Finite Group Databasebeta

← #27300840

Subgroups — #27300840

Order 13568.

396 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 121 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×#16810730C211yesmaximal, characteristic
2C2×#16810446C211yesmaximal, characteristic
3C2×#16810439C211yesmaximal, characteristic
4#20275469C211yesmaximal, characteristic
5#20272820C211yesmaximal, characteristic
6#20272794C211yesmaximal, characteristic
7C53×#2141C211yesmaximal, characteristic
8C2×Q8×Dic53—21no—
9C2×#13133157C2211yescharacteristic
10#16810730—22no—
11#16810730—22no—
12C2×#13133153C2211yescharacteristic
13C2×#13133149C2211yescharacteristic
14#16810650—21no—
15C2×#13133141—21no—
16C2×#13133127C2211yescharacteristic
17#16810446—22no—
18#16810439—22no—
19#16810382—21no—
20C22×Q16×C53C2211yescharacteristic
21C2×M32×C53C2211yescharacteristic
22C53×#427C2211yescharacteristic
23C53×#358C414yescomplemented
24C2×#11719917—21no—
25C22×#88322D411yescharacteristic
26Q8×Dic53—24no—
27Q8×Dic53—42no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53—41no—
30C2×C4×Dic53—41no—
31C2×C4×Dic53D411yescharacteristic
32#13133157—21no—
33#13133157—21no—
34C2×#11719908—21no—
35#13133153—21no—
36#13133153—21no—
37#13133149—21no—
38#13133149D412yes—
39C4⋊#88322—42no—
40#13133141—24no—
41#13133127—21no—
42#13133127—21no—
43C22×Q8×C53—21no—
44C2×Q16×C53—21no—
45C2×Q16×C53—21no—
46C2×Q16×C53—21no—
47C2×Q16×C53C2×C412yes—
48C2×Q16×C53D412yes—
49C2×Q16×C53Q812yes—
50C22×C8×C53C2311yescharacteristic
51#117×C53—21no—
52M32×C53—21no—
53M32×C53—21no—
54C2×C16×C53C2×C412yes—
55#107×C53C2×C412yes—
56#106×C53—21no—
57C2×Q8×C53—22no—
58C2×Q8×C53—22no—
59C2×Q8×C53—22no—
60C2×Q8×C53—41no—
61C2×Q8×C53—41no—
62C2×Q8×C53—41no—
63C22×C4×C53—41no—
64C22×C4×C53C2×D411yescharacteristic
65C22×Dic53—21no—
66C22×Dic53—41no—
67Q16×C53—22no—
68Q16×C53—22no—
69Q16×C53—22no—
70Q16×C53C4⋊C414yes—
71C2×C8×C53C22×C411yescharacteristic
72C2×C8×C53C2×D411yescharacteristic
73C2×C8×C53C2×D411yescharacteristic
74C2×C8×C53C2×Q811yescharacteristic
75C2×C8×C53#4111yescharacteristic
76C2×C8×C53#4111yescharacteristic
77C4⋊C4×C53—22no—
78C22⋊C4×C53—41no—
79C42×C53—41no—
80C4⋊Dic53—22no—
81C4⋊Dic53—44no—
82C4⋊Dic53—41no—
83C4×Dic53—21no—
84C4×Dic53—44no—
85C4×Dic53SD1612yes—
86#11719917—41no—
87#11719917—41no—
88C2×#88322—21no—
89C2×#88322—21no—
90C2×#88322—21no—
91C2×#88322—41no—
92C16×C53—22no—
93#11719908—41no—
94C23×C53AGL(1,ℤ/8)11yescharacteristic
95Q8×C53—24no—
96Q8×C53—42no—
97Q8×C53—42no—
98Q8×C53—42no—
99C2×C4×C53—42no—
100C2×C4×C53—42no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53C4×D411yescharacteristic
104C2×C4×C53#12011yescharacteristic
105C2×C4×C53C22⋊Q811yescharacteristic
106C2×C4×C53C2×SD1611yescharacteristic
107C2×C4×C53#13511yescharacteristic
108C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
109C2×Dic53—22no—
110C2×Dic53—42no—
111C2×Dic53—42no—
112C2×Dic53—82no—
113C8×C53C2×C4⋊C411yescharacteristic
114C8×C53C4×D411yescharacteristic
115C8×C53C22⋊Q811yescharacteristic
116C8×C53#12311yescharacteristic
117#88322—41no—
118#88322—81no—
119#12957—531nomaximal, Sylow
120C22×C53—22no—
121C22×C53#44211yescharacteristic
122C22×C53#45011yescharacteristic
123C22×C53#47811yescharacteristic
124C4×C53—44no—
125C4×C53—82no—
126C4×C53#38511yescharacteristic
127C4×C53C4×SD1611yescharacteristic
128C4×C53#44911yescharacteristic
129C4×C53#47611yescharacteristic
130Dic53—42no—
131Dic53—82no—
132C2×#441—531no—
