Finite Group Databasebeta

← #27285070

Subgroups — #27285070

Order 13568.

584 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 103 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#20285161C211yesmaximal, characteristic
2#20284746C212yesmaximal, complemented
3#20273113C212yesmaximal, complemented
4#20272546C211yesmaximal, characteristic, complemented
5C53×#1985C211yesmaximal, characteristic, complemented
6C2×#13133239—22no—
7#16811173C2212yes—
8#16811017—22no—
9#16811012—22no—
10#16810941—22no—
11#16810908—22no—
12#16810823—22no—
13#16810762—22no—
14C4×#11719924—24no—
15#16810676C2212yes—
16#16810350—24no—
17C2×C4×Q8×C53C2211yescharacteristic
18C2×C4⋊C8×C53C2211yescharacteristic
19C2×#97×C53C2211yescharacteristic
20C53×#336—22no—
21C2×#11719945—22no—
22#13133239—44no—
23#13133239—44no—
24C2×Dic212—22no—
25C2×#11719913—24no—
26C2×#11719924—24no—
27C2×D424—22no—
28#13133217D412yes—
29#13133214D412yes—
30#13133205—22no—
31#13133191—22no—
32#13133191—22no—
33C4×D212—22no—
34C4×D212—22no—
35#13133148—24no—
36C4⋊#88322—22no—
37#13133141—24no—
38#13133137—24no—
39C22⋊#88322—22no—
40C4×#88322D414yes—
41C22×Q8×C53D411yescharacteristic
42C22×C8×C53—21no—
43C22×C8×C53D412yes—
44C4×Q8×C53—22no—
45C4×Q8×C53—22no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53D411yescharacteristic
48C2×C42×C53—21no—
49C2×C42×C53C2311yescharacteristic
50C4⋊C8×C53—22no—
51#97×C53—22no—
52C2×Q8×C53—22no—
53C2×Q8×C53—42no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53C2×D411yescharacteristic
57C22×C4×C53C2×D411yescharacteristic
58C22×C4×C53#4111yescharacteristic
59C2×C22⋊D53—42no—
60#11719945—44no—
61#11719945—44no—
62C2×D212—22no—
63C2×D212—22no—
64C2×D212—22no—
65C2×C4×D53—42no—
66C2×Dic106—22no—
67C2×C8×C53—22no—
68C2×C8×C53—22no—
69C2×C8×C53—41no—
70C2×C8×C53—42no—
71C2×C8×C53SD1614yescomplemented
72C4⋊C4×C53—22no—
73C4⋊C4×C53—42no—
74C42×C53—41no—
75C42×C53—41no—
76C42×C53C2×D412yes—
77C42×C53C2×D412yes—
78#11719924—48no—
79#11719921—42no—
80#11719921—42no—
81C4⋊Dic53—22no—
82C4⋊Dic53—22no—
83C2×#88322—24no—
84Dic212—44no—
85D424—44no—
86#11719913—48no—
87C23×C53#12111yescharacteristic
88C22×D53—42no—
89Q8×C53—24no—
90Q8×C53—42no—
91C2×C4×C53—22no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53—41no—
96C2×C4×C53—42no—
97C2×C4×C53—41no—
98C2×C4×C53#12011yescharacteristic
99C2×C4×C53#12012yes—
100C2×C4×C53#12112yes—
101C2×C4×C53#12711yescharacteristic
102C2×C4×C53C2×SD1612yes—
103C2×C4×C53AGL(1,ℤ/8)12yes—
104C22⋊D53—84no—
105C2×Dic53—42no—
106D212—44no—
107D212—44no—
108D212—44no—
109C4×D53—82no—
110C4×D53—82no—
111Dic106—44no—
112C8×C53—24no—
113C8×C53—81no—
114#88322—44no—
115#10034—531nomaximal, Sylow
116C22×C53—21no—
117C22×C53—21no—
118C22×C53#39211yescharacteristic
119C22×C53#46012yes—
120C2×D53—82no—
121C2×D53—82no—
122C2×D53—82no—
123C4×C53—22no—
124C4×C53—22no—
125C4×C53—42no—
126C4×C53—81no—
127C4×C53#45012yes—
128C4×C53#49211yescharacteristic
129C4×C53#49611yescharacteristic
130Dic53—82no—
131#3150—531no—
132#3103—532no—
133#1985D5311yescharacteristic, complemented
134#1480—532no—
135#1275—531no—
136C2×C53—22no—
137C2×C53#201011yescharacteristic
138C2×C53#201411yescharacteristic
139C2×C53#201711yescharacteristic
