Finite Group Databasebeta

← #27284709

Subgroups — #27284709

Order 13568.

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

normal subgroups only

This group has 544 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#20285466C211yesmaximal, characteristic
2#20284655C212yesmaximal, complemented
3#20273074C212yesmaximal, complemented
4#20272592C211yesmaximal, characteristic, complemented
5C53×#1818C211yesmaximal, characteristic, complemented
6C2×#13133191C2212yes—
7#16810947—24no—
8#16810941—22no—
9#16810937—22no—
10#16810936—22no—
11#16810859—22no—
12#16810759—22no—
13#16810666—22no—
14#16810627—22no—
15#16810587C2212yes—
16#16810346—24no—
17C2×#117×C53C2211yescharacteristic
18C2×C22⋊C8×C53C2211yescharacteristic
19C2×C22⋊C8×C53C2211yescharacteristic
20C53×#336—22no—
21C22×D212D412yes—
22C22×C22⋊D53—22no—
23C2×D424—22no—
24C2×#11719917—22no—
25#13133191—22no—
26#13133191—22no—
27#13133191—22no—
28#13133191—22no—
29C2×C4⋊Dic53D412yes—
30C2×#11719921—22no—
31#13133151—22no—
32#13133150—24no—
33#13133148—24no—
34#13133139—24no—
35#13133137—24no—
36#13133137—24no—
37C22⋊#88322—22no—
38C22⋊#88322D414yes—
39C23×C4×C53C2311yescharacteristic
40C22×C8×C53—21no—
41C22×C8×C53D412yes—
42#117×C53—22no—
43#117×C53—22no—
44C2×C4⋊C4×C53D411yescharacteristic
45C2×C4⋊C4×C53D411yescharacteristic
46C2×C22⋊C4×C53—21no—
47C2×C42×C53—21no—
48C22⋊C8×C53—22no—
49C22⋊C8×C53—22no—
50C24×C53#4111yescharacteristic
51C23×D53—22no—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—42no—
55C22×C4×C53C2×D412yes—
56C22×C4×C53C2×D411yescharacteristic
57C22×C4×C53C2×D411yescharacteristic
58C22×C4×C53C2×D412yes—
59C2×C22⋊D53—42no—
60C2×C22⋊D53—42no—
61C2×C22⋊D53—84no—
62C22×Dic53—22no—
63C2×D212—22no—
64C2×D212—22no—
65C2×D212—44no—
66C2×C8×C53—22no—
67C2×C8×C53—42no—
68C2×C8×C53—42no—
69C2×C8×C53—41no—
70C4⋊C4×C53—22no—
71C4⋊C4×C53—22no—
72C22⋊C4×C53—42no—
73C42×C53—41no—
74C42×C53—41no—
75#11719921—42no—
76#11719921—42no—
77C4⋊Dic53—22no—
78C4⋊Dic53—22no—
79#11719917—44no—
80C2×#88322—24no—
81D424—82no—
82C23×C53—22no—
83C23×C53—41no—
84C23×C53—41no—
85C23×C53#12112yes—
86C23×C53#12111yescharacteristic
87C22×D53—42no—
88C22×D53—44no—
89C22×D53—82no—
90C22×D53—82no—
91C2×C4×C53—24no—
92C2×C4×C53—22no—
93C2×C4×C53—41no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53—42no—
99C2×C4×C53#12011yescharacteristic
100C2×C4×C53#12012yes—
101C2×C4×C53#12711yescharacteristic
102C22⋊D53—84no—
103C22⋊D53—84no—
104C2×Dic53—42no—
105C2×Dic53—82no—
106D212—44no—
107D212—44no—
108C8×C53—42no—
109C8×C53—81no—
110#88322—44no—
111#9977—531nomaximal, Sylow
112C22×C53—21no—
113C22×C53—21no—
114C22×C53—22no—
115C22×C53—22no—
116C22×C53—22no—
117C22×C53—82no—
118C22×C53#39211yescharacteristic
119C22×C53#47212yes—
120C2×D53—82no—
121C2×D53—82no—
122C2×D53—82no—
123C2×D53—164no—
124C4×C53—22no—
125C4×C53—21no—
126C4×C53—81no—
127C4×C53—81no—
128C4×C53#45812yes—
129Dic53—82no—
130#3186—531no—
131#3094—532no—
132#1818D5311yescharacteristic, complemented
133#1471—532no—
134#1293—531no—
135C2×C53—22no—
136C2×C53—42no—
137C2×C53#201711yescharacteristic
138C2×C53#201811yescharacteristic
