Finite Group Databasebeta

← #27281954

Subgroups — #27281954

Order 13568.

454 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 111 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#20283800C211yesmaximal, characteristic
2#20280077C211yesmaximal, characteristic
3#20273143C212yesmaximal
4#20272575C212yesmaximal
5C53×#1801C211yesmaximal, characteristic
6C2×C4×Dic106C2211yescharacteristic
7#16810754—24no—
8#16810680—22no—
9#16810676C2212yes—
10#16810562C412yes—
11#16810543C412yes—
12#16810530—21no—
13#16810519—21no—
14#16810507—22no—
15#16810489C2211yescharacteristic
16#16810353—24no—
17C2×C4×Q8×C53C2211yescharacteristic
18C53×#423C2212yes—
19C2×#101×C53—21no—
20C53×#343—21no—
21C22×Dic106C2×C411yescharacteristic
22C4×Dic106—22no—
23C4×Dic106—22no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×#11719919—22no—
27C2×#11719919—21no—
28C2×C4×Dic53—21no—
29C4⋊#88322—24no—
30C4⋊#88322—22no—
31C22⋊#88322—22no—
32#13133128—24no—
33C4×#88322D414yes—
34#13133124—22no—
35#13133124—22no—
36#13133124—22no—
37#13133124—21no—
38#13133124C2×C412yes—
39C22×Q8×C53D411yescharacteristic
40C2×M16×C53—22no—
41C4×Q8×C53—22no—
42C4×Q8×C53—22no—
43C2×C4⋊C4×C53C2×C412yes—
44C2×C4⋊C4×C53D411yescharacteristic
45C2×C42×C53D412yes—
46C2×C42×C53C2311yescharacteristic
47C4⋊C8×C53—22no—
48C4⋊C8×C53—22no—
49#101×C53—42no—
50C2×Q8×C53—22no—
51C2×Q8×C53—21no—
52C2×Q8×C53—21no—
53C2×Q8×C53—22no—
54C22×C4×C53C22⋊C412yes—
55C22×C4×C53C22⋊C412yes—
56C22×C4×C53C22×C411yescharacteristic
57C22×C4×C53C2×D411yescharacteristic
58C22×C4×C53C2×D411yescharacteristic
59C22×Dic53—21no—
60C22×Dic53—22no—
61C22×Dic53—21no—
62C2×Dic106—22no—
63C2×Dic106—42no—
64M16×C53—42no—
65M16×C53—42no—
66C2×C8×C53—42no—
67C4⋊C4×C53—24no—
68C4⋊C4×C53—22no—
69C42×C53—22no—
70C42×C53—22no—
71C42×C53C2×D412yes—
72C42×C53#4112yes—
73C4⋊Dic53—22no—
74C4⋊Dic53—41no—
75C4⋊Dic53—41no—
76#11719919—44no—
77#11719919—41no—
78#11719919—41no—
79C4×Dic53—42no—
80C2×#88322—24no—
81C23×C53C2×C22⋊C411yescharacteristic
82Q8×C53—42no—
83Q8×C53—42no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53—42no—
91C2×C4×C53C4×D412yes—
92C2×C4×C53#12011yescharacteristic
93C2×C4×C53#12112yes—
94C2×C4×C53#12311yescharacteristic
95C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
96C2×C4×C53#13712yes—
97C2×C4×C53#13711yescharacteristic
98C2×Dic53—22no—
99C2×Dic53—41no—
100C2×Dic53—41no—
101C2×Dic53—42no—
102C2×Dic53—42no—
103C2×Dic53—42no—
104C2×Dic53—41no—
105C2×Dic53—42no—
106Dic106—24no—
107Dic106—42no—
108C8×C53—82no—
109#88322—44no—
110#9488—531nomaximal, Sylow
111C22×C53—21no—
112C22×C53—21no—
113C22×C53#38611yescharacteristic
114C22×C53#41711yescharacteristic
115C22×C53#41911yescharacteristic
116C4×C53—22no—
117C4×C53—21no—
118C4×C53—21no—
119C4×C53—42no—
120C4×C53—42no—
121C4×C53#46812yes—
122C4×C53#47012yes—
123Dic53—42no—
124Dic53—82no—
125Dic53—81no—
126#2983—531no—
127#2581—531no—
128#1801D5311yescharacteristic
129#1487—532no—
