Finite Group Databasebeta

← #20282931

Subgroups — #20282931

Order 6784.

400 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16810563C211yesmaximal, characteristic, complemented
2#16810559C211yesmaximal, characteristic, complemented
3#16810559C211yesmaximal, characteristic, complemented
4#16810557C211yesmaximal, characteristic, complemented
5#16810556C211yesmaximal, characteristic, complemented
6#16810548C211yesmaximal, characteristic, complemented
7#16810548C211yesmaximal, characteristic, complemented
8#16810546C211yesmaximal, characteristic, complemented
9#16810546C211yesmaximal, characteristic, complemented
10#16810537C211yesmaximal, characteristic
11#16810537C211yesmaximal, characteristic
12#16810519C211yesmaximal, characteristic, complemented
13#16810516C211yesmaximal, characteristic, complemented
14#16810516C211yesmaximal, characteristic, complemented
15C53×#394C211yesmaximal, characteristic
16C2×C4⋊Dic53C2211yescharacteristic
17C2×C4⋊Dic53C2211yescharacteristic
18C2×C4⋊Dic53C2211yescharacteristic
19C2×C4⋊Dic53C2211yescharacteristic
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2211yescharacteristic
22C2×#11719919C2211yescharacteristic
23C2×#11719919C2211yescharacteristic
24C2×#11719919C2211yescharacteristic
25C2×#11719919C2211yescharacteristic
26C2×C22⋊Dic53C2211yescharacteristic
27C2×C22⋊Dic53C2211yescharacteristic
28C2×C22⋊Dic53C2211yescharacteristic
29C2×C22⋊Dic53C2211yescharacteristic
30C2×C22⋊Dic53C2211yescharacteristic
31C2×C22⋊Dic53C2211yescharacteristic
32C2×C22⋊Dic53C2211yescharacteristic
33C2×C22⋊Dic53C2211yescharacteristic
34C2×C4×Dic53C2211yescharacteristic
35#13133124C2211yescharacteristic
36#13133124C2211yescharacteristic
37#13133124C2211yescharacteristic
38#13133124C2211yescharacteristic
39#13133124C2211yescharacteristic
40#13133124C2211yescharacteristic
41#13133124C2211yescharacteristic
42#13133124C2211yescharacteristic
43#13133124C2211yescharacteristic
44C22×D4×C53C2211yescharacteristic
45C2×C4⋊C4×C53C2211yescharacteristic
46C2×C22⋊C4×C53C2211yescharacteristic
47C2×C22⋊C4×C53C2211yescharacteristic
48C2×C22⋊C4×C53C2211yescharacteristic
49C2×C22⋊C4×C53C2211yescharacteristic
50#95×C53C2211yescharacteristic
51C24×C53C2311yescharacteristic
52C24×C53C2311yescharacteristic
53C2×D4×C53—41no—
54C2×D4×C53—41no—
55C2×D4×C53—41no—
56C22×C4×C53C2311yescharacteristic
57C22×C4×C53C2311yescharacteristic
58C22×C4×C53C2311yescharacteristic
59C22×C4×C53C2311yescharacteristic
60C22×C4×C53C2311yescharacteristic
61C22×Dic53C2311yescharacteristic
62C22×Dic53C2311yescharacteristic
63C22×Dic53C2311yescharacteristic
64C22×Dic53C2311yescharacteristic
65C22×Dic53C2311yescharacteristic
66C22×Dic53C2311yescharacteristic
67C22×Dic53C2311yescharacteristic
68C22×Dic53C2311yescharacteristic
69C4⋊C4×C53—41no—
70C22⋊C4×C53—22no—
71C22⋊C4×C53—22no—
72C22⋊C4×C53—41no—
73C22⋊C4×C53—41no—
74C22⋊Dic53—41no—
75C22⋊Dic53—41no—
76C22⋊Dic53—41no—
77C22⋊Dic53—41no—
78C22⋊Dic53—41no—
79C22⋊Dic53—41no—
80C22⋊Dic53—41no—
81C22⋊Dic53—41no—
82C4⋊Dic53—22no—
83C4⋊Dic53—41no—
84C4⋊Dic53—41no—
85C4⋊Dic53—41no—
86#11719919—41no—
87#11719919—41no—
88#11719919—41no—
89#11719919—41no—
90#11719919—41no—
91#11719919—41no—
92C4×Dic53—41no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—21no—
107C23×C53C2411yescharacteristic
108D4×C53—44no—
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#4112yes—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—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—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—41no—
162C22×C53C2×#4111yescharacteristic
163C22×C53H(5,𝔽2)11yescharacteristic
164C22×C53H(5,𝔽2)11yescharacteristic
165C22×C53H(5,𝔽2)11yescharacteristic
166C22×C53H(5,𝔽2)11yescharacteristic
167C22×C532−1+411yescharacteristic
168C22×C532−1+411yescharacteristic
169C4×C53—22no—
