Finite Group Databasebeta

← #28311230

Subgroups — #28311230

Order 13568.

412 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 82 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#20291006C212yesmaximal
2#20290599C212yesmaximal, complemented
3#20289807C212yesmaximal, complemented
4#20283431C211yesmaximal, characteristic
5D53⋊#108—22no—
6#16811609C2214yes—
7#16811533—22no—
8#16811531—22no—
9#16811492—22no—
10C2×#13133223C2211yescharacteristic
11#16811169C2211yescharacteristic
12#16810694—22no—
13#16810692—22no—
14#16810646C411yescharacteristic
15#16810591C2211yescharacteristic
16#16810590C411yescharacteristic
17C2×#13133127C411yescharacteristic
18C2×#131×C53C411yescharacteristic
19D53⋊M16—24no—
20#13133288—42no—
21#13133287—42no—
22#13133278—24no—
23#13133277—24no—
24C2×#11719945C2311yescharacteristic
25#13133223—42no—
26C2×#11719917C2×C411yescharacteristic
27C2×#11719912D412yes—
28C22×#88322C2×C411yescharacteristic
29C2×C8×D53D412yes—
30#13133215D411yescharacteristic
31#13133213Q811yescharacteristic
32#13133206Q811yescharacteristic
33#13133204D411yescharacteristic
34#13133189—22no—
35C4×C22⋊D53—22no—
36C2×C4×Dic53C2×C411yescharacteristic
37#13133146—22no—
38C22⋊#88322—22no—
39#13133129—22no—
40#13133127—21no—
41#13133127—21no—
42C2×#41×C53C2×C411yescharacteristic
43#131×C53—42no—
44C2×M16×C53—21no—
45C2×M16×C53C2×C411yescharacteristic
46C22×C8×C53—21no—
47C22×C8×C53C2×C411yescharacteristic
48#41×C53—41no—
49#41×C53—41no—
50C2×Q8×C53C4⋊C411yescharacteristic
51C2×D4×C53—21no—
52C2×D4×C53C4⋊C411yescharacteristic
53C22×C4×C53—21no—
54C22×C4×C53C22×C411yescharacteristic
55C2×C22⋊D53—22no—
56C2×C22⋊D53#4112yes—
57C22×Dic53—21no—
58#11719945—41no—
59#11719945—41no—
60C2×D212C2×Q811yescharacteristic
61C2×C4×D53C2×D412yes—
62C2×Dic106C2×D411yescharacteristic
63#11719940—44no—
64#11719936—44no—
65D53⋊C8—44no—
66M16×C53—42no—
67M16×C53—41no—
68C2×C8×C53—22no—
69C2×C8×C53—21no—
70C2×C8×C53—42no—
71C2×C8×C53C22⋊C411yescharacteristic
72C2×C8×C53C22⋊C411yescharacteristic
73C22⋊Dic53—22no—
74#11719921—22no—
75#11719919—22no—
76C4×Dic53—21no—
77C4×Dic53C4⋊C411yescharacteristic
78C4×Dic53C4⋊C411yescharacteristic
79#11719917—41no—
80C2×#88322—21no—
81C2×#88322—41no—
82C2×#88322C22⋊C412yes—
83#11719912—42no—
84C8×D53—22no—
85C8×D53—22no—
86C23×C53—21no—
87C23×C53#11711yescharacteristic
88C22×D53C22⋊Q812yes—
89Q8×C53—41no—
90D4×C53—41no—
91D4×C53—42no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53C2×C22⋊C411yescharacteristic
98C2×C4×C53C2×C4⋊C411yescharacteristic
99C22⋊D53—42no—
100C22⋊D53—82no—
101C2×Dic53—21no—
102C2×Dic53—21no—
103C2×Dic53—81no—
104C2×Dic53#12112yes—
105D212—41no—
106C4×D53—22no—
107C4×D53—22no—
108Dic106—41no—
109#88324—84no—
110C8×C53—21no—
111C8×C53—41no—
112C8×C53#9912yes—
113#88322—22no—
114#88322—41no—
115#12810—531nomaximal, Sylow
116C22×C53—21no—
117C22×C53—41no—
118C22×C53—42no—
119C22×C53—41no—
120C22×C53#38011yescharacteristic
121C2×D53—22no—
122C2×D53—42no—
123C4×C53—21no—
124C4×C53—41no—
125C4×C53C2×#9911yescharacteristic
126C4×C53#41011yescharacteristic
127Dic53—22no—
128Dic53—41no—