140D53—162no—
141C2×#135—1062no—
142#532—532no—
143C2×C4×Q8C2×D5311yescharacteristic
144#497—1062no—
145#496—1062no—
146#481—1062no—
147#475—1062no—
148#460—1062no—
149#448—1062no—
150C4×SD16—1064no—
151#431—532no—
152C2×C4⋊C8C2×D5311yescharacteristic
153C2×#97C2×D5311yescharacteristic
154#336—22no—
155D4⋊C8—1064no—
156C53#1003411yescharacteristic, complemented, Sylow
157C2×#41—1062no—
158C22×Q8D21211yescharacteristic
159#135—2124no—
160#135—2124no—
161C2×Q16—1062no—
162C2×SD16—1064no—
163C2×SD16—1064no—
164C2×D8—1062no—
165C22×C8—21no—
166C22×C8C22⋊D5312yes—
167C4⋊Q8—532no—
168#127—532no—
169#124—1062no—
170#121—1062no—
171#121—1062no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×D4—1062no—
175C4×D4—1062no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4D21211yescharacteristic
178C2×C42—21no—
179C2×C42C22×D5311yescharacteristic
180#106—1064no—
181C4⋊C8—22no—
182C4⋊C8—1062no—
183#103—1064no—
184#102—1064no—
185C22⋊C8—1062no—
186#97—22no—
187C4×C8—534no—
188#41—2124no—
189#41—2124no—
190C2×Q8—22no—
191C2×Q8—42no—
192C2×Q8—1062no—
193C2×D4—1062no—
194C2×D4—1062no—
195C2×D4—1062no—
196C2×D4—2122no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—2122no—
200C22×C4C2×D21211yescharacteristic
201C22×C4#1171994511yescharacteristic
202C22×C4C2×C22⋊D5311yescharacteristic
203Q16—2124no—
204SD16—2128no—
205SD16—2128no—
206D8—2124no—
207C2×C8—22no—
208C2×C8—22no—
209C2×C8—41no—
210C2×C8—42no—
211C2×C8—1064no—
212C2×C8#1171992414yescomplemented
213C4⋊C4—22no—
214C4⋊C4—42no—
215C4⋊C4—1062no—
216C4⋊C4—1062no—
217C22⋊C4—2122no—
218C22⋊C4—2122no—
219C42—41no—
220C42—41no—
221C42D4×D5312yes—
222C42D4×D5312yes—
223C23—2122no—
224C23#1313319111yescharacteristic
225Q8—24no—
226Q8—42no—
227Q8—2124no—
228D4—4244no—
229D4—2124no—
230D4—2124no—
231D4—2124no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—4242no—
240C2×C4—4242no—
241C2×C4—2122no—
242C2×C4#1313318511yescharacteristic
243C2×C4#1313318712yes—
244C2×C4#1313319012yes—
245C2×C4#1313321311yescharacteristic
246C2×C4C2×#1171992412yes—
247C2×C4#1313324212yes—
248C8—24no—
249C8—81no—
250C8—2124no—
251C22—21no—
252C22—21no—
253C22—4242no—
254C22—4242no—
255C22—4242no—
256C22#1681053211yescharacteristic
257C22#1681082312yes—
258C4—22no—
259C4—22no—
260C4—42no—
261C4—81no—
262C4—4242no—
263C4#1681077312yes—
264C4#1681099711yescharacteristic
265C4#1681101311yescharacteristic
266C2—22no—
267C2—8482no—
268C2#2027578211yescharacteristic
269C2#2027580211yescharacteristic
270C2#2027581911yescharacteristic
subgroupG/H
centerC2×C4#13133213
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1985C2
socleC22×C53#392
prime radical#20285161C2
hypercenter#1985D53
hypocenterC53#10034
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1985

Complements

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

NH ≅ G/Ntype
2. #20284746C2semidirect
3. #20273113C2semidirect
4. #20272546C2semidirect
5. C53×#1985C2semidirect
71. C2×C8×C53SD16semidirect
133. #1985D53semidirect
156. C53#10034semidirect
212. C2×C8#11719924semidirect

Maximal subgroups

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

subgroupindex
#202851612
#202847462
#202731132
#202725462
C53×#19852
#1003453