Finite Group Databasebeta

← #20285944

Subgroups — #20285944

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#16811155C211yesmaximal, characteristic
3#16811031C212yesmaximal, complemented
4#16810981C212yesmaximal, complemented
5#16810960C212yesmaximal, complemented
6#16810940C212yesmaximal
7#16810909C211yesmaximal, characteristic, complemented
8#16810905C211yesmaximal, characteristic, complemented
9#16810903C211yesmaximal, characteristic, complemented
10#16810679C211yesmaximal, characteristic, complemented
11C53×#474C211yesmaximal, characteristic, complemented
12C2×Q8×D53—21no—
13#13133217C2211yescharacteristic
14#13133216—21no—
15#13133216C2212yes—
16#13133215—21no—
17#13133210—21no—
18#13133208C2211yescharacteristic
19#13133204C2211yescharacteristic
20#13133199—21no—
21#13133197C2211yescharacteristic
22#13133188C2211yescharacteristic
23C4×Dic106C2211yescharacteristic
24C4×C22⋊D53—21no—
25#13133176—21no—
26C4⋊C4×D53—21no—
27C4⋊C4×D53C2212yes—
28C42×D53C2211yescharacteristic
29#13133150C2212yes—
30#13133149C2212yes—
31#13133148C2212yes—
32C4⋊#88322C2211yescharacteristic
33#13133142C2212yes—
34#13133140C2212yes—
35#13133138C2212yes—
36#13133129C2212yes—
37#13133127C2212yes—
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—
47Q8×D53—44no—
48C2×C4×D53—22no—
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×C53C2312yes—
58C4⋊C4×C53C2311yescharacteristic
59C22⋊C4×C53—21no—
60C42×C53C2311yescharacteristic
61C22⋊Dic53—21no—
62C22⋊Dic53—21no—
63C22⋊Dic53—21no—
64#11719921—21no—
65C4⋊Dic53—21no—
66C4⋊Dic53—21no—
67C4⋊Dic53—21no—
68C4⋊Dic53C2312yes—
69C4⋊Dic53C2311yescharacteristic
70#11719919—21no—
71#11719919—22no—
72#11719919—22no—
73#11719919—21no—
74C4×Dic53—21no—
75C4×Dic53D411yescharacteristic
76C4×Dic53D411yescharacteristic
77C4×Dic53C2311yescharacteristic
78C2×#88322C2312yes—
79C23×C53—21no—
80C22×D53C2×D411yescharacteristic
81Q8×C53—42no—
82D4×C53—21no—
83D4×C53—21no—
84C2×C4×C53—22no—
85C2×C4×C53—21no—
86C2×C4×C53—42no—
87C2×C4×C53C2×D411yescharacteristic
88C2×C4×C53C2×D411yescharacteristic
89C2×C4×C53C2411yescharacteristic
90C22⋊D53—42no—
91C2×Dic53—22no—
92C2×Dic53—21no—
93C2×Dic53—21no—
94C2×Dic53C2×D411yescharacteristic
95C2×Dic53C2×D411yescharacteristic
96C2×Dic53C2×D411yescharacteristic
97C4×D53—22no—
98C4×D53—22no—
99C4×D53—21no—
100C4×D53—21no—
101C4×D53—44no—
102Dic106—21no—
103Dic106—21no—
104Dic106—44no—
105Dic106—42no—
106C8×C53—22no—
107#88322—22no—
108C22×C53—41no—
109C22×C53—41no—
110C22×C53—41no—
111C22×C53C22×D411yescharacteristic
112C2×D53—21no—
113C2×D53—21no—
114C2×D53#13712yes—
115C4×C53—21no—
116C4×C53—42no—
117C4×C53—41no—
118C4×C53AGL(1,ℤ/8)12yes—
119C4×C532−1+411yescharacteristic
120C4×C532−1+411yescharacteristic
121Dic53—21no—
122Dic53—21no—
123Dic53—21no—
124Dic53—41no—
