Finite Group Databasebeta

← #27296537

Subgroups — #27296537

Order 13568.

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

normal subgroups only

This group has 390 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#20286060C211yesmaximal, characteristic, complemented
2#20276644C212yesmaximal
3#20272726C212yesmaximal, complemented
4#20272655C211yesmaximal, characteristic, complemented
5C53×#2054C211yesmaximal, characteristic
6#16811180—21no—
7C2×#13133161—22no—
8#16810919—22no—
9#16810864—21no—
10#16810864C2212yes—
11#16810834—22no—
12#16810424—22no—
13C22⋊#11719908C2212yes—
14C22×D8×C53C2211yescharacteristic
15C2×C4⋊C8×C53C2211yescharacteristic
16C2×#102×C53—21no—
17C53×#390—21no—
18C53×#340C2211yescharacteristic
19C2×Dic212—21no—
20C2×Dic212D412yes—
21#13133217—21no—
22#13133217—21no—
23#13133209—41no—
24#13133196—22no—
25#13133196—41no—
26C4×Dic106—22no—
27C4×Dic106—41no—
28#13133176—41no—
29#13133161—44no—
30C2×#11719908—22no—
31#13133151—21no—
32#13133151D412yes—
33#13133142—22no—
34#13133142—22no—
35C22×D4×C53—21no—
36C2×D8×C53—22no—
37C2×D8×C53—42no—
38C22×C8×C53D411yescharacteristic
39C22×C8×C53C2311yescharacteristic
40C2×C4⋊C4×C53—21no—
41C2×C22⋊C4×C53—41no—
42C2×C22⋊C4×C53—41no—
43C2×C42×C53D411yescharacteristic
44C4⋊C8×C53—22no—
45#102×C53—42no—
46C24×C53—41no—
47C2×D4×C53—22no—
48C2×D4×C53—41no—
49C2×D4×C53—41no—
50C2×D4×C53—41no—
51C2×D4×C53—41no—
52C2×D4×C53—41no—
53C22×C4×C53—41no—
54C22×C4×C53D811yescharacteristic
55C22×C4×C53D811yescharacteristic
56C22×C4×C53C2×D411yescharacteristic
57C2×Dic106—22no—
58C2×Dic106—41no—
59D8×C53—24no—
60D8×C53—42no—
61C2×C8×C53—21no—
62C2×C8×C53—41no—
63C2×C8×C53—41no—
64C2×C8×C53C2×D412yes—
65C4⋊C4×C53—41no—
66C4⋊C4×C53—41no—
67C22⋊C4×C53—82no—
68C22⋊C4×C53—82no—
69C42×C53—21no—
70C42×C53—21no—
71C22⋊Dic53—42no—
72C4⋊Dic53—22no—
73C4⋊Dic53—41no—
74C4⋊Dic53—41no—
75C4⋊Dic53—41no—
76#11719919—42no—
77#11719919—42no—
78C4×Dic53—42no—
79Dic212—42no—
80Dic212—81no—
81#11719908—42no—
82C23×C53—42no—
83C23×C53—81no—
84C23×C53—81no—
85C23×C53—81no—
86C23×C53—81no—
87C23×C53—81no—
88C23×C53—81no—
89C23×C53C2×D811yescharacteristic
90D4×C53—42no—
91D4×C53—42no—
92D4×C53—81no—
93D4×C53—81no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—41no—
100C2×C4×C53—81no—
101C2×C4×C53—81no—
102C2×C4×C53—81no—
103C2×C4×C53SD3212yescomplemented
104C2×C4×C53#12011yescharacteristic
105C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
106C2×Dic53—42no—
107C2×Dic53—42no—
108Dic106—42no—
109Dic106—42no—
110Dic106—82no—
111C8×C53—22no—
112C8×C53—41no—
113#12159—531nomaximal, Sylow
114C22×C53—21no—
115C22×C53—21no—
116C22×C53—82no—
117C22×C53—82no—
118C22×C53—82no—
119C22×C53—161no—
120C22×C53—161no—
121C22×C53—161no—
122C22×C53—161no—
123C22×C53#44711yescharacteristic
124C22×C53C2×SD3211yescharacteristic
125C22×C53#51011yescharacteristic
126C4×C53—21no—
127C4×C53—22no—
128C4×C53—41no—
129C4×C53—161no—
130C4×C53#45411yescharacteristic
131C4×C53B(4,𝔽2)11yescharacteristic
132Dic53—82no—
133Dic53—82no—
134#3249—531no—
135#2187—532no—
136#2054D5311yescharacteristic
137#1348—532no—
138#1322—531no—
139C2×C53—21no—
140C2×C53—21no—
141C2×C53—162no—
