Finite Group Databasebeta

← #20285908

Subgroups — #20285908

Order 6784.

366 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 111 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#16811179C211yesmaximal, characteristic, complemented
2#16811171C211yesmaximal, characteristic
3#16811033C211yesmaximal, characteristic, complemented
4#16811032C211yesmaximal, characteristic, complemented
5#16810987C211yesmaximal, characteristic, complemented
6#16810946C212yesmaximal, complemented
7#16810945C212yesmaximal, complemented
8#16810911C212yesmaximal, complemented
9#16810901C212yesmaximal, complemented
10#16810678C211yesmaximal, characteristic, complemented
11C53×#490C211yesmaximal, characteristic, complemented
12C2×#11719947—21no—
13#13133217C2211yescharacteristic
14#13133216C2211yescharacteristic
15#13133216C2211yescharacteristic
16#13133209C2211yescharacteristic
17#13133204—21no—
18#13133195—21no—
19#13133195—21no—
20#13133188—21no—
21#13133188C2212yes—
22#13133182—21no—
23#13133182—21no—
24D4×Dic53—21no—
25D4×Dic53C2212yes—
26#13133175C2211yescharacteristic
27#13133175C2211yescharacteristic
28C42×D53C2211yescharacteristic
29#13133151C2212yes—
30#13133150C2212yes—
31#13133149C2212yes—
32#13133148C2212yes—
33#13133146C2212yes—
34#13133138C2212yes—
35#13133138C2212yes—
36#13133129C2212yes—
37#13133128C2211yescharacteristic
38#127×C53C2211yescharacteristic
39#125×C53C2211yescharacteristic
40#102×C53C2212yes—
41#102×C53C2212yes—
42#97×C53C2211yescharacteristic
43C2×D4×C53—21no—
44C2×D4×C53—21no—
45C2×D4×C53C2312yes—
46C2×C22⋊D53—22no—
47C22×Dic53—22no—
48#11719947—44no—
49C2×C4×D53D411yescharacteristic
50C2×C4×D53D411yescharacteristic
51C2×C4×D53C2311yescharacteristic
52C2×Dic106—21no—
53C2×Dic106—21no—
54C2×Dic106—21no—
55C2×C8×C53C2312yes—
56C4⋊C4×C53—21no—
57C4⋊C4×C53—21no—
58C4⋊C4×C53C2311yescharacteristic
59C4⋊C4×C53C2311yescharacteristic
60C42×C53C2311yescharacteristic
61C22⋊Dic53—22no—
62C22⋊Dic53—22no—
63#11719921—21no—
64#11719921—21no—
65C4⋊Dic53—21no—
66C4⋊Dic53C2311yescharacteristic
67C4⋊Dic53C2311yescharacteristic
68C4⋊Dic53C2312yes—
69#11719919—21no—
70#11719919—21no—
71#11719919—21no—
72#11719919—21no—
73C4×Dic53—21no—
74C4×Dic53—21no—
75C4×Dic53D411yescharacteristic
76C4×Dic53D411yescharacteristic
77C4×Dic53C2311yescharacteristic
78C2×#88322C2312yes—
79C23×C53—22no—
80C22×D53C2×D411yescharacteristic
81D4×C53—24no—
82D4×C53—42no—
83D4×C53—42no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53C2×D411yescharacteristic
87C2×C4×C53C2×D411yescharacteristic
88C2×C4×C53C2411yescharacteristic
89C22⋊D53—44no—
90C2×Dic53—22no—
91C2×Dic53—21no—
92C2×Dic53—21no—
93C2×Dic53—44no—
94C2×Dic53C2×D411yescharacteristic
95C2×Dic53C2×D411yescharacteristic
96C2×Dic53C2×D411yescharacteristic
97C4×D53—22no—
98C4×D53—22no—
99C4×D53—41no—
100Dic106—42no—
101Dic106—42no—
102Dic106—42no—
103C8×C53—22no—
104#88322—22no—
105C22×C53—42no—
106C22×C53—42no—
107C22×C53—42no—
108C22×C53C22×D411yescharacteristic
109C2×D53—21no—
110C2×D53—21no—
111C2×D53—21no—
112C4×C53—21no—
113C4×C53—21no—
114C4×C53—41no—
115C4×C53—41no—
116C4×C532−1+412yes—
