Finite Group Databasebeta

← #27273362

Subgroups — #27273362

Order 13568.

418 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 83 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#20281645C211yesmaximal, characteristic
2#20276279C212yesmaximal, complemented
3#20276248C211yesmaximal, characteristic
4#20272537C212yesmaximal, complemented
5C53×#1270C211yesmaximal, characteristic
6C2×#13133198—21no—
7C2×#13133148—21no—
8C2×#13133138—21no—
9#16810553—21no—
10#16810548—22no—
11#16810521C2212yes—
12#16810498—22no—
13#16810493C2211yescharacteristic
14#16810386—22no—
15#16810386—22no—
16#16810378C2211yescharacteristic
17C53×#390C2211yescharacteristic
18C53×#336C2212yes—
19C23×Dic53—21no—
20C22×#88322—21no—
21#13133198—42no—
22#13133198—42no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—22no—
26C2×C4⋊Dic53D412yes—
27C2×C4⋊Dic53D412yes—
28C2×#11719919—22no—
29C2×C22⋊Dic53—22no—
30C2×C22⋊Dic53—21no—
31#13133148—42no—
32#13133138—42no—
33#13133124—22no—
34#13133124—21no—
35#13133124—22no—
36C22×D4×C53D411yescharacteristic
37C22×C8×C53—22no—
38C2×C4⋊C4×C53D411yescharacteristic
39C2×C22⋊C4×C53—22no—
40C2×C42×C53C2311yescharacteristic
41C24×C53—21no—
42C2×D4×C53—21no—
43C2×D4×C53—41no—
44C2×D4×C53—41no—
45C2×D4×C53SD1612yes—
46C22×C4×C53—21no—
47C22×C4×C53C2×D411yescharacteristic
48C22×C4×C53C2×D412yes—
49C22×Dic53—22no—
50C22×Dic53—22no—
51C22×Dic53—42no—
52C22×Dic53—41no—
53C22×Dic53—81no—
54C22×Dic53—81no—
55C2×C8×C53—42no—
56C2×C8×C53—42no—
57C2×C8×C53—42no—
58C4⋊C4×C53—41no—
59C22⋊C4×C53—42no—
60C22⋊C4×C53—42no—
61C42×C53—41no—
62C22⋊Dic53—44no—
63C22⋊Dic53—42no—
64C4⋊Dic53—22no—
65C4⋊Dic53—22no—
66C4⋊Dic53—42no—
67C4⋊Dic53—41no—
68C4⋊Dic53—41no—
69C4⋊Dic53—44no—
70C4⋊Dic53—81no—
71#11719919—42no—
72#11719919—42no—
73C2×#88322—22no—
74C2×#88322—41no—
75C2×#88322—41no—
76C23×C53—22no—
77C23×C53—42no—
78C23×C53—41no—
79C23×C53—42no—
80C23×C53—41no—
81C23×C53#12011yescharacteristic
82D4×C53—42no—
83D4×C53—81no—
84C2×C4×C53—21no—
85C2×C4×C53—22no—
86C2×C4×C53—21no—
87C2×C4×C53—22no—
88C2×C4×C53—42no—
89C2×C4×C53—41no—
90C2×C4×C53C2×SD1611yescharacteristic
91C2×C4×C53#13512yes—
92C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
93C2×C4×C53#13712yes—
94C2×Dic53—42no—
95C2×Dic53—42no—
96C2×Dic53—42no—
97C2×Dic53—42no—
98C2×Dic53—42no—
99C2×Dic53—42no—
100C2×Dic53—84no—
101C2×Dic53—81no—
102C2×Dic53—82no—
103C8×C53—82no—
104#88322—42no—
105#7824—531nomaximal, Sylow
106C22×C53—42no—
107C22×C53—44no—
108C22×C53—82no—
109C22×C53—81no—
110C22×C53—81no—
111C22×C53#45011yescharacteristic
112C22×C53#45212yes—
113C22×C53#45311yescharacteristic
114C22×C53#45412yes—
115C22×C53#45811yescharacteristic
116C4×C53—21no—
117C4×C53—21no—
118C4×C53—22no—
119C4×C53—22no—
120C4×C53—81no—
121Dic53—82no—
122Dic53—82no—
123Dic53—161no—
124#2724—531no—
125#2100—532no—
126#2094—531no—
127#1271—532no—
128#1270D5311yescharacteristic
129C2×C53—82no—
