Finite Group Databasebeta

← #27284741

Subgroups — #27284741

Order 13568.

602 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 81 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#20284651C212yesmaximal, complemented
2#20274790C211yesmaximal, characteristic, complemented
3#20273244C212yesmaximal, complemented
4#20272623C211yesmaximal, characteristic, complemented
5C53×#2815C211yesmaximal, characteristic
6C2×#13133176C2211yescharacteristic
7#16810944—24no—
8#16810939—22no—
9#16810934—22no—
10#16810663—22no—
11#16810628—22no—
12#16810587C2212yes—
13#16810486C2211yescharacteristic
14#16810389—22no—
15#16810357—24no—
16C2×#121×C53C2212yes—
17C53×#397—22no—
18C53×#392—22no—
19C53×#392—22no—
20C53×#381C2211yescharacteristic
21C2×#11719917—22no—
22#13133176—22no—
23#13133176—22no—
24C2×C4⋊Dic53D412yescomplemented
25C2×C22⋊Dic53—21no—
26C2×C22⋊Dic53—21no—
27C2×C4×Dic53—21no—
28#13133150—24no—
29#13133148—24no—
30#13133138—24no—
31#13133138—24no—
32C22⋊#88322—22no—
33C22⋊#88322D414yescomplemented
34#13133124—22no—
35C22×D4×C53—22no—
36C22×D4×C53—22no—
37C22×D4×C53D412yes—
38C23×C4×C53C2311yescharacteristic
39#121×C53—22no—
40#121×C53—22no—
41#121×C53—22no—
42#121×C53—22no—
43C2×C4⋊C4×C53D412yes—
44C2×C22⋊C4×C53—24no—
45C2×C22⋊C4×C53—22no—
46C2×C22⋊C4×C53D412yes—
47#95×C53—22no—
48#95×C53—22no—
49C24×C53—22no—
50C24×C53C2×D411yescharacteristic
51C2×D4×C53—22no—
52C2×D4×C53—22no—
53C2×D4×C53—44no—
54C2×D4×C53—42no—
55C2×D4×C53—42no—
56C2×D4×C53—84no—
57C2×D4×C53—84no—
58C2×D4×C53—82no—
59C22×C4×C53—22no—
60C22×C4×C53—22no—
61C22×C4×C53—42no—
62C22×C4×C53C2×D412yes—
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53C2×D412yes—
65C22×C4×C53#4111yescharacteristic
66C22×Dic53—22no—
67C22×Dic53—41no—
68C4⋊C4×C53—22no—
69C4⋊C4×C53—22no—
70C22⋊C4×C53—42no—
71C22⋊C4×C53—42no—
72C22⋊C4×C53—42no—
73C22⋊C4×C53—84no—
74C22⋊Dic53—41no—
75C22⋊Dic53—41no—
76C22⋊Dic53—42no—
77C4⋊Dic53—22no—
78C4⋊Dic53—22no—
79C4×Dic53—42no—
80#11719917—44no—
81C2×#88322—24no—
82C23×C53—22no—
83C23×C53—42no—
84C23×C53—42no—
85C23×C53—44no—
86C23×C53—42no—
87C23×C53—42no—
88C23×C53—44no—
89C23×C53#12011yescharacteristic
90C23×C53#12111yescharacteristic
91C23×C53#12711yescharacteristic
92D4×C53—44no—
93D4×C53—82no—
94D4×C53—84no—
95D4×C53—84no—
96D4×C53—88no—
97C2×C4×C53—24no—
98C2×C4×C53—41no—
99C2×C4×C53—42no—
100C2×C4×C53—41no—
101C2×C4×C53—44no—
102C2×C4×C53—44no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53#12012yes—
106C2×C4×C53#12112yes—
107C2×Dic53—42no—
108C2×Dic53—82no—
109C2×Dic53—82no—
110C2×Dic53—81no—
111#88322—44no—
112#9983—531nomaximal, Sylow
113C22×C53—22no—
114C22×C53—22no—
115C22×C53—22no—
116C22×C53—21no—
117C22×C53—21no—
118C22×C53—82no—
119C22×C53—84no—
120C22×C53—84no—
121C22×C53—82no—
122C22×C53—82no—
123C22×C53—82no—
124C22×C53#39211yescharacteristic
125C22×C53#45812yes—
126C4×C53—22no—
127C4×C53—22no—
128C4×C53—82no—
129C4×C53—82no—
130Dic53—82no—
131Dic53—161no—
132#3094—532no—
133#2815D5311yescharacteristic
134#1820—531no—
135#1510—532no—
136#1306—531no—
