Finite Group Databasebeta

← #20285892

Subgroups — #20285892

Order 6784.

332 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 113 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#16811155C211yesmaximal, characteristic
3#16811024C212yesmaximal, complemented
4#16810976C212yesmaximal, complemented
5#16810966C212yesmaximal, complemented
6#16810945C212yesmaximal
7#16810919C211yesmaximal, characteristic, complemented
8#16810913C211yesmaximal, characteristic, complemented
9#16810905C211yesmaximal, characteristic, complemented
10#16810675C211yesmaximal, characteristic, complemented
11C53×#474C211yesmaximal, characteristic, complemented
12#13133217C2211yescharacteristic
13#13133210—21no—
14#13133208C2211yescharacteristic
15#13133208C2212yes—
16#13133204C2211yescharacteristic
17#13133199—21no—
18#13133197—21no—
19#13133197C2211yescharacteristic
20#13133193—21no—
21#13133188C2211yescharacteristic
22Q8×Dic53—21no—
23C4×Dic106C2211yescharacteristic
24#13133182—21no—
25C4×C22⋊D53—21no—
26#13133176—21no—
27#13133175C2212yes—
28C42×D53C2211yescharacteristic
29#13133151C2212yes—
30#13133148C2212yes—
31#13133148C2212yes—
32C4⋊#88322C2211yescharacteristic
33#13133142C2212yes—
34#13133140C2212yes—
35#13133138C2212yes—
36#13133129C2212yes—
37C8×Dic53C2212yes—
38C4⋊Q8×C53C2211yescharacteristic
39C4×D4×C53C2211yescharacteristic
40#107×C53C2212yes—
41C4⋊C8×C53C2211yescharacteristic
42#102×C53C2212yes—
43C2×Q8×C53—21no—
44C2×D4×C53C2311yescharacteristic
45C22×C4×C53—21no—
46C2×C22⋊D53—21no—
47C2×C4×D53D411yescharacteristic
48C2×C4×D53D411yescharacteristic
49C2×C4×D53C2311yescharacteristic
50C2×Dic106—21no—
51C2×Dic106C2311yescharacteristic
52C2×C8×C53C2312yes—
53C4⋊C4×C53—21no—
54C4⋊C4×C53C2312yes—
55C4⋊C4×C53C2311yescharacteristic
56C22⋊C4×C53—21no—
57C42×C53C2311yescharacteristic
58C22⋊Dic53—21no—
59C22⋊Dic53—21no—
60C22⋊Dic53—21no—
61#11719921—22no—
62#11719921—21no—
63C4⋊Dic53—21no—
64C4⋊Dic53—21no—
65C4⋊Dic53—21no—
66C4⋊Dic53—22no—
67C4⋊Dic53C2312yes—
68C4⋊Dic53C2311yescharacteristic
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—42no—
81D4×C53—21no—
82D4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—22no—
85C2×C4×C53—42no—
86C2×C4×C53C2×D411yescharacteristic
87C2×C4×C53C2×D411yescharacteristic
88C2×C4×C53C2411yescharacteristic
89C22⋊D53—42no—
90C2×Dic53—22no—
91C2×Dic53—21no—
92C2×Dic53—21no—
93C2×Dic53C2×D411yescharacteristic
94C2×Dic53C2×D411yescharacteristic
95C2×Dic53C2×D411yescharacteristic
96C4×D53—22no—
97C4×D53—21no—
98C4×D53—21no—
99C4×D53—41no—
100Dic106—21no—
101Dic106—21no—
102Dic106—42no—
103C8×C53—22no—
104#88322—22no—
105C22×C53—41no—
106C22×C53—41no—
107C22×C53—41no—
108C22×C53C22×D411yescharacteristic
109C2×D53—21no—
110C2×D53—21no—
111C2×D53—21no—
112C4×C53—21no—
113C4×C53—41no—
114C4×C53—42no—
115C4×C53#13512yes—
116C4×C532−1+411yescharacteristic
117C4×C532−1+411yescharacteristic
118Dic53—21no—
119Dic53—21no—
120Dic53—21no—
121Dic53—41no—
122Dic53—42no—
123Dic53—41no—
