Finite Group Databasebeta

← #20285946

Subgroups — #20285946

Order 6784.

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

normal subgroups only

This group has 362 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
1C4⋊Q8×D53C211yesmaximal, characteristic, complemented
2#16811158C211yesmaximal, characteristic
3#16811033C211yesmaximal, characteristic, complemented
4#16810985C211yesmaximal, characteristic, complemented
5#16810983C211yesmaximal, characteristic, complemented
6#16810960C212yesmaximal, complemented
7#16810943C212yesmaximal
8#16810907C212yesmaximal, complemented
9#16810902C212yesmaximal, complemented
10#16810678C211yesmaximal, characteristic, complemented
11C53×#501C211yesmaximal, characteristic, complemented
12C2×Q8×D53—21no—
13#13133217C2211yescharacteristic
14#13133216—21no—
15#13133216C2212yes—
16#13133215—21no—
17#13133212—21no—
18#13133205C2211yescharacteristic
19#13133202—21no—
20#13133197C2211yescharacteristic
21#13133192C2211yescharacteristic
22#13133182C2211yescharacteristic
23#13133179—21no—
24#13133179C2211yescharacteristic
25#13133175—21no—
26C4⋊C4×D53—21no—
27C4⋊C4×D53C2212yes—
28C42×D53C2211yescharacteristic
29#13133150C2212yes—
30#13133148C2212yes—
31#13133146C2212yes—
32#13133142C2212yes—
33#13133140C2212yes—
34#13133139C2212yes—
35#13133137C2212yes—
36#13133129C2212yes—
37#13133128C2211yescharacteristic
38C4⋊Q8×C53C2211yescharacteristic
39#125×C53C2211yescharacteristic
40#107×C53C2212yes—
41#106×C53C2212yes—
42#97×C53C2211yescharacteristic
43C2×Q8×C53—21no—
44Q8×D53—44no—
45C2×D212—21no—
46C2×D212C2311yescharacteristic
47C2×C4×D53—22no—
48C2×C4×D53—21no—
49C2×C4×D53D411yescharacteristic
50C2×C4×D53D411yescharacteristic
51C2×C4×D53C2311yescharacteristic
52C2×Dic106—21no—
53C2×Dic106—22no—
54C2×Dic106C2311yescharacteristic
55C2×C8×C53C2312yes—
56C4⋊C4×C53—21no—
57C4⋊C4×C53—21no—
58C4⋊C4×C53—21no—
59C4⋊C4×C53C2311yescharacteristic
60C4⋊C4×C53C2312yes—
61C4⋊C4×C53C2311yescharacteristic
62C42×C53C2311yescharacteristic
63#11719921—21no—
64#11719921—21no—
65#11719921—21no—
66#11719921—21no—
67#11719921—21no—
68C4⋊Dic53—21no—
69C4⋊Dic53—21no—
70C4⋊Dic53C2312yes—
71#11719919—22no—
72#11719919—22no—
73#11719919—21no—
74C4×Dic53—21no—
75C4×Dic53D411yescharacteristic
76C4×Dic53D411yescharacteristic
77C4×Dic53C2311yescharacteristic
78C2×#88322C2312yes—
79C22×D53—21no—
80C22×D53C2×D411yescharacteristic
81Q8×C53—42no—
82C2×C4×C53—22no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53C2×D411yescharacteristic
86C2×C4×C53C2×D411yescharacteristic
87C2×C4×C53C2411yescharacteristic
88C2×Dic53—22no—
89C2×Dic53—21no—
90C2×Dic53C2×D411yescharacteristic
91C2×Dic53C2×D411yescharacteristic
92C2×Dic53C2×D411yescharacteristic
93D212—22no—
94D212—42no—
95C4×D53—22no—
96C4×D53—22no—
97C4×D53—22no—
98C4×D53—44no—
99C4×D53—42no—
100Dic106—22no—
101Dic106—42no—
102Dic106—44no—
103C8×C53—22no—
104#88322—22no—
105C22×C53C22×D411yescharacteristic
106C2×D53—21no—
107C2×D53—21no—
108C2×D53—41no—
109C2×D53—41no—
110C2×D53—41no—
111C2×D53#13712yes—
112C4×C53—21no—
113C4×C53—21no—
114C4×C53—42no—
