Finite Group Databasebeta

← #20280438

Subgroups — #20280438

Order 6784.

932 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 932 conjugacy classes of subgroups — too many to draw. Only its 153 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×#13133190C211yesmaximal, characteristic, complemented
3C2×#13133189C212yesmaximal, complemented
4#16810536C211yesmaximal, characteristic, complemented
5#16810535C211yesmaximal, characteristic, complemented
6#16810532C212yesmaximal, complemented
7#16810531C211yesmaximal, characteristic, complemented
8#16810531C212yesmaximal, complemented
9#16810482C211yesmaximal, characteristic, complemented
10#16810481C212yesmaximal, complemented
11C53×#392C211yesmaximal, characteristic, complemented
12C2×#11719947—28no—
13C2×#11719947—24no—
14C22×Dic106C2211yescharacteristic
15C22×D212C2211yescharacteristic
16C22×C22⋊D53C2212yes—
17C22×C22⋊D53C2211yescharacteristic
18C22×C22⋊D53C2211yescharacteristic
19C22×C22⋊D53C2212yes—
20C23×Dic53C2212yes—
21C22×C4×D53C2211yescharacteristic
22#13133190—24no—
23#13133189—28no—
24C2×#11719919C2212yes—
25C2×#11719919C2211yescharacteristic
26C2×C22⋊Dic53C2212yes—
27C2×C22⋊Dic53C2212yes—
28C2×#11719921C2211yescharacteristic
29C2×#11719921C2212yes—
30C2×#11719921C2212yes—
31#13133124C2211yescharacteristic
32#13133124C2212yes—
33#13133124C2212yes—
34C22×D4×C53C2211yescharacteristic
35C22×D4×C53C2212yes—
36C2×C22⋊C4×C53C2211yescharacteristic
37C2×C22⋊C4×C53C2212yes—
38#95×C53C2211yescharacteristic
39C24×C53C2311yescharacteristic
40C24×C53C2312yes—
41C23×D53C2311yescharacteristic
42C23×D53C2311yescharacteristic
43C2×D4×C53—24no—
44C2×D4×C53—24no—
45C2×D4×C53—22no—
46C2×D4×C53—42no—
47C2×D4×C53—42no—
48C22×C4×C53C2312yes—
49C22×C4×C53C2311yescharacteristic
50C22×C4×C53C2311yescharacteristic
51C2×C22⋊D53—24no—
52C2×C22⋊D53—24no—
53C2×C22⋊D53—24no—
54C2×C22⋊D53—22no—
55C2×C22⋊D53—24no—
56C2×C22⋊D53—24no—
57C2×C22⋊D53—22no—
58C2×C22⋊D53—42no—
59C2×C22⋊D53—42no—
60C2×C22⋊D53—42no—
61C22×Dic53—24no—
62C22×Dic53—24no—
63C22×Dic53D418yescomplemented
64C22×Dic53C2311yescharacteristic
65C22×Dic53C2312yes—
66C22×Dic53C2312yes—
67C22×Dic53C2311yescharacteristic
68#11719947—232no—
69C2×D212—22no—
70C2×D212—42no—
71C2×C4×D53—24no—
72C2×C4×D53D414yescomplemented
73C2×Dic106—22no—
74C2×Dic106—24no—
75C22⋊C4×C53—22no—
76C22⋊C4×C53—42no—
77C22⋊Dic53—24no—
78C22⋊Dic53—42no—
79#11719921—22no—
80#11719921—24no—
81#11719921—42no—
82#11719919—24no—
83#11719919—22no—
84C23×C53—22no—
85C23×C53—22no—
86C23×C53—22no—
87C23×C53—22no—
88C23×C53—22no—
89C23×C53—22no—
90C23×C53—21no—
91C23×C53—22no—
92C23×C53—22no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53C2×D414yes—
96C23×C53C2411yescharacteristic
97C22×D53—22no—
98C22×D53—22no—
99C22×D53—21no—
100C22×D53—21no—
101C22×D53—21no—
102C22×D53—22no—
103C22×D53—22no—
104C22×D53—21no—
105C22×D53—21no—
106C22×D53C2×D412yes—
107D4×C53—28no—
108D4×C53—48no—
109C2×C4×C53—22no—
110C2×C4×C53—21no—
111C2×C4×C53—22no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—22no—
115C2×C4×C53C2×D412yes—
116C22⋊D53—216no—
117C22⋊D53—44no—
118C22⋊D53—48no—
119C22⋊D53—44no—
120C2×Dic53—22no—
121C2×Dic53—22no—
122C2×Dic53—216no—
123C2×Dic53—21no—
124C2×Dic53—22no—
125C2×Dic53—22no—
126C2×Dic53—22no—
127C2×Dic53—22no—
128C2×Dic53C2×D414yes—
129C2×Dic53C2×D414yes—
130C2×Dic53C2×D412yes—
131D212—44no—
132C4×D53—28no—
133Dic106—28no—
134C22×C53—24no—
135C22×C53—24no—
136C22×C53—24no—
137C22×C53—42no—
138C22×C53—42no—
139C22×C53—42no—
140C22×C53—42no—
141C22×C53—41no—
142C22×C53—41no—
143C22×C53—42no—
144C22×C53—42no—
145C22×C53—41no—
146C22×C53C22×D411yescharacteristic
147C22×C53C22×D412yes—
148C22×C53H(5,𝔽2)12yes—
149C22×C53H(5,𝔽2)11yescharacteristic
150C22×C532−1+411yescharacteristic
151C2×D53—22no—
152C2×D53—22no—
153C2×D53—22no—
154C2×D53—42no—
155C2×D53—42no—
156C2×D53—41no—
157C2×D53—41no—
158C2×D53—42no—
159C2×D53—42no—