125Dic53—42no—
126Dic53—41no—
127#3239—531nomaximal, Sylow
128C2×C53—81no—
129C2×C53#53611yescharacteristic
130C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
131C2×C53C2×#13711yescharacteristic
132D53—22no—
133C2×C4⋊Q8—531no—
134#529—531no—
135#501—532no—
136#489—532no—
137#484—532no—
138#481—532no—
139#475—531no—
140#475—531no—
141#474—531no—
142#474D5311yescharacteristic, complemented
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×Q8—531no—
158C4×D4—1061no—
159C4×D4C2×D5311yescharacteristic
160#117—1061no—
161C2×C4⋊C4—532no—
162C2×C4⋊C4—1061no—
163C2×C42—531no—
164#107—532no—
165#107C2×D5312yes—
166#106—532no—
167#106—532no—
168C4⋊C8—531no—
169C4⋊C8C2×D5311yescharacteristic
170#103—532no—
171#103—532no—
172#102—532no—
173#102C2×D5312yes—
174C22⋊C8—532no—
175#97—532no—
176C2×Q8—21no—
177C2×Q8—531no—
178C2×Q8—1062no—
179C2×Q8—1061no—
180C2×Q8—2124no—
181C2×D4—1061no—
182C2×D4C22×D5311yescharacteristic
183C22×C4—21no—
184C22×C4—531no—
185C22×C4—531no—
186C22×C4—531no—
187C22×C4—1062no—
188C2×C8—532no—
189C2×C8C22×D5312yes—
190C4⋊C4—21no—
191C4⋊C4—532no—
192C4⋊C4—531no—
193C4⋊C4—1062no—
194C4⋊C4—1062no—
195C4⋊C4—1061no—
196C4⋊C4—1061no—
197C4⋊C4—1061no—
198C4⋊C4—1061no—
199C4⋊C4—1061no—
200C4⋊C4C22×D5312yes—
201C4⋊C4C22×D5311yescharacteristic
202C22⋊C4—21no—
203C22⋊C4—1061no—
204C22⋊C4—1061no—
205C22⋊C4—1061no—
206C22⋊C4—1061no—
207C42—531no—
208C42—531no—
209C42—531no—
210C42—1061no—
211C42C22×D5311yescharacteristic
212C23—21no—
213C23—531no—
214Q8—42no—
215Q8—1061no—
216Q8—1061no—
217Q8—2124no—
218Q8—2122no—
219D4—21no—
220D4—21no—
221D4—2122no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—42no—
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—1061no—
234C2×C4—1062no—
235C2×C4—2124no—
236C2×C4D4×D5311yescharacteristic
237C2×C4D4×D5311yescharacteristic
238C2×C4C23×D5311yescharacteristic
239C8—22no—
240C8—1062no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22—532no—
245C22—1061no—
246C22—1061no—
247C22C2×D4×D5311yescharacteristic
248C4—21no—
249C4—41no—
250C4—42no—
251C4—1061no—
252C4—1061no—
253C4—1061no—
254C4—2122no—
255C4—2121no—
256C4—2121no—
257C4#1313324112yes—
258C4#1313326411yescharacteristic
259C4#1313326511yescharacteristic
260C2—81no—
261C2—1062no—
262C2#1681119611yescharacteristic
263C2C2×#1313324111yescharacteristic
264C2#137×D5311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#474C2
socleC22×C53C22×D4
hypercenter#474D53
hypocenterC53#3239
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. C4⋊Q8×D53C2semidirect
3. #16811031C2semidirect
4. #16810981C2semidirect
5. #16810960C2semidirect
7. #16810909C2semidirect
8. #16810905C2semidirect
9. #16810903C2semidirect
10. #16810679C2semidirect
11. C53×#474C2semidirect
142. #474D53semidirect
144. C53#3239semidirect