Finite Group Databasebeta

← #27284944

Subgroups — #27284944

Order 13568.

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

normal subgroups only

This group has 566 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#20284742C212yesmaximal
2#20278008C211yesmaximal, characteristic, complemented
3#20273143C212yesmaximal, complemented
4#20272544C211yesmaximal, characteristic, complemented
5C53×#1963C211yesmaximal, characteristic
6C2×C4×Dic106C2212yes—
7#16810939—22no—
8#16810905—22no—
9#16810820—22no—
10#16810761—22no—
11C4×#11719923—24no—
12#16810676C2212yes—
13#16810555—22no—
14#16810544—22no—
15C4×C4⋊Dic53—22no—
16#16810353—24no—
17C53×#532C2211yescharacteristic
18C53×#423C2211yescharacteristic
19C2×#104×C53—22no—
20C53×#376C2211yescharacteristic
21C22×Dic106D412yes—
22C2×#11719923—24no—
23C4×Dic106—22no—
24C4×Dic106—22no—
25C4×Dic106—22no—
26C4×Dic106—22no—
27C2×C4⋊Dic53—24no—
28C2×C4⋊Dic53D412yes—
29C2×#11719919—22no—
30C2×C4×Dic53—22no—
31#13133148—24no—
32C4⋊#88322—22no—
33#13133140—24no—
34#13133138—24no—
35C22⋊#88322—22no—
36C4×#88322D414yes—
37#13133124—22no—
38#13133124—22no—
39C2×#41×C53—21no—
40C2×M16×C53—21no—
41C4⋊Q8×C53D411yescharacteristic
42#127×C53D411yescharacteristic
43#124×C53—21no—
44#121×C53—22no—
45C4×D4×C53—22no—
46C2×C42×C53D412yes—
47C2×C42×C53C2311yescharacteristic
48C4⋊C8×C53—22no—
49#104×C53—44no—
50#95×C53—22no—
51#41×C53—42no—
52#41×C53—42no—
53C2×Q8×C53—21no—
54C2×D4×C53—22no—
55C2×D4×C53—21no—
56C2×D4×C53—41no—
57C22×C4×C53—22no—
58C22×C4×C53—41no—
59C22×C4×C53C2×D412yes—
60C22×C4×C53C2×D411yescharacteristic
61C22×Dic53—22no—
62C22×Dic53—22no—
63C2×Dic106—22no—
64C2×Dic106—22no—
65C2×Dic106—44no—
66M16×C53—42no—
67C2×C8×C53—41no—
68C4⋊C4×C53—22no—
69C22⋊C4×C53—42no—
70C42×C53—22no—
71C42×C53SD1614yescomplemented
72C42×C53C2×D411yescharacteristic
73C42×C53C2×D412yes—
74C42×C53#4111yescharacteristic
75#11719923—48no—
76C4⋊Dic53—22no—
77C4⋊Dic53—22no—
78C4⋊Dic53—48no—
79#11719919—42no—
80#11719919—42no—
81C4×Dic53—42no—
82C4×Dic53—42no—
83C2×#88322—24no—
84C23×C53—41no—
85C23×C53#12011yescharacteristic
86Q8×C53—42no—
87D4×C53—44no—
88D4×C53—42no—
89D4×C53—82no—
90C2×C4×C53—24no—
91C2×C4×C53—42no—
92C2×C4×C53—44no—
93C2×C4×C53—41no—
94C2×C4×C53—81no—
95C2×C4×C53—81no—
96C2×C4×C53#12012yes—
97C2×C4×C53#12112yes—
98C2×C4×C53#12111yescharacteristic
99C2×C4×C53#12711yescharacteristic
100C2×C4×C53C2×SD1612yes—
101C2×C4×C53#13712yes—
102C2×Dic53—24no—
103C2×Dic53—42no—
104C2×Dic53—42no—
105C2×Dic53—44no—
106C2×Dic53—42no—
107Dic106—24no—
108Dic106—24no—
109Dic106—42no—
110Dic106—42no—
111C8×C53—81no—
112#88322—44no—
113#10014—531nomaximal, Sylow
114C22×C53—22no—
115C22×C53—82no—
116C22×C53—81no—
117C22×C53#39211yescharacteristic
118C22×C53#44812yes—
119C4×C53—22no—
120C4×C53—22no—
121C4×C53—44no—
122C4×C53—81no—
123C4×C53#46112yes—
124C4×C53#49211yescharacteristic
125C4×C53#49711yescharacteristic
126Dic53—44no—
127Dic53—82no—
128#3103—532no—
129#2399—531no—
130#1963D5311yescharacteristic
131#1487—532no—
132#1274—531no—
133C2×C53—22no—
134C2×C53—161no—
135C2×C53#201012yes—
