Finite Group Databasebeta

← #20283969

Subgroups — #20283969

Order 6784.

806 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×#11719945C211yesmaximal, characteristic, complemented
2C2×#13133245C212yesmaximal, complemented
3C2×#13133191C211yesmaximal, characteristic, complemented
4#16810770C214yesmaximal, complemented
5#16810764C212yesmaximal, complemented
6#16810759C212yesmaximal, complemented
7#16810625C212yesmaximal, complemented
8C53×#407C211yesmaximal, characteristic, complemented
9C2×#11719945—21no—
10C2×#11719945—21no—
11C2×#11719945—28no—
12C2×#11719945C2212yes—
13C2×#11719945C2212yescomplemented
14C22×Dic106C2211yescharacteristic, complemented
15C22×D212C2211yescharacteristic, complemented
16C22×D212C2211yescharacteristic, complemented
17C22×C22⋊D53—21no—
18C22×C22⋊D53—21no—
19C22×C4×D53—21no—
20#13133245—28no—
21C2×#11719913C2214yescomplemented
22C2×D424C2214yescomplemented
23#13133191—21no—
24#13133191—21no—
25#13133191C2212yes—
26#13133191C2212yescomplemented
27C2×C4⋊Dic53C2211yescharacteristic, complemented
28C2×#11719921—21no—
29#13133142C2214yescomplemented
30#13133139C2214yescomplemented
31C23×C4×C53C2211yescharacteristic
32C2×M16×C53C2212yescomplemented
33C22⋊C8×C53C2212yescomplemented
34C22⋊C8×C53C2212yescomplemented
35C24×C53D411yescharacteristic
36C23×D53—21no—
37C23×D53—21no—
38C22×C4×C53—24no—
39C22×C4×C53D411yescharacteristic
40C22×C4×C53D411yescharacteristic
41C22×C4×C53D411yescharacteristic
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2311yescharacteristic
45C2×C22⋊D53—28no—
46C2×C22⋊D53—22no—
47C2×C22⋊D53—22no—
48C2×C22⋊D53—22no—
49C2×C22⋊D53—22no—
50C2×C22⋊D53—44no—
51C22×Dic53—21no—
52C22×Dic53—21no—
53#11719945—28no—
54#11719945—28no—
55#11719945—24no—
56#11719945—24no—
57#11719945—24no—
58#11719945D418yescomplemented
59C2×D212—21no—
60C2×D212—24no—
61C2×D212—24no—
62C2×D212—21no—
63C2×D212C2312yes—
64C2×D212C2312yes—
65C2×C4×D53—28no—
66C2×C4×D53—22no—
67C2×C4×D53—22no—
68C2×Dic106—21no—
69C2×Dic106—24no—
70C2×Dic106C2312yes—
71M16×C53—24no—
72C2×C8×C53C2314yes—
73#11719921—22no—
74#11719921—22no—
75C4⋊Dic53—21no—
76C4⋊Dic53C2312yes—
77D424—28no—
78#11719913—28no—
79C23×C53—22no—
80C23×C53—22no—
81C23×C53—41no—
82C23×C53C2×D411yescharacteristic
83C23×C53C2×D411yescharacteristic
84C23×C53C2×D411yescharacteristic
85C22×D53—24no—
86C22×D53—22no—
87C22×D53—22no—
88C22×D53—42no—
89C22×D53—42no—
90C22×D53—42no—
91C22×D53—42no—
92C22×D53—42no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53C2×D411yescharacteristic
99C2×C4×C53C2×D411yescharacteristic
100C2×C4×C53C2×D411yescharacteristic
101C2×C4×C53C2×D414yes—
102C2×C4×C53C2411yescharacteristic
103C22⋊D53—28no—
104C22⋊D53—28no—
105C22⋊D53—44no—
106C22⋊D53—44no—
107C2×Dic53—24no—
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53—42no—
111D212—24no—
112D212—22no—
113D212—24no—
114D212—42no—
115D212C2×D414yescomplemented
116C4×D53—28no—
117C4×D53—28no—
118Dic106—24no—
119Dic106—22no—
120Dic106C2×D414yescomplemented
121C8×C53—24no—
122C22×C53—21no—
123C22×C53—21no—
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—21no—
127C22×C53—21no—
128C22×C53—21no—
129C22×C53—44no—
130C22×C53#12012yes—
131C22×C53AGL(1,ℤ/8)12yescomplemented
132C22×C53C22×D411yescharacteristic
133C2×D53—24no—
134C2×D53—48no—
135C2×D53—42no—
136C2×D53—42no—
137C2×D53—42no—
138C2×D53—42no—
139C2×D53—42no—
140C2×D53—84no—
141C4×C53—21no—
142C4×C53—21no—
143C4×C53#12012yes—
144C4×C53C22×D412yes—
145Dic53—24no—
146Dic53—42no—
147#3006—531nomaximal, Sylow
148C2×C53—22no—
149C2×C53—22no—
150C2×C53—41no—
151C2×C53C2×#12011yescharacteristic
152C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
153C2×C53#57511yescharacteristic
154D53—44no—
155D53—82no—
156C22×#41—531no—
157C2×AGL(1,ℤ/8)—532no—
158C2×#121—531no—
159#449—534no—
160#448—532no—
