Finite Group Databasebeta

← #20282656

Subgroups — #20282656

Order 6784.

480 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16810564C211yesmaximal, characteristic, complemented
2#16810563C211yesmaximal, characteristic, complemented
3#16810536C211yesmaximal, characteristic, complemented
4#16810535C211yesmaximal, characteristic, complemented
5#16810534C211yesmaximal, characteristic, complemented
6#16810530C211yesmaximal, characteristic, complemented
7#16810527C211yesmaximal, characteristic, complemented
8#16810523C211yesmaximal, characteristic
9#16810500C211yesmaximal, characteristic
10#16810499C211yesmaximal, characteristic
11#16810498C211yesmaximal, characteristic, complemented
12#16810497C211yesmaximal, characteristic, complemented
13#16810497C211yesmaximal, characteristic, complemented
14#16810495C211yesmaximal, characteristic, complemented
15C53×#393C211yesmaximal, characteristic, complemented
16C22×Dic106C2211yescharacteristic
17C22×C22⋊D53C2211yescharacteristic
18C23×Dic53C2211yescharacteristic
19C22×C4×D53C2211yescharacteristic
20C2×C4⋊Dic53C2211yescharacteristic
21C2×C4⋊Dic53C2211yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×#11719919C2211yescharacteristic
25C2×C22⋊Dic53C2211yescharacteristic
26C2×C22⋊Dic53C2211yescharacteristic
27C2×#11719921C2211yescharacteristic
28C2×#11719921C2211yescharacteristic
29C2×#11719921C2211yescharacteristic
30C2×#11719921C2211yescharacteristic
31C2×#11719921C2211yescharacteristic
32C2×C4×Dic53C2211yescharacteristic
33#13133124C2211yescharacteristic
34#13133124C2211yescharacteristic
35#13133124C2211yescharacteristic
36#13133124C2211yescharacteristic
37#13133124C2211yescharacteristic
38#13133124C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44C22×Q8×C53C2211yescharacteristic
45C2×C22⋊C4×C53C2211yescharacteristic
46C2×C22⋊C4×C53C2211yescharacteristic
47C2×C22⋊C4×C53C2211yescharacteristic
48#95×C53C2211yescharacteristic
49#95×C53C2211yescharacteristic
50#95×C53C2211yescharacteristic
51C24×C53C2311yescharacteristic
52C23×D53C2311yescharacteristic
53C2×Q8×C53—22no—
54C2×Q8×C53—41no—
55C2×Q8×C53—41no—
56C22×C4×C53C2311yescharacteristic
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×C4×C53C2311yescharacteristic
62C2×C22⋊D53—41no—
63C2×C22⋊D53—41no—
64C2×C22⋊D53Q814yescomplemented
65C22×Dic53—41no—
66C22×Dic53—41no—
67C22×Dic53—41no—
68C22×Dic53C2311yescharacteristic
69C22×Dic53C2311yescharacteristic
70C22×Dic53C2311yescharacteristic
71C22×Dic53C2311yescharacteristic
72C22×Dic53C2311yescharacteristic
73C22×Dic53C2311yescharacteristic
74C22×Dic53C2311yescharacteristic
75C2×C4×D53—41no—
76C2×C4×D53—41no—
77C2×C4×D53—41no—
78C2×Dic106—22no—
79C2×Dic106—41no—
80C2×Dic106—41no—
81C22⋊C4×C53—22no—
82C22⋊C4×C53—41no—
83C22⋊C4×C53—41no—
84C22⋊Dic53—22no—
85C22⋊Dic53—41no—
86#11719921—22no—
87#11719921—22no—
88#11719921—41no—
89#11719921—41no—
90#11719921—41no—
91C4⋊Dic53—41no—
92C4⋊Dic53—41no—
93#11719919—41no—
94#11719919—41no—
95#11719919—41no—
96C4×Dic53—41no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53C2×Q812yes—
104C23×C53C2411yescharacteristic
105C22×D53—21no—
106C22×D53—21no—
107C22×D53—21no—
108C22×D53—21no—
109C22×D53—21no—
110C22×D53—21no—
111C22×D53C2×Q812yes—
112Q8×C53—44no—
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×C4×C53—21no—
130C2×C4×C53#4112yes—
131C22⋊D53—44no—
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—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—44no—
152C2×Dic53C2×Q812yes—
153C2×Dic53#4112yes—
154C4×D53—44no—
155Dic106—44no—
156C22×C53—22no—
157C22×C53—22no—
158C22×C53—22no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53—41no—
163C22×C53C22×Q811yescharacteristic
164C22×C53C2×#4111yescharacteristic
165C22×C53C2×#4111yescharacteristic
166C22×C53H(5,𝔽2)11yescharacteristic
167C22×C53H(5,𝔽2)11yescharacteristic
168C22×C53H(5,𝔽2)11yescharacteristic
169C22×C53H(5,𝔽2)11yescharacteristic
170C2×D53—22no—
171C2×D53—22no—
172C2×D53—22no—
173C2×D53—41no—
174C2×D53—41no—
175C2×D53—41no—
176C2×D53—41no—
177C4×C53—22no—
178C4×C53—41no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53—41no—
182C4×C53—41no—
183Dic53—22no—
184Dic53—22no—
185Dic53—41no—
186Dic53—41no—
187Dic53—41no—
188Dic53—41no—
189Dic53—41no—
190#2804—531nomaximal, Sylow