136C2×C53#201311yescharacteristic
137#532C2×D5311yescharacteristic
138C2×C4×Q8—532no—
139#481—1062no—
140#475—1062no—
141#460—1062no—
142#448—1062no—
143C4×SD16—1064no—
144#431—532no—
145#423C2×D5311yescharacteristic
146C2×#104—22no—
147#398—1062no—
148#395—1062no—
149C4×C4⋊C4—1062no—
150#376C2×D5311yescharacteristic
151Q8⋊C8—1064no—
152C53#1001411yescharacteristic, complemented, Sylow
153C2×#41—21no—
154C22×Q8—532no—
155C2×SD16—1064no—
156C2×M16—21no—
157C4⋊Q8D21211yescharacteristic
158#127D21211yescharacteristic
159#124—21no—
160#121—22no—
161C4×Q8—1062no—
162C4×Q8—1062no—
163C4×Q8—1062no—
164C4×Q8—1062no—
165C4×D4—22no—
166C2×C4⋊C4—532no—
167C2×C4⋊C4—1064no—
168C2×C4⋊C4—1062no—
169C2×C42—1062no—
170C2×C42C22⋊D5312yes—
171C2×C42C22×D5311yescharacteristic
172#106—1064no—
173C4⋊C8—22no—
174C4⋊C8—1062no—
175#104—44no—
176#103—1064no—
177#102—1064no—
178C22⋊C8—1062no—
179C4×C8—534no—
180#95—22no—
181#95—1062no—
182#95—1062no—
183#41—42no—
184#41—42no—
185C2×Q8—21no—
186C2×Q8—1062no—
187C2×Q8—1062no—
188C2×Q8—2124no—
189C2×D4—22no—
190C2×D4—21no—
191C2×D4—41no—
192C22×C4—22no—
193C22×C4—41no—
194C22×C4—1062no—
195C22×C4—1062no—
196C22×C4D4×D5312yes—
197C22×C4C2×C22⋊D5311yescharacteristic
198SD16—2128no—
199M16—42no—
200C2×C8—41no—
201C2×C8—1064no—
202C4⋊C4—22no—
203C4⋊C4—1062no—
204C4⋊C4—1062no—
205C4⋊C4—2128no—
206C4⋊C4—2122no—
207C4⋊C4—2122no—
208C22⋊C4—42no—
209C42—22no—
210C42—2122no—
211C42—2122no—
212C42#1171992314yescomplemented
213C42C2×D21211yescharacteristic
214C42#1171994511yescharacteristic
215C42D4×D5312yes—
216C23—41no—
217C23#1313318511yescharacteristic
218Q8—42no—
219Q8—1064no—
220Q8—1064no—
221Q8—2122no—
222Q8—2122no—
223D4—44no—
224D4—42no—
225D4—82no—
226C2×C4—24no—
227C2×C4—42no—
228C2×C4—44no—
229C2×C4—41no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—1064no—
233C2×C4—2124no—
234C2×C4—2122no—
235C2×C4—2122no—
236C2×C4—2122no—
237C2×C4#1313318712yes—
238C2×C4#1313319012yes—
239C2×C4#1313319111yescharacteristic
240C2×C4#1313321311yescharacteristic
241C2×C4C2×#1171992312yes—
242C2×C4#1313324712yes—
243C8—81no—
244C8—2124no—
245C22—22no—
246C22—82no—
247C22—81no—
248C22#1681053211yescharacteristic
249C22#1681076112yes—
250C4—22no—
251C4—22no—
252C4—44no—
253C4—81no—
254C4—4242no—
255C4—2124no—
256C4#1681082812yes—
257C4#1681099511yescharacteristic
258C4#1681101811yescharacteristic
259C2—22no—
260C2—161no—
261C2#2027578112yes—
262C2#2027579711yescharacteristic
subgroupG/H
centerC2×C4#13133213
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1963C2
socleC22×C53#392
prime radicalC53×#532C22
hypercenter#1963D53
hypocenterC53#10014
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1963

Complements

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

NH ≅ G/Ntype
2. #20278008C2semidirect
3. #20273143C2semidirect
4. #20272544C2semidirect
71. C42×C53SD16semidirect
152. C53#10014semidirect
212. C42#11719923semidirect

Maximal subgroups

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

subgroupindex
#202847422
#202780082
#202731432
#202725442
C53×#19632
#1001453