Finite Group Databasebeta

← #20286709

Subgroups — #20286709

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#16811283C211yesmaximal, characteristic, complemented
2#16811274C211yesmaximal, characteristic, complemented
3#16810984C211yesmaximal, characteristic, complemented
4#16810938C211yesmaximal, characteristic, complemented
5#16810917C211yesmaximal, characteristic, complemented
6#16810871C211yesmaximal, characteristic, complemented
7#16810860C211yesmaximal, characteristic, complemented
8#16810841C211yesmaximal, characteristic, complemented
9#16810770C211yesmaximal, characteristic, complemented
10#16810763C211yesmaximal, characteristic, complemented
11#16810722C211yesmaximal, characteristic, complemented
12#16810697C211yesmaximal, characteristic, complemented
13#16810622C211yesmaximal, characteristic, complemented
14#103×D53C211yesmaximal, characteristic, complemented
15C53×#484C211yesmaximal, characteristic, complemented
16C2×#11719945—21no—
17C2×#11719945C2211yescharacteristic
18C2×Q8×D53C2211yescharacteristic
19C2×#11719913C2211yescharacteristic
20C2×#11719924C2211yescharacteristic
21C2×D424C2211yescharacteristic
22C2×#11719917C2211yescharacteristic
23C2×C8×D53C2211yescharacteristic
24#13133216C2211yescharacteristic
25#13133206C2211yescharacteristic
26#13133203—21no—
27#13133202—21no—
28#13133200—21no—
29#13133195—21no—
30#13133195—21no—
31#13133192C2211yescharacteristic
32#13133192C2211yescharacteristic
33#13133191C2211yescharacteristic
34#13133190—21no—
35#13133182C2211yescharacteristic
36#13133179—21no—
37#13133179C2211yescharacteristic
38C4×C22⋊D53C2211yescharacteristic
39C4⋊C4×D53C2211yescharacteristic
40#13133150C2211yescharacteristic
41#13133146C2211yescharacteristic
42#13133142C2211yescharacteristic
43#13133141C2211yescharacteristic
44#13133140C2211yescharacteristic
45#13133140C2211yescharacteristic
46#13133139C2211yescharacteristic
47#13133137C2211yescharacteristic
48#13133137C2211yescharacteristic
49#13133137C2211yescharacteristic
50#13133129C2211yescharacteristic
51#13133127C2211yescharacteristic
52C22⋊Q8×C53C2211yescharacteristic
53C2×C4⋊C4×C53C2211yescharacteristic
54#107×C53C2211yescharacteristic
55#107×C53C2211yescharacteristic
56#103×C53C2211yescharacteristic
57#103×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic
59C2×Q8×C53C2311yescharacteristic
60C22×C4×C53—21no—
61C22×C4×C53C2311yescharacteristic
62C2×C22⋊D53—21no—
63C2×C22⋊D53—21no—
64C2×C22⋊D53D411yescharacteristic
65Q8×D53—22no—
66Q8×D53—41no—
67#11719945—22no—
68#11719945—41no—
69#11719945—44no—
70C2×D212—21no—
71C2×D212—21no—
72C2×D212C2311yescharacteristic
73C2×D212C2311yescharacteristic
74C2×C4×D53—21no—
75C2×C4×D53—21no—
76C2×C4×D53—21no—
77C2×C4×D53—21no—
78C2×C4×D53C2311yescharacteristic
79C2×Dic106—21no—
80C2×Dic106—21no—
81C2×Dic106C2311yescharacteristic
82C2×C8×C53C2311yescharacteristic
83C2×C8×C53C2311yescharacteristic
84C4⋊C4×C53—21no—
85C4⋊C4×C53—21no—
86C4⋊C4×C53C2311yescharacteristic
87C4⋊C4×C53C2311yescharacteristic
88C4⋊C4×C53C2311yescharacteristic
89C22⋊C4×C53—21no—
90C22⋊Dic53D411yescharacteristic
91#11719924—41no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921—21no—
96#11719921—21no—
97#11719921—21no—
98#11719921D411yescharacteristic
99C4⋊Dic53C2311yescharacteristic
100#11719919—21no—
101#11719919—21no—
102#11719919—21no—
103#11719919D411yescharacteristic
104C4×Dic53—21no—
105C4×Dic53C2311yescharacteristic
106#11719917—41no—
107C2×#88322C2311yescharacteristic
108C2×#88322C2311yescharacteristic
109D424—41no—
110#11719913—22no—
111C8×D53—41no—
112C23×C53C2×D411yescharacteristic
113C22×D53—21no—
114C22×D53—21no—
115C22×D53C2×D411yescharacteristic
116Q8×C53—21no—
117Q8×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—42no—
125C2×C4×C53C2×D411yescharacteristic
126C2×C4×C53C2411yescharacteristic
127C22⋊D53—22no—
128C22⋊D53—42no—
129C22⋊D53—42no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53C2×D411yescharacteristic
133C2×Dic53C2×D411yescharacteristic
134C2×Dic53C2×D411yescharacteristic
135D212—21no—
136D212—21no—
137D212—21no—
138D212—21no—
139D212—42no—
140D212—42no—
141C4×D53—21no—
142C4×D53—21no—
143C4×D53—42no—
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—21no—
156C22×C53—21no—
157C22×C53—21no—
158C22×C53C22×D411yescharacteristic
159C2×D53—21no—
160C2×D53—21no—
161C2×D53—41no—
162C2×D53—41no—
