Finite Group Databasebeta

← #28312470

Subgroups — #28312470

Order 13568.

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

normal subgroups only

This group has 388 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
2#20289855C212yesmaximal, complemented
3#20289854C212yesmaximal, complemented
4#20285091C211yesmaximal, characteristic
5C2×#13133276—22no—
6#16811528—22no—
7#16811528C2214yes—
8C2×#13133214C2211yescharacteristic
9#16810989C411yescharacteristic, complemented
10#16810989C411yescharacteristic, complemented
11#16810989C2211yescharacteristic
12#16810989C2211yescharacteristic
13#16810971—22no—
14#16810627—22no—
15C4×#11719917C411yescharacteristic, complemented
16C53×#533C411yescharacteristic, complemented
17C2×C22⋊#6157—42no—
18#13133276—24no—
19#13133276—42no—
20C22×D212—41no—
21C22×D212D412yes—
22C2×#11719917—21no—
23#13133214—41no—
24#13133214C2×C411yescharacteristic
25#13133214D412yes—
26#13133214C2311yescharacteristic
27#13133137—22no—
28#13133137—22no—
29#13133137—22no—
30#13133137—22no—
31#13133128C2×C411yescharacteristic, complemented
32#13133128C2×C411yescharacteristic, complemented
33C4×#88322—21no—
34C4⋊Q8×C53—21no—
35#125×C53C2×C411yescharacteristic, complemented
36#125×C53C2×C411yescharacteristic, complemented
37C22⋊Q8×C53—41no—
38C4×Q8×C53—41no—
39#117×C53—21no—
40C2×C42×C53C2×C411yescharacteristic
41C23×D53—22no—
42C22×#6157—82no—
43C2×Q8×C53—41no—
44C22×C4×C53—21no—
45C22×C4×C53C22⋊C411yescharacteristic
46C2×D212—22no—
47C2×D212—42no—
48C2×D212—42no—
49C2×D212—41no—
50C2×D212—42no—
51C2×D212—41no—
52C2×D212—84no—
53C2×D212C22⋊C412yes—
54C2×D212C2×D412yes—
55C22⋊#6157—44no—
56C22⋊#6157—82no—
57C4⋊C4×C53—21no—
58C4⋊C4×C53—21no—
59C4⋊C4×C53—41no—
60C4⋊C4×C53—41no—
61C4⋊C4×C53—41no—
62C22⋊C4×C53—41no—
63C42×C53—21no—
64C42×C53—41no—
65C42×C53C22⋊C411yescharacteristic
66C42×C53C22×C411yescharacteristic
67#11719917—42no—
68C2×#88322—21no—
69C2×#88322—21no—
70C23×C53#9911yescharacteristic
71C22×D53—22no—
72C22×D53—22no—
73C22×D53—42no—
74C22×D53—42no—
75C22×D53—84no—
76C2×#6157—84no—
77C2×#6157—164no—
78Q8×C53—82no—
79C2×C4×C53—22no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—42no—
83C2×C4×C53—41no—
84C2×C4×C53—41no—
85C2×C4×C53#9911yescharacteristic
86C2×C4×C53C2×C22⋊C411yescharacteristic
87D212—42no—
88D212—42no—
89D212—82no—
90D212—84no—
91D212—82no—
92D212—82no—
93#88322—41no—
94#88322—41no—
95#13595—531nomaximal, Sylow
96C22×C53—41no—
97C22×C53#35412yes—
98C22×C53C2×#9911yescharacteristic
99C2×D53—42no—
100C2×D53—42no—
101C2×D53—84no—
102C2×D53—82no—
103C2×D53—82no—
104C2×D53—162no—
105#6157—164no—
106C4×C53—22no—
107C4×C53—21no—
108C4×C53—21no—
109C4×C53—41no—
110C4×C53—81no—
111C4×C53—81no—
112#3138—531no—
113C2×#353—532no—
114#1412—532no—
115#1412—532no—
116C2×C53—21no—
117C2×C53—22no—
