Finite Group Databasebeta

← #27296554

Subgroups — #27296554

Order 13568.

354 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 67 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#20286060C211yesmaximal, characteristic
2#20276649C212yesmaximal
3#20272726C212yesmaximal
4#20272655C211yesmaximal, characteristic
5C53×#2056C211yesmaximal, characteristic
6#16811180—21no—
7C2×#13133163—22no—
8#16810919—22no—
9#16810864—21no—
10#16810864C2212yes—
11#16810834—22no—
12#16810427—22no—
13C22⋊#11719908C2212yes—
14C22×Q16×C53C2211yescharacteristic
15C2×C4⋊C8×C53C2211yescharacteristic
16C2×#103×C53—21no—
17C53×#391—21no—
18C53×#340C2211yescharacteristic
19C2×Dic212—21no—
20C2×Dic212D412yes—
21#13133217—21no—
22#13133217—21no—
23#13133209—41no—
24#13133196—22no—
25#13133196—41no—
26C4×Dic106—22no—
27C4×Dic106—41no—
28#13133176—41no—
29#13133163—44no—
30C2×#11719908—22no—
31#13133151—21no—
32#13133151D412yes—
33#13133142—22no—
34#13133142—22no—
35C22×Q8×C53—21no—
36C2×Q16×C53—22no—
37C2×Q16×C53—42no—
38C22×C8×C53D411yescharacteristic
39C22×C8×C53C2311yescharacteristic
40C2×C4⋊C4×C53—21no—
41C2×C42×C53D411yescharacteristic
42C4⋊C8×C53—22no—
43#103×C53—42no—
44#95×C53—41no—
45#95×C53—41no—
46C2×Q8×C53—22no—
47C2×Q8×C53—41no—
48C2×Q8×C53—41no—
49C2×Q8×C53—41no—
50C2×Q8×C53—41no—
51C2×Q8×C53—41no—
52C22×C4×C53—41no—
53C22×C4×C53—41no—
54C22×C4×C53D811yescharacteristic
55C22×C4×C53D811yescharacteristic
56C22×C4×C53C2×D411yescharacteristic
57C2×Dic106—22no—
58C2×Dic106—41no—
59Q16×C53—24no—
60Q16×C53—42no—
61C2×C8×C53—21no—
62C2×C8×C53—41no—
63C2×C8×C53—41no—
64C2×C8×C53C2×D412yes—
65C4⋊C4×C53—41no—
66C4⋊C4×C53—41no—
67C42×C53—21no—
68C42×C53—21no—
69C22⋊Dic53—42no—
70C4⋊Dic53—22no—
71C4⋊Dic53—41no—
72C4⋊Dic53—41no—
73C4⋊Dic53—41no—
74#11719919—42no—
75#11719919—42no—
76C4×Dic53—42no—
77Dic212—42no—
78Dic212—81no—
79#11719908—42no—
80C23×C53C2×D811yescharacteristic
81Q8×C53—42no—
82Q8×C53—42no—
83Q8×C53—81no—
84Q8×C53—81no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—41no—
91C2×C4×C53—42no—
92C2×C4×C53—81no—
93C2×C4×C53—81no—
94C2×C4×C53—81no—
95C2×C4×C53—81no—
96C2×C4×C53—81no—
97C2×C4×C53Q3212yescomplemented
98C2×C4×C53#12011yescharacteristic
99C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
100C2×Dic53—42no—
101C2×Dic53—42no—
102Dic106—42no—
103Dic106—42no—
104Dic106—82no—
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—161no—
118C4×C53#45411yescharacteristic
119C4×C53B(4,𝔽2)11yescharacteristic
120Dic53—82no—
121Dic53—82no—
122#3249—531no—
123#2188—532no—
124#2056D5311yescharacteristic
125#1348—532no—
126#1322—531no—
127C2×C53—21no—
128C2×C53—21no—
129C2×C53#159811yescharacteristic
130C2×C53#218811yescharacteristic
131C2×C53#219111yescharacteristic
132C22×Q16C2×D5311yescharacteristic
133#533—1061no—
134C2×Q32—1062no—
135#477—1062no—
136#467—532no—
137#467—1061no—
138#462—1062no—
139C2×C4⋊C8C2×D5311yescharacteristic
140C2×#103—21no—
141#391—21no—
142#358—1062no—
143C22⋊C16—532no—
144#340C2×D5311yescharacteristic
145C53#1216211yescharacteristic, complemented, Sylow
146C22×Q8—21no—
147C2×Q16—22no—
148C2×Q16—42no—
149C2×Q16—532no—
150C2×Q16—1061no—
151C22×C8C22⋊D5311yescharacteristic
152C22×C8C22×D5311yescharacteristic
153C4⋊Q8—1061no—
154C4⋊Q8—1061no—
155#125—2121no—
156C22⋊Q8—1062no—
157C22⋊Q8—2121no—
158C4×Q8—1062no—
159C4×Q8—2121no—
160#117—2121no—
161C2×C4⋊C4—21no—
162C2×C42C22⋊D5311yescharacteristic
163Q32—2124no—
164C2×C16—1062no—
165#107—532no—
166#107—1061no—
167C4⋊C8—22no—
168#103—42no—
169#103—1062no—
170#103—1062no—
171#95—41no—
172#95—41no—
173C2×Q8—22no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—1062no—
180C2×Q8—2121no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4#1171992211yescharacteristic
184C22×C4#1171992211yescharacteristic
185C22×C4C2×C22⋊D5311yescharacteristic
186Q16—24no—
187Q16—42no—
188Q16—4241no—
189Q16—2122no—
190C2×C8—21no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8D4×D5312yes—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—1062no—
197C4⋊C4—2121no—
198C4⋊C4—2121no—
199C4⋊C4—2122no—
200C4⋊C4—2121no—
201C4⋊C4—2122no—
202C22⋊C4—2122no—
203C42—21no—
204C42—21no—
205C42—2122no—
206C16—2122no—
207C23C2×#1171992211yescharacteristic
208Q8—42no—
209Q8—42no—
210Q8—81no—
211Q8—81no—
212Q8—4242no—
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—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—2122no—
228C2×C4—2122no—
229C2×C4#1313316312yescomplemented
230C2×C4#1313318511yescharacteristic
231C2×C4#1313324211yescharacteristic
232C8—22no—
233C8—41no—
234C22—21no—
235C22—21no—
236C22#1681075611yescharacteristic
237C22C2×#1313316311yescharacteristic
238C22#1681106311yescharacteristic
239C4—22no—
240C4—21no—
241C4—41no—
242C4—82no—
243C4—161no—
244C4—4242no—
245C4—4242no—
246C4#1681079511yescharacteristic
247C4#1681080911yescharacteristic
248C2—21no—
249C2—21no—
250C2#2027364911yescharacteristic
251C2#2027664811yescharacteristic
252C2#2027666511yescharacteristic
subgroupG/H
centerC22#16810756
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
FittingC53×#2056C2
socleC22×C53#447
prime radicalC23×C53C2×D8
hypercenter#2056D53
hypocenterC53#12162
LCSG ▷ C22×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #2056

Complements

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

NH ≅ G/Ntype
97. 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
#202860602
#202766492
#202727262
#202726552
C53×#20562
#1216253