191C2×C53—42no—
192C2×C53#53811yescharacteristic
193C2×C53#53811yescharacteristic
194C2×C53#54311yescharacteristic
195C2×C53#55411yescharacteristic
196C2×C53#55411yescharacteristic
197C2×C53#55911yescharacteristic
198C2×C53H(3,𝔽4)11yescharacteristic
199D53—42no—
200#400—531no—
201#400—531no—
202#393—531no—
203#393—531no—
204#393—531no—
205#393D5311yescharacteristic, complemented
206#391—531no—
207#391—531no—
208#390—531no—
209#386—531no—
210#386—531no—
211#385—531no—
212#385—531no—
213#385—531no—
214#385—531no—
215C53#280411yescharacteristic, complemented, Sylow
216C22×Q8—531no—
217C22×Q8C2×D5311yescharacteristic
218C22×D4—531no—
219C23×C4—531no—
220C23×C4—531no—
221C2×C4⋊C4—531no—
222C2×C4⋊C4—531no—
223C2×C4⋊C4—531no—
224C2×C4⋊C4—531no—
225C2×C4⋊C4—531no—
226C2×C22⋊C4—531no—
227C2×C22⋊C4—531no—
228C2×C22⋊C4—531no—
229C2×C22⋊C4—531no—
230C2×C22⋊C4—531no—
231C2×C22⋊C4—531no—
232C2×C22⋊C4—531no—
233C2×C22⋊C4C2×D5311yescharacteristic
234C2×C22⋊C4C2×D5311yescharacteristic
235C2×C22⋊C4C2×D5311yescharacteristic
236C2×C42—531no—
237#95—531no—
238#95—531no—
239#95—531no—
240#95—531no—
241#95—531no—
242#95—531no—
243#95—531no—
244#95—531no—
245#95—531no—
246#95—531no—
247#95—531no—
248#95C2×D5311yescharacteristic
249#95C2×D5311yescharacteristic
250#95C2×D5311yescharacteristic
251C24—531no—
252C24C22×D5311yescharacteristic
253C2×Q8—22no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—1062no—
257C2×Q8—2121no—
258C2×Q8—2121no—
259C2×D4—534no—
260C2×D4—2121no—
261C2×D4—2121no—
262C22×C4—531no—
263C22×C4—531no—
264C22×C4—531no—
265C22×C4—531no—
266C22×C4—531no—
267C22×C4—531no—
268C22×C4—531no—
269C22×C4—2121no—
270C22×C4—2121no—
271C22×C4—2121no—
272C22×C4—2121no—
273C22×C4—2121no—
274C22×C4—2121no—
275C22×C4C22×D5311yescharacteristic
276C22×C4C22×D5311yescharacteristic
277C22×C4C22×D5311yescharacteristic
278C22×C4C22×D5311yescharacteristic
279C22×C4C22×D5311yescharacteristic
280C22×C4C22×D5311yescharacteristic
281C4⋊C4—2121no—
282C4⋊C4—2121no—
283C4⋊C4—2121no—
284C4⋊C4—2121no—
285C4⋊C4—2121no—
286C22⋊C4—22no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—1062no—
290C22⋊C4—1062no—
291C22⋊C4—1062no—
292C22⋊C4—2121no—
293C22⋊C4—2121no—
294C22⋊C4—2121no—
295C22⋊C4—2121no—
296C42—2121no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—532no—
304C23—1061no—
305C23—1061no—
306C23—1061no—
307C23—1061no—
308C23—1061no—
309C23—1061no—
310C23Q8×D5312yes—
311C23C23×D5311yescharacteristic
312Q8—44no—
313Q8—2124no—
314D4—2124no—
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—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—532no—
333C2×C4—532no—
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—1061no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—1061no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—1061no—
350C2×C4—1061no—
351C2×C4—1061no—
352C2×C4—1061no—
353C2×C4—2124no—
354C2×C4—2124no—
355C2×C4#1171994712yes—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—1062no—
364C22—1062no—
365C22—1062no—
366C22—2121no—
367C22—2121no—
368C22—2121no—
369C22—2121no—
370C22C2×Q8×D5311yescharacteristic
371C22#41×D5311yescharacteristic
372C22C2×#1171994711yescharacteristic
373C22#1313326211yescharacteristic
374C22#1313326211yescharacteristic
375C22#1313326311yescharacteristic
376C22#1313326311yescharacteristic
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—1062no—
384C4—1062no—
385C4—2121no—
386C4—2121no—
387C4—2121no—
388C4—2121no—
389C4—2121no—
390C2—42no—
391C2—2122no—
392C2#1681120711yescharacteristic
393C2#1681120811yescharacteristic
394C2#1681124811yescharacteristic
395C2#1681132711yescharacteristic
396C2#1681133111yescharacteristic
397C2#1681135511yescharacteristic
398C2#1681136511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#393C2
hypercenter#393D53
hypocenterC53#2804
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. #16810564C2semidirect
2. #16810563C2semidirect
3. #16810536C2semidirect
4. #16810535C2semidirect
5. #16810534C2semidirect
6. #16810530C2semidirect
7. #16810527C2semidirect
11. #16810498C2semidirect
12. #16810497C2semidirect
13. #16810497C2semidirect
14. #16810495C2semidirect
15. C53×#393C2semidirect
64. C2×C22⋊D53Q8semidirect
205. #393D53semidirect
215. C53#2804semidirect