Finite Group Databasebeta

← #20280783

Subgroups — #20280783

Order 6784.

442 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16810557C211yesmaximal, characteristic, complemented
2#16810552C211yesmaximal, characteristic, complemented
3#16810550C211yesmaximal, characteristic, complemented
4#16810541C211yesmaximal, characteristic
5#16810541C211yesmaximal, characteristic
6#16810536C211yesmaximal, characteristic, complemented
7#16810536C211yesmaximal, characteristic, complemented
8#16810527C211yesmaximal, characteristic, complemented
9#16810515C211yesmaximal, characteristic, complemented
10#16810515C211yesmaximal, characteristic, complemented
11#16810506C211yesmaximal, characteristic, complemented
12#16810501C211yesmaximal, characteristic
13#16810497C211yesmaximal, characteristic, complemented
14#16810470C211yesmaximal, characteristic, complemented
15C53×#401C211yesmaximal, characteristic, complemented
16C22×Dic106C2211yescharacteristic
17C22×D212C2211yescharacteristic
18C22×C4×D53C2211yescharacteristic
19C2×C4⋊Dic53C2211yescharacteristic
20C2×C4⋊Dic53C2211yescharacteristic
21C2×#11719919C2211yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×#11719921C2211yescharacteristic
25C2×#11719921C2211yescharacteristic
26C2×#11719921C2211yescharacteristic
27C2×#11719921C2211yescharacteristic
28C2×#11719921C2211yescharacteristic
29C2×#11719921C2211yescharacteristic
30C2×#11719921C2211yescharacteristic
31C2×#11719921C2211yescharacteristic
32C2×#11719921C2211yescharacteristic
33C2×#11719921C2211yescharacteristic
34C2×#11719921C2211yescharacteristic
35C2×C4×Dic53C2211yescharacteristic
36C2×C4×Dic53C2211yescharacteristic
37#13133124C2211yescharacteristic
38#13133124C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44#95×C53C2211yescharacteristic
45#95×C53C2211yescharacteristic
46#95×C53C2211yescharacteristic
47#95×C53C2211yescharacteristic
48#95×C53C2211yescharacteristic
49#95×C53C2211yescharacteristic
50#95×C53C2211yescharacteristic
51C23×D53C2311yescharacteristic
52C23×D53C2311yescharacteristic
53C22×C4×C53C2311yescharacteristic
54C22×C4×C53C2311yescharacteristic
55C22×C4×C53C2311yescharacteristic
56C22×C4×C53C2311yescharacteristic
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×Dic53C2311yescharacteristic
61C22×Dic53C2311yescharacteristic
62C22×Dic53C2311yescharacteristic
63C22×Dic53C2311yescharacteristic
64C22×Dic53C2311yescharacteristic
65C22×Dic53C2311yescharacteristic
66C2×D212—22no—
67C2×D212—41no—
68C2×D212—41no—
69C2×C4×D53—22no—
70C2×C4×D53—41no—
71C2×C4×D53—41no—
72C2×Dic106—22no—
73C2×Dic106—41no—
74C2×Dic106—41no—
75#11719921—22no—
76#11719921—22no—
77#11719921—22no—
78#11719921—22no—
79#11719921—22no—
80#11719921—41no—
81#11719921—41no—
82#11719921—41no—
83#11719921—41no—
84#11719921—41no—
85#11719921—41no—
86C4⋊Dic53—22no—
87C4⋊Dic53—41no—
88#11719919—41no—
89#11719919—41no—
90#11719919—41no—
91C4×Dic53—41no—
92C4×Dic53—41no—
93C23×C53C2411yescharacteristic
94C22×D53—21no—
95C22×D53—21no—
96C22×D53—21no—
97C22×D53—21no—
98C22×D53—21no—
99C22×D53—21no—
100C22×D53—21no—
101C22×D53—21no—
102C22×D53—21no—
103C22×D53—21no—
104C22×D53—21no—
105C22×D53—21no—
106C22×D53—21no—
107C22×D53#4112yes—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—21no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53#4112yes—
146C2×Dic53#4112yes—
147D212—44no—
148C4×D53—44no—
149Dic106—44no—
150C22×C53C2×#4111yescharacteristic
151C22×C53C2×#4111yescharacteristic
152C22×C53C2×#4111yescharacteristic
153C22×C53H(5,𝔽2)11yescharacteristic
154C22×C53H(5,𝔽2)11yescharacteristic
155C22×C53H(5,𝔽2)11yescharacteristic
156C22×C532−1+411yescharacteristic
157C2×D53—22no—
158C2×D53—22no—
159C2×D53—22no—
160C2×D53—41no—
161C2×D53—41no—
162C2×D53—41no—
163C2×D53—41no—
164C2×D53—41no—
165C2×D53—41no—
166C2×D53—41no—
167C2×D53—41no—
168C2×D53—41no—
169C2×D53—41no—
170C2×D53—41no—
171C4×C53—41no—
172C4×C53—41no—
173C4×C53—41no—
174C4×C53—41no—
175C4×C53—41no—
176C4×C53—41no—
177C4×C53—41no—
178Dic53—22no—
179Dic53—22no—
180Dic53—41no—
181Dic53—41no—
182Dic53—41no—
183Dic53—41no—
184#2653—531nomaximal, Sylow
185C2×C53#52911yescharacteristic
186C2×C53#53811yescharacteristic
