Finite Group Databasebeta

← #20286713

Subgroups — #20286713

Order 6784.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811281C211yesmaximal, characteristic, complemented
2#16811278C211yesmaximal, characteristic, complemented
3#16810982C211yesmaximal, characteristic, complemented
4#16810957C211yesmaximal, characteristic, complemented
5#16810935C211yesmaximal, characteristic, complemented
6#16810919C211yesmaximal, characteristic, complemented
7#16810866C211yesmaximal, characteristic, complemented
8#16810856C211yesmaximal, characteristic, complemented
9#16810765C211yesmaximal, characteristic, complemented
10#16810760C211yesmaximal, characteristic, complemented
11#16810723C211yesmaximal, characteristic, complemented
12#16810693C211yesmaximal, characteristic, complemented
13#16810621C211yesmaximal, characteristic, complemented
14#103×D53C211yesmaximal, characteristic, complemented
15C53×#463C211yesmaximal, characteristic, complemented
16C2×#11719945—21no—
17C2×#11719947C2211yescharacteristic
18C2×Q8×D53C2211yescharacteristic
19C2×#11719923C2211yescharacteristic
20C2×#11719912C2211yescharacteristic
21C2×C8×D53C2211yescharacteristic
22#13133217C2211yescharacteristic
23#13133203—21no—
24#13133199—21no—
25#13133199—21no—
26#13133196—21no—
27#13133194—21no—
28#13133193C2211yescharacteristic
29#13133190—21no—
30#13133188C2211yescharacteristic
31#13133188C2211yescharacteristic
32#13133188C2211yescharacteristic
33C4×Dic106C2211yescharacteristic
34C4×D212C2211yescharacteristic
35#13133177—21no—
36C4⋊C4×D53C2211yescharacteristic
37C4⋊C4×D53C2211yescharacteristic
38#13133151C2211yescharacteristic
39#13133151C2211yescharacteristic
40#13133150C2211yescharacteristic
41#13133142C2211yescharacteristic
42#13133142C2211yescharacteristic
43#13133141C2211yescharacteristic
44#13133141C2211yescharacteristic
45#13133140C2211yescharacteristic
46#13133138C2211yescharacteristic
47#13133138C2211yescharacteristic
48#13133138C2211yescharacteristic
49#13133129C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133128C2211yescharacteristic
52C2×SD16×C53C2211yescharacteristic
53C2×D8×C53C2211yescharacteristic
54#124×C53C2211yescharacteristic
55C4×D4×C53C2211yescharacteristic
56C4⋊C8×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58#102×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C2×D4×C53C2311yescharacteristic
61C2×D4×C53C2311yescharacteristic
62C22×C4×C53—21no—
63C2×C22⋊D53—21no—
64C2×C22⋊D53—21no—
65C2×C22⋊D53—21no—
66C2×C22⋊D53—21no—
67C22×Dic53—21no—
68Q8×D53—22no—
69Q8×D53—41no—
70#11719947—22no—
71#11719947—41no—
72#11719945—44no—
73C2×D212D411yescharacteristic
74C2×C4×D53—21no—
75C2×C4×D53—21no—
76C2×C4×D53—21no—
77C2×C4×D53C2311yescharacteristic
78C2×C4×D53C2311yescharacteristic
79C2×Dic106—21no—
80C2×Dic106—21no—
81C2×Dic106C2311yescharacteristic
82SD16×C53—41no—
83D8×C53—41no—
84C2×C8×C53C2311yescharacteristic
85C2×C8×C53C2311yescharacteristic
86C4⋊C4×C53C2311yescharacteristic