117Dic53—21no—
118Dic53—21no—
119Dic53—41no—
120Dic53—42no—
121Dic53—41no—
122Dic53AGL(1,ℤ/8)12yes—
123#3235—531nomaximal, Sylow
124C2×C53—82no—
125C2×C53#53611yescharacteristic
126C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
127C2×C53#57511yescharacteristic
128D53—41no—
129#533—531no—
130#532—531no—
131#501—531no—
132#501—531no—
133#490—531no—
134#490D5311yescharacteristic, complemented
135#482—532no—
136#482—532no—
137#476—532no—
138#474—532no—
139#432—531no—
140C53#323511yescharacteristic, complemented, Sylow
141C2×#41—1061no—
142C4⋊Q8—531no—
143C4⋊Q8—531no—
144C4⋊Q8—531no—
145#127C2×D5311yescharacteristic
146#125—531no—
147#125C2×D5311yescharacteristic
148#124—1061no—
149C22⋊Q8—1061no—
150C22⋊Q8—1061no—
151#121—532no—
152#121—1061no—
153C4×Q8—1061no—
154C4×Q8—1061no—
155C4×D4—532no—
156C4×D4—1061no—
157#117—531no—
158#117—531no—
159C2×C42—531no—
160#107—532no—
161#107—532no—
162#106—532no—
163#106—532no—
164C4⋊C8—532no—
165#102—532no—
166#102—532no—
167#102C2×D5312yes—
168#102C2×D5312yes—
169C22⋊C8—532no—
170#97—531no—
171#97C2×D5311yescharacteristic
172#41—2124no—
173C2×Q8—1061no—
174C2×Q8—1061no—
175C2×Q8—1061no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—1062no—
179C2×D4C22×D5312yes—
180C22×C4—531no—
181C22×C4—531no—
182C22×C4—531no—
183C22×C4—1062no—
184C2×C8—532no—
185C2×C8C22×D5312yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—531no—
189C4⋊C4—531no—
190C4⋊C4—532no—
191C4⋊C4—1061no—
192C4⋊C4—1061no—
193C4⋊C4—1061no—
194C4⋊C4—1061no—
195C4⋊C4—1061no—
196C4⋊C4C22×D5311yescharacteristic
197C4⋊C4C22×D5311yescharacteristic
198C22⋊C4—1061no—
199C22⋊C4—1061no—
200C22⋊C4—1062no—
201C22⋊C4—1062no—
202C42—531no—
203C42—531no—
204C42—531no—
205C42—1061no—
206C42—1061no—
207C42C22×D5311yescharacteristic
208C23—22no—
209C23—531no—
210Q8—2122no—
211Q8—2122no—
212Q8—2122no—
213D4—24no—
214D4—42no—
215D4—42no—
216D4—2124no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—531no—
220C2×C4—531no—
221C2×C4—531no—
222C2×C4—1062no—
223C2×C4—1062no—
224C2×C4—1061no—
225C2×C4—1061no—
226C2×C4—1062no—
227C2×C4—2121no—
228C2×C4—2124no—
229C2×C4D4×D5311yescharacteristic
230C2×C4D4×D5311yescharacteristic
231C2×C4C23×D5311yescharacteristic
232C8—22no—
233C8—1062no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—1061no—
238C22—1061no—
239C22—1061no—
240C22C2×D4×D5311yescharacteristic
241C4—21no—
242C4—21no—
243C4—41no—
244C4—41no—
245C4—532no—
246C4—1061no—
247C4—1061no—
248C4—2122no—
249C4—2121no—
250C4—2121no—
251C4#1313326412yes—
252C2—82no—
253C2—2121no—
254C2#1681119311yescharacteristic
255C2AGL(1,ℤ/8)×D5311yescharacteristic
256C2#1681141911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#490C2
socleC22×C53C22×D4
hypercenter#490D53
hypocenterC53#3235
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #16811179C2semidirect
3. #16811033C2semidirect
4. #16811032C2semidirect
5. #16810987C2semidirect
6. #16810946C2semidirect
7. #16810945C2semidirect
8. #16810911C2semidirect
9. #16810901C2semidirect
10. #16810678C2semidirect
11. C53×#490C2semidirect
134. #490D53semidirect
140. C53#3235semidirect