Finite Group Databasebeta

← #20284892

Subgroups — #20284892

Order 6784.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16811240C211yesmaximal, characteristic, complemented
2#16811098C211yesmaximal, characteristic, complemented
3#16810965C211yesmaximal, characteristic, complemented
4#16810944C211yesmaximal, characteristic, complemented
5#16810911C211yesmaximal, characteristic, complemented
6#16810901C211yesmaximal, characteristic, complemented
7#16810836C211yesmaximal, characteristic, complemented
8#16810836C211yesmaximal, characteristic, complemented
9#16810774C211yesmaximal, characteristic, complemented
10#16810735C211yesmaximal, characteristic, complemented
11#16810720C211yesmaximal, characteristic, complemented
12#16810714C211yesmaximal, characteristic, complemented
13D8×Dic53C211yesmaximal, characteristic, complemented
14#16810681C211yesmaximal, characteristic, complemented
15C53×#447C211yesmaximal, characteristic
16C2×#11719923C2211yescharacteristic
17C2×#11719923C2211yescharacteristic
18#13133216C2211yescharacteristic, complemented
19#13133210—21no—
20#13133204—21no—
21#13133204C2211yescharacteristic, complemented
22#13133198—21no—
23#13133196C2211yescharacteristic
24#13133194—21no—
25#13133182C2211yescharacteristic, complemented
26D4×Dic53—21no—
27D4×Dic53—21no—
28D4×Dic53C2211yescharacteristic
29D4×Dic53C2211yescharacteristic
30D4×Dic53C2211yescharacteristic
31D4×Dic53C2211yescharacteristic
32D4×Dic53C2211yescharacteristic, complemented
33#13133176C2211yescharacteristic
34#13133176C2211yescharacteristic
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—21no—
37#13133151C2211yescharacteristic, complemented
38#13133150C2211yescharacteristic, complemented
39#13133149C2211yescharacteristic, complemented
40#13133148C2211yescharacteristic, complemented
41#13133146C2211yescharacteristic, complemented
42#13133146C2211yescharacteristic, complemented
43#13133142C2211yescharacteristic, complemented
44#13133142C2211yescharacteristic, complemented
45#13133138C2211yescharacteristic
46#13133138C2211yescharacteristic
47#13133138C2211yescharacteristic, complemented
48#13133138C2211yescharacteristic, complemented
49C22⋊#88322C2211yescharacteristic
50#13133127C2211yescharacteristic, complemented
51C8×Dic53C2211yescharacteristic, complemented
52C22×D4×C53C2211yescharacteristic
53C2×D8×C53C2211yescharacteristic
54C2×D8×C53C2211yescharacteristic
55#121×C53C2211yescharacteristic
56#102×C53C2211yescharacteristic
57#102×C53C2211yescharacteristic
58C22⋊C8×C53C2211yescharacteristic
59C24×C53—21no—
60C2×D4×C53—22no—
61C2×D4×C53—22no—
62C2×D4×C53—21no—
63C2×D4×C53—21no—
64C2×D4×C53C2311yescharacteristic
65C2×D4×C53C2311yescharacteristic
66C2×D4×C53C2311yescharacteristic
67C22×C4×C53C2311yescharacteristic
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53—21no—
73C2×Dic106—21no—
74C2×Dic106C2311yescharacteristic
75D8×C53—22no—
76D8×C53—22no—
77C2×C8×C53C2311yescharacteristic
78C2×C8×C53C2311yescharacteristic
79C4⋊C4×C53C2311yescharacteristic
80C22⋊C4×C53—21no—
81C22⋊Dic53—21no—
