Finite Group Databasebeta

← #20285529

Subgroups — #20285529

Order 6784.

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

normal subgroups only

This group has 466 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#16811134C211yesmaximal, characteristic, complemented
2#16810953C211yesmaximal, characteristic, complemented
3#16810944C212yesmaximal, complemented
4#16810936C211yesmaximal, characteristic, complemented
5#16810874C212yesmaximal, complemented
6#16810867C212yesmaximal, complemented
7#16810774C211yesmaximal, characteristic, complemented
8#16810766C212yesmaximal, complemented
9#16810757C211yesmaximal, characteristic, complemented
10#16810587C211yesmaximal, characteristic, complemented
11C2×#121×C53C211yesmaximal, characteristic, complemented
12C2×#11719945C2211yescharacteristic, complemented
13C2×#11719923C2212yescomplemented
14C2×#11719922C2212yescomplemented
15C2×#11719917C2212yescomplemented
16#13133200—21no—
17#13133196C2211yescharacteristic
18#13133196C2211yescharacteristic, complemented
19#13133191C2211yescharacteristic
20#13133191C2211yescharacteristic, complemented
21#13133186—21no—
22C4×C22⋊D53—21no—
23C4×C22⋊D53—21no—
24#13133176C2211yescharacteristic, complemented
25#13133150C2212yescomplemented
26#13133148C2212yescomplemented
27#13133140C2212yescomplemented
28#13133138C2212yescomplemented
29#13133138C2212yescomplemented
30#13133137C2212yescomplemented
31C22⋊#88322C2211yescharacteristic
32C22⋊#88322C2211yescharacteristic
33C22⋊#88322C2212yescomplemented
34C22×D4×C53—21no—
35C22×D4×C53C2211yescharacteristic
36C23×C4×C53C2211yescharacteristic
37#121×C53—21no—
38#121×C53—21no—
39#121×C53C2212yes—
40#121×C53C2212yes—
41C2×C4⋊C4×C53C2211yescharacteristic
42C2×C22⋊C4×C53—21no—
43C24×C53—21no—
44C24×C53D411yescharacteristic
45C2×D4×C53—22no—
46C2×D4×C53—22no—
47C2×D4×C53—21no—
48C2×D4×C53—42no—
49C2×D4×C53—44no—
50C2×D4×C53C2312yes—
51C22×C4×C53—21no—
52C22×C4×C53—42no—
53C22×C4×C53D411yescharacteristic
54C22×C4×C53D411yescharacteristic
55C22×C4×C53D411yescharacteristic
56C22×C4×C53C2311yescharacteristic
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C2×C22⋊D53—21no—
60C2×C22⋊D53—21no—
61C2×C22⋊D53—21no—
62#11719945—42no—
63C2×D212C2311yescharacteristic
64C2×C4×D53—21no—
65C2×Dic106C2311yescharacteristic
66C4⋊C4×C53—21no—
67C4⋊C4×C53C2312yes—
68C22⋊C4×C53—22no—
69C22⋊C4×C53—22no—
70C22⋊Dic53—21no—
71C22⋊Dic53—21no—
72C22⋊Dic53—21no—
73#11719923—42no—
74#11719922—42no—
75#11719921—21no—
76#11719921—21no—
77C4⋊Dic53C2311yescharacteristic
78C4⋊Dic53C2311yescharacteristic
79#11719919—21no—
80#11719919—21no—
81C4×Dic53—21no—
82#11719917—42no—
83C2×#88322C2312yes—
84C2×#88322C2312yes—
85C23×C53—22no—
86C23×C53—22no—
87C23×C53—41no—
88C23×C53—41no—
89C23×C53—42no—
90C23×C53—42no—
91C23×C53—42no—
92C23×C53C2×D411yescharacteristic
93C23×C53C2×D411yescharacteristic
94C23×C53C2×D411yescharacteristic
95C22×D53—21no—
96D4×C53—22no—
97D4×C53—22no—
98D4×C53—42no—
99D4×C53—44no—
100D4×C53—44no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—22no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—42no—
109C2×C4×C53C2×D411yescharacteristic
110C2×C4×C53C2×D411yescharacteristic
111C2×C4×C53C2×D411yescharacteristic
112C2×C4×C53C2411yescharacteristic
113C22⋊D53—42no—
114C22⋊D53—42no—
115C22⋊D53—42no—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—21no—
119D212—22no—
120C4×D53—42no—
121Dic106—22no—
122#88322—22no—
123#88322—22no—
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—21no—
127C22×C53—21no—
128C22×C53—21no—
129C22×C53—21no—
130C22×C53—21no—
131C22×C53—21no—
132C22×C53—44no—
133C22×C53—42no—
134C22×C53—42no—
135C22×C53—42no—
136C22×C53—84no—
137C22×C53AGL(1,ℤ/8)12yes—
138C22×C53C22×D411yescharacteristic
139C2×D53—41no—
140C2×D53—41no—
141C2×D53—41no—
142C4×C53—21no—
143C4×C53—21no—
144C4×C53—21no—
145C4×C53—42no—
146C4×C53H(5,𝔽2)12yes—
147Dic53—41no—
148Dic53—41no—
149Dic53—41no—
