Finite Group Databasebeta

← #20285107

Subgroups — #20285107

Order 6784.

424 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 121 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#16811172C211yesmaximal, characteristic
2C4⋊Q8×D53C211yesmaximal, characteristic, complemented
3#16810985C211yesmaximal, characteristic, complemented
4#16810983C211yesmaximal, characteristic, complemented
5#16810982C211yesmaximal, characteristic, complemented
6#16810972C212yesmaximal, complemented
7#16810968C212yesmaximal, complemented
8#16810623C212yesmaximal, complemented
9#16810621C212yesmaximal, complemented
10#16810575C211yesmaximal, characteristic, complemented
11C53×#489C211yesmaximal, characteristic, complemented
12C2×#11719949C2211yescharacteristic
13C2×#11719947C2211yescharacteristic
14C2×Q8×D53—21no—
15C2×#11719912C2212yes—
16#13133217C2211yescharacteristic
17#13133216—21no—
18#13133216C2212yes—
19#13133215—21no—
20#13133215C2211yescharacteristic
21#13133213C2211yescharacteristic
22#13133205C2211yescharacteristic
23#13133190—21no—
24#13133190—21no—
25#13133177—21no—
26#13133177—21no—
27C4⋊C4×D53—21no—
28C4⋊C4×D53C2212yes—
29C42×D53C2211yescharacteristic
30#13133142C2212yes—
31#13133141C2212yes—
32#13133140C2212yes—
33#13133139C2212yes—
34#13133138C2212yes—
35#13133137C2212yes—
36#13133128C2211yescharacteristic
37C8×Dic53C2212yes—
38C4⋊Q8×C53C2211yescharacteristic
39#124×C53C2211yescharacteristic
40#103×C53C2212yes—
41#102×C53C2212yes—
42#97×C53C2211yescharacteristic
43C2×Q8×C53—21no—
44C2×Q8×C53C2311yescharacteristic
45C2×D4×C53C2311yescharacteristic
46C2×C22⋊D53—21no—
47C2×C22⋊D53—21no—
48C2×C22⋊D53—21no—
49C22×Dic53—21no—
50#11719949—24no—
51Q8×D53—44no—
52#11719947—24no—
53C2×D212—21no—
54C2×D212C2311yescharacteristic
55C2×C4×D53—22no—
56C2×C4×D53—21no—
57C2×C4×D53D411yescharacteristic
58C2×C4×D53D411yescharacteristic
59C2×C4×D53C2311yescharacteristic
60C2×Dic106—21no—
61C2×Dic106—22no—
62C2×Dic106C2311yescharacteristic
63C2×C8×C53C2312yes—
64C4⋊C4×C53—21no—
65C4⋊C4×C53C2312yes—
66C22⋊C4×C53—21no—
67C22⋊C4×C53—21no—
68C42×C53C2311yescharacteristic
69#11719921—21no—
70#11719921—21no—
71C4⋊Dic53—21no—
72C4⋊Dic53C2312yes—
73#11719919—22no—
74#11719919—22no—
75#11719919—21no—
76#11719919—21no—
77C4×Dic53D411yescharacteristic
78C4×Dic53D411yescharacteristic
79C4×Dic53C2311yescharacteristic
80C2×#88322C2312yes—
81#11719912—24no—
82C23×C53—21no—
83C22×D53—21no—
84C22×D53C2×D411yescharacteristic
85Q8×C53—22no—
86Q8×C53—42no—
87D4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53—21no—
90C2×C4×C53C2×D411yescharacteristic
91C2×C4×C53C2×D411yescharacteristic
92C2×C4×C53C2411yescharacteristic
93C22⋊D53—42no—
94C22⋊D53—42no—
95C22⋊D53—42no—
96C2×Dic53—22no—
97C2×Dic53—21no—
98C2×Dic53—42no—
99C2×Dic53C2×D411yescharacteristic
100C2×Dic53C2×D411yescharacteristic
101C2×Dic53C2×D411yescharacteristic
102D212—22no—
103D212—42no—
104C4×D53—22no—
105C4×D53—22no—
106C4×D53—44no—
107C4×D53—42no—
108C4×D53#4114yes—
109Dic106—22no—
110Dic106—42no—
111Dic106—44no—
112C8×C53—22no—
113#88322—22no—
114C22×C53—41no—
115C22×C53—41no—
116C22×C53—41no—
117C22×C53C22×D411yescharacteristic
118C2×D53—21no—
119C2×D53—21no—
120C2×D53—41no—
121C2×D53—41no—
122C2×D53—41no—
123C2×D53#12412yes—
124C4×C53—21no—
125C4×C53—21no—
126C4×C53—41no—
127C4×C53—42no—
