Finite Group Databasebeta

← #20283965

Subgroups — #20283965

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×#13133243C212yesmaximal, complemented
3#16810766C214yesmaximal, complemented
4#16810761C212yesmaximal, complemented
5#16810756C212yesmaximal, complemented
6#16810622C212yesmaximal, complemented
7#16810587C211yesmaximal, characteristic, complemented
8C2×#121×C53C211yesmaximal, characteristic, complemented
9C2×#11719945—21no—
10C2×#11719945—21no—
11C2×#11719945—28no—
12C2×#11719945C2212yes—
13C2×#11719945C2212yescomplemented
14C22×Dic106C2211yescharacteristic, complemented
15C22×D212C2211yescharacteristic, complemented
16C22×C22⋊D53—21no—
17C22×C4×D53—21no—
18#13133243—28no—
19C2×#11719923C2214yescomplemented
20C2×#11719922C2214yescomplemented
21C2×#11719917C2212yescomplemented
22#13133140C2214yescomplemented
23#13133137C2214yescomplemented
24C22⋊#88322C2212yescomplemented
25C22⋊#88322C2212yescomplemented
26C22×D4×C53—21no—
27C22×D4×C53C2211yescharacteristic, complemented
28C23×C4×C53C2211yescharacteristic
29#121×C53—21no—
30#121×C53—21no—
31#121×C53C2212yes—
32#121×C53C2212yescomplemented
33C2×C4⋊C4×C53C2211yescharacteristic, complemented
34C2×C22⋊C4×C53—21no—
35C24×C53—21no—
36C24×C53D411yescharacteristic
37C23×D53—21no—
38C2×D4×C53—22no—
39C2×D4×C53—22no—
40C2×D4×C53—24no—
41C2×D4×C53—21no—
42C2×D4×C53—44no—
43C2×D4×C53C2312yes—
44C22×C4×C53—24no—
45C22×C4×C53—21no—
46C22×C4×C53D411yescharacteristic
47C22×C4×C53D411yescharacteristic
48C22×C4×C53D411yescharacteristic
49C22×C4×C53C2311yescharacteristic
50C22×C4×C53C2311yescharacteristic
51C22×C4×C53C2311yescharacteristic
52C2×C22⋊D53—28no—
53C2×C22⋊D53—22no—
54C2×C22⋊D53—22no—
55C22×Dic53—21no—
56#11719945—28no—
57#11719945—28no—
58#11719945—24no—
59#11719945—24no—
60#11719945—24no—
61#11719945D418yescomplemented
62C2×D212—21no—
63C2×D212—24no—
64C2×D212C2312yes—
65C2×C4×D53—28no—
66C2×C4×D53—22no—
67C2×C4×D53—22no—
68C2×Dic106—21no—
69C2×Dic106—24no—
70C2×Dic106C2312yes—
71C4⋊C4×C53—21no—
72C4⋊C4×C53C2312yes—
73C22⋊C4×C53—22no—
74C22⋊C4×C53—22no—
75#11719923—28no—
76#11719922—28no—
77#11719917—24no—
78C2×#88322C2314yes—
79C23×C53—22no—
80C23×C53—22no—
81C23×C53—22no—
82C23×C53—41no—
83C23×C53—42no—
84C23×C53—42no—
85C23×C53—42no—
86C23×C53C2×D411yescharacteristic
87C23×C53C2×D411yescharacteristic
88C23×C53C2×D411yescharacteristic
89C22×D53—24no—
90C22×D53—22no—
91C22×D53—42no—
92C22×D53—42no—
93D4×C53—24no—
94D4×C53—44no—
95D4×C53—44no—
96D4×C53—42no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—42no—
104C2×C4×C53C2×D414yes—
105C2×C4×C53C2×D411yescharacteristic
106C2×C4×C53C2×D411yescharacteristic
107C2×C4×C53C2×D411yescharacteristic
108C2×C4×C53C2411yescharacteristic
109C22⋊D53—28no—
110C22⋊D53—28no—
111C2×Dic53—24no—
112C2×Dic53—22no—
113D212—24no—
114D212—22no—
115D212C2×D414yescomplemented
116C4×D53—28no—
117C4×D53—28no—
118Dic106—24no—
119Dic106—22no—
120Dic106C2×D414yescomplemented
121#88322—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—42no—
131C22×C53—42no—
132C22×C53—42no—
133C22×C53—84no—
134C22×C53#12012yes—
135C22×C53AGL(1,ℤ/8)12yescomplemented
136C22×C53C22×D411yescharacteristic
137C2×D53—24no—
138C2×D53—48no—
139C2×D53—42no—
140C2×D53—42no—
141C4×C53—21no—
142C4×C53—21no—
143C4×C53—42no—
144C4×C53#12012yes—
145C4×C53C22×D412yes—
146Dic53—24no—
147#3006—531nomaximal, Sylow
148C2×C53—22no—
149C2×C53—22no—
150C2×C53—41no—
151C2×C53—82no—
152C2×C53C2×#12011yescharacteristic
153C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
154C2×C53#57511yescharacteristic
155D53—44no—
156C22×#41—531no—
157C2×AGL(1,ℤ/8)—532no—
158C2×#121D5311yescharacteristic, complemented
159#449—534no—
160#448—532no—