130#1286—532no—
131C2×C53—21no—
132C2×C53—21no—
133C2×C53#189111yescharacteristic
134C2×C53#189411yescharacteristic
135C2×C53#191211yescharacteristic
136C2×C4×Q8—531no—
137C2×C4×Q8C2×D5311yescharacteristic
138#446—1064no—
139#432—1062no—
140#431—532no—
141#423C2×D5312yes—
142C2×#101—21no—
143#400—532no—
144#395—532no—
145#391—1061no—
146#389—1061no—
147#387—1062no—
148#382—531no—
149#343—21no—
150Q8⋊C8—1064no—
151C53#948811yescharacteristic, complemented, Sylow
152C22×Q8—531no—
153C22×Q8C22⋊D5311yescharacteristic
154C2×M16—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—1062no—
158C4×Q8—1062no—
159C2×C4⋊C4—531no—
160C2×C4⋊C4—1061no—
161C2×C4⋊C4—1061no—
162C2×C4⋊C4—1062no—
163C2×C4⋊C4C4×D5312yes—
164C2×C4⋊C4C22⋊D5311yescharacteristic
165C2×C42—1061no—
166C2×C42D21212yes—
167C2×C42C22×D5311yescharacteristic
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—1064no—
171C4⋊C8—1062no—
172#101—42no—
173C22⋊C8—1062no—
174#97—1064no—
175C4×C8—534no—
176#95—532no—
177#95—1061no—
178#95—1062no—
179#95—1062no—
180#95—1062no—
181C2×Q8—22no—
182C2×Q8—21no—
183C2×Q8—22no—
184C2×Q8—21no—
185C2×Q8—1062no—
186C2×Q8—2122no—
187C22×C4—1061no—
188C22×C4—1062no—
189C22×C4—1061no—
190C22×C4#1171992112yes—
191C22×C4#1171992112yes—
192C22×C4C2×C4×D5311yescharacteristic
193C22×C4C2×D21211yescharacteristic
194C22×C4C2×C22⋊D5311yescharacteristic
195M16—42no—
196M16—42no—
197C2×C8—42no—
198C2×C8—1064no—
199C4⋊C4—24no—
200C4⋊C4—22no—
201C4⋊C4—1062no—
202C4⋊C4—2124no—
203C4⋊C4—2121no—
204C4⋊C4—2121no—
205C4⋊C4—2121no—
206C4⋊C4—2121no—
207C42—22no—
208C42—22no—
209C42—2122no—
210C42D4×D5312yes—
211C42#1171994712yes—
212C23C2×#1171992111yescharacteristic
213Q8—42no—
214Q8—42no—
215Q8—1064no—
216Q8—2122no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—1062no—
225C2×C4—2122no—
226C2×C4—2122no—
227C2×C4—2122no—
228C2×C4—2121no—
229C2×C4—2122no—
230C2×C4—2121no—
231C2×C4—2121no—
232C2×C4#1313317712yes—
233C2×C4#1313318711yescharacteristic
234C2×C4#1313318912yes—
235C2×C4#1313320111yescharacteristic
236C2×C4#1313324311yescharacteristic
237C2×C4#1313324712yes—
238C2×C4#1313324811yescharacteristic
239C8—82no—
240C8—2124no—
241C22—21no—
242C22—21no—
243C22#1681050011yescharacteristic
244C22#1681062211yescharacteristic
245C22#1681063111yescharacteristic
246C4—21no—
247C4—21no—
248C4—22no—
249C4—42no—
250C4—42no—
251C4—4242no—
252C4—4241no—
253C4—2122no—
254C4#1681086712yes—
255C4#1681087912yes—
256C2—21no—
257C2—21no—
258C2#2027513111yescharacteristic
259C2#2027514611yescharacteristic
260C2#2027525011yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1801C2
socleC22×C53#386
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1801D53
hypocenterC53#9488
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
151. C53#9488semidirect

Maximal subgroups

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

subgroupindex
#202838002
#202800772
#202731432
#202725752
C53×#18012
#948853