130C2×C53#161311yescharacteristic
131C2×C53#161512yes—
132C2×C53#163011yescharacteristic
133C2×C53#163512yes—
134C2×C53#165711yescharacteristic
135C2×#123—1061no—
136C2×#106—1061no—
137C2×#102—1061no—
138#398—1061no—
139#397—1062no—
140#390C2×D5311yescharacteristic
141#389—532no—
142#385—1062no—
143#384—531no—
144#340—1062no—
145#340—1062no—
146#336—531no—
147#336C2×D5312yes—
148C53#782411yescharacteristic, complemented, Sylow
149C22×D4D21211yescharacteristic
150C23×C4—1061no—
151C22×C8—22no—
152C22×C8—1061no—
153#123—2122no—
154#123—2122no—
155C2×C4⋊C4—532no—
156C2×C4⋊C4—532no—
157C2×C4⋊C4—1061no—
158C2×C4⋊C4—1062no—
159C2×C4⋊C4—1061no—
160C2×C4⋊C4—1062no—
161C2×C4⋊C4D21211yescharacteristic
162C2×C22⋊C4—22no—
163C2×C22⋊C4—1062no—
164C2×C22⋊C4—1061no—
165C2×C42C22×D5311yescharacteristic
166#106—2122no—
167#102—2122no—
168#95—1062no—
169#95—1061no—
170#95—1062no—
171C24—21no—
172C2×D4—21no—
173C2×D4—41no—
174C2×D4—41no—
175C2×D4#1171991312yes—
176C22×C4—21no—
177C22×C4—1062no—
178C22×C4—1062no—
179C22×C4—4241no—
180C22×C4—4241no—
181C22×C4—2121no—
182C22×C4—2122no—
183C22×C4C2×D21211yescharacteristic
184C22×C4D4×D5312yes—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—1062no—
189C2×C8—2121no—
190C2×C8—2121no—
191C4⋊C4—41no—
192C4⋊C4—1062no—
193C4⋊C4—1062no—
194C4⋊C4—4241no—
195C4⋊C4—2121no—
196C4⋊C4—2121no—
197C4⋊C4—2124no—
198C4⋊C4—2122no—
199C4⋊C4—2122no—
200C4⋊C4—2122no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—2122no—
204C22⋊C4—2124no—
205C42—41no—
206C23—22no—
207C23—42no—
208C23—41no—
209C23—42no—
210C23—41no—
211C23#1313318711yescharacteristic
212D4—42no—
213D4—81no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—4241no—
221C2×C4—4242no—
222C2×C4—4244no—
223C2×C4—2122no—
224C2×C4—2122no—
225C2×C4—2122no—
226C2×C4—2122no—
227C2×C4—2122no—
228C2×C4—2122no—
229C2×C4C2×#1171991311yescharacteristic
230C2×C4#1313323612yes—
231C2×C4#1313324511yescharacteristic
232C2×C4#1313324612yes—
233C8—82no—
234C8—2122no—
235C22—42no—
236C22—44no—
237C22—82no—
238C22—81no—
239C22—81no—
240C22#1681077611yescharacteristic
241C22#1681078612yes—
242C22#1681079311yescharacteristic
243C22#1681079712yes—
244C22#1681081411yescharacteristic
245C4—21no—
246C4—21no—
247C4—22no—
248C4—22no—
249C4—81no—
250C4—8481no—
251C4—4242no—
252C4—4242no—
253C2—82no—
254C2#2027373711yescharacteristic
255C2#2027374712yes—
256C2#2027383711yescharacteristic
257C2#2027386812yes—
258C2#2027401411yescharacteristic
subgroupG/H
centerC23#13133187
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1270C2
socleC23×C53#120
prime radicalC22×D4×C53D4
hypercenter#1270D53
hypocenterC53#7824
LCSG ▷ C2×C42×C53 ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #20276279C2semidirect
4. #20272537C2semidirect
148. C53#7824semidirect

Maximal subgroups

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

subgroupindex
#202816452
#202762792
#202762482
#202725372
C53×#12702
#782453