Finite Group Databasebeta

← #27296555

Subgroups — #27296555

Order 13568.

354 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 71 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#20276648C212yesmaximal
2#20276038C211yesmaximal, characteristic
3#20272727C212yesmaximal
4#20272655C211yesmaximal, characteristic
5C53×#3249C211yesmaximal, characteristic
6C22×Dic212C2211yescharacteristic
7C2×#13133163—22no—
8C2×#13133142—21no—
9#16810529—21no—
10#16810426—22no—
11C22⋊#11719908C2212yes—
12#16810386C2211yescharacteristic
13C53×#533—21no—
14C53×#477—22no—
15C53×#467—21no—
16C53×#467C2212yes—
17C53×#462—22no—
18C2×C4⋊C8×C53C2211yescharacteristic
19C22×Dic106—21no—
20C2×Dic212—22no—
21C2×Dic212—42no—
22C2×C4⋊Dic53—21no—
23#13133163—44no—
24C2×#11719908—22no—
25#13133142—42no—
26#13133124—41no—
27#13133124—41no—
28C2×Q16×C53—21no—
29C2×Q16×C53D412yes—
30C22×C8×C53D411yescharacteristic
31C22×C8×C53C2311yescharacteristic
32C4⋊Q8×C53—21no—
33C4⋊Q8×C53—21no—
34#125×C53—41no—
35C22⋊Q8×C53—22no—
36C22⋊Q8×C53—41no—
37C4×Q8×C53—22no—
38C4×Q8×C53—41no—
39#117×C53—41no—
40C2×C42×C53D411yescharacteristic
41#107×C53—21no—
42#107×C53D412yes—
43C4⋊C8×C53—22no—
44#103×C53—22no—
45#103×C53—22no—
46C2×Q8×C53—22no—
47C2×Q8×C53—41no—
48C22×C4×C53D811yescharacteristic
49C22×C4×C53D811yescharacteristic
50C22×C4×C53C2×D411yescharacteristic
51C22×Dic53—41no—
52C22×Dic53—41no—
53C2×Dic106—22no—
54C2×Dic106—41no—
55C2×Dic106—41no—
56C2×Dic106—41no—
57C2×Dic106—41no—
58C2×Dic106—41no—
59Q16×C53—42no—
60Q16×C53—81no—
61C2×C8×C53—21no—
62C2×C8×C53—41no—
63C2×C8×C53—41no—
64C2×C8×C53C2×D412yes—
65C4⋊C4×C53—22no—
66C4⋊C4×C53—41no—
67C4⋊C4×C53—41no—
68C4⋊C4×C53—42no—
69C4⋊C4×C53—41no—
70C4⋊C4×C53—42no—
71C22⋊C4×C53—42no—
72C42×C53—21no—
73C42×C53—21no—
74C42×C53—42no—
75C4⋊Dic53—41no—
76C4⋊Dic53—41no—
77Dic212—24no—
78Dic212—42no—
79#11719908—42no—
80C23×C53C2×D811yescharacteristic
81Q8×C53—42no—
82Q8×C53—42no—
83Q8×C53—82no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—41no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53Q3212yescomplemented
93C2×C4×C53#12011yescharacteristic
94C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
95C2×Dic53—42no—
96C2×Dic53—81no—
97C2×Dic53—81no—
98C2×Dic53—81no—
99C2×Dic53—81no—
100C2×Dic53—81no—
101Dic106—42no—
102Dic106—42no—
103Dic106—81no—
104Dic106—81no—
105C8×C53—22no—
106C8×C53—41no—
107#12162—531nomaximal, Sylow
108C22×C53—21no—
109C22×C53—21no—
110C22×C53#44711yescharacteristic
111C22×C53C2×Q3211yescharacteristic
112C22×C53#51011yescharacteristic
113C4×C53—21no—
114C4×C53—22no—
115C4×C53—41no—
116C4×C53—82no—
117C4×C53—82no—
118C4×C53#45411yescharacteristic
119C4×C53B(4,𝔽2)11yescharacteristic
120Dic53—82no—
121Dic53—161no—
122#3249D5311yescharacteristic
123#2188—532no—
124#2056—531no—
125#1348—532no—
126#1322—531no—
127C2×C53—21no—
128C2×C53—21no—
129C2×C53#159811yescharacteristic
