Finite Group Databasebeta

← #20285898

Subgroups — #20285898

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#16811160C211yesmaximal, characteristic
3#16811023C212yesmaximal, complemented
4#16810979C212yesmaximal, complemented
5#16810962C212yesmaximal, complemented
6#16810949C212yesmaximal
7#16810918C211yesmaximal, characteristic, complemented
8#16810914C211yesmaximal, characteristic, complemented
9#16810900C211yesmaximal, characteristic, complemented
10#16810675C211yesmaximal, characteristic, complemented
11C53×#475C211yesmaximal, characteristic, complemented
12#13133217C2211yescharacteristic
13#13133212—21no—
14#13133208C2211yescharacteristic
15#13133208C2212yes—
16#13133206C2211yescharacteristic
17#13133202—21no—
18#13133197—21no—
19#13133193—21no—
20#13133193C2211yescharacteristic
21#13133192C2211yescharacteristic
22Q8×Dic53—21no—
23#13133182—21no—
24C4×D212—21no—
25C4×D212C2211yescharacteristic
26#13133175C2212yes—
27#13133174—21no—
28C42×D53C2211yescharacteristic
29#13133150C2212yes—
30#13133150C2212yes—
31#13133149C2212yes—
32C4⋊#88322C2211yescharacteristic
33#13133141C2212yes—
34#13133139C2212yes—
35#13133137C2212yes—
36#13133129C2212yes—
37C8×Dic53C2212yes—
38C4⋊Q8×C53C2211yescharacteristic
39C4×Q8×C53C2211yescharacteristic
40#106×C53C2212yes—
41C4⋊C8×C53C2211yescharacteristic
42#103×C53C2212yes—
43C2×Q8×C53—21no—
44C2×Q8×C53C2311yescharacteristic
45C2×D212—21no—
46C2×D212C2311yescharacteristic
47C2×C4×D53—21no—
48C2×C4×D53D411yescharacteristic
49C2×C4×D53D411yescharacteristic
50C2×C4×D53C2311yescharacteristic
51C2×Dic106—21no—
52C2×C8×C53C2312yes—
53C4⋊C4×C53—21no—
54C4⋊C4×C53—21no—
55C4⋊C4×C53C2312yes—
56C4⋊C4×C53C2311yescharacteristic
57C42×C53—21no—
58C42×C53C2311yescharacteristic
59#11719921—22no—
60#11719921—21no—
61#11719921—21no—
62#11719921—21no—
63#11719921—21no—
64#11719921—21no—
65C4⋊Dic53—21no—
66C4⋊Dic53—21no—
67C4⋊Dic53—22no—
68C4⋊Dic53C2312yes—
69C4⋊Dic53C2311yescharacteristic
70#11719919—21no—
71#11719919—21no—
72#11719919—22no—
73C4×Dic53—22no—
74C4×Dic53D411yescharacteristic
75C4×Dic53D411yescharacteristic
76C4×Dic53C2311yescharacteristic
77C2×#88322C2312yes—
78C22×D53—21no—
79C22×D53C2×D411yescharacteristic
80Q8×C53—21no—
81Q8×C53—21no—
82Q8×C53—42no—
83C2×C4×C53—21no—
84C2×C4×C53—22no—
85C2×C4×C53—21no—
86C2×C4×C53C2×D411yescharacteristic
87C2×C4×C53C2×D411yescharacteristic
88C2×C4×C53C2411yescharacteristic
89C2×Dic53—22no—
90C2×Dic53—21no—
91C2×Dic53C2×D411yescharacteristic
92C2×Dic53C2×D411yescharacteristic
93C2×Dic53C2×D411yescharacteristic
94D212—21no—
95D212—21no—
96D212—42no—
97C4×D53—22no—
98C4×D53—21no—
99C4×D53—21no—
100C4×D53—41no—
101C4×D53—42no—
102Dic106—42no—
103C8×C53—22no—
104#88322—22no—
105C22×C53C22×D411yescharacteristic
106C2×D53—21no—
107C2×D53—21no—
108C2×D53—21no—
109C2×D53—41no—
110C2×D53—41no—
111C2×D53—41no—
112C4×C53—21no—
113C4×C53—41no—
114C4×C53—42no—
