Finite Group Databasebeta

← #20285537

Subgroups — #20285537

Order 6784.

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

normal subgroups only

This group has 410 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
1C2×#13133193C211yesmaximal, characteristic
2C22⋊Q8×D53C211yesmaximal, characteristic
3#16810954C211yesmaximal, characteristic, complemented
4#16810952C212yesmaximal, complemented
5#16810951C211yesmaximal, characteristic, complemented
6#16810880C212yesmaximal, complemented
7#16810878C212yesmaximal, complemented
8#16810779C211yesmaximal, characteristic, complemented
9#16810777C212yesmaximal, complemented
10#16810585C211yesmaximal, characteristic
11C53×#451C211yesmaximal, characteristic, complemented
12C2×Q8×D53C2211yescharacteristic
13C22×C4×D53C2211yescharacteristic
14C2×#11719925C2212yescomplemented
15C2×#11719912C2212yes—
16#13133197C2211yescharacteristic
17#13133196C2211yescharacteristic
18#13133195—21no—
19#13133194—21no—
20#13133193—21no—
21#13133193—21no—
22#13133193C2212yes—
23#13133193C2211yescharacteristic
24#13133193C2212yes—
25C2×C4⋊Dic53C2211yescharacteristic
26C2×#11719919—21no—
27C4⋊C4×D53—21no—
28C4⋊C4×D53C2211yescharacteristic
29C2×#11719921—21no—
30C22⋊C4×D53—21no—
31#13133149C2212yescomplemented
32#13133148C2212yescomplemented
33#13133142C2212yescomplemented
34#13133141C2212yescomplemented
35#13133141C2212yescomplemented
36C22⋊#88322C2211yescharacteristic
37#13133129C2212yes—
38C22×Q8×C53C2211yescharacteristic
39C2×Q16×C53C2212yescomplemented
40C22⋊Q8×C53C2211yescharacteristic
41#103×C53C2212yescomplemented
42C22⋊C8×C53C2211yescharacteristic
43C23×D53D411yescharacteristic
44C2×Q8×C53—21no—
45C2×Q8×C53—42no—
46C2×Q8×C53C2312yes—
47C2×Q8×C53C2311yescharacteristic
48C22×C4×C53—21no—
49C22×C4×C53C2311yescharacteristic
50C22×Dic53—21no—
51C22×Dic53D411yescharacteristic
52Q8×D53—42no—
53C2×C4×D53—21no—
54C2×C4×D53—21no—
55C2×C4×D53—42no—
56C2×C4×D53D411yescharacteristic
57C2×C4×D53D411yescharacteristic
58C2×C4×D53C2311yescharacteristic
59C2×C4×D53C2311yescharacteristic
60C2×Dic106—21no—
61C2×Dic106C2311yescharacteristic
62Q16×C53—42no—
63C2×C8×C53C2312yes—
64C4⋊C4×C53—21no—
65C4⋊C4×C53C2311yescharacteristic
66C22⋊C4×C53—21no—
67C22⋊Dic53—21no—
68#11719925—42no—
69#11719921—21no—
70#11719921—21no—
71#11719921—22no—
72#11719921—22no—
73C4⋊Dic53—21no—
74C4⋊Dic53—21no—
75C4⋊Dic53C2311yescharacteristic
76C4⋊Dic53C2312yes—
77#11719919—21no—
78#11719919—21no—
79#11719919—21no—
80#11719919—22no—
81#11719919—22no—
82C2×#88322C2312yes—
83#11719912—42no—
84C23×C53C2×D411yescharacteristic
85C22×D53—22no—
86C22×D53—22no—
87C22×D53—41no—
88C22×D53C2×D411yescharacteristic
89C22×D53C2×D411yescharacteristic
90Q8×C53—22no—
91Q8×C53—22no—
92Q8×C53—22no—
93Q8×C53—42no—
94C2×C4×C53—22no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—42no—
99C2×C4×C53C2×D411yescharacteristic
100C2×C4×C53C2411yescharacteristic
101C2×Dic53—21no—
102C2×Dic53—21no—
103C2×Dic53—22no—
104C2×Dic53—41no—
105C2×Dic53—42no—
106C2×Dic53C2×D411yescharacteristic
107C2×Dic53C2×D411yescharacteristic
108C4×D53—22no—
109C4×D53—22no—
110C4×D53—41no—
111C4×D53—41no—
112C4×D53—42no—
113C4×D53—42no—
114Dic106—22no—
115Dic106—42no—
116C8×C53—22no—
117#88322—22no—
118C22×C53—21no—
119C22×C53—21no—
120C22×C53#13712yes—
121C22×C53C22×D411yescharacteristic
122C2×D53—21no—
123C2×D53—21no—
124C2×D53—21no—
125C2×D53—21no—
126C2×D53—21no—
127C2×D53—44no—
128C2×D53#13712yes—
129C4×C53—21no—
130C4×C53—41no—
131C4×C53—42no—
132C4×C53—41no—
133C4×C53H(5,𝔽2)12yes—
134Dic53—21no—
135Dic53—21no—
