Finite Group Databasebeta

← #28312433

Subgroups — #28312433

Order 13568.

474 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C2×#16811528C212yesmaximal, complemented
2#20289848C212yesmaximal, complemented
3#20289846C212yesmaximal, complemented
4#20285168C211yesmaximal, characteristic
5C2×#13133276—22no—
6#16811528—22no—
7#16811528C2214yes—
8C2×#13133242—22no—
9C2×#13133214C2211yescharacteristic
10#16811013—22no—
11#16810999C411yescharacteristic, complemented
12#16810999C2211yescharacteristic
13#16810997C411yescharacteristic, complemented
14#16810997C2211yescharacteristic
15C4×#11719917C411yescharacteristic, complemented
16C53×#532C411yescharacteristic, complemented
17C2×C22⋊#6157—42no—
18#13133276—24no—
19#13133276—42no—
20C22×D212—41no—
21C22×D212D412yes—
22#13133242—44no—
23#13133242—44no—
24C2×#11719924—22no—
25C2×#11719924—22no—
26C2×#11719922—22no—
27C2×#11719922—22no—
28C2×#11719917—21no—
29#13133214—41no—
30#13133214C2×C411yescharacteristic
31#13133214D412yes—
32#13133214C2311yescharacteristic
33#13133128—21no—
34C4×#88322C2×C411yescharacteristic, complemented
35C4×#88322C2×C411yescharacteristic, complemented
36C2×#41×C53—21no—
37C4⋊Q8×C53C2×C411yescharacteristic, complemented
38#127×C53C2×C411yescharacteristic, complemented
39#124×C53—21no—
40#121×C53—41no—
41C4×D4×C53—41no—
42C2×C42×C53C2×C411yescharacteristic
43C23×D53—22no—
44C22×#6157—82no—
45#41×C53—42no—
46#41×C53—42no—
47C2×Q8×C53—21no—
48C2×D4×C53—21no—
49C2×D4×C53—41no—
50C2×D4×C53—41no—
51C22×C4×C53—21no—
52C22×C4×C53—41no—
53C22×C4×C53C22⋊C411yescharacteristic
54C2×D212—22no—
55C2×D212—42no—
56C2×D212—42no—
57C2×D212—41no—
58C2×D212—42no—
59C2×D212—41no—
60C2×D212—84no—
61C2×D212C22⋊C412yes—
62C2×D212C2×D412yes—
63C22⋊#6157—44no—
64C22⋊#6157—82no—
65C4⋊C4×C53—41no—
66C22⋊C4×C53—42no—
67C42×C53—21no—
68C42×C53C22⋊C411yescharacteristic
69C42×C53C22×C411yescharacteristic
70#11719924—44no—
71#11719924—44no—
72#11719922—44no—
73#11719922—44no—
74#11719917—24no—
75C2×#88322—21no—
76C2×#88322—21no—
77C23×C53—41no—
78C23×C53#9911yescharacteristic
79C22×D53—22no—
80C22×D53—22no—
81C22×D53—42no—
82C22×D53—42no—
83C22×D53—84no—
84C2×#6157—84no—
85C2×#6157—164no—
86Q8×C53—42no—
87D4×C53—42no—
88D4×C53—82no—
89D4×C53—82no—
90C2×C4×C53—22no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—42no—
94C2×C4×C53—41no—
95C2×C4×C53—81no—
96C2×C4×C53—81no—
97C2×C4×C53#9911yescharacteristic
98C2×C4×C53C2×C22⋊C411yescharacteristic
99D212—42no—
100D212—42no—
101D212—82no—
102D212—82no—
103D212—82no—
104D212—84no—
105#88322—22no—
106#88322—22no—
107#13584—531nomaximal, Sylow
108C22×C53—41no—
109C22×C53—82no—
110C22×C53—81no—
111C22×C53#35312yes—
112C22×C53C2×#9911yescharacteristic
113C2×D53—42no—
114C2×D53—42no—
115C2×D53—84no—
116C2×D53—82no—
117C2×D53—82no—
118C2×D53—162no—
119#6157—164no—
120C4×C53—22no—
121C4×C53—21no—
122C4×C53—21no—
123C4×C53—41no—
124C4×C53—81no—
125#3151—531no—
126C2×#353—532no—
127#1409—532no—
128#1408—532no—
129C2×C53—21no—
130C2×C53—22no—
131C2×C53—161no—
132C2×C53C2×#35311yescharacteristic
133D53—82no—
