Finite Group Databasebeta

← #20284956

Subgroups — #20284956

Order 6784.

418 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#16811227C211yesmaximal, characteristic, complemented
2#16811095C211yesmaximal, characteristic, complemented
3#16810946C211yesmaximal, characteristic, complemented
4#16810945C211yesmaximal, characteristic, complemented
5#16810924C211yesmaximal, characteristic, complemented
6#16810923C211yesmaximal, characteristic, complemented
7#16810815C211yesmaximal, characteristic, complemented
8#16810755C211yesmaximal, characteristic, complemented
9#16810755C211yesmaximal, characteristic, complemented
10#16810735C211yesmaximal, characteristic, complemented
11#16810732C211yesmaximal, characteristic, complemented
12D8×Dic53C211yesmaximal, characteristic, complemented
13#16810703C211yesmaximal, characteristic, complemented
14#16810683C211yesmaximal, characteristic, complemented
15C53×#459C211yesmaximal, characteristic, complemented
16C2×D4×D53C2211yescharacteristic
17C2×#11719922C2211yescharacteristic
18C2×#11719922C2211yescharacteristic
19#13133209C2211yescharacteristic, complemented
20#13133198—21no—
21#13133198—21no—
22#13133189—21no—
23#13133189—21no—
24#13133188C2211yescharacteristic
25#13133188C2211yescharacteristic
26C4×D212C2211yescharacteristic
27D4×Dic53C2211yescharacteristic
28D4×Dic53C2211yescharacteristic, complemented
29D4×Dic53C2211yescharacteristic, complemented
30#13133179C2211yescharacteristic
31C4×C22⋊D53—21no—
32#13133177—21no—
33#13133175C2211yescharacteristic
34#13133174C2211yescharacteristic
35C2×#11719921—21no—
36C22⋊C4×D53—21no—
37#13133151C2211yescharacteristic, complemented
38#13133150C2211yescharacteristic, complemented
39#13133149C2211yescharacteristic, complemented
40#13133148C2211yescharacteristic, complemented
41C4⋊#88322C2211yescharacteristic, complemented
42#13133139C2211yescharacteristic
43#13133139C2211yescharacteristic
44#13133138C2211yescharacteristic
45#13133138C2211yescharacteristic
46#13133138C2211yescharacteristic, complemented
47#13133138C2211yescharacteristic, complemented
48#13133129C2211yescharacteristic
49#13133129C2211yescharacteristic
50#13133127C2211yescharacteristic, complemented
51C8×Dic53C2211yescharacteristic, complemented
52C2×D8×C53C2211yescharacteristic
53C2×D8×C53C2211yescharacteristic
54#127×C53C2211yescharacteristic, complemented
55C4×D4×C53C2211yescharacteristic
56C4⋊C8×C53C2211yescharacteristic, complemented
57#102×C53C2211yescharacteristic, complemented
58#102×C53C2211yescharacteristic, complemented
59C23×D53—21no—
60C2×D4×C53—21no—
61C2×D4×C53—21no—
62C2×D4×C53C2311yescharacteristic
63C2×D4×C53C2311yescharacteristic
64C2×D4×C53C2311yescharacteristic
65C22×C4×C53—21no—
66C2×C22⋊D53—21no—
67C2×C22⋊D53—21no—
68C2×C22⋊D53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—21no—
72D4×D53—22no—
73D4×D53—22no—
74C2×D212C2311yescharacteristic
75C2×C4×D53—21no—
76C2×C4×D53—21no—
77C2×C4×D53C2311yescharacteristic
78D8×C53—22no—
79D8×C53—22no—
80C2×C8×C53C2311yescharacteristic
81C2×C8×C53C2311yescharacteristic
82C4⋊C4×C53C2311yescharacteristic
83C22⋊C4×C53—21no—
84C42×C53C2311yescharacteristic
85C22⋊Dic53—21no—
86C22⋊Dic53—21no—
87C22⋊Dic53—21no—
88C22⋊Dic53—21no—