187C2×C53#53811yescharacteristic
188C2×C53#53811yescharacteristic
189C2×C53#53911yescharacteristic
190C2×C53#54211yescharacteristic
191C2×C53#54211yescharacteristic
192D53—42no—
193D53—81no—
194#401D5311yescharacteristic, complemented
195#398—531no—
196#397—531no—
197#397—531no—
198#394—531no—
199#394—531no—
200#393—531no—
201#393—531no—
202#391—531no—
203#388—531no—
204#388—531no—
205#387—531no—
206#386—531no—
207#385—531no—
208C4×C22⋊C4—531no—
209C53#265311yescharacteristic, complemented, Sylow
210C22×Q8—531no—
211C22×D4—531no—
212C23×C4—531no—
213C2×C4⋊C4—531no—
214C2×C4⋊C4—531no—
215C2×C4⋊C4—531no—
216C2×C4⋊C4—531no—
217C2×C4⋊C4—531no—
218C2×C22⋊C4—531no—
219C2×C22⋊C4—531no—
220C2×C22⋊C4—531no—
221C2×C22⋊C4—531no—
222C2×C22⋊C4—531no—
223C2×C22⋊C4—531no—
224C2×C22⋊C4—531no—
225C2×C22⋊C4—531no—
226C2×C22⋊C4—531no—
227C2×C22⋊C4—531no—
228C2×C22⋊C4—531no—
229C2×C42—531no—
230C2×C42—531no—
231#95—531no—
232#95—531no—
233#95—531no—
234#95—531no—
235#95—531no—
236#95—531no—
237#95—531no—
238#95C2×D5311yescharacteristic
239#95C2×D5311yescharacteristic
240#95C2×D5311yescharacteristic
241#95C2×D5311yescharacteristic
242#95C2×D5311yescharacteristic
243#95C2×D5311yescharacteristic
244#95C2×D5311yescharacteristic
245C24—531no—
246C24—531no—
247C2×Q8—1062no—
248C2×Q8—2121no—
249C2×Q8—2121no—
250C2×D4—1062no—
251C2×D4—2121no—
252C2×D4—2121no—
253C22×C4—531no—
254C22×C4—531no—
255C22×C4—531no—
256C22×C4—531no—
257C22×C4—531no—
258C22×C4—531no—
259C22×C4—1062no—
260C22×C4—2121no—
261C22×C4—2121no—
262C22×C4C22×D5311yescharacteristic
263C22×C4C22×D5311yescharacteristic
264C22×C4C22×D5311yescharacteristic
265C22×C4C22×D5311yescharacteristic
266C22×C4C22×D5311yescharacteristic
267C22×C4C22×D5311yescharacteristic
268C22×C4C22×D5311yescharacteristic
269C4⋊C4—1062no—
270C4⋊C4—2121no—
271C4⋊C4—2121no—
272C4⋊C4—2121no—
273C4⋊C4—2121no—
274C22⋊C4—1062no—
275C22⋊C4—1062no—
276C22⋊C4—1062no—
277C22⋊C4—1062no—
278C22⋊C4—1062no—
279C22⋊C4—2121no—
280C22⋊C4—2121no—
281C22⋊C4—2121no—
282C22⋊C4—2121no—
283C22⋊C4—2121no—
284C22⋊C4—2121no—
285C42—2121no—
286C42—2121no—
287C23—532no—
288C23—1061no—
289C23—1061no—
290C23—1061no—
291C23—1061no—
292C23—1061no—
293C23—1061no—
294C23—1061no—
295C23—1061no—
296C23—1061no—
297C23—1061no—
298C23—1061no—
299C23—1061no—
300C23—1061no—
301C23C23×D5311yescharacteristic
302Q8—2124no—
303D4—2124no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—532no—
326C2×C4—532no—
327C2×C4—1061no—
328C2×C4—1061no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—1061no—
332C2×C4—1061no—
333C2×C4—1061no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—1061no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—1061no—
341C2×C4—1061no—
342C2×C4—1061no—
343C2×C4—2124no—
344C22—1062no—
345C22—1062no—
346C22—1062no—
347C22—2121no—
348C22—2121no—
349C22—2121no—
350C22—2121no—
351C22—2121no—
352C22—2121no—
353C22—2121no—
354C22—2121no—
355C22—2121no—
356C22—2121no—
357C22—2121no—
358C22#41×D5311yescharacteristic
359C22#41×D5311yescharacteristic
360C22#41×D5311yescharacteristic
361C22#1313326311yescharacteristic
362C22#1313326311yescharacteristic
363C22#1313326311yescharacteristic
364C22#1313326411yescharacteristic
365C4—41no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—1062no—
373C4—1062no—
374C4—2121no—
375C4—2121no—
376C4—2121no—
377C4—2121no—
378C2—4241no—
379C2—2122no—
380C2#1681115611yescharacteristic
381C2#1681121211yescharacteristic
382C2#1681121211yescharacteristic
383C2#1681121211yescharacteristic
384C2#1681121411yescharacteristic
385C2#1681123911yescharacteristic
386C2#1681123911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#401C2
hypercenter#401D53
hypocenterC53#2653
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #401

Complements

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

NH ≅ G/Ntype
1. #16810557C2semidirect
2. #16810552C2semidirect
3. #16810550C2semidirect
6. #16810536C2semidirect
7. #16810536C2semidirect
8. #16810527C2semidirect
9. #16810515C2semidirect
10. #16810515C2semidirect
11. #16810506C2semidirect
13. #16810497C2semidirect
14. #16810470C2semidirect
15. C53×#401C2semidirect
194. #401D53semidirect
209. C53#2653semidirect