118C2×C53C2×#35411yescharacteristic
119D53—82no—
120D53—82no—
121#533#615711yescharacteristic, complemented
122C2×#127—531no—
123#490—531no—
124#490—531no—
125#490—531no—
126#490—531no—
127#486—1062no—
128#418—1062no—
129C2×#99—1062no—
130C4×M16—531no—
131#353—534no—
132#353—1062no—
133C53#1359511yescharacteristic, complemented, Sylow
134C22×D4—532no—
135C22×D4—2121no—
136C2×M16—1061no—
137C4⋊Q8—21no—
138#127—532no—
139#127—531no—
140#127—531no—
141#127—2121no—
142#125C2×#615711yescharacteristic, complemented
143#125C2×#615711yescharacteristic, complemented
144C22⋊Q8—41no—
145C4×Q8—41no—
146#117—21no—
147C2×C22⋊C4—2122no—
148C2×C42C2×#615711yescharacteristic
149#102—1062no—
150#102—1062no—
151#102—1062no—
152#102—1062no—
153#99—1064no—
154#99—2122no—
155#97—531no—
156#97—531no—
157C4×C8—1061no—
158C24—1062no—
159C2×Q8—41no—
160C2×D4—532no—
161C2×D4—532no—
162C2×D4—1062no—
163C2×D4—4244no—
164C2×D4—2122no—
165C2×D4—2121no—
166C2×D4—2121no—
167C2×D4—2122no—
168C2×D4—2122no—
169C22×C4—21no—
170C22×C4—4242no—
171C22×C4C22⋊#615711yescharacteristic
172M16—2122no—
173C2×C8—1061no—
174C2×C8—1061no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C22⋊C4—41no—
181C22⋊C4—4242no—
182C22⋊C4—2124no—
183C42—21no—
184C42—41no—
185C42C22⋊#615711yescharacteristic
186C42C22×#615711yescharacteristic
187C23—1062no—
188C23—1062no—
189C23—4244no—
190C23—2122no—
191C23—2122no—
192C23#1313327511yescharacteristic
193Q8—82no—
194D4—4244no—
195D4—4242no—
196D4—4242no—
197D4—4242no—
198D4—2122no—
199D4—2122no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—8484no—
207C2×C4—4244no—
208C2×C4#1313327511yescharacteristic
209C2×C4C2×C22⋊#615711yescharacteristic
210C8—2121no—
211C8—2121no—
212C22—41no—
213C22—8482no—
214C22—4242no—
215C22—4242no—
216C22—4244no—
217C22—2122no—
218C22—2122no—
219C22#1681152912yes—
220C22C2×#1313327511yescharacteristic
221C4—22no—
222C4—21no—
223C4—21no—
224C4—41no—
225C4—81no—
226C4—81no—
227C4—8484no—
228C2—21no—
229C2—22no—
230C2—4242no—
231C2—4242no—
232C2C2×#1681152911yescharacteristic
subgroupG/H
centerC2C2×#16811529
commutatorC42×C53C22×C4
FrattiniC42C22×#6157
FittingC53×#533C4
socleC2×C53C2×#354
prime radicalC2×#13133214C22
cosocle#13133214C23
hypercenter#533#6157
hypocenterC53#13595
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #20289855C2semidirect
3. #20289854C2semidirect
9. #16810989C4semidirect
10. #16810989C4semidirect
15. C4×#11719917C4semidirect
16. C53×#533C4semidirect
31. #13133128C2×C4semidirect
32. #13133128C2×C4semidirect
35. #125×C53C2×C4semidirect
36. #125×C53C2×C4semidirect
121. #533#6157semidirect
133. C53#13595semidirect
142. #125C2×#6157semidirect
143. #125C2×#6157semidirect

Maximal subgroups

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

subgroupindex
C2×#168115282
#202898552
#202898542
#202850912
#1359553