82C22⋊Dic53—21no—
83C22⋊Dic53—21no—
84C22⋊Dic53—21no—
85C22⋊Dic53—21no—
86C22⋊Dic53—21no—
87C22⋊Dic53—21no—
88C22⋊Dic53—21no—
89C22⋊Dic53—21no—
90C22⋊Dic53D411yescharacteristic
91C22⋊Dic53D411yescharacteristic
92#11719923—22no—
93#11719923—22no—
94C4⋊Dic53D411yescharacteristic
95C4⋊Dic53D411yescharacteristic
96C4⋊Dic53C2311yescharacteristic
97C4⋊Dic53C2311yescharacteristic
98C4⋊Dic53C2311yescharacteristic
99#11719919—21no—
100#11719919—21no—
101#11719919—21no—
102C4×Dic53—21no—
103C4×Dic53—21no—
104C4×Dic53C2311yescharacteristic
105C4×Dic53C2311yescharacteristic
106C2×#88322C2311yescharacteristic
107C2×#88322C2311yescharacteristic
108C23×C53—24no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—81no—
113C23×C53—81no—
114C23×C53C2×D411yescharacteristic
115D4×C53—24no—
116D4×C53—21no—
117D4×C53—21no—
118D4×C53—21no—
119D4×C53—21no—
120D4×C53—81no—
121D4×C53#4112yes—
122D4×C53#4112yes—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53C2×D411yescharacteristic
127C2×C4×C53C2411yescharacteristic
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—81no—
133C2×Dic53—81no—
134C2×Dic53—81no—
135C2×Dic53—81no—
136C2×Dic53—81no—
137C2×Dic53C2×D411yescharacteristic
138C2×Dic53C2×D411yescharacteristic
139C2×Dic53C2×D411yescharacteristic
140C2×Dic53C2×D411yescharacteristic
141Dic106—21no—
142Dic106—21no—
143Dic106—81no—
144C8×C53—21no—
145C8×C53—21no—
146#88322—21no—
147#88322—21no—
148C22×C53—21no—
149C22×C53—22no—
150C22×C53—22no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—84no—
161C22×C53C22×D411yescharacteristic
162C4×C53—21no—
163C4×C53—41no—
164C4×C53C2×#4111yescharacteristic
165C4×C53C2×#4111yescharacteristic
166Dic53—21no—
167Dic53—21no—
168Dic53—21no—
169Dic53—21no—
170Dic53—41no—
171Dic53—41no—
172Dic53—41no—
173Dic53—41no—
174#3118—531nomaximal, Sylow
175C2×C53—41no—
176C2×C53—42no—
177C2×C53—42no—
178C2×C53—81no—
179C2×C53#52511yescharacteristic
180C2×C53#57611yescharacteristic
181C2×C53#57711yescharacteristic
182#542—531no—
183#518—531no—
184#485—531no—
185#482—531no—
186#476—531no—
187#474—531no—
188#463—531no—
189#463—531no—
190#450—531no—
191#447D5311yescharacteristic
192#442—531no—
193#440—531no—
194C4×SD16—531no—
195C4×D8—531no—
196#433—531no—
197C53#311811yescharacteristic, complemented, Sylow
198C22×D4C2×D5311yescharacteristic
199C2×SD16—531no—
200C2×SD16—531no—
201C2×D8C2×D5311yescharacteristic
202C2×D8C2×D5311yescharacteristic
203C4⋊Q8—531no—
204#126—1061no—
205#124—531no—
206#124—1061no—
207#123—1061no—
208C22⋊Q8—531no—
209C22⋊Q8—1061no—
210#121C2×D5311yescharacteristic
211C4×Q8—531no—
212C4×D4—531no—
213C4×D4—531no—
214C4×D4—531no—
215C4×D4—531no—
216C4×D4—531no—
217C4×D4—1061no—
218C4×D4—1061no—
219#117—531no—
220#117—531no—
221C2×C22⋊C4—1061no—
222C2×C22⋊C4—1061no—
223#107—531no—
224#107—531no—
225#106—531no—
226#106—531no—
227C4⋊C8—531no—
228C4⋊C8—531no—
229#103—531no—
230#103—531no—
231#102—531no—
232#102—531no—
233#102—531no—
234#102—531no—
235#102C2×D5311yescharacteristic
236#102C2×D5311yescharacteristic
237C22⋊C8—531no—
238C22⋊C8C2×D5311yescharacteristic
