Finite Group Databasebeta

← #20285464

Subgroups — #20285464

Order 6784.

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

normal subgroups only

This group has 520 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
1C2×#13133188C211yesmaximal, characteristic, complemented
2#121×D53C211yesmaximal, characteristic, complemented
3#16810937C211yesmaximal, characteristic, complemented
4#16810935C212yesmaximal, complemented
5#16810934C211yesmaximal, characteristic, complemented
6#16810858C212yesmaximal, complemented
7#16810856C212yesmaximal, complemented
8#16810757C211yesmaximal, characteristic, complemented
9#16810755C212yesmaximal, complemented
10C22⋊C8×D53C211yesmaximal, characteristic, complemented
11C53×#447C211yesmaximal, characteristic, complemented
12C22×C22⋊D53—21no—
13C2×D4×D53—21no—
14C2×D4×D53C2211yescharacteristic, complemented
15C22×C4×D53C2211yescharacteristic
16C2×#11719922C2212yescomplemented
17C2×C8×D53C2212yescomplemented
18#13133192C2211yescharacteristic, complemented
19#13133191C2211yescharacteristic
20#13133190—21no—
21#13133189—21no—
22#13133188—21no—
23#13133188—21no—
24#13133188C2212yes—
25#13133188C2211yescharacteristic, complemented
26#13133188C2212yescomplemented
27C2×C4⋊Dic53C2211yescharacteristic, complemented
28C4⋊C4×D53C2211yescharacteristic, complemented
29C2×C22⋊Dic53—21no—
30C22⋊C4×D53—21no—
31#13133151C2212yescomplemented
32#13133150C2212yescomplemented
33#13133139C2212yescomplemented
34#13133138C2212yescomplemented
35#13133138C2212yescomplemented
36C22⋊#88322C2211yescharacteristic, complemented
37#13133129C2212yescomplemented
38C22×D4×C53C2211yescharacteristic, complemented
39C2×D8×C53C2212yescomplemented
40#121×C53C2211yescharacteristic
41#102×C53C2212yescomplemented
42C22⋊C8×C53C2211yescharacteristic, complemented
43C24×C53—21no—
44C23×D53—21no—
45C23×D53D411yescharacteristic
46C2×D4×C53—21no—
47C2×D4×C53—21no—
48C2×D4×C53—42no—
49C2×D4×C53C2312yes—
50C2×D4×C53C2311yescharacteristic
51C22×C4×C53C2311yescharacteristic
52C2×C22⋊D53—21no—
53C2×C22⋊D53—22no—
54C2×C22⋊D53—22no—
55C2×C22⋊D53—21no—
56C2×C22⋊D53—21no—
57C2×C22⋊D53—44no—
58C22×Dic53—21no—
59C22×Dic53D411yescharacteristic
60D4×D53—42no—
61D4×D53—44no—
62C2×D212—21no—
63C2×D212C2311yescharacteristic
64C2×C4×D53—21no—
65C2×C4×D53—42no—
66C2×C4×D53D411yescharacteristic
67C2×C4×D53D411yescharacteristic
68C2×C4×D53C2311yescharacteristic
69C2×C4×D53C2311yescharacteristic
70D8×C53—42no—
71C2×C8×C53C2312yes—
72C4⋊C4×C53C2311yescharacteristic
73C22⋊C4×C53—21no—
74C22⋊Dic53—21no—
75C22⋊Dic53—22no—
76C22⋊Dic53—22no—
77#11719922—42no—
78#11719921—21no—
79#11719921—21no—
80C4⋊Dic53—21no—
81C4⋊Dic53C2311yescharacteristic
82C4⋊Dic53C2312yes—
83#11719919—21no—
84C2×#88322C2312yes—
85C8×D53—42no—
86C23×C53—21no—
87C23×C53—22no—
88C23×C53—42no—
89C23×C53—42no—
90C23×C53—42no—