134D53—82no—
135C2×AGL(1,ℤ/8)—1062no—
136#532#615711yescharacteristic, complemented
137C2×#127—531no—
138#496—1062no—
139#493—531no—
140#493—531no—
141#492—531no—
142#492—531no—
143C2×#99—1062no—
144C4×M16—531no—
145#353—534no—
146#353—1062no—
147C53#1358411yescharacteristic, complemented, Sylow
148C2×#41—21no—
149C22×D4—532no—
150C22×D4—2121no—
151AGL(1,ℤ/8)—2124no—
152AGL(1,ℤ/8)—2124no—
153C2×SD16—1062no—
154C2×SD16—1062no—
155C2×D8—1062no—
156C2×D8—1062no—
157C2×M16—1061no—
158C4⋊Q8C2×#615711yescharacteristic, complemented
159#127—532no—
160#127—531no—
161#127—531no—
162#127—2121no—
163#127C2×#615711yescharacteristic, complemented
164#124—21no—
165#121—41no—
166C4×D4—41no—
167C2×C22⋊C4—2122no—
168C2×C42C2×#615711yescharacteristic
169#99—1064no—
170#99—2122no—
171#97—1061no—
172C4×C8—531no—
173C4×C8—531no—
174C24—1062no—
175#41—42no—
176#41—42no—
177C2×Q8—21no—
178C2×D4—21no—
179C2×D4—41no—
180C2×D4—41no—
181C2×D4—532no—
182C2×D4—532no—
183C2×D4—1062no—
184C2×D4—4244no—
185C2×D4—2121no—
186C2×D4—2121no—
187C2×D4—2122no—
188C2×D4—2122no—
189C2×D4—2122no—
190C22×C4—21no—
191C22×C4—41no—
192C22×C4—4242no—
193C22×C4C22⋊#615711yescharacteristic
194SD16—2124no—
195SD16—2124no—
196D8—2124no—
197D8—2124no—
198M16—1064no—
199C2×C8—1061no—
200C2×C8—1061no—
201C4⋊C4—41no—
202C22⋊C4—42no—
203C22⋊C4—4242no—
204C22⋊C4—2124no—
205C42—21no—
206C42C22⋊#615711yescharacteristic
207C42C22×#615711yescharacteristic
208C23—41no—
209C23—1062no—
210C23—1062no—
211C23—4244no—
212C23—2122no—
213C23—2122no—
214C23#1313327511yescharacteristic
215Q8—42no—
216D4—42no—
217D4—82no—
218D4—82no—
219D4—4242no—
220D4—4242no—
221D4—4242no—
222D4—4244no—
223D4—2122no—
224D4—2122no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—8484no—
233C2×C4—4244no—
234C2×C4#1313327511yescharacteristic
235C2×C4C2×C22⋊#615711yescharacteristic
236C8—1062no—
237C8—1062no—
238C22—41no—
239C22—82no—
240C22—81no—
241C22—8482no—
242C22—4242no—
243C22—4242no—
244C22—4244no—
245C22—2122no—
246C22—2122no—
247C22#1681152712yes—
248C22C2×#1313327511yescharacteristic
249C4—22no—
250C4—21no—
251C4—21no—
252C4—41no—
253C4—81no—
254C4—8484no—
255C2—21no—
256C2—22no—
257C2—161no—
258C2—4242no—
259C2—4242no—
260C2C2×#1681152711yescharacteristic
subgroupG/H
centerC2C2×#16811527
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
FittingC53×#532C4
socleC2×C53C2×#353
prime radical#20285168C2
cosocle#13133214C23
hypercenter#532#6157
hypocenterC53#13584
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C2×#16811528C2semidirect
2. #20289848C2semidirect
3. #20289846C2semidirect
11. #16810999C4semidirect
13. #16810997C4semidirect
15. C4×#11719917C4semidirect
16. C53×#532C4semidirect
34. C4×#88322C2×C4semidirect
35. C4×#88322C2×C4semidirect
37. C4⋊Q8×C53C2×C4semidirect
38. #127×C53C2×C4semidirect
136. #532#6157semidirect
147. C53#13584semidirect
158. C4⋊Q8C2×#6157semidirect
163. #127C2×#6157semidirect

Maximal subgroups

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

subgroupindex
C2×#168115282
#202898482
#202898462
#202851682
#1358453