136Dic53—41no—
137Dic53—41no—
138Dic53—42no—
139#3195—531nomaximal, Sylow
140C2×C53—22no—
141C2×C53#53411yescharacteristic
142C2×C53C2×#13711yescharacteristic
143C2×C53C2×#13711yescharacteristic
144D53—22no—
145D53—41no—
146C2×C22⋊Q8—531no—
147C2×C22⋊Q8—531no—
148#483—531no—
149#483—532no—
150#483—531no—
151#470—532no—
152#470—532no—
153#451—531no—
154#451—532no—
155#451D5311yescharacteristic, complemented
156#407—531no—
157C53#319511yescharacteristic, complemented, Sylow
158C22×Q8—531no—
159C22×Q8C2×D5311yescharacteristic
160C23×C4—531no—
161C2×Q16—532no—
162C2×Q16C2×D5312yescomplemented
163C2×M16—532no—
164C22⋊Q8—532no—
165C22⋊Q8—531no—
166C22⋊Q8—532no—
167C22⋊Q8—531no—
168C22⋊Q8—531no—
169C22⋊Q8—1061no—
170C22⋊Q8—1061no—
171C22⋊Q8—1061no—
172C22⋊Q8—1061no—
173C22⋊Q8C2×D5311yescharacteristic
174C2×C4⋊C4—531no—
175C2×C4⋊C4—531no—
176C2×C4⋊C4—1061no—
177C2×C4⋊C4—1061no—
178C2×C22⋊C4—1061no—
179C2×C22⋊C4—1061no—
180#106—532no—
181#106—532no—
182#103—532no—
183#103—532no—
184#103—532no—
185#103C2×D5312yescomplemented
186C22⋊C8—531no—
187C22⋊C8—532no—
188C22⋊C8C2×D5311yescharacteristic
189C24—531no—
190C2×Q8—21no—
191C2×Q8—42no—
192C2×Q8—531no—
193C2×Q8—1061no—
194C2×Q8—2122no—
195C2×Q8C22×D5311yescharacteristic
196C2×Q8C22×D5312yes—
197C22×C4—21no—
198C22×C4—531no—
199C22×C4—531no—
200C22×C4—531no—
201C22×C4—531no—
202C22×C4—531no—
203C22×C4—1061no—
204C22×C4—1061no—
205C22×C4—1061no—
206C22×C4—2122no—
207C22×C4C22×D5311yescharacteristic
208Q16—42no—
209Q16—2122no—
210M16—2122no—
211C2×C8—532no—
212C2×C8C22×D5312yes—
213C4⋊C4—21no—
214C4⋊C4—531no—
215C4⋊C4—532no—
216C4⋊C4—1061no—
217C4⋊C4—1061no—
218C4⋊C4—1062no—
219C4⋊C4—1062no—
220C4⋊C4—1061no—
221C4⋊C4—1061no—
222C4⋊C4—1061no—
223C4⋊C4C22×D5311yescharacteristic
224C22⋊C4—21no—
225C22⋊C4—1061no—
226C22⋊C4—1061no—
227C22⋊C4—1062no—
228C22⋊C4—1062no—
229C22⋊C4—1061no—
230C23—531no—
231C23—531no—
232C23—1062no—
233C23—1062no—
234C23—2121no—
235C23D4×D5311yescharacteristic
236Q8—22no—
237Q8—22no—
238Q8—22no—
239Q8—42no—
240Q8—1062no—
241Q8—2122no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—531no—
248C2×C4—531no—
249C2×C4—1062no—
250C2×C4—1062no—
251C2×C4—1061no—
252C2×C4—1062no—
253C2×C4—1061no—
254C2×C4—2121no—
255C2×C4—2121no—
256C2×C4—2122no—
257C2×C4—2122no—
258C2×C4—2121no—
259C2×C4—2122no—
260C2×C4D4×D5311yescharacteristic
261C2×C4C23×D5311yescharacteristic
262C8—22no—
263C8—1062no—
264C22—21no—
265C22—21no—
266C22—532no—
267C22—1061no—
268C22—1061no—
269C22—1061no—
270C22—1061no—
271C22—1061no—
272C22—2124no—
273C22#1313324912yes—
274C22C2×D4×D5311yescharacteristic
275C4—21no—
276C4—41no—
277C4—42no—
278C4—41no—
279C4—1061no—
280C4—1061no—
281C4—2121no—
282C4—2122no—
283C4—2121no—
284C4#1313326212yes—
285C2—22no—
286C2—1062no—
287C2—2121no—
288C2#1681118311yescharacteristic
289C2#137×D5311yescharacteristic
290C2C2×#1313324911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#451C2
socleC22×C53C22×D4
hypercenter#451D53
hypocenterC53#3195
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #16810954C2semidirect
4. #16810952C2semidirect
5. #16810951C2semidirect
6. #16810880C2semidirect
7. #16810878C2semidirect
8. #16810779C2semidirect
9. #16810777C2semidirect
11. C53×#451C2semidirect
14. C2×#11719925C22semidirect
31. #13133149C22semidirect
32. #13133148C22semidirect
33. #13133142C22semidirect
34. #13133141C22semidirect
35. #13133141C22semidirect
39. C2×Q16×C53C22semidirect
41. #103×C53C22semidirect
155. #451D53semidirect
157. C53#3195semidirect
162. C2×Q16C2×D53semidirect
185. #103C2×D53semidirect