Finite Group Databasebeta

← #27286338

Subgroups — #27286338

Order 13568.

542 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 103 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#20285656C212yesmaximal
2#20285239C211yesmaximal, characteristic, complemented
3#20273278C212yesmaximal, complemented
4#20272560C211yesmaximal, characteristic, complemented
5C53×#1961C211yesmaximal, characteristic
6#16811180C2212yes—
7#16811032—22no—
8#16811032—22no—
9#16810995—22no—
10#16810969—22no—
11#16810820—24no—
12#16810774—24no—
13#16810676C2212yes—
14#16810661—22no—
15#16810359—24no—
16C2×#127×C53C2211yescharacteristic
17C2×#102×C53—22no—
18C2×#97×C53C2211yescharacteristic
19C53×#382C2211yescharacteristic
20C2×#11719923—28no—
21#13133217D412yes—
22#13133217D412yes—
23#13133209—22no—
24#13133196—22no—
25#13133196—22no—
26C4×Dic106—22no—
27C4×Dic106—22no—
28#13133176—22no—
29#13133151—24no—
30#13133149—22no—
31#13133149—22no—
32C4⋊#88322—22no—
33#13133138—28no—
34C22⋊#88322D414yes—
35C4×#88322—22no—
36C22×D4×C53—21no—
37C22×D4×C53—22no—
38C22×C8×C53D412yes—
39#127×C53—22no—
40#127×C53—21no—
41#127×C53D412yes—
42C2×C4⋊C4×C53—21no—
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—41no—
51C2×D4×C53—42no—
52C2×D4×C53—42no—
53C2×D4×C53—42no—
54C2×D4×C53—84no—
55C22×C4×C53—41no—
56C22×C4×C53C2×D412yes—
57C22×C4×C53C2×D411yescharacteristic
58C2×Dic106—22no—
59C2×Dic106—22no—
60C2×C8×C53—22no—
61C2×C8×C53—22no—
62C2×C8×C53SD1614yescomplemented
63C4⋊C4×C53—42no—
64C42×C53C2×D411yescharacteristic
65C42×C53C2×D412yes—
66C42×C53#4111yescharacteristic
67C22⋊Dic53—42no—
68#11719923—88no—
69C4⋊Dic53—22no—
70C4⋊Dic53—22no—
71C4⋊Dic53—22no—
72C4⋊Dic53—22no—
73#11719919—42no—
74#11719919—42no—
75C4×Dic53—42no—
76C2×#88322—24no—
77C23×C53—42no—
78C23×C53—84no—
79C23×C53—81no—
80C23×C53—81no—
81C23×C53#12711yescharacteristic
82D4×C53—48no—
83D4×C53—44no—
84D4×C53—84no—
85D4×C53—82no—
86C2×C4×C53—42no—
87C2×C4×C53—82no—
88C2×C4×C53—81no—
89C2×C4×C53#12011yescharacteristic
90C2×C4×C53#12012yes—
91C2×C4×C53#12112yes—
92C2×C4×C53#12111yescharacteristic
93C2×C4×C53C2×SD1612yes—
94C2×C4×C53#13712yes—
95C2×Dic53—42no—
96C2×Dic53—42no—
97Dic106—44no—
98Dic106—82no—
99C8×C53—24no—
100#88322—44no—
101#10220—531nomaximal, Sylow
102C22×C53—22no—
103C22×C53—84no—
104C22×C53—82no—
105C22×C53—162no—
106C22×C53—162no—
107C22×C53#39211yescharacteristic
108C22×C53#49211yescharacteristic
109C22×C53#49711yescharacteristic
110C4×C53—22no—
111C4×C53—22no—
112C4×C53—161no—
113C4×C53#44812yes—
114C4×C53#46112yes—
115Dic53—82no—
116Dic53—82no—
117#3208—532no—
118#3162—531no—
119#1961D5311yescharacteristic
120#1518—532no—
121#1281—531no—
122C2×C53—21no—
123C2×C53—21no—
124C2×C53—162no—
125C2×C53#200812yes—
126C2×C53#201311yescharacteristic
127#533—532no—