89C22⋊Dic53—21no—
90#11719922—22no—
91#11719922—22no—
92#11719921—21no—
93#11719921—21no—
94#11719921—21no—
95#11719921D411yescharacteristic
96#11719921D411yescharacteristic
97C4⋊Dic53C2311yescharacteristic
98C4⋊Dic53C2311yescharacteristic
99C4⋊Dic53C2311yescharacteristic
100#11719919—21no—
101#11719919—21no—
102#11719919D411yescharacteristic
103#11719919D411yescharacteristic
104C4×Dic53—21no—
105C4×Dic53C2311yescharacteristic
106C2×#88322C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C22×D53—21no—
112C22×D53—24no—
113C22×D53—81no—
114C22×D53—81no—
115C22×D53C2×D411yescharacteristic
116D4×C53—21no—
117D4×C53—21no—
118D4×C53—21no—
119D4×C53—21no—
120D4×C53—21no—
121D4×C53—81no—
122D4×C53—81no—
123D4×C53#4112yes—
124C2×C4×C53—21no—
125C2×C4×C53—81no—
126C2×C4×C53C2×D411yescharacteristic
127C2×C4×C53C2×D411yescharacteristic
128C2×C4×C53C2411yescharacteristic
129C22⋊D53—24no—
130C22⋊D53—81no—
131C22⋊D53—81no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—81no—
136C2×Dic53—81no—
137C2×Dic53—81no—
138C2×Dic53C2×D411yescharacteristic
139C2×Dic53C2×D411yescharacteristic
140C2×Dic53C2×D411yescharacteristic
141D212—21no—
142D212#4112yes—
143C4×D53—21no—
144C4×D53—21no—
145C4×D53—81no—
146C4×D53—81no—
147C8×C53—21no—
148C8×C53—21no—
149#88322—21no—
150#88322—21no—
151C22×C53—22no—
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×C53C22×D411yescharacteristic
161C2×D53—21no—
162C2×D53—22no—
163C2×D53—21no—
164C2×D53—21no—
165C2×D53—41no—
166C2×D53—41no—
167C2×D53—84no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—41no—
171C4×C53C2×#4111yescharacteristic
172C4×C53C2×#4111yescharacteristic
173Dic53—21no—
174Dic53—21no—
175Dic53—21no—
176Dic53—41no—
177Dic53—41no—
178Dic53—41no—
179#3122—531nomaximal, Sylow
180C2×C53—42no—
181C2×C53—81no—
182C2×C53—81no—
183C2×C53#52511yescharacteristic
184C2×C53#57611yescharacteristic
185C2×C53#57611yescharacteristic
186D53—41no—
187D53—42no—
188#540—531no—
189#518—531no—
190#482—531no—
191#482—531no—
192#478—531no—
193#478—531no—
194#459—531no—
195#459D5311yescharacteristic, complemented
196#447—531no—
197#447—531no—
198#442—531no—
199#442—531no—
200C4×D8—531no—
201C4×D8—531no—
202#433—531no—
203C53#312211yescharacteristic, complemented, Sylow
204C22×D4—531no—
205C2×D8—531no—
206C2×D8—531no—
207C2×D8C2×D5311yescharacteristic
208C2×D8C2×D5311yescharacteristic
209#127C2×D5311yescharacteristic, complemented
210#125—531no—
211#123—1061no—
212#123—1061no—
213#121—531no—
214#121—531no—
215#121—1061no—
216#121—1061no—
217C4×D4—531no—
218C4×D4—531no—
219C4×D4—531no—
220C4×D4—531no—
221C4×D4—531no—
222C4×D4—1061no—
223C4×D4—1061no—
224C4×D4C2×D5311yescharacteristic
225#117—531no—
226#117—531no—
227C2×C22⋊C4—1061no—
228C2×C22⋊C4—1061no—
229#107—531no—
230#107—531no—
231#106—531no—
232#106—531no—
233C4⋊C8—531no—
234C4⋊C8C2×D5311yescharacteristic, complemented
235#102—531no—
236#102—531no—
237#102—531no—
238#102—531no—
239#102—531no—
240#102—531no—
241#102C2×D5311yescharacteristic, complemented
242#102C2×D5311yescharacteristic, complemented
243C22⋊C8—531no—
244C22⋊C8—531no—
245#97—531no—
246C4×C8—531no—
247C24—1061no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—531no—
251C2×D4—1061no—
252C2×D4—1062no—
