Finite Group Databasebeta

← #27297312

Subgroups — #27297312

Order 13568.

396 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 67 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#20276901C212yesmaximal, complemented
3#20276143C211yesmaximal, characteristic, complemented
4#20272725C212yesmaximal, complemented
5C53×#1314C211yesmaximal, characteristic, complemented
6C2×#13133191—21no—
7#16810937—22no—
8#16810859—21no—
9#16810859C2212yes—
10#16810759—22no—
11C2×#13133151C2211yescharacteristic, complemented
12#16810483—21no—
13#16810451—22no—
14#16810425—22no—
15#16810386—21no—
16#16810386C2211yescharacteristic, complemented
17C2×C22⋊C8×C53C2211yescharacteristic
18C53×C22⋊C16C2212yescomplemented
19C22×D212—21no—
20C22×C22⋊D53—41no—
21C2×D424—21no—
22C2×D424D412yes—
23#13133191—22no—
24#13133191—22no—
25#13133191—41no—
26#13133191—41no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53—21no—
30C2×C22⋊Dic53—41no—
31C2×#11719921—41no—
32#13133151—21no—
33#13133151—22no—
34#13133151D412yes—
35#13133139—22no—
36#13133139—22no—
37#13133124—41no—
38C23×C4×C53D411yescharacteristic
39C22×C8×C53D411yescharacteristic
40C22×C8×C53C2311yescharacteristic
41C2×C16×C53—22no—
42C22⋊C8×C53—22no—
43C24×C53D811yescharacteristic
44C23×D53—41no—
45C22×C4×C53—21no—
46C22×C4×C53—21no—
47C22×C4×C53—41no—
48C22×C4×C53—41no—
49C22×C4×C53D811yescharacteristic
50C22×C4×C53C2×D411yescharacteristic
51C2×C22⋊D53—42no—
52C2×C22⋊D53—42no—
53C2×C22⋊D53—82no—
54C2×C22⋊D53—82no—
55C22×Dic53—41no—
56C22×Dic53—41no—
57C22×Dic53—41no—
58C2×D212—22no—
59C2×D212—41no—
60C2×D212—81no—
61C2×D212—81no—
62C2×C8×C53—21no—
63C2×C8×C53—41no—
64C2×C8×C53—41no—
65C2×C8×C53C2×D412yes—
66C22⋊Dic53—82no—
67#11719921—42no—
68#11719921—42no—
69C4⋊Dic53—22no—
70C4⋊Dic53—41no—
71C4⋊Dic53—41no—
72C4⋊Dic53—41no—
73C4⋊Dic53—41no—
74C4⋊Dic53—41no—
75D424—42no—
76D424—81no—
77C16×C53—42no—
78C23×C53—21no—
79C23×C53—21no—
80C23×C53—41no—
81C23×C53—41no—
82C23×C53SD3212yes—
83C23×C53C2×D811yescharacteristic
84C22×D53—42no—
85C22×D53—82no—
86C22×D53—82no—
87C22×D53—82no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53—81no—
95C2×C4×C53#12011yescharacteristic
96C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
97C22⋊D53—84no—
98C22⋊D53—82no—
99C22⋊D53—82no—
100C2×Dic53—42no—
101C2×Dic53—82no—
102C2×Dic53—81no—
103C2×Dic53—81no—
104C2×Dic53—81no—
105C2×Dic53—81no—
106C2×Dic53—81no—
107C2×Dic53—81no—
108D212—42no—
109D212—42no—
110D212—82no—
111C8×C53—22no—
112C8×C53—41no—
113#12281—531nomaximal, Sylow
114C22×C53—21no—
115C22×C53—21no—
116C22×C53—22no—
117C22×C53—42no—
118C22×C53—41no—
119C22×C53—41no—
120C22×C53—81no—
121C22×C53#44711yescharacteristic
122C22×C53C2×SD3211yescharacteristic
123C22×C53#50911yescharacteristic
124C2×D53—82no—
125C2×D53—82no—
126C2×D53—82no—
127C2×D53—164no—
128C4×C53—21no—
129C4×C53—41no—
130C4×C53#45511yescharacteristic
131C4×C53#45811yescharacteristic
132Dic53—82no—
133Dic53—161no—
134Dic53—161no—
135#3186—531no—
136#2233—532no—
137#2076—531no—
138#1347—532no—
139#1314D5311yescharacteristic, complemented
140C2×C53—21no—
141C2×C53—21no—
142C2×C53—42no—
143C2×C53#160711yescharacteristic
144C2×C53#218711yescharacteristic
145C2×C53#219011yescharacteristic
146D53—162no—
147C2×#121—1061no—
