Finite Group Databasebeta

← #20284249

Subgroups — #20284249

Order 6784.

504 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#13133241C211yesmaximal, characteristic, complemented
2C2×#13133236C211yesmaximal, characteristic, complemented
3#16811186C211yesmaximal, characteristic, complemented
4#16811102C211yesmaximal, characteristic, complemented
5#16810846C211yesmaximal, characteristic, complemented
6#16810844C211yesmaximal, characteristic, complemented
7#16810838C211yesmaximal, characteristic, complemented
8#16810836C211yesmaximal, characteristic, complemented
9#16810822C211yesmaximal, characteristic, complemented
10#16810820C211yesmaximal, characteristic, complemented
11#16810816C211yesmaximal, characteristic, complemented
12#16810648C211yesmaximal, characteristic, complemented
13#16810621C211yesmaximal, characteristic, complemented
14#102×D53C211yesmaximal, characteristic, complemented
15C53×#459C211yesmaximal, characteristic, complemented
16C2×#11719947C2211yescharacteristic
17C2×#11719947C2211yescharacteristic
18C2×D4×D53C2211yescharacteristic
19#13133241—24no—
20#13133236—24no—
21C2×Dic212C2211yescharacteristic, complemented
22C2×#11719913C2211yescharacteristic
23C2×#11719923C2211yescharacteristic
24C2×#11719923C2211yescharacteristic, complemented
25C2×#11719923C2211yescharacteristic, complemented
26C2×#11719922C2211yescharacteristic
27C2×#11719912C2211yescharacteristic
28C2×C8×D53C2211yescharacteristic
29#13133213C2211yescharacteristic
30#13133205C2211yescharacteristic
31#13133204C2211yescharacteristic
32#13133189—21no—
33#13133189—21no—
34#13133185—21no—
35#13133185—21no—
36C4×Dic106C2211yescharacteristic, complemented
37#13133182C2211yescharacteristic, complemented
38D4×Dic53C2211yescharacteristic, complemented
39C4×C22⋊D53—21no—
40#13133177—21no—
41#13133174C2211yescharacteristic
42#13133173—21no—
43C2×#11719919—21no—
44C4⋊C4×D53C2211yescharacteristic
45C4⋊#88322C2211yescharacteristic, complemented
46#13133146C2211yescharacteristic, complemented
47#13133146C2211yescharacteristic, complemented
48#13133142C2211yescharacteristic, complemented
49#13133140C2211yescharacteristic, complemented
50#13133139C2211yescharacteristic
51#13133138C2211yescharacteristic
52#13133138C2211yescharacteristic, complemented
53#13133137C2211yescharacteristic
54C2×D8×C53C2211yescharacteristic
55C2×D8×C53C2211yescharacteristic, complemented
56#127×C53C2211yescharacteristic
57C4×D4×C53C2211yescharacteristic, complemented
58C4⋊C8×C53C2211yescharacteristic, complemented
59#102×C53C2211yescharacteristic
60#102×C53C2211yescharacteristic, complemented
61C23×D53—21no—
62C2×D4×C53—21no—
63C2×D4×C53—21no—
64C2×D4×C53C2311yescharacteristic
65C2×D4×C53C2311yescharacteristic
66C2×D4×C53C2311yescharacteristic
67C22×C4×C53—21no—
68C2×C22⋊D53—21no—
69C2×C22⋊D53—21no—
70C2×C22⋊D53—21no—
71C2×C22⋊D53—21no—
72C2×C22⋊D53—21no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76#11719947—22no—
77#11719947—22no—
78#11719947—22no—
79#11719947D414yescomplemented
80D4×D53—22no—
81D4×D53—22no—
82C2×D212C2311yescharacteristic
83C2×C4×D53—21no—
84C2×C4×D53C2311yescharacteristic
85C2×Dic106C2311yescharacteristic
86C2×Dic106C2311yescharacteristic
87D8×C53—22no—
88D8×C53—22no—
89C2×C8×C53C2311yescharacteristic
90C2×C8×C53C2311yescharacteristic
91C4⋊C4×C53C2311yescharacteristic
92C22⋊C4×C53—21no—
93C42×C53C2311yescharacteristic
94C22⋊Dic53—21no—
95C22⋊Dic53—21no—
96C22⋊Dic53—21no—
97#11719923—22no—
98#11719923—22no—
99#11719923—22no—
