Finite Group Databasebeta

← #27296814

Subgroups — #27296814

Order 13568.

532 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 63 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#20275879C211yesmaximal, characteristic
2#20273705C212yesmaximal, complemented
3#20272715C212yesmaximal, complemented
4#20272643C211yesmaximal, characteristic
5C53×#2483C211yesmaximal, characteristic
6#16811249C2211yescharacteristic
7#16810953—22no—
8#16810479—21no—
9#16810390C2211yescharacteristic
10#16810389—21no—
11#16810359—22no—
12#16810347C2212yes—
13C22×C22⋊C4×C53—21no—
14C53×#397—21no—
15C53×#397C2212yes—
16C53×#385—22no—
17C53×#381—22no—
18C53×#379C2211yescharacteristic
19#13133196—22no—
20#13133194—42no—
21C2×C22⋊Dic53—21no—
22C2×C22⋊Dic53—21no—
23C2×C22⋊Dic53—42no—
24#13133148—42no—
25#13133140—42no—
26C22⋊#88322—22no—
27C25×C53D411yescharacteristic
28C23×C4×C53—21no—
29C23×C4×C53—21no—
30C2×C4⋊C4×C53—21no—
31C2×C4⋊C4×C53D412yes—
32C2×C22⋊C4×C53—22no—
33C2×C22⋊C4×C53—22no—
34C2×C22⋊C4×C53—22no—
35C2×C22⋊C4×C53—42no—
36C2×C22⋊C4×C53—42no—
37C2×C22⋊C4×C53D411yescharacteristic
38C2×C22⋊C4×C53C2311yescharacteristic
39#95×C53—21no—
40#95×C53—22no—
41#95×C53—22no—
42#95×C53D412yes—
43C24×C53—21no—
44C24×C53—21no—
45C24×C53—44no—
46C24×C53—41no—
47C24×C53—41no—
48C24×C53SD1612yes—
49C24×C53C2×D411yescharacteristic
50C22×C4×C53—22no—
51C22×C4×C53—21no—
52C22×C4×C53—22no—
53C22×C4×C53—42no—
54C22×C4×C53—42no—
55C22×C4×C53—44no—
56C22×C4×C53—81no—
57C22×C4×C53—81no—
58C22×C4×C53C2×D412yes—
59C22×Dic53—22no—
60C22×Dic53—41no—
61C2×Dic106—22no—
62C4⋊C4×C53—42no—
63C4⋊C4×C53—81no—
64C22⋊C4×C53—41no—
65C22⋊C4×C53—44no—
66C22⋊C4×C53—44no—
67C22⋊C4×C53—42no—
68C22⋊C4×C53—42no—
69C22⋊C4×C53—41no—
70C22⋊Dic53—41no—
71C22⋊Dic53—41no—
72C22⋊Dic53—41no—
73C22⋊Dic53—41no—
74C22⋊Dic53—84no—
75C4⋊Dic53—22no—
76#11719919—42no—
77C2×#88322—42no—
78C23×C53—24no—
79C23×C53—22no—
80C23×C53—21no—
81C23×C53—21no—
82C23×C53—42no—
83C23×C53—44no—
84C23×C53—42no—
85C23×C53—42no—
86C23×C53—44no—
87C23×C53—41no—
88C23×C53—41no—
89C23×C53—44no—
90C23×C53—81no—
91C23×C53—84no—
92C23×C53—82no—
93C23×C53#12011yescharacteristic
94C23×C53C2×SD1611yescharacteristic
95C23×C53AGL(1,ℤ/8)11yescharacteristic
96C2×C4×C53—22no—
97C2×C4×C53—42no—
98C2×C4×C53—44no—
99C2×C4×C53—42no—
100C2×C4×C53—42no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53—41no—
104C2×C4×C53—42no—
105C2×C4×C53—84no—
106C2×C4×C53—84no—
107C2×Dic53—42no—
108C2×Dic53—82no—
109C2×Dic53—81no—
110C2×Dic53—82no—
111Dic106—82no—
112#88322—82no—
113#12203—531nomaximal, Sylow
114C22×C53—22no—
115C22×C53—24no—
116C22×C53—21no—
117C22×C53—21no—
118C22×C53—41no—
119C22×C53—41no—
120C22×C53—42no—
121C22×C53—44no—
122C22×C53—44no—
123C22×C53—42no—
124C22×C53—42no—
125C22×C53—44no—
126C22×C53—81no—
127C22×C53—84no—
128C22×C53—82no—
129C22×C53#45011yescharacteristic
130C22×C53#45511yescharacteristic
131C22×C53B(4,𝔽2)11yescharacteristic
132C4×C53—42no—
133C4×C53—81no—
134C4×C53—82no—
135C4×C53—82no—
136Dic53—82no—
137Dic53—161no—
138#2483D5311yescharacteristic
139#2029—531no—