115C4×C53—41no—
116C4×C53—41no—
117C4×C532−1+412yes—
118Dic53—21no—
119Dic53—21no—
120Dic53—41no—
121Dic53—42no—
122Dic53AGL(1,ℤ/8)12yes—
123#3239—531nomaximal, Sylow
124C2×C53#53611yescharacteristic
125C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
126C2×C53C2×#13711yescharacteristic
127D53—22no—
128D53—81no—
129C2×C4⋊Q8—531no—
130#529—531no—
131#501—531no—
132#501D5311yescharacteristic, complemented
133#489—531no—
134#489—531no—
135#484—532no—
136#481—532no—
137#475—532no—
138#474—532no—
139#432—531no—
140C53#323911yescharacteristic, complemented, Sylow
141C22×Q8—1061no—
142C4⋊Q8—532no—
143C4⋊Q8—531no—
144C4⋊Q8—1061no—
145C4⋊Q8—1061no—
146C4⋊Q8C2×D5311yescharacteristic
147#126—1061no—
148#125C2×D5311yescharacteristic
149#124—531no—
150#123—1061no—
151C22⋊Q8—531no—
152#121—531no—
153C4×Q8—531no—
154C4×D4—531no—
155C4×D4—1061no—
156#117—1061no—
157C2×C4⋊C4—532no—
158C2×C4⋊C4—1061no—
159C2×C42—531no—
160#107—532no—
161#107C2×D5312yes—
162#106—532no—
163#106C2×D5312yes—
164C4⋊C8—532no—
165#103—532no—
166#103—532no—
167#102—532no—
168#102—532no—
169C22⋊C8—532no—
170#97—531no—
171#97C2×D5311yescharacteristic
172C2×Q8—21no—
173C2×Q8—531no—
174C2×Q8—1062no—
175C2×Q8—1061no—
176C2×Q8—2124no—
177C2×D4—531no—
178C2×D4—1061no—
179C22×C4—531no—
180C22×C4—531no—
181C22×C4—531no—
182C22×C4—1061no—
183C22×C4—1062no—
184C2×C8—532no—
185C2×C8C22×D5312yes—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—532no—
190C4⋊C4—1062no—
191C4⋊C4—1062no—
192C4⋊C4—1061no—
193C4⋊C4—1061no—
194C4⋊C4—1061no—
195C4⋊C4C22×D5312yes—
196C4⋊C4C22×D5311yescharacteristic
197C4⋊C4C22×D5311yescharacteristic
198C22⋊C4—1061no—
199C22⋊C4—1061no—
200C22⋊C4—1061no—
201C22⋊C4—1061no—
202C22⋊C4—1061no—
203C42—531no—
204C42—531no—
205C42—531no—
206C42—1061no—
207C42C22×D5311yescharacteristic
208C23—531no—
209C23—1061no—
210Q8—42no—
211Q8—1062no—
212Q8—2122no—
213Q8—2124no—
214D4—1062no—
215D4—2122no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—531no—
220C2×C4—531no—
221C2×C4—531no—
222C2×C4—1062no—
223C2×C4—1062no—
224C2×C4—1062no—
225C2×C4—1061no—
226C2×C4—1062no—
227C2×C4—2122no—
228C2×C4—2124no—
229C2×C4D4×D5311yescharacteristic
230C2×C4D4×D5311yescharacteristic
231C2×C4C23×D5311yescharacteristic
232C8—22no—
233C8—1062no—
234C22—532no—
235C22—1061no—
236C22—1061no—
237C22—2121no—
238C22—2121no—
239C22—2121no—
240C22C2×D4×D5311yescharacteristic
241C4—21no—
242C4—21no—
243C4—41no—
244C4—42no—
245C4—41no—
246C4—532no—
247C4—1061no—
248C4—1061no—
249C4—2122no—
250C4—2121no—
251C4#1313326512yes—
252C2—1062no—
253C2—4241no—
254C2#1681119811yescharacteristic
255C2AGL(1,ℤ/8)×D5311yescharacteristic
256C2#137×D5311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#501C2
socleC22×C53C22×D4
hypercenter#501D53
hypocenterC53#3239
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D53C2semidirect
3. #16811033C2semidirect
4. #16810985C2semidirect
5. #16810983C2semidirect
6. #16810960C2semidirect
8. #16810907C2semidirect
9. #16810902C2semidirect
10. #16810678C2semidirect
11. C53×#501C2semidirect
132. #501D53semidirect
140. C53#3239semidirect