100#11719922—22no—
101#11719921—21no—
102#11719921—21no—
103#11719921D411yescharacteristic
104#11719921D411yescharacteristic
105C4⋊Dic53C2311yescharacteristic
106#11719919—21no—
107#11719919—21no—
108#11719919—21no—
109#11719919D411yescharacteristic
110#11719919D411yescharacteristic
111C4×Dic53—21no—
112C4×Dic53—21no—
113C4×Dic53C2311yescharacteristic
114C2×#88322C2311yescharacteristic
115C2×#88322C2311yescharacteristic
116Dic212—22no—
117#11719913—22no—
118#11719912—22no—
119C8×D53—22no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—21no—
123C22×D53—21no—
124C22×D53—42no—
125C22×D53—81no—
126C22×D53—81no—
127C22×D53C2×D411yescharacteristic
128D4×C53—21no—
129D4×C53—21no—
130D4×C53—21no—
131D4×C53—21no—
132D4×C53—21no—
133D4×C53—81no—
134D4×C53—81no—
135D4×C53C2×D412yes—
136C2×C4×C53—21no—
137C2×C4×C53—81no—
138C2×C4×C53C2×D411yescharacteristic
139C2×C4×C53C2×D411yescharacteristic
140C2×C4×C53C2411yescharacteristic
141C22⋊D53—24no—
142C22⋊D53—42no—
143C22⋊D53—42no—
144C22⋊D53—81no—
145C22⋊D53—81no—
146C2×Dic53—21no—
147C2×Dic53—24no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—42no—
151C2×Dic53—81no—
152C2×Dic53C2×D411yescharacteristic
153C2×Dic53C2×D411yescharacteristic
154C2×Dic53C2×D411yescharacteristic
155D212—21no—
156D212—21no—
157C4×D53—81no—
158C4×D53C2×D412yes—
159C4×D53#4112yes—
160Dic106—21no—
161Dic106—21no—
162Dic106—21no—
163Dic106C2×D412yes—
164C8×C53—21no—
165C8×C53—21no—
166#88322—21no—
167#88322—21no—
168C22×C53—22no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53—41no—
176C22×C53—41no—
177C22×C53C22×D411yescharacteristic
178C2×D53—21no—
179C2×D53—21no—
180C2×D53—41no—
181C2×D53—41no—
182C2×D53—41no—
183C2×D53—84no—
184C2×D53#12112yes—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—41no—
188C4×C53C22×D411yescharacteristic
189C4×C53C2×#4111yescharacteristic
190Dic53—22no—
191Dic53—21no—
192Dic53—21no—
193Dic53—41no—
194Dic53—41no—
195Dic53#12112yes—
196#3045—531nomaximal, Sylow
197C2×C53—42no—
198C2×C53—81no—
199C2×C53—81no—
200C2×C53C2×#12111yescharacteristic
201C2×C53#57611yescharacteristic
202C2×C53#57711yescharacteristic
203D53—22no—
204D53—81no—
205C2×AGL(1,ℤ/8)—531no—
206C2×#135—531no—
207#535—531no—
208#520—531no—
209#464—531no—
210#464—531no—
211#463—531no—
212#463—531no—
213#460—531no—
214#460—531no—
215#459—531no—
216#459D5311yescharacteristic, complemented
217#424—531no—
218#417—531no—
219C2×#102—531no—
220C53#304511yescharacteristic, complemented, Sylow
221C2×#41—531no—
222C2×#41—531no—
223C22×D4—531no—
224AGL(1,ℤ/8)—1064no—
225#135—1064no—
226C2×Q16—531no—
227C2×SD16—531no—
228C2×SD16—531no—
229C2×SD16—531no—
230C2×SD16—531no—
231C2×D8—531no—
232C2×D8C2×D5311yescharacteristic
233C2×D8C2×D5311yescharacteristic, complemented
234C2×M16—531no—
235C22×C8—531no—
236#127—531no—
237#127C2×D5311yescharacteristic
238#124—531no—
239#124—531no—
240#121—1061no—
241#121—1061no—
242#120—1061no—
243#120—1061no—
244C4×Q8—531no—
245C4×Q8—531no—
246C4×D4—531no—
247C4×D4—1061no—
248C4×D4—1061no—
249C4×D4C2×D5311yescharacteristic, complemented
250#117—531no—
251#117—1061no—
252C2×C4⋊C4—531no—
253C2×C4⋊C4—1061no—
254C4⋊C8—531no—
255C4⋊C8—531no—
256C4⋊C8—531no—
257C4⋊C8C2×D5311yescharacteristic, complemented
258#103—531no—
259#103—531no—
260#102—531no—
261#102—531no—
262#102—531no—
263#102—531no—
264#102C2×D5311yescharacteristic
265#102C2×D5311yescharacteristic, complemented
266C24—1061no—
267#41—534no—
268#41—1062no—
269#41—1062no—
270#41—1062no—
271C2×Q8—531no—