161#447—532no—
162#417—532no—
163#407—531no—
164C53#300611yescharacteristic, complemented, Sylow
165C2×#41—532no—
166C2×#41—532no—
167C2×#41—1068no—
168C2×#41—1061no—
169C2×#41—1061no—
170C22×Q8—531no—
171C22×D4—21no—
172C22×D4—531no—
173C22×D4—1061no—
174C22×D4C2×D5311yescharacteristic, complemented
175C23×C4—1061no—
176C23×C4C2×D5311yescharacteristic
177AGL(1,ℤ/8)—1068no—
178C2×SD16—534no—
179C2×D8—534no—
180C2×M16—532no—
181#121—21no—
182#121—21no—
183#121C2×D5312yes—
184#121C2×D5312yescomplemented
185C2×C4⋊C4C2×D5311yescharacteristic, complemented
186C2×C22⋊C4—21no—
187#103—534no—
188#102—534no—
189C22⋊C8—532no—
190C22⋊C8—532no—
191C24—21no—
192C24—1061no—
193C24C22⋊D5311yescharacteristic
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—22no—
204C2×D4—22no—
205C2×D4—24no—
206C2×D4—21no—
207C2×D4—44no—
208C2×D4—532no—
209C2×D4—1064no—
210C2×D4—1068no—
211C2×D4—1062no—
212C2×D4—1062no—
213C2×D4—1061no—
214C2×D4C22×D5312yes—
215C22×C4—24no—
216C22×C4—21no—
217C22×C4—1068no—
218C22×C4—1061no—
219C22×C4—1062no—
220C22×C4—1062no—
221C22×C4C22⋊D5311yescharacteristic
222C22×C4C22⋊D5311yescharacteristic
223C22×C4C22⋊D5311yescharacteristic
224C22×C4C22×D5311yescharacteristic
225C22×C4C22×D5311yescharacteristic
226C22×C4C22×D5311yescharacteristic
227SD16—1068no—
228D8—1068no—
229M16—1064no—
230C2×C8—534no—
231C4⋊C4—21no—
232C4⋊C4C22×D5312yes—
233C22⋊C4—22no—
234C22⋊C4—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—41no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—1064no—
243C23—1062no—
244C23—2122no—
245C23—2122no—
246C23C2×C22⋊D5311yescharacteristic
247C23C2×C22⋊D5311yescharacteristic
248C23C2×C22⋊D5311yescharacteristic
249Q8—534no—
250Q8—1064no—
251Q8—1062no—
252D4—24no—
253D4—44no—
254D4—44no—
255D4—42no—
256D4—534no—
257D4—1064no—
258D4—1068no—
259D4—1068no—
260D4—1062no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—42no—
268C2×C4—1068no—
269C2×C4—1064no—
270C2×C4—1062no—
271C2×C4—1068no—
272C2×C4D4×D5314yes—
273C2×C4C2×C22⋊D5311yescharacteristic
274C2×C4C2×C22⋊D5311yescharacteristic
275C2×C4C2×C22⋊D5311yescharacteristic
276C2×C4C23×D5311yescharacteristic
277C8—1064no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—44no—
286C22—42no—
287C22—42no—
288C22—42no—
289C22—84no—
290C22—1064no—
291C22—2128no—
292C22—2122no—
293C22—2122no—
294C22#1313318512yes—
295C22#1313324312yescomplemented
296C22C22×C22⋊D5311yescharacteristic
297C4—21no—
298C4—21no—
299C4—42no—
300C4—1064no—
301C4#1313318512yes—
302C4C2×D4×D5312yes—
303C2—22no—
304C2—22no—
305C2—41no—
306C2—82no—
307C2—2124no—
308C2C2×#1313318511yescharacteristic
309C2C2×#1313324311yescharacteristic
310C2#1681142111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×#121×C53C2
socleC22×C53C22×D4
hypercenterC2×#121D53
hypocenterC53#3006
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#11719945C2semidirect
2. C2×#13133243C2semidirect
3. #16810766C2semidirect
4. #16810761C2semidirect
5. #16810756C2semidirect
6. #16810622C2semidirect
7. #16810587C2semidirect
8. C2×#121×C53C2semidirect
13. C2×#11719945C22semidirect
14. C22×Dic106C22semidirect
15. C22×D212C22semidirect
19. C2×#11719923C22semidirect
20. C2×#11719922C22semidirect
21. C2×#11719917C22semidirect
22. #13133140C22semidirect
23. #13133137C22semidirect
24. C22⋊#88322C22semidirect
25. C22⋊#88322C22semidirect
27. C22×D4×C53C22semidirect
32. #121×C53C22semidirect
33. C2×C4⋊C4×C53C22semidirect
61. #11719945D4semidirect
115. D212C2×D4semidirect
120. Dic106C2×D4semidirect
135. C22×C53AGL(1,ℤ/8)semidirect
158. C2×#121D53semidirect
164. C53#3006semidirect
174. C22×D4C2×D53semidirect
184. #121C2×D53semidirect
185. C2×C4⋊C4C2×D53semidirect
295. C22#13133243semidirect