Finite Group Databasebeta

← #20285952

Subgroups — #20285952

Order 6784.

328 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 109 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#16811178C211yesmaximal, characteristic, complemented
2#16811152C211yesmaximal, characteristic
3#16811032C211yesmaximal, characteristic, complemented
4#16811029C211yesmaximal, characteristic, complemented
5#16810982C211yesmaximal, characteristic, complemented
6#16810963C212yesmaximal, complemented
7#16810946C212yesmaximal
8#16810928C212yesmaximal, complemented
9#16810921C212yesmaximal, complemented
10#16810678C211yesmaximal, characteristic, complemented
11C53×#489C211yesmaximal, characteristic, complemented
12#13133217C2211yescharacteristic
13#13133210—21no—
14#13133209C2211yescharacteristic
15#13133208C2212yes—
16#13133199—21no—
17#13133197—21no—
18#13133193—21no—
19#13133193C2211yescharacteristic
20#13133188C2211yescharacteristic
21Q8×Dic53—21no—
22Q8×Dic53C2211yescharacteristic
23#13133182—21no—
24D4×Dic53C2211yescharacteristic
25#13133177—21no—
26#13133175C2212yes—
27#13133173—21no—
28C42×D53C2211yescharacteristic
29#13133151C2212yes—
30#13133150C2212yes—
31#13133149C2212yes—
32#13133148C2212yes—
33#13133146C2212yes—
34#13133141C2212yes—
35#13133138C2212yes—
36#13133129C2212yes—
37#13133128C2211yescharacteristic
38C4⋊Q8×C53C2211yescharacteristic
39#124×C53C2211yescharacteristic
40#103×C53C2212yes—
41#102×C53C2212yes—
42#97×C53C2211yescharacteristic
43C2×Q8×C53—21no—
44C2×Q8×C53C2311yescharacteristic
45C2×D4×C53C2311yescharacteristic
46C2×C22⋊D53—21no—
47C22×Dic53—21no—
48C2×C4×D53D411yescharacteristic
49C2×C4×D53D411yescharacteristic
50C2×C4×D53C2311yescharacteristic
51C2×Dic106—21no—
52C2×C8×C53C2312yes—
53C4⋊C4×C53—21no—
54C4⋊C4×C53C2312yes—
55C22⋊C4×C53—21no—
56C22⋊C4×C53—21no—
57C42×C53C2311yescharacteristic
58C22⋊Dic53—21no—
59C22⋊Dic53—21no—
60#11719921—22no—
61#11719921—21no—
62C4⋊Dic53—21no—
63C4⋊Dic53—22no—
64C4⋊Dic53—21no—
65C4⋊Dic53C2312yes—
66C4⋊Dic53C2311yescharacteristic
67C4⋊Dic53C2311yescharacteristic
68#11719919—21no—
69#11719919—21no—
70#11719919—21no—
71#11719919—22no—
72C4×Dic53—21no—
73C4×Dic53—22no—
74C4×Dic53D411yescharacteristic
75C4×Dic53D411yescharacteristic
76C4×Dic53C2311yescharacteristic
77C2×#88322C2312yes—
78C23×C53—21no—
79C22×D53C2×D411yescharacteristic
80Q8×C53—22no—
81Q8×C53—42no—
82D4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—21no—
85C2×C4×C53C2×D411yescharacteristic
86C2×C4×C53C2×D411yescharacteristic
87C2×C4×C53C2411yescharacteristic
88C22⋊D53—42no—
89C2×Dic53—22no—
90C2×Dic53—21no—
91C2×Dic53—21no—
92C2×Dic53—42no—
93C2×Dic53C2×D411yescharacteristic
94C2×Dic53C2×D411yescharacteristic
95C2×Dic53C2×D411yescharacteristic
96C4×D53—22no—
97C4×D53—41no—
98C4×D53—41no—
99Dic106—42no—
100C8×C53—22no—
101#88322—22no—
102C22×C53—41no—
103C22×C53—41no—
104C22×C53—41no—
105C22×C53C22×D411yescharacteristic
106C2×D53—21no—
107C2×D53—21no—
108C2×D53—21no—
109C4×C53—21no—
110C4×C53—21no—
111C4×C53—41no—