148#480—1062no—
149#466—532no—
150#466—1061no—
151#447—1062no—
152C2×#107—531no—
153C2×C22⋊C8C2×D5311yescharacteristic
154#380—1061no—
155#367—1062no—
156#357—1062no—
157C22⋊C16C2×D5312yescomplemented
158#340—531no—
159#340—1061no—
160C53#1228111yescharacteristic, complemented, Sylow
161C22×D4—1061no—
162C22×D4—2121no—
163C23×C4D21211yescharacteristic
164C2×D8—532no—
165C2×D8—1061no—
166C22×C8D21211yescharacteristic
167C22×C8C22×D5311yescharacteristic
168#121—1062no—
169#121—1062no—
170#121—2121no—
171#121—2121no—
172C2×C4⋊C4—1061no—
173C2×C4⋊C4—1061no—
174C2×C4⋊C4—1061no—
175C2×C22⋊C4—2121no—
176C2×C22⋊C4—2121no—
177C2×C16—22no—
178#107—532no—
179#107—1061no—
180#107—1062no—
181#102—1062no—
182#102—1062no—
183C22⋊C8—22no—
184#95—2121no—
185C24—2121no—
186C24D42411yescharacteristic
187C2×D4—1062no—
188C2×D4—4241no—
189C2×D4—4241no—
190C2×D4—4242no—
191C2×D4—4242no—
192C2×D4—2121no—
193C2×D4—2122no—
194C2×D4—2122no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—2121no—
200C22×C4—2121no—
201C22×C4—2121no—
202C22×C4D42411yescharacteristic
203C22×C4C2×D21211yescharacteristic
204D8—4241no—
205D8—2122no—
206C2×C8—21no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8D4×D5312yes—
210C4⋊C4—1062no—
211C4⋊C4—2121no—
212C4⋊C4—2121no—
213C4⋊C4—2121no—
214C4⋊C4—2121no—
215C4⋊C4—2121no—
216C22⋊C4—4242no—
217C22⋊C4—2122no—
218C22⋊C4—2122no—
219C16—42no—
220C23—21no—
221C23—21no—
222C23—41no—
223C23—41no—
224C23—4242no—
225C23—4242no—
226C23—4242no—
227C23—2122no—
228C23#1313316212yes—
229C23C2×D42411yescharacteristic
230D4—4242no—
231D4—4244no—
232D4—4242no—
233D4—4242no—
234D4—2122no—
235D4—2122no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—81no—
243C2×C4—4241no—
244C2×C4—4241no—
245C2×C4—4241no—
246C2×C4—4241no—
247C2×C4—4241no—
248C2×C4—4241no—
249C2×C4—4242no—
250C2×C4—2122no—
251C2×C4#1313318711yescharacteristic
252C2×C4#1313324511yescharacteristic
253C8—22no—
254C8—41no—
255C22—21no—
256C22—21no—
257C22—22no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—81no—
262C22—8484no—
263C22—4242no—
264C22—4242no—
265C22—4242no—
266C22#1681075911yescharacteristic
267C22C2×#1313316211yescharacteristic
268C22#1681106111yescharacteristic
269C4—21no—
270C4—41no—
271C4—8481no—
272C4—8481no—
273C4—4242no—
274C4#1681080311yescharacteristic
275C4#1681081411yescharacteristic
276C2—21no—
277C2—21no—
278C2—42no—
279C2—8482no—
280C2#2027370311yescharacteristic
281C2#2027664611yescharacteristic
282C2#2027666311yescharacteristic
subgroupG/H
centerC22#16810759
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
FittingC53×#1314C2
socleC22×C53#447
prime radical#20285466C2
hypercenter#1314D53
hypocenterC53#12281
LCSG ▷ C22×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #1314

Complements

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

NH ≅ G/Ntype
2. #20276901C2semidirect
3. #20276143C2semidirect
4. #20272725C2semidirect
5. C53×#1314C2semidirect
11. C2×#13133151C22semidirect
16. #16810386C22semidirect
18. C53×C22⋊C16C22semidirect
139. #1314D53semidirect
157. C22⋊C16C2×D53semidirect
160. C53#12281semidirect

Maximal subgroups

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

subgroupindex
#202854662
#202769012
#202761432
#202727252
C53×#13142
#1228153