Finite Group Databasebeta

← #20283970

Subgroups — #20283970

Order 6784.

754 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 141 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×#11719947C211yesmaximal, characteristic
2C2×#13133246C212yesmaximal, complemented
3C22⋊Q8×D53C211yesmaximal, characteristic, complemented
4#16810786C212yesmaximal, complemented
5#16810782C212yesmaximal, complemented
6#16810781C211yesmaximal, characteristic, complemented
7#16810778C211yesmaximal, characteristic, complemented
8#16810774C211yesmaximal, characteristic, complemented
9#16810621C212yesmaximal, complemented
10#16810585C211yesmaximal, characteristic, complemented
11C53×#450C211yesmaximal, characteristic, complemented
12C2×#11719947—28no—
13C2×#11719947—21no—
14C2×#11719947—21no—
15C2×#11719947C2212yes—
16C2×#11719947C2212yes—
17C22×Dic106C2211yescharacteristic
18C2×Q8×D53C2211yescharacteristic, complemented
19C22×C22⋊D53—21no—
20C23×Dic53—21no—
21C22×C4×D53C2211yescharacteristic
22#13133246—28no—
23C2×Dic212C2212yescomplemented
24C2×#11719925C2212yescomplemented
25C2×#11719923C2212yescomplemented
26C2×#11719912C2212yescomplemented
27#13133197C2211yescharacteristic, complemented
28#13133196C2211yescharacteristic
29#13133195—21no—
30#13133194—21no—
31#13133193C2211yescharacteristic, complemented
32C4⋊C4×D53—21no—
33C4⋊C4×D53C2211yescharacteristic, complemented
34C22⋊C4×D53—21no—
35#13133142C2212yescomplemented
36#13133140C2212yescomplemented
37#13133138C2212yescomplemented
38C22⋊#88322C2211yescharacteristic
39#13133129C2212yescomplemented
40C22×D4×C53C2211yescharacteristic
41C2×SD16×C53C2212yescomplemented
42C22⋊Q8×C53C2211yescharacteristic
43#102×C53C2212yescomplemented
44C22⋊C8×C53C2211yescharacteristic
45C24×C53—21no—
46C23×D53D411yescharacteristic
47C2×Q8×C53C2311yescharacteristic
48C2×D4×C53—24no—
49C2×D4×C53—21no—
50C2×D4×C53C2312yes—
51C22×C4×C53C2311yescharacteristic
52C2×C22⋊D53—28no—
53C2×C22⋊D53—22no—
54C2×C22⋊D53—22no—
55C22×Dic53—28no—
56C22×Dic53—21no—
57C22×Dic53—22no—
58C22×Dic53—22no—
59C22×Dic53D411yescharacteristic
60Q8×D53—24no—
61#11719947—28no—
62#11719947—28no—
63#11719947—24no—
64#11719947—24no—
65#11719947—24no—
66#11719947D418yes—
67C2×C4×D53—24no—
68C2×C4×D53—21no—
69C2×C4×D53—21no—
70C2×C4×D53D411yescharacteristic
71C2×C4×D53D411yescharacteristic
72C2×C4×D53C2311yescharacteristic
73C2×C4×D53C2311yescharacteristic
74C2×Dic106—21no—
75C2×Dic106—24no—
76C2×Dic106—21no—
77C2×Dic106C2311yescharacteristic
78C2×Dic106C2312yes—
79SD16×C53—24no—
80C2×C8×C53C2312yes—
81C4⋊C4×C53—21no—
82C4⋊C4×C53C2311yescharacteristic
83C22⋊C4×C53—21no—
84C22⋊Dic53—21no—
85#11719925—24no—
86#11719923—24no—
87#11719921—21no—
88#11719921—21no—
89C4⋊Dic53—21no—
90C4⋊Dic53C2311yescharacteristic
91#11719919—21no—
92#11719919—21no—
93#11719919—21no—