272C2×Q8—531no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—531no—
276C2×D4—1062no—
277C2×D4—1062no—
278C2×D4—1061no—
279C2×D4—1061no—
280C2×D4—1061no—
281C2×D4—1061no—
282C2×D4—1061no—
283C2×D4C22×D5311yescharacteristic
284C2×D4C22×D5311yescharacteristic
285C2×D4C22×D5311yescharacteristic
286C22×C4—21no—
287C22×C4—531no—
288C22×C4—1061no—
289C22×C4—1061no—
290C22×C4—1061no—
291C22×C4—1061no—
292Q16—1062no—
293SD16—1062no—
294SD16—1062no—
295SD16—1062no—
296SD16—1062no—
297D8—22no—
298D8—22no—
299D8—1062no—
300M16—1062no—
301C2×C8—531no—
302C2×C8—531no—
303C2×C8—1062no—
304C2×C8C22×D5311yescharacteristic
305C2×C8C22×D5311yescharacteristic
306C4⋊C4—531no—
307C4⋊C4—531no—
308C4⋊C4—531no—
309C4⋊C4—1061no—
310C4⋊C4—1061no—
311C4⋊C4—1061no—
312C4⋊C4C22×D5311yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—531no—
315C22⋊C4—531no—
316C22⋊C4—1061no—
317C22⋊C4—1061no—
318C22⋊C4—1061no—
319C22⋊C4—1061no—
320C22⋊C4—1061no—
321C42—531no—
322C42—1061no—
323C42—1061no—
324C42C22×D5311yescharacteristic
325C23—21no—
326C23—21no—
327C23—21no—
328C23—531no—
329C23—1061no—
330C23—4241no—
331C23—4241no—
332C23—2122no—
333Q8—532no—
334Q8—1061no—
335Q8—1061no—
336Q8—1061no—
337D4—21no—
338D4—21no—
339D4—21no—
340D4—21no—
341D4—21no—
342D4—81no—
343D4—81no—
344D4—1064no—
345D4—1061no—
346D4—1061no—
347D4—4241no—
348D4—4241no—
349D4—2122no—
350D4—2122no—
351D4D4×D5312yes—
352C2×C4—21no—
353C2×C4—81no—
354C2×C4—531no—
355C2×C4—532no—
356C2×C4—532no—
357C2×C4—531no—
358C2×C4—531no—
359C2×C4—1064no—
360C2×C4—1061no—
361C2×C4—1061no—
362C2×C4—1061no—
363C2×C4—4241no—
364C2×C4—4241no—
365C2×C4—2122no—
366C2×C4D4×D5311yescharacteristic
367C2×C4D4×D5311yescharacteristic
368C2×C4C23×D5311yescharacteristic
369C8—21no—
370C8—21no—
371C8—1061no—
372C8—1061no—
373C22—22no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—532no—
383C22—1061no—
384C22—1061no—
385C22—4244no—
386C22—2121no—
387C22—2121no—
388C22—2121no—
389C22C2×D4×D5311yescharacteristic
390C4—21no—
391C4—21no—
392C4—41no—
393C4—532no—
394C4—1062no—
395C4—1061no—
396C4—1061no—
397C4—2121no—
398C4—2121no—
399C4C2×D4×D5311yescharacteristic
400C4#41×D5311yescharacteristic
401C2—42no—
402C2—81no—
403C2—81no—
404C2—1062no—
405C2—4241no—
406C2#121×D5311yescharacteristic
407C2#1681142411yescharacteristic
408C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#459C2
socleC22×C53C22×D4
hypercenter#459D53
hypocenterC53#3045
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#13133241C2semidirect
2. C2×#13133236C2semidirect
3. #16811186C2semidirect
4. #16811102C2semidirect
5. #16810846C2semidirect
6. #16810844C2semidirect
7. #16810838C2semidirect
8. #16810836C2semidirect
9. #16810822C2semidirect
10. #16810820C2semidirect
11. #16810816C2semidirect
12. #16810648C2semidirect
13. #16810621C2semidirect
14. #102×D53C2semidirect
15. C53×#459C2semidirect
21. C2×Dic212C22semidirect
24. C2×#11719923C22semidirect
25. C2×#11719923C22semidirect
36. C4×Dic106C22semidirect
37. #13133182C22semidirect
38. D4×Dic53C22semidirect
45. C4⋊#88322C22semidirect
46. #13133146C22semidirect
47. #13133146C22semidirect
48. #13133142C22semidirect
49. #13133140C22semidirect
52. #13133138C22semidirect
55. C2×D8×C53C22semidirect
57. C4×D4×C53C22semidirect
58. C4⋊C8×C53C22semidirect
60. #102×C53C22semidirect
79. #11719947D4semidirect
216. #459D53semidirect
220. C53#3045semidirect
233. C2×D8C2×D53semidirect
249. C4×D4C2×D53semidirect
257. C4⋊C8C2×D53semidirect
265. #102C2×D53semidirect