenterC53#12181
LCSG ▷ C2×C4⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2039

Complements

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

NH ≅ G/Ntype
1. #20279957C2semidirect
3. #20272741C2semidirect
4. #20272661C2semidirect
147. C53#12181semidirect

Maximal subgroups

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

subgroupindex
#202799572
#202737522
#202727412
#202726612
C53×#20392
#1218153