87C22⋊C4×C53—21no—
88C22⋊C4×C53—21no—
89C22⋊C4×C53—21no—
90C42×C53C2311yescharacteristic
91C22⋊Dic53—21no—
92C22⋊Dic53—21no—
93C22⋊Dic53—21no—
94#11719923—22no—
95#11719921—21no—
96#11719921D411yescharacteristic
97#11719921D411yescharacteristic
98C4⋊Dic53—21no—
99C4⋊Dic53D411yescharacteristic
100C4⋊Dic53C2311yescharacteristic
101C4⋊Dic53C2311yescharacteristic
102C4⋊Dic53C2311yescharacteristic
103#11719919—21no—
104#11719919—21no—
105#11719919—21no—
106#11719919—21no—
107C4×Dic53—21no—
108C2×#88322C2311yescharacteristic
109C2×#88322C2311yescharacteristic
110#11719912—41no—
111C8×D53—41no—
112C23×C53—21no—
113C23×C53—21no—
114C22×D53C2×D411yescharacteristic
115C22×D53C2×D411yescharacteristic
116Q8×C53—21no—
117Q8×C53—21no—
118D4×C53—21no—
119D4×C53—21no—
120D4×C53—21no—
121D4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—42no—
125C2×C4×C53C2×D411yescharacteristic
126C2×C4×C53C2×D411yescharacteristic
127C2×C4×C53C2411yescharacteristic
128C22⋊D53—42no—
129C22⋊D53—42no—
130C22⋊D53—42no—
131C22⋊D53—42no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—42no—
137C2×Dic53C2×D411yescharacteristic
138C2×Dic53C2×D411yescharacteristic
139D212—22no—
140C4×D53—21no—
141C4×D53—21no—
142C4×D53—21no—
143C4×D53—21no—
144C4×D53—42no—
145C4×D53—42no—
146C4×D53—42no—
147Dic106—21no—
148Dic106—42no—
149Dic106—42no—
150Dic106#4112yescomplemented
151C8×C53—21no—
152C8×C53—21no—
153#88322—21no—
154#88322—21no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53C22×D411yescharacteristic
162C2×D53—21no—
163C2×D53—21no—
164C2×D53—21no—
165C2×D53—21no—
166C2×D53—21no—
167C2×D53AGL(1,ℤ/8)12yes—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—41no—
171C4×C53—41no—
172C4×C53C2×#4111yescharacteristic
173C4×C53H(5,𝔽2)11yescharacteristic
174Dic53—22no—
175Dic53—21no—
176Dic53—21no—
177Dic53—41no—
178Dic53—41no—
179Dic53—41no—
180#3308—531nomaximal, Sylow
181C2×C53—81no—
182C2×C53—81no—
183C2×C53#54611yescharacteristic
184C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C53#57811yescharacteristic
186D53—22no—
187D53—41no—
188#548—531no—
189#547—531no—
190#489—531no—
191#484—531no—
192#480—531no—
193#477—531no—
194#468—531no—
195#466—531no—
196#463D5311yescharacteristic, complemented
197#449—531no—
198#448—531no—
199#440—531no—
200#435—531no—
201#417—531no—
202C2×#103—531no—
203C53#330811yescharacteristic, complemented, Sylow
204C2×#41—531no—
205C2×#41—1061no—
206C22×Q8—531no—
207C2×SD16—531no—
208C2×SD16C2×D5311yescharacteristic
209C2×D8C2×D5311yescharacteristic
210C2×M16—531no—
211C22×C8—531no—
212C4⋊Q8—531no—
213#124—1061no—
214#124C2×D5311yescharacteristic
215#123—1061no—
216#123—1061no—
217C22⋊Q8—531no—
218C22⋊Q8—1061no—
219C22⋊Q8—1061no—
220#121—531no—
221#121—531no—
222#121—531no—
223#121—1061no—
224C4×Q8—531no—
225C4×D4—531no—
226C4×D4—1061no—
227C4×D4C2×D5311yescharacteristic
228C2×C4⋊C4—531no—
229C2×C4⋊C4—531no—
230#107—531no—
231#107—531no—
232#107—531no—
233C4⋊C8C2×D5311yescharacteristic