137C2×C53—22no—
138C2×C53—41no—
139C2×C53—41no—
140C2×C53—162no—
141C2×C53#200812yes—
142C2×C53#202211yescharacteristic
143C2×#121C2×D5312yes—
144C2×#117—531no—
145#482—1064no—
146#481—1062no—
147#480—1062no—
148#428—1062no—
149#418—1062no—
150#407—532no—
151#397—22no—
152#392—22no—
153#392—22no—
154#381—531no—
155#381C2×D5311yescharacteristic
156#342—1062no—
157#327—1064no—
158C53#998311yescharacteristic, complemented, Sylow
159C22×D4—22no—
160C22×D4—22no—
161C22×D4C22⋊D5312yes—
162C23×C4C22×D5311yescharacteristic
163C2×M16—1062no—
164#121—22no—
165#121—22no—
166#121—22no—
167#121—22no—
168#117—1062no—
169#117—1062no—
170C2×C4⋊C4—532no—
171C2×C4⋊C4C22⋊D5312yes—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—1061no—
175C2×C22⋊C4—1061no—
176C2×C22⋊C4C22⋊D5312yes—
177C2×C42—1061no—
178#107—1064no—
179#106—1064no—
180#102—1064no—
181#102—1064no—
182C22⋊C8—534no—
183C22⋊C8—1062no—
184#95—22no—
185#95—22no—
186#95—1062no—
187C24—22no—
188C24C2×C22⋊D5311yescharacteristic
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—44no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—84no—
195C2×D4—84no—
196C2×D4—82no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—42no—
200C22×C4—1062no—
201C22×C4—2121no—
202C22×C4D4×D5311yescharacteristic
203C22×C4D4×D5312yes—
204C22×C4#1171994711yescharacteristic
205C22×C4C2×C22⋊D5312yes—
206M16—2124no—
207C2×C8—1064no—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—1062no—
211C4⋊C4—1062no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—84no—
216C22⋊C4—2122no—
217C22⋊C4—2121no—
218C22⋊C4—2121no—
219C42—2122no—
220C23—22no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—44no—
225C23—42no—
226C23—44no—
227C23#1313318611yescharacteristic
228C23#1313318811yescharacteristic
229C23#1313321311yescharacteristic
230D4—44no—
231D4—82no—
232D4—84no—
233D4—84no—
234D4—88no—
235C2×C4—24no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—4241no—
244C2×C4—4242no—
245C2×C4—4242no—
246C2×C4—2122no—
247C2×C4#1313318512yes—
248C2×C4#1313318912yes—
249C8—2124no—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—21no—
254C22—21no—
255C22—82no—
256C22—84no—
257C22—84no—
258C22—82no—
259C22—82no—
260C22—82no—
261C22#1681053111yescharacteristic
262C22#1681081312yes—
263C4—22no—
264C4—22no—
265C4—82no—
266C4—82no—
267C4—8481no—
268C4—4242no—
269C2—22no—
270C2—41no—
271C2—41no—
272C2—162no—
273C2#2027576812yes—
274C2#2027584411yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
Fitting, prime radicalC53×#2815C2
socleC22×C53#392
hypercenter#2815D53
hypocenterC53#9983
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2815

Complements

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

NH ≅ G/Ntype
1. #20284651C2semidirect
2. #20274790C2semidirect
3. #20273244C2semidirect
4. #20272623C2semidirect
24. C2×C4⋊Dic53D4semidirect
33. C22⋊#88322D4semidirect
158. C53#9983semidirect

Maximal subgroups

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

subgroupindex
#202846512
#202747902
#202732442
#202726232
C53×#28152
#998353