94C2×#88322C2312yes—
95Dic212—24no—
96#11719912—24no—
97C23×C53—24no—
98C23×C53—22no—
99C23×C53—42no—
100C23×C53—42no—
101C23×C53C2×D411yescharacteristic
102C22×D53—22no—
103C22×D53—22no—
104C22×D53—41no—
105C22×D53C2×D411yescharacteristic
106C22×D53C2×D411yescharacteristic
107Q8×C53—22no—
108D4×C53—24no—
109D4×C53—22no—
110D4×C53C2×D414yes—
111C2×C4×C53—22no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53C2×D411yescharacteristic
115C2×C4×C53C2411yescharacteristic
116C22⋊D53—28no—
117C22⋊D53—28no—
118C2×Dic53—24no—
119C2×Dic53—28no—
120C2×Dic53—22no—
121C2×Dic53—22no—
122C2×Dic53—28no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53C2×D411yescharacteristic
126C2×Dic53C2×D411yescharacteristic
127C4×D53—22no—
128C4×D53—22no—
129C4×D53—22no—
130C4×D53—42no—
131C4×D53—42no—
132C4×D53C2×D414yes—
133Dic106—24no—
134Dic106—22no—
135Dic106—22no—
136Dic106—42no—
137Dic106C2×D414yes—
138C8×C53—22no—
139#88322—22no—
140C22×C53—21no—
141C22×C53—21no—
142C22×C53—24no—
143C22×C53—48no—
144C22×C53—42no—
145C22×C53—42no—
146C22×C53#13712yescomplemented
147C22×C53C22×D411yescharacteristic
148C2×D53—21no—
149C2×D53—21no—
150C2×D53—21no—
151C2×D53—21no—
152C2×D53—21no—
153C2×D53—44no—
154C2×D53#12012yes—
155C4×C53—21no—
156C4×C53—41no—
157C4×C53—41no—
158C4×C53C22×D412yes—
159Dic53—24no—
160Dic53—21no—
161Dic53—41no—
162Dic53—41no—
163Dic53#12012yes—
164#3007—531nomaximal, Sylow
165C2×C53—22no—
166C2×C53—44no—
167C2×C53C2×#12011yescharacteristic
168C2×C53C2×#13711yescharacteristic
169C2×C53#57511yescharacteristic
170D53—22no—
171D53—41no—
172C22×#41—531no—
173C2×#137—532no—
174C2×C22⋊Q8—531no—
175#452—532no—
176#452—532no—
177#451—531no—
178#451—531no—
179#450—531no—
180#450D5311yescharacteristic, complemented
181#417—532no—
182#407—531no—
183C53#300711yescharacteristic, complemented, Sylow
184C2×#41—532no—
185C2×#41—532no—
186C2×#41—1061no—
187C2×#41—1061no—
188C2×#41—1068no—
189C22×Q8—531no—
190C22×Q8—531no—
191C22×D4—1061no—
192C22×D4C2×D5311yescharacteristic
193C23×C4—531no—
194C23×C4—1061no—
195#137—1068no—
196C2×Q16—532no—
197C2×Q16—532no—
198C2×SD16—532no—
199C2×SD16C2×D5312yescomplemented
200C2×M16—532no—
201C22⋊Q8—531no—
202C22⋊Q8—531no—
203C22⋊Q8—531no—
204C22⋊Q8—1061no—
205C22⋊Q8—1061no—
206C22⋊Q8C2×D5311yescharacteristic
207C2×C4⋊C4—531no—
208C2×C4⋊C4—1061no—
209C2×C22⋊C4—1061no—
210#103—532no—
211#103—532no—
212#102—532no—
213#102C2×D5312yescomplemented
214C22⋊C8—531no—
215C22⋊C8—532no—
216C22⋊C8C2×D5311yescharacteristic
217C24—21no—
218C24—531no—
219#41—538no—
220#41—1064no—
221#41—1064no—
222#41—1068no—
223#41—1068no—
224#41—1064no—
225C2×Q8—532no—
226C2×Q8—531no—
227C2×Q8—1061no—
228C2×Q8—1064no—
229C2×Q8—1064no—
230C2×Q8—1061no—
231C2×Q8C22×D5311yescharacteristic
232C2×D4—24no—
233C2×D4—21no—
234C2×D4—1062no—
235C2×D4—1062no—