112C4×C53—42no—
113C4×C532−1+412yes—
114Dic53—21no—
115Dic53—21no—
116Dic53—21no—
117Dic53—42no—
118Dic53—41no—
119Dic53—41no—
120#3240—531nomaximal, Sylow
121C2×C53—81no—
122C2×C53#53611yescharacteristic
123C2×C53#57511yescharacteristic
124C2×C53#57511yescharacteristic
125D53—41no—
126#533—531no—
127#529—531no—
128#501—531no—
129#501—531no—
130#489—531no—
131#489D5311yescharacteristic, complemented
132#485—532no—
133#482—532no—
134#479—532no—
135#478—532no—
136#432—531no—
137C53#324011yescharacteristic, complemented, Sylow
138C4⋊Q8—531no—
139C4⋊Q8C2×D5311yescharacteristic
140#126—1061no—
141#125—532no—
142#125—531no—
143#124C2×D5311yescharacteristic
144#123—1061no—
145C22⋊Q8—531no—
146C22⋊Q8—1061no—
147C22⋊Q8—1061no—
148#121—531no—
149C4×Q8—531no—
150C4×Q8—1061no—
151C4×Q8—1061no—
152C4×D4—531no—
153C4×D4—1061no—
154#117—532no—
155#117—1061no—
156C2×C42—531no—
157#107—532no—
158#107—532no—
159#106—532no—
160#106—532no—
161C4⋊C8—532no—
162#103—532no—
163#103C2×D5312yes—
164#102—532no—
165#102C2×D5312yes—
166C22⋊C8—532no—
167#97—531no—
168#97C2×D5311yescharacteristic
169C2×Q8—21no—
170C2×Q8—1061no—
171C2×Q8C22×D5311yescharacteristic
172C2×D4—1061no—
173C2×D4C22×D5311yescharacteristic
174C22×C4—531no—
175C22×C4—531no—
176C22×C4—531no—
177C22×C4—1061no—
178C2×C8—532no—
179C2×C8C22×D5312yes—
180C4⋊C4—21no—
181C4⋊C4—532no—
182C4⋊C4—531no—
183C4⋊C4—531no—
184C4⋊C4—1061no—
185C4⋊C4—1061no—
186C4⋊C4—1062no—
187C4⋊C4—1062no—
188C4⋊C4—1061no—
189C4⋊C4—1061no—
190C4⋊C4—1061no—
191C4⋊C4C22×D5312yes—
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—1062no—
195C22⋊C4—1061no—
196C22⋊C4—1061no—
197C22⋊C4—1061no—
198C42—531no—
199C42—531no—
200C42—531no—
201C42—1062no—
202C42—1061no—
203C42C22×D5311yescharacteristic
204C23—21no—
205C23—531no—
206Q8—22no—
207Q8—42no—
208Q8—2122no—
209D4—22no—
210D4—2122no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—531no—
214C2×C4—531no—
215C2×C4—531no—
216C2×C4—1062no—
217C2×C4—1061no—
218C2×C4—1062no—
219C2×C4—1061no—
220C2×C4—2121no—
221C2×C4—2121no—
222C2×C4—2122no—
223C2×C4D4×D5311yescharacteristic
224C2×C4D4×D5311yescharacteristic
225C2×C4C23×D5311yescharacteristic
226C8—22no—
227C8—1062no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—1061no—
232C22—1061no—
233C22—1061no—
234C22C2×D4×D5311yescharacteristic
235C4—21no—
236C4—21no—
237C4—42no—
238C4—41no—
239C4—1061no—
240C4—1061no—
241C4—1061no—
242C4—2121no—
243C4—2121no—
244C4—2122no—
245C4#1313326412yes—
246C2—81no—
247C2—2121no—
248C2#1681119311yescharacteristic
249C2#1681141911yescharacteristic
250C2#1681142211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#489C2
socleC22×C53C22×D4
hypercenter#489D53
hypocenterC53#3240
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #16811178C2semidirect
3. #16811032C2semidirect
4. #16811029C2semidirect
5. #16810982C2semidirect
6. #16810963C2semidirect
8. #16810928C2semidirect
9. #16810921C2semidirect
10. #16810678C2semidirect
11. C53×#489C2semidirect
131. #489D53semidirect
137. C53#3240semidirect