115C4×C53—41no—
116C4×C53#13512yes—
117C4×C532−1+411yescharacteristic
118C4×C532−1+411yescharacteristic
119Dic53—21no—
120Dic53—21no—
121Dic53—21no—
122Dic53—41no—
123Dic53—42no—
124#3233—531nomaximal, Sylow
125C2×C53#53611yescharacteristic
126C2×C53C2×#13511yescharacteristic
127C2×C53#57511yescharacteristic
128D53—41no—
129D53—81no—
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#475D5311yescharacteristic, complemented
139#474—531no—
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—1061no—
154C4×Q8—1061no—
155C4×Q8C2×D5311yescharacteristic
156C4×D4—531no—
157C4×D4—1061no—
158#117—532no—
159#117—1061no—
160C2×C42—531no—
161#107—532no—
162#107—532no—
163#106—532no—
164#106C2×D5312yes—
165C4⋊C8—531no—
166C4⋊C8C2×D5311yescharacteristic
167#103—532no—
168#103C2×D5312yes—
169#102—532no—
170#102—532no—
171C22⋊C8—532no—
172C4×C8—532no—
173C2×Q8—21no—
174C2×Q8—1061no—
175C2×Q8C22×D5311yescharacteristic
176C2×D4—531no—
177C2×D4—1061no—
178C22×C4—531no—
179C22×C4—531no—
180C22×C4—531no—
181C22×C4—1061no—
182C2×C8—532no—
183C2×C8C22×D5312yes—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—532no—
187C4⋊C4—531no—
188C4⋊C4—1061no—
189C4⋊C4—1061no—
190C4⋊C4—1061no—
191C4⋊C4—1062no—
192C4⋊C4—1062no—
193C4⋊C4—1061no—
194C4⋊C4C22×D5312yes—
195C4⋊C4C22×D5311yescharacteristic
196C22⋊C4—1061no—
197C22⋊C4—1061no—
198C22⋊C4—1061no—
199C22⋊C4—1062no—
200C22⋊C4—1061no—
201C22⋊C4—1061no—
202C42—21no—
203C42—531no—
204C42—531no—
205C42—531no—
206C42—1062no—
207C42C22×D5311yescharacteristic
208C23—531no—
209C23—1061no—
210Q8—21no—
211Q8—21no—
212Q8—42no—
213Q8—2122no—
214D4—1061no—
215D4—1061no—
216D4—2122no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—531no—
221C2×C4—531no—
222C2×C4—531no—
223C2×C4—1062no—
224C2×C4—1061no—
225C2×C4—1061no—
226C2×C4—1062no—
227C2×C4—1061no—
228C2×C4—2121no—
229C2×C4—2122no—
230C2×C4D4×D5311yescharacteristic
231C2×C4D4×D5311yescharacteristic
232C2×C4C23×D5311yescharacteristic
233C8—22no—
234C8—1062no—
235C22—1061no—
236C22—1061no—
237C22—1061no—
238C22—2121no—
239C22—2121no—
240C22—2121no—
241C22C2×D4×D5311yescharacteristic
242C4—21no—
243C4—42no—
244C4—41no—
245C4—41no—
246C4—1061no—
247C4—1061no—
248C4—1061no—
249C4—2121no—
250C4—2122no—
251C4#1313323812yes—
252C4#1313326411yescharacteristic
253C4#1313326511yescharacteristic
254C2—4241no—
255C2—2121no—
256C2#1681119611yescharacteristic
257C2C2×#1313323811yescharacteristic
258C2#1681142211yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#475C2
socleC22×C53C22×D4
hypercenter#475D53
hypocenterC53#3233
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
1. #16811178C2semidirect
3. #16811023C2semidirect
4. #16810979C2semidirect
5. #16810962C2semidirect
7. #16810918C2semidirect
8. #16810914C2semidirect
9. #16810900C2semidirect
10. #16810675C2semidirect
11. C53×#475C2semidirect
138. #475D53semidirect
141. C53#3233semidirect