236C2×D4—1068no—
237C2×D4C22×D5312yes—
238C22×C4—531no—
239C22×C4—531no—
240C22×C4—531no—
241C22×C4—531no—
242C22×C4—531no—
243C22×C4—1064no—
244C22×C4—1061no—
245C22×C4—1061no—
246C22×C4—1062no—
247C22×C4—1062no—
248C22×C4—1068no—
249C22×C4—1061no—
250C22×C4C22×D5311yescharacteristic
251Q16—1064no—
252Q16—1064no—
253SD16—24no—
254SD16—1064no—
255M16—1064no—
256C2×C8—532no—
257C2×C8C22×D5312yes—
258C4⋊C4—21no—
259C4⋊C4—531no—
260C4⋊C4—1061no—
261C4⋊C4—1061no—
262C4⋊C4—1061no—
263C4⋊C4—1061no—
264C4⋊C4C22×D5311yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—1061no—
267C22⋊C4—1061no—
268C22⋊C4—1061no—
269C23—24no—
270C23—22no—
271C23—42no—
272C23—42no—
273C23—531no—
274C23—531no—
275C23—1062no—
276C23—1062no—
277C23—2121no—
278C23D4×D5311yescharacteristic
279Q8—22no—
280Q8—534no—
281Q8—1064no—
282Q8—1062no—
283Q8—1062no—
284Q8—2122no—
285D4—24no—
286D4—22no—
287D4—1068no—
288D4—1068no—
289D4D4×D5314yes—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—531no—
294C2×C4—534no—
295C2×C4—531no—
296C2×C4—1062no—
297C2×C4—1062no—
298C2×C4—1062no—
299C2×C4—1068no—
300C2×C4—1064no—
301C2×C4—1068no—
302C2×C4—1062no—
303C2×C4—1062no—
304C2×C4—1061no—
305C2×C4—1061no—
306C2×C4—2122no—
307C2×C4—2122no—
308C2×C4D4×D5311yescharacteristic
309C2×C4C23×D5311yescharacteristic
310C8—22no—
311C8—1062no—
312C22—21no—
313C22—21no—
314C22—24no—
315C22—48no—
316C22—42no—
317C22—42no—
318C22—532no—
319C22—1061no—
320C22—1061no—
321C22—1061no—
322C22—1061no—
323C22—1061no—
324C22—2124no—
325C22#1313324612yescomplemented
326C22C2×D4×D5311yescharacteristic
327C4—21no—
328C4—41no—
329C4—41no—
330C4—532no—
331C4—1061no—
332C4—1064no—
333C4—2121no—
334C4—2121no—
335C4C2×D4×D5312yes—
336C2—22no—
337C2—44no—
338C2—1062no—
339C2—2121no—
340C2#120×D5311yescharacteristic
341C2C2×#1313324611yescharacteristic
342C2#1681141911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#450C2
socleC22×C53C22×D4
hypercenter#450D53
hypocenterC53#3007
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
2. C2×#13133246C2semidirect
3. C22⋊Q8×D53C2semidirect
4. #16810786C2semidirect
5. #16810782C2semidirect
6. #16810781C2semidirect
7. #16810778C2semidirect
8. #16810774C2semidirect
9. #16810621C2semidirect
10. #16810585C2semidirect
11. C53×#450C2semidirect
18. C2×Q8×D53C22semidirect
23. C2×Dic212C22semidirect
24. C2×#11719925C22semidirect
25. C2×#11719923C22semidirect
26. C2×#11719912C22semidirect
27. #13133197C22semidirect
31. #13133193C22semidirect
33. C4⋊C4×D53C22semidirect
35. #13133142C22semidirect
36. #13133140C22semidirect
37. #13133138C22semidirect
39. #13133129C22semidirect
41. C2×SD16×C53C22semidirect
43. #102×C53C22semidirect
146. C22×C53#137semidirect
180. #450D53semidirect
183. C53#3007semidirect
199. C2×SD16C2×D53semidirect
213. #102C2×D53semidirect
325. C22#13133246semidirect