Finite Group Databasebeta

← #20278361

Subgroups — #20278361

Order 6784.

1088 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 161 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×D4×D53C211yesmaximal, characteristic
2C2×#13133203C212yesmaximal, complemented
3C2×#13133195C211yesmaximal, characteristic, complemented
4#16810541C211yesmaximal, characteristic, complemented
5#16810540C211yesmaximal, characteristic, complemented
6#16810537C211yesmaximal, characteristic, complemented
7#16810523C212yesmaximal, complemented
8#16810478C212yesmaximal, complemented
9#16810478C212yesmaximal, complemented
10#16810470C211yesmaximal, characteristic, complemented
11C53×#394C211yesmaximal, characteristic, complemented
12C24×D53C2212yes—
13C22×Dic106C2211yescharacteristic, complemented
14C22×D212C2211yescharacteristic
15C22×C22⋊D53C2212yes—
16C2×D4×D53—28no—
17C2×D4×D53—24no—
18C22×C4×D53C2211yescharacteristic
19#13133203—28no—
20#13133195—24no—
21C2×#11719919C2211yescharacteristic, complemented
22C2×#11719919C2211yescharacteristic, complemented
23C2×C22⋊Dic53C2212yescomplemented
24C2×C22⋊Dic53C2212yescomplemented
25C2×#11719921C2211yescharacteristic
26C2×#11719921C2211yescharacteristic
27C2×#11719921C2212yes—
28C2×#11719921C2212yescomplemented
29C2×#11719921C2212yescomplemented
30C2×#11719921C2212yescomplemented
31C2×#11719921C2212yescomplemented
32C2×C4×Dic53C2212yescomplemented
33#13133124C2211yescharacteristic, complemented
34C22×D4×C53C2211yescharacteristic
35C2×C4⋊C4×C53C2211yescharacteristic, complemented
36C2×C22⋊C4×C53C2212yescomplemented
37C2×C22⋊C4×C53C2212yescomplemented
38#95×C53C2211yescharacteristic, complemented
39C24×C53C2312yes—
40C23×D53—24no—
41C23×D53—24no—
42C23×D53—24no—
43C23×D53—24no—
44C23×D53—24no—
45C23×D53—42no—
46C23×D53—42no—
47C23×D53C2312yes—
48C23×D53C2311yescharacteristic
49C2×D4×C53—24no—
50C2×D4×C53—22no—
51C22×C4×C53C2312yes—
52C22×C4×C53C2311yescharacteristic
53C22×C4×C53C2311yescharacteristic
54C22×C4×C53C2311yescharacteristic
55C2×C22⋊D53—24no—
56C2×C22⋊D53—24no—
57C2×C22⋊D53D418yes—
58C22×Dic53C2312yes—
59C22×Dic53C2311yescharacteristic
60C22×Dic53C2311yescharacteristic
61C22×Dic53C2311yescharacteristic
62D4×D53—232no—
63C2×D212—24no—
64C2×D212—22no—
65C2×C4×D53—24no—
66C2×C4×D53D414yes—
67C2×Dic106—22no—
68C2×Dic106—24no—
69C4⋊C4×C53—22no—
70C22⋊C4×C53—24no—
71C22⋊C4×C53—42no—
72C22⋊Dic53—24no—
73C22⋊Dic53—42no—
74#11719921—22no—
75#11719921—22no—
76#11719921—24no—
77#11719921—24no—
78#11719921—24no—
79#11719921—42no—
80#11719921—42no—
81#11719919—22no—
82#11719919—22no—
83C4×Dic53—24no—
84C23×C53—22no—
85C23×C53—22no—
86C23×C53—22no—
87C23×C53—22no—
88C23×C53—22no—
89C23×C53—22no—
90C23×C53C2×D414yes—
91C23×C53C2411yescharacteristic
92C22×D53—22no—
93C22×D53—21no—
94C22×D53—21no—
95C22×D53—216no—
96C22×D53—22no—
97C22×D53—22no—
