Finite Group Databasebeta

← #20285949

Subgroups — #20285949

Order 6784.

362 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C4⋊Q8×D53C211yesmaximal, characteristic, complemented
2#16811160C211yesmaximal, characteristic
3#16811030C212yesmaximal, complemented
4#16810984C212yesmaximal, complemented
5#16810957C212yesmaximal, complemented
6#16810943C212yesmaximal
7#16810908C211yesmaximal, characteristic, complemented
8#16810904C211yesmaximal, characteristic, complemented
9#16810900C211yesmaximal, characteristic, complemented
10#16810679C211yesmaximal, characteristic, complemented
11C53×#475C211yesmaximal, characteristic, complemented
12C2×Q8×D53—21no—
13#13133217C2211yescharacteristic
14#13133216—21no—
15#13133216C2212yes—
16#13133215—21no—
17#13133212—21no—
18#13133208C2211yescharacteristic
19#13133206C2211yescharacteristic
20#13133202—21no—
21#13133193C2211yescharacteristic
22#13133192C2211yescharacteristic
23C4×D212—21no—
24C4×D212C2211yescharacteristic
25#13133174—21no—
26C4⋊C4×D53—21no—
27C4⋊C4×D53C2212yes—
28C42×D53C2211yescharacteristic
29#13133151C2212yes—
30#13133150C2212yes—
31#13133148C2212yes—
32C4⋊#88322C2211yescharacteristic
33#13133141C2212yes—
34#13133139C2212yes—
35#13133137C2212yes—
36#13133129C2212yes—
37#13133127C2212yes—
38C4⋊Q8×C53C2211yescharacteristic
39C4×Q8×C53C2211yescharacteristic
40#106×C53C2212yes—
41C4⋊C8×C53C2211yescharacteristic
42#103×C53C2212yes—
43C2×Q8×C53—21no—
44C2×Q8×C53C2311yescharacteristic
45Q8×D53—44no—
46C2×D212—21no—
47C2×D212C2311yescharacteristic
48C2×C4×D53—22no—
49C2×C4×D53—21no—
50C2×C4×D53D411yescharacteristic
51C2×C4×D53D411yescharacteristic
52C2×C4×D53C2311yescharacteristic
53C2×Dic106—21no—
54C2×Dic106—22no—
55C2×C8×C53C2312yes—
56C4⋊C4×C53—21no—
57C4⋊C4×C53—21no—
58C4⋊C4×C53C2312yes—
59C4⋊C4×C53C2311yescharacteristic
60C42×C53—21no—
61C42×C53C2311yescharacteristic
62#11719921—21no—
63#11719921—21no—
64#11719921—21no—
65#11719921—21no—
66#11719921—21no—
67C4⋊Dic53—21no—
68C4⋊Dic53—21no—
69C4⋊Dic53C2312yes—
70C4⋊Dic53C2311yescharacteristic
71#11719919—22no—
72#11719919—22no—
73#11719919—21no—
74#11719919—21no—
75C4×Dic53D411yescharacteristic
76C4×Dic53D411yescharacteristic
77C4×Dic53C2311yescharacteristic
78C2×#88322C2312yes—
79C22×D53—21no—
80C22×D53C2×D411yescharacteristic
81Q8×C53—21no—
82Q8×C53—21no—
83Q8×C53—42no—
84C2×C4×C53—22no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53C2×D411yescharacteristic
88C2×C4×C53C2×D411yescharacteristic
89C2×C4×C53C2411yescharacteristic
90C2×Dic53—22no—
91C2×Dic53—21no—
92C2×Dic53C2×D411yescharacteristic
93C2×Dic53C2×D411yescharacteristic
94C2×Dic53C2×D411yescharacteristic
95D212—21no—
96D212—21no—
97D212—42no—
98C4×D53—22no—
99C4×D53—22no—
100C4×D53—21no—
101C4×D53—21no—
102C4×D53—44no—
103C4×D53—42no—
104Dic106—44no—
105Dic106—42no—
106C8×C53—22no—
107#88322—22no—
108C22×C53C22×D411yescharacteristic
109C2×D53—21no—
110C2×D53—21no—
111C2×D53—41no—
112C2×D53—41no—
113C2×D53—41no—
114C2×D53#13712yes—
115C4×C53—21no—
116C4×C53—41no—
117C4×C53—42no—
118C4×C53—41no—
119C4×C53AGL(1,ℤ/8)12yes—
120C4×C532−1+411yescharacteristic
121C4×C532−1+411yescharacteristic