234#103—531no—
235#103—531no—
236#103—531no—
237#103—531no—
238#103—531no—
239#103C2×D5311yescharacteristic
240#102—531no—
241#102—531no—
242#102—531no—
243#102C2×D5311yescharacteristic
244C22⋊C8—531no—
245C22⋊C8—531no—
246#97—531no—
247#41—1062no—
248#41—2124no—
249#41—2121no—
250C2×Q8—531no—
251C2×Q8—1061no—
252C2×Q8—1061no—
253C2×Q8—1062no—
254C2×Q8—2121no—
255C2×Q8C22×D5311yescharacteristic
256C2×D4—531no—
257C2×D4—1061no—
258C2×D4—1061no—
259C2×D4—1061no—
260C2×D4—1061no—
261C2×D4C22×D5311yescharacteristic
262C2×D4C22×D5311yescharacteristic
263C22×C4—21no—
264C22×C4—531no—
265C22×C4—531no—
266C22×C4—1061no—
267C22×C4—1061no—
268C22×C4—1061no—
269C22×C4—1061no—
270SD16—41no—
271SD16—1062no—
272D8—41no—
273M16—2121no—
274C2×C8—531no—
275C2×C8—531no—
276C2×C8—2121no—
277C2×C8C22×D5311yescharacteristic
278C2×C8C22×D5311yescharacteristic
279C4⋊C4—531no—
280C4⋊C4—531no—
281C4⋊C4—531no—
282C4⋊C4—531no—
283C4⋊C4—1061no—
284C4⋊C4—1061no—
285C4⋊C4—1061no—
286C4⋊C4—1061no—
287C4⋊C4—1061no—
288C4⋊C4C22×D5311yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—531no—
293C22⋊C4—531no—
294C22⋊C4—1061no—
295C22⋊C4—1061no—
296C22⋊C4—1061no—
297C22⋊C4—1061no—
298C42—1061no—
299C42C22×D5311yescharacteristic
300C23—21no—
301C23—21no—
302C23—531no—
303C23—531no—
304Q8—21no—
305Q8—21no—
306Q8—532no—
307Q8—1061no—
308Q8—2122no—
309Q8—2122no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—1062no—
315D4—2122no—
316D4—2122no—
317D4—2122no—
318D4—2122no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—531no—
323C2×C4—531no—
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—2122no—
333C2×C4—2122no—
334C2×C4—2122no—
335C2×C4—2122no—
336C2×C4D4×D5311yescharacteristic
337C2×C4D4×D5311yescharacteristic
338C2×C4C23×D5311yescharacteristic
339C8—21no—
340C8—21no—
341C8—1061no—
342C8—1061no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—532no—
350C22—1061no—
351C22—1061no—
352C22—1061no—
353C22—1061no—
354C22—1061no—
355C22C2×D4×D5311yescharacteristic
356C4—21no—
357C4—21no—
358C4—41no—
359C4—41no—
360C4—1061no—
361C4—1061no—
362C4—1062no—
363C4—2121no—
364C4—2121no—
365C4—2121no—
366C4#41×D5311yescharacteristic
367C4#1313326211yescharacteristic
368C2—81no—
369C2—81no—
370C2—1062no—
371C2—2121no—
372C2#1681127011yescharacteristic
373C2AGL(1,ℤ/8)×D5311yescharacteristic
374C2#1681143711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#463C2
socleC22×C53C22×D4
hypercenter#463D53
hypocenterC53#3308
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #16811281C2semidirect
2. #16811278C2semidirect
3. #16810982C2semidirect
4. #16810957C2semidirect
5. #16810935C2semidirect
6. #16810919C2semidirect
7. #16810866C2semidirect
8. #16810856C2semidirect
9. #16810765C2semidirect
10. #16810760C2semidirect
11. #16810723C2semidirect
12. #16810693C2semidirect
13. #16810621C2semidirect
14. #103×D53C2semidirect
15. C53×#463C2semidirect
150. Dic106#41semidirect
196. #463D53semidirect
203. C53#3308semidirect