163C2×D53—41no—
164C2×D53—41no—
165C2×D53—41no—
166C2×D53—41no—
167C2×D53AGL(1,ℤ/8)12yes—
168C4×C53—21no—
169C4×C53—41no—
170C4×C53—41no—
171C4×C53—41no—
172C4×C53—41no—
173C4×C53C2×#4111yescharacteristic
174C4×C53H(5,𝔽2)11yescharacteristic
175Dic53—22no—
176Dic53—21no—
177Dic53—21no—
178Dic53—21no—
179Dic53—41no—
180#3308—531nomaximal, Sylow
181C2×C53—41no—
182C2×C53#54611yescharacteristic
183C2×C53C2×AGL(1,ℤ/8)11yescharacteristic
184C2×C53#57811yescharacteristic
185D53—22no—
186D53—81no—
187D53—81no—
188#548—531no—
189#547—531no—
190#489—531no—
191#484D5311yescharacteristic, complemented
192#480—531no—
193#477—531no—
194#468—531no—
195#466—531no—
196#463—531no—
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×SD16—531no—
209C2×D8—531no—
210C2×M16—531no—
211C22×C8—531no—
212C4⋊Q8—531no—
213#124—531no—
214#124—1061no—
215#123—1061no—
216#123—1061no—
217C22⋊Q8—1061no—
218C22⋊Q8—1061no—
219C22⋊Q8C2×D5311yescharacteristic
220#121—531no—
221#121—531no—
222#121—531no—
223#121—1061no—
224C4×Q8—531no—
225C4×D4—531no—
226C4×D4—531no—
227C4×D4—1061no—
228C2×C4⋊C4—531no—
229C2×C4⋊C4C2×D5311yescharacteristic
230#107—531no—
231#107C2×D5311yescharacteristic
232#107C2×D5311yescharacteristic
233C4⋊C8—531no—
234#103—531no—
235#103—531no—
236#103—531no—
237#103—531no—
238#103C2×D5311yescharacteristic
239#103C2×D5311yescharacteristic
240#102—531no—
241#102—531no—
242#102—531no—
243#102—531no—
244C22⋊C8—531no—
245C22⋊C8C2×D5311yescharacteristic
246#97—531no—
247#41—1062no—
248#41—2121no—
249#41—2124no—
250C2×Q8—531no—
251C2×Q8—1061no—
252C2×Q8—1061no—
253C2×Q8—1062no—
254C2×Q8—2121no—
255C2×Q8C22×D5311yescharacteristic
256C2×D4—531no—
257C2×D4—531no—
258C2×D4—531no—
259C2×D4—1061no—
260C2×D4—1061no—
261C2×D4—1061no—
262C2×D4—1061no—
263C22×C4—21no—
264C22×C4—531no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4—1061no—
268C22×C4—1061no—
269C22×C4C22×D5311yescharacteristic
270SD16—1062no—
271SD16—2121no—
272D8—2121no—
273M16—2121no—
274C2×C8—531no—
275C2×C8—531no—
276C2×C8—2121no—
277C2×C8C22×D5311yescharacteristic
278C2×C8C22×D5311yescharacteristic
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—531no—
282C4⋊C4—531no—
283C4⋊C4—1061no—
284C4⋊C4—1061no—
285C4⋊C4—1061no—
286C4⋊C4C22×D5311yescharacteristic
287C4⋊C4C22×D5311yescharacteristic
288C4⋊C4C22×D5311yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—531no—
291C22⋊C4—531no—
292C22⋊C4—1061no—
293C22⋊C4—1061no—
294C22⋊C4—1061no—
295C22⋊C4—1061no—
296C22⋊C4—1061no—
297C22⋊C4—1061no—
298C42—531no—
299C42—1061no—
300C23—531no—
301C23—1061no—
302C23—1061no—
303C23D4×D5311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—532no—
307Q8—1061no—
308Q8—2122no—
309Q8—2122no—
310D4—1062no—
311D4—1061no—
312D4—1061no—
313D4—1061no—
314D4—1061no—
315D4—2122no—
316D4—2122no—
317D4—2122no—
318D4—2122no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—42no—
326C2×C4—531no—
327C2×C4—531no—
328C2×C4—531no—
329C2×C4—1061no—
330C2×C4—1061no—
331C2×C4—1061no—
332C2×C4—1061no—
333C2×C4—2122no—
334C2×C4—2122no—
335C2×C4—2122no—
336C2×C4—2122no—
337C2×C4D4×D5311yescharacteristic
338C2×C4C23×D5311yescharacteristic
339C8—21no—
340C8—21no—
341C8—1061no—
342C8—1061no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—532no—
347C22—1061no—
348C22—1061no—
349C22—2121no—
350C22—2121no—
351C22—2121no—
352C22—2121no—
353C22—2121no—
354C22—2121no—
355C22C2×D4×D5311yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—1061no—
362C4—1062no—
363C4—1061no—
364C4—1061no—
365C4—2121no—
366C4#41×D5311yescharacteristic
367C4#1313326311yescharacteristic
368C2—41no—
369C2—1062no—
370C2—4241no—
371C2—4241no—
372C2#1681126511yescharacteristic
373C2AGL(1,ℤ/8)×D5311yescharacteristic
374C2#1681143611yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#484C2
socleC22×C53C22×D4
hypercenter#484D53
hypocenterC53#3308
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. #16811283C2semidirect
2. #16811274C2semidirect
3. #16810984C2semidirect
4. #16810938C2semidirect
5. #16810917C2semidirect
6. #16810871C2semidirect
7. #16810860C2semidirect
8. #16810841C2semidirect
9. #16810770C2semidirect
10. #16810763C2semidirect
11. #16810722C2semidirect
12. #16810697C2semidirect
13. #16810622C2semidirect
14. #103×D53C2semidirect
15. C53×#484C2semidirect
150. Dic106#41semidirect
191. #484D53semidirect
203. C53#3308semidirect