91C23×C53C2×D411yescharacteristic
92C22×D53—21no—
93C22×D53—41no—
94C22×D53—41no—
95C22×D53—42no—
96C22×D53—42no—
97C22×D53—42no—
98C22×D53D814yescomplemented
99C22×D53C2×D411yescharacteristic
100C22×D53C2×D411yescharacteristic
101D4×C53—22no—
102D4×C53—22no—
103D4×C53—22no—
104D4×C53—42no—
105D4×C53—42no—
106C2×C4×C53—22no—
107C2×C4×C53—21no—
108C2×C4×C53C2×D411yescharacteristic
109C2×C4×C53C2411yescharacteristic
110C22⋊D53—42no—
111C22⋊D53—42no—
112C22⋊D53—42no—
113C22⋊D53—44no—
114C22⋊D53—44no—
115C2×Dic53—21no—
116C2×Dic53—22no—
117C2×Dic53—41no—
118C2×Dic53—42no—
119C2×Dic53C2×D411yescharacteristic
120C2×Dic53C2×D411yescharacteristic
121D212—22no—
122D212—42no—
123C4×D53—22no—
124C4×D53—22no—
125C4×D53—41no—
126C4×D53—41no—
127C4×D53—42no—
128C8×C53—22no—
129#88322—22no—
130C22×C53—21no—
131C22×C53—21no—
132C22×C53—41no—
133C22×C53—42no—
134C22×C53—42no—
135C22×C53—41no—
136C22×C53—41no—
137C22×C53—42no—
138C22×C53—84no—
139C22×C53C2×D812yes—
140C22×C53C22×D411yescharacteristic
141C2×D53—21no—
142C2×D53—21no—
143C2×D53—21no—
144C2×D53—21no—
145C2×D53—44no—
146C2×D53—41no—
147C2×D53—41no—
148C2×D53—41no—
149C2×D53—84no—
150C2×D53C2×D812yes—
151C2×D53C2×D812yes—
152C4×C53—21no—
153C4×C53—41no—
154C4×C53H(5,𝔽2)12yes—
155Dic53—21no—
156Dic53—21no—
157Dic53—41no—
158Dic53—42no—
159#3186—531nomaximal, Sylow
160C2×C53—22no—
161C2×C53—81no—
162C2×C53—82no—
163C2×C53#53411yescharacteristic
164C2×C53C22×D811yescharacteristic
165C2×C53#57511yescharacteristic
166D53—22no—
167D53—22no—
168D53—81no—
169C2×#121—531no—
170C2×#121—531no—
171#480—531no—
172#480—532no—
173#480—531no—
174#466—532no—
175#466—532no—
176#447—531no—
177#447—532no—
178#447D5311yescharacteristic, complemented
179C2×C22⋊C8—531no—
180C53#318611yescharacteristic, complemented, Sylow
181C22×D4—531no—
182C22×D4—1061no—
183C22×D4—1061no—
184C22×D4C2×D5311yescharacteristic, complemented
185C23×C4—531no—
186C2×D8—532no—
187C2×D8C2×D5312yescomplemented
188C22×C8—532no—
189#121—532no—
190#121—531no—
191#121—532no—
192#121—531no—
193#121—531no—
194#121—1061no—
195#121—1061no—
196#121—1061no—
197#121—1061no—
198#121C2×D5311yescharacteristic
199C2×C4⋊C4—531no—
200C2×C4⋊C4—531no—
201C2×C22⋊C4—1061no—
202C2×C22⋊C4—1061no—
203#107—532no—
204#107—532no—
205#102—532no—
206#102—532no—
207#102—532no—
208#102C2×D5312yescomplemented
209C22⋊C8—531no—
210C22⋊C8—532no—
211C22⋊C8C2×D5311yescharacteristic, complemented
212C24—21no—
213C24—531no—
214C24—1061no—
215C2×D4—21no—
216C2×D4—21no—
217C2×D4—42no—
218C2×D4—531no—
219C2×D4—1061no—
220C2×D4—1061no—
221C2×D4—1061no—
222C2×D4—1062no—
223C2×D4—1062no—
224C2×D4—1061no—
225C2×D4—2122no—
226C2×D4—2124no—
227C2×D4—2124no—
228C2×D4C22×D5311yescharacteristic
229C2×D4C22×D5312yes—
230C22×C4—531no—
231C22×C4—531no—
232C22×C4—531no—
233C22×C4—531no—
234C22×C4—531no—
235C22×C4—1061no—