130C2×C53#218811yescharacteristic
131C2×C53#219111yescharacteristic
132C22×Q16—531no—
133#533—21no—
134C2×Q32—1062no—
135#477—22no—
136#467—21no—
137#467C2×D5312yes—
138#462—22no—
139C2×C4⋊C8C2×D5311yescharacteristic
140C2×#103—1061no—
141#391—1061no—
142#358—1062no—
143C22⋊C16—532no—
144#340—531no—
145C53#1216211yescharacteristic, complemented, Sylow
146C22×Q8—1061no—
147C2×Q16—21no—
148C2×Q16—1062no—
149C2×Q16—2122no—
150C2×Q16C22⋊D5312yes—
151C22×C8C22⋊D5311yescharacteristic
152C22×C8C22×D5311yescharacteristic
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155#125—41no—
156C22⋊Q8—22no—
157C22⋊Q8—41no—
158C4×Q8—22no—
159C4×Q8—41no—
160#117—41no—
161C2×C4⋊C4—1061no—
162C2×C42C22⋊D5311yescharacteristic
163Q32—2124no—
164C2×C16—1062no—
165#107—21no—
166#107C22⋊D5312yes—
167C4⋊C8—22no—
168#103—22no—
169#103—22no—
170#103—2122no—
171#95—2121no—
172#95—2121no—
173C2×Q8—22no—
174C2×Q8—41no—
175C2×Q8—1062no—
176C2×Q8—2121no—
177C2×Q8—2121no—
178C2×Q8—2121no—
179C2×Q8—2121no—
180C2×Q8—2121no—
181C22×C4—2121no—
182C22×C4—2121no—
183C22×C4#1171992211yescharacteristic
184C22×C4#1171992211yescharacteristic
185C22×C4C2×C22⋊D5311yescharacteristic
186Q16—42no—
187Q16—81no—
188Q16—1064no—
189Q16—2122no—
190C2×C8—21no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8C2×C22⋊D5312yes—
194C4⋊C4—22no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—41no—
199C4⋊C4—42no—
200C4⋊C4—2121no—
201C4⋊C4—2121no—
202C22⋊C4—42no—
203C42—21no—
204C42—21no—
205C42—42no—
206C16—2122no—
207C23C2×#1171992211yescharacteristic
208Q8—42no—
209Q8—42no—
210Q8—82no—
211Q8—4241no—
212Q8—4241no—
213Q8—2122no—
214Q8—2122no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—4241no—
224C2×C4—4241no—
225C2×C4—4241no—
226C2×C4—4241no—
227C2×C4—4241no—
228C2×C4—2122no—
229C2×C4#1313316312yescomplemented
230C2×C4#1313318611yescharacteristic
231C2×C4#1313324311yescharacteristic
232C8—22no—
233C8—41no—
234C22—21no—
235C22—21no—
236C22#1681075711yescharacteristic
237C22C2×#1313316311yescharacteristic
238C22#1681106411yescharacteristic
239C4—22no—
240C4—21no—
241C4—41no—
242C4—82no—
243C4—82no—
244C4—8481no—
245C4—4242no—
246C4#1681079611yescharacteristic
247C4#1681081011yescharacteristic
248C2—21no—
249C2—21no—
250C2#2027365011yescharacteristic
251C2#2027664911yescharacteristic
252C2#2027666611yescharacteristic
subgroupG/H
centerC22#16810757
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
FittingC53×#3249C2
socleC22×C53#447
prime radicalC23×C53C2×D8
hypercenter#3249D53
hypocenterC53#12162
LCSG ▷ C22×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #3249

Complements

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

NH ≅ G/Ntype
92. C2×C4×C53Q32semidirect
145. C53#12162semidirect
229. C2×C4#13133163semidirect

Maximal subgroups

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

subgroupindex
#202766482
#202760382
#202727272
#202726552
C53×#32492
#1216253