239#97—531no—
240C4×C8—531no—
241C24—21no—
242C2×Q8—531no—
243C2×Q8—1061no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4C22×D5311yescharacteristic
249C2×D4C22×D5311yescharacteristic
250C2×D4C22×D5311yescharacteristic
251C22×C4—1061no—
252C22×C4—1061no—
253C22×C4—1061no—
254C22×C4—1061no—
255C22×C4—1061no—
256C22×C4C22×D5311yescharacteristic
257SD16—1062no—
258SD16—1062no—
259D8—22no—
260D8—22no—
261C2×C8—531no—
262C2×C8—531no—
263C2×C8C22×D5311yescharacteristic
264C2×C8C22×D5311yescharacteristic
265C4⋊C4—531no—
266C4⋊C4—531no—
267C4⋊C4—531no—
268C4⋊C4—531no—
269C4⋊C4—531no—
270C4⋊C4—1061no—
271C4⋊C4—1061no—
272C4⋊C4—1061no—
273C4⋊C4C22×D5311yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—531no—
276C22⋊C4—531no—
277C22⋊C4—1061no—
278C22⋊C4—1061no—
279C22⋊C4—1061no—
280C22⋊C4—1061no—
281C22⋊C4—1061no—
282C22⋊C4—1061no—
283C22⋊C4—1061no—
284C22⋊C4—1061no—
285C22⋊C4—1061no—
286C42—531no—
287C42—531no—
288C42—1061no—
289C42—1061no—
290C23—24no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—81no—
295C23—81no—
296C23D4×D5311yescharacteristic
297Q8—1061no—
298Q8—1061no—
299Q8—4241no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—21no—
304D4—21no—
305D4—81no—
306D4#1171994712yes—
307D4#1171994712yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—531no—
312C2×C4—531no—
313C2×C4—531no—
314C2×C4—531no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—1061no—
318C2×C4—1061no—
319C2×C4—4241no—
320C2×C4—4241no—
321C2×C4—4241no—
322C2×C4—4241no—
323C2×C4—4241no—
324C2×C4D4×D5311yescharacteristic
325C2×C4C23×D5311yescharacteristic
326C8—21no—
327C8—21no—
328C8—1061no—
329C8—1061no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—84no—
343C22C2×D4×D5311yescharacteristic
344C4—21no—
345C4—41no—
346C4—1061no—
347C4—1061no—
348C4—1061no—
349C4—1061no—
350C4—2121no—
351C4—2121no—
352C4—2121no—
353C4—2121no—
354C4C2×#1171994711yescharacteristic
355C4C2×#1171994711yescharacteristic
356C2—41no—
357C2—42no—
358C2—42no—
359C2—81no—
360C2#1681112911yescharacteristic
361C2#1681142411yescharacteristic
362C2#1681143311yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#447C2
socleC22×C53C22×D4
hypercenter#447D53
hypocenterC53#3118
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #16811240C2semidirect
2. #16811098C2semidirect
3. #16810965C2semidirect
4. #16810944C2semidirect
5. #16810911C2semidirect
6. #16810901C2semidirect
7. #16810836C2semidirect
8. #16810836C2semidirect
9. #16810774C2semidirect
10. #16810735C2semidirect
11. #16810720C2semidirect
12. #16810714C2semidirect
13. D8×Dic53C2semidirect
14. #16810681C2semidirect
18. #13133216C22semidirect
21. #13133204C22semidirect
25. #13133182C22semidirect
32. D4×Dic53C22semidirect
37. #13133151C22semidirect
38. #13133150C22semidirect
39. #13133149C22semidirect
40. #13133148C22semidirect
41. #13133146C22semidirect
42. #13133146C22semidirect
43. #13133142C22semidirect
44. #13133142C22semidirect
47. #13133138C22semidirect
48. #13133138C22semidirect
50. #13133127C22semidirect
51. C8×Dic53C22semidirect
197. C53#3118semidirect