98C22×D53—22no—
99C22×D53—22no—
100C22×D53—22no—
101C22×D53—22no—
102C22×D53—48no—
103C22×D53—48no—
104C22×D53—48no—
105C22×D53—48no—
106C22×D53—48no—
107C22×D53—48no—
108C22×D53C2×D414yes—
109C22×D53C2×D412yes—
110C22×D53#4114yes—
111D4×C53—28no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—21no—
115C2×C4×C53—22no—
116C2×C4×C53—21no—
117C2×C4×C53—22no—
118C2×C4×C53—21no—
119C2×C4×C53—22no—
120C2×C4×C53—21no—
121C2×C4×C53C2×D412yes—
122C22⋊D53—216no—
123C2×Dic53—21no—
124C2×Dic53—22no—
125C2×Dic53—22no—
126C2×Dic53—21no—
127C2×Dic53—22no—
128C2×Dic53—21no—
129C2×Dic53—22no—
130C2×Dic53—21no—
131C2×Dic53C2×D414yes—
132C2×Dic53C2×D412yes—
133D212—28no—
134C4×D53—28no—
135Dic106—44no—
136C22×C53—24no—
137C22×C53—24no—
138C22×C53—24no—
139C22×C53—42no—
140C22×C53—42no—
141C22×C53—42no—
142C22×C53—42no—
143C22×C53C22×D411yescharacteristic
144C22×C53C22×D412yes—
145C22×C53C2×#4112yes—
146C22×C53H(5,𝔽2)11yescharacteristic
147C22×C53H(5,𝔽2)11yescharacteristic
148C2×D53—22no—
149C2×D53—22no—
150C2×D53—24no—
151C2×D53—24no—
152C2×D53—24no—
153C2×D53—24no—
154C2×D53—24no—
155C2×D53—432no—
156C2×D53—42no—
157C2×D53—42no—
158C2×D53—42no—
159C2×D53—42no—
160C2×D53#12414yescomplemented
161C4×C53—22no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53—42no—
165Dic53—41no—
166Dic53—41no—
167Dic53—42no—
168Dic53#12014yes—
169#2435—531nomaximal, Sylow
170C2×C53—44no—
171C2×C53C2×#12011yescharacteristic
172C2×C53C2×#12411yescharacteristic
173C2×C53#53511yescharacteristic
174C2×C53#54612yes—
175C2×C53#54612yes—
176D53—24no—
177D53—44no—
178C23×D4—531no—
179C2×#124—532no—
180C2×C22⋊Q8—531no—
181#394—531no—
182#394—531no—
183#394—531no—
184#394D5311yescharacteristic, complemented
185#390—532no—
186#379—532no—
187#379—532no—
188C4×C22⋊C4—531no—
189C53#243511yescharacteristic, complemented, Sylow
190C25—532no—
191C22×Q8—531no—
192C22×D4—532no—
193C22×D4—531no—
194C22×D4—1064no—
195C22×D4—1068no—
196C22×D4C2×D5311yescharacteristic
197C23×C4—531no—
198#124—1068no—
199C22⋊Q8—1064no—
200C2×C4⋊C4—531no—
201C2×C4⋊C4—531no—
202C2×C4⋊C4C2×D5311yescharacteristic, complemented
203C2×C22⋊C4—532no—
204C2×C22⋊C4—531no—
205C2×C22⋊C4—531no—
206C2×C22⋊C4—532no—
207C2×C22⋊C4—532no—
208C2×C22⋊C4—532no—
209C2×C22⋊C4—532no—
210C2×C22⋊C4—532no—
211C2×C22⋊C4—532no—
212C2×C22⋊C4C2×D5312yescomplemented
213C2×C22⋊C4C2×D5312yescomplemented
214C2×C42—532no—
215#95—531no—
216#95C2×D5311yescharacteristic, complemented
217C24—531no—
218C24—532no—
219C24—1064no—
220C24—1064no—
221C24—1064no—
222C24—1064no—
223C24—1064no—
224C24—2122no—
225C24—2122no—
226C24C22×D5312yes—
227C2×Q8—1062no—
228C2×Q8—1064no—
229C2×D4—24no—
230C2×D4—22no—
231C2×D4—538no—
232C2×D4—1064no—
233C2×D4—1064no—
234C2×D4—1064no—
235C2×D4—10632no—
236C2×D4—1062no—
237C22×C4—531no—
238C22×C4—534no—
239C22×C4—532no—
240C22×C4—531no—
241C22×C4—531no—