140#1608—532no—
141#1344—532no—
142#1316—531no—
143C2×C53—21no—
144C2×C53—21no—
145C2×C53—41no—
146C2×C53—44no—
147C2×C53—41no—
148C2×C53#160111yescharacteristic
149C2×C53#219712yes—
150#543—531no—
151C22×C22⋊C4—21no—
152#483—1062no—
153#397—21no—
154#397C2×D5312yes—
155#385—22no—
156#381—22no—
157#379—1061no—
158#379C2×D5311yescharacteristic
159#342—531no—
160#342—1061no—
161#328—1062no—
162#323—532no—
163C53#1220311yescharacteristic, complemented, Sylow
164C25C22⋊D5311yescharacteristic
165C23×C4—21no—
166C23×C4—21no—
167C22⋊Q8—1062no—
168C22⋊Q8—2122no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4C22⋊D5312yes—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—1061no—
177C2×C22⋊C4—1061no—
178C2×C22⋊C4—2122no—
179C2×C22⋊C4C22⋊D5311yescharacteristic
180C2×C22⋊C4C22×D5311yescharacteristic
181#106—2122no—
182#103—2122no—
183C22⋊C8—1062no—
184#95—21no—
185#95—22no—
186#95—22no—
187#95C22⋊D5312yes—
188C24—21no—
189C24—21no—
190C24—41no—
191C24—44no—
192C24—41no—
193C24#1171992312yes—
194C24C2×C22⋊D5311yescharacteristic
195C2×Q8—1062no—
196C22×C4—22no—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—44no—
202C22×C4—81no—
203C22×C4—81no—
204C22×C4—1062no—
205C22×C4—2121no—
206C22×C4C2×C22⋊D5312yes—
207C2×C8—2122no—
208C4⋊C4—42no—
209C4⋊C4—81no—
210C4⋊C4—1062no—
211C4⋊C4—2122no—
212C22⋊C4—41no—
213C22⋊C4—44no—
214C22⋊C4—44no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—41no—
218C22⋊C4—4244no—
219C22⋊C4—2121no—
220C22⋊C4—2121no—
221C22⋊C4—2121no—
222C22⋊C4—2121no—
223C23—24no—
224C23—22no—
225C23—21no—
226C23—21no—
227C23—42no—
228C23—42no—
229C23—44no—
230C23—42no—
231C23—44no—
232C23—41no—
233C23—41no—
234C23—44no—
235C23—81no—
236C23—84no—
237C23—82no—
238C23#1313318611yescharacteristic
239C23C2×#1171992311yescharacteristic
240C23#1313324311yescharacteristic
241Q8—4242no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—84no—
252C2×C4—84no—
253C2×C4—4241no—
254C2×C4—4242no—
255C2×C4—4242no—
256C2×C4—2122no—
257C8—4242no—
258C22—22no—
259C22—24no—
260C22—21no—
261C22—21no—
262C22—41no—
263C22—41no—
264C22—42no—
265C22—44no—
266C22—44no—
267C22—42no—
268C22—42no—
269C22—44no—
270C22—81no—
271C22—84no—
272C22—82no—
273C22#1681077411yescharacteristic
274C22#1681080111yescharacteristic
275C22#1681081011yescharacteristic
276C4—42no—
277C4—82no—
278C4—81no—
279C4—82no—
280C4—8481no—
281C4—4242no—
282C2—21no—
283C2—21no—
284C2—41no—
285C2—44no—
286C2—41no—
287C2#2027366711yescharacteristic
288C2#2027670512yes—
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C22⋊C4×C53C23
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2483C2
socleC22×C53B(4,𝔽2)
prime radicalC25×C53D4
hypercenter#2483D53
hypocenterC53#12203
LCSG ▷ C2×C22⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2483

Complements

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

NH ≅ G/Ntype
2. #20273705C2semidirect
3. #20272715C2semidirect
163. C53#12203semidirect

Maximal subgroups

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

subgroupindex
#202758792
#202737052
#202727152
#202726432
C53×#24832
#1220353