122Dic53—21no—
123Dic53—21no—
124Dic53—21no—
125Dic53—42no—
126Dic53—41no—
127#3239—531nomaximal, Sylow
128C2×C53#53611yescharacteristic
129C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
130C2×C53C2×#13711yescharacteristic
131D53—22no—
132D53—81no—
133C2×C4⋊Q8—531no—
134#529—531no—
135#501—532no—
136#489—532no—
137#484—532no—
138#481—532no—
139#475—531no—
140#475D5311yescharacteristic, complemented
141#474—531no—
142#474—531no—
143#432—531no—
144C53#323911yescharacteristic, complemented, Sylow
145C22×Q8—1061no—
146C4⋊Q8—532no—
147C4⋊Q8—531no—
148C4⋊Q8—1061no—
149C4⋊Q8—1061no—
150C4⋊Q8C2×D5311yescharacteristic
151#126—1061no—
152#125—531no—
153#124—531no—
154#123—1061no—
155C22⋊Q8—531no—
156#121—531no—
157C4×Q8C2×D5311yescharacteristic
158C4×D4—531no—
159C4×D4—1061no—
160#117—1061no—
161C2×C4⋊C4—532no—
162C2×C4⋊C4—1061no—
163C2×C42—531no—
164#107—532no—
165#107—532no—
166#106—532no—
167#106C2×D5312yes—
168C4⋊C8—531no—
169C4⋊C8C2×D5311yescharacteristic
170#103—532no—
171#103C2×D5312yes—
172#102—532no—
173#102—532no—
174C22⋊C8—532no—
175#97—532no—
176C2×Q8—21no—
177C2×Q8—1062no—
178C2×Q8—1061no—
179C2×Q8—2124no—
180C2×Q8C22×D5311yescharacteristic
181C2×D4—531no—
182C2×D4—1061no—
183C22×C4—531no—
184C22×C4—531no—
185C22×C4—531no—
186C22×C4—1062no—
187C22×C4—1061no—
188C2×C8—532no—
189C2×C8C22×D5312yes—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—532no—
193C4⋊C4—531no—
194C4⋊C4—1062no—
195C4⋊C4—1062no—
196C4⋊C4—1061no—
197C4⋊C4—1061no—
198C4⋊C4—1061no—
199C4⋊C4—1061no—
200C4⋊C4C22×D5312yes—
201C4⋊C4C22×D5311yescharacteristic
202C22⋊C4—1061no—
203C22⋊C4—1061no—
204C22⋊C4—1061no—
205C22⋊C4—1061no—
206C22⋊C4—1061no—
207C42—21no—
208C42—531no—
209C42—531no—
210C42—531no—
211C42C22×D5311yescharacteristic
212C23—531no—
213C23—1061no—
214Q8—21no—
215Q8—21no—
216Q8—42no—
217Q8—2124no—
218Q8—2122no—
219D4—1061no—
220D4—1061no—
221D4—2122no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—531no—
226C2×C4—531no—
227C2×C4—531no—
228C2×C4—1062no—
229C2×C4—1062no—
230C2×C4—1061no—
231C2×C4—1061no—
232C2×C4—1061no—
233C2×C4—1062no—
234C2×C4—2124no—
235C2×C4—2122no—
236C2×C4D4×D5311yescharacteristic
237C2×C4D4×D5311yescharacteristic
238C2×C4C23×D5311yescharacteristic
239C8—22no—
240C8—1062no—
241C22—532no—
242C22—1061no—
243C22—1061no—
244C22—2121no—
245C22—2121no—
246C22—2121no—
247C22C2×D4×D5311yescharacteristic
248C4—21no—
249C4—42no—
250C4—41no—
251C4—41no—
252C4—1061no—
253C4—1061no—
254C4—1061no—
255C4—2121no—
256C4—2122no—
257C4#1313324412yes—
258C4#1313326411yescharacteristic
259C4#1313326511yescharacteristic
260C2—1062no—
261C2—4241no—
262C2#1681119611yescharacteristic
263C2C2×#1313324411yescharacteristic
264C2#137×D5311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#475C2
socleC22×C53C22×D4
hypercenter#475D53
hypocenterC53#3239
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. C4⋊Q8×D53C2semidirect
3. #16811030C2semidirect
4. #16810984C2semidirect
5. #16810957C2semidirect
7. #16810908C2semidirect
8. #16810904C2semidirect
9. #16810900C2semidirect
10. #16810679C2semidirect
11. C53×#475C2semidirect
140. #475D53semidirect
144. C53#3239semidirect