161#447—532no—
162#417—532no—
163#407D5311yescharacteristic, complemented
164C53#300611yescharacteristic, complemented, Sylow
165C2×#41—532no—
166C2×#41—532no—
167C2×#41—1068no—
168C2×#41—1061no—
169C2×#41—1061no—
170C22×Q8—531no—
171C22×D4—531no—
172C22×D4—531no—
173C22×D4—1061no—
174C22×D4—1061no—
175C23×C4—1061no—
176C23×C4C2×D5311yescharacteristic
177AGL(1,ℤ/8)—1068no—
178C2×SD16—534no—
179C2×D8—534no—
180C2×M16C2×D5312yescomplemented
181#121—532no—
182#121—532no—
183#121—1061no—
184#121—1061no—
185C2×C4⋊C4—531no—
186C2×C22⋊C4—1061no—
187#103—534no—
188#102—534no—
189C22⋊C8C2×D5312yescomplemented
190C22⋊C8C2×D5312yescomplemented
191C24—1061no—
192C24—1061no—
193C24D21211yescharacteristic
194#41—538no—
195#41—1064no—
196#41—1064no—
197#41—1068no—
198#41—1068no—
199#41—1064no—
200C2×Q8—532no—
201C2×Q8—1064no—
202C2×Q8—1061no—
203C2×D4—532no—
204C2×D4—532no—
205C2×D4—1064no—
206C2×D4—1068no—
207C2×D4—1062no—
208C2×D4—1062no—
209C2×D4—1062no—
210C2×D4—1062no—
211C2×D4—1061no—
212C2×D4—1061no—
213C2×D4—1064no—
214C2×D4—2124no—
215C22×C4—24no—
216C22×C4—1068no—
217C22×C4—1061no—
218C22×C4—1062no—
219C22×C4—1062no—
220C22×C4—1061no—
221C22×C4D21211yescharacteristic
222C22×C4D21211yescharacteristic
223C22×C4D21211yescharacteristic
224C22×C4C22×D5311yescharacteristic
225C22×C4C22×D5311yescharacteristic
226C22×C4C22×D5311yescharacteristic
227SD16—1068no—
228D8—1068no—
229M16—24no—
230C2×C8C22×D5314yes—
231C4⋊C4—532no—
232C4⋊C4—1061no—
233C22⋊C4—1062no—
234C22⋊C4—1062no—
235C23—22no—
236C23—22no—
237C23—41no—
238C23—1064no—
239C23—1062no—
240C23—1062no—
241C23—2122no—
242C23—2122no—
243C23—2122no—
244C23—2122no—
245C23—2122no—
246C23C2×D21211yescharacteristic
247C23C2×D21211yescharacteristic
248C23C2×D21211yescharacteristic
249Q8—534no—
250Q8—1064no—
251Q8—1062no—
252D4—534no—
253D4—1064no—
254D4—1068no—
255D4—1068no—
256D4—1062no—
257D4—1064no—
258D4—2124no—
259D4—2124no—
260D4—2122no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—1068no—
267C2×C4—1064no—
268C2×C4—1062no—
269C2×C4—1068no—
270C2×C4—1062no—
271C2×C4—2122no—
272C2×C4C2×D21211yescharacteristic
273C2×C4C2×D21211yescharacteristic
274C2×C4C2×D21211yescharacteristic
275C2×C4D4×D5314yes—
276C2×C4C23×D5311yescharacteristic
277C8—24no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—44no—
286C22—1064no—
287C22—4244no—
288C22—2128no—
289C22—2122no—
290C22—2122no—
291C22—2122no—
292C22—2122no—
293C22—2122no—
294C22#1313318712yes—
295C22#1313324512yescomplemented
296C22C22×D21211yescharacteristic
297C4—21no—
298C4—21no—
299C4—1064no—
300C4—2122no—
301C4#1313318712yes—
302C4C2×D4×D5312yes—
303C2—22no—
304C2—22no—
305C2—41no—
306C2—4242no—
307C2—2124no—
308C2C2×#1313318711yescharacteristic
309C2C2×#1313324511yescharacteristic
310C2#1681142311yescharacteristic
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#407C2
socleC22×C53C22×D4
hypercenter#407D53
hypocenterC53#3006
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. C22×#11719945C2semidirect
2. C2×#13133245C2semidirect
3. C2×#13133191C2semidirect
4. #16810770C2semidirect
5. #16810764C2semidirect
6. #16810759C2semidirect
7. #16810625C2semidirect
8. C53×#407C2semidirect
13. C2×#11719945C22semidirect
14. C22×Dic106C22semidirect
15. C22×D212C22semidirect
16. C22×D212C22semidirect
21. C2×#11719913C22semidirect
22. C2×D424C22semidirect
26. #13133191C22semidirect
27. C2×C4⋊Dic53C22semidirect
29. #13133142C22semidirect
30. #13133139C22semidirect
32. C2×M16×C53C22semidirect
33. C22⋊C8×C53C22semidirect
34. C22⋊C8×C53C22semidirect
58. #11719945D4semidirect
115. D212C2×D4semidirect
120. Dic106C2×D4semidirect
131. C22×C53AGL(1,ℤ/8)semidirect
163. #407D53semidirect
164. C53#3006semidirect
180. C2×M16C2×D53semidirect
189. C22⋊C8C2×D53semidirect
190. C22⋊C8C2×D53semidirect
295. C22#13133245semidirect