Finite Group Databasebeta

← #20278029

Subgroups — #20278029

Order 6784.

1056 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 159 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, complemented
2C2×#13133185C212yesmaximal, complemented
3C2×C22⋊C4×D53C211yesmaximal, characteristic, complemented
4#16810536C211yesmaximal, characteristic, complemented
5#16810535C211yesmaximal, characteristic, complemented
6#16810531C211yesmaximal, characteristic, complemented
7#16810503C212yesmaximal, complemented
8#16810500C212yesmaximal, complemented
9#16810499C212yesmaximal, complemented
10#16810485C211yesmaximal, characteristic, complemented
11C53×#392C211yesmaximal, characteristic, complemented
12C24×D53C2211yescharacteristic
13C2×#11719947—28no—
14C2×#11719947—24no—
15C22×Dic106C2211yescharacteristic, complemented
16C22×C22⋊D53C2212yes—
17C22×C22⋊D53C2212yes—
18C23×Dic53C2212yescomplemented
19C22×C4×D53C2211yescharacteristic
20C22×C4×D53C2211yescharacteristic
21#13133185—28no—
22C2×C22⋊Dic53C2211yescharacteristic
23C2×C22⋊Dic53C2212yes—
24C2×C22⋊Dic53C2212yescomplemented
25C2×#11719921C2212yes—
26C2×#11719921C2211yescharacteristic
27C2×#11719921C2212yes—
28C2×#11719921C2211yescharacteristic
29C22⋊C4×D53—24no—
30#13133124C2212yescomplemented
31#13133124C2212yescomplemented
32#13133124C2212yescomplemented
33#13133124C2211yescharacteristic, complemented
34C22×D4×C53C2212yes—
35C22×D4×C53C2211yescharacteristic, complemented
36C2×C22⋊C4×C53C2211yescharacteristic
37C2×C22⋊C4×C53C2212yescomplemented
38#95×C53C2211yescharacteristic, complemented
39C24×C53C2312yes—
40C24×C53C2311yescharacteristic
41C23×D53—22no—
42C23×D53—22no—
43C23×D53—22no—
44C23×D53—24no—
45C23×D53—42no—
46C23×D53C2311yescharacteristic
47C23×D53C2311yescharacteristic
48C2×D4×C53—24no—
49C2×D4×C53—24no—
50C2×D4×C53—24no—
51C2×D4×C53—22no—
52C2×D4×C53—42no—
53C22×C4×C53C2312yes—
54C22×C4×C53C2311yescharacteristic
55C22×C4×C53C2311yescharacteristic
56C2×C22⋊D53—24no—
57C2×C22⋊D53—24no—
58C2×C22⋊D53—24no—
59C2×C22⋊D53—24no—
60C2×C22⋊D53—42no—
61C2×C22⋊D53D418yescomplemented
62C22×Dic53—24no—
63C22×Dic53—24no—
64C22×Dic53—24no—
65C22×Dic53C2312yes—
66C22×Dic53C2311yescharacteristic
67C22×Dic53C2311yescharacteristic
68C22×Dic53C2312yes—
69#11719947—232no—
70C2×C4×D53—24no—
71C2×C4×D53—22no—
72C2×C4×D53—42no—
73C2×C4×D53D414yescomplemented
74C2×Dic106—22no—
75C2×Dic106—24no—
76C22⋊C4×C53—22no—
77C22⋊C4×C53—42no—
78C22⋊Dic53—22no—
79C22⋊Dic53—24no—
80C22⋊Dic53—42no—
81#11719921—24no—
82#11719921—22no—
83#11719921—22no—
84#11719921—24no—
85C23×C53—22no—
86C23×C53—22no—
87C23×C53—22no—
88C23×C53—21no—
89C23×C53—22no—
90C23×C53—22no—
91C23×C53—21no—
92C23×C53—22no—
93C23×C53—22no—
94C23×C53—22no—
95C23×C53C2×D414yes—
96C23×C53#4112yescomplemented
97C23×C53C2411yescharacteristic
98C22×D53—21no—
99C22×D53—21no—
100C22×D53—22no—
101C22×D53—28no—