139C2×C53#202211yescharacteristic
140D53—162no—
141C2×#121—532no—
142C2×#117C2×D5311yescharacteristic
143#482—1064no—
144#481—1062no—
145#480—1062no—
146#480—1062no—
147#466—1062no—
148#447—1062no—
149#428—1062no—
150#418—1062no—
151#407—532no—
152C2×C22⋊C8C2×D5311yescharacteristic
153C2×C22⋊C8C2×D5311yescharacteristic
154#336—22no—
155#323—1064no—
156C53#997711yescharacteristic, complemented, Sylow
157C22×D4—532no—
158C22×D4—1062no—
159C23×C4C22×D5311yescharacteristic
160C2×D8—1062no—
161C2×M16—1062no—
162C22×C8—21no—
163C22×C8C22⋊D5312yes—
164#121—1062no—
165#121—1062no—
166#121—1062no—
167#121—1062no—
168#117—22no—
169#117—22no—
170C2×C4⋊C4—532no—
171C2×C4⋊C4D21211yescharacteristic
172C2×C4⋊C4D21211yescharacteristic
173C2×C22⋊C4—21no—
174C2×C22⋊C4—1062no—
175C2×C42—21no—
176#107—1064no—
177#107—1062no—
178#106—1064no—
179#102—1064no—
180#102—1064no—
181#102—1064no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—534no—
185C22⋊C8—1062no—
186C24—1062no—
187C24#1171994511yescharacteristic
188C2×D4—1062no—
189C2×D4—1062no—
190C2×D4—4244no—
191C2×D4—2124no—
192C2×D4—2122no—
193C2×D4—2122no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—42no—
197C22×C4—1062no—
198C22×C4C2×D21211yescharacteristic
199C22×C4D4×D5312yes—
200C22×C4D4×D5312yes—
201C22×C4C2×C22⋊D5311yescharacteristic
202D8—4242no—
203M16—2124no—
204C2×C8—22no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—41no—
208C2×C8—1064no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—1062no—
212C4⋊C4—1062no—
213C22⋊C4—42no—
214C22⋊C4—2122no—
215C22⋊C4—2122no—
216C42—41no—
217C42—41no—
218C23—22no—
219C23—41no—
220C23—41no—
221C23—4242no—
222C23—4242no—
223C23—2124no—
224C23—2122no—
225C23#1313319012yes—
226C23#1313319111yescharacteristic
227D4—4244no—
228D4—4244no—
229D4—2124no—
230D4—2124no—
231C2×C4—22no—
232C2×C4—24no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—4242no—
240C2×C4—2122no—
241C2×C4#1313318511yescharacteristic
242C2×C4#1313318712yes—
243C2×C4#1313321311yescharacteristic
244C8—42no—
245C8—81no—
246C8—2124no—
247C22—21no—
248C22—21no—
249C22—22no—
250C22—22no—
251C22—22no—
252C22—82no—
253C22—8484no—
254C22—4242no—
255C22—4242no—
256C22—4242no—
257C22#1681053211yescharacteristic
258C22#1681089312yes—
259C4—21no—
260C4—22no—
261C4—81no—
262C4—81no—
263C4—4242no—
264C4#1681081212yes—
265C2—22no—
266C2—42no—
267C2—8482no—
268C2#2027581911yescharacteristic
269C2#2027582411yescharacteristic
270C2#2027584611yescharacteristic
subgroupG/H
centerC22#16810532
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#1818C2
socleC22×C53#392
prime radical#20285466C2
hypercenter#1818D53
hypocenterC53#9977
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1818

Complements

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

NH ≅ G/Ntype
2. #20284655C2semidirect
3. #20273074C2semidirect
4. #20272592C2semidirect
5. C53×#1818C2semidirect
132. #1818D53semidirect
156. C53#9977semidirect

Maximal subgroups

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

subgroupindex
#202854662
#202846552
#202730742
#202725922
C53×#18182
#997753