128C2×#127C2×D5311yescharacteristic
129#501—1062no—
130#501—1062no—
131#492—1062no—
132#486—1062no—
133#460—1064no—
134#450—1064no—
135#431—532no—
136#427—1062no—
137C2×#102—22no—
138C2×#97C2×D5311yescharacteristic
139#382C2×D5311yescharacteristic
140#328—1064no—
141C53#1022011yescharacteristic, complemented, Sylow
142C22×D4—22no—
143C22×D4—21no—
144C2×SD16—1068no—
145C22×C8C22⋊D5312yes—
146C4⋊Q8—532no—
147C4⋊Q8—532no—
148#127—21no—
149#127—22no—
150#127D21212yes—
151#125—1062no—
152C22⋊Q8—1062no—
153C22⋊Q8—1062no—
154C4×Q8—1062no—
155C4×Q8—1062no—
156#117—1062no—
157C2×C4⋊C4—21no—
158C2×C42C22×D5311yescharacteristic
159#107—1064no—
160#106—1062no—
161#106—1062no—
162C4⋊C8—1062no—
163#102—44no—
164#102—1068no—
165C22⋊C8—534no—
166#97—22no—
167C4×C8—1062no—
168#95—22no—
169C24—41no—
170C2×Q8—1062no—
171C2×Q8—1062no—
172C2×D4—24no—
173C2×D4—22no—
174C2×D4—41no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—84no—
179C22×C4—41no—
180C22×C4D4×D5312yes—
181C22×C4C2×C22⋊D5311yescharacteristic
182SD16—4248no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—1064no—
186C2×C8#1171992314yescomplemented
187C4⋊C4—42no—
188C4⋊C4—1062no—
189C4⋊C4—1062no—
190C4⋊C4—1062no—
191C4⋊C4—1062no—
192C4⋊C4—2122no—
193C4⋊C4—2122no—
194C22⋊C4—2122no—
195C42—2122no—
196C42C2×D21211yescharacteristic
197C42#1171994511yescharacteristic
198C42D4×D5312yes—
199C23—42no—
200C23—84no—
201C23—81no—
202C23—81no—
203C23#1313321311yescharacteristic
204Q8—4242no—
205Q8—2124no—
206D4—48no—
207D4—44no—
208D4—84no—
209D4—82no—
210C2×C4—42no—
211C2×C4—82no—
212C2×C4—81no—
213C2×C4—2122no—
214C2×C4—2122no—
215C2×C4#1313318511yescharacteristic
216C2×C4#1313318712yes—
217C2×C4#1313319012yes—
218C2×C4#1313319111yescharacteristic
219C2×C4C2×#1171992312yes—
220C2×C4#1313324712yes—
221C8—24no—
222C8—2124no—
223C22—22no—
224C22—84no—
225C22—82no—
226C22—162no—
227C22—162no—
228C22#1681053211yescharacteristic
229C22#1681099511yescharacteristic
230C22#1681101811yescharacteristic
231C4—22no—
232C4—22no—
233C4—161no—
234C4—4242no—
235C4—4242no—
236C4#1681076112yes—
237C4#1681082812yes—
238C2—21no—
239C2—21no—
240C2—162no—
241C2#2027577012yes—
242C2#2027579711yescharacteristic
subgroupG/H
centerC22#16810532
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1961C2
socleC22×C53#392
prime radicalC2×#127×C53C22
hypercenter#1961D53
hypocenterC53#10220
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1961

Complements

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

NH ≅ G/Ntype
2. #20285239C2semidirect
3. #20273278C2semidirect
4. #20272560C2semidirect
62. C2×C8×C53SD16semidirect
141. C53#10220semidirect
186. C2×C8#11719923semidirect

Maximal subgroups

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

subgroupindex
#202856562
#202852392
#202732782
#202725602
C53×#19612
#1022053