102C22×D53—22no—
103C22×D53—21no—
104C22×D53—21no—
105C22×D53—22no—
106C22×D53—48no—
107C22×D53—48no—
108C22×D53—44no—
109C22×D53—44no—
110C22×D53C2×D412yes—
111C22×D53C2×D414yes—
112C22×D53#4112yescomplemented
113D4×C53—28no—
114D4×C53—48no—
115C2×C4×C53—22no—
116C2×C4×C53—21no—
117C2×C4×C53—22no—
118C2×C4×C53—22no—
119C2×C4×C53—22no—
120C2×C4×C53—22no—
121C2×C4×C53C2×D412yes—
122C22⋊D53—216no—
123C22⋊D53—48no—
124C2×Dic53—216no—
125C2×Dic53—22no—
126C2×Dic53—22no—
127C2×Dic53—22no—
128C2×Dic53—21no—
129C2×Dic53—22no—
130C2×Dic53—22no—
131C2×Dic53—22no—
132C2×Dic53—22no—
133C2×Dic53C2×D414yes—
134C2×Dic53C2×D412yes—
135C4×D53—28no—
136C4×D53—44no—
137Dic106—28no—
138C22×C53—24no—
139C22×C53—22no—
140C22×C53—24no—
141C22×C53—24no—
142C22×C53—24no—
143C22×C53—41no—
144C22×C53—42no—
145C22×C53—41no—
146C22×C53—42no—
147C22×C53—42no—
148C22×C53—42no—
149C22×C53—42no—
150C22×C53C22×D411yescharacteristic
151C22×C53C22×D412yes—
152C22×C53C2×#4111yescharacteristic
153C22×C53C2×#4111yescharacteristic
154C22×C53H(5,𝔽2)12yes—
155C2×D53—22no—
156C2×D53—22no—
157C2×D53—24no—
158C2×D53—24no—
159C2×D53—22no—
160C2×D53—24no—
161C2×D53—416no—
162C2×D53—41no—
163C2×D53—41no—
164C2×D53—42no—
165C2×D53#12014yes—
166C4×C53—22no—
167C4×C53—41no—
168C4×C53—42no—
169Dic53—24no—
170Dic53—22no—
171Dic53—41no—
172Dic53—42no—
173#2404—531nomaximal, Sylow
174C2×C53—42no—
175C2×C53—44no—
176C2×C53C2×#12011yescharacteristic
177C2×C53#52511yescharacteristic
178C2×C53#53511yescharacteristic
179C2×C53#54612yes—
180C2×C53#54612yes—
181D53—24no—
182D53—42no—
183C22×#41—531no—
184C2×#120—532no—
185C22×C22⋊C4—531no—
186#393—531no—
187#393—531no—
188#392—531no—
189#392D5311yescharacteristic, complemented
190#386—532no—
191#386—532no—
192#386—532no—
193#381—531no—
194C53#240411yescharacteristic, complemented, Sylow
195C25—531no—
196C2×#41—1064no—
197C2×#41—1068no—
198C22×Q8—531no—
199C22×D4—532no—
200C22×D4—532no—
201C22×D4C2×D5312yes—
202C22×D4C2×D5311yescharacteristic, complemented
203C23×C4—531no—
204C23×C4—532no—
205C23×C4—531no—
206#120—1068no—
207C2×C22⋊C4—531no—
208C2×C22⋊C4—532no—
209C2×C22⋊C4—531no—
210C2×C22⋊C4—531no—
211C2×C22⋊C4—532no—
212C2×C22⋊C4—532no—
213C2×C22⋊C4—532no—
214C2×C22⋊C4—1064no—
215C2×C22⋊C4C2×D5311yescharacteristic
216C2×C22⋊C4C2×D5312yescomplemented
217#95—532no—
218#95—531no—
219#95—532no—
220#95—532no—
221#95C2×D5311yescharacteristic, complemented
222C24—531no—
223C24—531no—
224C24—1062no—
225C24—1062no—
226C24—1064no—
227C24—1062no—
228C24—2122no—
229C24C22×D5312yes—
230C24C22×D5311yescharacteristic
231#41—10632no—
232C2×Q8—1064no—
233C2×Q8—1062no—
234C2×D4—24no—
235C2×D4—24no—
236C2×D4—24no—
237C2×D4—22no—
238C2×D4—42no—
239C2×D4—538no—
240C2×D4—1064no—
241C2×D4—1064no—
242C2×D4—1064no—
243C2×D4—1064no—
244C2×D4—2122no—
245C22×C4—531no—
246C22×C4—532no—
247C22×C4—534no—
248C22×C4—532no—
249C22×C4—531no—