128C4×C53C2×#4112yes—
129Dic53—41no—
130Dic53—42no—
131Dic53#12412yes—
132Dic53AGL(1,ℤ/8)12yes—
133Dic53#13712yes—
134#3141—531nomaximal, Sylow
135C2×C53—81no—
136C2×C53C2×#12411yescharacteristic
137C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C53C2×#13711yescharacteristic
139D53—22no—
140D53—81no—
141#532—531no—
142C2×C4⋊Q8—531no—
143#489—531no—
144#489—531no—
145#489—531no—
146#489D5311yescharacteristic, complemented
147#487—532no—
148#486—532no—
149#417—532no—
150#417—532no—
151C4×M16—531no—
152C53#314111yescharacteristic, complemented, Sylow
153C2×#41—531no—
154C2×#41—531no—
155C22×Q8—1061no—
156C2×M16—532no—
157C4⋊Q8—532no—
158C4⋊Q8—531no—
159C4⋊Q8—531no—
160C4⋊Q8—1061no—
161C4⋊Q8—1061no—
162C4⋊Q8C2×D5311yescharacteristic
163#127—531no—
164#124—531no—
165#124C2×D5311yescharacteristic
166#121—1061no—
167#121—1061no—
168C4×D4—1061no—
169C4×D4—1061no—
170C2×C4⋊C4—532no—
171C2×C4⋊C4—1061no—
172C2×C42—531no—
173#103—532no—
174#103—532no—
175#103—532no—
176#103C2×D5312yes—
177#102—532no—
178#102—532no—
179#102—532no—
180#102C2×D5312yes—
181#97—531no—
182#97C2×D5311yescharacteristic
183C4×C8—532no—
184#41—1064no—
185#41—1064no—
186C2×Q8—21no—
187C2×Q8—531no—
188C2×Q8—1062no—
189C2×Q8—1061no—
190C2×Q8—2124no—
191C2×Q8C22×D5311yescharacteristic
192C2×D4—531no—
193C2×D4—1061no—
194C2×D4—1061no—
195C2×D4—1061no—
196C2×D4—1061no—
197C2×D4C22×D5311yescharacteristic
198C22×C4—531no—
199C22×C4—531no—
200C22×C4—531no—
201C22×C4—1061no—
202C22×C4—1062no—
203C22×C4—1061no—
204M16—1064no—
205C2×C8—532no—
206C2×C8C22×D5312yes—
207C4⋊C4—21no—
208C4⋊C4—532no—
209C4⋊C4—1062no—
210C4⋊C4—1062no—
211C4⋊C4—1061no—
212C4⋊C4—1061no—
213C4⋊C4—1061no—
214C4⋊C4C22×D5312yes—
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—1061no—
218C22⋊C4—1061no—
219C42—531no—
220C42—531no—
221C42—531no—
222C42C22×D5311yescharacteristic
223C23—21no—
224C23—531no—
225C23—1061no—
226Q8—22no—
227Q8—42no—
228Q8—1062no—
229Q8—2122no—
230Q8—2124no—
231D4—22no—
232D4—1062no—
233D4—2122no—
234D4—2122no—
235D4—2122no—
236D4—2122no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—531no—
240C2×C4—531no—
241C2×C4—531no—
242C2×C4—534no—
243C2×C4—1062no—
244C2×C4—1062no—
245C2×C4—1061no—
246C2×C4—1062no—
247C2×C4—2122no—
248C2×C4—2124no—
249C2×C4—2122no—
250C2×C4D4×D5311yescharacteristic
251C2×C4D4×D5311yescharacteristic
252C2×C4C23×D5311yescharacteristic
253C8—22no—
254C8—1062no—
255C22—41no—
256C22—41no—
257C22—41no—
258C22—532no—
259C22—1061no—
260C22—1061no—
261C22—2121no—
262C22—2121no—
263C22—2121no—
264C22C2×D4×D5311yescharacteristic
265C4—21no—
266C4—21no—
267C4—41no—
268C4—42no—
269C4—532no—
270C4—532no—
271C4—532no—
272C4—2121no—
273C4—2122no—
274C4#41×D5312yes—
275C2—81no—
276C2—1062no—
277C2—4241no—
278C2#124×D5311yescharacteristic
279C2AGL(1,ℤ/8)×D5311yescharacteristic
280C2#137×D5311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#489C2
socleC22×C53C22×D4
hypercenter#489D53
hypocenterC53#3141
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D53C2semidirect
3. #16810985C2semidirect
4. #16810983C2semidirect
5. #16810982C2semidirect
6. #16810972C2semidirect
7. #16810968C2semidirect
8. #16810623C2semidirect
9. #16810621C2semidirect
10. #16810575C2semidirect
11. C53×#489C2semidirect
146. #489D53semidirect
152. C53#3141semidirect