129Dic53—41no—
130#2932—531no—
131#2132—532no—
132#1813—532no—
133#1392—532no—
134C2×C53—21no—
135C2×C53—41no—
136C2×C53—81no—
137C2×C53#179911yescharacteristic
138D53—42no—
139C2×#131—531no—
140C2×#131#615711yescharacteristic
141#532—531no—
142#435—1062no—
143#435—1062no—
144C2×#108—1062no—
145#423—531no—
146#413—534no—
147#408—531no—
148#408—531no—
149C2×#97—531no—
150#356—1062no—
151#355—1062no—
152#338—1062no—
153C53#1281011yescharacteristic, complemented, Sylow
154C2×#41—531no—
155C2×#41C2×#615711yescharacteristic
156#131—42no—
157#131—2122no—
158C2×M16—21no—
159C2×M16—531no—
160C2×M16—532no—
161C2×M16—1064no—
162C2×M16C2×#615711yescharacteristic
163C22×C8—21no—
164C22×C8—532no—
165C22×C8—531no—
166C22×C8C2×#615711yescharacteristic
167C4⋊Q8—531no—
168#127—531no—
169#124—531no—
170#124—531no—
171#121—1062no—
172C4×D4—1062no—
173C2×C42—531no—
174#108—2122no—
175#108—2122no—
176C4⋊C8—1062no—
177#101—1064no—
178#100—1064no—
179C22⋊C8—1062no—
180C22⋊C8—1062no—
181#97—1061no—
182#97—1061no—
183#41—41no—
184#41—41no—
185#41—2121no—
186#41—2121no—
187C2×Q8—531no—
188C2×Q8C4⋊#615711yescharacteristic
189C2×D4—21no—
190C2×D4—532no—
191C2×D4—531no—
192C2×D4—1062no—
193C2×D4C4⋊#615711yescharacteristic
194C22×C4—21no—
195C22×C4—532no—
196C22×C4—1061no—
197C22×C4C22×#615711yescharacteristic
198M16—42no—
199M16—41no—
200M16—2122no—
201M16—2121no—
202M16—2124no—
203M16—2124no—
204C2×C8—22no—
205C2×C8—21no—
206C2×C8—42no—
207C2×C8—532no—
208C2×C8—1062no—
209C2×C8—1062no—
210C2×C8—1061no—
211C2×C8—2124no—
212C2×C8—2121no—
213C2×C8C22⋊#615711yescharacteristic
214C2×C8C22⋊#615711yescharacteristic
215C4⋊C4—1062no—
216C22⋊C4—1062no—
217C22⋊C4—1062no—
218C42—531no—
219C42—531no—
220C42—1061no—
221C23—21no—
222C23—532no—
223C23#1313329611yescharacteristic
224Q8—41no—
225Q8—2121no—
226D4—41no—
227D4—42no—
228D4—4242no—
229D4—2121no—
230D4—2122no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—532no—
237C2×C4—1062no—
238C2×C4—1062no—
239C2×C4—1061no—
240C2×C4—1061no—
241C2×C4—4241no—
242C2×C4C2×C22⋊#615711yescharacteristic
243C2×C4C2×C4⋊#615711yescharacteristic
244C8—21no—
245C8—41no—
246C8—1062no—
247C8—4244no—
248C8—2121no—
249C8#1313327512yes—
250C22—21no—
251C22—41no—
252C22—42no—
253C22—41no—
254C22—1062no—
255C22—2122no—
256C22#1681156511yescharacteristic
257C4—21no—
258C4—41no—
259C4—1062no—
260C4—2121no—
261C4—2121no—
262C4C2×#1313327511yescharacteristic
263C4#1681160211yescharacteristic
264C2—21no—
265C2—41no—
266C2—81no—
267C2—2122no—
268C2#2029057511yescharacteristic
subgroupG/H
centerC4#16811602
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#131×C53C4
socleC2×C53#1799
prime radical#16811169C22
cosocleC2×#11719945C23
hypercenterC2×#131#6157
hypocenterC53#12810
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C8 ◁ C22×C8 ◁ C2×#131

Complements

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

NH ≅ G/Ntype
2. #20290599C2semidirect
3. #20289807C2semidirect
153. C53#12810semidirect

Maximal subgroups

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

subgroupindex
#202910062
#202905992
#202898072
#202834312
#1281053