242C22×C4—1064no—
243C22×C4C22×D5312yes—
244C22×C4C22×D5311yescharacteristic
245C22×C4C22×D5311yescharacteristic
246C22×C4C22×D5311yescharacteristic
247C4⋊C4—22no—
248C4⋊C4—1062no—
249C4⋊C4—1062no—
250C22⋊C4—24no—
251C22⋊C4—42no—
252C22⋊C4—1062no—
253C22⋊C4—1062no—
254C22⋊C4—1064no—
255C22⋊C4—1064no—
256C22⋊C4—1064no—
257C22⋊C4—1064no—
258C22⋊C4—2122no—
259C22⋊C4—2122no—
260C22⋊C4—2122no—
261C42—1064no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—532no—
269C23—534no—
270C23—534no—
271C23—10616no—
272C23—1062no—
273C23—1061no—
274C23—1061no—
275C23—1062no—
276C23—1062no—
277C23—1062no—
278C23—1062no—
279C23—1062no—
280C23—1062no—
281C23—2128no—
282C23—2128no—
283C23—2128no—
284C23—2128no—
285C23—2128no—
286C23—2128no—
287C23D4×D5314yes—
288C23C23×D5311yescharacteristic
289Q8—2124no—
290D4—28no—
291D4—1068no—
292D4—10616no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—534no—
303C2×C4—532no—
304C2×C4—1068no—
305C2×C4—1062no—
306C2×C4—1062no—
307C2×C4—1061no—
308C2×C4—1061no—
309C2×C4—1062no—
310C2×C4—1061no—
311C2×C4—1062no—
312C2×C4—1061no—
313C2×C4D4×D5312yes—
314C22—24no—
315C22—24no—
316C22—24no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—534no—
322C22—1064no—
323C22—1062no—
324C22—1062no—
325C22—1064no—
326C22—1064no—
327C22—1064no—
328C22—1064no—
329C22—21232no—
330C22—2122no—
331C22—2122no—
332C22—2122no—
333C22—2122no—
334C22C2×D4×D5312yes—
335C22C2×D4×D5311yescharacteristic
336C22#41×D5312yes—
337C22#1313326211yescharacteristic
338C22#1313326311yescharacteristic
339C4—22no—
340C4—41no—
341C4—41no—
342C4—42no—
343C4—534no—
344C4—2122no—
345C4—2121no—
346C4—2121no—
347C2—44no—
348C2—1064no—
349C2—2124no—
350C2#120×D5311yescharacteristic
351C2#124×D5311yescharacteristic
352C2#1681118911yescharacteristic
353C2#1681126212yes—
354C2#1681126512yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#394C2
hypercenter#394D53
hypocenterC53#2435
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#13133203C2semidirect
3. C2×#13133195C2semidirect
4. #16810541C2semidirect
5. #16810540C2semidirect
6. #16810537C2semidirect
7. #16810523C2semidirect
8. #16810478C2semidirect
9. #16810478C2semidirect
10. #16810470C2semidirect
11. C53×#394C2semidirect
13. C22×Dic106C22semidirect
21. C2×#11719919C22semidirect
22. C2×#11719919C22semidirect
23. C2×C22⋊Dic53C22semidirect
24. C2×C22⋊Dic53C22semidirect
28. C2×#11719921C22semidirect
29. C2×#11719921C22semidirect
30. C2×#11719921C22semidirect
31. C2×#11719921C22semidirect
32. C2×C4×Dic53C22semidirect
33. #13133124C22semidirect
35. C2×C4⋊C4×C53C22semidirect
36. C2×C22⋊C4×C53C22semidirect
37. C2×C22⋊C4×C53C22semidirect
38. #95×C53C22semidirect
160. C2×D53#124semidirect
184. #394D53semidirect
189. C53#2435semidirect
202. C2×C4⋊C4C2×D53semidirect
212. C2×C22⋊C4C2×D53semidirect
213. C2×C22⋊C4C2×D53semidirect
216. #95C2×D53semidirect