124#3233—531nomaximal, Sylow
125C2×C53—81no—
126C2×C53#53611yescharacteristic
127C2×C53C2×#13511yescharacteristic
128C2×C53#57511yescharacteristic
129D53—41no—
130#533—531no—
131#529—531no—
132#499—532no—
133#488—532no—
134#485—532no—
135#482—532no—
136#477—531no—
137#476—531no—
138#475—531no—
139#474D5311yescharacteristic, complemented
140#431—531no—
141C53#323311yescharacteristic, complemented, Sylow
142C4⋊Q8—531no—
143C4⋊Q8C2×D5311yescharacteristic
144#126—1061no—
145#125—531no—
146#125—532no—
147#124—531no—
148#123—1061no—
149C22⋊Q8—531no—
150C22⋊Q8—1061no—
151C22⋊Q8—1061no—
152#121—531no—
153C4×Q8—531no—
154C4×Q8—1061no—
155C4×Q8—1061no—
156C4×D4—1061no—
157C4×D4C2×D5311yescharacteristic
158#117—532no—
159#117—1061no—
160C2×C42—531no—
161#107—532no—
162#107C2×D5312yes—
163#106—532no—
164#106—532no—
165C4⋊C8—531no—
166C4⋊C8C2×D5311yescharacteristic
167#103—532no—
168#103—532no—
169#102—532no—
170#102C2×D5312yes—
171C22⋊C8—532no—
172C4×C8—532no—
173C2×Q8—21no—
174C2×Q8—531no—
175C2×Q8—1061no—
176C2×D4—1061no—
177C2×D4C22×D5311yescharacteristic
178C22×C4—21no—
179C22×C4—531no—
180C22×C4—531no—
181C22×C4—531no—
182C2×C8—532no—
183C2×C8C22×D5312yes—
184C4⋊C4—21no—
185C4⋊C4—532no—
186C4⋊C4—531no—
187C4⋊C4—1061no—
188C4⋊C4—1061no—
189C4⋊C4—1061no—
190C4⋊C4—1062no—
191C4⋊C4—1062no—
192C4⋊C4—1061no—
193C4⋊C4—1061no—
194C4⋊C4C22×D5312yes—
195C4⋊C4C22×D5311yescharacteristic
196C22⋊C4—21no—
197C22⋊C4—1062no—
198C22⋊C4—1061no—
199C22⋊C4—1061no—
200C22⋊C4—1061no—
201C22⋊C4—1061no—
202C42—531no—
203C42—531no—
204C42—531no—
205C42—1062no—
206C42—1061no—
207C42C22×D5311yescharacteristic
208C23—21no—
209C23—531no—
210Q8—42no—
211Q8—1061no—
212Q8—1061no—
213Q8—2122no—
214D4—21no—
215D4—21no—
216D4—2122no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—42no—
220C2×C4—531no—
221C2×C4—531no—
222C2×C4—531no—
223C2×C4—1062no—
224C2×C4—1061no—
225C2×C4—1061no—
226C2×C4—1061no—
227C2×C4—1061no—
228C2×C4—1062no—
229C2×C4—2121no—
230C2×C4D4×D5311yescharacteristic
231C2×C4D4×D5311yescharacteristic
232C2×C4C23×D5311yescharacteristic
233C8—22no—
234C8—1062no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—1061no—
239C22—1061no—
240C22—1061no—
241C22C2×D4×D5311yescharacteristic
242C4—21no—
243C4—41no—
244C4—42no—
245C4—1061no—
246C4—1061no—
247C4—1061no—
248C4—2121no—
249C4—2122no—
250C4—2121no—
251C4#1313323612yes—
252C4#1313326411yescharacteristic
253C4#1313326511yescharacteristic
254C2—81no—
255C2—2121no—
256C2#1681119611yescharacteristic
257C2C2×#1313323611yescharacteristic
258C2#1681142211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#474C2
socleC22×C53C22×D4
hypercenter#474D53
hypocenterC53#3233
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #16811178C2semidirect
3. #16811024C2semidirect
4. #16810976C2semidirect
5. #16810966C2semidirect
7. #16810919C2semidirect
8. #16810913C2semidirect
9. #16810905C2semidirect
10. #16810675C2semidirect
11. C53×#474C2semidirect
139. #474D53semidirect
141. C53#3233semidirect