142C2×C53#159811yescharacteristic
143C2×C53#218511yescharacteristic
144C2×C53#219011yescharacteristic
145C22×D8C2×D5311yescharacteristic
146#533—1061no—
147C2×SD32—1062no—
148#477—1062no—
149#467—532no—
150#467—1061no—
151#462—1062no—
152C2×C4⋊C8C2×D5311yescharacteristic
153C2×#102—21no—
154#390—21no—
155#357—1062no—
156C22⋊C16—532no—
157#340C2×D5311yescharacteristic
158C53#1215911yescharacteristic, complemented, Sylow
159C22×D4—21no—
160C2×Q16—532no—
161C2×Q16—1061no—
162C2×D8—22no—
163C2×D8—42no—
164C22×C8C22⋊D5311yescharacteristic
165C22×C8C22×D5311yescharacteristic
166C4⋊Q8—1061no—
167C4⋊Q8—1061no—
168#125—2121no—
169C22⋊Q8—1062no—
170C22⋊Q8—2121no—
171C4×Q8—1062no—
172C4×Q8—2121no—
173#117—2121no—
174C2×C4⋊C4—21no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C42C22⋊D5311yescharacteristic
178SD32—2124no—
179C2×C16—1062no—
180#107—532no—
181#107—1061no—
182C4⋊C8—22no—
183#103—1062no—
184#103—1062no—
185#102—42no—
186C24—41no—
187C2×Q8—1062no—
188C2×Q8—2121no—
189C2×D4—22no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4—41no—
195C22×C4—41no—
196C22×C4#1171992211yescharacteristic
197C22×C4#1171992211yescharacteristic
198C22×C4C2×C22⋊D5311yescharacteristic
199Q16—4241no—
200Q16—2122no—
201D8—24no—
202D8—42no—
203C2×C8—21no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8D4×D5312yes—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—1062no—
210C4⋊C4—2121no—
211C4⋊C4—2121no—
212C4⋊C4—2122no—
213C4⋊C4—2121no—
214C4⋊C4—2122no—
215C22⋊C4—82no—
216C22⋊C4—82no—
217C22⋊C4—2122no—
218C42—21no—
219C42—21no—
220C42—2122no—
221C16—2122no—
222C23—42no—
223C23—81no—
224C23—81no—
225C23—81no—
226C23—81no—
227C23—81no—
228C23—81no—
229C23C2×#1171992211yescharacteristic
230Q8—4242no—
231Q8—2122no—
232Q8—2122no—
233D4—42no—
234D4—42no—
235D4—81no—
236D4—81no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—2122no—
247C2×C4—2122no—
248C2×C4#1313316112yescomplemented
249C2×C4#1313318511yescharacteristic
250C2×C4#1313324211yescharacteristic
251C8—22no—
252C8—41no—
253C22—21no—
254C22—21no—
255C22—82no—
256C22—82no—
257C22—82no—
258C22—161no—
259C22—161no—
260C22—161no—
261C22—161no—
262C22#1681075611yescharacteristic
263C22C2×#1313316111yescharacteristic
264C22#1681106311yescharacteristic
265C4—22no—
266C4—21no—
267C4—41no—
268C4—161no—
269C4—4242no—
270C4—4242no—
271C4#1681079511yescharacteristic
272C4#1681080911yescharacteristic
273C2—21no—
274C2—21no—
275C2—162no—
276C2#2027364911yescharacteristic
277C2#2027663111yescharacteristic
278C2#2027666011yescharacteristic
subgroupG/H
centerC22#16810756
commutator, cosocleC22×C8×C53C23
FrattiniC22×C8C22×D53
FittingC53×#2054C2
socleC22×C53#447
prime radicalC22×D8×C53C22
hypercenter#2054D53
hypocenterC53#12159
LCSG ▷ C22×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #2054

Complements

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

NH ≅ G/Ntype
1. #20286060C2semidirect
3. #20272726C2semidirect
4. #20272655C2semidirect
103. C2×C4×C53SD32semidirect
158. C53#12159semidirect
248. C2×C4#13133161semidirect

Maximal subgroups

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

subgroupindex
#202860602
#202766442
#202727262
#202726552
C53×#20542
#1215953