250C22×C4—1064no—
251C22×C4—1062no—
252C22×C4—1064no—
253C22×C4—1064no—
254C22×C4—1064no—
255C22×C4—2122no—
256C22×C4C22×D5312yes—
257C22×C4C22×D5311yescharacteristic
258C22×C4C22×D5311yescharacteristic
259C22⋊C4—22no—
260C22⋊C4—42no—
261C22⋊C4—1062no—
262C22⋊C4—1064no—
263C22⋊C4—1064no—
264C22⋊C4—1062no—
265C22⋊C4—1064no—
266C22⋊C4—1062no—
267C22⋊C4—2122no—
268C23—21no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—21no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—534no—
279C23—532no—
280C23—532no—
281C23—1068no—
282C23—1061no—
283C23—1061no—
284C23—1062no—
285C23—1061no—
286C23—1062no—
287C23—1061no—
288C23—1062no—
289C23—2128no—
290C23—2124no—
291C23—2128no—
292C23—2124no—
293C23D4×D5314yes—
294C23#1171994712yescomplemented
295C23C23×D5311yescharacteristic
296Q8—1068no—
297D4—28no—
298D4—48no—
299D4—10616no—
300D4—2128no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—532no—
308C2×C4—534no—
309C2×C4—1068no—
310C2×C4—10616no—
311C2×C4—1062no—
312C2×C4—1062no—
313C2×C4—1062no—
314C2×C4—1061no—
315C2×C4—1062no—
316C2×C4—1062no—
317C2×C4—1062no—
318C2×C4—1062no—
319C2×C4—2124no—
320C2×C4D4×D5312yes—
321C22—24no—
322C22—22no—
323C22—24no—
324C22—24no—
325C22—24no—
326C22—41no—
327C22—42no—
328C22—41no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—534no—
334C22—1062no—
335C22—1062no—
336C22—1064no—
337C22—1064no—
338C22—1062no—
339C22—1064no—
340C22—21216no—
341C22—2121no—
342C22—2121no—
343C22—2122no—
344C22C2×D4×D5311yescharacteristic
345C22C2×D4×D5312yes—
346C22#41×D5311yescharacteristic
347C22C2×#1171994711yescharacteristic
348C22#1313326212yes—
349C4—22no—
350C4—41no—
351C4—42no—
352C4—1062no—
353C4—1064no—
354C4—2121no—
355C4—2122no—
356C2—42no—
357C2—44no—
358C2—1064no—
359C2—2122no—
360C2#120×D5311yescharacteristic
361C2#1681113311yescharacteristic
362C2#1681118611yescharacteristic
363C2#1681126012yes—
364C2#1681126212yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#392C2
hypercenter#392D53
hypocenterC53#2404
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C22×#11719947C2semidirect
2. C2×#13133185C2semidirect
3. C2×C22⋊C4×D53C2semidirect
4. #16810536C2semidirect
5. #16810535C2semidirect
6. #16810531C2semidirect
7. #16810503C2semidirect
8. #16810500C2semidirect
9. #16810499C2semidirect
10. #16810485C2semidirect
11. C53×#392C2semidirect
15. C22×Dic106C22semidirect
18. C23×Dic53C22semidirect
24. C2×C22⋊Dic53C22semidirect
30. #13133124C22semidirect
31. #13133124C22semidirect
32. #13133124C22semidirect
33. #13133124C22semidirect
35. C22×D4×C53C22semidirect
37. C2×C22⋊C4×C53C22semidirect
38. #95×C53C22semidirect
61. C2×C22⋊D53D4semidirect
73. C2×C4×D53D4semidirect
96. C23×C53#41semidirect
112. C22×D53#41semidirect
189. #392D53semidirect
194. C53#2404semidirect
202. C22×D4C2×D53semidirect
216. C2×C22⋊C4C2×D53semidirect
221. #95C2×D53semidirect
294. C23#11719947semidirect