150#3194—531nomaximal, Sylow
151C2×C53—22no—
152C2×C53—41no—
153C2×C53—41no—
154C2×C53—82no—
155C2×C53#53411yescharacteristic
156C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
157C2×C53#57511yescharacteristic
158D53—81no—
159#525—531no—
160C2×#121D5311yescharacteristic, complemented
161#483—531no—
162#482—532no—
163#480—531no—
164#469—532no—
165#468—532no—
166#450—531no—
167#449—532no—
168#447—531no—
169#407—531no—
170C53#319411yescharacteristic, complemented, Sylow
171C2×#41—531no—
172C22×D4—21no—
173C22×D4C2×D5311yescharacteristic
174C23×C4C2×D5311yescharacteristic
175C2×SD16—532no—
176C2×D8—532no—
177C2×M16—532no—
178#123—1061no—
179C22⋊Q8—531no—
180C22⋊Q8—531no—
181#121—21no—
182#121—21no—
183#121—531no—
184#121—531no—
185#121C2×D5312yes—
186#121C2×D5312yes—
187#120—1061no—
188C4×D4—1061no—
189C4×D4—1061no—
190#117—531no—
191C2×C4⋊C4C2×D5311yescharacteristic
192C2×C22⋊C4—21no—
193#107—532no—
194#106—532no—
195#103—532no—
196#102—532no—
197#102—532no—
198#102—532no—
199C22⋊C8—531no—
200C22⋊C8—531no—
201C22⋊C8—532no—
202C24—21no—
203C24C22⋊D5311yescharacteristic
204#41—2122no—
205C2×Q8—531no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—42no—
210C2×D4—44no—
211C2×D4—531no—
212C2×D4—1061no—
213C2×D4—1061no—
214C2×D4—1061no—
215C2×D4C22×D5312yes—
216C22×C4—21no—
217C22×C4—42no—
218C22×C4—1061no—
219C22×C4C22⋊D5311yescharacteristic
220C22×C4C22⋊D5311yescharacteristic
221C22×C4C22⋊D5311yescharacteristic
222C22×C4C22×D5311yescharacteristic
223C22×C4C22×D5311yescharacteristic
224C22×C4C22×D5311yescharacteristic
225SD16—2122no—
226D8—2122no—
227M16—2122no—
228C2×C8—532no—
229C2×C8—532no—
230C4⋊C4—21no—
231C4⋊C4—531no—
232C4⋊C4—531no—
233C4⋊C4—1061no—
234C4⋊C4—1061no—
235C4⋊C4C22×D5312yes—
236C22⋊C4—22no—
237C22⋊C4—22no—
238C22⋊C4—1061no—
239C22⋊C4—1061no—
240C22⋊C4—1061no—
241C22⋊C4—1061no—
242C22⋊C4—1061no—
243C42—1061no—
244C23—22no—
245C23—22no—
246C23—41no—
247C23—41no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—1061no—
252C23C2×C22⋊D5311yescharacteristic
253C23C2×C22⋊D5311yescharacteristic
254C23C2×C22⋊D5311yescharacteristic
255Q8—1062no—
256D4—22no—
257D4—22no—
258D4—42no—
259D4—44no—
260D4—44no—
261D4—1062no—
262D4—2122no—
263D4—2122no—
264D4—2122no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—1061no—
274C2×C4—1061no—
275C2×C4—1061no—
276C2×C4—2122no—
277C2×C4C2×C22⋊D5311yescharacteristic
278C2×C4C2×C22⋊D5311yescharacteristic
279C2×C4C2×C22⋊D5311yescharacteristic
280C2×C4C23×D5311yescharacteristic
281C8—1062no—
282C8—1062no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—44no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—84no—
296C22—2121no—
297C22—2121no—
298C22—2121no—
299C22#1313324312yes—
300C22C22×C22⋊D5311yescharacteristic
301C4—21no—
302C4—21no—
303C4—21no—
304C4—42no—
305C4—2121no—
306C4—2121no—
307C4—2121no—
308C4#1313326212yes—
309C2—22no—
310C2—41no—
311C2—41no—
312C2—82no—
313C2—4241no—
314C2#1681118211yescharacteristic
315C2C2×#1313324311yescharacteristic
316C2#1681142111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC2×#121×C53C2
socleC22×C53C22×D4
hypercenterC2×#121D53
hypocenterC53#3194
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. #16811134C2semidirect
2. #16810953C2semidirect
3. #16810944C2semidirect
4. #16810936C2semidirect
5. #16810874C2semidirect
6. #16810867C2semidirect
7. #16810774C2semidirect
8. #16810766C2semidirect
9. #16810757C2semidirect
10. #16810587C2semidirect
11. C2×#121×C53C2semidirect
12. C2×#11719945C22semidirect
13. C2×#11719923C22semidirect
14. C2×#11719922C22semidirect
15. C2×#11719917C22semidirect
18. #13133196C22semidirect
20. #13133191C22semidirect
24. #13133176C22semidirect
25. #13133150C22semidirect
26. #13133148C22semidirect
27. #13133140C22semidirect
28. #13133138C22semidirect
29. #13133138C22semidirect
30. #13133137C22semidirect
33. C22⋊#88322C22semidirect
160. C2×#121D53semidirect
170. C53#3194semidirect