236C22×C4—1061no—
237C22×C4—2122no—
238C22×C4C22×D5311yescharacteristic
239D8—42no—
240D8—2122no—
241C2×C8—532no—
242C2×C8—2122no—
243C2×C8C22×D5312yes—
244C4⋊C4—531no—
245C4⋊C4—532no—
246C4⋊C4—1061no—
247C4⋊C4—1061no—
248C4⋊C4C22×D5311yescharacteristic
249C22⋊C4—21no—
250C22⋊C4—1061no—
251C22⋊C4—1061no—
252C22⋊C4—1062no—
253C22⋊C4—1062no—
254C22⋊C4—1061no—
255C23—21no—
256C23—22no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—531no—
261C23—531no—
262C23—534no—
263C23—1061no—
264C23—2121no—
265C23—2121no—
266C23—2122no—
267C23—2122no—
268C23—2122no—
269C23D4×D5311yescharacteristic
270D4—22no—
271D4—22no—
272D4—22no—
273D4—42no—
274D4—42no—
275D4—1062no—
276D4—2122no—
277D4—2122no—
278D4—2122no—
279D4—2124no—
280D4—2124no—
281D4—2122no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—531no—
285C2×C4—531no—
286C2×C4—1062no—
287C2×C4—1062no—
288C2×C4—1061no—
289C2×C4—1062no—
290C2×C4—2121no—
291C2×C4—2121no—
292C2×C4—2122no—
293C2×C4—2121no—
294C2×C4—2122no—
295C2×C4D4×D5311yescharacteristic
296C2×C4C23×D5311yescharacteristic
297C8—22no—
298C8—1062no—
299C22—21no—
300C22—21no—
301C22—41no—
302C22—42no—
303C22—42no—
304C22—41no—
305C22—42no—
306C22—41no—
307C22—84no—
308C22—532no—
309C22—532no—
310C22—1061no—
311C22—1061no—
312C22—1061no—
313C22—1061no—
314C22—4244no—
315C22—2124no—
316C22—2121no—
317C22—2121no—
318C22—2121no—
319C22D8×D5312yes—
320C22C2×D4×D5311yescharacteristic
321C4—21no—
322C4—41no—
323C4—1061no—
324C4—1061no—
325C4—2121no—
326C4—2122no—
327C4#1313326212yes—
328C2—22no—
329C2—82no—
330C2—81no—
331C2—1062no—
332C2—1062no—
333C2—4241no—
334C2#1681118311yescharacteristic
335C2C2×D8×D5311yescharacteristic
336C2#1681141911yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#447C2
socleC22×C53C22×D4
hypercenter#447D53
hypocenterC53#3186
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C2×#13133188C2semidirect
2. #121×D53C2semidirect
3. #16810937C2semidirect
4. #16810935C2semidirect
5. #16810934C2semidirect
6. #16810858C2semidirect
7. #16810856C2semidirect
8. #16810757C2semidirect
9. #16810755C2semidirect
10. C22⋊C8×D53C2semidirect
11. C53×#447C2semidirect
14. C2×D4×D53C22semidirect
16. C2×#11719922C22semidirect
17. C2×C8×D53C22semidirect
18. #13133192C22semidirect
25. #13133188C22semidirect
26. #13133188C22semidirect
27. C2×C4⋊Dic53C22semidirect
28. C4⋊C4×D53C22semidirect
31. #13133151C22semidirect
32. #13133150C22semidirect
33. #13133139C22semidirect
34. #13133138C22semidirect
35. #13133138C22semidirect
36. C22⋊#88322C22semidirect
37. #13133129C22semidirect
38. C22×D4×C53C22semidirect
39. C2×D8×C53C22semidirect
41. #102×C53C22semidirect
42. C22⋊C8×C53C22semidirect
98. C22×D53D8semidirect
178. #447D53semidirect
180. C53#3186semidirect
184. C22×D4C2×D53semidirect
187. C2×D8C2×D53semidirect
208. #102C2×D53semidirect
211. C22⋊C8C2×D53semidirect