170C4×C53—41no—
171C4×C53—41no—
172C4×C53—41no—
173C4×C53—41no—
174Dic53—41no—
175Dic53—41no—
176Dic53—41no—
177Dic53—41no—
178Dic53—41no—
179Dic53—41no—
180Dic53—41no—
181Dic53—41no—
182#2837—531nomaximal, Sylow
183C2×C53—81no—
184C2×C53—81no—
185C2×C53#53911yescharacteristic
186C2×C53#54011yescharacteristic
187C2×C53#54211yescharacteristic
188C2×C53#56011yescharacteristic
189C2×C53#56211yescharacteristic
190C2×C53#56211yescharacteristic
191C2×C53#56311yescharacteristic
192#400—531no—
193#399—531no—
194#399—531no—
195#398—531no—
196#398—531no—
197#397—531no—
198#397—531no—
199#396—531no—
200#396—531no—
201#394—531no—
202#394—531no—
203#394D5311yescharacteristic
204#389—531no—
205#388—531no—
206#388—531no—
207C53#283711yescharacteristic, complemented, Sylow
208C22×D4C2×D5311yescharacteristic
209C2×C4⋊C4—531no—
210C2×C4⋊C4—531no—
211C2×C4⋊C4—531no—
212C2×C4⋊C4—531no—
213C2×C4⋊C4—531no—
214C2×C4⋊C4—531no—
215C2×C4⋊C4—531no—
216C2×C4⋊C4—531no—
217C2×C4⋊C4—531no—
218C2×C4⋊C4—531no—
219C2×C4⋊C4C2×D5311yescharacteristic
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⋊C4C2×D5311yescharacteristic
229C2×C22⋊C4C2×D5311yescharacteristic
230C2×C22⋊C4C2×D5311yescharacteristic
231C2×C22⋊C4C2×D5311yescharacteristic
232C2×C42—531no—
233#95—531no—
234#95—531no—
235#95—531no—
236#95—531no—
237#95—531no—
238#95—531no—
239#95—531no—
240#95—531no—
241#95—531no—
242#95C2×D5311yescharacteristic
243C24C22×D5311yescharacteristic
244C24C22×D5311yescharacteristic
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C22×C4—531no—
249C22×C4—531no—
250C22×C4—531no—
251C22×C4—531no—
252C22×C4—531no—
253C22×C4—531no—
254C22×C4—531no—
255C22×C4—531no—
256C22×C4C22×D5311yescharacteristic
257C22×C4C22×D5311yescharacteristic
258C22×C4C22×D5311yescharacteristic
259C22×C4C22×D5311yescharacteristic
260C22×C4C22×D5311yescharacteristic
261C4⋊C4—41no—
262C4⋊C4—1062no—
263C4⋊C4—2121no—
264C4⋊C4—2121no—
265C4⋊C4—2121no—
266C4⋊C4—2121no—
267C4⋊C4—2121no—
268C4⋊C4—2121no—
269C4⋊C4—2121no—
270C4⋊C4—2121no—
271C4⋊C4—2121no—
272C22⋊C4—22no—
273C22⋊C4—22no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—2121no—
277C22⋊C4—2121no—
278C22⋊C4—2121no—
279C22⋊C4—2121no—
280C22⋊C4—2121no—
281C22⋊C4—2121no—
282C22⋊C4—2121no—
283C22⋊C4—2121no—
284C42—2121no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23C23×D5311yescharacteristic
300D4—44no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
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—1061no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—1061no—
326C2×C4—1061no—
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#1171994712yes—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22C2×#1171994711yescharacteristic
355C22#1313326211yescharacteristic
356C22#1313326211yescharacteristic
357C22#1313326211yescharacteristic
358C22#1313326311yescharacteristic
359C22#1313326411yescharacteristic
360C22#1313326411yescharacteristic
361C4—22no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—2121no—
367C4—2121no—
368C4—2121no—
369C4—2121no—
370C4—2121no—
371C4—2121no—
372C4—2121no—
373C4—2121no—
374C2—81no—
375C2—81no—
376C2#1681121811yescharacteristic
377C2#1681122611yescharacteristic
378C2#1681124011yescharacteristic
379C2#1681136211yescharacteristic
380C2#1681136711yescharacteristic
381C2#1681137011yescharacteristic
382C2#1681137611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#394C2
hypercenter#394D53
hypocenterC53#2837
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #16810563C2semidirect
2. #16810559C2semidirect
3. #16810559C2semidirect
4. #16810557C2semidirect
5. #16810556C2semidirect
6. #16810548C2semidirect
7. #16810548C2semidirect
8. #16810546C2semidirect
9. #16810546C2semidirect
12. #16810519C2semidirect
13. #16810516C2semidirect
14. #16810516C2semidirect
207. C53#2837semidirect