160C2×D53—41no—
161C4×C53—22no—
162C4×C53—41no—
163C4×C53—42no—
164Dic53—24no—
165Dic53—41no—
166Dic53—42no—
167Dic53#12014yes—
168#2615—531nomaximal, Sylow
169C2×C53—44no—
170C2×C53—81no—
171C2×C53C2×#12011yescharacteristic
172C2×C53#53511yescharacteristic
173C2×C53#53512yes—
174C2×C53#53712yes—
175C2×C53#53711yescharacteristic
176D53—42no—
177D53—81no—
178C22×#41—531no—
179C2×#121—532no—
180C2×#121—531no—
181#393—531no—
182#393—531no—
183#392—532no—
184#392—531no—
185#392—532no—
186#392D5311yescharacteristic, complemented
187#380—532no—
188#380—531no—
189C53#261511yescharacteristic, complemented, Sylow
190C2×#41—1068no—
191C2×#41—1064no—
192C22×Q8—531no—
193C22×D4—531no—
194C22×D4—532no—
195C22×D4—531no—
196C22×D4—532no—
197C22×D4—531no—
198C22×D4C2×D5311yescharacteristic
199C22×D4C2×D5312yes—
200C23×C4—532no—
201C23×C4—531no—
202#121—1064no—
203#121—1068no—
204C2×C4⋊C4—531no—
205C2×C4⋊C4—532no—
206C2×C22⋊C4—532no—
207C2×C22⋊C4—532no—
208C2×C22⋊C4—531no—
209C2×C22⋊C4—532no—
210C2×C22⋊C4—532no—
211C2×C22⋊C4C2×D5311yescharacteristic
212C2×C22⋊C4C2×D5312yes—
213#95—532no—
214#95—532no—
215#95—531no—
216#95C2×D5311yescharacteristic
217C24—531no—
218C24—531no—
219C24C22×D5311yescharacteristic
220C24C22×D5312yes—
221#41—10632no—
222C2×Q8—1064no—
223C2×Q8—1062no—
224C2×D4—24no—
225C2×D4—24no—
226C2×D4—22no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—1062no—
230C2×D4—1064no—
231C2×D4—1062no—
232C2×D4—1064no—
233C2×D4—1064no—
234C2×D4—1064no—
235C2×D4—1062no—
236C2×D4—1064no—
237C2×D4—2122no—
238C2×D4—2122no—
239C2×D4—2122no—
240C2×D4—2122no—
241C22×C4—531no—
242C22×C4—538no—
243C22×C4—532no—
244C22×C4—534no—
245C22×C4—532no—
246C22×C4—531no—
247C22×C4—1064no—
248C22×C4—1064no—
249C22×C4—1064no—
250C22×C4C22×D5311yescharacteristic
251C22×C4C22×D5312yes—
252C22×C4C22×D5311yescharacteristic
253C4⋊C4—1062no—
254C4⋊C4—1064no—
255C22⋊C4—22no—
256C22⋊C4—42no—
257C22⋊C4—1064no—
258C22⋊C4—1064no—
259C22⋊C4—1062no—
260C22⋊C4—2122no—
261C22⋊C4—2122no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—21no—
269C23—22no—
270C23—21no—
271C23—22no—
272C23—21no—
273C23—532no—
274C23—1062no—
275C23—1061no—
276C23—1062no—
277C23—1061no—
278C23—1061no—
279C23—1062no—
280C23—1062no—
281C23—1061no—
282C23—1061no—
283C23D4×D5314yes—
284C23C23×D5311yescharacteristic
285Q8—1068no—
286D4—28no—
287D4—48no—
288D4—10616no—
289D4—2124no—
290D4—2124no—
291D4—2128no—
292D4—2124no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—532no—
300C2×C4—534no—
301C2×C4—534no—
302C2×C4—1068no—
303C2×C4—10616no—
304C2×C4—1062no—
305C2×C4—1062no—
306C2×C4—1061no—
307C2×C4—1062no—
308C2×C4—1062no—
309C2×C4—1062no—
310C2×C4—1062no—
311C2×C4D4×D5312yes—
312C22—24no—
313C22—24no—
314C22—24no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—41no—
320C22—41no—
321C22—42no—
322C22—42no—
323C22—41no—
324C22—1062no—
325C22—1062no—
326C22—1062no—
327C22—2121no—
328C22—2121no—
329C22—2122no—
330C22—2122no—
331C22—2121no—
332C22—2122no—
333C22—2122no—
334C22C2×D4×D5311yescharacteristic
335C22C2×D4×D5312yes—
336C22#1313326212yes—
337C22#1313326211yescharacteristic
338C22#1313326411yescharacteristic
339C4—22no—
340C4—41no—
341C4—42no—
342C4—534no—
343C4—1064no—
344C4—2121no—
345C4—2122no—
346C2—44no—
347C2—81no—
348C2—4241no—
349C2—2122no—
350C2#120×D5311yescharacteristic
351C2#1681118611yescharacteristic
352C2#1681118812yes—
353C2#1681120012yes—
354C2#1681120211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#392C2
hypercenter#392D53
hypocenterC53#2615
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×#13133190C2semidirect
3. C2×#13133189C2semidirect
4. #16810536C2semidirect
5. #16810535C2semidirect
6. #16810532C2semidirect
7. #16810531C2semidirect
8. #16810531C2semidirect
9. #16810482C2semidirect
10. #16810481C2semidirect
11. C53×#392C2semidirect
63. C22×Dic53D4semidirect
72. C2×C4×D53D4semidirect
186. #392D53semidirect
189. C53#2615semidirect