253C2×D4—1062no—
254C2×D4—1061no—
255C2×D4—1061no—
256C2×D4C22×D5311yescharacteristic
257C2×D4C22×D5311yescharacteristic
258C2×D4C22×D5311yescharacteristic
259C22×C4—21no—
260C22×C4—531no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4—1061no—
264C22×C4—1061no—
265C22×C4—1061no—
266D8—22no—
267D8—22no—
268D8—1062no—
269D8—1062no—
270C2×C8—531no—
271C2×C8—531no—
272C2×C8C22×D5311yescharacteristic
273C2×C8C22×D5311yescharacteristic
274C4⋊C4—531no—
275C4⋊C4—531no—
276C4⋊C4—531no—
277C4⋊C4—531no—
278C4⋊C4—531no—
279C4⋊C4—1061no—
280C4⋊C4—1061no—
281C4⋊C4C22×D5311yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—531no—
284C22⋊C4—531no—
285C22⋊C4—1061no—
286C22⋊C4—1061no—
287C22⋊C4—1061no—
288C22⋊C4—1061no—
289C22⋊C4—1061no—
290C22⋊C4—1061no—
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C42—531no—
294C42—1061no—
295C42C22×D5311yescharacteristic
296C23—21no—
297C23—21no—
298C23—21no—
299C23—531no—
300C23—1064no—
301C23—1061no—
302C23—4241no—
303C23—4241no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—21no—
309D4—81no—
310D4—81no—
311D4—532no—
312D4—1064no—
313D4—1061no—
314D4—4241no—
315D4—4241no—
316D4#1171994712yes—
317C2×C4—21no—
318C2×C4—81no—
319C2×C4—531no—
320C2×C4—531no—
321C2×C4—531no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—1061no—
326C2×C4—1061no—
327C2×C4—4241no—
328C2×C4—4241no—
329C2×C4—4241no—
330C2×C4—4241no—
331C2×C4—4241no—
332C2×C4D4×D5311yescharacteristic
333C2×C4D4×D5311yescharacteristic
334C2×C4C23×D5311yescharacteristic
335C8—21no—
336C8—21no—
337C8—1061no—
338C8—1061no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—1061no—
349C22—1062no—
350C22—1061no—
351C22—1061no—
352C22—4244no—
353C22—2121no—
354C22—2121no—
355C22C2×D4×D5311yescharacteristic
356C4—21no—
357C4—21no—
358C4—41no—
359C4—1061no—
360C4—1061no—
361C4—1061no—
362C4—2121no—
363C4—2121no—
364C4—2121no—
365C4#41×D5311yescharacteristic
366C4C2×#1171994711yescharacteristic
367C2—42no—
368C2—81no—
369C2—81no—
370C2—2121no—
371C2—2122no—
372C2#1681113311yescharacteristic
373C2#1681142411yescharacteristic
374C2#1681142711yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#459C2
socleC22×C53C22×D4
hypercenter#459D53
hypocenterC53#3122
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. #16811227C2semidirect
2. #16811095C2semidirect
3. #16810946C2semidirect
4. #16810945C2semidirect
5. #16810924C2semidirect
6. #16810923C2semidirect
7. #16810815C2semidirect
8. #16810755C2semidirect
9. #16810755C2semidirect
10. #16810735C2semidirect
11. #16810732C2semidirect
12. D8×Dic53C2semidirect
13. #16810703C2semidirect
14. #16810683C2semidirect
15. C53×#459C2semidirect
19. #13133209C22semidirect
28. D4×Dic53C22semidirect
29. D4×Dic53C22semidirect
37. #13133151C22semidirect
38. #13133150C22semidirect
39. #13133149C22semidirect
40. #13133148C22semidirect
41. C4⋊#88322C22semidirect
46. #13133138C22semidirect
47. #13133138C22semidirect
50. #13133127C22semidirect
51. C8×Dic53C22semidirect
54. #127×C53C22semidirect
56. C4⋊C8×C53C22semidirect
57. #102×C53C22semidirect
58. #102×C53C22semidirect
195. #459D53semidirect
203. C53#3122semidirect
209. #127C2×D53semidirect
234. C4⋊C8C2×D53semidirect
241. #102C2×D53semidirect
242. #102C2×D53semidirect