133C2×#365—531no—
134C2×#362—531no—
135#2141D5311yescharacteristic
136#1946—531no—
137#1386—531no—
138#1375—531no—
139C2×C53—21no—
140C2×C53—21no—
141C2×C53#186411yescharacteristic
142C2×C53#218012yes—
143C22×Q16C2×D5311yescharacteristic
144C2×C4×Q8—1061no—
145C2×M32—531no—
146C2×M32C2×D5311yescharacteristic
147#441—1062no—
148#441—1062no—
149C2×#108—531no—
150#427C2×D5311yescharacteristic
151C2×#106—531no—
152#424—1061no—
153C2×#103—1061no—
154C2×#97—531no—
155#365—1062no—
156#362—1062no—
157#358Dic5314yescomplemented
158#339—1061no—
159C53#1295711yescharacteristic, complemented, Sylow
160C22×Q8—21no—
161C2×Q16—21no—
162C2×Q16—21no—
163C2×Q16—21no—
164C2×Q16Dic10612yes—
165C2×Q16D21212yes—
166C2×Q16C2×Dic5312yes—
167C2×M16—1061no—
168C22×C8—531no—
169C22×C8C22×D5311yescharacteristic
170C4×Q8—1064no—
171C4×Q8—2122no—
172#117—21no—
173C2×C4⋊C4—1061no—
174C2×C4⋊C4—2121no—
175C2×C42—531no—
176C2×C42—2121no—
177M32—21no—
178M32—21no—
179M32—1061no—
180M32—1061no—
181C2×C16—1061no—
182C2×C16C2×Dic5312yes—
183#108—1061no—
184#108—1061no—
185#107C2×Dic5312yes—
186#106—21no—
187#106—532no—
188#106—1061no—
189C4⋊C8—2122no—
190#103—1064no—
191#97—1061no—
192#97—1061no—
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—41no—
199C22×C4—41no—
200C22×C4—1061no—
201C22×C4—2121no—
202C22×C4D4×D5311yescharacteristic
203Q16—22no—
204Q16—22no—
205Q16—22no—
206Q16C4⋊Dic5314yes—
207M16—2121no—
208M16—2121no—
209C2×C8—1061no—
210C2×C8—1061no—
211C2×C8—1061no—
212C2×C8—2121no—
213C2×C8C2×Dic10611yescharacteristic
214C2×C8C2×D21211yescharacteristic
215C2×C8D4×D5311yescharacteristic
216C2×C8#1171994711yescharacteristic
217C2×C8#1171994711yescharacteristic
218C2×C8C22×Dic5311yescharacteristic
219C4⋊C4—22no—
220C4⋊C4—1062no—
221C4⋊C4—2124no—
222C4⋊C4—2121no—
223C22⋊C4—41no—
224C42—41no—
225C42—532no—
226C42—1061no—
227C42—2124no—
228C16—22no—
229C16—2121no—
230C23#1313324111yescharacteristic
231Q8—24no—
232Q8—42no—
233Q8—42no—
234Q8—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—1062no—
240C2×C4—4242no—
241C2×C4—2122no—
242C2×C4—2122no—
243C2×C4D4×Dic5311yescharacteristic
244C2×C4#1313318711yescharacteristic
245C2×C4#1313319411yescharacteristic
246C2×C4SD16×D5311yescharacteristic
247C2×C4#1313323811yescharacteristic
248C2×C4#1313324111yescharacteristic
249C8—4241no—
250C8—2121no—
251C8C2×C4⋊Dic5311yescharacteristic
252C8D4×Dic5311yescharacteristic
253C8#1313319411yescharacteristic
254C8#1313320111yescharacteristic
255C22—22no—
256C22#1681073511yescharacteristic
257C22#1681077511yescharacteristic
258C22#1681092511yescharacteristic
259C4—44no—
260C4—82no—
261C4—4242no—
262C4—2122no—
263C4#1681049811yescharacteristic
264C4SD16×Dic5311yescharacteristic
265C4#1681076911yescharacteristic
266C4#1681091111yescharacteristic
267C2—21no—
268C2—21no—
269C2#2027497811yescharacteristic
270C2#2027660712yes—
subgroupG/H
centerC22#16810735
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#2141C2
socleC22×C53#442
prime radicalC23×C53AGL(1,ℤ/8)
cosocleC22×C8×C53C23
hypercenter#2141D53
hypocenterC53#12957
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2141

Complements

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

NH ≅ G/Ntype
23. C53×#358C4semidirect
157. #358Dic53semidirect
159. C53#12957semidirect

Maximal subgroups

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

subgroupindex
C2×#168107302
C2×#168104462
C2×#168104392
#202754692
#202728202
#202727942
C53×#21412
#1295753